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General Relativity and Gravitation, Vol. 34, No. 8, August 2002 (
C
°
2002)
Cosmology with Curvature-Saturated Gravitational
Lagrangian R
/
p
1
+
l
4
R
2
Hagen Kleinert
1
and Hans-J ¨urgen Schmidt
2
Received January 11, 2002
We argue that the Lagrangian for gravity should remain bounded at large curvature, and
interpolate between the weak-field tested Einstein-Hilbert Lagrangian
L
EH
=
R
/
16
π
G
and a pure cosmological constant for large R with the curvature-saturated ansatz
L
cs
=
L
EH
/
√
1
+
l
4
R
2
, where l is a length parameter expected to be a few orders
of magnitude above the Planck length. The curvature-dependent effective gravitational
constant defined by d
L
/
d R
=
1
/
16
π
G
eff
is G
eff
=
G
√
1
+
l
4
R
2
3
, and tends to infinity
for large R, in contrast to most other approaches where G
eff
→
0. The theory possesses
neither ghosts nor tachyons, but it fails to be linearization stable. In a curvature sat-
urated cosmology, the coordinates with ds
2
=
a
2
[da
2
/
B(a)
−
d x
2
−
d y
2
−
d z
2
] are
most convenient since the curvature scalar becomes a linear function of B(a). Cosmo-
logical solutions with a singularity of type R
→ ±∞
are possible which have a bounded
energy-momentum tensor everywhere; such a behaviour is excluded in Einstein’s theory.
In synchronized time, the metric is given by
ds
2
=
dt
2
−
t
6
/
5
(d x
2
+
d y
2
+
d z
2
)
.
On the technical side we show that two different conformal transformations make
L
cs
asymptotically equivalent to the Gurovich-ansatz
L
= |
R
|
4
/
3
on the one hand, and to
Einstein’s theory with a minimally coupled scalar field with self-interaction on the
other.
KEY WORDS: Cosmology; effective gravitational constant; fourth-order gravity.
1
Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, D-14195 Berlin, Germany;
e-mail: kleinert@physik.fu-berlin.de, http://www.physik.fu-berlin.de/˜kleinert
2
Institut f¨ur Mathematik, Universit¨at Potsdam, PF 601553, D-14415 Potsdam, Germany; e-mail:
hjschmi@rz.uni-potsdam.de, http://www.physik.fu-berlin.de/˜hjschmi
1295
0001-7701/02/0800-1295/0
C
°
2002 Plenum Publishing Corporation
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1. INTRODUCTION
According to an old idea by Sakharov [1], the gravitational properties of
spacetime are caused by the bending stiffness of all quantum fields in a spacetime
of scalar curvature R. This idea of induced gravity has inspired many subsequent
theories of gravitation, from Adler’s [2] proposal to consider Einstein gravity as a
symmetry breaking effect in quantum field theory to the modern induced gravity
derived from string fluctuations [3]. Whatever the precise mechanism, any induced
gravity will lead to a Lagrangian which is bounded at large R, and may also go to
zero. The latter case would be analogous to the elastic stiffness of solids, which is
constant for small distortions, but vanishes after the solid cracks.
In this paper we investigate the physical consequences of a simple Lagrangian
which goes to a constant at large R, thus interpolating between the Einstein-
Hilbert Lagrangian for small R and a pure cosmological constant for large R. This
Lagrangian will be referred to as curvature-saturated and reads
L
cs
=
1
16
π
G
R
√
1
+
l
4
R
2
.
(1.1)
The length parameter l may range from an order of the Planck length l
P
or a
few orders of magnitude larger than l
P
. Applying standard methods and those of
Refs. [4–8], we shall derive the cosmological consequences of the saturation and
compare our ansatz with others.
One of the motivations for a renewed interest in a more detailed consideration
of cosmology with non-linear curvature terms comes from M-theory, see Ref. [9]
“Brane new world.” In [9] a conformal anomaly is considered, which turns out to
have analogous consequences as Starobinsky’s anomaly-driven inflation with R–
and R
2
-terms, see e.g. Refs. [10] for the older results. Ref. [11] contains the latest
results concerning the effective
3
-term in such models.
Our own direct motivation to tackle the model discussed below was as follows:
We tried to make the analogy proposed in [1] more closer than done by others;
the analogy with solid state physics is this one: For small forces, the resistance
to bending is proportional to this force, but after a certain threshold – defined by
cracking the solid – the resistance vanishes.
A similar line of reasoning was deduced in Ref. [12]: There the finite-size
effects from the closed Friedmann universe to the quantum states of fields have been
calculated. Instead of continuous distribution of the energy levels of the quantum
fields, one has a discrete spectrum. Qualitatively, the result is: If the radius a of the
spatial part of spacetime shrinks close to zero, which is almost the same as very
large R, then the spacings between the energy levels become larger and larger, and
after a certain threshold, all fields will be in the ground state. This behaviour shall
be represented by an effective action. The concrete form of the corresponding
effective Lagrangian is not yet fully determined (that shall be the topic of later
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l
4
R
2
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work), but preliminarily we found out that the behaviour for large R will quite
probably be of a Lagrangian bounded by a special effective
3
; so we have chosen
one of the easiest analytic functions possessing this large-R behaviour together
with the correct weak-field shape.
The paper is organized as follows: In Sec. 2 we calculate the consequences
of the effective Lagrangian
L
cs
.
In Sec. 3 we investigate the consequences of the R-dependence of the effective
gravitational constant defined by
1
16
π
G
eff
≡
d
L
d R
,
(1.2)
which is
G
eff
=
G
p
1
+
l
4
R
2
3
(1.3)
for
L
=
L
cs
and tends to infinity as R
→ ±∞
.
Then we apply two different conformal transformations to
L
cs
. One of them,
presented in Sec. 4, makes
L
cs
asymptotically equivalent to the Gurovich-ansatz
[13], [14]
L
=
R
16
π
G
+
c
1
|
R
|
4
/
3
.
(1.4)
The other transformation, by the Bicknell theorem given in Sec. 5, establishes a
conformal relation to Einstein’s theory, with a minimally coupled scalar field. In
the literature, see [15] and the references cited there, only the second of these
conformal transformations has so far been used. The physical consequences of
these three theories are, of course, quite different since the metrics are not related
to each other by coordinate transformations.
Our approach differs fundamentally from that derived from the limiting cur-
vature hypothesis (LCH) in Refs. [16], where the gravitational Lagrangian reads
L
=
R
+
3
2
¡p
1
−
R
2
/3
2
−
1
¢
(1.5)
whose derivative with respect to R diverges for R
→
3
. This divergence was
supposed to prevent a curvature singularity, a purpose not completely reached by
the model presented in the first of Refs. [16] because other curvature invariants
may still diverge. (Let us note for completeness: In the second of Refs. [16], a
more detailed version of the LCH is presented which covers also the bounding
of the other curvature invariants; it is restricted to isotropic cosmological models.
For more general space-times one faces the problem that sometimes a curva-
ture singularity exists, but all polynomial curvature invariants remain bounded
there.)
In contrast to Eq. (1.5), our model favors high curvature values.
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It turns out that the use of synchronized or conformal time is not optimal for
our problem. We therefore use a new time coordinate which we call curvature time
for the spatially flat Friedmann model. The general properties of this coordinate
choice are described in Sec. 6.
In Sec. 7 we study the consequences of curvature-saturation for some cosmo-
logical models using the coordinates of Sec. 6. In Sec. 8, finally, we summarize
our results and compare with the related papers [17] to [31].
2. FIELD EQUATIONS OF CURVATURE-SATURATED GRAVITY
The curvature-saturated Lagrangian (1.1) interpolates between the Einstein-
Hilbert Lagrangian
L
EH
=
R
16
π
G
,
(2.1)
which is experimentally confirmed at weak fields, and a pure cosmological constant
at strong fields
L
=
±
1
16
π
Gl
2
.
(2.2)
The R dependence is plotted in Fig. 1.
The usual gravitational constant is obtained from the derivative of the
Einstein-Hilbert Lagrangian:
1
16
π
G
=
d
L
EH
d R
.
(2.3)
Figure 1. Curvature-Saturated Lagrangian
L
cs
as a function of the curvature
scalar R.
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From our curvature-saturated Lagrangian (1.1) we obtain, with this derivative, the
effective gravitational constant (1.3). The definition (2.3) is motivated as follows:
If one considers the Newtonian limit for a general Lagrangian
L
(R) which may
contain a nonvanishing cosmological constant, the potential between two point
masses contains a Newtonian 1
/
r -part plus a Yukawa-like part exp(
−
r
/
r
Y
) stem-
ming from the nonlinearities of the Lagrangian; the details are given in Appendix A.
At distances much larger than r
Y
, but much smaller than 1
/
√
R, only the 1
/
r -term
survives, and the coupling strength of the 1
/
r -term is given by the effective gravi-
tational constant G
eff
. For a recent version to deduce such weak-field expressions,
see Ref. [24].
For a general Lagrangian
L
(R) such as (1.1), the calculation of the field
equation is somewhat tedious, since the Palatini formalism which simplifies the
calculation in Einstein’s theory is no longer applicable. Recall that in this, metric
and the affine connection are varied independently, the latter being identified with
the Christoffel symbol only at the end.
Here the following indirect procedure leads rather efficiently to the correct
field equations. Let
L
0
≡
d
L
d R
,
L
00
≡
d
2
L
d R
2
,
(2.4)
and form the covariant energy-momentum tensor of the gravitational field which
is given by the variational derivative of
L
with respect to the metric g
ab
:
2
ab
≡
2
√
−
g
δ
L
√
−
g
δ
g
ab
,
(2.5)
where g denotes the determinant of g
ab
. For dimensional reasons,
2
ab
has
the following structure
2
ab
=
α
L
0
R
ab
+
β
L
0
Rg
ab
+
γ
L
g
ab
+
δ
¤
L
0
g
ab
+
²
L
0
;ab
(2.6)
with the 5 real constants
α . . . ²
. These constants can be uniquely determined up
to one overall constant factor by the covariant conservation law
2
ab
;b
=
0
.
(2.7)
The overall factor is fixed by the Einstein limit l
→
0 of the theory, where
2
ab
=
(R
ab
−
1
2
Rg
ab
)
/
8
π
G. In this way we derive the following form of the covariantly
conserved energy-momentum tensor of the gravitational field
2
ab
=
1
16
π
G
(2
L
0
R
ab
−
L
g
ab
+
2
¤
L
0
g
ab
−
2
L
0
;ab
)
.
(2.8)
The calculation is straightforward, if one is careful to distinguish between (
¤
L
0
)
;a
and
¤
(
L
0
;a
), which differ by a multiple of the curvature scalar.
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Inserting our curvature-saturated Lagrangian (1.1) into (2.4) and omitting the
subscript, we have
L
=
R
2
(1
+
l
4
R
2
)
−
1
/
2
,
L
0
=
d
L
d R
=
1
2
(1
+
l
4
R
2
)
−
3
/
2
,
(2.9)
and find from (2.8)
2
ab
=
1
8
π
G
½
R
ab
(1
+
l
4
R
2
)
3
/
2
−
Rg
ab
2(1
+
l
4
R
2
)
1
/
2
+
g
ab
¤
·
1
(1
+
l
4
R
2
)
3
/
2
¸
−
·
1
(1
+
l
4
R
2
)
3
/
2
¸
;ab
)
.
(2.10)
Setting l
=
0 reduces this to 1
/
16
π
G times the Einstein tensor. The trace of (2.10)
is
2
a
a
=
1
8
π
G
½
R
+
2l
4
R
3
(1
+
l
4
R
2
)
3
/
2
−
3
¤
·
1
(1
+
l
4
R
2
)
3
/
2
¸¾
.
(2.11)
According to Einstein’s equation,
2
ab
has to be equal to the energy momentum
tensor of the matter T
ab
, i.e., T
ab
=
2
ab
. Equation (2.11) implies that in the vac-
uum, the only constant curvature scalar is R
=
0, such that this model does not
possess a de Sitter solution. Further, we can see from Eq. (2.10), that a curvature
singularity does not necessarily imply a divergence of energy-momentum, but may
be compensated by the infinity of G
eff
.
3. EFFECTIVE GRAVITATIONAL CONSTANT
AND WEAK-FIELD BEHAVIOR
Let us compare the effective gravitational constant G
eff
of our curvature-
saturated model with those of other models discussed in the literature. From (1.3)
we see that G
eff
has the weak-field expansion
G
eff
=
G
µ
1
+
3
2
l
4
R
2
+ · · ·
¶
,
(3.1)
and the strong-field expansion
G
eff
=
Gl
6
|
R
|
3
µ
1
+
3
2l
4
R
2
+ · · ·
¶
.
(3.2)
The full R-behavior is plotted in Fig. 2.
The weak-field expansion of
L
cs
is given by
L
cs
=
R
16
π
G
√
1
+
l
4
R
2
=
R
16
π
G
+
∞
X
k
=
1
b
k
R
2k
+
1
(3.3)
with real coefficients b
k
, where b
1
= −
l
4
/
32
π
G.
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l
4
R
2
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Figure 2. Effective gravitational constant as a function of the curvature scalar.
As one can see, the quadratic term is absent, so that the linearized field equa-
tion coincides with the linearized Einstein equation. Thus we encounter neither
ghosts nor tachyons; for details see Appendix B.
There is, however, a price to pay for it. The theory has lost linearization
stability of the solutions. This latter property has the following consequences: If
one performs a weak-field expansion
g
i j
=
η
i j
+
∞
X
m
=
1
²
m
g
(m)
i j
(3.4)
around flat spacetime to solve the field equation, one has to use the terms up to
the order m
=
2 to get the complete weak-field part of the set of solutions. With
this peculiarity, we obtain a well-posed Cauchy problem for the gravity theory
following from the Lagrangian
L
cs
.
Let us now compare our theory with others available in the literature. Let
L
α,
n
(R)
=
R
16
π
G
+
α
R
n
(3.5)
with some number n
>
1 and constant
α
6=
0. In analogy with Eq. (1.2) we calculate
the effective gravitational constant from
1
16
π
G
eff
=
d
L
α,
n
d R
=
1
16
π
G
+
α
n R
n
−
1
(3.6)
such that
G
eff
=
G
1
+
16
πα
n R
n
−
1
G
,
(3.7)
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Figure 3. Effective gravitational constant G
eff
for
L
α,
3
with
α >
0 as a function of R.
i.e., G
eff
→
0 as R
→ ±∞
. For n
=
2, more exactly: for all even natural numbers
n, we meet an additional peculiarity that G
eff
can diverge for finite values of R
already. Such values of R
=
R
crit
are called critical [4]. For n
=
2 we get
R
crit
= −
1
32
απ
G
,
(3.8)
and this is the region where G
eff
changes its sign, as shown in Figures 3 and 4.
At critical values of the curvature scalar, the Cauchy problem fails to be a well-
posed one.
4. CONFORMAL DUALITY
In Ref. [8], a duality transformation relating between different types of non-
linear Lagrangians has been found. In the present notation it implies the following
relation. Let
ˆg
ab
=
L
0
2
g
ab
(4.1)
be the conformally transformed metric with
L
0
6=
0, which is fulfilled by our
Lagrangian (1.1). Then the conformally transformed curvature scalar equals
ˆ
R
=
3R
L
0
2
−
4
L
L
0
3
,
(4.2)
and the associated Lagrangian is
ˆ
L
=
2R
L
0
3
−
3
L
L
0
4
.
(4.3)
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/
p
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l
4
R
2
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Figure 4. Effective gravitational constant G
eff
for
L
α,
2
with
α >
0 as a function of R.
We easily verify that ˆ
L
0
L
0
=
1. Then one can prove that g
ab
solves the vacuum
field equation following from
L
(R) if and only if ˆg
ab
of Eq. (4.1) solves the
corresponding equation for ˆ
L
( ˆ
R) of Eq. (4.3).
Example: For
L
=
R
k
+
1
we find, up to an inessential constant factor, ˆ
L
=
ˆ
R
ˆk
+
1
with ˆk
=
1
/
(2
−
1
/
k), such that for a purely quadratic theory with
L
=
R
2
,
also ˆ
L
=
ˆ
R
2
. For our curvature-saturated model
L
→
const. we should expect
a behavior with k
→ −
1, i.e., ˆk
→
1
/
3, this leads to ˆ
L
∼
ˆ
R
4
/
3
, which is the
Gurovich-model [13], cf. Eq. (1.4).
Let us study this in more detail. To simplify the expressions we use, in this
subsection only, reduced units with 16
π
G
=
1 to best exhibit the fixed point l
=
0
of this transformation making it an identity transformation if applied to Einstein’s
theory where k
=
1. In the present units, Eqs. (2.9) have to be multiplied by 2 and
become
L
=
R(1
+
l
4
R
2
)
−
1
/
2
,
L
0
=
d
L
d R
=
(1
+
l
4
R
2
)
−
3
/
2
.
(4.4)
Inserting these into (4.1)–(4.3), we obtain
ˆg
ab
=
g
ab
(1
+
l
4
R
2
)
3
(4.5)
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and
ˆ
R
= −
R(1
+
l
4
R
2
)
3
(1
−
4l
4
R
2
)
.
(4.6)
For small R we have
ˆ
R
= −
R(1
−
l
4
R
2
+ · · ·
)
,
(4.7)
and for large
|
R
|
ˆ
R
=
4l
16
R
9
µ
1
+
11
4l
4
R
2
+ · · ·
¶
.
(4.8)
The inverse function R( ˆ
R) of (4.6) is not expressible in closed form, but its small-
and large-curvature expansion can be calculated from (4.7) and (4.8)
R
= −
ˆ
R(1
+
l
4
ˆ
R
2
+ · · ·
)
,
R
=
Ã
ˆ
R
4l
16
!
1
/
9
"
1
−
11
36l
4
µ
4l
16
ˆ
R
¶
2
/
9
+ · · ·
#
(4.9)
From Eq. (4.3) we see that
ˆ
L
= −
R(1
+
l
4
R
2
)
9
/
2
(1
−
3l
4
R
2
)
(4.10)
where R( ˆ
R) has to be inserted. For large R we use the right-hand equation in (4.9)
and obtain the limiting behavior
ˆ
L
=
3l
22
Ã
ˆ
R
4l
16
!
4
/
3
"
1
−
51
6l
4
µ
4l
16
ˆ
R
¶
2
/
9
+ · · ·
#
.
(4.11)
5. BICKNELL’S THEOREM
Bicknell’s theorem [25], in the form described in Ref. [4], relates Lagrangians
of the type (2.9) to Einstein’s theory coupled minimally to a scalar field
φ
with a
certain interaction potential ˜
V (
φ
). This Lagrangian is given by
L
EH
+
1
2
φ
,
i
φ
,
i
−
˜
V (
φ
)
.
(5.1)
The relation of ˜
V (
φ
) with
L
(R) is expressed most simply by defining a field with a
different normalization
ψ
=
√
2
/
3
φ
, in terms of which the potential ˜
V (
φ
)
=
V (
ψ
)
reads
V (
ψ
)
=
L
(R)e
−
2
ψ
−
R
2
e
−
ψ
,
(5.2)
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l
4
R
2
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with R being the inverse function of
ψ
=
ln [2
L
0
(R)]
.
(5.3)
The metric in the transformed Lagrangian (5.1) is
˜g
ab
=
e
ψ
g
ab
.
(5.4)
For our particular Lagrangian (2.9) we have from (5.3):
ψ
= −
3
2
ln (1
+
l
4
R
2
)
.
(5.5)
Now we restrict our attention to the range R
>
0 where
ψ <
0; the other sign can
be treated analogously. Then (5.5) is inverted to
R
=
1
l
2
p
e
−
2
ψ/
3
−
1
,
(5.6)
such that (5.2) becomes
V (
ψ
)
=
1
2l
2
(e
−
5
ψ/
3
−
e
−
ψ
)
p
e
−
2
ψ/
3
−
1
.
(5.7)
In the range under consideration, this is a positive and monotonously increasing
function of
−
ψ
(see Fig. 5), with the large-
φ
behavior
V
=
1
2l
2
e
−
2
ψ
.
(5.8)
This is the typical exponential potential for power-law inflation. As mentioned at
the end of Section 2, no exact de Sitter inflation exists. For
ψ
→
0, also V (
ψ
)
→
0
like 4
√
2
/
3
ψ
3
/
2
.
Figure 5. Potential V (
ψ
) associated with curvature-
saturated action via Bicknell’s theorem.
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If V (
ψ
) has a quadratic minimum at some
ψ
0
with positive value V
0
=
V (
ψ
0
),
then there exists a stable de Sitter inflationary phase. As a pleasant feature, the
potential V (
ψ
) has no maximum which have given rise to tachyons.
From Eq. (5.5) one can see that for weak fields,
ψ
∼
R
2
, whereas a R
+
R
2
-
theory has
ψ
∼
R. In other words: In our model it is a better approximation to
assume the conformal factor e
ψ
to be approximately constant for weak fields than
in R
+
R
2
-theories, since at the level keeping only terms linear in R the two metrics
g
ab
and ˜g
ab
in (5.4) coincide.
6. FRIEDMANN MODELS IN CURVATURE TIME
The expanding spatially flat Friedmann model may be parametrized with the
help of curvature time a
>
0 as follows:
ds
2
=
a
2
·
da
2
B(a)
−
d x
2
−
d y
2
−
d z
2
¸
,
(6.1)
where B(a) is an arbitrary positive function determining R as
R
= −
3
a
3
d B
da
,
(6.2)
depending only on the first derivative of B(a). This is a special feature of (6.1)
since, in general, the curvature scalar depends on the second derivative of the metric
components. Note also the linear dependence of R on B
0
≡
d B
/
da, in contrast
to the usual nonlinear dependence of the curvature scalar on the first derivative of
the metric coefficients.
Let us recall some facts on Friedmann models in curvature time and exhibit
the corresponding transformation to synchronized time.
6.1. From Curvature Time to Synchronized Time
The spatially flat Friedmann model in synchronized time has the metric
ds
2
=
dt
2
−
a
2
(t )(d x
2
+
d y
2
+
d z
2
)
.
(6.3)
Metric (6.1) goes over to metric (6.3) via
dt
=
a da
√
B(a)
,
(6.4)
such that
t
=
t (a)
=
Z
a da
√
B(a)
.
(6.5)
The inverse function a(t ) provides us with the desired transformation.
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6.2. From Synchronized Time to Curvature Time
Consider a(t ) in an expanding model with
˙
a
≡
da
dt
>
0
.
(6.6)
Then we can invert a(t ) to t(a), and have
B(a)
=
a
2
[ ˙a(t (a)]
2
.
(6.7)
From this relation we understand why R depends on the first derivative of B only:
B itself contains a derivative of a, and R is known to contain up to second order
derivatives of a(t ).
6.3. Examples
Let a(t )
=
t
n
, i.e., t
=
a
1
/
n
, ˙a(t )
=
nt
n
−
1
, ˙a(t (a))
=
na
1
−
1
/
n
. Then Eq. (6.7)
yields
B(a)
=
n
2
a
4
−
2
/
n
.
(6.8)
Let further a(t)
=
e
H t
, H
=
const.
>
0, ˙a
=
H a. Then
B(a)
=
H
2
a
4
.
(6.9)
Obviously, Eq. (6.9) is a limiting form of Eq. (6.8) for n
→ ∞
. Equation (6.1) with
B(a) from (6.9) represents a vacuum solution of Einstein’s theory with
3
-term
where
3
=
3H
2
, namely the de Sitter spacetime.
Let us also give some examples for the direct use the curvature time:
1. From Eq. (6.2) we see that R
=
0 implies B
≡
const., corresponding to
n
=
1
2
in Eq. (6.8), i.e., a
=
t
1
/
2
in synchronized time. This is the usual
Friedmann radiation model.
2. Also from Eq. (6.2), a constant R
6=
0 implies B
=
C
1
+
C
2
a
4
with con-
stants C
1
and C
2
, C
2
6=
0. For C
1
=
0
,
C
2
=
H
2
, this represents the de
Sitter spacetime Eq. (6.9).
3. The dust-model in synchronized coordinates is given by a
=
t
2
/
3
, i.e., with
Eq. (6.8) we get
B(a)
=
4
9
a
,
(6.10)
such that B
0
=
const. Together with Eq. (6.2), this leads to
R a
3
=
const,
(6.11)
ensuring mass conservation, because R is proportional to the mass density,
and the pressure is negligible for dust.
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6.4. The Variational Derivative
For the metric (6.1) we have
√
−
g
≡
p
−
det g
i j
=
a
4
√
B
.
(6.12)
The Lagrangian for Einstein’s theory with
3
-term reads
L
=
(R
+
2
3
)
√
−
g
.
(6.13)
With (6.2) and (6.7) we get from (6.13)
L
=
µ
2
3
−
3B
0
a
3
¶
a
4
B
−
1
/
2
.
(6.14)
The vanishing of the variational derivative
δ
L
δ
B
≡
∂
L
∂
B
−
µ
∂
L
∂
B
0
¶
0
=
0
(6.15)
gives B
=
H
2
a
4
with
3
=
3H
2
, i.e., the usual de Sitter spacetime. No integration
is necessary, since the derivative of B cancels. Intermediate expressions are
∂
L
∂
B
=
µ
2
3
−
3B
0
a
3
¶
a
4
µ
−
1
2
¶
B
−
3
/
2
,
(6.16)
∂
L
∂
B
0
= −
3a B
−
1
/
2
,
µ
∂
L
∂
B
0
¶
0
= −
3B
−
1
/
2
+
3
2
a B
0
B
−
3
/
2
.
(6.17)
6.5. Remaining Coordinate-Freedom
Translations in t do not change the form of the metric (6.3). This freedom
is related to the fact that the integration constant in the integral (6.5) remains
undetermined; this coordinate freedom has no analog in the metric in curvature
time Eq. (6.1).
The metric (6.1) has the following property: It remains unchanged under
multiplication of a
4
and B by the same positive constant. Such a constant factor
appears if we multiply the spatial coordinates by a constant factor. In synchronized
coordinates this property means that not a itself, but only the Hubble parameter
H (t ) :
=
˙a
a
(6.18)
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has an invariant meaning. By the same token, not B(a) itself, but only B(a)
/
a
4
has an invariant meaning. In fact, from Eq. (6.7) we see that
B
a
4
=
H
2
.
(6.19)
7. COSMOLOGICAL SOLUTIONS
Here we recall some formulas of Ref. [5], and present some new results for
the curvature-saturated Lagrangian.
7.1. Solutions for Lagrangian R
m
For the Lagrangian
L
=
R
m
, we obtain the following exact solutions for a
closed Friedmann universe:
ds
2
=
dt
2
−
t
2
2m
2
−
2m
−
1
d
σ
2
(
+
)
,
(7.1)
where d
σ
2
(
+
)
is the metric of the unit 3-sphere.
Analogously, for the open model
ds
2
=
dt
2
−
t
2
2m
−
2m
2
+
1
d
σ
2
(
+
)
.
(7.2)
Of course, both expressions are valid for positive denominators only.
For the spatially flat Friedmann model, it proves useful to employ the cosmic
scale factor a itself as a time-like coordinate.
ds
2
=
a
2
[Q
2
(a)da
2
−
d x
2
−
d y
2
−
d z
2
]
.
(7.3)
This coordinate is meaningful as long as the Hubble parameter is different from
zero, so that we cover only time intervals where the universe is either expanding
or contracting. Possibly existing maxima or minima of the cosmic scale factor as
seen in synchronized time can, however, been dealt by a suitable limiting process
and patching. The curvature scalar reads now
R
=
6
a
3
Q
3
d Q
da
,
(7.4)
and to reduce the order of the field equation it proves useful to define
P(a)
=
d ln Q
da
.
(7.5)
Then the field equation is fulfilled if
0
=
m(m
−
1)
d P
da
+
(m
−
1)(1
−
2m)P
2
+
m(4
−
3m)
P
a
.
(7.6)
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Therefore, the spatially flat Friedmann models can be solved in closed form, but
not always in synchronized coordinates.
7.2. Solutions for Lagrangian
L
cs
In the context of our curvature-saturated model, we shall restrict ourselves
to the expanding spatially flat Friedmann model. The field equation written in
synchronized or conformal time—the two most often used time coordinates used
for this purpose—have the disadvantage that the number of terms is quite large,
and that even in the simplest case
L
=
1
2
R
2
we cannot give closed-form solutions,
apart from the trivial solutions R
≡
0 having the same geometry as the radiation
universe (a
=
√
t in synchronized time t ) and the de Sitter universe (a
=
e
t
in
synchronized time t). So, we prefer to work in the less popular coordinates (7.3).
In principle, the field equation should be of fourth order, but we shall reduce it to
second order.
To find the field equation for a spatially flat Friedmann model with our
Lagrangian, it is useful to consider first a general nonlinear Lagrangian and spe-
cialize to
L
cs
afterwards. To simplify (7.4), we define instead of Q(a) the function
B(a)
=
Q(a)
−
2
>
0 as a new dependent function. Then (7.3) reads
ds
2
=
a
2
·
da
2
B(a)
−
d x
2
−
d y
2
−
d z
2
¸
(7.7)
and (7.4) goes over to
R
= −
3
a
3
d B
da
.
(7.8)
Thus, B itself does not appear explicitly, and only first, and not second derivatives
are present. The geometric origin of this property is the same as in Schwarzschild
coordinates—one integration constant is lost in the definition of the coordinates,
and this makes curvature depend only on the first derivative of the metric.
From the 10 vacuum field equations (2.10) only the 00-component is essential;
it is the constraint equation, therefore it has one order less than the full field
equation, but if the constraint is fulfilled always, then all other components are
fulfilled, too.
3
Together with Eq. (7.8) we should now expect that the fourth order
field equation (2.10) can be reduced to one single second order equation for B(a),
where hopefully, B itself no more appears.
The equation
2
00
=
0 is via (2.4) and (2.8) equivalent to
0
=
3
L
0
µ
2B
−
a
d B
da
¶
−
a
4
L
−
18a B
L
00
d
da
µ
1
a
3
d B
da
¶
,
(7.9)
3
This behavior is known already from the Friedmann equation in General Relativity: Energy density
is proportional to the square of the Hubble parameter which contains only a first derivative.
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R
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which is much simpler than the analogous equation in synchronous time, as ob-
served here for the first time.
Before we insert our Lagrangian
L
cs
into (7.9), let us cross check its validity
by solving known problems: If
L
00
vanishes identically, then
L
0
is a constant, and
we return to Einstein’s theory. The case B
≡
const. gives the radiation universe,
while B
=
a
4
is the exact de Sitter solution. For the Lagrangian
L
=
1
2
R
2
with
L
0
=
R and
L
00
=
1, and Eq. (7.9) reduces to
0
=
a ˙
B
2
−
4a B ¨
B
+
8B ˙
B
,
(7.10)
where a dot denotes differentiation with respect to a. Again, B
=
a
4
is the exact
de Sitter solution. Defining
β
=
ln B and z
=
a ˙
β
, Eq. (7.10) goes over in
4a ˙z
=
3z(4
−
z)
.
(7.11)
With
α
=
ln a we arrive at
4
d z
d
α
=
3z(4
−
z)
,
(7.12)
which can be solved in closed form. Qualitatively it is clear that z
=
4, i.e., the de
Sitter solution, represents an attractor. Solving Eq. (7.12) we obtain in the region
0
<
z
<
4:
z
=
2
+
2 tanh
µ
3
2
α
¶
,
(7.13)
showing explicitly that z
→
4 for
α
→ ∞
. The metric can be calculated from
˙
β
=
2
a
µ
1
+
a
3
−
1
a
3
+
1
¶
,
(7.14)
using the identity
tanh ln x
=
x
2
−
1
x
2
+
1
.
(7.15)
After these preparations we are ready to deal with our Lagrangian
L
cs
. We
insert
L
and
L
0
from Eq. (4.4), and
L
00
= −
3l
4
R(1
+
l
4
R
2
)
−
5
/
2
(7.16)
into Eq. (7.9) and obtain, after setting l
=
1, the simple expression
54a
9
B ˙
B
d
da
(a
−
3
˙
B)
=
a
5
(a
6
+
9 ˙
B
2
)(2B
−
a ˙
B)
+
˙
B(a
6
+
9 ˙
B
2
)
2
.
(7.17)
In these coordinates, the flat Minkowski spacetime does not exist, and the
radiation universe R
=
0 is not a solution. This is why B
=
const. yields no
solution to Eq. (7.17). Also, as was known from the beginning: the de Sitter
spacetime B
=
a
4
is not an exact solution here. However, in the nearby-region
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where the Lagrangian is well approximated by a quadratic function in R with
a nonvanishing linear term, the behavior of the solutions is quite similar to that
of R
+
R
2
-models, where no exact de Sitter solution exists, but a quasi de Sitter
solution represents a transient attractor with sufficient long duration to solve the
known cosmological problems. These calculations have been presented at different
places, most explicitly in Ref. [6]. After this phase, the universe goes to the weak-
field behavior, where our model behaves as usual.
The main departure of our model from the usual one is in the region of large
curvature scalar, where
|
˙
B
|
is large compared to a
3
. To find out the behavior of the
solutions in this limit, we compare the leading terms in Eq. (7.7) and see that ¨
B is
proportional to ˙
B
4
, where the coefficient of proportionality is positive and slowly
varying. Thus, we find approximately B(a)
≈
a
2
/
3
for small a. This implies the
existence of a big-bang singularity, but with a different behavior: From Eq. (7.7)
we obtain
ds
2
=
a
2
·
da
2
a
2
/
3
−
d x
2
−
d y
2
−
d z
2
¸
,
(7.18)
which corresponds in synchronized time to the behavior
ds
2
=
dt
2
−
t
6
/
5
(d x
2
+
d y
2
+
d z
2
)
,
(7.19)
this being a good approximation to the exact metric for small t , differing from the
usual big-bang behavior in almost all other models. Further details of our model
will be presented elsewhere.
7.3. The Cosmological Singularity
Here we present the argument with the singularity behaviour mentioned at the
end of section 2: In our model, differently from Einstein’s theory, the divergence
of the curvature does not necessarily imply the divergence of any part of the
energy-momentum-tensor. Let us concentrate on the trace. The r.h.s. of Eq. (2.11)
reads
1
8
π
G
½
R
+
2l
4
R
3
(1
+
l
4
R
2
)
3
/
2
−
3
¤
·
1
(1
+
l
4
R
2
)
3
/
2
¸¾
and this expression must be equal to the trace T of the energy-momentum tensor.
In Einstein’s theory, R
→ ±∞
necessarily implies T
→ ±∞
, whereas here, T
may remain finite even if R
→ ∞
.
Detailed numerical calculations would support this qualitative picture, how-
ever, we postpone such calculations until we have a more strictly physically mo-
tivated form of the Lagrangian.
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R
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8. DISCUSSION
We have argued that the gravitational action
A
has a decreasing dependence
on R for increasing
|
R
|
. Such a behavior is expected from the spacetime stiffness
caused by the vacuum fluctuations of all quantum fields in the universe.
Our model does not have the tachyonic disease of R
+
R
2
models studies by
Stelle [17] and others [18].
Since our model has an action which interpolates between Einstein’s action
and a pure cosmological term, it promises to have interesting observable conse-
quences which may explain some of the experimental cosmological data.
The heat-kernel expansion of the effective action in a curved background is
closely related to the Seeley-Gilkey coefficients [19], and for higher loop expansion
also higher powers of curvature appear: To get the n-loop approximation one has
to add terms until
∼
R
n
+
1
, a behavior which also happens in the string effective
action [20]. So, if one cuts this procedure at a certain value of n, one gets always
as leading term for high curvature values a term like
∼
R
n
+
1
. However, the n-loop
approximations need not converge to the correct result if one simply takes n
→ ∞
in the n-loop-result. In fact, what we have used in the present paper is such an
example:
L
cs
=
R
16
π
G
√
1
+
l
4
R
2
=
R
16
π
G
+
∞
X
k
=
1
b
k
R
2k
+
1
(8.1)
with some real constants b
k
, where
b
1
= −
l
4
32
π
G
(8.2)
but the Taylor expansion on the right hand side diverges for R
>
l
−
2
. So, the
Taylor expansion is useful for small R-values only, and for large values R we need
a correct analytical continuation.
Prigogine et al. have proposed in Eq. (18) of Ref. [21] a model where the effec-
tive gravitational constant depends on the Hubble parameter of a Friedmann model.
Though this ansatz depends on the special 3
+
1-decomposition of spacetime, it
shares some similarities with the model discussed here. More recent developments
how to find a well-founded gravitational action from considering quantum effects
can been found in [22] and [23].
Quite recently, see for instance [26], accelerated expansion models of the
universe have been discussed and compared with new observations. We postpone
the comparison of our model with these observations to later work. A continuation
of the present paper is [30], where the inclusion of matter is explicitly done, and
thus the present qualitative results are substantiated.
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APPENDIX A: NEWTONIAN LIMIT IN A NONFLAT BACKGROUND
The Newtonian limit of a theory of gravity is defined as follows: It is the weak-
field slow-motion limit for fields whose energy-momentum tensor is dominated
by its zero-zero component in comoving time. Usually, the limit is formed in a flat
background, and sometimes, this is assumed to be a necessary assumption. This is,
however, not true, and we show here briefly how to calculate the Newtonian limit
in a nonflat background, Moreover, our approach is different from what is usually
called Newtonian cosmology. To have a concrete example, we take the background
as a de Sitter spacetime. Recent progress for calculating the Newtonian limit in
the presence of an effective cosmological constant can be seen in [31].
The slow-motion assumption allows us to work with static spacetime and the
matter, assuming the energy-momentum tensor to be
T
i j
=
ρ δ
0
i
δ
0
j
,
(A1)
where
ρ
is the energy density, and time is assumed to be synchronized. The de
Sitter spacetime in its static form can be given as
ds
2
= −
(1
−
kr
2
)dt
2
+
dr
2
1
−
kr
2
+
r
2
d
Ä
2
,
(A2)
where x
0
=
t
,
x
1
=
r
,
x
2
=
χ,
x
3
=
θ
and d
Ä
2
=
d
χ
2
+
sin
2
χ
d
θ
2
is the metric
of the 2-sphere. In this Appendix, we have changed the signature of the metric
from (
+ − −−
), which is usual in cosmology, to (
− + ++
), which leads to the
standard definition of the Laplacian.
The parameter k characterizes the following physical situations: For k
=
0,
we have the usual flat background. By setting k
=
0 we can therefore compare
the results with the well-known ones. The case k
>
0 corresponds to a positive
cosmological constant
3
. In the calculations, we must observe that the time co-
ordinate t fails to be a synchronized for k
6=
0, but it is obvious from the context
how to obtain the synchronized time from it.
In the coordinates (A2), there is a horizon at r
=
r
0
≡
1
√
k
. So, our approach
makes sense in the interval 0
<
r
<
r
0
. However, r
0
shall be quite large in com-
parison with the system under consideration, so that we do not meet a problem
here.
Now, the following ansatz seems appropriate:
ds
2
= −
(1
−
kr
2
)(1
−
2
ϕ
)dt
2
+
µ
dr
2
1
−
kr
2
+
r
2
d
Ä
2
¶
(1
+
2
ψ
)
,
(A3)
where
ϕ
and
ψ
depend on the spatial coordinates only. The weak-field assumption
allows us to make linearization with respect to
ϕ
and
ψ
. An extended matter
configuration can be obtained by superposition of point particles, so we only need
to solve the problem for a
δ
-source at r
=
0. This one is spherically symmetric, so
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we may assume
ϕ
=
ϕ
(r ) and
ψ
=
ψ
(r ) in Eq. (A3). For the metric components
we get:
g
00
= −
(1
−
kr
2
)(1
−
2
ϕ
)
,
g
11
=
1
+
2
ψ
1
−
kr
2
,
g
22
=
r
2
(1
+
2
ψ
)
,
(A4)
g
33
=
g
22
·
sin
2
χ.
The inverted components are up to linear order in
ϕ
and
ψ
:
g
00
= −
1
+
2
ϕ
1
−
kr
2
,
g
11
=
(1
−
kr
2
)(1
−
2
ψ
)
,
g
22
=
1
−
2
ψ
r
2
,
(A5)
g
33
=
g
22
sin
−
2
χ,
which gives the Christoffel symbols
0
0
01
= −
ϕ
0
−
kr
1
−
kr
2
,
(A6)
0
1
00
=
(1
−
kr
2
)[
−
kr
+
2kr (
ϕ
+
ψ
)
−
ϕ
0
(1
−
kr
2
)]
,
(A7)
0
1
11
=
ψ
0
+
kr
1
−
kr
2
,
(A8)
0
2
12
=
0
3
13
=
ψ
0
+
1
r
,
(A9)
0
1
22
= −
r (1
−
kr
2
)
−
ψ
0
r
2
(1
−
kr
2
)
,
(A10)
0
1
33
=
sin
2
χ 0
1
22
,
(A11)
0
3
32
=
cot
χ,
(A12)
0
2
33
= −
sin
χ
cos
χ,
(A13)
and the Ricci tensor reads
R
00
= −
3k(1
−
kr
2
)
−
ϕ
00
(1
−
kr
2
)
2
−
2
ϕ
0
r
(1
−
kr
2
)
+
6k(
ϕ
+
ψ
)(1
−
kr
2
)
+
kr (1
−
kr
2
)(5
ϕ
0
−
ψ
0
)
,
(A14)
R
11
= −
2
ψ
00
+
ϕ
00
−
2
r
ψ
0
+
3k
1
−
kr
2
+
kr
1
−
kr
2
(
ψ
0
−
3
ϕ
0
)
,
(A15)
R
22
=
3kr
2
−
ψ
00
r
2
(1
−
kr
2
)
−
ψ
0
(2r
−
4kr
3
)
+
(
ϕ
0
−
ψ
0
)(r
−
kr
3
)
,
(A16)
R
33
=
R
22
·
sin
2
χ.
(A17)
Before we discuss these equations, we consider two obvious limits:
For k
=
0, we see that R
00
= −
ϕ
00
−
2
ϕ
0
/
r
= −
1ϕ
, leading to the usual
Newtonian limit
1ϕ
= −
4
π
G
ρ
.
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For
ϕ
=
ψ
=
0 we get for the Ricci tensor:
R
0
0
=
R
1
1
=
R
2
2
=
R
3
3
=
3k
,
(A18)
and thus the de Sitter spacetime with R
=
12k for k
>
0.
Returning to the general case we have
R
2
=
6k
−
12k
ψ
+
(
ϕ
00
−
2
ψ
00
)(1
−
kr
2
)
+
2
r
ϕ
0
−
5kr
ϕ
0
−
4
r
ψ
0
+
7kr
ψ
0
(A19)
and then
R
0
0
−
R
2
= −
3k
+
6k
ψ
+
2
ψ
00
(1
−
kr
2
)
−
6kr
ψ
0
+
4
r
ψ
0
.
(A20)
The other components have a similar structure and can be calculated easily from the
above equations. The first term of the r.h.s.,
−
3k, will be compensated by the
3
-
term. The usual gauging to
ψ
→
0 and
ϕ
→
0 as r
→ ∞
is no more possible
because for r
>
r
0
our approximation is no more valid. As an alternative gauge we
add such constant values to
ψ
and
ϕ
that they are approximately zero in the region
under consideration. So we may disregard the term 6k
ψ
. All remaining terms
with k can be obtained from those without k by multiplying with factors of the
type 1
+
²
where
²
≈
kr
2
, k
=
1
/
r
2
0
, with r
0
being of the order of magnitude of
the world radius. In a first approximation, this gives only a small correction to the
gravitational constant. In a second approximation, there are deviations from the
1
/
r -behavior.
An analogous discussion for the Lagrangian R
+
l
2
R
2
tells us that in a range
where l
¿
r
¿
r
0
, the potential behaves like (1
−
c
1
e
−
r
/
l
)
/
r , as in flat space.
APPENDIX B: THE ABSENCE OF GHOSTS AND TACHYONS
Here we show in more details what has been stated after Eq. (3.3). In the
conformally transformed picture with a scalar field, the absence of tachyons (i.e.,
particles with wrong sign in front of the potential term) becomes clear from the
form of the potential. For checking ghosts (i.e., particles with wrong sign in front
of the kinetic term) we have to go a little more into the details: In Stelle [27]
the particle content of fourth order gravity with terms up to quadratic order has
been determined, and the existence/absence of ghosts and tachyons has been given
in dependence on the free constants of the theory. In the first of Refs. [4], the
analogous calculation as in [27] has been done for a term R
3
added to the Einstein–
Hilbert-Lagragian. Let us give here the argument for general n
≥
3: If R
n
is in
L
,
then the term R
n
−
1
and its derivatives are in the corresponding expression after
variational derivative with respect to the metric. In the result, all terms represent
products of at least n
−
1 small quantities; because of n
≥
3 these are always at
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least two factors; thus, they all vanish in the linearization about the Minkowski
space–time.
Now, one might be tempted to require the analogous linearization properties
for a Friedmann–Robertson–Walker background. However, linearization around
other than flat space–times is not at all a trivial task, see [28], even for Einstein’s
theory: For the closed Friedmann model, Einstein’s theory is linearization unstable,
for spatially flat models it is stable, and for the open Friedmann model the result
is – contrary to other claims in the older literature – not yet known. We face the
further problem that linearization around the de Sitter space-time is complicated
to determine, because the same geometry can be locally represented as a spatially
flat as well as a closed Friedmann model. So, we leave the question of linearization
stability with non-flat background of our model unanswered.
Another type of reasoning was given quite recently: In [29] the possibility
has been discussed that the contributions to the Lagrangian coming of gravitons
on the one hand and of gravitinos on the other may cancel each other to avoid the
ghost problem.
ACKNOWLEDGMENTS
H.-J. S. gratefully acknowledges financial support from DFG and from the
HSP III-program. We thank V. Gurovich and the colleagues of the Free University
Berlin, where this work has been done, especially M. Bachmann and A. Pelster,
for valuable comments.
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