# An Almost-FLRW Universe as the Averaged Geometry in Macroscopic Gravity

Anish<EMAIL_ADDRESS>Department of Physics, The University
of Texas at Dallas, Richardson, TX 75080, USA  Mustapha
<EMAIL_ADDRESS>Department of Physics, The University of Texas at
Dallas, Richardson, TX 75080, USA

###### Abstract

It is well-known that spacetime averaging is an operation that does not
commute with building the Einstein tensor. In the framework of Macroscopic
gravity (MG), a covariant averaging procedure, this non-commutativity gives
averaged field equations with an additional correction term known as back-
reaction. It is important to explore whether such a term, even if known to be
small, may or may not cause any systematic effect for precision cosmology. In
this work, we explore the application of the MG formalism to an almost
Friedmann-Lemaître-Robertson-Walker (FLRW) model. Namely, we find solutions to
the field equations of MG taking the averaged universe to be almost-FLRW
modelled using a linearly perturbed FLRW metric. We study several solutions
with different functional forms of the metric perturbations including plane
waves ansatzes. We find that back-reaction terms are present not only at the
background level but also at perturbed level, reflecting the non-linear nature
of the averaging process. Thus, the averaging effect can extend to both the
expansion and the growth of structure in the universe.

Keywords: Averaging problem, macroscopic gravity, back-reaction, cosmological
perturbation theory

##  1 Introduction

When building a cosmological model in the framework of general relativity
(GR), an effective averaging of the real discrete matter distribution is
carried out on the right hand side of the Einstein field equations (EFEs),
while the left hand side – the Einstein tensor – is assumed to describe the
gravitational behaviour of this matter distribution without being explicitly
averaged [1, 2, 3]. Though this has been a standard practice in cosmology, it
has been put into question [4, 5]. An obvious approach to deal with this issue
would be to take an explicit average of the EFEs. Indeed, a number of authors
have argued for the use of field equations that are averaged [6, 4, 7, 5, 1,
2, 3].

The construction of such averaged field equations, however, is not
straightforward. It is difficult to define a mathematically rigorous
formulation to calculate covariant volume averages of tensor fields on the
pseudo-Riemannian space-time of general relativity. The difficulties are
exacerbated due to the non-linear nature of Einstein field equations. This
means that the operation of taking the average and calculating Einstein tensor
do not commute. In other words, the average of Einstein tensor for a
microscopic geometry is not equal to the Einstein tensor for the averaged
(macroscopic) geometry. It was suggested in [1, 6], and has been well accepted
since (see [8, 9, 10] and the references therein), that any form of averaged
field equations of gravity built from EFEs should have a correction term that
will account for this non-commutativity. This correction, arising in the
macroscopic description of gravity, is referred to as back-reaction. The need
for an averaged theory of gravity; the difficulties in formulating such a
theory; and given such a theory, the task of calculating and interpreting the
back-reaction, together constitute what is known as the averaging problem in
general relativity and cosmology.

There is a long history of approaches taken towards solving this problem [6,
4, 7, 5, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. With various
approaches at hand, some debate has ensued as well. A few years back, there
was a debate regarding the relevance of the back-reaction for cosmology
between Green and Wald, and Buchert et. al [61]. Green and Wald argued that
back-reaction is irrelevant for the large scale dynamics (see [54] and
subsequent work), however, Buchert et al. criticised their formalism to be not
general enough. Green and Wald put forth a rebuttal to this [62] but did point
out that their definition of back-reaction is different from that of the other
approaches. Recently, the two averaging approaches and their relation to
space-time averaging was compared for simple exact space-times (specifically,
spherically symmetric and plane symmetric space-times) in [63]. Let us clarify
that in this work, we do not take sides in this debate and it is outside the
scope of our work to comment on the validity of any of the approaches
involved. It seems to us that even if back-reaction cannot have large
dynamical effects, it remains to be determined either from theory or
observations if back-reaction can have any percent or sub-percent level
contributions to observational cosmology like other astrophysical systematics.
In this work, we push the framework of macroscopic gravity further to make it
comparable to observations (as done in other previous papers [64, 65, 66, 67])
by analysing almost-FLRW models within macroscopic gravity.

Macroscopic Gravity (MG) [38, 39, 40, 41, 42, 43, 44] is a covariant and exact
(non-perturbative) approach to calculate volume averages of arbitrary tensor
fields on an $n$-dimensional differentiable manifold. The averages are
calculated by the virtue of bilocal averaging operators and Lie dragging the
averaging regions [38, 39, 40, 41, 42, 43, 44]. Applying this procedure to the
$4$-dimensional pseudo-Riemannian geometry of GR yields the macroscopic
Einstein field equations (mEFEs). In these equations, the effect of averaging
(the back-reaction) is characterised by a tensorial correction term,
$\boldsymbol{C}$. The averaged field equations can be written in the form of
EFEs by taking this correction term on the right hand side of the averaged
equations. Then, this term can be regarded as a geometrical correction to the
averaged matter-energy distribution [38, 39, 40, 41, 42, 43, 44].

This correction term is made of contractions of a tensor called the connection
correlation tensor, $\boldsymbol{Z}$. The connection correlation tensor
satisfies (by construction) certain algebraic and differential conditions. By
imposing constraints on the structure of $\boldsymbol{Z}$ and assuming a
macroscopic geometry, these conditions can be solved to find the value of the
correction term $\boldsymbol{C}$ in terms of the components of
$\boldsymbol{Z}$ and the (assumed) metric coefficients. This has been done for
exact geometries like FLRW [68, 65, 69], Bianchi Type-I [66] and static
spherically symmetric (Schwarzschild) [70]. In [71, 72], the authors took the
microscopic geometry to be Lemaître-Tolman-Bondi (LTB) and averaged it using
the MG procedure. In [65, 73, 74, 75, 76], perturbations around the exact-FLRW
solutions were considered. The FLRW solution in MG was constrained using
observational data in [64, 66, 67] and it was found that the back-reaction is
small and is only significant at the level of sub-percent systematics in
precision cosmology.

In this paper, we find solutions to the field equations in MG taking the
averaged universe to be almost-FLRW. We consider that after applying the
averaging procedure of MG to the small scale lumpy universe, we reach a scale
where the inhomogeneities are almost smoothed out, but not completely. We
calculate the MG correction term for this scale. It has indeed been argued in
[77, 78, 79, 80] that the averaging process should be multi-scale. That is,
the effect of averaging at any scale should be the cumulative effect of all
the coarse-graining done to reach the smoothness at that scale. The almost-
FLRW geometry is conventionally modelled by taking perturbations around the
exact-FLRW universe. To this end, we solve the field equations of MG with the
macroscopic geometry given by a linearly perturbed FLRW metric. The back-
reaction at this scale can then be considered to be due to all the higher
order perturbations that have been averaged over.

Linear perturbations around FLRW background in the context of MG have been
previously analysed in [65, 73, 74, 75, 76, 66, 67]. There, it was assumed
that $\boldsymbol{Z}$ can be split into zeroth, ${}^{(0)}\boldsymbol{Z}$, and
first order, ${}^{(1)}\boldsymbol{Z}$, terms. Then, the constraint equations
for ${}^{(1)}\boldsymbol{Z}$ [66] were solved separately. In this work, we
take a rather passive approach. We do not split $\boldsymbol{Z}$ beforehand.
Instead, considering the averaged geometry of the universe to be linearly
perturbed FLRW, we explicitly calculate $\boldsymbol{Z}$ in terms of the
metric coefficients just like one calculates other quantities (Einstein
tensor, connection, etc.) in standard cosmological perturbation theory.

We restrict ourselves to scalar perturbations and find several solutions with
various assumptions on their functional form and space/time dependencies. We
find that when considered as arbitrary functions, only severely restricted
form of inhomogeneous perturbations lead to non-trivial back-reaction terms.
However, more general solutions can be found by taking an ansatz on the
functional form of the perturbations. We find that $\boldsymbol{C}$ can be
split in terms that are zeroth and first order in perturbations. Therefore,
when working with almost-FLRW geometry, the back-reaction has a component that
enters the field equations at first order. That is, the back-reaction modifies
the evolution of not just the background, but also the perturbations around
it. We also find that, in some particular cases, the back-reaction terms can
lead to a non-zero slip parameter even in the absence of anisotropic stress in
the matter-energy content of the universe.

The paper is arranged as follows: in section 2, we present an overview of
macroscopic gravity averaging scheme and describe the constraints on the
connection correlation tensor. Then, in section 3, we explain how to solve
these constraints in a systematic manner to find the correction tensor
$\boldsymbol{C}$ in terms of the metric coefficients. Then, we present our
results for the MG correction term with an almost-FLRW metric in section 4.
Readers familiar with the MG theory and its solutions can skip directly to
this section. We calculate the correction term for plane wave perturbations in
section 5. In section 6, we present the full field equations for an almost-
FLRW metric. Section 7 summarises the results in this paper with some remarks.

The notation and convention used is as follows: tensors associated with the
microscopic geometry are denoted by lowercase letters and those with the
macroscopic geometry by uppercase letters. Greek indices run from $0$ to $3$
and Latin indices from $1$ to $3$. Angular brackets
$\left\langle\cdots\right\rangle$ denote the averaging operation or sometimes
averaged quantities. Covariant differentiation with respect to the macroscopic
connection is denoted by $\|$. Indices with round brackets $(~{})$/square
brackets $[~{}]$ are symmetrised/anti-symmetrised; and underlined indices are
not included in (anti-)symmetrisation. The sign convention followed is the
‘Landau-Lifshitz Space-like Convention (LLSC)’ [81]. That is, the signature of
the metric is taken to be Lorentzian $(-,+,+,+)$, the Riemann curvature tensor
is defined as,
${r^{\mu}}_{\alpha\nu\beta}~{}=~{}2\partial_{[\nu}{\gamma^{\mu}}_{\underline{\alpha}\beta]}+2{\gamma^{\epsilon}}_{\alpha[\beta}{\gamma^{\mu}}_{\underline{\epsilon}\nu]}$
and $r_{\mu\nu}~{}=~{}{r^{\alpha}}_{\mu\alpha\nu}$ is the Ricci tensor. The
Ricci scalar is defined as
$r~{}=~{}{r^{\mu}}_{\mu}~{}=~{}\textnormal{{g}}^{\mu\nu}r_{\mu\nu}$. Finally,
the units are taken such that $G=1=c$, i.e., $\kappa=8\pi$.

##  2 Macroscopic Gravity Formalism

Using the concepts of macroscopic electrodynamics [82, 83, 84], a covariant
averaging procedure was introduced in [38, 39, 40, 41, 42, 43, 44] which can
be used in general relativity. It is a generalisation of the averaging in
Minkowski space-time and is valid for arbitrary classical tensor fields on any
differentiable manifold [38, 39, 40, 41, 42, 43, 44]. Applying this procedure
to the $4$-dimensional pseudo-Riemannian geometry of GR leads to an averaged
theory of gravity – Macroscopic Gravity.

Let there be a geometric object, $p^{\alpha}_{\beta}(x)$, defined on an
$n$-dimensional differentiable metric manifold
$(\mathcal{M},\textnormal{{g}}_{\alpha\beta})$. Then, the space-time averaged
value of this object over a compact region $\Sigma\subset\mathcal{M}$ with a
volume $n$-form around a supporting point $x\in\Sigma$, is defined as,

$\left\langle
p^{\alpha}_{\beta}(x)\right\rangle=\dfrac{\int_{\Sigma}\mathcal{A}^{\alpha}_{\mu^{\prime}}(x,x^{\prime})p^{\mu^{\prime}}_{\nu^{\prime}}(x^{\prime})\mathcal{A}^{\nu^{\prime}}_{\beta}(x^{\prime},x)\sqrt{-\textnormal{{g}}^{\prime}}\
\textrm{d}^{n}x^{\prime}}{\int_{\Sigma}\sqrt{-\textnormal{{g}}^{\prime}}\
\textrm{d}^{n}x^{\prime}}$ (2.1)

where $\int_{\Sigma}\sqrt{-\textnormal{{g}}^{\prime}}\
\textrm{d}^{n}x^{\prime}$ is the volume ($V_{\Sigma}$) of the region $\Sigma$
and
$\textnormal{{g}}^{\prime}=\det\left[\textnormal{{g}}_{\alpha\beta}(x^{\prime})\right]$.
The integration is done over all the points $x^{\prime}\in\Sigma$. The
integrand
$\mathcal{A}^{\alpha}_{\mu^{\prime}}(x,x^{\prime})p^{\mu^{\prime}}_{\nu^{\prime}}(x^{\prime})\mathcal{A}^{\nu^{\prime}}_{\beta}(x^{\prime},x)$
is called the bilocal extension of the object $p^{\alpha}_{\beta}(x)$; and the
objects $\mathcal{A}^{\alpha}_{\mu^{\prime}}(x,x^{\prime})$ and
$\mathcal{A}^{\nu^{\prime}}_{\beta}(x^{\prime},x)$ are called the bilocal
averaging operators. In the next subsections, we describe come of the
underlying concepts and definition in macroscopic gravity

###  2.1 Basic Definitions

The average of the microscopic Riemann curvature tensor,
$\left\langle{r^{\alpha}}_{\beta\rho\sigma}\right\rangle$ is written as
${R^{\alpha}}_{\beta\rho\sigma}$ and it is assumed to behave like a curvature
tensor itself [38, 39, 40, 41, 42, 43, 44]. It can be shown to satisfy the
algebraic and differential properties of the Riemann curvature tensor.
Further, bilocal objects defined as,
${\mathcal{F}^{\alpha}}_{\beta\rho}(x,x^{\prime})\equiv\mathcal{A}^{\alpha}_{\epsilon^{\prime}}\left(\partial_{\rho}\mathcal{A}^{\epsilon^{\prime}}_{\beta}+\nabla_{\sigma^{\prime}}\mathcal{A}^{\epsilon^{\prime}}_{\beta}\mathcal{A}^{\sigma^{\prime}}_{\rho}\right)$,
behave as connection coefficients at $x$ and hence, can be considered to be
the bilocal extension of the microscopic connection coefficients,
${\gamma^{\alpha}}_{\beta\rho}$. They have the following coincidence limit,

$\lim_{x^{\prime}\to
x}{\mathcal{F}^{\alpha}}_{\beta\rho}={\gamma^{\alpha}}_{\beta\rho}$ (2.2)

The average of these objects,
${\overline{\mathcal{F}}^{\alpha}}_{\beta\rho}\equiv\frac{\int{\mathcal{F}^{\alpha}}_{\beta\rho}\sqrt{-\textnormal{{g}}^{\prime}}\
\textrm{d}^{n}x^{\prime}}{\int\sqrt{-\textnormal{{g}}^{\prime}}\
\textrm{d}^{n}x^{\prime}}=\left\langle{\mathcal{F}^{\alpha}}_{\beta\rho}\right\rangle$,
is interpreted as the affine connection coefficients of the averaged space-
time [38, 39]. The curvature tensor, ${M^{\alpha}}_{\beta\rho\sigma}$,
associated with these connection coefficients is the “macroscopic” curvature
tensor, written as,

${M^{\alpha}}_{\beta\rho\sigma}={R^{\alpha}}_{\beta\rho\sigma}+2\left\langle{\mathcal{F}^{\delta}}_{\beta[\rho}{\mathcal{F}^{\alpha}}_{\underline{\delta}\sigma]}\right\rangle-2\left\langle{\mathcal{F}^{\delta}}_{\beta[\rho}\right\rangle\left\langle{\mathcal{F}^{\alpha}}_{\underline{\delta}\sigma]}\right\rangle$
(2.3)

where
${R^{\alpha}}_{\beta\rho\sigma}=\left\langle{r^{\alpha}}_{\beta\rho\sigma}\right\rangle$
is the average of the microscopic Riemann curvature tensor.

One can calculate another connection, ${\Pi^{\alpha}}_{\beta\rho}$, associated
with the curvature tensor ${R^{\alpha}}_{\beta\rho\sigma}$. Therefore, we have
two connections associated with two curvature tensors which are related to
each other by equation (2.3). The difference between these two connections is
captured by another object, namely, the affine deformation tensor,
${A^{\alpha}}_{\beta\rho}=\left\langle{\mathcal{F}^{\alpha}}_{\beta\rho}\right\rangle-{\Pi^{\alpha}}_{\beta\rho}$.
This is a consequence of the non-metricity of
${R^{\alpha}}_{\beta\rho\sigma}$, i.e., the average of microscopic metric,
$\left\langle\textnormal{{g}}_{\alpha\beta}\right\rangle$, is not a metric
tensor, and,

$G_{\alpha\beta\lvert\rho}\neq 0$ (2.4)

where, $G_{\alpha\beta}$ is the metric tensor associated with the averaged
manifold, and $\lvert$ denotes the covariant derivative with respect to the
connection ${\Pi^{\alpha}}_{\beta\rho}$. That is, we have a macroscopic
metric, $G_{\alpha\beta}$, associated with the macroscopic space-time which is
not compatible with the non-Riemannian curvature,
${R^{\alpha}}_{\beta\rho\sigma}$, and its associated connection
${\Pi^{\alpha}}_{\beta\rho}$, while the average of the microscopic metric
tensor, $\left\langle\textnormal{{g}}_{\alpha\beta}\right\rangle$, is not the
metric associated with the macroscopic space-time, and is not a metric tensor.

Further, looking at the construction of both the curvature tensors in terms of
their respective connections, equation (2.3) takes the form,

${A^{\alpha}}_{\beta[\rho\|\sigma]}-{A^{\epsilon}}_{\beta[\rho}{A^{\alpha}}_{\underline{\epsilon}\sigma]}=-{\frac{1}{2}}{Q^{\alpha}}_{\beta\rho\sigma}$
(2.5)

where,

${Q^{\alpha}}_{\beta\rho\sigma}=2\left\langle{\mathcal{F}^{\delta}}_{\beta[\rho}{\mathcal{F}^{\alpha}}_{\underline{\delta}\sigma]}\right\rangle-2\left\langle{\mathcal{F}^{\delta}}_{\beta[\rho}\right\rangle\left\langle{\mathcal{F}^{\alpha}}_{\underline{\delta}\sigma]}\right\rangle$
(2.6a) The tensor, ${Q^{\alpha}}_{\beta\rho\sigma}$, in its mathematical
construction, is like a curvature tensor, built from the affine deformation
tensor ${A^{\alpha}}_{\beta\rho}$ in the same way
${M^{\alpha}}_{\beta\rho\sigma}$ is built from its associated connection. From
equations (2.3) and (2.6a), we can see the that it measures the difference
between the two curvature tensors – a curvature deformation tensor,
${Q^{\alpha}}_{\beta\rho\sigma}={M^{\alpha}}_{\beta\rho\sigma}-{R^{\alpha}}_{\beta\rho\sigma}$
(2.6b)

Both the tensors, ${Q^{\alpha}}_{\beta\rho\sigma}$ and
${M^{\alpha}}_{\beta\rho\sigma}$, follow the algebraic properties of a Riemann
curvature tensor. The Ricci tensors
$M_{\alpha\beta}={M^{\epsilon}}_{\alpha\epsilon\beta}$ and
$R_{\alpha\beta}={R^{\epsilon}}_{\alpha\epsilon\beta}$ are symmetric. The
Ricci tensor equivalent for ${Q^{\alpha}}_{\beta\rho\sigma}$ can be written as
$Q_{\alpha\beta}={Q^{\epsilon}}_{\alpha\epsilon\beta}$.

Moreover, both the curvature tensors of the averaged geometry satisfy the
differential Bianchi identities with respect to their corresponding
connections. The averaging of the differential Bianchi identities for the
microscopic curvature tensor gives the following relations,

${R^{\alpha}}_{\beta[\rho\sigma,\epsilon]}+\left\langle{r^{\gamma}}_{\beta[\rho\sigma}{\mathcal{F}^{\alpha}}_{\underline{\gamma}\epsilon]}\right\rangle-\left\langle{r^{\alpha}}_{\gamma[\rho\sigma}{\mathcal{F}^{\gamma}}_{\underline{\beta}\epsilon]}\right\rangle=0$
(2.7)

To solve the above equations completely, one needs a splitting rule for the
averaging operation present in the second and third terms. Such a rule can be
constructed by using an object called the connection correlation, which is
defined as [38, 39],

${Z^{\alpha}}_{\beta[\gamma}{{}^{\mu}}_{\underline{\nu}\sigma]}=\left\langle{\mathcal{F}^{\alpha}}_{\beta[\gamma}{\mathcal{F}^{\mu}}_{\underline{\nu}\sigma]}\right\rangle-\left\langle{\mathcal{F}^{\alpha}}_{\beta[\gamma}\right\rangle\left\langle{\mathcal{F}^{\mu}}_{\underline{\nu}\sigma]}\right\rangle$
(2.8)

Using equations (2.6a) and (2.8), we see that,

${Q^{\alpha}}_{\beta\rho\sigma}=2{Z^{\epsilon}}_{\beta[\rho}{{}^{\alpha}}_{\underline{\epsilon}\sigma]}$
(2.9)

Therefore, to summarise, when the averaging procedure in equation (2.1) is
applied to the 4-dimensional Riemannian manifold in general relativity, the
averaged manifold is characterised by a metric, two equi-affine symmetric
connections, two curvature tensors associated with these connections and
correlation tensors constructed to solve differential Bianchi identities for
these curvature tensors. The number of correlation tensors is finite and
restricted by the dimensionality of the manifold.

###  2.2 The Macroscopic Einstein Field Equations

The microscopic Einstein field equations are given by,

${e^{\epsilon}}_{\gamma}={r^{\epsilon}}_{\gamma}-{\frac{1}{2}}\delta^{\epsilon}_{\gamma}\textnormal{{g}}^{\mu\nu}r_{\mu\nu}=\kappa{t^{\epsilon}}_{\gamma}$
(2.10)

where, ${e^{\epsilon}}_{\gamma}$ is the Einstein tensor333We use
$\boldsymbol{e}/\boldsymbol{E}$ to denote the Einstein tensor, instead of
$\boldsymbol{g}/\boldsymbol{G}$, to avoid confusion with the metric tensor.
and
${r^{\epsilon}}_{\gamma}=\textnormal{{g}}^{\alpha\epsilon}r_{\alpha\gamma}$ is
the Ricci tensor. The matter-energy content of the microscopic manifold is
described by the microscopic energy-momentum tensor ${t^{\epsilon}}_{\gamma}$.
Then, the average of Einstein field equations,

$\left\langle{e^{\epsilon}}_{\gamma}\right\rangle=\left\langle\textnormal{{g}}^{\beta\epsilon}r_{\beta\gamma}\right\rangle-{\frac{1}{2}}\delta^{\epsilon}_{\gamma}\left\langle\textnormal{{g}}^{\mu\nu}r_{\mu\nu}\right\rangle=\kappa\left\langle{t^{\epsilon}}_{\gamma}\right\rangle$
(2.11)

yields the macroscopic Einstein field equations (mEFEs), which are given by,

${E^{\epsilon}}_{\gamma}=\left\langle\textnormal{{g}}^{\beta\epsilon}\right\rangle
M_{\beta\gamma}-{\frac{1}{2}}\delta^{\epsilon}_{\gamma}\left\langle\textnormal{{g}}^{\mu\nu}\right\rangle
M_{\mu\nu}=\kappa\left\langle{t^{\epsilon}}_{\gamma}\right\rangle+\left({Z^{\epsilon}}_{\mu\nu\gamma}-{\frac{1}{2}}\delta^{\epsilon}_{\gamma}Q_{\mu\nu}\right)\left\langle\textnormal{{g}}^{\mu\nu}\right\rangle$
(2.12)

where
${Z^{\epsilon}}_{\mu\nu\gamma}=2{Z^{\epsilon}}_{\mu[\alpha}{{}^{\alpha}}_{\underline{\nu}\gamma]}$
is a Ricci-tensor-like object for the connection correlation and
$Q_{\mu\nu}={Q^{\epsilon}}_{\mu\epsilon\nu}$. Therefore, in the theory of
macroscopic gravity, averaging out the Einstein field equations introduces
additional terms constituting various traces (contractions) of the connection
correlation.

The appearance of these additional terms in equation (2.12) can be considered
as a correction arising from the change in the geometric structure of the
space-time due to averaging. We will call this term, the MG correction term,
and denote it as,

${C^{\epsilon}}_{\gamma}=\left({Z^{\epsilon}}_{\mu\nu\gamma}-\frac{1}{2}\delta^{\epsilon}_{\gamma}Q_{\mu\nu}\right)\left\langle\textnormal{{g}}^{\mu\nu}\right\rangle$
(2.13)

Further, we can introduce a macroscopic energy-momentum tensor,
${T^{\epsilon}}_{\gamma}$, which is defined as,

${T^{\epsilon}}_{\gamma}=\left\langle{t^{\epsilon}}_{\gamma}\right\rangle+{{}^{\rm(MG)}T^{\epsilon}}_{\gamma}$
(2.14)

where
${{}^{\rm(MG)}T^{\epsilon}}_{\gamma}=\frac{1}{\kappa}{C^{\epsilon}}_{\gamma}$.
Using this definition, the mEFEs take the following elegant form,

$\left\langle\textnormal{{g}}^{\beta\epsilon}\right\rangle
M_{\beta\gamma}-{\frac{1}{2}}\delta^{\epsilon}_{\gamma}\left\langle\textnormal{{g}}^{\mu\nu}\right\rangle
M_{\mu\nu}=\kappa{T^{\epsilon}}_{\gamma}$ (2.15)

###  2.3 Macroscopic Gravity Objects

####  2.3.1 The Connection Correlation Tensor

The connection correlation tensor is an important object that exists on the
averaged space-time and has the following algebraic and differential
properties [38, 39, 40, 41, 42, 43, 44],

The anti-symmetry in the third and sixth indices (noted hereafter as Z1),
$\displaystyle{Z^{\alpha}}_{\beta\gamma}{{}^{\mu}}_{\nu\sigma}$
$\displaystyle=-{Z^{\alpha}}_{\beta\sigma}{{}^{\mu}}_{\nu\gamma}$ (2.16b) The
anti-symmetry in interchange of the index pairs (noted hereafter as Z2),
$\displaystyle{Z^{\alpha}}_{\beta\gamma}{{}^{\mu}}_{\nu\sigma}$
$\displaystyle=-{Z^{\mu}}_{\nu\gamma}{{}^{\alpha}}_{\beta\sigma}$ (2.16c) The
algebraic cyclic identity (noted hereafter as Z3),
$\displaystyle{Z^{\alpha}}_{\beta[\gamma}{{}^{\mu}}_{\nu\sigma]}$
$\displaystyle=0$ (2.16d) The equi-affinity property (noted hereafter as Z4),
$\displaystyle{Z^{\epsilon}}_{\epsilon\gamma}{{}^{\mu}}_{\nu\sigma}$
$\displaystyle=0$ (2.16e) The differential constraint (noted hereafter as Z5),
$\displaystyle{Z^{\alpha}}_{\beta[\gamma}{{}^{\mu}}_{\underline{\nu}\sigma\|\lambda]}$
$\displaystyle=0$ (2.16f) The integrability condition (noted hereafter as Z6),
$\displaystyle{Z^{\epsilon}}_{\beta[\gamma}{{}^{\mu}}_{\underline{\nu}\sigma}{M^{\alpha}}_{\underline{\epsilon}\lambda\rho]}-{Z^{\alpha}}_{\epsilon[\gamma}{{}^{\mu}}_{\underline{\nu}\sigma}{M^{\epsilon}}_{\underline{\beta}\lambda\rho]}$
$\displaystyle+{Z^{\alpha}}_{\beta[\gamma}{{}^{\epsilon}}_{\underline{\nu}\sigma}{M^{\mu}}_{\underline{\epsilon}\lambda\rho]}-{Z^{\alpha}}_{\beta[\gamma}{{}^{\mu}}_{\underline{\epsilon}\sigma}{M^{\epsilon}}_{\underline{\nu}\lambda\rho]}=0$
(2.16g) The quadratic constraint (noted hereafter as Z7),
${Z^{\delta}}_{\beta[\gamma}{{}^{\theta}}_{\underline{\kappa}\pi}{Z^{\alpha}}_{\underline{\delta}\epsilon}{{}^{\mu}}_{\underline{\nu}\sigma]}+{Z^{\delta}}_{\beta[\gamma}{{}^{\mu}}_{\underline{\nu}\sigma}{Z^{\theta}}_{\underline{\kappa}\pi}{Z^{\alpha}}_{\underline{\delta}\epsilon]}+{Z^{\alpha}}_{\beta[\gamma}{{}^{\delta}}_{\underline{\nu}\sigma}{Z^{\mu}}_{\underline{\delta}\epsilon}{{}^{\theta}}_{\underline{\kappa}\pi]}+{Z^{\alpha}}_{\beta[\gamma}{{}^{\mu}}_{\underline{\delta}\epsilon}{Z^{\theta}}_{\underline{\kappa}\pi}{{}^{\delta}}_{\underline{\nu}\sigma]}\\\
+{Z^{\alpha}}_{\beta[\gamma}{{}^{\theta}}_{\underline{\delta}\epsilon}{Z^{\mu}}_{\underline{\nu}\sigma}{{}^{\delta}}_{\underline{\kappa}\pi]}+{Z^{\alpha}}_{\beta[\gamma}{{}^{\delta}}_{\underline{\kappa}\pi}{Z^{\theta}}_{\underline{\delta}\epsilon}{{}^{\mu}}_{\underline{\nu}\sigma]}=0$
(2.16h)

####  2.3.2 The Affine Deformation Tensor

The affine deformation tensor, through its definition, has the following field
equations [38, 39, 40, 41, 42, 43, 44],

${A^{\alpha}}_{\beta[\rho\|\sigma]}-{A^{\epsilon}}_{\beta[\rho}{A^{\alpha}}_{\underline{\epsilon}\sigma]}=-{\frac{1}{2}}{Q^{\alpha}}_{\beta\rho\sigma}$
(2.17a)

As a result of equation (2.16f), the field equations for the non-Riemannian
macroscopic curvature tensor become
${R^{\alpha}}_{\beta\rho\sigma\|\lambda}=0$. Using equation (2.6b), this takes
the form,

${A^{\alpha}}_{\epsilon[\rho}{M^{\epsilon}}_{\underline{\beta}\sigma\lambda]}-{A^{\alpha}}_{\epsilon[\rho}{Q^{\epsilon}}_{\underline{\beta}\sigma\lambda]}-{A^{\epsilon}}_{\beta[\rho}{M^{\alpha}}_{\underline{\epsilon}\sigma\lambda]}+{A^{\epsilon}}_{\beta[\rho}{Q^{\alpha}}_{\underline{\epsilon}\sigma\lambda]}=0$
(2.17b)

The macroscopic gravity equations (2.12), (2.16) and (2.17) constitute a
system of coupled first and second order non-linear partial differential
equations for the connection correlation, affine deformation tensor and a (not
yet assumed) macroscopic metric. Given a macroscopic metric, these equations
can be solved to calculate components of the connection correlation and hence
the additional terms in the mEFEs. Several such solutions have been found with
interesting implications.

##  3 Solutions to the Macroscopic Einstein Field Equations

It is worth recalling here that there are two common approaches to averaging
in general relativity and cosmology.

The first is what comes to ones’ mind when considering averaging in a general
context and similar to simple settings like in Newtonian gravity for example.
That is, one thinks about taking a tensor or a scalar and apply to it a direct
and explicit averaging operation. This also has been done in general
relativity – one can take a given microscope metric and explicitly calculate
the average of this metric. This approach has it pros and cons but it is in
general not trivial to find microscopic metrics or covariant averaging schemes
to work with. For example, [71, 72] applied this direct approach to average
spherically symmetric microscopic geometries. They worked with the volume
preserving coordinates [40] in which the averaging bi-vectors are simply the
Kronecker delta and hence the process of averaging, in this case, is
simplified.

The second approach that we followed here is where one takes an ansatz on the
macroscopic geometry and finds the back-reaction using the algebraic and
differential constraints on the connection correlation. As mentioned earlier,
when one uses a metric to describe the universe as a whole, we are
intrinsically taking an ansatz on the macroscopic metric. Therefore, this
approach follows the same way of model-building with the mEFEs as is done with
just the EFEs. This can be viewed as an inverse approach to the problem and
complementary to the other method described above. This approach has been
widely used to analyse models within macroscopic gravity [68, 69, 70, 65, 66,
67] and we have followed it in this work.

The procedure to find the solutions to the mEFEs has been described
extensively in the literature. Taking specific forms of $\boldsymbol{Z}$, one
can solve the constraint equations Z1-Z7 to determine the connection
correlation completely. The solution for an FLRW macroscopic metric was first
presented in [68] and further elaborated upon in [69]. A systematic approach
to solving the MG equations has been presented in [42, 69, 66]. However, it is
useful describe it here briefly. In the next two sections, we work following
the same path as in [69].

The connection correlation is a six rank tensor with 4096 components. These
components are constrained through its various algebraic and differential
properties discussed in the previous section. A step by step approach to
solving for the components of this tensor is as follows,

  1. 1.

Properties Z1 and Z2 (equations (2.16b) and (2.16c)) reduce the number of
independent components in $\boldsymbol{Z}$ to 720.

  2. 2.

Property Z3 (equation (2.16d)) reduces this number to 470.

  3. 3.

Property Z4 (equation (2.16e)) further reduces this to 396.

Now, we are left with is properties Z5, Z6, Z7. It was found in [69] that this
property is identically satisfied if we assume the ‘electric part’ of
$\boldsymbol{Z}$ to be zero. That is,

${{{ZE^{\alpha}}_{\beta}}^{\gamma}}_{\delta\mu}={{{Z^{\alpha}}_{\beta\mu}}^{\gamma}}_{\delta\nu}u^{\nu}=0$
(3.1)

where, $u^{\nu}$ is the time like unit vector field admitted by the
macroscopic geometry.

  1. 4.

Setting the electric part of $\boldsymbol{Z}$ to zero further reduces its
independent components to 121.

The steps described above do not depend on the macroscopic geometry. In other
words, for any given macroscopic geometry, the connection correlation will
have, at most, 121 independent components. Now, we are left with properties Z6
and Z7. Solving these requires a macroscopic metric and specifying the
functional form of $\boldsymbol{Z}$. We do this, first here with an exact-FLRW
metric and then, a linearly perturbed FLRW metric in the next sections.

The line element for the FLRW space-time is given by,

$\textrm{d}s^{2}=a^{2}(\tau)\left(-\textrm{d}\tau^{2}+\delta_{ij}\textrm{d}x^{i}\textrm{d}x^{j}\right)$
(3.2)

where, $\tau\equiv\int\frac{\textrm{d}t}{a(t)}$ is the conformal time.

Starting with the 121 independent components, constraint Z6 reduces this
number to 69. Applying constraint Z5 makes all of these 69 components
independent of time. Further, taking $\boldsymbol{Z}$ to be invariant under
the $\mathcal{G}_{6}$ group of motions as the metric, reduces the number of
independent components to 3. These 3 components are arbitrary constants. We
will label these variables as $b_{1}$, $b_{2}$ and $b_{3}$. The MG correction
term has only one of these variables and is found to be,

$\displaystyle{C^{0}}_{0}$ $\displaystyle=\frac{\mathcal{B}}{a^{2}(\tau)}$
(3.3a) $\displaystyle{C^{0}}_{i}$ $\displaystyle=0={C^{i}}_{0}$ (3.3b)
$\displaystyle{C^{i}}_{j}$
$\displaystyle=\frac{1}{3}\frac{\mathcal{B}}{a^{2}(\tau)}\delta^{i}_{j}$
(3.3c)

In the matrix form, we have,

${C^{\epsilon}}_{\gamma}=\frac{\mathcal{B}}{a(\tau)^{2}}\begin{bmatrix}&1&0&0&0&\\\
&0&\frac{1}{3}&0&0&\\\ &0&0&\frac{1}{3}&0&\\\
&0&0&0&\frac{1}{3}&\end{bmatrix}$ (3.4)

where, $\mathcal{B}=15b_{2}$. The effect of the MG correction term is that of
an additional spatial curvature (as previously found in [68, 69, 42, 65, 66])
in the sense that it evolves as $a^{-2}$ in the Friedmann equations.

Considering the average of the microscopic matter distribution to be a perfect
fluid, we have,

$^{{\rm(fluid)}}\left\langle{t^{\epsilon}}_{\gamma}\right\rangle=(\bar{\rho}+\bar{p})\bar{u}^{\epsilon}\bar{u}_{\gamma}+\bar{p}\delta^{\epsilon}_{\gamma}$
(3.5)

where, $\bar{\rho}\equiv\bar{\rho}(\tau)$ and $\bar{p}\equiv\bar{p}(\tau)$ are
the energy density and pressure for the averaged matter-energy content,
respectively; and $\bar{u}~{}=~{}\left[\frac{1}{a},0,0,0\right]$ is the
averaged 4-velocity of the fluid. Then, writing the mEFEs (equation (2.15))
explicitly gives us the modified Friedmann equations,

$\displaystyle 3\mathcal{H}^{2}$
$\displaystyle=8\pi\bar{\rho}a^{2}-\mathcal{B}$ (3.6a) $\displaystyle
2\mathcal{H}^{\prime}+\mathcal{H}^{2}$
$\displaystyle=-8\pi\bar{p}a^{2}-\frac{1}{3}\mathcal{B}$ (3.6b)

where, $\mathcal{H}\equiv\frac{a^{\prime}}{a}$ and the ‘prime’ represents
differentiation with respect to the conformal time, $\tau$.

##  4 Solutions for an Almost-FLRW Macroscopic Metric

We will now describe new results calculating the MG correction term with an
almost-FLRW macroscopic geometry, given by a linearly perturbed spatially flat
FLRW metric. The line element for such a geometry, in the so called
(conformal) Newtonian gauge, is given by,

$ds^{2}=a^{2}(\tau)\left[-\left(1+2\Psi\right)\textrm{d}\tau^{2}+\left(1-2\Phi\right)\delta_{ij}\textrm{d}x^{i}\textrm{d}x^{j}\right]$
(4.1)

where, $\Psi$ and $\Phi$ are the gauge invariant variables characterising the
first order scalar perturbations. It is worth noting here that we are using
the metric in equation (4.1) as an ansatz for the macroscopic metric.
Analysing cosmological perturbations explicitly within the framework of
macroscopic gravity is a non trivial exercise. The issue of gauge invariance
becomes important and several assumptions are required to do cosmological
perturbation theory consistently within MG [73, 74]. These issues are not
applicable here since we are taking the perturbed FLRW geometry itself as the
macroscopic geometry. That is, equation (4.1) describes the geometry of the
universe after being averaged. This is consistent with other analyses of
cosmological perturbation theory with MG in the literature [65, 66, 67].

To find the back-reaction within this macroscopic geometry, we take the
components of connection correlation to be space-dependent i.e.,
$\boldsymbol{Z}\equiv~{}\boldsymbol{Z}(x,y,z)$. We solve the constraint
equations Z1-Z7 to derive the form of the the connection correlation and hence
the MG correction tensor. Solving Z1 to Z4 and taking the electric part of
$\boldsymbol{Z}$ to be zero leaves us with 121 independent components (they do
not depend on the macroscopic geometry). Then we apply constraints Z5 and Z6
assuming the macroscopic geometry to be the one in equation (4.1). We do this
for multiple cases with various assumptions on the functional form of the
perturbations. We will list the results below one by one.

###  4.1 Perturbations as Arbitrary Functions of Time

This is the case of spatially homogeneous and isotropic perturbations. This is
the same as taking an exact-FLRW metric to be the macroscopic metric.
Therefore, this is equivalent to taking $\Psi=0=\Phi$. Obviously, solving the
constraint equations (2.16) gives the same results as for the exact-FLRW case
and the MG correction tensor is the same as that found in equation (3.3), with
a rescaled time and the scale factor. This is the case that was previously
analysed in [65]. We will go beyond this case in the next sections.

###  4.2 Perturbations as Arbitrary Functions of Time and One Space Coordinate

In this case, the constraint equations need to be solved explicitly with a
perturbed metric. We do this to find that constraint Z6 reduces the
independent components from $121$ to $3$ while Z7 is trivially satisfied, and
does not reduce the independent components further. We label these variables
as $f_{1},f_{2},f_{3}$. For a given choice of space coordinate dependence of
the perturbations, these variables are arbitrary functions of the two other
spatial coordinates that the perturbations do not depend on. Only two of these
three components are there in the MG correction term. Characterising the sub-
cases here by a variable $N\in\\{1,2,3\\}$, our results for the MG correction
term can be written as,

$\displaystyle{C^{0}}_{0}$
$\displaystyle=\frac{\mathcal{B}}{a^{2}(\tau)\left\\{1-2\Phi(\tau,q_{N})\right\\}}$
(4.2a) $\displaystyle{C^{0}}_{i}$ $\displaystyle=0={C^{i}}_{0}$ (4.2b)
$\displaystyle{C^{i}}_{j}$
$\displaystyle=\frac{\mathcal{B}}{a^{2}(\tau)\left\\{1-2\Phi(\tau,q_{N})\right\\}}\delta^{i}_{j}\qquad$
$\displaystyle\ i=N$ (4.2c) $\displaystyle{C^{i}}_{j}$ $\displaystyle=0$
$\displaystyle\ i\neq N$ (4.2d)

In the matrix form, we have,

${C^{\epsilon}}_{\gamma}=\frac{\mathcal{B}}{a(\tau)^{2}\left\\{1-2\Phi(\tau,q_{N})\right\\}}\begin{bmatrix}&1&0&0&0&\\\
&0&&&&\\\ &0&&\delta^{N}_{j}&&\\\ &0&&&&\end{bmatrix}$ (4.3)

where, we have,

$\mathcal{B}\ ;\;q_{N}=\begin{cases}2f_{1}(y,z)+2f_{2}(y,z)\ ;\;x\quad\
&N=1\\\ 2f_{1}(x,z)+2f_{2}(x,z)\ ;\;y\quad\ &N=2\\\ 2f_{1}(x,y)+2f_{2}(x,y)\
;\;z\quad\ &N=3\end{cases}$ (4.4)

This is a very restrictive form of the perturbations but the only one (other
than homogeneous perturbations) that gives an explicit solution for the back-
reaction. In all other more general cases, we recover a trivial solution to
the constraint equations due to limitations in computing other non-trivial
solutions.

##  5 Solutions for Plane Wave Perturbations

The results in the section above are novel, however, as has been found in
other work [65], to find more general solutions with the perturbations as
functions of time and two or more spatial coordinates, one needs to specify a
functional form of the perturbations. Following the commonly used ansatz in
cosmological perturbation theory [85, 86, 87, 88, 89], we use plane wave
perturbations, $\Phi\sim\phi(\tau)\textrm{e}^{{\textrm{i}}k_{j}x_{j}}$, where,
$\phi(\tau)$ is an arbitrary function of time and $k_{j}$ is the wave-vector
$(k_{1},k_{2},k_{3})$. Since we have only taken scalar perturbations in our
macroscopic metric (equation (4.1)), these plane waves will be acoustic waves,
since the scalar perturbations correspond to matter density perturbations in
the universe. Working with this ansatz, we analyse all the previously
described cases. Now, in all these cases, constraint Z6 reduces the
independent components in $\boldsymbol{Z}$ from $121$ to $3$ while Z7 is
trivially satisfied, and does not reduce the independent components further.
These components are arbitrary functions of space. We label these functions as
$f_{1},\ f_{2},\ f_{3}$. Note that these are not necessarily the same
functions as those in the previous sections. Only two of these three functions
are present in the MG correction term which takes the form,

$\displaystyle{C^{0}}_{0}$ $\displaystyle=\frac{\mathcal{B}_{W}\
K}{a^{2}(\tau)\left(1-2\Phi\right)}k_{i}k_{i}$ (5.1a)
$\displaystyle{C^{i}}_{0}$ $\displaystyle=0={C^{0}}_{i}$ (5.1b)
$\displaystyle{C^{i}}_{j}$ $\displaystyle=\frac{\mathcal{B}_{W}\
K}{a^{2}(\tau)\left(1-2\Phi\right)}k_{i}k_{j}$ (5.1c)

In the matrix form, we have,

${C^{\epsilon}}_{\gamma}=\frac{\mathcal{B}_{W}\
K}{a(\tau)^{2}(1-2\Phi)}\begin{bmatrix}&k_{i}k_{i}&0&0&0&\\\ &0&&&&\\\
&0&&k_{i}k_{j}&&\\\ &0&&&&\end{bmatrix}$ (5.2)

where,
$\mathcal{B}_{W}=2f_{1}(\tilde{v},\tilde{w})+2f_{2}(\tilde{v},\tilde{w})$,
$\tilde{v}~{}=~{}\frac{k_{1}y-k_{2}x}{k1}$,
$\tilde{w}~{}=~{}\frac{k_{1}z-k_{3}x}{k1}$, and,
$K=\frac{k_{j}k_{j}}{k_{1}^{2}(k_{1}^{2}+k_{2}^{2})}$.

When we consider only the real part of the plane wave, i.e,
$\Phi\sim\phi(\tau)\cos(k_{i}x_{i})$, our results remain the same as in
equation (5.1), with the perturbation term in the denominator given by a
cosine function instead of an exponential.

##  6 Macroscopic Einstein Field Equations for an Almost-FLRW Metric

In the previous two sections, we presented our results for the MG correction
term for an almost-FLRW universe. Here, we present the full field equations
for an almost-FLRW metric including this correction. The MG correction leads
to an extra term in both the zeroth and first order equations. This modifies
the dynamical evolution of background as well as perturbations.

The macroscopic metric is the one in equation (4.1). The average of
microscopic energy-momentum tensor is that of a perturbed perfect fluid, given
by,

$^{{\rm(pfluid)}}\left\langle{t^{\epsilon}}_{\gamma}\right\rangle=\\{(\bar{\rho}+\delta\rho)+(\bar{p}+\delta
p)\\}u^{\epsilon}u_{\gamma}+(\bar{p}+\delta p)\delta^{\epsilon}_{\gamma}$
(6.1)

where, $\delta\rho\equiv\delta\rho(\tau,x,y,z)$ and $\delta p\equiv\delta
p(\tau,x,y,z)$ are the perturbations to the energy density and pressure. The
4-velocity of the perturbed fluid is given by,
$u^{\nu}=~{}\frac{1}{a}\left[\left(1-\Psi\right),\delta u^{i}\right]$, where,
$\delta u^{i}$ is sometimes called the peculiar velocity. Note that the order
of the perturbations in the energy-momentum tensor is same as the metric
perturbations, i.e., $\Psi\sim\Phi\sim\delta\rho\sim\delta
p\sim\mathcal{O}(\epsilon)$ (say), where, $\epsilon<<1$, is a small parameter.

We present the mEFEs by calculating the Einstein tensor for the given metric,
equating it to the right hand side in equation (2.12) and then restricting all
the terms to be only up to first (linear) order in perturbations. This gives
the following equations,

$\displaystyle
3\mathcal{H}^{2}+2\partial^{i}\partial_{i}\Phi-6\mathcal{H}(\Phi^{\prime}+\mathcal{H}\Psi)$
$\displaystyle=8\pi\bar{\rho}a^{2}+8\pi\delta\rho
a^{2}-\mathcal{B}-2\mathcal{B}\Phi$ (6.2a)
$\displaystyle\partial_{i}\left(\Phi^{\prime}+\mathcal{H}\Psi\right)$
$\displaystyle=4\pi(\bar{\rho}+\bar{p})\delta u_{i}a^{3}$ (6.2b)
$\displaystyle\partial^{i}\partial_{j}(\Psi-\Phi)$
$\displaystyle=-{C^{i}}_{j}a^{2}\qquad\qquad\qquad i\neq j$ (6.2c)
$2\mathcal{H}^{\prime}+\mathcal{H}^{2}-2\Phi^{\prime\prime}-4\mathcal{H}\Phi^{\prime}-2\mathcal{H}\Psi^{\prime}-2(2\mathcal{H}^{\prime}+\mathcal{H}^{2})\Psi+\frac{2}{3}\partial^{i}\partial_{i}(\Psi-\Phi)\\\
=-8\pi\bar{p}a^{2}-8\pi\delta
pa^{2}-\frac{1}{3}\mathcal{B}-\frac{2}{3}\mathcal{B}\Phi$ (6.2d)

where, $\mathcal{B}=2(f_{1}+f_{2})$, and $f_{1},f_{2}$ are the arbitrary
functions of spatial coordinates arising in the MG correction term.

Then, at first order, we have,

$\displaystyle\partial^{i}\partial_{i}\Phi-3\mathcal{H}(\Phi^{\prime}+\mathcal{H}\Psi)$
$\displaystyle=4\pi\delta\rho a^{2}-\mathcal{B}\Phi$ (6.3a)
$\displaystyle\partial_{i}\left(\Phi^{\prime}+\mathcal{H}\Psi\right)$
$\displaystyle=4\pi(\bar{\rho}+\bar{p})\delta u_{i}a^{3}$ (6.3b)
$\displaystyle\partial^{i}\partial_{j}(\Psi-\Phi)$
$\displaystyle=-{C^{i}}_{j}a^{2}$ (6.3c)
$\displaystyle\Phi^{\prime\prime}+2\mathcal{H}\Phi^{\prime}+\mathcal{H}\Psi^{\prime}+(2\mathcal{H}^{\prime}+\mathcal{H}^{2})\Psi$
$\displaystyle-\frac{1}{3}\partial^{i}\partial_{i}(\Phi-\Psi)=4\pi\delta
pa^{2}+\frac{1}{3}\mathcal{B}\Phi$ (6.3d)

The arbitrary functions involved in the almost-FLRW solutions can be assumed
to be equal to the constants involved in the exact-FLRW solutions to get the
above first order equations. The full set of field equations is used so that
the terms at the zeroth and the first orders separate out as they should.
Equations (6.3) give us the dynamical evolution equations for the scalar part
of the gauge invariant metric perturbations. The equations above tell us that
the geometric correction in MG affects both the background evolution as well
as the perturbations around this background.

In an expanding universe, the second and third terms on the left hand side of
equation (6.3a) can be neglected on sub-Hubble scales [85, 86, 87] (they are
suppressed by a factor of $\sim\lambda H$), and hence, it again reduces to the
Poisson equation (with a modified source). In the usual cosmological
perturbation theory with GR, the right hand side of equation (6.3c) is equal
to zero in the absence of anisotropic stress in the fluid (equation (6.1)).
This is interpreted as the two potentials being equal $(\Phi=\Psi)$ at late
times, that is, the ‘slip parameter’ $(\Phi-\Psi)$ being zero. However, in MG,
even with zero anisotropic stress, a non-zero slip parameter could exist,
solely due to the correction tensor. In other words, one of the effects
averaging in cosmology could lead to is that of a non-zero slip parameter (an
effective non-zero anisotropic stress) even when the anisotropic stress in the
energy-momentum tensor is taken to be zero. In our analysis here, we find that
the cases with plane wave perturbations depending on two or more spatial
coordinates lead to this effect. For all the other cases, the slip parameter
is still zero.

##  7 Conclusion

In this paper, we analysed almost-FLRW cosmological models within the
framework of macroscopic gravity. We noted that the effects due to averaging
appear, not only when considering the largest scales (where the geometry is
exact-FLRW), but also at slightly smaller length scales (where the geometry is
almost-FLRW). When going from small lumpy scales to the largest smooth scales,
one can consider that the geometry at the intermediate scales is represented
by some particular corresponding metrics. Then, at a given scale, one would
have different values of the back-reaction related to the corresponding
metric. The back-reaction at every such scale would be an integrated effect of
all the ‘coarse graining’ employed to reach the level of smoothness
represented by the metric at this scale.

We took the macroscopic metric to be almost-FLRW by using an FLRW metric and
linear perturbations around it. We computed the macroscopic gravity field
equations including the back-reaction for such a metric and restricted all the
terms to be up to first order in perturbations. In doing so, we analysed
several cases with various functional forms and spatial/temporal dependencies
of the perturbations and found the macroscopic gravity correction term. These
functional dependencies correspond to different coordinate ansatzes.
Therefore, it should be noted that explicit coordinate ansatzes might lead to
some coordinate specific effects. In particular, in case of the plane wave
ansatz, the wave vector can always be chosen to be oriented such that the
plane wave is a function of only one spatial coordinate. Hence, effects of
averaging in this case might be explicit only in specific coordinate choice
and thus is of a limited scope, but nevertheless, was necessary in order to be
able to analyse (macroscopic) metric perturbations since the mathematical
complexity of MG makes it difficult to do so in general terms. In sum, using
this approach, we were able to show that the use of special functional forms
of perturbations can lead to general solutions with back-reaction terms not
only at the background level but also at the perturbed level confirming
explicitly the non-linearity of the averaging procedure resulting from its
non-commutativity with the construction of the Einstein tensor.

## Acknowledgements

We would like to thank Robert J. van den Hoogen, Timothy Clifton and Alan
Coley for useful comments. The results in this paper were obtained by
modification and expansion of a code originally written by Tharake Wijenayake
and Mustapha Ishak [66], using the computer algebra software Maple and the
openly available package GRTensor. MI acknowledges that this material is based
upon work supported in part by the Department of Energy, Office of Science,
under Award Number DE-SC0022184.

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