# Local Limit of Nonlocal Gravity: Cosmological Perturbations

Javad Tabatabaei1<EMAIL_ADDRESS>Abdolali Banihashemi1
<EMAIL_ADDRESS>Shant Baghram1<EMAIL_ADDRESS>Bahram
Mashhoon2,3<EMAIL_ADDRESS>1Department of Physics, Sharif University
of Technology, Tehran 11365-9161, Iran
2School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
Tehran 19395-5531, Iran
3Department of Physics and Astronomy, University of Missouri, Columbia,
Missouri 65211, USA

###### Abstract

We explore the cosmological implications of the local limit of nonlocal
gravity, which is a classical generalization of Einstein’s theory of
gravitation within the framework of teleparallelism. An appropriate solution
of this theory is the modified Cartesian flat cosmological model. The main
purpose of this paper is to study linear perturbations about the orthonormal
tetrad frame field adapted to the standard comoving observers in this model.
The observational viability of the perturbed model is examined using all
available data regarding the cosmic microwave background. The implications of
the linearly perturbed modified Cartesian flat model are examined and it is
shown that the model is capable of alleviating the $H_{0}$ tension.

Gravitation, Teleparallelism, Cosmology

###### pacs:

04.20.Cv, 04.50.Kd, 98.80.Jk

##  I Introduction

Cosmological observations can, in general, be well described by the standard
$\Lambda$CDM model, where the cosmological constant $\Lambda$ is assumed to be
the cause of the accelerated expansion of the universe (“dark energy”) and the
cold dark matter (CDM) is supposed to be the main component of the cosmic web.
The universe, in terms of its energy content, consists of about $70\%$ dark
energy, about $25\%$ dark matter and about $5\%$ visible matter. On the other
hand, dark aspects of the universe provide the impetus to modify Einstein’s
theory of gravitation Einstein .

There have been many attempts at modifications of Einstein’s general
relativity (GR). For a review of the latest developments in modified gravity
theories see Sotiriou:2008rp ; DeFelice:2010aj ; MGC ). We are interested in a
classical nonlocal generalization of GR; that is, nonlocal gravity (NLG)
theory, which modifies one of the fundamental pillars of relativity theory,
namely, the _locality hypothesis_. The gravitational nonlocality is
implemented within the framework of the teleparallel equivalent of GR. This
framework presents many similarities between gravitation and electromagnetism
(EM). The constitutive relations in EM, which incorporate the effect of medium
on the EM fields, are in general nonlocal. In a similar way, nonlocality could
be introduced into the teleparallel equivalent of GR. In electrodynamics,
nonlocal constitutive relations could be approximated by simple local
relations. We follow this approach in the present paper and consider the local
limit of nonlocal gravity and investigate its cosmological implications. The
vast amount of cosmological data from cosmic microwave background (CMB)
Planck:2018vyg  to large-scale structure (LSS) survey data SDSS:2005xqv  are
essential to studying the nature of gravitation beyond the Milky Way
Weinberg:2013agg  and also to constrain modified gravity models Song:2006ej ;
Zhang:2007nk ; Baghram:2009fr ; Samushia:2012iq . The modified Friedmann-
Lemaître-Robertson-Walker (FLRW) models were studied in the local limit of NLG
in Tabatabaei:2022tbq  and the modified Cartesian flat model was singled out
as being the most plausible. The purpose of the present work is to study
linear perturbations of the modified Cartesian flat cosmological model. We
find the prediction of our model for the CMB and the growth of structure in
the linear regime. We use the available data to constrain the parameters of
our proposed model. Furthermore, we discuss the implications of this model in
connection with the Hubble constant tension DiValentino:2021izs .

The structure of this work is as follows: In Section II, we review the
theoretical background of the local limit of nonlocal gravity first proposed
in Tabatabaei:2022tbq . We briefly describe the teleparallel extension of
general relativity and present the field equations of nonlocal gravity (NLG)
and its local limit. The modified Cartesian flat model is a particular
solution of the field equations of the local limit of NLG and its properties
are studied in Section III. In Section IV, linear perturbations of this
background cosmological model are studied in the Newtonian gauge. We show in
detail the scalar-vector-tensor decomposition of the perturbations and also
write the perturbed field and conservation equations in the synchronous gauge
to check the model with the cosmic microwave background (CMB) data. This is an
essential step as the CAMB code, which is a Boltzmann equation solver based on
the gravity model, works in the synchronous gauge camb . In Section V, we
present our results and find the best-fitting parameters of our model. The
data we employ in this section involve the cosmic microwave background (CMB)
temperature anisotropy angular power spectrum, nearby Supernovae type Ia
(SNeIa) and baryon acoustic oscillation (BAO). Finally, Section VI is devoted
to a discussion of our results.

##  II Local Limit of Nonlocal gravity: Theoretical Background

Nonlocal gravity (NLG) is a classical nonlocal generalization of general
relativity (GR) based on its teleparallel equivalent Hehl:2008eu ; Hehl:2009es
. For a comprehensive account of NLG, see BMB . The teleparallel equivalent of
general relativity (TEGR) is a tetrad theory; that is, the framework of GR is
supplemented with a latticework of preferred orthonormal tetrad frame fields
that are _parallel_ via the Weitzenböck connection. To see how this comes
about, consider a gravitational field with spacetime metric

$ds^{2}=g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\,,$ (1)

where we have a preferred set of observers with adapted orthonormal tetrads
$e^{\mu}{}_{\hat{\alpha}}(x)$, namely,

$g_{\mu\nu}(x)\,e^{\mu}{}_{\hat{\alpha}}(x)\,e^{\nu}{}_{\hat{\beta}}(x)=\eta_{\hat{\alpha}\hat{\beta}}\,.$
(2)

In our convention, Greek indices run from 0 to 3, while Latin indices run from
1 to 3; moreover, the signature of the metric is +2 and $\eta_{\alpha\beta}$
is the Minkowski metric tensor given by diag$(-1,1,1,1)$. We use units such
that $c=1$, unless specified otherwise. Furthermore, hatted indices specify
the tetrad axes in the local tangent space, while indices without hats are
normal spacetime indices.

It is natural to use the preferred tetrad frame field to define the
_Weitzenböck connection_ We

$\Gamma^{\mu}_{\alpha\beta}=e^{\mu}{}_{\hat{\rho}}~{}\partial_{\alpha}\,e_{\beta}{}^{\hat{\rho}}\,,$
(3)

which is nonsymmetric and curvature free. Let $\nabla$ denote covariant
differentiation with respect to the Weitzenböck connection; then,
$\nabla_{\nu}\,e_{\mu}{}^{\hat{\alpha}}=0$, which means that the spacetime
becomes a parallelizable manifold via the Weitzenböck connection. In this
framework of teleparallelism, distant vectors are parallel if they have the
same components with respect to the local preferred frame field Itin:2018dru ;
Maluf:2013gaa ; Aldrovandi:2013wha . The torsion tensor of the Weitzenböck
connection is given by

$C_{\mu\nu}{}^{\alpha}=\Gamma^{\alpha}_{\mu\nu}-\Gamma^{\alpha}_{\nu\mu}=e^{\alpha}{}_{\hat{\beta}}\Big{(}\partial_{\mu}e_{\nu}{}^{\hat{\beta}}-\partial_{\nu}e_{\mu}{}^{\hat{\beta}}\Big{)}\,,$
(4)

which represents the gravitational field in the context of teleparallelism. It
is interesting to compare and contrast the Weitzenböck connection with the
Levi-Civita connection of GR given by the Christoffel symbols

${{}^{0}}\Gamma^{\mu}_{\alpha\beta}=\frac{1}{2}g^{\mu\nu}(g_{\nu\alpha,\beta}+g_{\nu\beta,\alpha}-g_{\alpha\beta,\nu})\,.$
(5)

This torsion-free connection has Riemannian curvature
${{}^{0}}R_{\alpha\beta\gamma\delta}$ that in GR represents the gravitational
field. We employ a left superscript “0” to designate geometric quantities
directly related to the Levi-Civita connection. Einstein’s gravitational field
equations are then given by

${{}^{0}}G_{\mu\nu}+\Lambda\,g_{\mu\nu}=\kappa\,T_{\mu\nu}\,,$ (6)

where $T_{\mu\nu}$ is the symmetric energy-momentum tensor of matter,
$\kappa:=8\pi G/c^{4}$ and the Einstein tensor ${{}^{0}}G_{\mu\nu}$ is defined
as

${{}^{0}}G_{\mu\nu}:={{}^{0}}R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\,{{}^{0}}R\,,$
(7)

where ${{}^{0}}R_{\mu\nu}={{}^{0}}R^{\alpha}{}_{\mu\alpha\nu}$ is the Ricci
tensor and ${{}^{0}}R={{}^{0}}R^{\mu}{}_{\mu}$ is the scalar curvature.

On a given manifold, the difference between two connections is always a
tensor. Indeed, the _contorsion_ tensor is defined by

$K_{\mu\nu}{}^{\alpha}={{}^{0}}\Gamma^{\alpha}_{\mu\nu}-\Gamma^{\alpha}_{\mu\nu}\,.$
(8)

Let us note that $\nabla_{\nu}\,g_{\alpha\beta}=0$, which follows from the
orthonormality of the preferred tetrad frame field. Therefore, the Weitzenböck
connection is metric compatible; indeed, we have two metric-compatible
connections in our extended GR framework. Moreover, the metric-compatibility
of the Weitzenböck connection can be used to relate contorsion to torsion,
namely,

$K_{\mu\nu\rho}=\frac{1}{2}\,(C_{\mu\rho\nu}+C_{\nu\rho\mu}-C_{\mu\nu\rho})\,.$
(9)

The torsion tensor is antisymmetric in its first two indices; hence, the
contorsion tensor is antisymmetric in its last two indices. In this extended
GR framework, the torsion of the Weitzenböck connection, $C_{\mu\nu\rho}$, and
the curvature of the Levi-Civita connection, ${{}^{0}}R_{\mu\nu\rho\sigma}$,
are complementary features of the gravitational field.

The Levi-Civita connection is the sum of the contorsion tensor plus the
Weitzenböck connection; therefore, one can reformulate GR in terms of the
torsion tensor. The result is TEGR, the teleparallel equivalent of GR, which
is the subject of the next subsection.

###  II.1 Teleparallel Equivalent of General Relativity

Within the framework of teleparallelism, one can write the Einstein tensor as
BMB

$\displaystyle{{}^{0}}G_{\mu\nu}=\frac{\kappa}{\sqrt{-g}}\Big{[}e_{\mu}{}^{\hat{\gamma}}\,g_{\nu\alpha}\,\frac{\partial}{\partial
x^{\beta}}\,\mathfrak{H}^{\alpha\beta}{}_{\hat{\gamma}}-\Big{(}C_{\mu}{}^{\rho\sigma}\,\mathfrak{H}_{\nu\rho\sigma}-\frac{1}{4}\,g_{\mu\nu}\,C^{\alpha\beta\gamma}\,\mathfrak{H}_{\alpha\beta\gamma}\Big{)}\Big{]}\,,$
(10)

where the auxiliary torsion field $\mathfrak{H}_{\mu\nu\rho}$ is simply
proportional to the auxiliary torsion tensor $\mathfrak{C}_{\mu\nu\rho}$ by

$\mathfrak{H}_{\mu\nu\rho}:=\frac{\sqrt{-g}}{\kappa}\,\mathfrak{C}_{\mu\nu\rho}\,.$
(11)

Here, as in GR, $g:=\det(g_{\mu\nu})$ and
$\sqrt{-g}=\det(e_{\mu}{}^{\hat{\alpha}})$. The auxiliary torsion tensor is
antisymmetric in its first two indices, just like the torsion tensor, and is
defined by

$\mathfrak{C}_{\alpha\beta\gamma}:=C_{\alpha}\,g_{\beta\gamma}-C_{\beta}\,g_{\alpha\gamma}+K_{\gamma\alpha\beta}\,,$
(12)

where $C_{\mu}$,

$C_{\mu}:=C^{\alpha}{}_{\mu\alpha}=-C_{\mu}{}^{\alpha}{}_{\alpha}\,,$ (13)

is the torsion vector.

It is now possible to write Einstein’s field equation (6) in the form

$\frac{\partial}{\partial
x^{\nu}}\,\mathfrak{H}^{\mu\nu}{}_{\hat{\alpha}}+\frac{\sqrt{-g}}{\kappa}\,\Lambda\,e^{\mu}{}_{\hat{\alpha}}=\sqrt{-g}\,(T_{\hat{\alpha}}{}^{\mu}+\mathbb{T}_{\hat{\alpha}}{}^{\mu})\,,$
(14)

which is therefore the TEGR field equation. Here, $\mathbb{T}_{\mu\nu}$,

$\sqrt{-g}\,\mathbb{T}_{\mu\nu}:=C_{\mu\rho\sigma}\,\mathfrak{H}_{\nu}{}^{\rho\sigma}-\frac{1}{4}g_{\mu\nu}\,C_{\rho\sigma\delta}\,\mathfrak{H}^{\rho\sigma\delta}\,,$
(15)

is the trace-free energy-momentum tensor of the gravitational field. From the
TEGR field Eq. (14), we can derive the law of conservation of total energy-
momentum tensor; that is,

$\frac{\partial}{\partial
x^{\mu}}\,\Big{[}\sqrt{-g}\,(T_{\hat{\alpha}}{}^{\mu}+\mathbb{T}_{\hat{\alpha}}{}^{\mu}-\frac{\Lambda}{\kappa}\,e^{\mu}{}_{\hat{\alpha}})\Big{]}=0\,,$
(16)

which follows from taking the partial derivative $\partial/\partial x^{\mu}$
of Eq. (14) and using the antisymmetry of
$\mathfrak{H}^{\mu\nu}{}_{\hat{\alpha}}$ in its first two indices.

It can be shown that TEGR is the gauge theory of the 4-parameter Abelian group
of spacetime translations Cho ; BlHe . This nonlinear theory is in some ways
similar to Maxwell’s electrodynamics in a medium HeOb . To illustrate this
point, consider the torsion tensor in the form

$C_{\mu\nu}{}^{\hat{\alpha}}=e_{\rho}{}^{\hat{\alpha}}C_{\mu\nu}{}^{\rho}=\partial_{\mu}e_{\nu}{}^{\hat{\alpha}}-\partial_{\nu}e_{\mu}{}^{\hat{\alpha}}\,,$
(17)

where we have an analogue of the electromagnetic field strength
$(\mathbf{E},\mathbf{B})\mapsto
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ defined in terms of
the vector potential $A_{\mu}=e_{\mu}{}^{\hat{\alpha}}$ for each
${\hat{\alpha}}={\hat{0}},{\hat{1}},{\hat{2}},{\hat{3}}$. In the same way, the
electromagnetic excitation $(\mathbf{D},\mathbf{H})\mapsto H_{\mu\nu}$ is
similar to $\mathfrak{H}_{\mu\nu}{}^{\hat{\alpha}}$, while the TEGR field
equation (14) is similar to Maxwell’s equation
$\partial_{\nu}H^{\mu\nu}=4\pi\,J^{\mu}$, where $J^{\mu}$ is the free electric
current 4-vector. In Maxwell’s theory, the constitutive relation connects the
excitation $H_{\mu\nu}$ to the field strength $F_{\mu\nu}$; similarly, we have
in TEGR the local constitutive relation

$\mathfrak{H}_{\alpha\beta\gamma}=\frac{\sqrt{-g}}{\kappa}\,\mathfrak{C}_{\alpha\beta\gamma}=\frac{\sqrt{-g}}{\kappa}\,\left[\frac{1}{2}(C_{\gamma\beta\alpha}+C_{\alpha\beta\gamma}-C_{\gamma\alpha\beta})+C_{\alpha}\,g_{\beta\gamma}-C_{\beta}\,g_{\alpha\gamma}\right]\,,$
(18)

since it connects $\mathfrak{H}_{\alpha\beta\gamma}$ to
$C_{\alpha\beta\gamma}$.

The physical link established here between TEGR and the electrodynamics of
media will be of basic importance as we seek to modify GR and hence TEGR; that
is, we can use our intuition regarding electrodynamics to guide us in the
teleparallel extension of GR. In particular, Maxwell’s basic equations remain
the same when we describe the electrodynamics of different media, only the
constitutive relation changes from one medium to another. We follow this same
approach as we seek to extend TEGR; that is, we keep the TEGR field equation
(14), but modify the constitutive relation (18).

###  II.2 Teleparallel Extension of General Relativity

As explained in the previous section, we change the constitutive relation (18)
in order to modify GR. A simple generic way to do this would be to introduce a
tensor $N_{\mu\nu\rho}=-N_{\nu\mu\rho}$ related to the torsion tensor in Eq.
(18); that is, the new modified constitutive relation is given by

$\mathcal{H}_{\mu\nu\rho}=\frac{\sqrt{-g}}{\kappa}(\mathfrak{C}_{\mu\nu\rho}+N_{\mu\nu\rho})\,,$
(19)

where we recover TEGR for $N_{\mu\nu\rho}\to 0$ and
$\mathcal{H}_{\mu\nu\rho}\to\mathfrak{H}_{\alpha\beta\gamma}$. For the
modified field equation, we simply replace $\mathfrak{H}_{\mu\nu\rho}$ in Eqs.
(14)–(15) by $\mathcal{H}_{\mu\nu\rho}$, namely,

$\frac{\partial}{\partial
x^{\nu}}\,\mathcal{H}^{\mu\nu}{}_{\hat{\alpha}}+\frac{\sqrt{-g}}{\kappa}\,\Lambda\,e^{\mu}{}_{\hat{\alpha}}=\sqrt{-g}\,(T_{\hat{\alpha}}{}^{\mu}+\mathcal{T}_{\hat{\alpha}}{}^{\mu})\,,$
(20)

where $\mathcal{T}_{\mu\nu}$ is the new trace-free energy-momentum tensor of
the gravitational field given by

$\kappa\,\mathcal{T}_{\mu\nu}=\kappa\,\mathbb{T}_{\mu\nu}+Q_{\mu\nu}\,.$ (21)

Here, $Q_{\mu\nu}$ is a trace-free tensor proportional to $N_{\mu\nu\rho}$,

$Q_{\mu\nu}:=C_{\mu\rho\sigma}N_{\nu}{}^{\rho\sigma}-\frac{1}{4}\,g_{\mu\nu}\,C_{\delta\rho\sigma}N^{\delta\rho\sigma}\,.$
(22)

The conservation law of the total energy-momentum now takes the form

$\frac{\partial}{\partial
x^{\mu}}\,\Big{[}\sqrt{-g}\,(T_{\hat{\alpha}}{}^{\mu}+\mathcal{T}_{\hat{\alpha}}{}^{\mu}-\frac{\Lambda}{\kappa}\,e^{\mu}{}_{\hat{\alpha}})\Big{]}=0\,.$
(23)

We have presented here a simple constitutive framework for the teleparallel
extension of GR. Einstein’s GR is based on the metric tensor $g_{\mu\nu}$,
which is basically determined by the GR field equations. On the other hand,
the modified field equations should determine the 16 components of the
fundamental orthonormal tetrad frame field $e^{\mu}{}_{\hat{\alpha}}$. Of
these, 10 fix the components of the metric tensor $g_{\mu\nu}$ via the
orthonormality condition, while the other 6 are local Lorentz degrees of
freedom representing boosts and rotations.

It is interesting to derive the modified Einstein field equations in this
general case. Using Eqs. (18) and (19), we find

$\mathfrak{H}_{\mu\nu\rho}=\mathcal{H}_{\mu\nu\rho}-\frac{\sqrt{-g}}{\kappa}N_{\mu\nu\rho}\,,$
(24)

which we plug into the Einstein tensor (10). Employing Eq. (20), the 16
modified GR field equations take the form

$^{0}G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa
T_{\mu\nu}+\mathbb{R}_{\mu\nu}\,,\qquad\mathbb{R}_{\mu\nu}:=Q_{\mu\nu}-\mathcal{N}_{\mu\nu}\,,$
(25)

where the tensor $\mathcal{N}_{\mu\nu}$ is defined by

$\mathcal{N}_{\mu\nu}:=g_{\nu\alpha}e_{\mu}{}^{\hat{\gamma}}\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{\beta}}\,(\sqrt{-g}N^{\alpha\beta}{}_{\hat{\gamma}})\,.$ (26)

The modified GR field equations can be naturally split into symmetric and
antisymmetric parts. In fact, we have the 10 modified Einstein equations

$^{0}G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa
T_{\mu\nu}+Q_{(\mu\nu)}-\mathcal{N}_{(\mu\nu)}\,$ (27)

and the 6 constraint equations

$Q_{[\mu\nu]}=\mathcal{N}_{[\mu\nu]}\,.$ (28)

To go forward, we need to specify the exact relationship between
$N_{\mu\nu\rho}$ and $C_{\mu\nu\rho}$. By using this approach, a nonlocal
relation has thus far led to nonlocal gravity (NLG) theory that is patterned
after the nonlocal electrodynamics of media Hehl:2008eu ; Hehl:2009es ; BMB .
However, the local limit of this nonlocal relation is the main focus of the
present investigation.

####  II.2.1 Nonlocal Gravity (NLG)

Let us assume that on the basis of measurements performed by the preferred
observers, the components of $N_{\mu\nu\rho}$ are related to those of the
torsion tensor $C_{\mu\nu\rho}$ via Puetzfeld:2019wwo ; Mashhoon:2022ynk

$N_{\hat{\mu}\hat{\nu}\hat{\rho}}(x)=\int\mathcal{K}(x,x^{\prime})\,X_{\hat{\mu}\hat{\nu}\hat{\rho}}(x^{\prime})\sqrt{-g(x^{\prime})}\,d^{4}x^{\prime}\,,$
(29)

where the scalar $\mathcal{K}(x,x^{\prime})$ is the fundamental causal kernel
of NLG and

$X_{\hat{\mu}\hat{\nu}\hat{\rho}}=\mathfrak{C}_{\hat{\mu}\hat{\nu}\hat{\rho}}+\check{p}\,(\check{C}_{\hat{\mu}}\,\eta_{\hat{\nu}\hat{\rho}}-\check{C}_{\hat{\nu}}\,\eta_{\hat{\mu}\hat{\rho}})\,.$
(30)

A detailed explanation for the choice of $X_{\mu\nu\rho}$ is contained in BMB
, which should be consulted for an extensive treatment of the physical basis
of NLG. In Eq. (30), $\check{C}_{\mu}$ is the torsion pseudovector defined via
the Levi-Civita tensor $\epsilon_{\alpha\beta\gamma\delta}$ by

$\check{C}_{\mu}:=\frac{1}{3!}C^{\alpha\beta\gamma}\,\epsilon_{\alpha\beta\gamma\mu}\,,$
(31)

while $\check{p}\neq 0$ is a constant dimensionless parameter.

The kernel $\mathcal{K}(x,x^{\prime})$ must be ultimately determined by
observation. In this connection, linearized NLG has been treated in detail in
BMB  and the Newtonian regime of the theory has been compared with
observational data Rahvar:2014yta ; Chicone:2015coa ; Roshan:2021ljs ;
Roshan:2022zov ; Roshan:2022ypk . Within the nonlinear regime of NLG, it has
been possible to show that de Sitter spacetime is not an exact solution
Mashhoon:2022ynk ; indeed, the only known exact solution of NLG is the trivial
solution, namely, we find the Minkowski spacetime in the absence of
gravitation.

####  II.2.2 Local Limit of Nonlocal gravity

In dealing with the nonlocal electrodynamics of media, it is often convenient
to introduce local quantities for the polarizability and magnetizability of
the medium and write the corresponding local constitutive relations as
$\mathbf{D}=\epsilon(x)\mathbf{E}$ and $\mathbf{B}=\mu(x)\mathbf{H}$, where
the electric permittivity $\epsilon(x)$ and magnetic permeability $\mu(x)$ are
characteristics of the medium. For NLG, a similar procedure would involve the
4D Dirac delta function; that is,

$\mathcal{K}(x,x^{\prime}):=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x^{\prime})\,.$
(32)

Here, $S(x)$ is a dimensionless scalar _susceptibility_ that is characteristic
of the spacetime under consideration. This function, as with the nonlocal
kernel, must ultimately be determined on the basis of observational data.

The constitutive relation in this case is given by

$\mathcal{H}_{\mu\nu\rho}=\frac{\sqrt{-g}}{\kappa}[(1+S)\,\mathfrak{C}_{\mu\nu\rho}+S\,\check{p}\,(\check{C}_{\mu}\,g_{\nu\rho}-\check{C}_{\nu}\,g_{\mu\rho})]\,,$
(33)

which reduces to the case of TEGR for $S=0$. If $S(x)\neq 0$, we have a local
generalization of GR with a new scalar function $S(x)$. Moreover, we must have
$1+S>0$; otherwise, GR will not be a limiting case of the new theory. In
conformity with the electrodynamics of media, $S(x)$ must be compatible with
the nature of the background spacetime.

In recent papers, we have employed the local limit of NLG to study FLRW
cosmological models Tabatabaei:2022tbq ; Tabatabaei:2023qxw . The modified
theory under consideration is a tetrad theory; therefore, we must pay
particular attention to the tetrad frame of the model from which the FLRW
metric is obtained via orthonormality. The spatially homogeneous and isotropic
FLRW models are dynamic and time dependent; hence, we expect on physical
grounds that $dS/dt\neq 0$. It turns out that this requirement is satisfied
only in the case of the modified Cartesian flat model Tabatabaei:2022tbq .
That is, the adapted orthonormal tetrad frame of the standard preferred
comoving observers should be along the Cartesian coordinate axes of the
background spatially Euclidean FLRW spacetime.

The rest of this paper is devoted to the cosmological implications of the
modified Cartesian flat model. We recall the main properties of the model in
the following section and in Section IV we describe the corresponding linear
perturbations in detail. We note that the cosmological implications of
teleparallel theories of gravity have been the subject of recent
investigations Bahamonde:2021gfp ; Heisenberg:2022mbo .

##  III Modified Cartesian Flat Cosmological Model

Consider a spatially flat FLRW model with preferred observers that are
spatially at rest and have adapted orthonormal tetrad axes that are parallel
to the Cartesian coordinate directions, namely,

$e^{\mu}{}_{\hat{\alpha}}=\frac{1}{a(\eta)}\,\delta^{\mu}_{\alpha}\,,\qquad
e_{\mu}{}^{\hat{\alpha}}=a(\eta)\,\delta_{\mu}^{\alpha}\,.$ (34)

Here $a(\eta)$ is the scale factor and $\eta$ is the standard _conformal time_
$x^{0}=\eta$ such that $x^{\mu}=(\eta,x^{i})$. The corresponding metric is
thus conformally flat,

$ds^{2}=a^{2}(\eta)\,\eta_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\,.$ (35)

The conformal time $\eta$ is related to _cosmic time_ $t$ via
$dt=a(\eta)d\eta$. Using standard conventions, we can write

$\dot{a}:=\frac{da}{dt}=\frac{1}{a(\eta)}\,\frac{da(\eta)}{d\eta}:=\frac{a^{\prime}}{a}=\mathcal{H}(\eta)=aH(t)\,,$
(36)

where prime is derivative with respect to conformal time and $H$ is the Hubble
parameter.

The torsion tensor for our Cartesian flat model is given by

$C_{\alpha\beta\gamma}=aa^{\prime}(\delta^{0}_{\alpha}\,\eta_{\beta\gamma}-\delta^{0}_{\beta}\,\eta_{\alpha\gamma})\,.$
(37)

We find the torsion vector, the contorsion tensor and the auxiliary torsion
tensor are

$C_{\alpha}=-3\frac{a^{\prime}}{a}\,\delta_{\alpha 0}\,,\qquad
K_{\alpha\beta\gamma}=C_{\beta\gamma\alpha}\,,\qquad\mathfrak{C}_{\alpha\beta\gamma}=-2\,C_{\alpha\beta\gamma}\,.$
(38)

Moreover, $N_{\alpha\beta\gamma}=-2S(\eta)\,C_{\alpha\beta\gamma}$, since
$\check{C}_{\alpha}=0$ in this case.

The Einstein tensor for our conformally flat spacetime is diagonal; its
components are given by ${}^{0}G_{0i}=0$ and

$^{0}G_{00}=3\left(\frac{a^{\prime}}{a}\right)^{2}\,,\qquad^{0}G_{ij}=-\left[2\frac{a^{\prime\prime}}{a}-\left(\frac{a^{\prime}}{a}\right)^{2}\right]\,\delta_{ij}\,.$
(39)

The source of the gravitational field is a perfect fluid of energy density
$\rho(\eta)$, pressure $P(\eta)$ and energy-momentum tensor

$T_{\mu\nu}=\rho\,u_{\mu}\,u_{\nu}+P\,(g_{\mu\nu}+u_{\mu}\,u_{\nu})\,,$ (40)

where $u^{\mu}$ is the 4-velocity vector of the perfect fluid. We assume, as
in the standard models, that the particles of the fluid are comoving with the
preferred observers and are thus spatially at rest, namely,

$u^{\mu}=e^{\mu}{}_{\hat{0}}=\frac{1}{a(\eta)}\delta^{\mu}_{0}\,,\qquad
u_{\mu}=a(\eta)\eta_{\mu 0}\,.$ (41)

As demonstrated before Tabatabaei:2022tbq ; Tabatabaei:2023qxw , field Eqs.
(27) and (28) are satisfied in this case with $S(\eta)$. More specifically, we
find

$\frac{3}{a^{2}}(1+S)\left(\frac{a^{\prime}}{a}\right)^{2}=\Lambda+8\pi
G\rho\,,$ (42)
$2(1+S)\left(\frac{a^{\prime}}{a}\right)^{\prime}+(1+S)\left(\frac{a^{\prime}}{a}\right)^{2}=a^{2}(\Lambda-8\pi
GP)-2\frac{dS}{d\eta}\frac{a^{\prime}}{a}\,,$ (43)

respectively. As expected, the standard GR results are recovered when $S=0$.
Furthermore, it has been shown that the modified Cartesian flat model is
inconsistent with the existence of the cosmological constant $\Lambda$, just
as in NLG; therefore, we must resort to dynamic dark energy in order to
account for the accelerated expansion of the universe Tabatabaei:2022tbq ;
Tabatabaei:2023qxw . Employing $\mathcal{H}=a^{\prime}/a$, the unperturbed
equations for our modified model become

$3(1+S)\mathcal{H}^{2}=\kappa a^{2}\rho\,,$ (44)
$2(1+S)\mathcal{H}^{\prime}+(1+S)\mathcal{H}^{2}=-\kappa
a^{2}P-2\frac{dS}{d\eta}\mathcal{H}\,.$ (45)

##  IV Cosmological Perturbations

In this section, we treat the linear perturbation theory of our modified
Cartesian flat cosmological model. In the standard GR case, one chooses a
perturbed FLRW metric in a specific gauge and finds the Einstein equations in
terms of the first-order perturbed metric components. In the absence of a
cosmological constant, GR implies $\delta G_{\mu\nu}=\kappa\delta T_{\mu\nu}$,
which results in the relativistic Poisson equation Amendola:2015ksp . On the
other hand, for the local limit of nonlocal gravity (NLG), we perturb the
tetrad frame field. The local theory under consideration here in general
allows the possibility of parity-violating solutions proportional to the
constant dimensionless parameter $\check{p}$. To simplify matters, in this
first treatment of cosmological perturbations, we simply ignore this
possibility by working with the local limit of NLG with $\check{p}=0$ in Eq.
(33).

In our perturbation analysis, the perturbed susceptibility function can be a
function of position as well as time and can be written as

$S(\eta,\mathbf{x})=\bar{S}(\eta)+\delta S(\eta,\mathbf{x})\,,$ (46)

where $\bar{S}(\eta)$ is the unperturbed value of this function. Henceforth,
we use a “bar” to indicate background unperturbed quantities. It proves useful
to employ Fourier transformation such that

$\widetilde{\delta S}(\eta,\mathbf{k})=\int
d^{3}x\,e^{-i\mathbf{k}\cdot\mathbf{x}}\,\delta S(\eta,\mathbf{x})\,.$ (47)

The resulting Fourier transform is a complex function; as usual, however, only
its real part has physical significance in our linear perturbation scheme.
Henceforth, we use a “tilde” to indicate quantities in Fourier space. In the
following subsections, we investigate the tetrad perturbations and the
corresponding perturbed field equations.

###  IV.1 Perturbation of tetrads and energy-momentum tensor

We write the perturbed tetrad frame field for our modified TEGR cosmological
model in the form

$e_{\mu}{}^{\hat{\alpha}}(x)=a(\eta)[\delta_{\mu}^{\alpha}+\psi_{\mu}{}^{\hat{\alpha}}(x)]\,,\qquad
e^{\mu}{}_{\hat{\alpha}}(x)=a^{-1}(\eta)[\delta^{\mu}_{\alpha}-\psi^{\mu}{}_{\hat{\alpha}}(x)]\,,$
(48)

where $\psi_{\mu}{}^{\hat{\alpha}}$ and $\psi^{\mu}{}_{\hat{\alpha}}$ are
treated to linear order away from the Minkowski background and, accordingly,
at this order the distinction between spacetime and tetrad indices can
henceforward be ignored. It follows from the tetrad orthonormality condition,
$e_{\mu}{}^{\hat{\alpha}}e_{\nu}{}^{\hat{\beta}}\eta_{\hat{\alpha}\hat{\beta}}=g_{\mu\nu}$,
that the metric tensor of the perturbed spacetime is given by

$g_{\mu\nu}=a^{2}(\eta)(\eta_{\mu\nu}+h_{\mu\nu})\,,\qquad
h_{\mu\nu}:=2\psi_{(\mu\nu)}\,.$ (49)

It remains to express $\psi_{\mu\nu}(\eta,\mathbf{x})$ in suitable form.

We employ the scalar-vector-tensor decomposition of cosmological perturbations
Golovnev:2018wbh ; Bahamonde:2020lsm ; Hohmann:2020vcv  and write

$\psi_{00}:=\psi\,,\qquad\psi_{i0}:=-(\partial_{i}\varpi+\gamma_{i})\,,\qquad\psi_{0i}:=\partial_{i}\zeta+v_{i}\,,$
(50)
$\psi_{ij}:=\delta_{ij}\,\phi+\partial_{i}\,\partial_{j}\varsigma+\epsilon_{ijk}\,\partial_{k}\mathfrak{s}+\partial_{j}c_{i}+\epsilon_{ijk}\,\mathfrak{v}^{k}+\frac{1}{2}\,\chi_{ij}\,,$
(51)

where $\psi$, $\varpi$, $\zeta$, $\phi$, and $\varsigma$ are scalar functions,
while $\gamma_{i}$, $v_{i}$, and $c_{i}$ are all divergenceless vector
functions. Moreover, $\mathfrak{s}$ and $\mathfrak{v}_{i}$ are pseudoscalar
and pseudovector functions, respectively. Finally, $\chi_{ij}$ is a symmetric
and traceless tensor that is also transverse, i.e.
$\delta^{ik}\,\partial_{k}\chi_{ij}=0$. With our choice of tetrad functions,
the metric is given by

$g_{00}=-a^{2}(\eta)(1-2\psi)\,,\quad
g_{0i}=a^{2}(\eta)[\partial_{i}(\zeta-\varpi)+v_{i}-\gamma_{i})]\,,$ (52)
$g_{ij}=a^{2}(\eta)[(1+2\phi)\delta_{ij}+2\partial_{i}\partial_{j}\varsigma+\partial_{i}\,c_{j}+\partial_{j}\,c_{i}+\chi_{ij}]\,.$
(53)

Except for the pseudoscalar ($\mathfrak{s}$) and pseudovector
($\mathfrak{v}_{i}$) terms, the other functions appear in the metric. The
gauge transformation of the tetrad functions has been discussed in
Golovnev:2018wbh . In what follows, we adopt the conformal Newtonian gauge in
which

$\varsigma=0\,,\qquad\zeta=\varpi\,,\qquad c_{i}=0\,.$ (54)

For the sake of simplicity, we discarded parity-violating terms in the field
equations at the outset. In this connection, it would make sense to ignore
$\mathfrak{s}$ and $\mathfrak{v}_{i}$ in our treatment. Indeed, it will turn
out that $\mathfrak{s}$ does not appear in the perturbed field equations at
all and, moreover, $\mathfrak{v}_{i}$ can be consistently neglected as
$\epsilon_{ijk}\partial_{j}\mathfrak{v}_{k}=0$.

It is also essential to determine the perturbed energy-momentum tensor (40).
In its unperturbed form,
$\bar{T}_{\mu\nu}=~{}$diag$[a^{2}(\eta)\,(\bar{\rho},\bar{P},\bar{P},\bar{P})]$.
In the perturbed energy-momentum tensor,
$T_{\mu\nu}=(\rho+P)u_{\mu}\,u_{\nu}+P\,g_{\mu\nu}$, we ignore heat transfer
and viscous shear stresses. The perturbed energy density and pressure are
given by $\rho=\bar{\rho}+\delta\rho$ and $P=\bar{P}+\delta{P}$. Here we
define the density contrast parameter as

$\delta:=\delta\rho/\bar{\rho}\,.$ (55)

Moreover, the perturbed 4-velocity vector $u^{\mu}$, $u^{\mu}u_{\mu}=-1$,
takes the form

$u^{\mu}=\frac{1}{a}(1+\psi,~{}~{}\mathfrak{u}^{i})\,,\qquad
u_{\mu}=a(-1+\psi,~{}~{}v_{i}-\gamma_{i}+\mathfrak{u}_{i})\,,$ (56)

where the spatial velocity perturbation $\mathfrak{u}^{i}$ is treated to
linear order and $\mathfrak{u}_{i}:=\delta_{ij}\,\mathfrak{u}^{j}$. Employing
the Helmholtz decomposition, we can write the vector field $\mathfrak{u}$ in
terms of its divergence-free $(\mathfrak{u}_{\perp})$ and curl-free
$(\mathfrak{u}_{\parallel})$ components, namely,

$\mathfrak{u}^{i}=\mathfrak{u}^{i}_{\perp}+\mathfrak{u}^{i}_{\parallel}\,,\qquad\nabla\cdot\mathfrak{u}_{\perp}=0\,,\qquad\nabla\times\mathfrak{u}_{\parallel}=0\,.$
(57)

The expansion parameter for the perturbed spatial velocity field is defined by

$\theta:=\partial_{i}\,\mathfrak{u}^{i}=\nabla\cdot\mathfrak{u}_{\parallel}\,.$
(58)

Regarding density and pressure, we define $w$,

$w:=\frac{\bar{P}}{\bar{\rho}}\,,$ (59)

which is normally assumed to be a constant. Moreover, we need to specify the
relation between $\delta P$ and $\delta\rho$; this relation, in the rest frame
of the fluid, is defined as the sound speed $c_{s}\geq 0$,

$c^{2}_{s}:=\frac{dP}{d\rho}|_{\rm{FluidRestFrame}}\,,$ (60)

which is a characteristic feature of the fluid under consideration. For dark
matter and baryons, we set $c_{s}=0$.

###  IV.2 Perturbed Field Equations

In this subsection, we study the perturbed field equations for our modified
TEGR cosmological model. The modified GR field equations can be naturally
split into symmetric and antisymmetric parts as in Eqs. (27) and (28). In
fact, we have the 10 modified perturbed Einstein equations

$\delta\,^{0}G_{\mu\nu}=\kappa\delta T_{\mu\nu}+\delta
Q_{(\mu\nu)}-\delta\mathcal{N}_{(\mu\nu)}\,$ (61)

and the 6 perturbed constraint equations

$\delta Q_{[\mu\nu]}=\delta\mathcal{N}_{[\mu\nu]}\,.$ (62)

Let us first consider the perturbed symmetric field Eq. (61). The $(00)$
component of this equation reads

$-\Delta\phi+3\mathcal{H}(\phi^{\prime}+\mathcal{H}\psi)=\frac{\kappa
a^{2}\delta\rho-3\mathcal{H}^{2}\delta S}{2(1+\bar{S})}\,,$ (63)

where $\Delta$ is the Laplace operator. Next, the three symmetric $(0i)$
components are

$\Delta(\gamma_{i}-v_{i})+6(1+w)\,\mathcal{H}^{2}\,(v_{i}-\gamma_{i}+\mathfrak{u}_{i})+\ss\,\epsilon_{ijk}\partial_{j}\mathfrak{v}_{k}-2\partial_{i}\left(2\,\mathcal{H}\psi+\mathcal{H}\frac{\delta
S}{1+\bar{S}}+2\phi^{\prime}+\ss\phi\right)=0\,.$ (64)

Here, we have introduced the notation $\ss:=\bar{S}^{\prime}/(1+\bar{S})$ for
the sake of simplicity. Finally, the six symmetric $(ij)$ components are

$\begin{split}&-4\delta_{ij}\left[\Phi+\mathcal{H}\,\Psi+\frac{1}{2}\kappa
a^{2}\frac{\delta
P}{1+\bar{S}}-\mathcal{H}\left(2\ss+3w\mathcal{H}\right)\frac{\delta
S}{1+\bar{S}}+2\mathcal{H}\frac{\delta S^{\prime}}{1+\bar{S}}\right]\\\
&-2(\partial_{i}\partial_{j}-\delta_{ij}\Delta)\left(\ss\,\zeta-\psi+\phi\right)+\partial_{i}\mathbb{V}_{j}+\partial_{j}\mathbb{V}_{i}\\\
&+\chi^{\prime\prime}_{ij}+\left(2\mathcal{H}+\ss\right)\chi^{\prime}_{ij}-\Delta\chi_{ij}=0\,.\end{split}$
(65)

Here, $\Phi$, $\Psi$ and $\mathbb{V}_{i}$ are given by

$\Phi=\phi^{\prime\prime}+\left(2\mathcal{H}+\ss\right)\phi^{\prime}\,,$ (66)
$\Psi=\psi^{\prime}-3w\mathcal{H}\psi\,,$ (67)
$\mathbb{V}_{i}=2\mathcal{H}(\gamma_{i}-v_{i})-\ss\,v_{i}+\gamma^{\prime}_{i}-v^{\prime}_{i}\,.$
(68)

For $S=0$ in the 10 modified perturbed Einstein equations, we recover the
standard GR results Amendola:2015ksp .

For the six antisymmetric equations (62), let us first consider the three
$[0i]$ relations, namely,

$2\mathcal{H}\partial_{i}\delta
S+\bar{S}^{\prime}\left(\epsilon_{ijk}\partial_{j}\mathfrak{v}_{k}-2\partial_{i}\phi\right)=0\,.$
(69)

The remaining three antisymmetric $[ij]$ relations are given by

$\bar{S}^{\prime}\,(\partial_{i}\gamma_{j}-\partial_{j}\gamma_{i})=0\,.$ (70)

For $S=0$, the constraint equations all vanish, as expected.

We now have the 16 perturbation equations of our modified TEGR model. To
discuss the physical content of these equations, let us start with the
simplest, namely, Eq. (70). We know that $\bar{S}^{\prime}\neq 0$ by
assumption; therefore, Eq. (70) implies $\gamma_{i}=\partial_{i}\xi$, where
$\xi$ is a scalar function that satisfies Laplace’s equation, $\Delta\xi=0$,
since the vector $\gamma_{i}$ is divergence-free in our decomposition. We
assume all of the functions that appear in our decomposition are sufficiently
smooth and bounded as would be required for our cosmological perturbation
scheme. A real harmonic function is a smooth function defined on an open
subset of the three-dimensional Euclidean space that satisfies Laplace’s
equation. According to Liouville’s theorem, a bounded harmonic function
defined over all of the three-dimensional Euclidean space must be constant.
Therefore, $\xi$ is a constant and

$\gamma_{i}=0\,.$ (71)

Next, let us take the divergence of Eq. (64). We recall that $\gamma_{i}=0$
and $v_{i}$ is divergence free in our decomposition; therefore,

$\Delta\left(2\,\mathcal{H}\,\psi+\mathcal{H}\frac{\delta
S}{1+\bar{S}}+2\phi^{\prime}+\ss\,\phi\right)=3(1+w)\,\mathcal{H}^{2}\,\theta.$
(72)

For $S=0$, we have the standard GR result, i.e.
$\Delta(\phi^{\prime}+\mathcal{H}\,\psi)=\kappa
a^{2}(1+w)\bar{\rho}\,\theta/2$ Amendola:2015ksp , where we have used the
unperturbed Eq. (44) in this case, namely, $3{\mathcal{H}}^{2}=\kappa
a^{2}\bar{\rho}$. Relation (72) is a dynamical equation for the scalar degree
of freedom in our perturbation scheme. It is also possible to extract the
vectorial part of Eq. (64) by taking its curl; we find,

$\nabla\times\mathbf{U}=0\,,\qquad U_{i}=\Delta
v_{i}-6(1+w)\,\mathcal{H}^{2}\,(v_{i}+\mathfrak{u}_{i}^{\perp})-\ss\,\epsilon_{ijk}\partial_{j}\mathfrak{v}_{k}\,,$
(73)

where we have used $\gamma_{i}=0$ and
$\nabla\times\mathfrak{u}_{\parallel}=0$. The vectors $v_{i}$ and
$\mathfrak{u}_{i}^{\perp}$ in $\mathbf{U}$ are divergence free; therefore, the
divergence of $\mathbf{U}$ vanishes. On the other hand, it follows from Eq.
(73) that $\mathbf{U}=\nabla f$ for a scalar field $f$. Thus, $\Delta f=0$; as
before, Liouville’s theorem implies that $f$ is constant and $\mathbf{U}=0$,
namely,

$\Delta
v_{i}-6(1+w)\,\mathcal{H}^{2}\,(v_{i}+\mathfrak{u}_{i}^{\perp})-\ss\,\epsilon_{ijk}\partial_{j}\mathfrak{v}_{k}=0\,.$
(74)

Let us now return to Eq. (65) and define

$\Theta_{ij}:=\chi^{\prime\prime}_{ij}+\left(2\mathcal{H}+\ss\right)\chi^{\prime}_{ij}-\Delta\chi_{ij}\,,$
(75) $\mathbb{S}_{1}:=\ss\,\zeta-\psi+\phi\,,$ (76)
$\mathbb{S}_{2}:=\Phi+\mathcal{H}\Psi+4\pi Ga^{2}\frac{\delta
P}{1+\bar{S}}-\mathcal{H}\left(2\ss+3w\mathcal{H}\right)\frac{\delta
S}{1+\bar{S}}+2\mathcal{H}\frac{\delta S^{\prime}}{1+\bar{S}}\,.$ (77)

Accordingly, Eq. (65) becomes:

$\mathbb{T}_{ij}+\partial_{i}\mathbb{V}_{j}+\partial_{j}\mathbb{V}_{i}-2(\partial_{i}\partial_{j}-\delta_{ij}\Delta)\mathbb{S}_{1}-4\delta_{ij}\mathbb{S}_{2}=0\,,$
(78)

where

$-\mathbb{V}_{i}=2\mathcal{H}v_{i}+\ss\,v_{i}+v^{\prime}_{i}\,.$ (79)

Here, $\mathbb{T}_{ij}$ is a symmetric, transverse and traceless tensor;
moreover, vector $\mathbb{V}$ is divergence free. We can rewrite Eq. (78) as

$\Theta_{ij}+\partial_{i}\mathbb{V}_{j}+\partial_{j}\mathbb{V}_{i}-2(\partial_{i}\partial_{j}-\frac{1}{3}\delta_{ij}\Delta)\mathbb{S}_{1}-4\delta_{ij}(\mathbb{S}_{2}-\frac{1}{3}\Delta\mathbb{S}_{1})=0\,.$
(80)

Calculating the trace of (80), we find

$\mathbb{S}_{2}-\frac{1}{3}\Delta\mathbb{S}_{1}=0\,.$ (81)

We are thus left with

$\Theta_{ij}+\partial_{i}\mathbb{V}_{j}+\partial_{j}\mathbb{V}_{i}-2(\partial_{i}\partial_{j}-\frac{1}{3}\delta_{ij}\Delta)\mathbb{S}_{1}=0\,.$
(82)

Taking the double divergence ($\partial_{i}\partial_{j}$) of Eq. (82) results
in

$\Delta^{2}\mathbb{S}_{1}=0\,.$ (83)

Let us assume that the cosmological perturbation is such that $\phi$, $\psi$
and $\zeta$ vanish at spatial infinity; then, applying Liouville’s theorem
twice results in $\mathbb{S}_{1}=0$ and subsequently $\mathbb{S}_{2}=0$ as
well. The remaining equation is

$\Theta_{ij}+\partial_{i}\mathbb{V}_{j}+\partial_{j}\mathbb{V}_{i}=0\,.$ (84)

Taking the divergence of this equation, one finds

$\Delta\mathbb{V}_{i}=0\,.$ (85)

Assuming that $v_{i}$ tends to zero at spatial infinity, Liouville’s theorem
and Eq. (79) imply $\mathbb{V}_{i}=0$ and this subsequently results in
$\Theta_{ij}=0$.

Finally, it is possible to decompose Eq. (69) into scalar and pseudo-vectorial
parts. Writing down the divergence of this equation gives
$\Delta(\mathcal{H}\delta S-\bar{S}^{\prime}\phi)=0$. It follows from
Liouville’s theorem that

$\mathcal{H}\,\delta S-\bar{S}^{\prime}\,\phi=0\,,$ (86)

assuming localized perturbations, as before, such that both $\phi$ and $\delta
S$ vanish at infinity. Substituting this result in Eq. (69), we obtain

$\epsilon_{ijk}\partial_{j}\mathfrak{v}_{k}=0\,,$ (87)

since $\bar{S}^{\prime}\neq 0$ by assumption. The pseudovector
$\mathfrak{v}_{k}$ appears in our perturbation equations only as in Eq. (87);
therefore, we can essentially ignore the pseudovector $\mathfrak{v}_{k}$ in
our subsequent analysis. Furthermore, the pseudoscalar term $\mathfrak{s}$
does not appear in the perturbed field equations and we have the freedom to
set it equal to zero. In connection with parity-violating terms, let us recall
that for simplicity we imposed at the outset the requirement that
$\check{p}=0$; if we relax this condition, the situation may be different and
should be reconsidered. In following, we will examine the equations of motion
in each sector.

####  IV.2.1 Scalar Sector

The remaining perturbation equations contain the scalar fields $\phi$, $\psi$,
$\zeta$, $\delta S$, $\delta\rho$ and $\delta P$. The first three appear in
the perturbed metric and are related by

$\phi-\psi+\ss\,\zeta=0\,.$ (88)

Of these, only $\zeta$ has no dynamical equation for its determination. It is
therefore possible to set $\zeta=0$ in a way that is completely consistent
with the remaining perturbations. Hence, $\phi=\psi$. Moreover, it follows
from Eq. (86) that

$\delta S=\frac{\bar{S}^{\prime}}{\mathcal{H}}\,\phi\,.$ (89)

Using these results, Eq. (63) turns into

$\phi^{\prime}+\left(-\frac{1}{3\mathcal{H}}\Delta+\mathcal{H}+\frac{1}{2}\ss\right)\phi=\frac{\kappa\,a^{2}}{6(1+\bar{S})\mathcal{H}}\delta\rho\,.$
(90)

From $\mathbb{S}_{2}=0$, we find (with $\phi=\psi$),

$\phi^{\prime\prime}+(3\mathcal{H}+\ss)\phi^{\prime}-(3w\mathcal{H}^{2}+3w\mathcal{H}\ss+2\ss^{2})\phi+\frac{1}{2}\kappa
a^{2}\frac{\delta P}{1+\bar{S}}+2\mathcal{H}\frac{\delta
S^{\prime}}{1+\bar{S}}=0\,.$ (91)

Here, $\delta P=c_{s}^{2}\,\delta\rho$ and $\delta S^{\prime}$ can be obtained
from differentiating Eq. (89) with respect to conformal time. Combining Eqs.
(89)–(90) in this way, we obtain one main equation for the temporal evolution
of scalar field $\phi$ with conformal time $\eta$.

On the other hand, with $\psi=\phi$ and $\mathcal{H}\,\delta
S=\bar{S}^{\prime}\phi$, Eq. (72) can be expressed as

$(\mathcal{H}+\ss)\,\Delta\phi\,+\Delta\phi^{\prime}=\frac{3}{2}(1+w)\,\mathcal{H}^{2}\,\theta\,.$
(92)

Substituting for $\phi^{\prime}$ the formula given in Eq. (90), we find

$2\Delta^{2}\,\phi+3\mathcal{H}\,\ss\,\Delta\phi=9(1+w)\,\mathcal{H}^{3}\,\theta-\frac{\kappa\,a^{2}}{1+\bar{S}}\Delta(\delta\rho)\,,$
(93)

or, in terms of spatial Fourier transforms as defined in Eq. (47),

$\tilde{\phi}(\eta,\mathbf{k})=\frac{9(1+w)\,\mathcal{H}^{3}\,\tilde{\theta}(\eta,\mathbf{k})+\kappa\,a^{2}\,k^{2}\,\widetilde{\delta\rho}(\eta,\mathbf{k})/(1+\bar{S})}{2k^{4}-3\mathcal{H}\,\ss
k^{2}}\,.$ (94)

####  IV.2.2 Vector Sector

The vector field $\mathbf{v}$ is now the only nonzero vectorial degree of
freedom in our model. It satisfies

$\Delta v_{i}-6(1+w)\,\mathcal{H}^{2}\,(v_{i}+\mathfrak{u}_{i}^{\perp})=0\,$
(95)

and

$v^{\prime}_{i}=-\left(2\mathcal{H}+\ss\right)\,v_{i}\,.$ (96)

As $2\mathcal{H}+\bar{S}^{\prime}/(1+\bar{S})$ is always a positive function
of conformal time, the vectorial degree of freedom has a decaying evolution
and eventually becomes negligible as the universe expands.

The components of the spatial velocity perturbation can be obtained from Eqs.
(64) and (95), namely,

$\frac{3}{2}(1+w)\mathcal{H}^{2}\,\mathfrak{u}_{i}^{\parallel}=\partial_{i}[(\mathcal{H}+\ss)\phi+\phi^{\prime}]\,,$
(97) $6(1+w)\mathcal{H}^{2}\,\mathfrak{u}_{i}^{\perp}=\Delta
v_{i}-6(1+w)\,\mathcal{H}^{2}\,v_{i}\,.$ (98)

Hence, $\mathfrak{u}_{i}^{\perp}$ decays in conformal time with the expansion
of the universe.

####  IV.2.3 Tensor Sector

The tensor field $\chi_{ij}$ occurs only in the perturbation Eq. (65). The
tensor part decouples from the other perturbation equations and we find
$\Theta_{ij}=0$. That is,

$\chi_{ij}^{{}^{\prime\prime}}+\left(2\mathcal{H}+\ss\right)\chi_{ij}^{{}^{\prime}}-\Delta\,\chi_{ij}=0\,.$
(99)

A similar equation for free gravitational waves in Minkowski spacetime was
studied in Tabatabaei:2022tbq  within the framework of the local limit of NLG.
In the present work, we are interested in the $H_{0}$ tension and ignore the
possibility of existence of gravitational waves in our model; therefore, we
assume $\chi_{ij}=0$ in the rest of this paper.

We have worked out the basic field equations governing the perturbations of
the tetrads and the energy-momentum tensor in order to determine the spectrum
of CMB temperature anisotropy. In the next subsection, we further develop the
dynamical equations governing $\delta$ and $\theta$.

###  IV.3 Continuity and Euler equations

From the solutions of the field Eqs. (44) and (45) for the background modified
Cartesian flat model, we find

$\bar{\rho}^{\prime}=-3\,(1+w)\,\bar{\rho}\,\mathcal{H}-\frac{3\mathcal{H}^{2}}{\kappa\,a^{2}}\,\bar{S}^{\prime}\,.$
(100)

This gives us the simple solution $\bar{\rho}\propto a^{-3(1+w)}/(1+\bar{S})$
Tabatabaei:2022tbq . Moving on to the perturbative part of the energy-momentum
content of our model, we note that it is possible to find two conservation
formulas for perturbed quantities in the energy-momentum tensor as well. In
principle the conservation equations are not independent of field equations
obtained in previous subsections. Indeed, the situation here is rather similar
to GR Amendola:2015ksp .

Let us define as in Eq. (23),

$\mathcal{E}^{\mu}{}_{\hat{\alpha}}:=\sqrt{-g}\,(T_{\hat{\alpha}}{}^{\mu}+\mathcal{T}_{\hat{\alpha}}{}^{\mu}-\frac{\Lambda}{\kappa}\,e^{\mu}{}_{\hat{\alpha}})\,;$
(101)

then, the continuity equation follows from
${\partial}_{\mu}\mathcal{E}^{\mu}{}_{\hat{0}}=0$, while
$\partial_{\mu}\mathcal{E}^{\mu}{}_{\hat{i}}=0$ results in the Euler equation
in this case. Using these results or the perturbation equations in the scalar
sector, we find the following equations for the density contrast
$\tilde{\delta}$ and volume expansion $\tilde{\theta}$ both in Fourier space:

$\tilde{\delta}^{\prime}+A_{1}\,\tilde{\delta}+A_{2}\,\tilde{\theta}=0\,,$
(102)
$\tilde{\theta}^{\prime}+B_{1}\,\tilde{\theta}+B_{2}\,\tilde{\delta}=0\,.$
(103)

Here, the functions $A_{1}$, $A_{2}$, $B_{1}$ and $B_{2}$ are defined as

$A_{1}=-\frac{18\,(1+w)\mathcal{H}^{3}+4k^{2}\ss+3\mathcal{H}^{2}\left(7-3w-6\,\delta
P/\delta\rho\right)\ss+12\,k^{2}\mathcal{H}(w-\delta
P/\delta\rho)-6\mathcal{H}\ss^{\prime}}{2(1+\bar{S})\mathfrak{K}}\,,$ (104)
$A_{2}=-a\frac{-4k^{4}+54(1+w)\mathcal{H}^{4}+12\mathcal{H}\,\ss(2\,k^{2}+3\mathcal{H}^{2})+9\,\mathcal{H}^{2}\left[2k^{2}(1+w)-\ss^{2}-2\ss^{\prime}\right]}{2\,k^{2}\,(1+\bar{S})\mathfrak{K}(1+w)^{-1}}\,,$
(105)
$B_{1}=-(2+3\,w)\mathcal{H}-\ss+\frac{3\mathcal{H}[2\,k^{2}+3(1+w)\mathcal{H}^{2}]}{\mathfrak{K}}\,,$
(106) $B_{2}=k^{2}\frac{-2k^{2}\,\,\delta
P/\delta\rho+3(1+w)\mathcal{H}^{2}+3(1+\delta
P/\delta\rho)\,\ss\,\mathcal{H}}{(1+w)\mathfrak{K}}\,,$ (107)

where

$\mathfrak{K}:=2k^{2}-3\,\ss\,\mathcal{H}\,.$ (108)

Solving Eqs. (102) and (103), we have the density contestant for each redshift
and wavenumber; accordingly, the matter power spectrum is known. To study the
dynamics of perturbations in this theory and find the angular power spectrum
of temperature anisotropy of CMB, four perturbed quantities $\phi$, $\psi$,
$\delta$ and $\theta$ are in general needed. These can be obtained by solving
equations (88), (90), (102) and (103). The CAMB can solve these equations in
the synchronous gauge. Therefore, in the next subsection, we write the above
equations in the synchronous gauge and Fourier space.

###  IV.4 Cosmological Perturbations in Synchronous Gauge

To confront our model with cosmological observations, the perturbed
gravitational field equations must be numerically integrated to determine the
temporal evolution of the model with the expansion of the universe. To compare
the result with observational data, it is convenient to use the synchronous
gauge. This gauge has been explored in Ma:1994dv . Therefore, we need to
transform the perturbation equations from the conformal Newtonian gauge to the
synchronous gauge in order to be able to use the CAMB-code Lewis:1999bs . This
code makes it possible to find the CMB angular power spectrum.

In the synchronous gauge, the metric takes the form

$ds^{2}=a^{2}(\eta)\left[-d\eta^{2}+(\delta_{ij}+\texttt{h}_{ij})dx^{i}\,dx^{j}\right]\,.$
(109)

Here, $\texttt{h}_{ij}$ has the Fourier decomposition

$\texttt{h}_{ij}(\eta,\mathbf{x})=\int{d^{3}k\,e^{i\,\mathbf{k}.\mathbf{x}}\left[\hat{k}_{i}\,\hat{k}_{j}\,\tilde{h}(\eta,\mathbf{k})+(\hat{k}_{i}\,\hat{k}_{j}-\frac{1}{3}\,\delta_{ij})\,6\,\tilde{\mathfrak{h}}(\eta,\mathbf{k})\right]}\,,$
(110)

where $\tilde{h}$ and $\tilde{\mathfrak{h}}$ are scalar functions. The
coordinate transformation that can take the conformal Newtonian gauge to
synchronous gauge is of the general form

$x^{\mu}\rightarrow\hat{x}^{\mu}=x^{\mu}+b^{\mu}(x^{\nu})\,.$ (111)

It can be shown that, using the proper vector field $b^{\mu}$, the perturbed
quantities $\tilde{\psi}$ and $\tilde{\phi}$ in the conformal Newtonian gauge
can be transformed to the synchronous gauge. We note that the quantities with
tilde are in Fourier space. The results are:

$\tilde{\psi}{}^{\rm{(syn)}}(\eta,\mathbf{k})=\frac{1}{2\,k^{2}}\left\\{\tilde{h}^{{}^{\prime\prime}}(\eta,\mathbf{k})+6\,\tilde{\mathfrak{h}}^{{}^{\prime\prime}}(\eta,\mathbf{k})+\frac{a^{\prime}}{a}\left[\tilde{h}^{\prime}(\eta,\mathbf{k})+6\,\tilde{\mathfrak{h}}(\eta,\mathbf{k})\right]\right\\}\,,$
(112)
$\tilde{\phi}{}^{\rm{(syn)}}(\eta,\mathbf{k})=\tilde{\mathfrak{h}}(\eta,\mathbf{k})-\frac{1}{2\,k^{2}}\,\frac{a^{\prime}}{a}\left[\tilde{h}^{\prime}(\eta,\mathbf{k})+6\,\tilde{\mathfrak{h}}^{\prime}(\eta,\mathbf{k})\right]\,.$
(113)

We also should transform energy momentum tensor components as

$T^{\mu}{}_{\nu}\ {}^{\rm{(syn)}}=\frac{\partial\hat{x}^{\mu}}{\partial
x^{\sigma}}\,\frac{\partial
x^{\rho}}{\partial\hat{x}^{\nu}}T^{\sigma}{}_{\rho}{}^{\rm{(con)}},$ (114)

where “syn” represents the synchronous gauge and “con” the conformal Newtonian
gauge, which will appear as superscript on perturbed quantities as below:

$\tilde{\delta}^{\rm{(syn)}}=\tilde{\delta}^{\rm{(con)}}-\Upsilon\frac{\bar{\rho}^{\prime}}{\bar{\rho}}\,,$
(115) $\widetilde{\delta P}^{\rm{(syn)}}=\widetilde{\delta
P}^{\rm{(con)}}-\Upsilon\bar{P}^{\prime}\,,$ (116)
$\tilde{\theta}^{\rm{(syn)}}=\tilde{\theta}^{\rm{(con)}}-\Upsilon k^{2}\,,$
(117)

Here,
$\Upsilon=\tfrac{1}{2\,k^{2}}\,(\tilde{h}^{\prime}+6\tilde{\mathfrak{h}}^{\prime})$.
Then, the resulting modified perturbation equations become

$k^{2}\tilde{\mathfrak{h}}-\frac{1}{2}\mathcal{H}\left(\tilde{h}^{\prime}-3\,\ss\,\tilde{\mathfrak{h}}\right)=-\frac{1}{2}\kappa\,a^{2}\frac{\widetilde{\delta\rho}^{(syn)}}{1+\bar{S}}\,,$
(118)
$k^{2}\left(\tilde{\mathfrak{h}}^{\prime}+\ss\,\tilde{\mathfrak{h}}\right)=-\frac{3}{2}(1+w)\tilde{\theta}^{(syn)}\mathcal{H}^{2}\,,$
(119)
$\tilde{h}^{{}^{\prime\prime}}+2\mathcal{H}\,\tilde{h}^{\prime}-\left(2k^{2}+3\,\ss\,\mathcal{H}-6\,\ss^{2}+\,6\ss^{\prime}\right)\tilde{\mathfrak{h}}=\frac{3\kappa\,a^{2}}{1+\bar{S}}\widetilde{\delta
P}^{(syn)}+\frac{18}{k^{2}}(1+w)\tilde{\theta}^{(syn)}\,\ss\,\mathcal{H}^{2}\,,$
(120)
$2k^{2}\,\tilde{\mathfrak{h}}=2\mathcal{H}\left(\tilde{h}^{\prime}+6\tilde{\mathfrak{h}}^{\prime}\right)+\tilde{h}^{{}^{\prime\prime}}+6\tilde{\mathfrak{h}}^{{}^{\prime\prime}}\,.$
(121)

Let us now express the density contrast $\tilde{\delta}$ and volume expansion
$\tilde{\theta}$ in this gauge, namely,

$\tilde{\delta}^{\prime(syn)}=-(1+w)\left(1-\frac{9}{2k^{2}}\mathcal{H}\ss\right)\tilde{\theta}^{(syn)}-\frac{(1+w)k^{2}+\mathcal{H}\ss+\ss^{2}-\ss^{\prime}}{\mathfrak{L}}\>\tilde{h}^{\prime}\\\
+\frac{\ss\left[2k^{2}-\frac{3}{2}\left(1-3w+6\frac{\delta
P}{\delta\rho}\right)\mathcal{H}^{2}\right]+3\mathcal{H}\left[2k^{2}\left(w-\frac{\delta
P}{\delta\rho}\right)+2\ss^{2}-\ss^{\prime}\right]}{\mathfrak{L}}\,\tilde{\delta}^{(syn)},$
(122)

where $\mathfrak{L}:=2k^{2}+3\,\ss\,\mathcal{H}$ and

$\tilde{\theta}^{(syn)\prime}=-\left((1-3w)\mathcal{H}-\ss\right)\tilde{\theta}^{(syn)}+\frac{k^{2}}{1+w}\frac{\delta
P}{\delta\rho}\,\tilde{\delta}^{(syn)}+\frac{1}{\mathcal{H}}\,\ss\,\frac{k^{2}}{1+w}\tilde{\mathfrak{h}}\>.$
(123)

Finally we need to relate the parameter $\delta P$ to $\delta\rho$. This is
usually done with defining the sound speed, $c_{s}^{2}\equiv\delta
P/\delta\rho|_{\rm{FluidRestFrame}}$. In the synchronous gauge, however, the
perturbed equation of state becomes

$\widetilde{\delta
P}=c_{s}^{2}\widetilde{\delta\rho}+3\mathcal{H}\bar{\rho}(1+w)(c_{s}^{2}-w)\frac{\tilde{\theta}}{k^{2}}\,.$
(124)

In this way, the required perturbation equations are in the synchronous gauge
and we can obtain the CMB angular power spectrum using the CAMB code.

Figure 1: Two dimensional posterior probabilities for the free parameters of
the modified Cartesian flat model with the susceptibility function $S$
described by Eq. (125). The model has been constructed within the framework of
the local limit of nonlocal gravity (NLG). These constraints are obtained by
using CMB, BAO and R19. This plot is produced by employing the publicly
available code, GetDist Lewis:2019xzd . The dark smaller regions show the
$1\sigma$ and the lighter broader regions indicate the $2\sigma$ confidence
levels.

##  V Results

The modified Cartesian flat cosmological model under consideration here is a
solution of the local limit of nonlocal gravity theory. We have worked out the
perturbations of this model in the synchronous gauge. Accordingly, by choosing
a specific susceptibility function $S(t)$, we can compare the theory with the
observational data. As proposed and investigated in Tabatabaei:2022tbq , we
choose

$S(z)=\alpha\left(1+z\right)^{\beta}\,,$ (125)

where $z$ is the cosmological redshift and $\alpha$ and $\beta$ are two extra
free parameters in addition to the ones in common with the $\Lambda$CDM model.
Hence, the set of free parameters of our modified model is as follows:

$\mathcal{P}=\\{\Omega_{b}\,h^{2},\Omega_{c}\,h^{2},100\,\Theta_{MC},\tau,n_{s},\ln[10^{10}A_{s}],\alpha,\beta\\}\,,$
(126)

where $\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ are the current physical
baryonic and dark matter densities respectively; $\Theta_{MC}$ is the acoustic
angular scale on the CMB sky; $\tau$ is the re-ionization optical depth,
$A_{s}$ is the amplitude of primordial power spectrum of scalar perturbations,
inherited from inflation; $n_{s}$ is the spectral index of those primordial
perturbations and describes their scale dependence; $\alpha$ shows the
deviation of our model away from $\Lambda$CDM and finally $\beta$ indicates
the dependence of $S$ on the cosmological redshift. In order to find the best-
fitting values for the free parameters, we need to use observational data and
define a so-called loss function $\chi^{2}$, which quantifies the deviation of
our model’s prediction from the data. Naturally, the goal is to minimize
$\chi^{2}$, which is a function of the free parameters of the model. Best-
fitting values for the free parameters are those which minimize $\chi^{2}$.
Usually, the loss function in cosmology is not an analytic function of the
free parameters and one cannot perform differentiation to find the minimum.
This is because the predictions of a model for an observable like CMB
temperature angular power spectrum consist of many numerical integrations and
there is no simple analytic form of the prediction as a function of free
parameters. In these cases, in order to have the minimum of the loss function,
one has to check every point in the discretized space of the free parameters,
$\mathcal{P}$, and find the point where the loss function is a minimum. This
procedure is resource intensive, especially when the dimension of
$\mathcal{P}$ (number of free parameters) is more than $\sim 5$. On the other
hand, the relevant region in $\mathcal{P}$, where the loss function is close
to its minimum, is a rather small zone and can be found quickly with a few
steps in a kind of biased random walk in $\mathcal{P}$. In such a random walk,
every step is biased toward the direction in which $\chi^{2}$ decreases. After
taking a sufficient number of such steps in $\mathcal{P}$, one can find the
best point; moreover, one has now access to a chain of points on which every
step had landed during the random walk. These chains can be used to show the
confidence level of the result. The statistically 1-2 $\sigma$ levels of
confidence are related to the points at which steps landed during the walk
within $68\%$ and $95\%$ confidence intervals. Accordingly, the density of
these points at each region, gives the relevance of that region. More dense
zones are more probable ones for the free parameters. The resulting
probability distribution for the free parameters is called the posterior
probability. Often, one projects the posterior probability onto one and two
dimensional subspaces of $\mathcal{P}$, in order to have a sensible view of it
and to infer the constraints on each parameter. In this regard, we made use of
($i$) the CMB temperature and polarization angular power spectra, reported by
the Planck team Planck:2015bpv , referred to as “CMB”; ($ii$) the measurement
of $H_{0}$ by Riess et al. Riess:2019cxk , referred to as “R19”; ($iii$)
distance to several baryon acoustic oscillations, serving as standard rulers
located at redshift $\lesssim 1$ Beutler:2011hx ; Ross:2014qpa ; Blake:2011en
, referred to as “BAO”. We put both background and perturbation equations into
the CAMB code camb ; Lewis:1999bs  to have the observational implications of
this model and using the CosmoMC code Lewis:2002ah ; Lewis:2013hha , we
sampled the space of free parameters and obtained the desired chain of points.
In figure 1 and table 1, the resulting posterior probabilities for these
parameters are shown. It is evident there is a positive correlation between
$\alpha$ and $H_{0}$. So in this model, CMB is more compatible with the local
measurements of the $H_{0}$. It is worth noticing that $\alpha$ is more than
1$\sigma$ away from zero, where our modified model reduces to $\Lambda$CDM. It
also seems that the data employed here is insensitive to the value of $\beta$
and this quantity has not been constrained, which shows that while the data
prefers $S$ to increase from $S=0$ at $z=\infty$ to $S=\alpha$ at the present
epoch, the shape of this transition is not significant.

Parameter |  Prior Values |  Posterior Values
---|---|---
$\Omega_{b}\,h^{2}$ | [0.005 , 0.1] | $0.02217\pm 0.00014$
$\Omega_{c}\,h^{2}$ | [0.001 , 0.99] | $0.1176\pm 0.0011$
100 $\Theta_{MC}$ | [0.5 , 10] | $1.04124\pm 0.00032$
$\tau$ | [0.01 , 0.8] | $0.107\pm 0.021$
$n_{s}$ | [0.8 , 1.2] | $0.9730^{+0.0043}_{-0.0049}$
$\ln[10^{10}A_{s}]$ | [2 , 4] | $3.142\pm 0.041$
$\alpha$ | [0 , 0.5] | $0.0034^{+0.0016}_{-0.0019}$
$\beta$ | [-4 , 0] | $---$
$H_{0}\ [\rm km/s/Mpc]$ | - | $68.42\pm 0.52$
|  |
Table 1: The best-fitting values and $68\%$ confidence level intervals for the
parameters of the modified model when CMB, BAO and R19 are used. The Prior
Values column, shows the interval in which every parameter is explored.

In table 1, we summarize the best-fitting parameters of our model with the
prior valuess we used and the 1$\sigma$ confidence levels.

##  VI Discussion

In this paper, linear perturbations about the modified Cartesian flat
cosmological solution of the local limit of nonlocal gravity theory are
derived and investigated. The modified model differs from $\Lambda$CDM through
a susceptibility function $S(t)$. By choosing this function as in Eq. (125), a
new cosmological model is specified. Using CMB temperature and polarization
angular power spectra, the viability of this model has been verified. Our
results show that this model fits the data well and is to some extent capable
of reconciling CMB data with the local measurements of the Hubble constant.

## Acknowledgments

The work of A. B. has been financially supported by the Iran Science Elites
Federation.

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