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Eur. Phys. J. C (2020) 80:32

https://doi.org/10.1140/epjc/s10052-019-7580-z

Regular Article - Theoretical Physics

Anisotropic massive Brans–Dicke gravity extension

of the standard CDM model

Özgür Akarsu

1,a

, Nihan Katırcı

1,b

,Ne¸se Özdemir

1,c

, J. Alberto Vázquez

2,d

Department of Physics, Istanbul Technical University, 34469 Istanbul, Maslak, Turkey

Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Apdo, Postal 48-3, 62251 Morelos, Cuernavaca, Mexico

Received: 27 September 2019 / Accepted: 21 December 2019 / Published online: 14 January 2020

Abstract We present an explicit detailed theoretical and

observational investigation of an anisotropic massive Brans–

Dicke (BD) gravity extension of the standard CDM model,

wherein the extension is characterized by two additional

degrees of freedom; the BD parameter, ω, and the present day

density parameter corresponding to the shear scalar, 

The BD parameter, determining the deviation from general

relativity (GR), by alone characterizes both the dynamics of

the effective dark energy (DE) and the redshift dependence

of the shear scalar. These two affect each other depending

on ω, namely, the shear scalar contributes to the dynamics

of the effective DE, and its anisotropic stress – which does

not exist in scalar ﬁeld models of DE within GR – controls

the dynamics of the shear scalar deviating from the usual

∝ (1 + z)

form in GR. We mainly conﬁne the current work

to non-negative ω values as it is the right sign – theoreti-

cally and observationally – for investigating the model as a

correction to the CDM. By considering the current cosmo-

logical observations, we ﬁnd that ω  250, 

 10

−23

and the contribution of the anisotropy of the effective DE

to this value is insigniﬁcant. We conclude that the simplest

anisotropic massive BD gravity extension of the standard

CDM model exhibits no signiﬁcant deviations from it all

the way to the Big Bang Nucleosynthesis. We also point out

the interesting features of the model in the case of nega-

tive ω values; for instance, the constraints on 

could

be relaxed considerably, the values of ω ∼−1 (relevant to

string theories) predict dramatically different dynamics for

the expansion anisotropy.

e-mail: akarsuo@itu.edu.tr

e-mail: nihan.katirci@itu.edu.tr

e-mail: nozdemir@itu.edu.tr

e-mail: javazquez@icf.unam.mx

1 Introduction

The base -cold dark matter (CDM) model, relying on the

inﬂationary paradigm [1–8], is the simplest and most suc-

cessful cosmological model to describe the dynamics and

the large scale structure in agreement with the most of the

currently available observational data [9–12]. It is today cred-

ited as the standard cosmological model, yet it is probably

not where the story has concluded but the hardest part has

just begun. It suffers from severe theoretical issues relating

to the cosmological constant  being responsible for the

late time acceleration of the Universe [13–19] and, on the

observational side, from tensions of various degrees of sig-

niﬁcance between some existing data sets [20–28]. Besides,

based on the most minimal a priori assumptions, model inde-

pendent reconstructions of the evolution of the dark energy

(DE) equation of state (EoS) parameter w [29–31] and also

model independent diagnoses [32,33] exhibit a dynamical

behaviour of w(z).

Nevertheless, even small deviations from/corrections to

the CDM mostly imply/require profound, and sometimes

highly non-trivial, modiﬁcations to the fundamental theories

of physics [34]. Indeed, we still do not have a promising and

concrete fundamental theory leading to DE models that are

more general than  and also can account for the small, but

signiﬁcant, deviations from the CDM model as persistently

suggested by the high precision data. Though, depending on

the characteristics of the DE models favored by observations,

we can decide whether it is more natural to consider DE as an

actual physical source [15,35]orasaneffective source orig-

inated from a modiﬁcation [36–40] to the standard theory of

gravity, i.e., general relativity (GR). For instance, eliminating

 by detecting w>−1 would not be by itself illuminating

to the nature of DE, but any detection of w<−1 would be

very illuminating to the nature of gravitation. For a perfect

barotropic ﬂuid, the adiabatic sound speed c

is the physical

propagation speed of perturbations, and therefore w<−1

123

32 Page 2 of 26 Eur. Phys. J. C (2020) 80 :32

(viz., phantom [41] or quintom [42,43] DE models) typically

accompanying by c

< 0, implies the instability of pertur-

bations (and/or ghost instabilities, see, e.g., [44]), whereas,

as shown in [45,46], scalar-tensor theories of gravity, such

as Brans–Dicke (BD) theory, can lead to an effective source

with w<−1 that does not correspond to the change of

the sign of c

, and hence perturbations can still be stable.

Similarly, any detection of DE with negative energy density

would also be hugely informative, such that this for an actual

physical source is of course physically ill, whereas it can be

an effective source so that negative energy density does not

lead to any pathology since this is not the true energy den-

sity. On the observational side, the constraints on the EoS

parameter of DE persistently indicate that w ≈−1, and so

do not exclude w<−1[12] and it has recently been shown

in [21,32,47] that DE models with energy densities passing

below zero at high redshifts ﬁt the data better and can address

the tensions relevant to Lyman-α forest measurements [20].

Such DE sources are indeed possible, in general, in modiﬁed

theories of gravity with an effective gravitational coupling

strength weaker in the past, e.g., in scalar-tensor theories, if

we collect all modiﬁcations to the usual Einstein ﬁeld equa-

tions to deﬁne aneffective DE. Consequently, it can be argued

that any detection of a deviation from CDM model implies

that the late time acceleration is not driven by 

and of DE

yielding w<−1 and/or ρ<0 implies that gravity is not

minimally coupled (which eliminate the actual perfect-ﬂuid

models of DE and stands as a strong sign in favor of the

effective DE models from modiﬁed gravity).

Generically, the effective DE models from modiﬁed grav-

ity induce non-zero anisotropic stresses,

which can be real-

ized if we relax the isotropic space assumption of the CDM

model along with the modiﬁcation to GR. Of course, we are

not able to observe anisotropic stresses directly, yet they can

reveal themselves through their effect on the evolution of the

expansion anisotropyas well as on the average expansion rate

of the Universe. Then, a natural question iswhethertheobser-

vations allow or suggest an anisotropic space, unless other-

wise anisotropic stresses would be irrelevant. Indeed, there

are some clues for questioning the spatially maximally sym-

metric Universe assumption, i.e., Robertson–Walker (RW)

background, of the CDM model. This has been mainly

motivated by hints of anomalies in the CMB distribution ﬁrst

observed on the full sky by the WMAP experiment [49–52]

and so remained in Planck experiment [53–56]. So far, the

local deviations from the statistically highly isotropic Gaus-

sianity of the CMB in some directions (the so called cold

spots) could not have been excluded at high conﬁdence lev-

The converse is not true, even if it turns out that DE exactly yields

w =−1.

See [48] for a list of well know anisotropic stresses and their effects

on the expansion anisotropy.

els [52,53,57,58]. Furthermore, it has been shown that the

CMB angular power spectrum has a quadrupole power lower

than expected from the best-ﬁt CDM model [59,60]. Sev-

eral explanations for this anomaly have been proposed [61–

70] including the anisotropic expansion of the Universe. On

the other hand, inﬂation (canonical) isotropizes the Universe

very efﬁciently [71,72], leaving a residual anisotropy that

is negligible for any practical application in the observable

Universe. This could be irrelevant if an anisotropic expansion

is developed only well after the matter-radiation decoupling,

for example during the domination of DE, say, by means of

its anisotropic pressure acting as a late-time source of not

insigniﬁcant anisotropy [69,73–83] (see also, e.g., [84–87]

for constraint studies on the anisotropy of DE). Indeed, the

CMB provides very tight constraints on the anisotropy at the

time of recombination [88–90] of the order of the quadrupole

temperature, i.e., (T /T )

=2

∼ 10

−5

. And, in the sim-

plest anisotropic generalization of the standard CDM (viz.,

replacing the spatially ﬂat RW metric by Bianchi type I

metric), the energy density corresponding to the expansion

anisotropy scales as the inverse of the square of the comoving

volume, ρ

∝ S

−6

, which implies an isotropization of the

expansion from the recombination up to the present, leading

to the typically derived upper bounds on the correspond-

ing density parameter today as 

∼ 10

−20

. However,

this is true if the anisotropic expansion is not generated by

any anisotropic source, say, by an anisotropic DE (effective

or actual), arising after decoupling [69,73–83]. Neverthe-

less, almost all of these strong constraints on the expansion

anisotropy in the late Universe, 

 10

−22

, assume the

non-existence of anisotropic sources (actual or effective) in

the late universe, and are derived in fact from the contribution

of the expansion anisotropy on the average expansion rate of

theUniverse[91–95]. On the other hand, direct observational

constraints, e.g., from SNIa data, on the expansion anisotropy

of the later Universe are much weaker, namely, on the order

of 

 10

−4

for z ∼ 0[96,97].

It is in fact always possible to imitate any modiﬁcation to

the Einstein ﬁeld equations as GR with an arbitrary mixture

of scalars, vectors and tensors [98,99]. Therefore, one may

think of abandoning any attempt to distinguish between the

actual physical DE and the effective DE from modiﬁed grav-

ities. However, a modiﬁed theory of gravity, e.g., the Brans–

Dicke theory, can modify both the average expansion rate of

the Universe and the evolution of the expansion anisotropy

in a distinctive way depending on its free parameter charac-

terizing the model, e.g., the Brans–Dicke parameter ω. So, of

course, owing to Occam’s razor, looking for these two con-

current modiﬁcations predicted by a speciﬁc modiﬁed the-

ory of gravity and testing them against the observational data

can provide us a strong reason for favoring or disfavoring the

modiﬁed theory of gravity under consideration over GR. In

this paper, for instance, the effective EoS parameter corre-

123

Eur. Phys. J. C (2020) 80 :32 Page 3 of 26 32

sponding to the expansion anisotropy, the present day EoS of

the effective DE and the redshift at which it evolves into the

phantom region range in speciﬁc intervals as 1 <w

≤

−1.14  w<−1 and 0.50 ≤ z

PDL

< 0.65 (see [100]for

a similar result) for positive values of the BD parameter ω.

Indeed, revealing the origin of the late time acceleration of

the Universe considering such distinctive features of the DE

models from modiﬁed theories of gravity is among the sci-

entiﬁc themes underlying some upcoming experiments (e.g.,

[101]).

Finally, we have also independent motivations for modify-

ing GR by fundamental theoretical physics. Namely, almost

all of the attempts to quantize GR introduce deviations from

it in the form of extra degrees of freedom, higher powers of

the curvature in the action, higher order derivatives in the

ﬁeld equations, or non-local terms. The low-energy limits of

the string theories typically yield BD gravity with ω =−1

and similarly d-brane models yield BD gravity with ω ∼−1

[102–105]. Indeed, among all possible alternatives to GR, it is

found natural by many to ﬁrst consider scalar-tensor theories

[106–110], which only add a (usually massive) scalar degree

of freedom, the BD-like scalar ﬁeld, to the two massless,

spin-2 polarizations (gravitons) contained in the metric ten-

sor. In relevance with our focus in this paper, the ﬁeld equa-

tions in such models can be regarded as effective Einstein

ﬁeld equations and so the terms originating from the BD-like

scalar ﬁeld and its derivatives, moved to the right hand side,

can always be interpreted as an effective ﬂuid. Within this

approach, the correspondence between general scalar-tensor

theories and this effective ﬂuid has been worked out explic-

itly ﬁrst in [111](seealso[112] for the preceding work), and

has shown that, in general, the corresponding effective ﬂuid

is imperfect. A recent work [113] extended and completed

the correspondence showing how a symmetry of BD theory

translates into a symmetry of this ﬂuid such that this corre-

spondence is valid for any spacetime geometries. Brans and

Dicke’s 1961 theory [106,110], motivated ﬁrst by the imple-

mentation of Mach’s principle in gravity, today provides a

prototype of scalar-tensor theories [107–109].

The original BD theory [106,110] has only one additional

constant parameter ω, determining the deviations from GR,

which is recovered in the limit of |ω|→∞. Theoretically

ω ≥−

is necessary to avoid the Jordan/scalar ﬁeld (ϕ)from

yielding negative energy density values in the Einstein frame

that leads, e.g., the Minkowski vacuum to be unstable. Obser-

vations on a wide range of scales further constrain BD the-

ory around GR. The tightest constraints, to date, are imposed

by the observations of gravitational phenomena in the Solar

System; speciﬁcally, ω  40000 at the 2σ conﬁdence level

(C.L.) from observations of radio signals from the Cassini

spacecraft as it passes behind the Sun [114]. It can be vastly

relaxed for the massive BD theory, i.e., when the Jordan

ﬁeld is accompanied by a potential U(ϕ), as in this case one

should consider the corresponding mass (M) of the Jordan

ﬁeld along with the BD parameter, viz., the {ω, M} parame-

ter region [115]. It is shown that ω = O(1) is allowed for the

Solar System constraints provided that M  2 × 10

−17

[115]. The constraints on M, however, are typically sub-

ject to the cosmological observations, since, in particular,

it becomes cosmologically signiﬁcant at substantially lower

scales, viz., at the Hubble mass scale M

 10

−33

eV, which

leaves the constraint from the Cassini spacecraft still valid.

The cosmological constraints on ω are complementary

to the local constraints as they probe different length and

time scales, as well as different epochs of the Universe. The

strongest cosmological constraint, which assumes U (ϕ) =

const. and considers CMB data from the ﬁrst Planck release

along with the constraints from Big Bang Nucleosynthesis

(BBN) light element abundances, gives ω>890 at the 2σ

C.L. [116]. The constraints from Planck 2015 (2013) and a

compilation of baryon acoustic oscillations (BAO) data are

ω>333 (ω>208) at the 2σ C.L., weakly dependent on

the exponent n > 0 for a power-law potential U (ϕ) ∝ ϕ

mimicking  in the late universe [117,118] (also see [119]

for a recent work considering a more complicated potential).

The BBN leads to ω  277 (for a universe containing dust,

radiation and conventional vacuum energy), which assumes

power-law solutions and, by using general solutions, can be

somewhat altered depending on the behaviour of the Jordan

ﬁeld in the early universe [120]. It is forecast that future

cosmological experiments will be able to constrain ω at a

level comparable to the local tests [121–123].

The typical mass scale of M ∼ 10

−33

eV imposed by cos-

mological observations is so small that the local constraint

ω  40000 from the Cassini spacecraft must be satisﬁed

at all times in all parts of the Universe, and leads to the

conclusion that the massive BD extension of the standard

CDM model must be phenomenologically very similar to it

throughout most of the history of the Universe, implying that

the cosmological features of BD theory would be observa-

tionally irrelevant. On the other hand, it is still worth studying

in detail and understanding the cosmological constructions

within massive BD theory by considering the cosmological

constraints on ω only, as BD theory is representative of some

more general gravity theories that can evade the local con-

straints (see [37] and references therein for further reading).

For instance, straightforwardly, it is possible to consider a

further extension of BD theory by allowing the BD param-

eter to be some functions of the Jordan ﬁeld, ω = ω(ϕ),in

addition to the presence of potential U (ϕ) and this leads to

the possibility of having an ω small enough to be signiﬁ-

cant in the past, while being large enough today to be con-

sistent with the tight constraints imposed by observations

of gravitational phenomena in the Solar System. Alterna-

tively, for instance, in the Horndeski theory (the most general

scalar–tensor theory having second-order ﬁeld equations in

123

32 Page 4 of 26 Eur. Phys. J. C (2020) 80 :32

four dimensions) [124,125], while on cosmological scales

BD theory emerges as an approximation, the derivative self-

interactions of the Horndeski scalar can be large enough on

small scales (higher curvature than cosmological environ-

ments) to screen the scalar resulting at the same time to the

recovery GR. For example, it was proposed in [126] that the

so called Galileon theories (a close cousin of BD theory)

could also explain the late time acceleration of the Universe

while evading solar system constraints. A different kind of

example is that, in the presence of extra dimensions, it is

possible to make BD theory (massive or massless) consis-

tent with gravitational tests (including solar system tests) for

|ω|=O(1) [127]. See also [128] for a recent overview on

cosmological probes of gravity for testing deviations from

GR in relevance with such modiﬁed gravity theories.

Motivated by the discussions above, we present a detailed

theoretical and observational investigation of the anisotropic

(LRS Bianchi type I metric) massive Brans–Dicke gravity

extension of the standard CDM model, which is char-

acterized by two additional free parameters, namely, the

BD parameter ω and the corresponding present day density

parameter of the expansion anisotropy 

. The role of

the cosmological constant is taken over by the Jordan ﬁeld

potential of the form U (ϕ) ∝ ϕ

and the physical ingredient

of the Universe is considered as in the standard CDM. We

conﬁne the present work, basically, to the non-negative val-

ues of the BD parameter since, here, we mainly consider the

extension as a correction to the standard CDM.

2 Anisotropic massive Brans–Dicke gravity extension of

the CDM model

We consider the BD action [106,110] written in the Jordan

frame in the following form:

JBD



√

−g



R − ω



∇

ϕ∇

ϕ +



+ S

Matter

(1)

where ϕ = ϕ(t) is the Jordan scalar ﬁeld (function of cosmic

time t only) and ω = const. is the Brans–Dicke parameter,

R is the Ricci scalar, g is the determinant of the metric g

μν

and S

Matter

is the matter action, which is independent of ϕ

so that the weak equivalence principle is satisﬁed. It is clear

from the way of writing the action that the term M

stands

as the bare mass-squared of the Jordan ﬁeld.

We assume

= const. so that, as can also be seen from the action,

We consider the bare mass as it is done in [100], where it formally

appears as the mass of a minimally coupled canonical scalar ﬁeld when

curvature scalar is dropped. On the other hand, when the effective mass

it stands like a cosmological constant as 2ωM

≡  and

thereby can drive accelerated expansion. Hence, switching to

massive BD from GR with a positive cosmological constant

provides us with an opportunity to construct CDM-type

cosmologies, such that the mass of the Jordan ﬁeld alone can

play the role of positive cosmological constant like in the

standard CDM cosmology provided that 2ωM

≡ >0,

and the Jordan ﬁeld ϕ varying slowly enough on the top

of this can account for small deviations from the standard

CDM model in a particular way, which in turn may lead to

an improved ﬁt to the observational data w.r.t. the standard

CDM model. In line with that, we intend to study the BD

extension of the standard CDM model as a correction, and

therefore we demand the term 2ωM

to be positive deﬁnite,

which requires ω>0 as long as we keep M

> 0 to avoid

the Jordan ﬁeld from having an imaginary mass.

Hence, in

this study, unless otherwise is mentioned, we carry out our

investigations by assuming ω>0, which is already stronger

than the assumption ω ≥−3/2 to avoid the Jordan ﬁeld from

yielding negative energy density values in the Einstein frame

that leads, for instance the Minkowski vacuum to be unstable.

We consider the simplest anisotropic generalization of the

spatially ﬂat and homogeneous spacetime, i.e., locally rota-

tionally symmetric (LRS) Bianchi type I metric, which can

be written as follows;

=−dt

+ S



4β

+ e

−2β

(dy

+ dz

)



, (2)

where S = S(t) is the mean scale factor. Here, the exponent

β = β(t) satisﬁes the relation

, where σ

= σ

and σ

are shear scalar and tensor, respectively. The scalar

curvature for this metric (2) may be written in terms of the

Footnote 3 continued

of the Jordan ﬁeld is considered there are various deﬁnitions in the

literature (see, e.g., [130] for a recent discussion and further references),

though one cannot say that these would not fail from a strict particle

physics point of view.

If this is not the case then 2ωM

=  will contribute to the ﬁeld

equations like a negative cosmological constant, which may be com-

pensated by a rapidly changing Jordan ﬁeld. Within the standard BD

gravity (i.e., massless BD gravity) in the presence of pressureless

source, there exist cosmological solutions with accelerating expansion

if −2 <ω<−1, and Jordan ﬁeld is real (i.e., the effective cosmologi-

cal gravitational coupling is positive deﬁnite) as well if −

<ω<−

leading to deceleration parameter in the range 0 > q > −1, and to

∝ (1 + z)

and ϕ

∝ (1 + z)

in the two boundaries of this range,

correspondingly [131]. Hence, obviously, if we consider ω<0and

> 0leadingto2ωM

< 0 (i.e., effectively negative cosmological

constant), then it will be necessary to conﬁne ourselves to the range

−

<ω<−

, and moreover the decelerating effect of the term

2ωM

< 0 would be compensated by bringing the value of the BD

parameter closer to −4/3, which in turn, implies a faster Jordan ﬁeld

[viz., for ω ∼−4/3 Jordan ﬁeld changes as fast as the pressureless

source, i.e., ϕ

∼ (1 + z)

] whereas we are looking for slowly chang-

ing Jordan ﬁeld, say, ϕ

∼ const. since we demand it to do only small

modiﬁcations on the CDM dynamics.

123

Eur. Phys. J. C (2020) 80 :32 Page 5 of 26 32

mean scale factor and shear scalar as R =−6





−σ

Throughout the paper a dot denotes derivative w.r.t. cosmic

time t.

We consider all types of matter distribution (namely, the

usual cosmological sources such as radiation, baryons, etc.)

as isotropic perfect ﬂuids, which are described by the follow-

ing energy-momentum tensor (EMT)

= diag[−ρ, p, p, p], (3)

where ρ and p are the energy density and pressure, respec-

tively.

The ﬁeld equations for the action (1) within the framework

of the metric (2) in the presence of (3) read:

−

− 2ω

˙ϕ

+ 6

˙ϕ

− 2ωM

ρ, (4)

−





˙σ +



+ 2

˙ϕ





+ (2ω + 2)

˙ϕ

+ 2

¨ϕ

+ 4

˙ϕ

− 2ωM

=−

p, (5)





˙σ +



+ 2

˙ϕ





+ (2ω + 2)

˙ϕ

+ 2

¨ϕ

+ 4

˙ϕ

− 2ωM

=−

p, (6)

for which the latter two correspond to the pressure equations

along the x-axis and the y- and z-axes, respectively. The

Klein–Gordon equation for the Jordan ﬁeld reads

¨ϕ

+ 3

˙ϕ

−

2ω





−

2ω

+ M

= 0. (7)

We consider the standard cosmological ﬂuids, namely, pres-

sureless ﬂuid (p

= 0) and radiation/relativistic ﬂuids

= ρ

/3), so that, here, ρ = ρ

+ ρ

and p = p

.Ifthe

Jordan ﬁeld is approximately constant then the ﬁrst Fried-

mann Eq. (4) becomes 3

≈

ρ +

+ 2ωM

(as like

in GR), where the effective cosmological gravitational cou-

pling strength is then given by G =

, where G =

2πϕ

and G

2πϕ

with zero subscript denoting today’s value

(throughout the paper).

We consider the conventional representation of the true

equation for scalar-tensor gravity interacting with matter in

the Einsteinian form with the constant G

= G(t

)

Note however that for bound systems in the quasi-static regime, e.g.,

our solar system, the effective Newton’s constant is G

2ω+3

2ω+4

[110], thus observers in a bound system which formed today would

measure the same cosmological and local gravitational strength if ϕ

2ω+4

2ω+3

μν

−

μν

= 8π G

μν,m

+ T

μν,DE

), (8)

8π G

= 4/ϕ

following Refs. [45,46], where it is discussed

that modiﬁed gravitytheories can be recast in the standard GR

form such that (8) where all the new geometrical terms are

grouped (on the r.h.s.) to form an effective DE contribution

denoted as T

μν,DE

(for a detailed discussion, see introduction

in Ref. [46]). We note that the shear scalar is kept on the left

hand side as it is a part of the Einstein tensor, namely, the

metric itself only. Accordingly, we deﬁne the effective DE in

the following way:

−

(ρ + ρ

(9)

−





˙σ +3



=−

( p + p

DE,x

(10)





˙σ + 3



=−

( p + p

DE,y

(11)

where ρ

is the energy density and p

DE,x

and p

DE,y

are the

principal pressures of the effective DE along the x-axis and

the y- and z-axes, respectively, and which read





−



ρ + 2ω

˙ϕ

− 6

˙ϕ

+ 2ωM



(12)

DE,x





−



p − 2



˙ϕ

+(2ω + 2)

˙ϕ

+ 2

¨ϕ

+ 4

˙ϕ

− 2ωM



, (13)

DE,y





−



p +



˙ϕ

+(2ω + 2)

˙ϕ

+ 2

¨ϕ

+ 4

˙ϕ

− 2ωM



. (14)

We note that the Jordan ﬁeld and shear scalar are coupled

and they together lead to an anisotropy in the pressure of the

effective DE, which may be represented by

p

= p

DE,y

− p

DE,x

√

˙ϕ

σ. (15)

This, in turn, leads to a modiﬁcation in the evolution of the

expansion anisotropy provided that ϕ is dynamical and the

space is not exactly isotropic as would be expected from a

realistic cosmological model (see, e.g., [132]). We can restate

this equation in terms of the cosmological gravitation cou-

pling strength as

123

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