Microsoft Word - DETF

iii

Executive Summary……………………………………………………………..1

II.

Dark Energy in Context…………..……………………………………………..5

III.

Goals and Methodology for Studying Dark Energy………………………….....7

IV.

Findings of the Dark Energy Task Force……………………………………….11

Recommendations of the Dark Energy Task Force………………………….....21

VI.

A Dark Energy Primer………………………………………………………….27

VII.

DETF Fiducial Model and Figure of merit…………………………..................39

VIII.

Staging Stage IV from the Ground and/or Space……...……..………………....45

IX.

DETF Technique Performance Projections…………………………..................53

BAO………………………….....................................................................54

CL…………………………........................................................................60

SN…………………………........................................................................65

WL………………………….......................................................................70

Table of models...........................................................................................77

Dark Energy Projects (Present and Future) ………………...…………….….....79

XI.

References…………………………………........................................................89

XII.

Acknowledgments…………………………........................................................91

XIII.

Technical Appendix...…………………………...................................................93

XIV.

Logistical Appendix…..………………………..................................................123

I. Executive Summary

Over the last several years scientists have accumulated conclusive evidence that the
Universe is expanding ever more rapidly. Within the framework of the standard
cosmological model, this implies that 70% of the universe is composed of a new,
mysterious dark energy, which unlike any known form of matter or energy, counters the
attractive force of gravity. Dark energy ranks as one of the most important discoveries in
cosmology, with profound implications for astronomy, high-energy theory, general
relativity, and string theory.

One possible explanation for dark energy may be Einstein’s famous cosmological
constant. Alternatively, dark energy may be an exotic form of matter called quintessence,
or the acceleration of the Universe may even signify the breakdown of Einstein’s Theory
of General Relativity. With any of these options, there are significant implications for
fundamental physics. The problem of understanding the dark energy is called out
prominently in major policy documents such as the

Quantum Universe Report

and

Connecting Quarks with the Cosmos

, and it is no surprise that it is featured as number

one in

Science

magazine’s list of the top ten science problems of our time.

To date, there are no compelling theoretical explanations for the dark energy. In the
absence of useful theoretical guidance, observational exploration must be the focus of our
efforts to understand what the Universe is made of.

Although there is currently conclusive observational evidence for the existence of dark
energy, we know very little about its basic properties. It is not at present possible, even
with the latest results from ground and space observations, to determine whether a
cosmological constant, a dynamical fluid, or a modification of general relativity is the
correct explanation. We cannot yet even say whether dark energy evolves with time.

Fortunately, the extraordinary scientific challenge of the dark energy has generated
outstanding ideas for an observational program that can greatly impact our understanding.
A properly executed dark energy program should have as its goals to

Determine as well as possible whether the accelerating expansion is consistent
with a cosmological constant.

Measure as well as possible any time evolution of the dark energy.

Search for a possible failure of general relativity through comparison of the effect
of dark energy on cosmic expansion with the effect of dark energy on the growth
of cosmological structures like galaxies or galaxy clusters.

To recommend a program to reach these goals, the Dark Energy Task Force first
requested input from the community. The community responded with fifty impressive
white papers outlining current and future research programs on dark energy. Second,
using these submissions and our own expertise, we performed extensive calculations so
different approaches could be compared side-by-side in a standardized and quantitative
manner. We then developed a quantitative “figure of merit” that is sensitive to the

properties of dark energy, including its evolution with time. Our extensive findings are
based on these calculations.

Using our figure of merit, we evaluated ongoing and future dark energy studies in four
areas represented in the white papers. These are based on observations of Baryon
Acoustic Oscillations, Galaxy Clusters, Supernova, and Weak Lensing.

One of our main findings is that no single technique can answer the outstanding questions
about dark energy: combinations of at least two of these techniques must be used to fully
realize the promise of future observations. Already there are proposals for major, long-
term (Stage IV

) projects incorporating these techniques that have the promise of

increasing our figure of merit by a factor of ten beyond the level it will reach with the
conclusion of current experiments. What is urgently needed is a commitment to fund a
program comprised of a selection of these projects. The selection should be made on the
basis of critical evaluations of their costs, benefits, and risks.

Success in reaching our ultimate goal will depend on the development of dark-energy
science. This is in its infancy. Smaller, faster programs (Stage III

) are needed to

provide the experience on which the long-term projects can build. These projects can
reduce systematic uncertainties that could otherwise impede the larger projects, and at the
same time make important advances in our knowledge of dark energy.

We recommend that the agencies work together to support a balanced program that
contains from the outset support for both the long-term projects and the smaller projects
that will have more immediate returns. We call for a

coordinated

program to attack one

of the most profound questions in the physical sciences. Our report provides a
quantitative basis for prioritizing near-term and long-term projects.

We are very fortunate that a wide range of new observations are possible that can drive
significant progress in this field. Many researchers from both particle physics and
astronomy are being drawn to these remarkable opportunities. It is a rare moment in the
history of science when such clear steps can be taken to address such a profound
problem.

In this Report we describe dark-energy research in

Stages

: Stage I represents dark-energy projects that

have been completed; Stage II represents ongoing projects relevant to dark-energy; Stage III comprises
near-term, medium-cost, currently proposed projects; Stage IV comprises a Large Survey Telescope (LST),
and/or the Square Kilometer Array (SKA), and/or a Joint Dark Energy (Space) Mission (JDEM).

II. Dark Energy in Context

Conclusive evidence from supernovae and other observations shows that the
expansion of the Universe, rather than slowing because of gravity, is increasingly
rapid. Within the standard cosmological framework, this must be due to a
substance that behaves as if it has negative pressure. This substance has been
termed “dark energy.” Experiments indicate that dark energy accounts for about
70% of the mass-energy in the Universe.

One possibility is that the Universe is permeated by an energy density, constant in
time and uniform in space. Such a “cosmological constant” (Lambda:

) was

originally postulated by Einstein, but later rejected when the expansion of the
Universe was first detected. General arguments from the scale of particle
interactions, however, suggest that if

is not zero, it should be very large, larger

by a truly enormous factor than what is measured. If dark energy is due to a
cosmological constant, its ratio of pressure to energy density (its equation of state)
is

= −1

at all times.

Another possibility is that the dark energy is some kind of dynamical fluid, not
previously known to physics. In this case the equation of state of the fluid would
likely not be constant, but would vary with time, or equivalently with redshift

with

(

−1

, the scale factor (or size) of the Universe relative to its current

scale or size. Different theories of dynamical dark energy are distinguished
through their differing predictions for the evolution of the equation of state.

The impact of dark energy (whether dynamical or a constant) on cosmological
observations can be expressed in terms

(

)

= P

(

, which is to be

measured through its influence on the large-scale structure and dynamics of the
Universe.

An alternative explanation of the accelerating expansion of the Universe is that
general relativity or the standard cosmological model is incorrect. We are driven
to consider this prospect by potentially deep problems with the other options. A
cosmological constant leaves unresolved one of the great mysteries of quantum
gravity and particle physics: If the cosmological constant is not zero, it would be
expected to be 10

120

times larger than is observed. A dynamical fluid picture

usually predicts new particles with masses thirty-five orders of magnitude smaller
than the electron mass. Such a small mass could imply the existence of a new
observable long-range force in nature in addition to gravity and electromagnetism.
Regardless of which (if any) of these options are realized, exploration of the
acceleration of the Universe’s expansion will profoundly change our
understanding of the composition and nature of the Universe.

It is not at present possible, even with the latest results from ground and space
observations, to determine whether a cosmological constant, a dynamical fluid, or

III.

Goals and Methodology for Studying Dark Energy

The goal is to determine the very nature of the dark energy that causes the
Universe to accelerate and seems to comprise most of the mass-energy of the
Universe.

Toward this goal, our observational program must

Determine as well as possible whether the accelerating expansion is
consistent with being due to a cosmological constant.

If the acceleration is not due to a cosmological constant, probe the
underlying dynamics by measuring as well as possible the time evolution
of the dark energy by determining the function

(

Search for a possible failure of general relativity through comparison of
the effect of dark energy on cosmic expansion with the effect of dark
energy on the growth of cosmological structures like galaxies or galaxy
clusters.

Since

(

) is a continuous function with an infinite number of values at

infinitesimally separated points,

(

) must be modeled using just a few

parameters whose values are determined by fitting to observations. No single
parameterization can represent all possibilities for

(

). We choose to

parameterize the equation of state as

(

)

= w

(

1−

)

, where

is the present

value of

and where

parameterizes the evolution of

(

). This simple

parameterization is most useful if dark energy is important at late times and
insignificant at early times.

The goals of a dark energy observational program may be reached through
measurement of the expansion history of the Universe [traditionally measured by
luminosity distance vs. redshift, angular-diameter distance vs. redshift, expansion
rate vs. redshift, and volume element vs. redshift], and through measurement of
the growth rate of structure, which is suppressed during epochs when the dark
energy dominates. All these measurements of dark energy properties can be
expressed in terms of the value of the dark energy density today,

, and its

evolution,

If the accelerating expansion is due instead to a failure of general

relativity, this could be revealed by finding discrepancies between the values of

(

) inferred from these two types of data.

In order to quantify progress in measuring the properties of dark energy we define
a dark-energy “figure of merit” formed from a combination of the uncertainties in

and

The

DETF figure of merit is the reciprocal of the area of the error

ellipse enclosing the 95% confidence limit in the w

–w

plane. Larger

figure of merit indicates greater accuracy.

The one-dimensional errors in

and

are correlated, and their product is not a

good indication of the power of a particular experiment. This is why the DETF
figure of merit is defined as the area contained within the 95% confidence limit
contours in the

–

plane (not the simple product of one-dimensional

uncertainties in

and

). In Section VII we discuss the DETF figure of merit.

We also discuss the utility of defining a pivot value of

, defined as

The pivot value of

is the value at the redshift for which

is best constrained by

a particular experiment; its variance is equal to the variance of

in a model

assuming

= 0

. We demonstrate that the figure of merit is the inverse of the

product of uncertainties in

and

The error in

reflects the ability of a single

experiment or a combination of experiments to test whether dark energy equation
of state is consistent

= −1

;

i.e.,

a cosmological constant.

The DETF dark-energy parameterization of

(

) and the associated figure of

merit serve as a robust, quantitative guide to the ability of an experimental
program to constrain a large, but not exhaustive, set of dark-energy models. Since
the nature of dark energy is so poorly understood, no single figure of merit is
appropriate for every eventuality. Particular experiments may excel at testing
dark-energy models that are poorly described by our parameterization and their
utility may not be reflected in out figure of merit. However, potential
shortcomings of the choice of any figure of merit must be evaluated in the larger
context, which includes the critical need to make side-by-side comparisons and
specific choices to move the field forward. In our judgment there is no better
choice of a figure of merit available at this time. We expect continuing
theoretical and experimental advances in our understanding of dark energy will
allow us to explore other figures of merit. We recognize that developments may
eventually lead to recognition by the community that some new measure better
meets the overall needs of the field.

We have made extensive use of statistical (Fisher-Matrix) techniques
incorporating information about cosmic microwave background (CMB) and
Hubble’s constant (

) to predict the future performance of possible dark-energy

projects, and combinations of these projects.

Our considerations for a dark-energy program follow developments in “Stages:”

Stage I represents what is now known.

Stage II represents the anticipated state of knowledge upon completion of
ongoing projects that are relevant to dark-energy.

Stage III comprises near-term, medium-cost, currently proposed projects.

Stage IV comprises a Large Survey Telescope (LST), and/or the Square
Kilometer Array (SKA), and/or a Joint Dark Energy (Space) Mission
(JDEM).

Just as dark-energy science has far-reaching implications for other fields of
physics, advances and discoveries in other fields of physics may point the way

IV.

Findings of the Dark-Energy Task Force

Four observational techniques dominate the White Papers received by the task
force. In alphabetical order:

Baryon Acoustic Oscillations (BAO)

are observed in large-scale surveys

of the spatial distribution of galaxies. The BAO technique is sensitive to
dark energy through its effect on the angular-diameter distance vs. redshift
relation and through its effect on the time evolution of the expansion rate.

Galaxy Cluster (CL)

surveys measure the spatial density and distribution

of galaxy clusters. The CL technique is sensitive to dark energy through
its effect on a combination of the angular-diameter distance vs. redshift
relation, the time evolution of the expansion rate, and the growth rate of
structure.

Supernova (SN)

surveys use Type Ia supernovae as standard candles to

determine the luminosity distance vs. redshift relation. The SN technique
is sensitive to dark energy through its effect on this relation.

Weak Lensing (WL)

surveys measure the distortion of background

images due to the bending of light as it passes by galaxies or clusters of
galaxies. The WL technique is sensitive to dark energy through its effect
on the angular-diameter distance vs. redshift relation and the growth rate
of structure.

Other techniques discussed in White Papers, such as using

-ray bursts or

gravitational waves from coalescing binaries as standard candles, merit further
investigation. At this time, they have not yet been practically implemented, so
it is difficult to predict how they might be part of a dark energy program. We
do note that if dark energy dominance is a recent cosmological phenomenon,
very high-redshift (

) probes will be of limited utility.

Different techniques have different strengths and weaknesses and are sensitive in
different ways to the dark energy properties and to other cosmological parameters.

Each of the four techniques can be pursued by multiple observational approaches,

e.g.,

radio, visible, near-infrared (NIR), and/or x-ray observations, and a single

experiment can study dark energy with multiple techniques. Individual missions
need not necessarily cover multiple techniques; combinations of projects can
achieve the same overall goals.

The techniques are at different levels of maturity:

The

BAO

technique has only recently been established. It is less affected

by astrophysical uncertainties than other techniques.

The

technique has the statistical potential to exceed the BAO and SN

techniques but at present has the largest systematic errors. Its eventual
accuracy is currently very difficult to predict and its ultimate utility as a
dark energy technique can only be determined through the development of

techniques that control systematics due to non-linear astrophysical
processes.

The

technique is

at present

the most powerful and best proven

technique for studying dark energy. If redshifts are determined by
multiband photometry, the power of the supernova technique depends
critically on the accuracy achieved for photo-

’s

. (Multiband photometry

measures the intensity of the object in several colors. A redshift
determined by multiband photometry is called photometric redshift,
or a photo-z.)

If spectroscopically measured redshifts are used, the power

of the experiment as reflected in the DETF figure of merit is much better
known, with the outcome depending on the uncertainties in supernova
evolution and in the astronomical flux calibration.

The

technique is also an emerging technique. Its eventual accuracy

will also be limited by systematic errors that are difficult to predict.

the

systematic errors are at or below the level asserted by the proponents, it is
likely to be the most powerful individual Stage-IV technique and also the
most powerful component in a multi-technique program.

A program that includes multiple techniques at Stage IV can provide an order of
magnitude increase in the DETF figure of merit. This would be a major advance
in our understanding of dark energy. A program that includes multiple techniques
at Stage III can provide a factor of three increase in the DETF figure of merit.
This would be a valuable advance in our understanding of dark energy. In the
absence of a persuasive theoretical explanation for dark energy, we must be
guided by ever more precise observations.

We find that no single observational technique is sufficiently powerful and well
established that it is guaranteed to achieve by itself an order of magnitude
increase in the DETF figure of merit. Combinations of the principal techniques
have substantially more statistical power, much greater ability to discriminate
among dark energy models, and more robustness to systematic errors than any
single technique. The case for multiple techniques is supported as well by the
critical need for confirmation of results from any single method. (The results for
various model combinations can be found at the end of Section IX.)

bin

atio

Technique #2

Technique #1

Illustration of the power of combining techniques. Technique #1 and Technique #2 have roughly
equal DETF figure of merit. When results are combined, the DETF figure of merit is
substantially improved.

Results on structure growth, obtainable from weak lensing or cluster observations,
provide additional information not obtainable from other techniques. In
particular, they allow for a consistency test of the basic paradigm: spatially
constant dark energy plus general relativity.

In our modeling we assume constraints on

from current data and constraints on

other cosmological parameters expected to come from further measurement of
CMB temperature and polarization anisotropies.

These data, though insensitive to

(

) on their own, contribute to our

knowledge of

(

) when combined with any of the dark energy techniques

we have considered.

Increased precision in a particular cosmological parameter may improve
dark-energy constraints from a single technique. Increased precision is
valuable for the important task of comparing dark energy results from
different techniques.

Increased precision in cosmological parameters tends not to improve significantly
the overall DETF figure of merit obtained from a multi-technique program.
Indeed, a multi-technique program would itself provide powerful new constraints
on cosmological parameters within the context of our parametric dark-energy
model.

10.

Setting the spatial curvature of the Universe to zero greatly strengthens the dark-
energy constraints from supernovae, but has a modest impact on the other
techniques once a dark-energy parameterization is selected. When techniques are
combined, setting the spatial curvature of the Universe to zero makes little
difference to constraints on parameterized dark energy, because the curvature is
one of the parameters well determined by a multi-technique approach.

Illustration of the sensitivity of dark energy constraints to prior assumptions about cosmological
parameters in the case of Stage IV space-based measurements with optimistic systematic errors.
The solid lines indicate the factor by which the DETF figure of merit increases with the
assumption that the spatial curvature of the Universe vanishes. There is a marked improvement
in the power of the SN technique with the assumption that the spatial curvature vanishes.
However, if the SN technique is combined with other techniques, e.g., the WL technique, the
improvement is modest. The dotted lines indicate the factor by which the DETF figure of merit
increases with the assumption that the uncertainty in the Hubble constant is

km s

−1

Mpc

−1

compared to the present uncertainty of

km s

−1

Mpc

− 1

. Reducing the uncertainty in H

makes at

most a 50% improvement on individual techniques at the Stage IV level. Space experiments are
illustrated here but results from ground Stage IV experiments are similar.

11.

Optical, NIR, and x-ray experiments with very large numbers of astronomical
targets will rely on photometrically determined redshifts. The ultimate accuracy
that can be attained for photo-

's is likely to determine the power of such

measurements. (Radio HI (neutral hydrogen) surveys produce precise redshifts as
part of the survey.)

12.

Our inability to forecast systematic error levels reliably is the biggest impediment
to judging the future capabilities of the techniques. Assessments of effectiveness
could be made more reliably with:

For

BAO–

Theoretical investigations of how far into the non-linear regime

the data can be modeled with sufficient reliability and further
understanding of galaxy bias on the galaxy power spectrum.

For

CL–

Combined lensing, Sunyaev-Zeldovich, and x-ray observations

of large numbers of galaxy clusters to constrain the relationship between
galaxy cluster mass and observables.

For

SN–

Detailed spectroscopic and photometric observations of about

500 nearby supernovae to study the variety of peak explosion magnitudes
and any associated observational signatures of effects of evolution,
metallicity, or reddening, as well as improvements in the system of
photometric calibrations.

For

WL–

Spectroscopic observations and multi-band imaging of tens to

hundreds of thousands of galaxies out to high redshifts and faint
magnitudes in order to calibrate the photometric redshift technique and
understand its limitations. It is also necessary to establish how well
corrections can be made for the intrinsic shapes and alignments of
galaxies, the effects of optics, (from the ground) the atmosphere, and the
anisotropies in the point-spread function.

13.

Six types of Stage-III projects have been considered. They include:

a BAO survey on a 4-m class telescope using photo-

’s.

a BAO survey on an 8-m class telescope employing spectroscopy.

a CL survey on a 4-m class telescope obtaining optical photo-

’s for

clusters detected in ground-based SZ surveys.

a SN survey on a 4-m class telescope using photo-

’s.

a SN survey on a 4-m class telescope employing spectroscopy from an 8-
m class telescope.

a WL survey on a 4-m class telescope using photo-

’s.

These projects are typically projected by proponents to cost in the range of tens of
millions of dollars. (Cost projections were not independently checked by the
DETF.)

14.

Our findings regarding Stage-III projects are

Only an incremental increase in knowledge of dark-energy parameters is
likely to result from a Stage-III BAO project using photo-

’s. The primary

benefit from a Stage-III BAO photo-

project would be in exploring

systematic photo-

uncertainties.

A modest increase in knowledge of dark-energy parameters is likely to
result from a Stage-III SN project using photo-

’s. Such a survey would

be valuable if it were to establish the viability of photometric
determination of supernova redshifts, types, and evolutionary effects.

A modest increase in knowledge of dark-energy parameters is likely to
result from any single Stage-III CL, WL, spectroscopic BAO, or
spectroscopic SN survey.

The SN, CL, or WL techniques could, individually, produce factor of two
improvements in the DETF figure of merit, if the systematic errors are
close to what the proponents claim.

If executed in combination, Stage-III projects would increase the DETF
figure of merit by a factor in the range of approximately three to five, with

15.

Four types of Stage-IV projects have been considered

an optical Large Survey Telescope (LST), using one or more of the four
techniques.

an optical/NIR Joint Dark Energy Mission (JDEM) satellite, using one or
more of the four techniques.

an x-ray JDEM satellite, which would study dark energy by the cluster
technique.

a radio Square Kilometer Array, which could probe dark energy by WL
and/or BAO techniques through a hemisphere-scale survey of 21-cm and
continuum emission. The very large range of frequencies currently
demanded by the SKA specifications would likely require more than one
type of antenna element. Our analysis is relevant to a lower frequency
system, specifically to frequencies below 1.5 GHz.

Each of these projects is projected by proponents to cost in the $0.3-1B range, but
dark energy is not the only (in some cases not even the primary) science that
would be done by these projects. (Cost projections were not independently
checked by the DETF.) According to the white papers received by the Task
Force, the technical capabilities needed to execute LST and JDEM are largely in
hand. (The Task Force is not constituted to undertake a study of the technical
issues.)

16.

Each of the Stage IV projects considered (LST, JDEM, and SKA) offers
compelling potential for advancing our knowledge of dark energy as part of a
multi-technique program.

17.

The Stage IV experiments have different risk profiles:

The SKA would likely have very low systematic errors, but it needs
technical advances to reduce its costs and risk. Particularly important is
the development of wide-field imaging techniques that will enable large
surveys. The effectiveness of an SKA survey for dark energy would also
depend on the number of galaxies it could detect, which is uncertain.

An optical/NIR JDEM can mitigate systematic errors because it would
likely obtain a wider spectrum of diagnostic data for SN, CL, and WL than
possible from the ground, and it has no systematics associated with
atmospheric influence, though it would incur the usual risks and costs of a
space-based mission.

LST would have higher systematic-error risk than an optical/NIR JDEM,
but could in many respects match the power of JDEM if systematic errors,
especially if those due to photo-

measurements, are small. An LST Stage

IV program could be effective only if photo-

uncertainties on very large

samples of galaxies can be made smaller than what has been achieved to
date.

18.

A mix of techniques is essential for a fully effective Stage IV program. The
technique mix may be comprised of elements of a ground-based program, or

To quantify our empirical knowledge of dark energy we form a figure of merit from a
product of observational uncertainties in parameters that describe the evolution of dark
energy. The DETF figure of merit is the reciprocal of the area of the error ellipse
enclosing the 95% confidence limit in the

–

plane. Larger figure of merit indicates

greater accuracy. (The DETF figure of merit is discussed in detail in Section VII.)

III. We recommend that the dark energy program include a
combination of techniques from one or more Stage III projects
designed to achieve, in combination, at least a factor of three gain over
Stage II in the DETF figure of merit, based on critical appraisals of
likely statistical and systematic uncertainties.

Our modeling indicates that a Stage III program can, in principle, reach this goal.
Moreover, such a program would help to determine systematic uncertainties and would
provide experience valuable to Stage IV planning and execution using the same
techniques. Significant progress understanding Stage IV systematic error levels should
be made as soon as possible. As much as possible these goals should be integrated with
the Stage III projects.

IV. We recommend that the dark energy program include a
combination of techniques from one or more Stage IV projects
designed to achieve, in combination, at least a factor of ten gain over
Stage II in the DETF figure of merit, based on critical appraisals of
likely statistical and systematic uncertainties. Because JDEM, LST,
and SKA all offer promising avenues to greatly improved
understanding of dark energy, we recommend continued research and
development investments to optimize the programs and to address
remaining technical questions and systematic-error risks.

Our modeling suggests that there are several combinations of Stage IV projects and
techniques capable, in principle, of reaching a factor of ten increase, by a ground-based
program, a space-based program, or a combination of ground-based and space-based
programs. Further improvements in our understanding of systematic error levels are
required to determine with confidence the overall and relative effectiveness of specific
combinations of Stage IV projects. Findings 12 and 17 discuss this issue in detail.

______________________

V. We recommend that high priority for near-term funding should be
given as well to projects that will improve our understanding of the
dominant systematic effects in dark energy measurements and,
wherever possible, reduce them, even if they do not immediately
increase the DETF figure of merit.

Among the projects that can contribute to this goal are

Improving knowledge of the precision and reliability attainable from near-infrared
and visible photometric redshifts for both galaxies and supernovae, through
statistically significant samples of spectroscopic measurements for a wide range
in redshift. The precision with which photometric redshifts can be measured will
impact many dark energy measurements. They are particularly critical for large-
scale weak lensing surveys, and they bound the potential of baryon-oscillation and
supernova surveys that forego spectroscopy. There must be a robust program to
develop the precision that will be required for experiments in Stages III and IV.

Demonstrating weak-lensing observations with low shear-measurement errors.
Future weak-lensing surveys will demand measurements of gravitational shear, in
the presence of optical and atmospheric distortions, that exceed currently
demonstrated accuracy. Development of the lensing methodology and testing on
large volumes of real and simulated image data are required.

Obtaining high-precision spectra and light curves of a large ensemble of Type Ia
SNe in the ultraviolet/visible/near-infrared to constrain, for example, systematic
effects due to reddening, metallicity, evolution, and photometric/spectroscopic
calibrations.

Establishing a high-precision photometric and spectrophotometric calibration
system in the ultraviolet, visible, and near-infrared. Precision photometric
redshifts,

-corrections, and luminosity distances cannot be achieved until the

fundamental calibration system is significantly improved.

Obtaining better estimation of the galaxy population that would be detectable in
21 cm by a SKA at high redshifts (

0.5

). Current plausible models show

considerable differences in the evolution of the HI luminosity function. This is
the primary uncertainty in our predictions of the performance of an SKA galaxy
survey as it determines the size and redshift distribution of the galaxy sample.

Better characterization of cluster mass-observable relations through joint x-ray,
SZ, and weak lensing studies and also via numerical simulations including the
effects of cooling, star-formation, and active galactic nuclei.

Supporting theoretical work on non-linear gravitational growth and its impacts on
baryon acoustic oscillation measurements, weak lensing error statistics, cluster
mass observables, simulations, and development of analysis techniques.

______________________

Because the dark energy program will employ a variety of techniques, a number of
experiments, and three funding agencies, management of the program poses special
challenges.

VI. We recommend that the community and the funding agencies
develop a coherent program of experiments designed to meet the goals
and criteria set out in these recommendations.

We propose a number of guidelines for the development of the program:

Individual proposals should not be reviewed in isolation. Decisions on projects
should take into account how they fit into the overall dark-energy program.

•

In judging Stage III proposals, in addition to contributing toward a factor
of three increase in the DETF figure of merit, significant weight should be
placed on their capacity to enhance the efforts to develop an optimal Stage
IV program.

In this regard, the timing of experiments is an issue. That is,

Stage III experiments will be of most value if they inform the planning
and/or execution of the Stage IV program.

•

In ranking proposed projects, the precision gain in an individual technique
is not necessarily the most important factor. When considered in
conjunction with other techniques, significant gains in precision for a
single technique may not be as valuable as more modest advances in
another technique. In evaluating projects, there is considerable
opportunity for trade-offs between different techniques.

•

Projects that combine multiple techniques are desirable. While multiple
techniques are crucial, the order of magnitude gain in the dark energy
figure of merit is unlikely to require that all four techniques be pursued
through Stage IV. As detailed in our report, combinations of three (or
possibly even two) techniques probably can achieve the stated goal.

It is incumbent on proponents of Stage III and IV projects to demonstrate that
they will be able to limit systematic uncertainties well enough to achieve the
claims they make for improving the measurements of dark energy parameters.

•

In modeling projected performance of Stage III and Stage IV projects, the
DETF concluded that systematic uncertainties will ultimately determine
the accuracy of our knowledge of dark energy. Critical assessment of the
potential systematic uncertainties is a necessary step in the evaluation of
these projects.

Potential gains from the Stage IV facilities beyond their dark energy studies
should be taken into account.

•

Each of the Stage IV facilities would offer enormous gains in knowledge of
the Universe beyond their dark-energy studies, at very little marginal cost.

A means of quantifying the increase in our understanding of dark energy from the
suite of experiments should be developed.

VI. A Dark Energy Primer

In General Relativity (GR), the growth of the Universe is described by a scale factor

(

defined so that at the present time

(

) = 1. The time evolution of the expansion in

GR obeys

(

)

= −

where

and

are the mean pressure and density of the contents of the Universe, and

the

cosmological constant

proposed and then discarded by Einstein. Remarkably, several

lines of evidence (described below) confirm that at the present time,

. This

acceleration immediately implies that either

The Universe is dominated by some particle or field (

dark energy

) that has

negative pressure, in particular

1/ 3;

< −

There is in fact a non-zero cosmological constant;

The theoretical basis for this equation, GR or the standard cosmological model, is
incorrect.

Any of these three explanations would require fundamental revision to the underpinning
theories of physics. It is of great interest to determine which of these three explanations
is correct.

The Observable Consequences of Dark Energy

Within the context of GR, a convenient expression of the equation for the expansion is

( )

⎛ ⎞ ≡

−

⎜ ⎟

⎝ ⎠

where

is the curvature. The value of

today,

, is the Hubble constant, 72

8 km s

-1

Mpc

-1

. From these two equations it follows that

3 (

)

= −

which holds separately for each contributor to the energy density. For non-relativistic
matter,

is of order (v

)

, and can be ignored, and the equation becomes

= −

= −3

(

) and

, where

is the density of non-relativistic matter

today. More generally, if

is constant, then

−

3(1

)

For non-relativistic matter, we define

and we define analogously

for the density of relativistic matter (and radiation), for

which

= 1/3

. To obtain an attractive equation we introduce

Ω = −

Now we can write

( )

3(1

)

−

⎛ ⎞

⎡

⎤

≡

+ Ω

⎜ ⎟

⎣

⎦

⎝ ⎠

The term

represents the cosmological constant if

= −1

. Otherwise, it represents

dark energy with constant

. This generalizes easily for non-constant

with the

replacement

[

]

3(1

)

exp 3

( )

w a

−

⎛

⎞

′

→

⎜

⎟

′

⎝

⎠

∫

The quantity

describes the current curvature of the universe. For

< 0, the Universe

is closed and finite; for

> 0 the Universe is open and potentially infinite; while for

= 0 the geometry of the Universe is Euclidean (flat).

The cosmic microwave background radiation (CMB) gives very good constraints on the
matter and radiation densities

and

, so it appears one could determine the

time history of the dark-energy density, modulo some uncertainty due to curvature, if
one could accurately measure the expansion history

(

). When a distant astronomical

source is observed, it is straightforward to determine the scale factor

at the time of

emission of the light, since all photon wavelengths stretch during the expansion; this is
quantified by the

redshift z

, with (1+

) =

−1

. The derivative

is more difficult,

however, since time is not directly observable. Most cosmological observations instead
quantify the

distance

to a given source at redshift

, which is closely related to the

expansion history since a photon on a radial path must satisfy

−

This implies that the distance to a source at redshift

, defined as

(

), is given by

( )

D z

a t

H z

′

−

∫

This procedure also can be used to express the coordinate

in terms of the redshift:

( )

1/ 2

r z

D z

H z

−

⎡

⎤

′

⎡

⎤

⎢

⎥

⎣

⎦

′

⎣

⎦

∫

where the function

[

] is given by

[ ]

sin

0
0

sinh

⎧

⎪

⎨

⎪

⎩

The coordinate

(

) has several measurable consequences. This function, or closely

related ones, determine: a) the apparent flux of an object of fixed luminosity (

standard-

candle method

); b) the apparent angular size or redshift extent of an object of fixed linear

size (

standard-ruler method

); or c) the apparent sky density of an object of known space

density. The distance functions related to each of these observations are given in the
table below. [Recall that

(

)

Ω −

measurable Definition

proper distance

( )

1/ 2

sin

sinh

r z

D z

r z

H z

r z

−

⎧

⎡

⎤

⎣

⎦

⎪

′

⎪

⎨

′

⎪

⎡

⎤

⎪

⎣

⎦

⎩

∫

luminosity distance

( ) ( )(

)

r z

angular diameter distance

( ) ( ) (

)

r z

volume element

( )

drd

−

Dark energy enters through the dependence of

(

) on dark energy. In turn, the

dependence of the expansion rate on dark energy results in a dark-energy dependence to

(

The existence of dark energy has a second observable consequence: it affects the growth
of density perturbations. Quantum fluctuations in the early Universe create density
fluctuations. These are measured in great detail as temperature fluctuations in the CMB
at redshift

= 1088.

Fig. VI-1: Fluctuations in the temperature of the early Universe, as measured by the
WMAP experiment.

In a static Universe, overdense regions will increase their density at an exponential rate,
but in our expanding Universe there is a competition between the expansion and
gravitational collapse. More rapid expansion – as induced by dark energy – retards the
growth of structure. GR provides the following relation, in linear perturbation theory,
between the

growth factor g

(

) and the expansion history of the Universe:

π ρ

Because the fluctuations at

= 1088 are accurately quantified by CMB measurements,

the amplitude of matter fluctuations provides an additional observable
manifestation of dark energy via the growth-redshift relation g(z).

Within the context of GR, this differential equation provides a one-to-one relation
between the two observable quantities

(

) and

(

). Inconsistency between these two

quantities would indicate that GR is incorrect on the largest observable scales in the
Universe (or that dark energy contributes to the growth of clustering in an unexpected

manner). If both quantities can be measured, the veracity of this relation can be checked,

permitting a test of the underlying GR theory

Figure IV-2 illustrates the effect of dark energy on the distance-redshift and growth-
redshift relations, highlighting the need for percent-level precision in these quantities if
we are to constrain the dark-energy equation of state to about 0.1 accuracy.

Fig. VI-2: The primary observables for dark-energy – the distance-redshift relation D

(

)

and the growth-redshift relation g

(

)

– are plotted vs. redshift for three cosmological

models. The green curve is an open-Universe model with no dark energy at all. The
black curve is the “concordance”

CDM model, which is flat and has a cosmological

constant, i.e., w

−1

. This model is consistent with all reliable present-day data. The

red curve is a dark-energy model with w

= −0.9

, for which other parameters have been

adjusted to match WMAP data. At left one sees that dark-energy models are easily
distinguished from non-dark-energy models. At right, we plot the ratios of each model to
the

CDM model, and it is apparent that distinguishing the w

= −0.9

model from

CDM

requires percent-level precision on the diagnostic quantities.

Four Astrophysical Approaches to Dark Energy Measurements

1. Type Ia Supernovae

Type Ia supernovae are believed to be the explosive disintegrations of white-dwarf stars
that accrete material to exceed the stability limit of 1.4 solar masses derived by
Chandrasekhar. Because the masses of these objects are nearly all the same, their
explosions are expected to serve as standard candles of known luminosity

, in which

case the relation

f = L

can be used to infer the luminosity distance

. Spectral

lines in the supernova light may be used to identify the redshift, as can spectral features
of the galaxy hosting the explosion.

Type Ia supernovae observed from the ground and the Hubble Space Telescope (HST)
have been used successfully to deduce the acceleration of the Universe after

z =

1, as

illustrated in Fig. IV-2. In practice one finds that Type Ia supernovae are not
homogeneous in luminosity. However, variations in luminosity appear to be correlated
with other, distance-independent, features of the events, such as the rest-frame duration
of the event or its spectral features. Thus Type Ia SNe are

standardizable

, to some yet-

unknown degree of precision. Theoretical modeling of SN explosions is extremely
difficult; it is not expected that this theory will ever deduce the absolute magnitude nor
the standardization process to the accuracy required for dark-energy study. Hence the
standardization process must be empirical, and its ultimate accuracy or evolution with
cosmic time are very difficult to predict.

Fig VI-3: Left: High-redshift supernovae observed from HST by Riess et al (2004).
Right: Cosmological results from the GOODS SNe (Riess et al. 2004). Upper panel:
distance (

5 log

const

.) vs. redshift; lower: constraints on present-day

acceleration

Other standard(izable)-candle sources may be available in the future: other types of SNe,
gamma-ray bursts, or gravity-wave sources. There is not yet evidence that any of them
will exceed the precision of Type Ia SNe over the critical 0 <

< 2 range in the coming

decade.

2. Baryon Acoustic Oscillations

From the moment inflation ends, the Universe is filled with an ionized plasma. Pressure
waves propagate in this baryon-photon fluid at the sound speed of

From any

initial density fluctuation, a expanding spherical perturbation propagates until the time,
approximately 370,000 years after the Big Bang, when electrons and protons combine to
form neutral hydrogen. At this moment the pressure waves cease to expand, and are
frozen into the matter distribution. The total propagation distance

, is called the

sound

horizon

, and the matter distribution is imprinted with this characteristic size. The physics

of these

baryon acoustic oscillations

(BAO) is well understood, and their manifestation

as wiggles in the CMB fluctuation spectrum is modeled to very high accuracy. The
value of

is found to be 148

3 Mpc, by the Wilkinson Microwave Anisotropy Probe

(WMAP) 3-year data (Spergel

et al.

2006). The sound horizon scale can thus serve as a

standard ruler for distance measurements. Indeed their presence in the CMB allows the
distance to

= 1088 to be determined to very high accuracy. If we consider that galaxies

roughly trace the (dark) matter distribution, then a survey of the galaxy density field
should reveal this characteristic scale.

The largest galaxy survey to date, the Sloan Digital Sky Survey, has yielded the first
detection of the BAO signal outside of the CMB, as illustrated in Fig. IV.3. The
identification of the horizon scale as a transverse angle determines the distance ratio

(

(modulo the curvature contribution), while its determination along the line of

sight determines

(

z)r

The density survey to find the BAO feature can use galaxies as the target, in optical, near-
IR, or 21-cm emission, or it may be possible to identify the BAO feature in the
distribution of neutral hydrogen at redshifts

> 5.

Fig. VI-4: The baryon acoustic oscillations are seen as wiggles in the power spectrum of
the CMB (left, Hinshaw et al. 2003), and have now been detected as a feature in the
correlation function of nearby galaxies using the Sloan Digital Sky Survey (right,
Eisenstein et al 2005).

3. Galaxy Cluster Counting

Clusters of galaxies are the largest structures in the Universe to have undergone
gravitational collapse, and they serve as markers for those locations which were endowed

with the highest density fluctuations in the early Universe. Analytic prediction is
possible for the

mass function

dN /

(

dM dV)

of these rare events per unit comoving

volume per unit cluster mass. Gravitational

-body modeling can produce even more

precise predictions of the mass function. One can in principle measure the abundance of
clusters on the sky,

dN /

(

dM d

)

This is sensitive to dark energy in two ways: First,

the comoving volume element depends on dark energy, so cluster counts depend upon the
expansion history. Second, the mass function itself is sensitive to the amplitude of
density fluctuations; in fact it is exponentially sensitive to the growth function

(

) at

fixed mass

Fig. VI-5: Galaxy clusters as viewed in three different spectral regimes: top left, an
optical view showing the concentration of yellowish member galaxies (SDSS); top right,
Sunyaev, Zel’dovich flux decrements at 30 GHz (Carlstrom, et al. 2001); bottom, x-ray
emission (Chandra Science Center). These images are not at a common scale.

Galaxy clusters can be and have been detected in several ways: originally, by the optical
detection of their member galaxies; then by the x-ray emission from the hot electrons
confined by the gravitational potential well; by the

Sunyaev-Zeldovich effect

, whereby

these hot electrons up-scatter the CMB photons, leaving an apparent deficit of low-
frequency CMB flux in their direction; and, most recently, by their weak gravitational
lensing effect on background galaxy images (see below). The main obstacle to cluster
counting is that none of the first three of these techniques measure mass directly Rather

they measure some proxy quantity such as galaxy counts, x-ray flux and/or temperature,
or the Sunyaev-Zeldovich decrement. The mass function is exponentially sensitive to
errors in the calibration of this mass-vs-observable relationship, just as it is exponentially
sensitive to the mass itself. These relations are harder to model than pure gravitational
growth because they involve complex baryonic physics,

e.g.,

hydrodynamics and galaxy

formation.

Weak Gravitational Lensing

Foreground mass concentrations deflect the photons from background sources on their
way to Earthbound observers, causing us to see the background source at a position
deflected from the “true” direction. The size of the deflection angle depends both on the
mass of the foreground deflector and upon the ratios of distances between observer, lens,
and source. Like cluster counting, gravitational lensing observations hence probe the
dark energy via both the expansion history,

(

)

and the growth history of density

fluctuations,

(

The deflection angles are not observable in general, because we are not at liberty to
remove the foreground lens structures to observe the unlensed position. In rare cases the
deflection is strong enough to deflect two distinct ray bundles to the observer, who will
then see two (or more) distinct images of the same source, and can deduce the deflection
angles. But in the more common and general case of

weak lensing

, we can measure the

gradient of the deflection angle because any anisotropy in this gradient makes circular
source galaxies look slightly elliptical. On a typical line of sight in the Universe, this

shear

amounts to about a 2% stretch along the preferred axis. Since most galaxies are far

from circular even in an unlensed view, it is not possible to deduce the lensing signal
from a single background galaxy image. However when large numbers of galaxies are
observed, the lensing signal can be discerned as a slight tendency for nearby galaxies to
have aligned shapes (the intrinsic galaxy shapes need not behave in this manner). The
signal-to-noise ratio for weak lensing can be very large if 10

-10

galaxy images are

surveyed, as planned for future projects.

This

cosmic shear

effect was first detected in 2000, because of the large volumes of deep

digital imaging that are necessary, and because the signal is very subtle and must be
carefully distinguished from image distortions caused by the atmosphere and telescope
optics. Levels of accuracy are advancing quickly but still far from those needed for the
best possible dark-energy measurements. But weak-lensing data is very rich. The
cosmic-shear patterns can be measured in many ways, especially if the source galaxies
can be divided by redshift. There are power spectra, cross-spectra for every pair of
source distances, cross-spectra between the shear patterns and the foreground galaxy
distribution, and non-Gaussian statistics such as bispectra. In addition, the peaks in the
shear field are a form of cluster counting. It is thus possible to diagnose and correct for
many sources of systematic error (but not all) using internal comparisons of different
weak-lensing statistics.

VII. The DETF Fiducial Model and Figure of merit

We wish to predict how well future projects would do in constraining dark energy
parameters.

The first step is to construct a cosmological model. With the choice of the equation of
state parameterization

(

)

= w

(

1−

)

, the dark energy cosmological parameters are

and

. (In general, for any component ‘

’,

is the present-day value of

where the critical density is

= 3

/ 8

, except that we define

= 1 − Ω

− Ω

Including the dark energy parameters, the DETF cosmological model is described by
eight cosmological parameters:

0 :

the present value of the dark energy equation of state parameter

: the rate of change of the dark energy equation of state parameter

: the present dark energy density

: the present matter density

: the present matter density in the form of baryons

: the Hubble constant

= (

)

1/2

, the rms primordial curvature fluctuation per

fold

evaluated at

= 0.05

Mpc

−1

: the spectral index of cosmological perturbations.

We do not assume a flat-space prior;

i.e.,

we do not set to zero the curvature contribution

(

= 1 − Ω

−

While current data are consistent with zero curvature, and

most inflation models predict |

| ~ 10

-5

, this remains a theoretical prejudice. Given that

the acceleration phenomenon was unanticipated by theory, it seems prudent to rely upon
observational constraints for curvature rather than accept the theoretical prejudice.

With regard to cosmological perturbations, we assume a pure power-law spectral index,
no massive neutrinos, and pure adiabatic perturbations. Allowing for such complications
(or others such as running of the spectral index) would weaken the derived dark-energy
constraints for some techniques. In general, such effects are minor, and more importantly
they tend to have very little impact on the

relative

merit of dark-energy constraints from

different experiments.

CMB temperature and polarization data provide constraints on the cosmological
parameters, and also provide the distance to last scattering. We model the data anticipated
from the Planck satellite mission as detailed in the Technical Appendix, and take these
CMB constraints as prior information for any dark-energy experiment.

We also assume as a prior the result on the Hubble constant from the Hubble Space
Telescope Key Project:

±8

km s

−1

Mpc

−1

[Freedman, et al. (2001)].

The dark-energy parameters and cosmological parameters for the DETF fiducial model
were chosen to be consistent with existing observations, including the first-year WMAP
results.

= −1.0

= 0.0

0.73

0.27

0.046

72 km s

−1

Mpc

−1

5.07 × 10

−5

= 0.17

= 1.0

The next step is to model the quality and quantity of the data expected for particular
experimental implementations of the four dark energy techniques. Each data model
incorporates information on anticipated statistical and systematic errors. In the Section
IX and in the Technical Appendix we give details about the DETF data models.

For the four techniques we examine in detail (BAO, CL, SN, and WL), we construct data
models describing the evolution of a Dark Energy Program in various stages:

Stage I represents what is now known.

Stage II represents the anticipated state of knowledge upon completion of ongoing
projects that are relevant to dark-energy.

Stage III comprises near-term, medium-cost, currently proposed projects.

Stage IV comprises a Large Survey Telescope (LST), and/or the Square
Kilometer Array (SKA), and/or a Joint Dark Energy (Space) Mission (JDEM).

We use Fisher-matrix techniques (described in the Technical Appendix) to predict how
well an individual model experiment would be able to restrict the dark energy parameters

, and

. This information can be expressed in terms of the standard deviations

(

), and

(

). Since in some sense theoretical predictions for

are off by

120 orders of magnitude, the DETF has not placed high priority on precision
measurements of

. Of more relevance is the precision in

and

. The information

may be presented in terms of a diagram in the

–

plane with a contour enclosing

some confidence level (C.L.) after marginalization over the other six cosmological
parameters and any other nuisance parameters specific to the experiment. An example is
given below.

All diagrams in this report will show contours enclosing 95% C.L., i.e.,

Δχ

= 6.17

for our assumption of Gaussian uncertainties in two dimensions.

−1

DETF

Fiducial Model

Contour enclosing

95% confidence

The DETF figure of merit is defined as the reciprocal of the area of the error ellipse in
the w

–w

plane that encloses the 95% C.L. contour. (We show in the Technical

Appendix that the area enclosed in the w

–w

plane is the same as the area enclosed in

the w

–w

plane.)

Note that if dark-energy uncertainties are dominated by a noise source that scales as

-0.5

for some quantity

, such as survey area or source counts, then the figure of merit will

scale as

Recall that a goal of a dark energy program is to test whether dark energy arises from a
simple cosmological constant, (

= −1,

= 0

). A given data model may do a better job

excluding

= −1

and

= 0 than is apparent from simply quoting

(

) and

(

This is because the effect of dark energy is generally not best constrained at the present
epoch (

= 0

;

= 1

). For each data model the constraint on

(

)

+ (1−

)

varies

with

. However there is some pivot value of

, denoted as

, where the uncertainty in

(

) is minimized for a given data model. The idea is illustrated in the figure below.

Each data model results in values for

〈

〉

= [

(

)]

〈

〉

[

(

)]

, and the

correlation

〈

〉

, which determine the error ellipse. With

(

1−

)

, the

(

)

= −1

(

)

uncertainty in

is least when

1−

= −

〈

〉

〈

〉

. We demonstrate in the

Technical Appendix that:

The errors on

and

are uncorrelated,

i.e.,

the error ellipse in the

–w

plane

is not tilted;

The area of the error ellipse in the

–w

plane is the same as that in the

–w

plane, so the DETF figure of merit is proportional to [

(

)×

(

)]

−1

;

The uncertainty

(

) is the same as the uncertainty that one would have in

the equation of state parameter

that was assumed constant in time.

−1

The DETF figure of merit, which is defined to be the reciprocal of the area in the
w

−

plane that encloses the 95% C.L. region, is also proportional to

[

)×

)]

−1

For each data model the results will be presented in tabular form and in the form of a
figure in the

–

plane with 95% C.L. contours of what our task force experts feel

will be reasonable optimistic and pessimistic estimates including systematic errors. Note
that the plots are

not

in the

−

plane where the figure of merit is defined. Because

and

are uncorrelated, the ellipses would not be more informative than the tabulated

data. We plot the

–

contours so that one can perhaps see how different

experiments break degeneracies with this additional parameter.

For each model there will be a table of possible origins of systematic errors and how well
a project has to perform to be within the systematic errors of the technique.

An example is given here for the data model CL-IIIp:

Dashed contours represent pessimistic projections and solid contours represent
optimistic projections.

MODEL

(

)

(

)

(

)

(

) [

(

)

(

)]

−1

CL-IIIp-o

0.256 0.774 0.022 0.672 0.037 35.21

CL-IIIp-p

0.698 2.106 0.047 0.670 0.078 6.11

Data models are denoted by
TECHNIQUE-STAGE+QUALIFIER-OPTIMISTIC/ PESSIMISTIC.

TECHNIQUE STAGE QUALIFIER OPTIMISTIC/PESSIMISTIC

BAO I

s spectroscopic survey o optimistic

CL II

p photometric survey p pessimistic

SN III

LST Large Survey Telescope

WL IV

SKA Square Kilometer Array

S Space

For each data model we present the assumptions regarding statistical and
systematic uncertainties. While the statistical performance is reasonably
straightforward, the key is systematic errors. Considerable effort and thought went
into our projections. It is absolutely crucial that any proposed project justify its
systematic error budget.

VIII. Staging Stage IV: Ground and Space Options

Stage IV of the dark-energy program will aim for full exploitation of the available
measurement techniques. In this Section we summarize the strengths and weaknesses of
the four most prominent measurement techniques and compare the three types of
observational platforms that have been proposed (space mission, ground-based Large
Survey Telescope, and Square Kilometer Array for radio observations). Each platform
has unique advantages: as a result, none of these three platforms can at present be judged
as redundant even if another one or even two were to be built. There are very strong
motivations, from dark-energy science and more general astrophysics, for continuing
development of all three projects. The relative benefits, risks, and costs of these projects
for dark-energy science should be much better known on a time scale of a few years, if
their development and supporting research on systematic errors are pursued aggressively.

Analysis of the four techniques:

•

Baryon Acoustic Oscillations (BAO) [Dark-energy Observables: D(z),

H(z)]

Strengths:

This is the method least affected by systematic uncertainties, and

for which we have the most reliable forecasts of resources required to
accomplish a survey of chosen accuracy. This method uses a standard ruler
understood from first principles and calibrated with CMB observations. The
BAO technique can constrain

(

) (from oscillations viewed transversely) and

H(z)

(from oscillations viewed radially) well at high

which complements

other techniques. If the dark-energy approximates a cosmological constant,
then it is unimportant at high

, so high-redshift measures are useful for

controlling curvature and testing the

CDM model independent of dark

energy. If dark energy is more prominent at high redshifts than in the

CDM

model, then high-

measures of

(

) and

(

) become useful for dark-energy

constraints.

Weaknesses:

This method is the one with the least statistical power to detect

departures from the fiducial

CDM model within the (

)

parameterization, since the most precise measurements are made at

> 1,

where dark energy is relatively unimportant if dark energy approximates a
cosmological constant. Relying on photometry in place of spectroscopy for
redshift determination probably sacrifices the ability to probe

(

) directly

and reduces the signal used to determine

(

). If

is determined

photometrically, errors in the redshift distributions must be controlled very
well in order to avoid significant biases in cosmological parameter estimates.

Potential Advantages of LST:

A survey that foregoes spectroscopy can

largely compensate for the increased statistical errors on

(

) by covering

very large amounts of sky. Obtaining high galaxy number densities, as is
possible with very deep imaging, means that one can retain only the galaxies
with the very best photometric redshifts, discarding the rest without
significantly increasing the statistical errors in

(

Potential Advantages of SKA:

21-cm detection of galaxies yields high-

precision redshifts without additional effort. An SKA with wide-field
capabilities can conduct such a spectroscopic survey over the full hemisphere.

Potential Advantages of Space Mission:

It is likely that low-background NIR

spectra can obtain redshifts more quickly than ground-based surveys, over
much of the interesting redshift range.

Steps to Sharpen Forecasts:

Uncertainty in the effect of nonlinear processes

on the galaxy power spectrum can be reduced with further theoretical and
numerical studies rather than the execution of a precursor survey. Further
development of the photometric-redshift technique is required just as in the
case of weak lensing. The redshift limit of a SKA BAO survey depends upon
the evolution of neutral hydrogen content of galaxies, which is poorly
determined at present.

•

Galaxy Cluster Counting (CL) [Dark-energy Observables:

(z)/H(z) and g(z)]

Strengths:

Galaxy-cluster abundances are sensitive to both the expansion and

growth histories of the Universe, in this case with extremely strong
dependence on the growth factor. There are multiple approaches to cluster
detection: the Sunyaev-Zeldovich (SZ) effect, x-ray emission, lensing shear,
and of course optical detection of the cluster galaxies. A large SZ cluster
survey (SPT) is already funded, and is the only funded project in our Stage III
class.

Weaknesses:

While

-body simulations will be able to predict the abundance

of clusters vs. mass and vs. lensing shear to high accuracy, the prediction of
SZ, x-ray, or galaxy counts is subject to substantial uncertainties in the
baryonic physics. Dark-energy constraints are very sensitive to errors in these
“mass-observable” relations, which are likely to dominate the error budget.
This method is the one for which our forecasts are least reliable, due to this
large astrophysical systematic effect.

Potential Advantages of LST:

LST can detect galaxy clusters via the effect of

their mass on shear patterns and also via the overdensities of the cluster
galaxies themselves. Deep weak-lensing observations would play a key role
for calibrating the mass-observable relation for optical (LST) observables as
well as SZ and x-ray observables of spatially overlapping SZ or x-ray surveys.

Potential Advantages of Space Mission:

x-ray cluster survey, of course,

requires a space mission. With an optical/NIR-imaging space mission,
lensing-selected cluster surveys benefit from in the same way as WL surveys
do, by offering lower noise levels for WL mapping due to higher density of
resolved background galaxies. We subsume consideration of lensing-selected
clusters into our WL category because any cosmic-shear survey is also a
cluster survey. A similar statement can be made for optically-selected galaxy
clusters.

Potential Advantages of SKA:

None recognized: cluster galaxies tend to be

deficient in neutral hydrogen, so cluster detection is not a strength of SKA.

Steps to Sharpen Forecasts:

“Self-calibration” methods can potentially

recover much of the information lost to the mass-observable uncertainties, but

their efficacy depends critically on the complexity/diversity of cluster baryon
evolution. A better understanding of cluster baryonic physics will likely result
from the SZ surveys about to commence. Weak-lensing observations of the
detected clusters in these surveys may help as well; more generally,
intercomparison of all four kinds of observables could constrain many of the
uncertain parameters in the mass-observable relations.

•

Supernovae (SN) [Dark-energy Observables:

D(z)]

Strengths:

The most established method and the one that currently contributes

the most to the constraint of dark energy. If Type Ia SN luminosities were
exactly standard(izable) over the full redshift range 0 <

< 2, then the

statistical precision of SN method would ultimately be limited only by the
accuracy to which we can establish the astronomical flux scale across
visible/NIR spectrum.

Weaknesses:

Changes in the population of Type Ia events and foreground

extinction over time will bias dark energy parameters unless they can be
identified by signatures in the colors/spectra/light curves of individual events.
Estimates of the systematic errors that will affect given surveys are very
difficult in the absence of a quantitative understanding of the diversity of SN
events and their foreground extinction, and the ways in which this diversity is
manifested in the spectral/temporal observables of each event.

Potential Advantages of LST:

High throughput enables discovery of SNe at

very high rate (tens of thousands per year) with densely sampled light curves
with high signal-to-noise ratios in optical bands at

< 1. These large numbers

of high signal-to-noise events would be useful in the search for further
parameters to improve supernovae as standard candles and control
evolutionary effects.

Potential Advantages of Space Mission:

NIR coverage offers light curves less

affected by extinction at low

, and rest-frame-visible light curves and spectra

for

>1. We expect unified, stable photometric calibration across the

visible/NIR to be better above the atmosphere as well. In the long term,
JWST observations of SNe at

> 2 may constrain evolution of SNe and

extinction.

Potential Advantages of SKA:

None – radio detection/follow-up has not been

proposed as a principal observational tool for SNe.

Steps to Sharpen Forecasts:

A large low-

SN Ia survey (many hundreds of

events) could quantify the diversity and key observational signatures of SNe
variation, well before high-

surveys are conducted. Forecasting the quality of

photometric redshifts requires better understanding of the variety of spectra in
the rest-frame wavelength range probed by optical observations of SNe in the
0 <

< 1 range. High-

surveys should ultimately be designed to measure all

the observables needed to diagnose and correct for this diversity.

•

Weak Gravitational Lensing (WL) [Dark-energy Observables:

D(z) and g(z)]

Strengths:

The method with the greatest potential for constraining dark

energy. The multitude of WL statistics (power spectra, cross-spectra,

bispectra,

etc

.) allows internal tests for, and correction of, many potential

systematic errors. Both expansion and growth history may be determined from
WL data. The theory of WL is still developing but there appear no
fundamental barriers to doing all the calculations needed to exploit the data.
WL surveys produce shear-selected galaxy-cluster counts and photo-

BAO

data at no additional cost.

Weaknesses:

WL is likely to be limited by systematic errors arising from

incomplete knowledge of the error distributions of photometric redshifts
(except for SKA survey). While NIR data are known to be of great utility in
producing reliable photo-

’s, their ultimate impact on photo-

’s and WL dark-

energy constraints cannot be ascertained without a quantitative understanding
of the diversity of galaxy spectra at modest redshifts. The methodology of WL
is progressing rapidly but not yet mature: it is not yet demonstrated that one
can measure galaxy shapes to the statistical limits, especially from the ground.

Potential Advantages of LST:

A ground-based telescope can be built with a

large collecting area and large field-of-view that would be prohibitively
expensive in space. This higher-throughput instrument could rapidly map a
full hemisphere of sky, reducing statistical errors, and enabling repeated
observations with varying observational parameters to evaluate and control
systematic errors.

Potential Advantages of Space Mission:

Deep wide-field NIR imaging would

improve photo-

accuracy and reliability, and extend the galaxy sample to

higher redshifts. Higher angular resolution triples the number density of
galaxies with measurable shapes. Absence of thermal, gravity-loading, and
atmospheric variations allows improved correction for instrumental effects on
galaxy shapes. Observations above the atmosphere may permit more accurate
photometric calibrations, improving photo-

’s.

Potential Advantages of SKA:

21-cm observations yield precise redshift

information for every detected galaxy. An SKA with wide-field capabilities
can conduct such a survey over the full hemisphere. Radio interferometric
observations measure galaxy shapes directly in Fourier space, making it much
simpler to understand and correct for instrumental effects – if the array
contains sufficiently long baselines to resolve the source galaxies’ neutral-
hydrogen emission. Atmospheric effects on radio imaging are much less
severe than in the visible/NIR.

Steps to Sharpen Forecasts:

Since shape measurement is an instrumental and

data-processing problem, rather than an astrophysical unknown, it can be
improved and evaluated with further simulation and testing. Experiments
with new-generation telescopes with improved monitoring and control of
wavefront quality can be used to better understand the limitations of ground-
based shape measurements. Improvements in forecasts of the resources
needed for photo-

training sets are possible by quantitative study of the

diversity of galaxy spectra in the SDSS, for example, but new, high-quality,
NIR imaging, deep spectroscopic surveys (or deep imaging in tens of narrow
bands) will be needed to fully understand the limitations of photo-

’s at

> 1.

Deep 21-cm observations to ascertain the HI size and flux distribution of

moderate-redshift galaxies would reduce the uncertainty in forecasts for a
SKA survey.

Advantages of an optical ground-based dark-energy experiment (LST):

Weak lensing is potentially the most powerful probe of dark energy. An
experiment that can measure weak lensing of source galaxies to high redshifts
over half of the sky, while designing for the reduction and control of systematic
errors, is thus a very attractive proposition. The combination of high sky
coverage and depth is enabled by the product of large collecting area with large
field-of-view (“throughput”); a ground-based telescope can offer substantially
larger throughput. The ultimate limit would be set by the extent to which the
systematics can be controlled. All WL statistics have greater constraining power
with greater sky coverage. In particular, the WL three-point statistics
(bispectrum) and peak-counting statistics (cluster counts), which have not been
incorporated into the DETF forecasts for WL, would gain strength with increased
survey area or volume.

A well-studied design exists for combining photo-

WL and BAO in an LST

mission. Combination with SNe and clusters may also be feasible. Indeed, a WL
survey might have as a byproduct a cluster catalog of substantial statistical power.
LST thus appears capable of providing multiple probes of DE, including sets that
can measure expansion and growth histories independently.

If an LST does not improve over the nominal systematics levels assumed for our
Stage III experiments, then its gain in our DE figure of merit is modest: a factor of
2.5 for WL+BAO relative to knowledge at the end of Stage II, a factor of 3.4 for
WL+BAO+SN. If we anticipate lower systematic errors for all techniques with
an LST, then the figure of merit of an LST for WL+BAO is 11 times better than
the nominal Stage II. A triple-method LST, WL+BAO+SN, would have 17 times
higher figure of merit than Stage II nominal results.

The huge numbers of SNe Ia (tens of thousands per year) with high signal-to-
noise well-sampled light curves in 5 optical bands at

< 1 would be useful for

better understanding the demography of SNe in ways that are complementary to a
JDEM SN survey, which would have much smaller numbers of supernovae,
though much richer data on each one.

An LST mission would make gains in many dimensions of observational phase
space, providing great opportunities for science beyond dark energy and the
potential to make completely unanticipated discoveries.

Performance of an LST mission depends critically on control of systematic errors
in both shape measurement and photometry. Potential LSTs should be judged by
their ability to control systematic errors.

We expect significant progress in understanding the limitations of photometric
redshifts and ground-based shape measurements. A clearer picture will emerge
over the next few years, especially if the “steps to sharpen forecasts” are pursued
vigorously in the interim.

Advantages of a space-based dark-energy experiment (JDEM):

A single space mission appears capable of providing multiple probes of DE,
including sets that can measure expansion and growth histories independently.
For example, a well-studied design exists for combining SN+WL on JDEM-class
missions, with significant optically-selected and lensing-selected cluster catalogs
as a potential byproduct. Missions have also been described that exploit BAO
simultaneously with other techniques. None of the DETF White Papers describe
missions that would simultaneously exploit three techniques to the levels assumed
in the DETF data models, but the JDEM Concept Studies will better explore the
breadth achievable in a single JDEM mission.

The primary virtue of a space mission is that it offers opportunities to reduce
significantly the systematic uncertainties associated with all methods (save BAO,
for which a statistical advantage may exist in space). As a result, there is less
down-side risk in a space mission than in a purely ground-based program. With
our pessimistic projections, a combining WL and SN would provide an
improvement of a factor of 7.7 over the Stage II results. Adding BAO, again with
pessimistic systematics would bring this to 8.8. With our optimistic projection of
systematics, WL+SN provides a factor of 13 increase over Stage II, while adding
BAO would bring this to 17.

JDEM may be, in this sense, the lower-risk step into Stage IV, because space-
based surveys are capable of collecting richer data sets that are more likely to
include the key pieces of information needed to ameliorate key systematic errors.

A JDEM with NIR capabilities could make use of SNe at higher redshifts than is
possible with ground-based visible or radio detection. NIR imaging would also
lead to a higher median redshift for WL source galaxies with good photometric
redshifts, which strengthens dark-energy constraints.

The choice of methods to include on JDEM should be weighted toward those that
can best improve systematic errors over their ground-based alternatives, those that
can be most effectively combined with ground-based Stage III and IV results, and
on those that can be combined into a single mission within the expected JDEM
budget.

In this regard an x-ray cluster survey stands alone since the other JDEM
approaches use widefield optical/NIR imaging/spectroscopy. It is also possible
that a visible/NIR WL survey would have as a byproduct a cluster catalog of
substantial statistical power.

While executing both JDEM and LST would certainly improve upon the
constraints obtained from one of the two alone, the degree of improvement is
difficult to forecast at this time due to the uncertainties in the systematic-error
levels of both.

Most proposed forms of JDEM observatory would greatly expand our capabilities
for astronomical investigations beyond dark energy, for example by providing
unique high-throughput, high-resolution, visible/NIR imaging capabilities.

We expect the Concept Studies that will soon be funded by NASA will serve to
refine estimates of both the cost and the benefit (

i.e.,

systematic-error reduction)

of a space mission. A clearer picture will emerge at this time, especially if the
“steps to sharpen forecasts” are pursued vigorously in the interim.

Advantages of a radio-interferometer dark-energy experiment (SKA):

An interferometric array with sufficiently large collecting area, baselines, field of
view, and correlator throughput can efficiently survey the 21-cm-emitting galaxy
population over half of the entire sky.

The combination of high-precision redshift information with stable
interferometric imaging can produce BAO and WL data that is unsurpassed in
statistical and systematic quality, over the volume that is accessible to the SKA.

The limiting redshift and galaxy density of a SKA survey are quite uncertain, due
to our ignorance of the 21-cm emission properties of moderate-redshift galaxies.

Current plans for the SKA envision an array capable of operation to much shorter
wavelengths than 21 cm. This higher-frequency capability may produce dark-
energy science through other investigations, for instance

constraints from

extragalactic masers, or weak-lensing observations of continuum-detected
sources.

The SKA would be a large multi-purpose facility, with dark-energy science as just
on of many primary goals. Studies of its design and cost are underway, and
continued attention to its capabilities for dark-energy science is well justified.

Baryon Acoustic Oscillations Data Models

Before the universe had cooled sufficiently for neutral atoms to persist, the plasma of
electrons, protons, other light nuclei, and photons was capable of propagating sound
waves. Each density fluctuation initially created in the plasma and dark matter
distribution was the source of a wavefront which expanded until the neutralization of the
plasma at redshift

~ 1000. The pattern of initial perturbations and expanding

wavefronts is seen in the CMB, and is ultimately imprinted on the matter distribution.
The primary manifestation of these

baryon acoustic oscillations (BAO)

is a feature at the

“sound horizon” length

, which is the distance traveled by the acoustic waves by the

time of plasma recombination. The sound horizon is known from CMB measurements,
thus providing a standard cosmic meter stick.

The meter stick can be measured both in an orientation transverse to the line-of-sight and
oriented along the line-of-sight. The sound horizon becomes apparent in the two-point
correlations between galaxies. In the transverse orientation, the angle subtended by the
sound horizon feature gives a measurement of

(

i.e.

the angular-diameter distance

to the redshift

at which it is observed. Measurement of the radial scale of the sound-

horizon feature gives

(

)

An advantage of BAO is that it does not require precision measurements of galaxy
magnitudes, though if photo-

’s are used then precision in galaxy colors is important. In

contrast to weak lensing, BAO does not require that galaxy images be resolved; only their
three-dimensional positions need be determined.

The statistical power of a BAO experiment to measure

(

), and hence dark

energy,

depends on the volume of sky surveyed, the range of

, and the precision with

which

is measured. The survey volume determines the level of sample variance in the

power spectrum or correlation function that is used to identify the sound-horizon scale. If
too few galaxies are used, shot noise will dominate the sample variance. We will assume
that future BAO surveys will be designed for a density

of surveyed galaxies to yield

3, so that shot noise is a minor contributor.

Measurements of BAO fall into two classes: those using spectroscopic or other high-
accuracy measurements of

and those that determine

photometrically. Results of both

sorts are now available from SDSS (Eisenstein

et al.

2005; Padmanabhan

et al.,

2005),

demonstrating the feasibility of the techniques. The spectroscopic and photometric
approaches have contrasting strengths. The large surveys using photo-

’s would have

little or no information in the radial direction. Spectroscopy of large numbers of distant
galaxies is, however, much more expensive than imaging them, so at fixed cost a photo-

survey can survey a larger volume than a spectroscopic survey. A SKA 21-cm survey
would have spectroscopic redshifts inherently over its full survey volume.

To compute the expected statistical uncertainties of BAO experiments, we first posit the
solid angle, redshift coverage, and redshift precision to be expected. The resultant

accuracy on the determination of

(

and

(

)

are taken from the estimates of Blake

et al. (2005), as described in greater detail in the Technical Appendix.

Systematic errors in the spectroscopic BAO method are more likely to arise from the
underlying theory than from the measurement process. The acoustic-oscillation
information at higher wave numbers is degraded by non-linear effects in the growth of
structure. Theoretical modeling will be limited by our understanding of “bias,” the
difference between the distribution of galaxies (the measured quantity) and the
distribution of matter (the predicted quantity). Bias and non-linearities in the

velocities

galaxies will degrade the accuracy of sound-horizon determinations in the radial
(redshift) direction.

Our pessimistic model for the cumulative effects of errors in the theory for non-linearities
and bias assumes independent 1% uncertainties in each

(

and

(

)

in each bin of

width 0.5 in

. These systematic errors are added in quadrature to the statistical errors.

An optimistic view is that future theoretical developments will reduce these uncertainties
well below future statistical errors.

There are additional issues to be addressed for photo-

BAO surveys. In practice, a large

number of galaxies will be studied both photometrically and spectroscopically and this
sample will be used to calibrate the photo-

’s. This process will have both statistical and

systematic errors. The dispersion

in difference between the true

and that inferred

photometrically (more precisely,

= Var(

phot

)/(1+

)

is a parameter in the statistical-

error model of Blake

et al.

, as it degrades the line-of-sight resolution of the acoustic

scale. The simplest systematic error is an overall bias in the photometric redshift scale.
We presume the bias is bounded by a spectroscopic survey of

= 1000 galaxies per

redshift bin, so that the bias has a prior uncertainty of

(1+

) /

√

In addition to a simple bias, catastrophic errors can occur when photometry gives an
ambiguous redshift as a consequence of the diversity of galaxy types. Our models specify
both the dispersion and the bias as a function of

, but do not explicitly include the

possibility of catastrophic error. More generally, we do not look at any non-Gaussian
distribution in the discrepancy between the true and measured

. We note that BAO

surveys require redshift information for only a small fraction of the galaxies in a given
volume in order to maintain

3. Hence a photo-

survey can choose to use only

those galaxies with the most reliable redshifts, making the calibration task easier than for
weak-lensing surveys.

The specific experimental configurations analyzed by the DETF are as follows:

We consider no

Stage II

BAO experiment.

The

Stage III spectroscopic

experiment would cover 2000 square degrees with 0.5 <

1.3, plus 300 square degrees with 2.3 <

<3.3. The interval between

of 1.3 and 2.3 is

less amenable to an efficient terrestrial spectroscopic redshift survey. This experiment

would obtain

spectra. While this ambitious program is included in Stage III, we

would not expect it to be completed for a long time, perhaps by 2016.

The

Stage III

photometric

BAO experiment would cover 4000 square degrees, with

from 0.5 to 1.4. We take photo-

dispersion

to be 0.01 in the optimistic alternative

and 0.05 in the pessimistic alternative. The RMS bias per redshift bin is taken to be

(1 +

) /

√

, with

= 1000.

For the

Stage IV ground-based (LST)

experiment, we consider a large photo-

survey,

covering 20,000 square degrees, with 0.2 <

< 3.5. The photo-

capabilities are again the

optimistic and pessimistic scenarios used for the Stage III photo-

BAO experiment.

The

Stage IV

SKA

measurement of BAO, which by its very nature has spectroscopically

measured

’s, assumes coverage of 20,000 square degrees and 0.01 <

< 1.5. The

redshift range corresponds to the median of three possible models given by Abdalla &
Rawlings (2005) for the evolution of the 21-cm luminosity function.

For the

Stage IV space-based

BAO experiment, we postulate spectroscopic coverage of

10,000 square degrees over 0.5 <

< 2.

Data Model

sky

range

(spectra per

bin)

BAO-IIIs 0.05

0.0075

0.5–1.3
2.3–3.3

… …

BAO-IIIp-o 0.1 0.5–1.4

0.01 1000

BAO-IIIp-p 0.1 0.5–1.4

0.05 1000

BAO-IVLST-o 0.5

0.2–3.5 0.01

1000

BAO-IVLST-p 0.5

0.2–3.5 0.05

1000

BAO-IVSKA 0.5 0.01–1.5 …

…

BAO-IVS-p 0.25 0.5–2.0 …

…

Cluster Data Models

The number of rare clusters of galaxies as a function of their mass

is exponentially

sensitive to the linear density field, and hence its rate of growth, as well as linearly
sensitive to the volume element. Thus, the CL method is sensitive to dark energy both
through

∝

(

)

(

) and through the growth rate of structure.

The statistical errors in a cluster-counting experiment are specified by the survey redshift
range, solid angle, and mass threshold. The latter two may be functions of redshift, and
one may specify the total number of detected clusters in lieu of one of these parameters.

We will assume that cosmological

-body simulations will in the future calibrate the

number density or mass function to the required accuracy (see below). The main
challenge for using cluster counts for dark energy tests is that the mass of a cluster is not
directly observable. A selection that is based on a threshold in an observable property of
clusters –

e.g.,

Sunyaev-Zel’dovich flux decrement, x-ray temperature and surface

brightness, optical richness, or lensing shear – will carry a corresponding selection
function in the mass domain. On the other hand it is this richness in the available
observables of a cluster that provides the opportunity to calibrate the selection
empirically and the checks against systematic errors in the modeling. Furthermore, a
cluster survey yields more dark-energy observables than the cluster abundance alone.
For example, the spatial clustering of clusters also contains BAO information.

Uncertainties in the mass selection propagate from uncertainties in the mean relationship
between the cluster observable and mass as well as uncertainties in the scatter, and, more
generally, the mass-observable distribution. Instrumental effects and contamination from
point sources can further distort the selection, especially near threshold. Given the
steepness of the mass function near the detection threshold, characterizing the mass
selection function is the main obstacle to extracting dark energy information from cluster
counts.

Our approach to forecasting the performance of cluster dark-energy surveys
correspondingly places the emphasis on determining the level of control on the mass
selection required to reach a given a given level of dark energy performance. For
illustrative purposes we examine two control functions: the mean of the mass-observable
relation and its variance. We allow these functions to evolve in redshift arbitrarily by
representing them as two independent nuisance parameters per redshift interval of

= 0.1

(see the Technical Appendix for details).

Our projections for optimistic and pessimistic levels of control on these parameters are
currently highly uncertain and

it is critical that proposed surveys demonstrate their

expected control of the mass-observable relation

. The pessimistic target reflects

determinations of the control parameters that are likely to be available internally through
“self-calibration” in any one given survey without multi-wavelength follow up. Self-
calibration techniques, still in development, use the shape of the mass function, and the
spatial clustering of clusters–both of which can be accurately calibrated as functions of

mass using

-body simulations–to constrain the mass-observable relation and

cosmological parameters simultaneously. Our approach to self calibration is to consider
it as one of many possible priors on the selection function. The optimistic target reflects
determinations that may be available through multiple mass determinations,

e.g.,

with

high resolution x-ray follow up that can determine the mass and physical properties of the
cluster gas, weak-lensing cluster-shear correlation functions for clusters near threshold, or
advances in hydrodynamic simulations. This optimistic target is highly uncertain. There
are projections in the literature that both exceed and fall short of this target by factors of a
few.

Our control parameters are illustrative but not exhaustive. For example, non-Gaussian
tails in the mass-observable relation and point-source contamination must be sufficiently
controlled such that the abundant low-mass clusters do not cause uncertainties for the
high-mass counts. For example (see the Technical Appendix) for typical surveys we
consider, uncertainties in the mass selection at

1/3

of the mean threshold must be less

in order to measure

10%

. Likewise, the mass function itself must be calibrated with

-body simulations of volumes at least as large as the planned survey volume. Accurate

cluster redshifts must be determined from follow-up observations,

e.g.,

optical

photometric redshifts. We have not included uncertainties from photometric redshifts
since the requirement per galaxy is less stringent here than in the other dark-energy
techniques: red cluster galaxies are well suited to the photo-

technique, and random

variance is reduced by the number of red galaxies available in each cluster.

Given our focus on systematic control, we do not attempt to model in detail the statistical
errors of any one proposed cluster detection method or survey. Instead we describe a
generic cluster survey that detects clusters to some mean mass threshold that is
independent of cosmology and redshift.

A more detailed calculation of a specific survey

may therefore achieve statistical errors that are moderately better or worse than our
fiducial projections

for a given total number of detected clusters, but we expect the

requirements for control over systematic errors to scale appropriately.

We note also that a cluster survey based on detection by gravitational shear will face
substantially different systematic errors than x-ray, SZ, or optical cluster surveys. In this
case the mass-observable relation is well defined, but it is the projected mass rather than
the virial mass which is detected, and the impact of this difference on dark-energy
constraints is not yet fully understood.

For

Stage II

, we model a

200

square-degree survey to a mean threshold of

−1

with a total of approximately

4,000

5,000

clusters. For the projection we take the

mean and variance of the mass-observable relation to be determined to

27%

in redshift

bins of

= 0.1.

This choice corresponds to degradation in errors in

of a factor of

= 3

from the purely statistical uncertainty.

For

Stage III

, we model a

4,000

square-degree survey to a mean threshold of

14.2

-1

with a total of about

30,000

clusters. For the pessimistic projection we take the

control parameters to be determined to

14%

, or

. For the optimistic projection we

take the control parameters to be determined to 2%, corresponding to

1.4

, or equal

statistical and systematic errors.

For

Stage IV

, we model a

20,000

square-degree survey to a mean threshold of

14.4

−1

with a total of about

30,000

clusters. For the pessimistic projection we take

the control parameters to be determined to

11%

, or

= 3

. For the optimistic projection

we take the control parameters to be determined to

1.6%,

corresponding to

= 1.4

, or

equal statistical and systematic errors.

Note that Stage IV is conservatively intended to represent a survey that sacrifices depth
for detailed measurements and control over systematic errors. Although we have kept
the optimistic and pessimistic degradation factors constant between Stages III and IV to
reflect the range of possibilities, we expect that a Stage IV survey will achieve a level
that is closer to the optimistic projection. Moreover, if systematic errors in Stage III are
demonstrably under control, then this Stage IV projection does not reflect an ultimate
limitation since the statistical errors can be further reduced by lowering the mass
threshold. Current projections in the literature employ up to 3 times the numbers
assumed here (see the Technical Appendix for the scaling of errors with cluster numbers).

Likewise, although we have kept the optimistic and pessimistic degradation factors
constant between Stages III and IV to reflect the range of possibilities we expect that a
Stage IV survey will achieve a level that is closer to the optimistic projection.

Data Model

sky

Mass

threshold
(

−1

)

Cluster
count

RMS error in
mean/variance
of mass, per

bin

Degradation
factor for

(

)

CL-II 0.005 10

4000 27% 3

CL-III-p 0.1

14.2

30,000

14% 3

CL-III-o 0.1

14.2

30,000

1.4

CL-IV-p 0.5

14.4

30,000

11% 3

CL-IV-o 0.5

14.4

30,000

1.6% 1.4

Supernovae Data Models

To date, Supernovae Type Ia (SNIa) provide the most direct indication of the accelerating
expansion of the universe. To a first approximation, these supernovae all have the same
intrinsic luminosity. Thus measuring both their redshift and their apparent peak flux
gives a direct measurement of their distance (more properly, their luminosity distance) as
a function of redshift

. In practice, the peak luminosities of SNIa are not identical, but

the variations do strongly correlate with the rate at which the supernovae decline in
brightness. The observed rates of decline yield corrections to the peak brightnesses,
reducing the dispersion in SNIa fluxes at fixed redshift. Measures of the SNe colors can
similarly reduce dispersion due to variable dust extinction in the host galaxies.
Obviously, calibration of the flux is crucial to precision measurements. In addition, each
supernova must be studied carefully enough to determine whether it is truly of type Ia.

For a SNIa survey with spectroscopic followup, the statistical uncertainties in

(

), and

hence in dark energy, are determined by the number of observed SNe, their redshift
distribution, and the standard deviation

in the SN absolute magnitude after all

corrections for decline rate and extinction. Undoubtedly there will be further luminosity
indicators discovered in the future as well. To the “intrinsic” dispersion

we must also

add measurement error, which will be denoted as

when it is treated as a distinct

quantity.

In addition to surveys covering redshift out to one or beyond, there is a need to anchor the
Hubble diagram at the lower end. For this, we postulate a sample of 500 supernovae at
low

but still in the Hubble flow. The low-

events are important because the

(standardized) absolute magnitude of the SNIa is not known

a priori

An alternative to spectroscopic followup is to rely on photometrically determined

’s.

Without the need for spectroscopy, it is possible to collect very much larger samples.
However, the cost is loss of resolution in

, the loss of some detail that would help reduce

dispersion, and possible contamination with supernovae of other types. For scenarios of
SNIa surveys using photo-

’s, their dispersion is taken as

(1+

), where

= 0.01

(0.05) in optimistic (pessimistic) projections.

Systematic errors in SNIa measures of

(

) will arise from two dominant sources, for

which neither the functional form nor the amplitude are straightforward to forecast. First,
wavelength-dependent errors in the astronomical flux scale propagate into

(

) as the

observed wavelength of the SNe redshifts through the visible and NIR spectrum. Second,
any shift with redshift in the properties of the SNe or their host extinction propagates into

(

), to the extent that it is not recognized and corrected through the use of decline rates

or other observables. If, for example, some portion of the 0.10-0.15 mag “random”
scatter in SNIa intrinsic luminosities is attributable to some physical variable that is
systematically different in high-redshift events, then dark-energy results are biased unless
this variable can be identified and measured in both nearby and distant events.

Given the ill-constrained nature of systematic errors in SNIa surveys, we adopt a simple
generic model for them. We first posit an unknown offset

in photometric calibration

between the nearby sample and the distant sample, and then place a Gaussian prior of
dispersion 0.01 mag on this offset. The possibility that there is an undiagnosed

dependent “evolution” in the brightness of supernovae or extinction properties is
simulated by adding to the expression for the observed magnitude a term

az + bz

where

the parameters

and

are drawn from independent Gaussian distributions with standard

deviations

= 0.01 /

√

2 (0.03 /

√

2) in optimistic (pessimistic) scenarios.

In addition, we suppose that the calibration of the photo-

’s would be done in bins of

with width

= 0.1. Each bin is susceptible to a bias, for which we take the prior

√

, with

= 100.

For

Stage II

, we take parameters representative of the Supernova Legacy Survey (SNLS)

or the ESSENCE survey. The redshifts are determined spectroscopically for 700
supernovae, with 0.1 <

< 1.0. These are supplemented by the postulated 500 nearby

supernovae. The dispersion in observed magnitude is the sum in quadrature of a fixed

= 0.15 and a second piece

, which is 0.02 up to

= 0.4 but then increases until it

reaches 0.3 at

= 1.

For

Stage III

, we consider both spectroscopic and photometric surveys. For the

spectroscopic survey we simply scale up the SNLS program to 2000 supernovae and
suppose that systematics can be reduced so that

= 0.12, while the

-dependent

measurement noise

, is unchanged. The model for the photometric survey uses the

same number of supernovae and the same distribution in

, but adds uncertainties

specified by

= 0.01 or 0.05.

For the

Space-based Stage IV

program, the

’s of the supernovae are assumed to be

measured spectroscopically. The 2000 supernova are distributed over the range

= 0.1 to

= 1.7.

is reduced to 0.10 independent of

with

=0.

For the

Ground-based Stage IV

program, we postulate 300,000 photometrically

measured supernovae. The same dispersion,

= 0.10, without a

-dependence, is used,

with photo-

errors represented by the choices

= 0.01 or 0.05.

Weak Lensing Data Models

The distortion pattern imparted on the images of distant light sources is sensitive to
distance ratios between observer, lens, and source redshifts, as well as to the growth
history of lensing mass structures between observer and source epochs. Schemes for
exploiting the lensing effects to extract dark-energy information are diverse in several
respects:

•

The source of background photons: in present investigations, these are galaxies, in
the range

where they can be detected in great abundance. Future

observations should also allow detection of lensing effects on CMB photons
originating at

1100

, and also photons emitted at

cm by neutral hydrogen in

the “reionization epoch” of

•

The wavelength of detection: galaxy shapes can be measured in large numbers
either by optical/NIR detection, or by interferometric observations in the radio.
CMB photons must of course be observed in the radio/mm regime, and

-cm

emission from reionization is redshifted well into the radio regime.

•

The source and extent of redshift knowledge for sources (and lenses): at the very
least, the distribution

of sources must be known to high accuracy to extract

dark-energy constraints. Much more powerful are surveys with redshift
information on a source-by-source basis, enabling “tomography” by examination
of the dependence of lens effects on source redshift. Low-precision photometric
redshifts are available for galaxies from broadband visible/NIR imaging. Much
higher precision and accuracy are available for the CMB, and for galaxies or
reionization-era observations in the

-cm line of hydrogen.

•

The signature of lensing to be detected: the shear of source shapes is presently the
dominant signature, but it may be possible to make use of lensing magnification
as well.

•

The statistics formed from the lensing signature(s):

The power spectrum (PS) of the lensing signal (including cross-spectra
between different source-redshift bins) is the primary observable for
Gaussian lensing fields.

The cross-correlation (CC) of the lensing signal with identified foreground
structure provides additional information, even in the Gaussian regime.

The bispectrum (BS) or other higher-order correlators of the lensing signal
carry significant information on scales where the mass distribution has
become non-linear and hence non-Gaussian.

Cluster counts (CL) can be produced by counting peaks in the density
inferred from the lensing signals.

The development of WL techniques is ongoing, and there is as yet no means for
estimating the total amount of dark-energy information available from WL data. We
restrict ourselves to the family of two-point statistics (PS and CC); additional WL
constraints will undoubtedly be available from the non-Gaussian BS and CL data, but the
theoretical tools for combining their information with the 2-point statistics are not yet in
hand. We also consider only shear measurement, not magnification, because the
experimental difficulties of the latter have not yet been surmounted, and the resultant

dark-energy constraints would be improved by less than about

. Only scenarios

including tomographic information are constructed, since such experiments would have
much stronger dark energy constraints and/or immunity to small errors in knowledge of
the redshift distribution.

All WL case studies presume that lensing is detected by the shear of typical galaxy
images. We examine cases in which these shapes are to be measured by broadband
visible/NIR imaging, with photometric redshift information; or by detection of

-cm

emission by a Square Kilometer Array with the continental-scale baselines required to
resolve galaxies at

The statistical power of such surveys is determined by these parameters:

•

The fraction

sky

of the full sky which is surveyed.

•

The noise density of the shear measurement

eff

, where

is the uncertainty

in each component of the shear

per perfectly-measured galaxy, and

eff

is the

sky density of perfectly-measured galaxies which would yield the same shear
noise as the (imperfectly) measured ensemble of galaxies.

•

The redshift distribution of source galaxies

dn/dz

. For the visible/NIR surveys,

dn/dz

is parameterized by the median source redshift

med

and we assume the form

(

)

(

)

1.5

exp

dn dz

z z

⎡

⎤

∝

−

⎣

⎦

. Values of

eff

, and

med

are

estimated from current results, analysis of deep HST imaging, and extrapolation
of redshift survey data to fainter magnitudes. For the SKA survey scenario we
adopt the range of redshift distributions specified by models A, B, and C of
Abdalla & Rawlings (2005).

•

The fiducial correlation coefficient

between the galaxy distribution and the dark-

matter distribution, which is relevant to the CC method (

cf.,

Bernstein 2006). We

assume

= 0.5

, which is in the range suggested by halo-model calculations (Hu &

Jain 2004).

Systematic errors will certainly be important in determining the dark-energy power of
WL surveys. We include the following systematic errors in our projections:

•

The theoretical power spectrum

(

k,z

) of dark matter will be calculated from

future

-body simulations, but baryonic physics will render these predictions

inexact. This “theory systematic” ultimately limits the utility of the PS statistic.
We presume that

(

k,z

) in each (

k,z

) bin will be uncertain by

= 0.5

of the

difference between baryonic and no-baryon power spectra as estimated by Zhan
& Knox (2004); see also Jing

et al.

(2006). Bins are

0.5

wide in log

(

), and

0.15

wide in ln(1+

•

The intrinsic correlations of galaxy shapes with each other and with local density
(Hirata & Seljak 2004) are left as free parameters in each (

k,z

) bin.

•

The shear measurement is assumed to be miscalibrated by a factor (1+

cal

) that

varies independently for each redshift bin. For a given scenario, a Gaussian prior
of width

(

cal

) is placed on the calibration factor of each redshift bin. The

pessimistic scenarios assume that this calibration uncertainty is no better than that

demonstrated by the best currently available methods in the STeP tests (Heymans

et al.

2006).

•

The bias and correlation coefficients between galaxies and dark matter are
presumed to take different, unknown values in each (

k,z

) bin. This again is

relevant to the CC statistic.

•

For the photo-

surveys, the dominant systematic errors are uncertainties in the

relation between photometric redshift and true redshift,

(

phot

). For WL

purposes, it is not necessary that this distribution be very narrow, but rather that it
be very well known (Ma, Huterer, & Hu 2006). We consider only the simplest
possible error in this distribution, namely that the mean

be biased with respect to

phot

for galaxies in some redshift bin. We quantify this by

ln(1+

)

, the RMS bias

in ln(1+

) for each bin of width

0.15

in ln(1+

). Pessimistic scenarios presume

that this does not improve beyond the performance attained for photo-

’s of

luminous red galaxies in the Sloan Digital Sky Survey (Padmanabhan

et al.

2005).

A full formulation of the WL forecasting methodology is given in the Technical
Appendix. The Table below gives the values of all parameters assumed for each WL
scenario studied by this Task Force. A full exploration of the response of WL dark-
energy constraints to choices of parameters is well beyond the scope of this document,
but we note a few points.

First, WL data offers a rich variety of statistics, with cross-spectra between every pair of
source-redshift bins, cross-correlations between every source and lens redshift bin, and a
range of angular scales for every such 2-point statistic. These respond to dark energy and
to systematic errors in different ways, making it possible to distinguish dark-energy
signals from systematic errors at mild penalties in statistical accuracy. The exception is
the photo-

biases (

cf.

Huterer

et al.

2006). Redshift errors lead us to make very precise

measures of distance and growth, but assign them to the incorrect redshift and hence mis-
characterize the dark energy. The SKA survey, with spectroscopic redshifts for each
galaxy, is immune to this error; however the source distribution is currently poorly
known.

Our neglect of BS and CL statistics, CMB or reionization-epoch lensing information, and
magnification information renders our forecasts conservative. On the other hand our
treatment of photometric-redshift errors is very simplistic,

e.g.,

Ma, Huterer, & Hu (2006)

demonstrate that the variance as well as bias of the photo-

estimator must be known to

high accuracy in order to avoid degradation of dark-energy constraints.

Data Model

sky

eff

(arcmin

-2

)

med

(

cal

)

ln(1+z)

WL-II 0.0042

0.25

15 1.0

0.02

WL-IIIp-p 0.1 0.25 15

1.0 0.01 0.01

WL-IIIp-o 0.1 0.25 15

1.0 0.01 0.003

WL-IVLST-p 0.5 0.25

1.0

0.01

WL-IVLST-o 0.5 0.25

1.2

0.001

WL-IVSKA-p

0.5

0.3 (b) (b) 0.0001

-0-

WL-IVSKA-o

0.5

0.3 (a) (a) 0.0001

-0-

WL-IVS-p 0.1 0.3 100 1.5

0.003 0.003

WL-IVS-o 0.1 0.3 100 1.5

0.001 0.001

(a)

dn/dz

from Abdalla & Rawlings Model A for 21-cm luminosity evolution.

(b)

dn/dz

from Abdalla & Rawlings Model B for 21-cm luminosity evolution.

Results for models

MODEL

(

)

(

)

(

)

(

) [

(

)

(

)]

−1

Stage II
(CL-II+SN-II+WL-II) 0.115 0.523 0.01 0.79 0.04 53.82
BAO-IIIp-o

0.911 3.569 0.06 0.76 0.26

1.06

BAO-IIIp-p

1.257 5.759 0.06 0.79 0.32

0.55

BAO-IIIs-o

0.424 1.099 0.04 0.63 0.11

8.04

BAO-IIIs-p

0.442 1.169 0.04 0.64 0.12

6.97

BAO-IVLST-o

0.489 1.383 0.04 0.65 0.09

7.78

BAO-IVLST-p

0.582 1.642 0.05 0.65 0.13

4.58

BAO-IVSKA-o

0.202 0.556 0.02 0.64 0.03

55.15

BAO-IVSKA-p

0.293 0.849 0.02 0.66 0.05

21.53

BAO-IVS-o

0.243 0.608 0.02 0.61 0.04

42.19

BAO-IVS-p

0.330 0.849 0.03 0.62 0.06

19.84

CL-II

1.089 3.218 0.05 0.67 0.18

1.76

CL-IIIp-o

0.256 0.774 0.02 0.67 0.04

35.21

CL-IIIp-p

0.698 2.106 0.05 0.67 0.08

6.11

CL-IVS-o

0.241 0.730 0.02 0.67 0.04

38.72

CL-IVS-p

0.730 2.175 0.05 0.67 0.07

6.23

SN-II

0.159 1.142 0.03 0.90 0.11

7.68

SN-IIIp-o

0.092 0.872 0.03 0.95 0.08

13.91

SN-IIIp-p

0.185 1.329 0.03 0.89 0.12

6.31

SN-IIIs

0.105 0.880 0.03 0.94 0.09

12.39

SN-IVLST-o

0.076 0.661 0.03 0.95 0.07

22.19

SN-IVLST-p

0.150 1.230 0.03 0.91 0.10

7.93

SN-IVS-o

0.074 0.683 0.02 0.93 0.05

27.01

SN-IVS-p

0.088 0.692 0.03 0.94 0.08

19.10

WL-II

0.560 1.656 0.05 0.67 0.12

4.89

WL-IIIp-o

0.189 0.513 0.02 0.64 0.05

42.96

WL-IIIp-p

0.277 0.758 0.03 0.65 0.07

19.55

WL-IVLST-o

0.055 0.142 0.01 0.63 0.02 453.60

WL-IVLST-p

0.187 0.495 0.02 0.64 0.06

32.04

WL-IVSKA-o

0.039 0.118 0.00 0.68 0.01 645.76

WL-IVSKA-p

0.195 0.723 0.01 0.73 0.03

39.84

WL-IVS-o

0.063 0.169 0.01 0.64 0.02 310.10

WL-IVS-p

0.103 0.249 0.01 0.60 0.03 131.72

MODEL:

(combined with

Figure of merit

Stage II)

(

)

(

)

(

)

(Ω

)

(Normalized to Stage II)

BAO-IIIp-o

0.103 0.461

0.035 0.010

1.1

BAO-IIIp-p

0.109 0.494

0.036 0.011

1.1

BAO-IIIs-o

0.091 0.376

0.034 0.009

1.5

BAO-IIIs-p

0.094 0.393

0.034 0.010

1.4

BAO-IVLST-o 0.092 0.376

0.033 0.009

1.5

BAO-IVLST-p 0.099 0.427

0.034 0.010

1.3

BAO-IVSKA-o 0.071 0.222

0.023 0.007

3.7

BAO-IVSKA-p 0.082 0.305

0.029 0.008

2.1

BAO-IVS-o

0.069 0.210

0.025 0.007

3.6

BAO-IVS-p

0.078 0.279

0.030 0.008

2.2

CL-IIIp-o

0.081 0.292

0.026 0.007

2.4

CL-IIIp-p

0.100 0.420

0.034 0.010

1.3

CL-IVS-o

0.080 0.287

0.026 0.007

2.5

CL-IVS-p

0.098 0.405

0.033 0.010

1.4

SN-IIIp-o

0.071 0.349

0.025 0.009

2.1

SN-IIIp-p

0.095 0.453

0.027 0.010

1.5

SN-IIIs

0.077 0.369

0.026 0.009

2.0

SN-IVLST-o

0.062 0.311

0.022 0.008

2.7

SN-IVLST-p

0.089 0.426

0.027 0.010

1.6

SN-IVS-o

0.060 0.271

0.020 0.006

3.5

SN-IVS-p

0.070 0.328

0.023 0.008

2.5

WL-IIIp-o

0.073 0.261

0.030 0.007

2.4

WL-IIIp-p

0.085 0.336

0.033 0.009

1.7

WL-IVLST-o 0.043 0.116

0.015 0.005 11.0

WL-IVLST-p 0.076 0.292

0.032 0.009

2.0

WL-IVSKA-o 0.035 0.106

0.013 0.004 13.8

WL-IVSKA-p 0.083 0.336

0.026 0.006

2.1

WL-IVS-o

0.046 0.133

0.017 0.005

8.1

WL-IVS-p

0.056 0.161

0.024 0.007

4.8

X. Dark-Energy Projects (Present and Future)

This inventory of present and future dark-energy observational projects is based

on the white papers received by the DETF and on publicly available descriptions. It
is necessarily incomplete as there are projects that have not reached these stages. The
survey may be particularly incomplete for projects funded and executed outside the
United States.

The targets for these projects are given as presented by their proposers. The DETF

was not charged with the analysis of specific projects, and presents these figures
without scrutiny or endorsement. We have not aimed to be complete. There are many
other ongoing projects in addition to those presented here, reflecting the current
interest and activity in this field.

Stage I

The case for dark energy currently comes from combinations of four types of

observations: Type Ia supernovae, anisotropies of the cosmic microwave background
radiation (combined with measurements of the Hubble constant or large-scale
structure), weak lensing, and baryon acoustic oscillations.

Type Ia Supernovae:

The primary evidence for the acceleration of the universe

came originally from studies of Type Ia supernovae (Riess

et al.

1998; Perlmutter

et al.

1999). Type Ia SNe remain the strongest evidence for acceleration, though

the need for dark energy can now be inferred from the other indicators. These
studies have established that supernovae at high redshift are fainter than their
local counterparts. Under the assumption of a flat universe, these results are
consistent with a cosmology in which approximately one third of the universe is
composed of matter, and two thirds dark energy. The most recent results are
consistent with the existence of a cosmological constant, where

= −1

(Astier

al.

2006; Sollerman

al.

2006).

CMB Anisotropies:

Studies of angular power spectrum of anisotropies in the

cosmic microwave background radiation from WMAP, in combination with the
Hubble Key Project or large-scale surveys of galaxies, have yielded a
cosmological parameters in excellent agreement with the supernova results
(Spergel

et al.

2003; Bennett

et al.

2003; Spergel

et al.

2006), both for the one-

year and three-year WMAP data.

Weak Lensing:

The largest weak lensing surveys to date cover 50-100 square

degrees in a single filter, and are dominated by statistical uncertainties (van
Waerbeke

et al.

2005, Jarvis

et al.

2006). The CFHT Legacy Survey currently

underway would survey 170 deg

in multiple colors, and has published

preliminary results from partial coverage, yielding

< −0.8

(68% confidence)

from weak-lensing data alone (Hoekstra et al. 2005). Jarvis et al. find

= −0.89

[+0.16,

−

0.21,95% CL] combining weak lensing, WMAP one-year data, and Type

Ia SNe, under the assumption that

is constant.

Baryon Acoustic Oscillations:

Eisenstein et al. (2005) have used about 50,000

luminous red galaxies over 3800 deg

from the Sloan Digital Sky Survey and

made the first clear detection of the acoustic peak in the correlation function at a
scale of 100

-1

Mpc. This technique already this method provides geometric

evidence for dark energy, particularly in combination with CMB data. The BAO
analysis of SDSS would presumably be extended to the full survey area of about
8000 deg

in the near future.

Stage II

The

Canada-France-Hawaii Telescope Supernova Legacy Survey

(CFHT-SNLS)

is a Canadian-French project using the 1-deg

camera on

the 3.5m CFHT telescope in Hawaii. The initial survey includes a “deep”
search for SNe (which doubles as a small, deep weak-lensing survey), and
a “wide” imaging weak lensing survey covering 170 deg

, both in multiple

filters. The SNe survey expects to discover 700 SNe in the 0.2-0.9
redshift range.

The

ESSENCE

project is a multi-year ground-based supernova survey

using the MOSAIC wide-field CCD camera on the Blanco 4-m telescope
at the Cerro Tololo Inter-American Observatory (CTIO). The goal is to
measure luminosity distances for a sample of about 200 SNIa at redshifts
in the 0.2-0.8 redshift range.

χ.

The

Sloan Digital Sky Survey-II (SDSS-II),

a three-year extension of the

original SDSS,

is using the wide-field, 2.5-meter telescope at Apache

Point Observatory to undertake a supernova survey. Scanning the SDSS
Southern equatorial stripe (about 2.5 deg wide by about120 deg long)
over the course of three 3-month campaigns (Sept-Nov. 2005-7), they are
obtaining multi-band lightcurves for about 200 Type Ia supernovae in the
redshift range

=0.1−0.3.

The

Center for Astrophysics Supernova Program

obtains low

dispersion spectra, UBVRI and recently JHK light curves for most
supernovae brighter than 18

mag and north of about 20 degrees. To date,

the sample of well-observed low-redshift supernovae totals almost 100. By
the year 2007, the sample should double.

The

Nearby Supernova Factory

is an experiment being carried out using

the large-area Yale-built QUEST camera at the 1.2m Samuel Oschin
Schmidt Telescope at Palomar. The SN Factory is designed to collect 300

or more SNIa in the redshift range 0.03-0.08, each followed up by about
15 optical spectra (3400-10,000Å) spaced by about 3-4 day intervals at the
Hawaii 2.2m telescope.

The

Katzman Automatic Imaging Telescope (KAIT),

located at Lick

Observatory on Mount Hamilton, is a 76-cm robotic telescope with a
dedicated to searching and obtaining multicolor photometry for nearby
supernovae. Since 1998, over 500 nearby supernovae have been
discovered by KAIT, the largest nearby sample to date. Funding is
provided by the National Science Foundation.

The

Carnegie Supernova Project (CSP)

is using the Las Campanas 1-m,

2.5-m and 6.5-m telescopes to follow up several ongoing supernova search
surveys (KAIT, CFHTLS, ESSENCE, SDSSII). The goal of the CSP is to
obtain an I-band rest-frame Hubble diagram for approximately 200
supernovae. Multicolor (10-color) photometry, as well as optical
spectroscopy is being obtained over the redshift range 0 <

< 0.1, and

near-infrared photometry is being obtained for supernovae in the range 0.1
<

< 0.7.

The Palomar

QUEST Survey

is a time variability sky survey using the

1.3m Samuel Oshin Schmidt Telescope with the Large Area (4

4 deg

)

Yale-Indiana QUEST CCD camera. The survey is planned to last five
years and cover 15,000 square degrees to a magnitude limit of about 21
with the aim of studying Type Ia and II supernovae as part of the Nearby
Supernova Factory, for the dark energy part of its program.

HST Searches for High Redshift Supernovae:

The

ACS HST Treasury

program and continued later HST searches have been used to discover
supernovae at

> 1. There are now 25 Type Ia supernovae known at

> 1,

and if Hubble continues to function, in the next several years, the sample
could be grown to more than 100.

PanSTARRS-1

is a 1.8m telescope survey being constructed at Haleakala

in Hawaii. The telescope is funded by the Air Force, with science
operations planned for the beginning of 2007. The telescope would have a
prototype 1.4-billion-pixel CCD camera and six filters. The dark energy
science goals potentially include supernovae, weak lensing, baryon-
oscillation, and cluster surveys, with the exact allocation of survey
resources currently undecided. It is a precursor experiment to
PanSTARRS-4 (a stage III project described below).

The

Parallel Imager for Southern Cosmological Observations

(PISCO)

is being built for use on the 6.5-meter Magellan telescope for the

purpose of obtaining simultaneous broadband images over a few

arcminute field of view for photometric redshifts for upcoming Sunyaev-
Zel’dovich surveys. The proposers expect the camera to be ready by
December, 2006.

The

South Pole Telescope (SPT)

is a 10-meter submillimeter-wave

telescope with a 1000-element bolometric focal plane array with channels
at 90, 150, 220 and 270 GHz. I

would conduct a deep, large solid angle

(4000 square degree) galaxy-cluster survey using the Sunyaev –Zel’dovich
effect. About 20,000 clusters with masses greater than 2 × 10

solar

masses are expected to be discovered. The project is funded by the NSF
Office of Polar Programs, and the survey is scheduled to start in spring
2007.

The

Atacama Cosmology Telescope (ACT)

would be located in the

Atacama Desert in Chile. It would map 200 square degrees of the
microwave sky at three frequencies (145 GHz, 220 Ghz and 265 GHz) at
arcminute angular resolution over 100 square degrees. Clusters with
masses greater than 3 × 10

solar masses would be discovered through the

Sunyaev-Zel’dovich effect, and optical spectroscopic redshifts would be
obtained for a subsample of 400 clusters. The millimeter bolometer array
camera is composed of three 32 × 32 arrays of transition edge sensing
bolometers. It is expected to be completed in 2006, with science
observations from mid-2007 through 2008. It is funded by the NSF.

The

XMM Cluster Survey (XCS)

proposes to use the XMM EPIC

camera to image about 500 deg

to discover thousands of clusters, 250 of

which have redshifts

> 1. Optical photometric redshift data are being

obtained from public archives, where possible (INT-Wide Field Survey,
XMM-ESO Imaging Survey, UBVRI CFHT data, SDSS imaging). With a
three-year time period, the proposers hope to optically image about 330
XMM cluster candidates.

The

Red-Sequence Cluster Survey 2 (RCS2)

is a shallow cluster-

counting and weak-lensing survey in 3 filters over 1000 deg

. The survey

is underway at the Canada-France-Hawaii telescope.

The

Deep Lensing Survey (DLS)

is a deep BVRz

′

imaging survey

covering 20 deg

in five 2 deg by 2 deg fields, with a primary focus on

weak lensing. It is being carried out at the CTIO 4-m with the Mosaic2
camera. As of May 2006, data-taking is almost complete, and preliminary
photometric redshift and shape catalogs have been produced.

The

Kilo-Degree Survey (KIDS)

is a 4-band, 1500 deg

optical survey

using the VLT Survey Telescope, with infrared follow-up on the VISTA
telescope. This European project is focused on weak gravitational lensing

but has potential for galaxy-cluster studies as well. It is pending approval
by the ESO TAC, to begin observations in the very near future.

The

DEEP2 Galaxy Redshift Survey

of about 50,000 galaxies at 0.7 <

< 1.3 over 3 deg

of sky is nearing completion at the Keck telescopes.

This would facilitate an optically-based census of galaxy clusters and
groups.

Stage III

The

Dark Energy Survey (DES)

would be a new 520 megapixel wide-

field camera (2.2 square degrees) mounted on the 4-m Blanco Telescope
of the Cerro Tololo Inter-American Observatory (CTIO) in Chile. This is
a US-led collaboration with collaborators from the UK and Spain. The
DES plans to obtain photometric redshifts in four bands. The planned
survey area is 5,000 square degrees. The techniques to study dark energy
would be baryon oscillations, clusters, supernovae, and weak lensing.
With the Cluster technique they plan to exploit the clusters detected by the
South Pole Telescope. The proponents plan a five-year survey and hope
to start observing in 2009.

The

Hobby-Eberly Telescope Dark Energy Experiment (HETDEX)

aims to measure BAO over two areas, each 100 square degrees, using
500,000 galaxies over the redshift range 1.8 <

< 3.7. The proposers

expect that the instrument can be completed within 3.5 years of full
funding, and that dark energy constraints would be provided by 2011.

The

Wide-Field Multi-Object Spectrograph (WFMOS)

is being

designed for the Subaru 8-m telescope at Mauna Kea. It would have a field
of view of about 1.5 degrees in diameter, and the capability of
simultaneously obtaining spectra for 4,000-5,000 objects, or about 20,000
objects per night. A redshift range of 0.5 <

< 1.3 for emission-line

galaxies would be targeted, and 2.3 <

< 3.3 for Lyman-break galaxies.

WFMOS is a second-generation Gemini instrument, proposed as part of
the ‘Aspen’ process. Its primary dark-energy goal is to measure baryon
acoustic oscillations, as well as to measure thousands of supernovae. A
precursor imaging survey would be required. It is expected by the
proposers that building and commissioning would occur in 2010-2012,
and the dark energy science survey in 2013-2016.

Pan-STARRS-4

is a large optical/near-IR survey telescope to be sited on

Mauna Kea in Hawaii, planned for 2009. It consists of an array of four
1.8m telescopes with a 7 degree field of view, each telescope equipped
with a 1.4 billion-pixel CCD camera and six filters. The dark energy
science goals include supernovae, weak lensing and clusters surveys. The
supernova survey aims to discover supernovae to

~ 1, covering

approximately 1200 square degrees with a cadence of about 4 days. A 3

steradian survey would be undertaken, useful for weak lensing and cluster
surveys. It is expected that the survey would continue for ten years. The
telescopes and instruments are funded by the Air Force, but support for
operations is not included.

One-Degree Imager (ODI)

at the WIYN 3.5m telescope at Kitt Peak is

currently being planned, and is largely funded. A baseline survey covering
9 square degrees to a depth of z ~ 27 and Y ~ 27 mag, with a time cadence
similar to LSST is planned. The details of an expanded survey are yet to
be determined. The dark energy science goals include supernovae and
weak lensing. It is expected to be commissioned in 2009.

The

One Thousand Points of Light Spectrograph

is a 1000-fiber

spectrograph to be prototyped at Lawrence Berkeley Lab. The goal is to
undertake a baryon oscillation experiment by surveying 1 million galaxies
at 0.7 <

< 1.2 on an existing 4-m class telescope, followed by 1 million

galaxies at 2.3 <

< 3 on a 10-m class telescope.

The

ALPACA

project proposes to install an 8-meter zenith-pointing

liquid-mirror telescope in Chile to conduct a 1000 deg

survey of a 3-

degree-wide strip circling the sky in five filters. The survey could begin in
2010 and would require three years. The survey would be well-suited to
SN and weak-lensing observations, and like other multicolor imaging
surveys would produce data of use for cluster counting and BAO
measurement as well.

The

Cluster Imaging eXperiment (CIX)

is a 64-pixel four spectral band

radiometer to be installed on the Large Millimeter Telescope (LMT). It
aims to use high-resolution (12 arcsecond) Sunyaev-Zel’dovich images to
measure masses, peculiar velocities and shapes of clusters, with the aim of
improving the understanding of systematic errors in SZ surveys.
Construction would be completed three years after the start of funding,
and the proposal for the camera has been submitted to the NSF ATI
program.

The

Cornell-Caltech Atacama Telescope (CCAT)

is a 25-m sub-

millimeter and millimeter-wave telescope to be located in the Atacama
Desert in Chile for high-angular-resolution thermal Sunyaev-Zel’dovich
measurements. A range of low- to high-mass clusters would be followed
up in detail to test SZ systematics and the relation between SZ flux and
cluster mass. The proposers hope CCAT would be operational in 2012.

Stage IV

The

Large Synoptic Survey Telescope (LSST)

would have a newly

constructed 8.4m telescope and a 3 Gigapixel camera. The survey would
scan a hemisphere (approximately 20,000 deg

) several times per month in

six colors. It would reach galaxies in the redshift range 0.5

3. It

would study dark energy through baryon oscillations, supernovae, and
weak-lensing techniques. The proponents hope to see first light in 2013,
with science runs in 2014.

The

Joint Dark Energy Mission

(JDEM) is a DOE/NASA effort aimed

toward a space mission to investigate dark energy. There are several
proposals that have been advanced, three of which submitted white papers
to the DETF: DESTINY, JEDI and SNAP. Other proposals may exist
and may have submitted proposals for NASA’s Mission Concept Design
competition, but they have not been described in open publications or in
confidential white papers submitted to the Task Force.

The

Dark Energy Space Telescope (DESTINY)

is a proposed

2m-class space telescope, which aims to measure near-infrared
grism spectrophotometry over the wavelength range 0.85

m <

1.7

m for high-redshift supernovae. Continuous observations of

several square degrees would yield an estimated 2500 SN1a with
0.5 <

< 1.7 in two years.

ii.

The

Joint Efficient Dark-energy Investigation (JEDI)

is a

proposed 2m space telescope with a one-degree field of view. It
would have simultaneous imaging and multi-slit (microshutter
array) spectroscopic capability to exploit the 0.8-4

m wavelength

range for measurement of supernovae, baryon oscillations and
weak lensing. It would discover 14,000 type Ia supernovae, and
survey over 10,000 and 1,000 square degrees for baryon
oscillations and weak lensing, respectively.

iii.

The

Supernova Acceleration Probe (SNAP)

is a 2m space

telescope concept with a 0.7 square-degree field of view and
optical and near-infrared imaging plus spectroscopy for the study
of Type Ia supernovae, weak lensing, and baryon oscillations.
About 2,000 extremely well-characterized, “Branch-normal” Type
Ia supernovae (out of 10,000 total) would be discovered out to a
redshift of 1.7. A 1,000 square-degree field would be covered for
weak lensing and baryon oscillations, with a larger survey possible
in an extended mission.

The

Square Kilometer Array (SKA)

is a proposed radio telescope with a

collecting area of order one square kilometer, capable of operating at a
wide range of frequencies and angular resolutions. Current baseline (goal)
specifications give a frequency range of 100 MHz - 25 GHz (60 MHz - 35
GHz) and an angular resolution of better than 0.02 arcsec at 1 GHz,
scaling with wavelength. Several dark energy experiments are planned for
the SKA: a neutral hydrogen survey of about 10

galaxies for the study of

baryon oscillations; shear statistics for about 10

continuum-detected

galaxies for weak lensing; and a determination of the Hubble constant
with about 1% accuracy from extragalactic maser sources. It is anticipated
that construction of Phase 1 (10% of the collecting area) would begin in
2011, first science with Phase 1 would begin in 2014, and the full SKA
would be operational in 2019.

Cluster Surveys:

The

10K X-Ray Cluster Survey

would use a proposed x-ray

telescope with large FOV mirrors to undertake a galaxy cluster
survey over 10,000 deg

out to a redshift of

~ 1.5. Optical

photometric redshifts would be obtained to identify a sample for
deeper follow-up x-ray observations of approximately 1,000
massive high-redshift clusters, for which masses would be
determined. Optical spectroscopic redshifts with 6.5-m class
telescopes would be obtained for clusters with x-ray follow-up.

ii.

NASA

Medium-Explorer Mission

has been proposed for an x-

ray telescope to undertake a survey of about 20,000 deg

out to

1.5 for a sample of about 100,000 galaxy clusters. A photometric
redshift survey all of these objects is also planned, with a
spectroscopic training set. The proponents say the mission could be
ready to be flown around 2011.

iii.

Constellation-X

would probe dark energy using X-ray

observations of galaxy clusters in two different ways: 1)
measurements of the X-ray gas mass fraction, with follow-up
observations of the Sunyaev-Zel’dovich effect and 2) using the
spectroscopic capability of Con-X to measure scaling relations
between X-ray measurements and mass. Short (~1ks) exposures of
2,000 of the most massive clusters out to redshifts 0 <

< 2 would

be obtained, and deeper (20-40ks) exposures of the 250-500 most
relaxed systems would be obtained to measure the X-ray gas mass
fraction. It is proposed that about 10-15% of the available time
over the first five years of the Con-X mission be aimed at dark
energy studies.

XIII. Technical Appendix

Our tools and methods .............................................................................................. 94

1.1

Fisher Matrix Overview.................................................................................... 94

1.2 Priors................................................................................................................. 95
1.3 Marginalization................................................................................................. 96
1.4

Adding two data sets......................................................................................... 96

1.5

The DETF Cosmological Parameters ............................................................... 96

1.6 Fiducial

Model .................................................................................................. 97

1.7 Pivot

Parameters ............................................................................................... 97

2 Supernova

data.......................................................................................................... 98

2.1 Statistical

Errors................................................................................................ 98

2.2 Systematic

Errors.............................................................................................. 99

Absolute Magnitude:................................................................................................. 99
Photo-z errors:........................................................................................................... 99
Quadratic

offset:.................................................................................................... 99

Step

offset: ........................................................................................................... 100

Specific SN models................................................................................................. 100

3.1 Near

Sample.................................................................................................... 100

3.2 Stage

II............................................................................................................ 100

3.3 Stage

IIIs ......................................................................................................... 100

3.4 Stage

IIIp......................................................................................................... 101

3.5 Stage

IVLST ................................................................................................... 101

3.6 Stage

IVS ........................................................................................................ 101

Baryon Oscillation data........................................................................................... 102

4.1 The

Formulas .................................................................................................. 102

4.2 Photometric-redshift

Surveys.......................................................................... 103

5 Specific

BAO

models ............................................................................................. 104

6 Clusters ................................................................................................................... 104

6.1 Statistical

Errors.............................................................................................. 104

6.2 Mass

Selection ................................................................................................ 105

6.3 Mass

Outliers .................................................................................................. 106

6.4 Fiducial

Surveys.............................................................................................. 107

7 Weak

Lensing ......................................................................................................... 108

7.1 Overview......................................................................................................... 108
7.2

Lensing Fisher Matrix..................................................................................... 108

7.3 Mass/galaxy

likelihood ................................................................................... 110

7.4 Likelihood

Function........................................................................................ 110

7.5

Lensing Fisher Matrix..................................................................................... 111

7.6

Power Spectrum Prior ..................................................................................... 111

7.7 Marginalization............................................................................................... 113
7.8 Power

Spectrum

Uncertainties........................................................................ 113

7.9

Projection onto Dark Energy .......................................................................... 114

7.10 A Note on Priors ............................................................................................. 114
7.11 Fiducial

Case................................................................................................... 115

8 PLANCK

priors ...................................................................................................... 115

Other technical issues ............................................................................................. 118

9.1 Combining

issues............................................................................................ 118

9.1.1 Nuisance

Parameters............................................................................... 118

9.1.2

Duplication of the supernova near sample.............................................. 118

9.2

Growth and transfer functions ........................................................................ 118

Bibliography for the Appendix ........................................................................... 119

1 Our tools and methods

1.1 Fisher Matrix Overview

Here we review the Fisher Matrix methods used by the DETF. These methods are
standard in many fields. First we consider a statistically simple case of a series of
measurements with Gaussian error distributions. Suppose we measure the quantity

when the remaining observables have the values

and suppose we put the values of

bins

b=1,…B

. Suppose in addition that the data should be described by a function

of the

bin

and some parameters

and that the expected variance in bin

, then we can

form

( ( )

)

−

∑∑

(1.1)

where

labels the events in bin

. If the parameters

give the true underlying

distribution

, then a Gaussian distribution of data values is:

( )

exp(

)

P y

∝

−

(1.2)

The problem, however, is to estimate parameters

given a realization of the data

Using Bayes’ theorem with uniform prior we have

P p

(

)

∝

(

) so that the

likelihood of a parameter estimate can be described as a Gaussian with the same

, now

viewed as a function of parameters. If we expand about the true values of the parameters,

, and average over realizations of the data,

( )

...

∂χ

δ δ

∂

∂ ∂

(1.3)

where the expectation values are taken at the true values

. The mean

value of the events in bin

is indeed

(

)

, so the second term vanishes. The distribution

of errors in the measured parameters is thus in the limit of high statistics proportional to

exp

∂χ

δ δ

∂ ∂

⎛

⎞

⎛

⎞

⎛

⎞

−

∝

−

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎝

⎠

⎝

⎠

⎝

⎠

(1.4)

where the Fisher matrix is

∂

σ ∂

∂

∑

(1.5)

and

is the average number of events in bin

. From this expression it follows that

( )

δ δ

−

(1.6)

In other words, the covariance matrix is simply the inverse of the Fisher matrix (and vice
versa).

More generally, if one can create a probability

(

) of the model parameters given a

set of observed data,

e.g.

by Bayesian methods, then one can define the Fisher matrix

components via

∂

∂ ∂

= −

and the Cramer-Rao theorem states that any unbiased estimator for the parameters will
deliver a covariance matrix on the parameters that is no better than

-1

. The Fisher

matrix therefore offers a best-case scenario for ones ability to constrain cosmology
parameters given a set of observations.

If we want to use some other set of parameters

the new Fisher matrix is

simply

( )

∂

σ ∂ ∂

σ ∂

∂

≡

∑

(1.7)

using the usual summation convention on

j,k

1.2 Priors

A Gaussian prior with width

can be placed on the

th parameter by adding to the

appropriate diagonal element of the Fisher matrix:

δ δ

→

(1.8)

which can also be written as

→ +

(1.9)

where in this case

is an extremely simple matrix (with a single non-zero diagonal

element).

If one wants to add a prior on some quantity that is not a single parameter of the Fisher
matrix one can work in variables where it is and then transform the diagonal ˆ

(where

the hat indicates that the new variables are used) back into the working variables using
the inverse of the transformation given in Eqn. (1.7). This generates a less trivial

given by

( )

ˆ ˆ

−

F M

(1.10)

1.3 Marginalization

On many occasions we need to produce a Fisher matrix in a smaller parameter space by
marginalizing over the undesired “nuisance” parameters. This amounts to integrating
over the nuisance parameters without assuming any additional priors on their values.
There is a simple way to do this: Invert

, remove the rows and columns that are being

marginalized over, and then invert the result to obtained the reduced Fisher matrix.

1.4 Adding two data sets

If one calculates the Fisher matrices

and

for two independent data sets

and

one can find the Fisher matrix for the combined probability distribution by adding:

A B

(1.11)

In general any marginalization over nuisance parameters must be done

after

summation

of the two Fisher matrices. If, however, the nuisance parameters of

are disjoint from

those of

, then the two data sets have independent probability distributions over the set

of nuisance parameters, and it is permissible to marginalize before summation.

The Fisher matrices produced by DETF to represent individual experiments have been
marginalized over all parameters other than the eight in Eqn. (1.12). We can therefore
legitimately sum these reduced matrices as long as we consider experiments with distinct
nuisance parameters. In many cases of interest this is incorrect,

e.g.

when combining two

different supernova experiments that may share nuisance parameters related to supernova
evolution. Care must be exercised about such combinations.

1.5 The DETF Cosmological Parameters

Because our goal is to estimate the precision with which various cosmological parameters
can be determined, the Fisher matrix provides exactly the tool needed: we simply invert it
to find the expected uncertainties and covariances.

For many of our calculations a convenient set of parameters is

{

}

, ,

w w

ω ω ω ω

∈

(1.12)

The

are equal to

for each component, i.e. their present-day mass-energy densities.

The dark-energy equation of state follows

(

(1-

)

. The power spectrum of

primordial scalar density fluctuations has slope

and normalization

defined with the

conventions of the WMAP publications (Verde et al. 2003). These parameters are
“natural” because the Friedmann equation looks simple in terms of these parameters

(

)

( )

exp 3 1+w +w ln

- w 1-a

h a

⎧

⎫

⎛

⎞

⎛ ⎞

⎨

⎬

⎜ ⎟

⎜

⎟

⎝ ⎠

⎝

⎠

⎩

⎭

(1.13)

where

(

)

ω ω ω

= =

, and current observations provide a prior constraint

0.72

0.08 (Freedman

et al

2001).

1.6 Fiducial Model

The fiducial model around which the parameters are perturbed has

0.380

0.722

0.146

0.278

0.024

72.5

0.87

7.91 10

4.16 10

−

Ω =

= −

Ω =

(1.14)

Note that

is not a free parameter for our calculations but is fixed by the CMB

temperature (and the standard assumption of three massless neutrinos).

1.7 Pivot Parameters

As discussed in Huterer and Turner, 2001 and Hu and Jain, 2004, the pivot point is the
value of

for which the uncertainty in

w(a)

is least. If we minimize

(

−

)

we find that the pivot

occurs when

−

= −

so we can take as our parameters

and

−

)

The “pivot” parameters are a linear transformation from

{

}

w w

space to

{

}

w w

1 1
0

−

⎛

⎞

∂

≡

= ⎜

⎟

∂

⎝

⎠

(1.15)

The DETF figure of merit is inversely proportional to the area of the error ellipse in the

–w

plane, that is to

det(F).

Since

the Fisher matrix in the

variables is

F’=M

and

det M=1

, it follows that the error ellipse in the

plane has the same

area.

2 Supernova data

The observables for SN data are apparent magnitudes

corrected using light curve

shapes or spectroscopic data to behave as standard candles with absolute magnitude

that

( )

The ( )

for the set of measured redshifts

{ }

are

( )

(

)

5log

(2.1)

( )

(

)

( )

(

)

sinh

( )

sin

⎧

⎪

⎨

⎪

⎪⎩

(2.2)

and

( )

a H a

η η

−

≡

∫

(2.3)

with

≡

Ω =

in Mpc

-1

The Fisher matrix is constructed by assigning Gaussian uncertainties of size

to the

corrected apparent magnitude

of each supernova. More commonly we will consider a

set of bins in redshift centered on the set of mean redshifts

{ }

with

SNe per bin.

The observables are the mean values of

in each bin but this time the statistical

magnitude uncertainty per bin is reduced by a factor

. The treatment of the

systematic errors is unchanged by binning.

2.1 Statistical Errors

The peak luminosities of Type Ia supernovae vary, even after other observable features of
the supernovae have been used to “standardize” the events. The uncertainty of the
corrected apparent magnitudes due solely to variation in the properties of SNe is denoted
as

, to which we add in quadrature a measurement uncertainty

. The assumed

values for each supernova data model are listed in Section 3.

2.2 Systematic Errors

Various systematic effects can be represented by additional nuisance parameters,
possibly applying a Gaussian prior to that parameter. The systematic-error nuisance
parameters are marginalized away to reduce the Fisher matrix to the cosmological
parameter set described in Section 1.5.

Absolute Magnitude:
The absolute magnitude

of the SNe is a nuisance parameter in all SN calculations.

This is equivalent to adding an additional parameter

off

to the distance moduli:

( )

off

→

. By incorporating the absolute magnitude into the definition of the

distance moduli, we can simply consider the

as observables rather than the apparent

magnitudes. No prior is assigned to

off

. A consequence of this is that SN

measurements do not determine

causing a degeneracy among

and

. For SN

measurements we can therefore take the basic parameter set to be

, w

, and

Including

would be redundant and would lead to a singular Fisher matrix.

Photo-z errors:
If the

{ }

are determined photometrically then the resultant uncertainties in redshift are

modeled in the following way: there is a nuisance parameter

, which is the bias in the

measurement of photo-z’s in each bin. The observables are then modeled as

(

)

. A prior

is assigned to each of the nuisance parameters:

(

)

(2.4)

Please note:

represents the number of available spectra for calibration of photometric

redshifts in the bin, and is

not

the number of detected SNe in the bin

. All our

calculations use

100

and

{

}

0.01,0.05

∈

Quadratic

offset:

The peak luminosity of supernovae might show some z-dependent effects, for example
due to evolution of the SN population or extinction properties, that are not fully corrected
by recourse to light-curve or spectroscopic information. This is represented with two
additional nuisance parameters

and

to give the evolution a quadratic redshift

dependence:

( )

→

( )

We apply equal, independent priors on each of these two parameters with

0.01 0.03

L Q

⎧

⎫

∈ ⎨

⎬

⎩

⎭

. (2.5)

These two options are our “optimistic” and “pessimistic” cases respectively.

100

Step

offset:

Each supernovae data model combines a collection of nearby supernovae with a
collection of distant supernovae obtained from separate experiments. We allow for an
offset between the photometric systems of the near and far samples. This is represented
with as an additional nuisance parameter

(“S” is for step). Specifically,

( )

→

( )

for the

near sample only.

We apply a prior to

with

0.01

mag.

3 Specific SN models

3.1 Near Sample

All specified SN data sets (except for STAGE I) include a “near sample” of 500 SNe
at

0.025

≈

. These SNe are always assumed to have spectroscopic redshifts and have the

same

as the far data.

3.2 Stage II

The Table defines the bins and number of supernovae in each bin. The per-event error is
given by adding in quadrature

=0.15 and

(

bin

≡

) from the Table. The

distribution is modeled on SNLS (Astier

et al

. 2006).

zmax

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.08

zmin

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.03

N_bin

104

111

104

100

500

sigma_bin

0.3

0.09

0.07 0.06

0.04

0.02

Note that the last bin is the “near sample” which overlaps (slightly) with the “deep”
sample.

3.3 Stage IIIs

This models a spectroscopic experiment identical to Stage II, but with number of SNe per
bin rescaled to give a total of 2001 SNe in the far sample:

zmax

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.08

zmin

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.03

N_bin

195

298

318

298

286

246

195

123

500

sigma_bin

0.3

0.09

0.07 0.06

0.04

0.02

101

3.4 Stage IIIp

The same distribution as Stage IIIs, but done with photo-z’s.

zmax

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.05

zmin

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.05

0.01

N_bin

195

298

318

298

286

246

195

123

500

In this case the random errors are described by

=0.12 mag,

=0. Additional photo-z

errors are modeled by the nuisance parameters discussed above, so that StageIIIp01 and
StageIIIp05 are produced according to the value of

for the photo-z errors. The last

bin is the “near sample” that does not accrue photo-z errors. Note that (for historical
reasons) the last two bins are slightly different than for the Stage II and Stage IIs cases.

3.5 Stage IVLST

This high-statistics, photo-z experiment is taken to represent LST experiments.

zmax

1.2

1.1

0.9

0.8

0.7

0.6

0.5

zmin

1.1

0.9

0.8

0.7

0.6

0.5

0.4

N_bin

#### 20543 25505 #### 31940 31646 30379 29520

Continued:
zmax

0.4

0.3

0.2

0.1

0.05

zmin

0.3

0.2

0.1

0.05

0.01

N_bin

#### 25652 20419 7221

500

In this case the random errors are described by

=0.10 mag,

=0. Additional photo-z

errors are modeled by the nuisance parameters discussed above, so that StageIVLST01
and StageIVLST05 are produced according to the value of

for the photo-z errors.

Again, the last bin is the “near sample” that does not accrue photo-z errors.

3.6 Stage IVS

This represents a supernova experiment carried out on JDEM.

zmax

1.7

1.6

1.5

1.4

1.3

1.2

1.1

zmin

1.6

1.5

1.4

1.3

1.2

1.1

0.9

N_bin

107

119

130

142

155

170

Continued:
zmax

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.08

zmin

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.03

N_bin

179

183

171

150

124

500

In this case the random errors are described by

=0.10 mag,

=0. This is similar to

the simulation of SNAP found in astro-ph/0304509 but with more SNe in the near sample
and with a smaller value of

102

4 Baryon Oscillation data

4.1 The Formulas

We model errors on BAO data using the fitting formulae presented by Blake et al (2006).
The observables are comoving angular diameter distance

(

)

and expansion rate

(

) ,

and the quantities

(

)

( )

and ln

(

)

( )

are the observables

that appear in (1.1).

(4.1)

(where

is defined in Eqn.(2.2)). The comoving angular diameter distance is

−

(4.2)

To simulate BAO data we consider bins in redshift space of width

centered on the

grid

. For each value of

we take ln

(

)

(

)

and

(

)

(

)

to be measured with

errors

( )

4
3

(4.3)

and

( )

(4.4)

where the commoving survey volume in the redshift bin is, for a survey spanning solid
angle

sky

( )

(

)

H z

( )

Sky

(4.5)

and erasure of the baryon features by non-linear evolution is factored in using

( )

⎛
⎝⎜

⎞
⎠⎟

⎧

⎨

⎪⎪

⎩

⎪

(4.6)

We use the parameters

103

0.0085

0.0148

2.16

Gpc

1 / 2

1.4

(4.7)

The values of

and

Sky

are specified by the particular survey being modeled.

Systematic errors in the BAO method are possible due to uncertainties in the theory of
non-linear evolution and galaxy biasing. These are modeled (for both types of
observable) as independent uncertainties in the (log of) the distance measures in each
redshift bin:

→

( )

(4.8)

with

0.01

0.5

(4.9)

4.2 Photometric-redshift Surveys

If photo-z’s are used, these formulae are slightly modified. The bin redshifts

at which

the distance measures are evaluated are, as for the SNe, taken to have biases

, and we

apply a Gaussian prior to each of these nuisance parameters with

( )

(4.10)

Here N

is intended to represent the number of calibrating galaxies. In calculations we

use

1000 and

∈

0.01,0.05

{

}

. Thus the assumed photo-z bias is, at

=1, in the

range 0.003 to 0.0006.
The photometric-redshift survey also surrenders nearly all information along the line of
sight, and degrades the accuracy transverse to the line of sight as well. Continuing to
follow Blake et al (2006), we replace

in Eqns. (4.3), (4.4) and (4.7) with

0.0123

( )

34.1

−

Mpc

−

( )

⎛

⎝

⎜

⎞

⎠

⎟

= ∞

(4.11)

104

5 Specific BAO models

Name

Omega_Sky (sq. deg) zmin

zmax

number of bins Photo-z***

IIIa Part 1**

2000

0.5

1.3

10 no

IIIa Part 2

300

2.3

3.3

10 no

IIIb

4000

0.5

1.4

9 yes

IVS (JDEM)

10000

0.5

10 no

IVG (SKA)

20000

0.01

1.5

10 no

IVG_2 (LST)

20000

0.2

3.5

32 yes

** Stage III BAO is comprised of two survey parts. These are combined in one Fisher
matrix.
*** “yes” here means photo-z systematic errors are implemented as described.

6 Clusters

6.1 Statistical Errors

The number density of rare clusters as a function of their mass

is exponentially

sensitive to the linear density field and hence its rate of growth as well as linearly
sensitive to the volume element. Cosmological

-body simulations can accurately

calibrate this mass function and suggest that the differential comoving number
density of clusters can be approximated by

3.82

0.3

exp

0.64

M d

−

⎡

⎤

−

⎢

⎥

⎣

⎦

(6.1)

Where

(

)

( )

;

M z

≡

, the linear density field variance in a region enclosing

/ 3

π ρ

at the mean matter density today

. Given this exponential

sensitivity, dark energy projections for clusters also depend sensitively on the
amplitude of structure today

(

and we have assumed a value of 0.91

throughout.

Take as the data the number of clusters

in the

th pixel. The pixels are defined

by their angular and redshift extent as well as selection criteria for the clusters in
the sample. These selection criteria can be characterized as a mass selection
function ( )

P M

such that

(

)

( )

V d

M P M

∫

(6.2)

105

Statistical fluctuations in the number counts arise from Poisson shot noise and
sample variance due to the large scale structure of the universe. The covariance
matrix of the counts becomes

(

) (

)

≡

−

(6.3)

where the sample variance is

( )

( ) ( ).

(2 )

d k

b N b N

P k

∫

(6.4)

( )

is the Fourier transform of the pixel window normalized such that

( ) 1

d xW

∫

. Here

is the average bias of the selected clusters

( ; ).

b M z

M d

∫

(6.5)

Cosmological

-body simulations calibrate the bias as a function of mass. We

adopt

(

)

( ; ) 1

b M z

δ σ

−

= +

⎡

⎤

⎢

⎥

⎣

⎦

(6.6)

with 0.75

, 0.3

, and

1.69

The Fisher matrix of the number counts is then

( )

μν

−

∂

∑

(6.7)

where

contains both cosmological parameters and nuisance parameters that

define the mass selection.

6.2 Mass Selection

The mass selection function ( )

P M

carries uncertainties since the mass is not a

direct observable. Observable quantities such as the SZ flux decrement, X-ray
temperature and gas mass have uncertainties in their mean relationship to cluster
mass as well as an unknown distribution around the mean. Finally instrumental
effects and inaccurate modeling of contaminating point sources can distort the
selection especially near threshold. Characterizing the mass selection function is
the main challenge for extracting the dark energy information in the number
counts.

To assess the importance of characterizing the mass selection function we
introduce two nuisance parameters per redshift interval of

= 0.1

( ) ln

( )

Erfc

( )

lnM

P M

⎡

⎤

−

⎢

⎥

⎢

⎥

⎣

⎦

(6.8)

106

that models an unknown mean and Gaussian variance in the mass observable
relations.

Priors on these mass selection parameters can come from hydrodynamic
cosmological simulations, hydrostatic equilibrium solutions involving a measured

-ray temperature profile, gas mass from the

-ray surface brightness, weak

lensing measurements of individual clusters, and statistical weak lensing
calibration through the cluster-mass correlation function. Low and intermediate
redshift observations currently suggest that these mass selection parameters are
known to better than the 10% level.

Priors can also come from the counts data themselves in a form of “self-
calibration.” Self-calibration may be especially useful for the high redshift
clusters where multiple wavelength measurements are more difficult to obtain.
There are at least two sources of information for self-calibration: the spatial
clustering of the clusters and their abundance as a function of the observable.
Given that cosmological simulations predict both the shape of the mass function
and the bias as a function of mass, the statistical properties of the number counts
will only be consistent for the correct mass selection function. The full Fisher
matrix of the counts as a function of spatial position and observable therefore
contains more information than we represent in Eqn. (6.7). Since this information
is mainly useful for calibrating the mass-observable relation, we absorb it into
priors on the nuisance parameters. Our pessimistic projections reflect the efficacy
of self-calibration. Note however that if the mass selection is fixed by external
calibration, this additional information in the counts can enhance the dark energy
constraints themselves.

6.3 Mass Outliers

Given the steepness of the mass function near threshold, non-Gaussian tails in the
mass-observable relations can easily contaminate the counts from the low mass
end. We do not attempt to model these effects in detail but instead provide a
rough translation of constraints based on our Gaussian distribution for a more
general case.
In order for tails in the mass selection function not to overwhelm the dark energy
signal, uncertainties in the mass selection function at some

must be

controlled at roughly the level

( )

(

)

P M

⎛

⎞

⎜

⎟

⎝

⎠

(6.9)

where

is the effective slope of the mass function

between

and

For example, near

14.2

h M

−

and a typical

= 0.7,

= 2. Thus the

selection function for clusters that are 1/3 of the threshold must yield less that

∼

1% contamination in order to measure

∼

10%.

107

Examples of effects that might produce outliers include unsubtracted point source
contamination and projection effects. Note however that point sources would
affect different cluster observables in different ways. For example for SZ-selected
clusters, point contamination fills in the decrement and scatters rare high mass
clusters down in effective mass whereas for

-ray-selected clusters, AGN can

bring the abundant low mass clusters into the sample unless the instrument
resolution is better than approximately 10”. Conversely,

-ray-selected clusters

are the most robust against chance projections of structure that can cause low
mass clusters to masquerade as high mass clusters whereas lensing-selected
clusters will be limited in their utility by these projection effects. Thus the
approximately 20000 5-

shear selected clusters from LST or SNAP may not

reflect the number of clusters that can be used for dark energy studies.

6.4 Fiducial Surveys

We model a generic cluster count survey that has mass selection function with a

constant

threshold mass

and

constant

scatter of

0.25

. This is

not

expected to model an

-ray or

selection in detail so that the results should be

taken qualitatively not quantitatively. We are mainly interested in making a
correspondence between a level of degradation in

( )

vs statistical errors,

( )

statistical

, and control over the mass selection. Our pessimistic

projection reflects control over the selection that should conservatively be
available through self-calibration or extrapolation of current data provided the
selection function varies slowly with mass and redshift. Our optimistic projection
reflects levels that are potentially achievable through multi-wavelength
observations and detailed modeling of the clusters. The number of clusters in
each category should be taken as an arbitrary normalizing point since there is a
currently unknown trade-off between decreasing statistical errors and increasing
systematic errors by including more objects identified in the survey. Statistical
errors will scale mainly with

-1/2
clusters

and our results can be adjusted accordingly

as estimates of the trade-off become more refined.

Stage II:

200 deg ;

;

2; 4000 to 5000 clusters.

max

h M

−

(

)

(

)

Pessimistic:

3 or

0.27 and

0.27 per

0.1.

σ σ

Δ =

(

)

(

)

Optimistic:

2 or

0.08 and

0.08 per

0.1.

σ σ

Δ =

Stage III:

14.2

4000 deg ;

;

2; 30,000 clusters.

max

h M

−

(

)

(

)

Pessimistic:

3 or

0.14 and

0.14 per

0.1.

σ σ

Δ =

108

(

)

(

)

Optimistic:

2 or

0.02 and

0.02 per

0.1.

σ σ

Δ =

Stage IV:

14.4

20000 deg ;

;

2; 30,000 clusters.

max

h M

−

(

)

(

)

Pessimistic:

3 or

0.11 and

0.11 per

0.1.

σ σ

Δ =

(

)

(

)

Optimistic:

2 or

0.016 and

0.016 per

0.1.

σ σ

Δ =

7 Weak Lensing

7.1 Overview

This Appendix details the method for predicting dark-energy constraints from future
weak-lensing experiments. The theory and forecasting methods for weak lensing are still
evolving rapidly (Mirlada-Escude 1991; Blandford et al.1991; Kaiser 1992; Jain and
Seljak 1997; Hu 1999; Jain and Taylor 2003; Bernstein and Jain 2004; Song and Knox
2004; Takada and Jain 2004; Zhang

et al.

2005; Knox

et al.

2005; Bernstein (2006);

Huterer et al. 2006) so the DETF calculations can be considered only a snapshot. Most,
but not all, of the statistical information and systematic effects believed to be important
have been included in this analysis: shear-shear 2-point correlations are analyzed,
assuming use of tomography to exploit redshift information; similarly for galaxy-shear 2-
point correlations. Systematic errors that are treated include redshift-dependent shear
calibration biases; photo-

biases; intrinsic galaxy shape correlations; intrinsic shape-

density correlations; and uncertainties in the theoretical power spectrum due to baryonic
physics. Bispectrum information is known to be important, but not included here (Takada
and Jain 2004); uncertain structure in the photo-

distributions beyond simple biases are

also known to be important, but not included here (Ma

et al.

2006). To date there are no

analyses in the literature that treat all these elements simultaneously.

7.2 Lensing Fisher Matrix

There are three fields of interest:

is the convergence that would be inferred from the

(E-mode) shear pattern of source galaxies at this redshift shell;

is the mass overdensity;

and

is an

estimate

of the mass overdensity that is derived from the galaxy distribution.

Note that this need not be the galaxy distribution itself, but more likely will involve an
attempt to assign halos to galaxies or groups.

All three are properly functions in the 3d continuum space (

)

θ ϕ

. To render our problem

discrete, we decompose the angular variables into spherical harmonic coordinates

, and

slice the depth variable into a finite set of shells indexed by

. Each depth bin is idealized

by a set of galaxies confined to a thin shell at redshift

with comoving angular diameter

109

distance

( )

D z

. Taking the limit of a large number of shells should converge back

to the continuum limit.

We will drop the spherical-harmonic indices

from our quantities for brevity, leaving

our three fields described by vectors

, and

over the distance bins. To make a

Fisher matrix, we need a likelihood function (

)

, ,

m g

, which we then must marginalize

over the unobservable

. This probability can be expressed as

(

)

(

) (

)

, , =

, .

m g

(7.1)

Note the implicit assumption here that likelihoods for different spherical harmonics are
independent.

The first likelihood term is straightforward, if we split the convergence into the
deterministic term from gravitational lensing, and a stochastic term from the intrinsic
orientations of galaxies:

lens

intrinsic

κ κ

(7.2)

lens

(7.3)

(

)(

)

2 (1

)

D D

ω χ

−

⎧

⎪

⎨

≤

⎪⎩

(7.4)

Here we take

[

]

[

]

= Ω

, measure distance in units of

100

2998

c H

Mpc, and

approximate the effect of curvature to first order in

. The comoving radial thickness of

the lens shell is

Δ = Δ

. We’ve also introduced the systematic-error

variables

, which describe a multiplicative (calibration) error on the shear measured on

galaxies in source shell

. The calibration error is assumed to be scale-independent.

The intrinsic galaxy shapes are usually assumed to have scale-independent variance
of

, where

is an effective number of sources on shell

. We allow,

however, for the possibility that galaxies have intrinsic alignment with each other and
potentially with the local mass distribution, such that

intrinsic

A m

. A non-zero

shear-mass correlation will produce the “GI” systematic effect described by Hirata and
Seljak (2004). We must allow this term to depend upon angular scale

and redshift.

Similarly, there may exist intrinsic shape correlations that do not correlate with the local
density, so we must consider Var( )

potentially to have slight scale-dependent

departures from the fiducial

. Leaving

( )

N l

as a free parameter will allow us to

marginalize over the intrinsic alignments termed “II” by Hirata and Seljak (2004).

The random shape noise will be Gaussian to high accuracy, with a covariance matrix that
is diagonal over multipole and redshift indices. The intrinsic correlations will likely be
weak, and will not cross distance bins, so the likelihood function of the total convergence
will be Gaussian to good approximation at all scales, given by:

110

1 2

(

)

exp

(

)

(

)

− /

−

⎧

⎫

⎡

⎤

−

⎨

⎬

⎣

⎦

⎩

⎭

Am N

(7.5)

Here diag(

)

, and we have placed the intrinsic-correlation terms

into the

otherwise empty diagonal of the geometry matrix

7.3 Mass/galaxy likelihood

Next we need to determine the joint likelihood (

)

m g

. We start with the assumption that

the relation is local in redshift, and our redshift shells are thick enough that each shell is
independent of every other. We define the bias and correlation

m g

≡

(7.6)

C C

≡

(7.7)

Note that this is the bias of the

mass

for a fixed

galaxy

estimator; the usual bias

parameter

specifies the converse, and is given by

r B

= /

Next we assume that the likelihood function for

and

is Gaussian, with each

multipole independent. This assumption is much less secure than the assumption
that (

)

is Gaussian, so may limit the regime of applicability. With the Gaussian

assumption, we have at each multipole

1 2

1 (

)

(

)

(2 )

exp

L m g

C C

− /

⎡

⎤

⎢

⎥

⎣

⎦

⎧

⎫

⎡

⎤

−

⎪

−

⎢

⎥

⎨

⎬

⎢

⎥

⎪

⎣

⎦

⎩

⎭

(7.8)

)

r C

≡ −

(7.9)

is the portion of the mass variance which is uncorrelated with the mass estimator

7.4 Likelihood Function

Multiplying Equations (7) and (10), then integrating over

gives the Gaussian

distribution

1 2

(

)

exp

(

)

(

)

− /

−

⎧

⎫

⎡

⎤

, =

−

⎨

⎬

⎣

⎦

⎩

⎭

ABg K

ABg

(7.10)

= +

AC A

(7.11)

Here the matrices

, ,

N C C B

are diag(

)

, etc., and we recall that this likelihood

applies to a single harmonic

is the covariance matrix for

if the mass estimators

are held fixed,

i.e.

only the components of the mass distribution that are uncorrelated

with

are considered stochastic.

111

7.5 Lensing Fisher Matrix

From this multivariate Gaussian distribution we may derive a Fisher matrix using the
formulae for zero-mean distribution given,

e.g.

, by Tegmark

et al.

(1997). We also

multiply this by the number of spherical harmonic modes in our bin of width

to get

sky

(

)

(

)

l l f

−

⎡

⎤

−

⎢

⎥

⎢

⎥

⎣

⎦

= Δ

C C

K K K K

AB C AB

(7.12)

with the commas in the subscripts denoting differentiation. The Fisher information nicely
separates into three parts: the first is the information that can be gleaned from the
variances of the mass estimator

i.e.,

the galaxy power spectrum. The second term is

Fisher information that would arise from adjusting parameters to minimize the

in the

fit of

to the estimated mass

, with the values of

taken as fixed and the matrix

taken as a known covariance for the

values. The third term is information gleaned

from the covariance of the

residuals to this fit, and looks just like the Fisher matrix for

pure shear power-spectrum tomography, except that the relevant mass power spectrum in
this term is

, the power that is uncorrelated with the galaxies, not the (larger) total

power of

We take the parameters of this Fisher matrix to be, most generally:

•

, and

for each of the

redshift bins and

bins in

, for a total

parameters.

•

The intrinsic-correlation parameters

and

at each redshift and

angular scale, another 2

free parameters;

•

Cosmological parameters

and

;

•

The angular-diameter distances

(

free parameters);

•

The shear-calibration errors

(

free parameters);

•

The scale factor

at each redshift shell (

free parameters), or

equivalently a redshift estimation error

with1

/ = + +

Note that

is expressible in terms of

, and

. With these parameters, all of

the derivatives required in Equation (7.14) are simple, sparse matrices.

7.6 Power Spectrum Prior

For a given cosmological theory we should be able to predict the 3-dimensional power
spectrum (

)

P k z

of mass overdensity. This prediction, however, may have some finite

uncertainty

log

due to the difficulties of predicting non-linear growth, especially at

high

and low

where baryonic physics may be important. We incorporate the

theoretical knowledge/ignorance into the Fisher matrix by noting that the Limber
approximation implies

(

)

B C

P l D z D

/ ,

(7.13)

at each bin in

and

. We therefore define a power-spectrum prior likelihood for the bin

to be

112

log

log(

)

log(

) log (

)

log

(

)

B C

P l D z

l D z

⎡

⎤

−

/ , +

⎣

⎦

−

/ ,

∑

(7.14)

From this likelihood we may produce a Fisher matrix. We will assume that the theory
uncertainty

log

is independent of all parameters, so we need only know the derivatives

of the logarithms in the numerator of the sum. The parameters of the Fisher matrix will
clearly include those in the lensing Fisher matrix, namely{

}

C B C D

δ ω

, , , , ,

, but must

also include any further parameters that will affect the power spectrum

. This would of

course include dark-energy parameters, or any parameters of extensions to General
Relativity.

In our current implementation we make the assumption that the full non-linear power
spectrum (

)

k z

is a function solely of the linear power spectrum

at the same era.

The linear power spectrum is in turn the product of the spectrum at an initial epoch (

e.g.

recombination) and a growth factor

since that epoch. All dark-energy dependence of

the non-linear spectrum is thence absorbed into a vector

of growth-function values

at galaxy shell

. In General Relativity the linear growth factor is scale-independent; we

could generalize to a scale-dependent growth factor given a suitable non-GR gravity
theory.

We adopt a power-law primordial power spectrum

(

)

A k k

−

, where

0 05

= .

Mpc

−

by the WMAP convention. The transfer function is a function of

and

(ignoring neutrinos); we use the formulation of Eisenstein and Hu (1999). The

non-linear power spectrum can hence be expressed as

(

)

l D G

/ , ;

where we take the

shorthand {

}

n A

ω ω ω

, , , ,

for the set of cosmological parameters that are

independent of any parameterization of dark energy.

Peacock and Dodds (1996) propose a mapping from the linear to non-linear power
spectrum. We instead use the Smith

et al

. (2003) prescription, in which the non-linear

power spectrum at redshift

is determined by the linear power spectrum at

but also

requires the matter density

( )

(for nearly-flat models). We ignore this subtlety,

fixing ( )

at the value for the fiducial cosmology, so as to preserve the simplicity that

all dark-energy dependence is implicit through

. We emphasize that this simplification

could be abandoned for a more complex description of the non-linear power spectrum if
desired, at the cost of additional Fisher-matrix parameters.

The power-spectrum theory prior is now a Fisher matrix over the same parameters as the
lensing Fisher matrix, with the addition of the linear-spectrum parameters

, and

, plus the growth factor vector

over the redshift shells.

113

7.7 Marginalization

Once we have summed the lensing Fisher matrix with the power-spectrum-theory Fisher
matrix for a given

bin, we can marginalize over the parameters {

}

, ,

C B C

at all

redshifts shells. We note that when

log

is small, the power-spectrum-theory Fisher

matrix can be large, causing roundoff error in the subsequent marginalization. It is
numerically advantageous to combine the addition of the prior and the marginalization
into a single operation. The power-spectrum Fisher matrix can be expressed as

−

where

log

diag(

)

Σ =

, and

is a matrix of the derivatives of the (log) power spectra

with respect to the Fisher-matrix parameters. If we divide the Fisher parameters into the
subvectors A that we wish to keep and B that will be marginalized away, then our
summed, marginalized Fisher matrix is

(

)

−

′

−

⎡

⎤

−

⎢

⎥

⎣

⎦

(7.15)

F F F

−

(7.16)

(

)(

) (

)

F F D

D F D

F F D

−

Σ +

−

(7.17)

This form avoids roundoff error as

log

→

In the cases

and

Σ =

, we also may use this expression to recover the usual

Fisher matrix for power-spectrum tomography.

7.8 Power Spectrum Uncertainties

Zhan and Knox (2004) quantify the effect of baryonic pressure support on the observed

lensing spectrum using a halo model. White (2004) estimates the effect of baryonic

cooling with a similar approach. These two effects have opposite signs but roughly
similar amplitudes. A numerical simulation by Jing

et al.

(2006) incorporates both effects

and confirms the amplitude of the baryonic effects on the power spectrum. Hu Zhan has
kindly provided a tabulation of log (

)

P k z

, the fractional change in power due to the

hot-baryon effect. We will assume here that

log

(

)

log (

)

k z

P k z

, =

, with the logic

that the accuracy of future power-spectrum calculations will be some fraction of the total
baryonic effect.

A rough fit to the Zhan data is

2 6

1 5log (

)

log (

) 0 012

( / )

Mpc

k k

P k z

k k

−

− .

⎧

⎪

, = .

⎨

⎪

≡

⎩

(7.18)

When

log

, we simply marginalize over {

}

C C B

, ,

without applying any power-

spectrum prior information. The Fisher matrix then becomes equivalent to that used in
Bernstein (2006) for analysis of pure cross-correlation cosmography.

114

7.9 Projection onto Dark Energy

After summing the lensing and power-spectrum-prior Fisher matrices, and marginalizing
over the shell powers and biases, we next eliminate the other

-dependent nuisance

parameters

and

associated with intrinsic galaxy alignments. At present we apply

no external prior constraints on these systematic variables,

i.e.

we presume they must be

completely self-calibrated from the lensing data.

The Fisher matrices for the different

bins may now be summed. Because the power-

spectrum information is degraded at highly non-linear

(

)

k z

, we have less concern over

the choice of maximum permissible

for the analysis. The non-Gaussian nature of power

at high

is of less concern because the Fisher matrix is automatically using only cross-

correlation information at high

, and the measurement uncertainties are primarily

traceable to shear noise—which remains Gaussian at all

—rather than the behavior of

the mass fluctuations.

We may now apply any desired prior to the shear-calibration factors

, then marginalize

these away. What remains is a Fisher matrix over the parameter vector {

}

D G

, ,

At this point one may add prior constraints on the redshift biases

. This Fisher matrix

is independent of the choice of dark-energy or gravity model, as long as the gravity
model preserves scale-independent linear growth and maintains the GR form for light
deflection.

We next may constrain any model for dark energy or gravity which is described by
additional cosmological parameters

. The dark-energy model will predict the distances

(

)

D z

; ,

and similarly the growth factors

(

)

G z

δ ω

; ,

. We

may therefore project the Fisher matrix onto a new set of variables {

}

, ,

, then

marginalize over redshift errors to obtain the final Fisher matrix over the cosmological
parameters

and the dark energy parameters.

7.10 A Note on Priors

Our formalism allows for prior constraints on systematic-error nuisance parameters
which represent functions of

, namely

and

. We assume that the priors are

Gaussian in

and log

, with standard deviations

and

log

In order for our results to be stable under a change in

step size, we must scale these

priors by the inverse square root of the bin width

. We specify input values

and

log

which refer to the values on shells of width

log

0 15

= .

; then each shell gets an

equal prior uncertainty

1 2

log

0 15

⎛

⎞

⎜

⎟

⎝

⎠

1 2

log

0 15

⎛

⎞

⎜

⎟

⎝

⎠

115

In other words we are assuming that these systematic errors, which are functions of

the continuum limit, have a fixed variance per unit redshift. Alternatively we could adopt
the approach of Huterer

et al.

(2006), and project the systematic errors onto the

coefficients of a (truncated) series of orthogonal functions of

Similarly there are systematic-error priors on several quantities that are functions of both

and angular scale

, namely the power-spectrum theoretical uncertainties and the

intrinsic-correlation strengths. If we choose finite priors, we must rescale them by the
square roots of both the

and the

bin widths. We use bins logarithmically spaced in

and refer all systematic errors to a standard bin of width 0.5 dex. We specify an input

and then adopt

1 2

log

(

)

log (

)

0 15 0 5log10

k z

P k z

⎛

⎞

, =

, .

⎜

⎟

⎝

⎠

(7.19)

A statement of the level of systematic error on a

-dependent quantity is always

meaningless without some accompanying description of the averaging width or
functional form to which it applies.

7.11 Fiducial Case

The fiducial

CDM cosmology common to all DETF models is presumed. The

Eisenstein and Hu (1999) transfer function and Smith

et al.

(2003) recipe for non-

linearity define the power spectrum (

)

P k z

. The fiducial value of

follows from

Equation (7.15) given a choice of redshift binning. We set the fiducial

at all

redshifts and scales—this choice does not affect the results. Of more importance is the
choice of fiducial correlation coefficient

between mass and its estimator. We take

0 5

= .

unless otherwise noted, which leads to fiducial

)

r C

= −

and

r C

Our discretized Fisher matrix should converge toward the true continuum value as the
shell width goes to zero. In practice we find convergence of parameter accuracies for

log

0 1

≤ .

, sufficiently large that our assumption of uncorrelated redshift shells is valid

for the multipoles

100

which provide most of the information.

8 PLANCK priors

The Planck Fisher matrix is initially calculated for a flat

CDM Universe with an

adiabatic power-law primordial scalar perturbation power spectrum. Tensor and
isocurvature perturbations are assumed to be zero, as are neutrino masses. Reionization is
assumed to occur in a step process at the redshift of

which we marginalize over. We

116

also let the primordial Helium mass fraction,

float and marginalize over that. After

calculating the Planck Fisher matrix in this space (which has

Ω = +

) we

then transform it to the full standard DETF parameter space.

We model the Planck dataset as foreground-free maps of CMB temperature and
polarization over 80% of the sky with homogeneous white noise. Each map has been
smoothed with a circular Gaussian beam. The beam size and noise levels are listed in the
Table.

Experiment

max

E B

max

(GHz)

(

Planck 2000

2500 100 9.2’

5.5

∞

143

7.1’

6 11

217

5.0’

13 27

Table 1. Experimental specifications. The

are the full-width at half maximum of the beam profiles. The

and

are temperature and polarization (Stokes parameters

and

) noise standard deviations in

a pixel of area

Although Planck has frequency bands from 30 up to 850 GHz we ignore these in our
analysis. We are crudely taking foregrounds into account by assuming that over 80% of
the sky these outer channels can be used to remove the foreground contributions to the
central channels without significant increase in the noise in the central channels, and
further by assuming that the other 20% of the sky is irretrievably contaminated by
galactic emission.

We use the formalism for calculating the Fisher matrix given CMB temperature and
polarization maps as laid out in

Zaldarriaga

et al.

1997 and Bond

et al.

1997. The

important quantities are the auto and cross angular power spectra of the two fields,
temperature (T) and E-mode polarization (E), and their derivatives with respect to the
cosmological parameters.

We discard polarization information at

because our assumptions about

foregrounds are almost certainly too optimistic for polarization on large angular scales.
These low

values are important for constraining the optical depth,

, to Thomson

scattering by electrons in the reionized inter-galactic medium. Forecasts that include
foreground modeling (Tegmark et al. 2000) indicate that Planck can determine

0 01

± .

so we include the appropriate prior on

in order to achieve ( ) 0 01

σ τ

= .

. Were

we to drop the prior, but include the polarization data at

we would find

( ) 0 005

σ τ

= .

which is not only better than what we expect because of foreground

contamination, but also better than we can expect due to uncertainty in the shape of the
reionization history (Holder et al. 2003).

117

We discard temperature data at

2000

to reduce our sensitivity to contributions from

“patchy” reionization (Knox et al. 1998, Santos et al. 2003) and residual point source
contamination. For similar reasons we discard polarization data at

2500

We first calculate the Fisher matrix assuming a flat universe with a cosmological
constant. That is, we fix

= −

, 0

and

Ω =

. This is because there is a strict

geometric degeneracy (at least for

20) between these three parameters and

. We do

the calculation in this manner to make sure the degeneracy is not artificially broken. It
may appear that we are artificially breaking it by fixing three of the parameters, but we
fix that later by putting the degeneracy back in by hand. Our parameter space is

{

}

Y z

ω ω θ

, , , ,

where

is the angular size of the sound horizon, which we use instead of

is the

primordial fraction of nuclear mass in Helium-4,

is the redshift of reionization,

assumed to occur instantaneously. The Helium-4 mass fraction and reionization redshift
are nuisance parameters that we marginalize over. The first five are the five parameters
(of the eight that we care about) that are well-determined by the CMB. Varying

, and

in a manner that leaves the above parameters fixed will not change the CMB

observables, except at very low

where there is large cosmic variance.

We calculate the Fisher matrix in the

parameter space, marginalize over

and

then transform the resulting 5-dimensional Fisher matrix to the eight-dimensional

parameter space where

{

}

, ,ln

w w

ω ω

With some parameter re-ordering, the Jacobian for this transformation is mostly trivial;

i.e.,

most of the Jacobian is the identity matrix. The only non-trivial parts are the

derivatives of

with respect to all the

parameters except for n

and ln

(since these

derivatives are zero). These derivatives we calculate numerically by finite difference.

By following the above procedure we have restored and rigorously enforced the
geometric degeneracy.

Our treatment of the CMB is conservative in the sense that we have ignored lensing
signals (particularly the lensing-induced polarization B modes) and low

signals (ISW)

that are sensitive to the dark energy and could potentially provide us with more
information about the dark energy. Including the ISW effect would not significantly
change the DETF figure of merit forecasts for any probe, or combination of probes, with
figure of merit greater than about 10. To exploit the dark-energy dependence of the
lensing-induced B modes in a significant manner requires higher resolution and
sensitivity than Planck (Acquaviva and Baccigalupi 2005).

118

9 Other technical issues

9.1 Combining issues

As discussed in Section 0, we combine model data sets by adding Fisher matrices
produced individually for each data model in our eight dimensional cosmological
parameter space. This leads us to neglect several effects that should be studied in further
work.

9.1.1 Nuisance Parameters

Some data models may have common nuisance parameters. Ideally these data models
should be combined in the higher dimensional parameter space that includes the common
nuisance parameters before marginalizing down to the eight standard parameters. As an
illustration, when we construct Stage 3 and Stage 4 figures of merit “normalized to stage
2” we include Fisher matrices from all Stage 2 data in all single and combined cases.
This means, for example, that supernovae form Stage 2 and Stage 4 might be combined,
but only after what are essentially the same evolution nuisance parameters are separately
marginalized out. We’ve investigated this particular effect and found that it typically
leads to errors of no worse than 10% in the figure of merit, although in one case we found
a 25% error. This effect could also be important for other combinations and further
investigations may even lead to significant new strategies for controlling systematic
errors (see for example Zhan, 2006).

9.1.2 Duplication of the supernova near sample

Each of our supernova data models includes a “near sample.” When two SN data models
are added using our methodology the near sample is included twice. We’ve checked and
found that a more careful calculation that makes sure the near sample only appears once
do not lead to significant changes to the figure of merit. The main reason for this is that
in our data models the uncertainties on the far sample are inherited by the near sample.
(Whether this is an accurate model of a realistic program of SN studies deserves further
scrutiny.) Thus when combing two or more SN data models there is one near sample
with smaller errors which dominates over the contributions from any other near sample.

9.2 Growth and transfer functions

Part of the construction of the weak lensing and cluster data models includes modeling
the perturbation spectrum growth in the linear regime using standard transfer function
techniques. These techniques involve approximating the evolution of perturbations after
some redshift (

≈

) as being scale independent, and also assumes that

∝

around

≈

to allow for the transition between two domains of approximation (see for

example Dodelson 2004). For clusters we additionally assume that the cluster abundance
depends on the linear power spectrum in a known and universal manner

Our investigations show that in some cases we model such high quality data that the
above approximations are not sufficiently valid to produce precise determinations of our
figure of merit. While we estimate that most single weak lensing data models to have 10-

124

National Science Foundation

and the

National Aeronautics and Space Administration

Professor Garth Illingworth
University of California Santa Cruz
Lick Observatory
Santa Cruz, California 95064

Dear Dr. Illingworth:

This letter is to request that the Astronomy and Astrophysics Advisory Committee
(AAAC), in cooperation with the High Energy Physics Advisory Panel (HEPAP) of NSF
and DOE, establish a Dark Energy Task Force as a joint sub-committee to advise NSF,
NASA and DOE on the future of dark energy research.

Background and Purpose

The NRC Report

Connecting Quarks with the Cosmos

poses eleven science questions

for the new century, among which the nature of dark energy is identified as “probably the
most vexing.” The report outlines a near-term program to constrain the properties of dark
energy, which includes the measurement of the apparent brightness of Type Ia
supernovae as a function of redshift, the study of the number density of galaxies and
clusters of galaxies as a function of redshift, and the use of weak gravitational lensing to
study the growth of structure in the universe. The report also recommends the
construction of two wide-field telescopes, one in space and one on the ground, to measure
much larger numbers of supernovae with control of systematics and to map gravitational
lensing over large scales.

In response, an NSTC interagency working group has established a federal strategy for
approaching the dark energy question (see

The Physics of the Universe

). The

recommended triple-pronged strategy covers measurements of weak lensing, Type Ia
supernovae and studies of the Sunyaev-Zel'dovich effect, primarily through a ground-
based large survey telescope (LST), a space-based Joint Dark Energy Mission (JDEM)
and coordinated ground-based CMB and space-based x-ray observations of galaxy
clusters.

The joint Dark Energy Task Force (DETF) will help the agencies to identify actions that
will optimize a near- and intermediate-term dark energy program and ensure rapid
progress in the development and implementation of a concerted effort towards
understanding the nature of dark energy.

http://www.nap.edu/books/0309074061/html/

www.ostp.gov/html/physicsoftheuniverse2.pdf

125

The DETF is asked to advise the agencies on the optimum near- and intermediate-term
programs to investigate dark energy and, in cooperation with agency efforts, to advance
the justification, specification and optimization of LST and JDEM.
The DETF is asked to address the following areas:

Summarize the existing program of funded projects by projected capabilities,
systematics, risks, required developments and progress-to-date.

Where possible, similarly summarize proposed and emergent approaches and
techniques for dark energy studies; that is, characterize these approaches and
techniques by the added value the projected capabilities would provide to the
investigation of dark energy.

Identify important steps, precursors, R&D and other projects that are required in
preparation for JDEM, LST and other existing or planned experiments.

If possible, identify any areas of dark energy parameter space that the existing or
proposed projects fail to address.

The DETF is not constituted, nor has available time, to review individual proposals to
determine their technical feasibility or likelihood of meeting performance goals. Rather,
in addressing the areas above the DETF is asked to advise on the coverage of parameter
space, to identify potential knowledge gaps that would preclude informed decisions about
projects, to identify unnecessary or duplicated efforts, and to quantify the sensitivity of
the determination of dark energy parameters to experimental performance goals such as
sky coverage, number of objects, image quality or other requirements. The DETF should
also comment on areas where expanded theoretical or modeling activity would be of
significant benefit.

Reporting

The DETF Chair is responsible for preparing the final report, in consultation with all
DETF members. In accordance with FACA rules, this report will be discussed
independently at the first meetings of the AAAC and the HEPAP following completion
of the report, before formal presentation to the agencies. We request that the DETF
prepare their report for submission to the committees with a target date of December
2005.

We thank you for your efforts and wish you success in this important endeavor.

Sincerely,

Michael S. Turner

Anne Kinney

Assistant Director, Directorate for

Director, Universe Division

127

U.S. Department of Energy

and the

National Science Foundation

Professor Frederick Gilman
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, Pennsylvania 15213

Dear Dr. Gilman:

This letter is to request that the High Energy Physics Advisory Panel (HEPAP), in
cooperation with the Astronomy and Astrophysics Advisory Committee (AAAC) of NSF
and NASA, establish a Dark Energy Task Force as a joint sub-committee to advise NSF,
NASA and DOE on the future of dark energy research.

Background and Purpose

The NRC Report

Connecting Quarks with the Cosmos

poses eleven science questions

The Physics of the Universe

). The

http://www.nap.edu/books/0309074061/html/

www.ostp.gov/html/physicsoftheuniverse2.pdf

128

progress in the development and implementation of a concerted effort towards
understanding the nature of dark energy.

Charge to the Task Force

Summarize the existing program of funded projects by projected capabilities,
systematics, risks, required developments and progress-to-date.

Identify important steps, precursors, R&D and other projects that are required in
preparation for JDEM, LST and other existing or planned experiments.

If possible, identify any areas of dark energy parameter space that the existing or
proposed projects fail to address.

Reporting

130

AGENDA

Dark Energy Task Force

22–23 March 2005

National Science Foundation

4201 Wilson Blvd., Arlington, VA

Stafford I Bldg., Room 1235

Tuesday, 22 March 2005

8:30 – 9:00

Coffee and Conversation

9:00 – 9:15

Welcome and Introductions

R. Kolb

Charge from the Agencies:

9:15 – 9:30

National Science Foundation

W. Van Citters

9:30 – 9:45

National Aeronautics and Space Administration

M. Salamon

9:45 – 10:00

U.S. Department of Energy

K. Turner

10:00 – 10:30 Discussion with the Agencies

10:30 – 11:00

Break

11:00 – 12:00 Committee Discussion—Scope and Procedure

R. Kolb

12:00 – 1:00

Box Lunch and Discussion

M. Turner

1:00 – 1:30

Discussions with AAAC Chair and HEPAP Chair G. Illingworth; F. Gilman

1:30 – 2:15

Review of Dark Energy Theory

A. Albrecht

Review of Current and Emerging Approaches:

2:15 – 2:45

Identify 1

-Order Matrix of Approaches vs.

Projects

R. Kolb

2:45 – 3:15

Break

3:15 – 4:00

Type Ia Supernovae

W. Freedman

4:00 – 5:00

Committee Discussion

5:00

Adjourn for the Day

TBD

Committee Dinner

132

AGENDA

Dark Energy Task Force

29–30 September 2005

National Science Foundation

Room 595, Stafford II Building

Thursday, 29 September 2005

9:00 – 9:15

Report from the Chair

9:15 – 10:15

Discussion of level 0 findings

10:15 – 10:30

Discussion of parameters to be used by technique working groups

10:30 – 10:45

Break

10:45 – 12:15

Working groups meet to complete reports on current status of dark-energy
techniques

12:15 – 1:15

Lunch

1:15 – 2:15

Reports on current status from technique working groups

2:15 – 3:15

Revision of current status reports or move on to “next step” goals

3:15 – 3:30

Break

3:30 – 5:30

Working groups continue to discuss “next step” goals

5:30

Adjourn for the Day

6:00

Optional committee dinner, TBD

Friday, 30 September 2005

9:00 – 9:30

Planning for October meeting at UC Davis

9:30 – 10:30

Discussion of DETF report outline

10:30 – 10:45

Break

10:45 – 12:30

Continued work of technique subgroups

12:30 – 1:30

Lunch

1:30 – 3:00

Wrap-up discussion

3:00

Adjourn

133

Dark Energy Task Force

30 June –1 July 2005

Fermi National Accelerator Laboratory

Room 1 East, Wilson Hall

Thursday, 30 June 2005

9:00 – 9:10

Welcome and Announcements

R. Kolb

9:10 – 9:40

Committee Discussion

9:40 – 10:20

JDEM concept: DESTINY

J. Morse

10:20 – 10:50

Break

10:50 – 11:30 JDEM concept: JEDI

A. Crotts

11:30 – 12:00

Update on JDEM SDT Activities

A. Albrecht

12:00 – 1:00

Lunch

Reconvene in Room 9 South East

1:00 – 1:40

JDEM concept: SNAP (videoconference)

S. Perlmutter/M. Levi/E.
Linder

1:40 – 1:50

Return to Room 1 East

1:50 – 2:30

LST concept: LSST

T. Tyson

2:30 – 3:10

LST concept: Pan-STARRS

G. Magnier

3:10 – 3:40

Dark Energy Survey (DES)

J. Frieman

3:40 – 4:10

Break

Techniques and Calibrations:

4:10 – 4:30

Clusters

J. Mohr

4:30 – 4:50

Weak Lensing

G. Bernstein

4:50 – 5:10

CMB Studies

W. Hu

5:10 – 5:20

Baryon Oscillations

L. Knox

5:20 – 5:30

Redshifted 21-cm Measurements

J. Hewitt

5:30 – 6:00

Committee Discussion

6:00

Adjourn for the Day

7:00

Committee Dinner, Chez Leon

135

Dark Energy Task Force Whitepapers

Beckwith, Andrew

projectbeckwith2@yahoo.com
Proposal for using mix of analytical work with data analysis of early CMB data obtained
from the JDEM NASA - DOE Investigation

Riess,

Adam

ariess@stsci.edu

Dark Energy Evolution from HST and SNe Ia at z > 1

Allen,

Steve

swa@stanford.edu
Probing Dark Energy with Constellation X

Clarke, Tracy

tclarke@ccs.nrl.navy.mil
Laying the Groundwork for Cluster Studies: The Long Wavelength Array

Kaiser,

Nick

kaiser@ifa.hawaii.edu
The Pan-STARRS Project

Baltay, Charles

cahrles.baltay@yale.edu

The Palomar QUEST Variability Survey

Vikhlinin, Alexey

avikhlinin@cfa.harvard.edu

Probing Dark Energy with Cluster Evolution in a 10,000 Square Degree ("10K")

Thompson, Rodger

thompson@as.arizona.edu

A Molecular Probe of Dark Energy

Miller, Chris

cmiller@ctio.noao.edu

The XMM Cluster Survey (XCS)

Aldering, Greg
aldering@panisse.lbl.gov
The Nearby Supernova Factory

Jahoda, Keith
keith@milkyway.gsfc.nasa.gov
An X-ray Galaxy Cluster Survey for Investigations of Dark Energy

141

Fermilab

F e r m i   N a t i o n a l   A c c e l e r a t o r   L a b o r a t o r y
P a r t i c l e   A s t r o p h y s i c s   C e n t e r
P . O . B o x   5 0 0   -     M S 2 0 9
B a t a v i a ,  I l l i n o i s  •  60510

June 6, 2006

Dr. Garth Illingworth

Chair, Astronomy and Astrophysics Advisory Committee

Dr. Mel Shochet

Chair, High Energy Physics Advisory Panel

Dear Garth, Dear Mel,

I am pleased to transmit to you the report of the Dark Energy Task Force.

The report is a comprehensive study of the dark energy issue, perhaps the most
compelling of all outstanding problems in physical science. In the Report, we
outline the crucial need for a vigorous program to explore dark energy as fully as
possible since it challenges our understanding of fundamental physical laws and
the nature of the cosmos.

We recommend that program elements include

Prompt critical evaluation of the benefits, costs, and risks of proposed long-term
projects.

Commitment to a program combining observational techniques from one or more
of these projects that will lead to a dramatic improvement in our understanding of
dark energy. (A quantitative measure for that improvement is presented in the
report.)

Immediately expanded support for long-term projects judged to be the most
promising components of the long-term program.

Expanded support for ancillary measurements required for the long-term program
and for projects that will improve our understanding and reduction of the
dominant systematic measurement errors.

An immediate start for nearer term projects designed to advance our knowledge of
dark energy and to develop the observational and analytical techniques that will
be needed for the long-term program.

Sincerely yours, on behalf of the Dark Energy Task Force,

Edward Kolb
Director, Particle Astrophysics Center, Fermi National Accelerator Laboratory
Professor of Astronomy and Astrophysics, The University of Chicago