各向异性Brans-Dicke引力扩展对ΛCDM模型的影响

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"我们对标准ΛCDM模型的各向异性块状Brans-Dicke（BD）重力扩展进行了深入的理论和观测研究，引入了两个新的自由度：BD参数ω和剪切标量的当前密度参数Ωσ2,0。此扩展允许我们探讨与广义相对论（GR）的偏离，特别是对暗能量动态和剪切标量红移行为的影响。" 在标准的ΛCDM模型中，宇宙被认为是以一种几乎均匀、各向同性的状态扩张。然而，这个模型并不排除局部的各向异性或不均匀性。Brans-Dicke理论是一种替代GR的理论，它引入了一个标量场，影响重力作用。在各向异性块状BD扩展中，这种标量场不仅影响重力，还可能产生非零的剪切张量，这在ΛCDM模型中是不存在的。 BD参数ω是一个关键的量，它度量了BD理论与GR的差异。在正的ω值下，理论更接近GR；而负的ω值则可能导致显著的偏离。在这种扩展模型中，ω不仅影响暗能量的动态，还与剪切标量的红移依赖性相互作用。剪切标量代表了宇宙扩张的不均匀性，其动力学受到ω的调控，并且其各向异性应力对暗能量的动力学有贡献，这是GR中的标量场模型所不具备的特性。观测数据表明，剪切标量的密度参数Ωσ2,0必须保持在相对较小的值，这可能暗示宇宙的各向异性程度相对较低。通过比较理论预测和当前的宇宙学观测，如宇宙微波背景辐射（CMB）、大尺度结构（LSS）以及超新星Ia的数据，我们可以对ω和Ωσ2,0进行约束。例如，研究发现ω大约需大于250，而Ωσ2,0则小于10^-6，以符合观测限制。此外，剪切标量的红移行为不同于GR中的典型形式，这可能会对早期宇宙的演化和结构形成产生影响。在BD理论中，剪切标量的演化不再简单地遵循(1+z)^6的形式，其中z是红移，这可能会改变我们对宇宙早期阶段的理解。总结来说，这个研究深化了我们对宇宙学模型的理解，尤其是关于暗能量和宇宙不均匀性的理解。通过拓展ΛCDM模型，我们可以探索更广泛的物理现象，包括重力理论的可能替代方案和宇宙的各向异性特征。这种扩展模型能够提供新的测试GR的途径，并可能帮助揭示暗能量的本质。然而，这样的理论必须经过严格的观测检验，以确保它们与现有宇宙学数据的一致性。

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

32 Page 6 of 26 Eur. Phys. J. C (2020) 80 :32

p

=−



8π G

σ. (16)

This effective anisotropic pressure (in the presence of shear)

induced by the Jordan ﬁeld in the Jordan frame is absent in

the Einstein frame (see, e.g., Ref. [132]).

3 Exact solution compatible with standard CDM

model

We followthe method, given in [100], for obtaining exactcos-

mological solutions of a scalar-tensor gravity theory compat-

ible with the CDM model, albeit we consider LRS Bianchi

type I spacetime rather than spatially ﬂat Robertson–Walker

spacetime.

6,7

To do so, we ﬁrst re-express the set of differen-

tial equations given by Eqs. (4)–(7)using

=−H (1 + z),

where H =

S/S is the average Hubble parameter and z is

the cosmic redshift we deﬁne in terms of S as z =−1 +

Accordingly, we re-express the energy density equation (4)



1 −

2ω

(1 + z)



− 2(1 + z)





−

− 2ωM

ρ, (17)

and obtain the pressure equation as

−



1 + z − (1 + z)





+3H



1 + (1 + z)





(1 + ω)





−

(1 + z)





− 2ωM

=−

p, (18)

We note that in the GR limit (say, ϕ = const.) the anisotropic model

under consideration here, (4)–(7), reduces mathematically to the stan-

dard CDM+stiff matter model in the RW framework [133]. However,

the role of the stiff matter with positive energy density in [133]isplayed

by the shear scalar here, so that one can straightforwardly utilize the

rich class of solutions given in [133] for the GR limit in our model.

This presents a good example that one could ﬁnd various solutions of

the model under consideration here but yet we focus on the solution

obtained by extending the method given in [100] to LRS Bianchi type

I metric.

We followed the method given in [100] for obtaining an exact cosmo-

logical solution of BD gravity theory for the expansion rate H (z),with

its CDM model counterpart up to a large redshift, viz., for pressure-

less matter in a spatially ﬂat, homogeneous and isotropic universe. A

more general exact solution of the same setup was given in [134](much

earlier than [100]), was nevertheless given for the scale factor in cosmic

time and very complicated for extracting an exact H (z) required for

observational analyses.

using Eqs. (5) and (6), the shear propagation equation as



−

1 + z

+ 2



= 0, (19)

by subtracting Eq. (5) from Eq. (6), and ﬁnally re-express the

scalar ﬁeld equation (7)as

−



ω(1 + z)



(1 + z)





−2ω



(1 + z)

+ 4ω(1 + z)



+ 6



− 2ωM

= 0, (20)

where



denotes derivative with respect to redshift (d/dz).

We note that Eqs. (18) and (20) have exactly the same

mathematical form, that is, a ﬁrst order linear differential

equation in H

, such as

(z)

+ B

(z)H

+ C

(z) + D

= 0, i = 1, 2, (21)

provided that we set p = p

= 0, viz., the universe is ﬁlled

with only pressureless matter. We next note that constants

= D

=−2ωM

, and that the shear scalar σ

is the term

that differs in our system of equations from the one given in

[100]. Fortunately, it contributes to (18) and (20)inthesame

way, namely, as C

(z) = C

(z) =

. In accordance with

these points, we assume A

(z) = A

(z) in (18) and (20) and

then solve for the rate of change of Jordan ﬁeld in z as



=−

2(1 +ω)

1 + z

, (22)

which in turn renders the coefﬁcients of H

identical, i.e,

(z) = B

(z), as well. Thus, integrating (22), it turns out

that the solution of the Jordan ﬁeld is

= ϕ

(1 + z)

−

1+ω

, (23)

where ϕ

= ϕ(z = 0) is the present time value of the Jordan

ﬁeld, and which in turn gives

G = G

(1 + z)

1+ω

. (24)

The integration of the shear propagation equation given in

Eq. (19)gives

= σ

(1 + z)





−4

, (25)

123

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