, ,

# On the intrinsically flat cosmological models in a lattice

Eduardo Bittencourt, Leandro G. Gomes, Grasiele B. Santos  Federal University
of Itajubá, Av. BPS, 1303, Itajubá-MG, 37500-903, Brazil
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this manuscript we investigate the intrinsically flat (space-flat)
spacetimes as viable cosmological models. We show that they have a natural
geometric structure which is suitable to describe inhomogeneous matter
distributions forming a periodic pattern throughout the space. We prove
theorems for their local representation and for existence and uniqueness of
the Einstein’s equations with these periodic boundary conditions. We also find
an interesting class of exact solutions, which illustrates the applicability
of such spacetimes in cosmology, with an early time behavior close to
homogeneity and isotropy and a late time aspect with peaks and voids in the
matter distribution.

##  1  Introduction

The concepts of group action and symmetry play a central role in the theory of
general relativity, for they are guidelines for probing the nonlinear
Einstein’s equations. These ideas are behind the mathematical description of
one of the most fundamental tenets in modern cosmology, known as the
cosmological principle: the universe is spatially homogeneous and isotropic on
the large-scale average (see [1] for a historical account of it). This context
of the standard cosmological model is mathematically characterized by the
presence of the maximal degree of spatial symmetry in the spacetime
$(\mathcal{M},\mathbf{g})$ [2], or at least its local version, which is
locally equivalent, but richer in the possibilities for the choice of the
global topology [3]. These are the Robertson-Walker (RW) spacetimes.

Some decades ago, Collins and Szafron [4, 5, 6] called the attention to the
cosmological applications of a more general concept, called intrinsic
symmetry. In a suitable context for our purposes here, let us take a splitting
into time plus space of the manifold $\mathcal{M}$, i.e.
$\mathcal{M}=\mathbb{R}\times\Sigma$, denote each space section by
$\Sigma_{t}:=\\{t\\}\times\Sigma$ and the restricted (Riemannian) space metric
by $\mathbf{h}(t):=\mathbf{g}_{|_{\Sigma_{t}}}$, with $\mathbf{g}$ as the
spacetime metric. A diffeomorphism $f:\mathcal{M}\to\mathcal{M}$ is an
intrinsic symmetry of this structure if it preserves the leaves of the
foliation, $f(\Sigma_{t})=\Sigma_{t}$, and is an isometry of $\mathbf{h}(t)$,
for every $t$. Note that $f$ does not need to be an isometry of the spacetime
metric $\mathbf{g}$. Therefore $\mathcal{M}$ could be “space-homogeneous” , in
the sense that each leaf is a homogeneous Riemannian space, without being
spatially homogeneous, where the intrinsic symmetries turn out as spacetime
isometries, just like the RW and Bianchi models [7].

In this paper we scrutinize the class of the space-flat spacetimes, which are
not necessarily spatially flat, and probe its potentiality as cosmological
models. As we shall see later, the theorems demonstrated along the text ensure
that such class encompass the following interesting properties:

  * •

For physically motivated equations of state, the Einstein’s equations simplify
enough in order to provide existence and uniqueness theorems using known
results from the theory of nonlinear partial differential equations.
Furthermore, the initial conditions can be chosen closely to the inhomogeneous
spatial distribution of matter observed today [8] (Theorem 1).

  * •

The time evolution is left apart as we use the scale factor as a time
variable. As a consequence, the Hubble parameter is a free function which is
able to fit, a priori, any late time observations of luminosity distances and
redshift relation, with no need of a cosmological constant (section 2.2).

  * •

The spatial distribution of matter forms a lattice in which the fundamental
domain (cosmological cell) is filled with an inhomogeneous energy density
under periodic boundary conditions.

  * •

The solutions are consistent with the standard viewpoint of the Universe
emerging from a highly uniform early phase, and as it expands, it becomes less
homogeneous (See Corollary 2).

The space-flat cosmological models should also be put in the class of the
spacetimes based on their less symmetrical structure, as compared to the RW’s
ones, offer different perspectives in describing the observed universe. In
particular, the prospect of using them to give a realistic alternative to dark
energy is feasible [9]. In this vein, there has been a number of attempts to
explain the apparent accelerated expansion in the standard model as a result
of inhomogeneities. One of the most accepted alternatives to dark energy
concerns the inhomogeneous Lemaître-Tolman-Bondi models, claiming that the
accelerated expansion is a fictitious effect due to the fact that we are
located close to the center of a void [10, 11]. Also, backreaction terms due
to small scale inhomogeneities could mimic an accelerated expansion on larger
scales [12, 13, 14]. Besides that, there is the idea of constructing
relativistic cosmological models with a discrete matter distribution, which is
not a new one. It can be achieved, for example, as using black-hole lattices
(see, for instance, reference [15]).

The paper is organized as follows: in the section 2 we obtain the first
results on space-flat Lorentzian manifolds, such as the theorem for their
local representation and the form that the Einstein’s equations take in the
adapted coordinates, as well as we set the boundary conditions suitable for
the cosmological applications. The section 3 is devoted to the main theorem on
the existence and uniqueness of solutions of Einstein’s equations with the
equation of state $p=(\gamma(t)-1)\rho$. As a consequence, we also prove an
“early time” (small scale factor) behavior close to the RW model. In the final
section, we present the concluding remarks. We followed the reference [7] for
the conventions of the spacetime indexes, the signature of the metric and the
definition of tensors in general.

##  2 The General Setting

###  2.1 General aspects of a space-flat spacetime

Let $\mathbf{u}$ be a unitary time-like vector field ($\mathbf{u}^{2}=-1$) in
a $m$-dimensional spacetime $(\mathcal{M},\mathbf{g})$ and denote its dual
$1$-form as $\mathbf{u}^{\flat}=\mathbf{g}(\mathbf{u},\cdot)$, that is,
$u^{\flat}_{\nu}=u_{\nu}$ is obtained by lowering the index of $u^{\nu}$.
Hence, the orthogonal distribution $\mathbf{u}^{\bot}$ is just the kernel
$\ker\mathbf{u}^{\flat}$, given by the vectors orthogonal to $\mathbf{u}$ at
each point in $\mathcal{M}$. We recall that $\mathbf{u}$ is vorticity-free [7]
if for all $\mathbf{X},\mathbf{Y}\in\mathbf{u}^{\bot}$ we have
$\mathbf{g}(\nabla_{\mathbf{X}}\mathbf{u},\mathbf{Y})=\mathbf{g}(\nabla_{\mathbf{Y}}\mathbf{u},\mathbf{X})$,
implying that $\mathbf{g}(\nabla_{\mathbf{X}}\mathbf{Y},\mathbf{u})$ is
symmetric. Since the Levi-Civita connection is torsion-free, this is
equivalent as demanding the orthogonal distribution to be integrable, a
property we will assume throughout the text. Each maximal integral submanifold
passing through $p\in\mathcal{M}$ is a Riemannian manifold with the metric
$\mathbf{h}=\mathbf{g}+\mathbf{u}^{\flat}\,{\scriptstyle\otimes}\,\mathbf{u}^{\flat}$.
It is called the space section at $p$ and denoted as $\Sigma_{p}$.

###### Definition 1

A spacetime $(\mathcal{M},\mathbf{g})$ is said to be _space-flat_ , or
_intrinsically flat_ , if there is a timelike, vorticity-free, vector field
$\mathbf{u}$ such that their space sections with the inherited geometry are
flat Riemannian manifolds.

Our first proposition is concerned with the local representation of the space-
flat spacetimes.

###### Proposition 1

In the vicinity of any point in a space-flat spacetime
$(\mathcal{M},\mathbf{g})$, with $\mathbf{u}$ the vector field as in the
definition 1, there are coordinates $(x^{\mu})=(t,x^{i})$, with
$i=1,\dots,m-1$, for which
$\mathbf{u}=e^{-\phi(x^{\mu})}\,\frac{\partial}{\partial t}$ and the
components of the space metric $\mathbf{h}$ are all time functions:

$\mathbf{g}=-\,e^{2\phi}\,dt\otimes dt+h_{ij}(t)\,dx^{i}\otimes dx^{j}\,.$ (1)

In particular, $\mathbf{u}$ is shear-free if, and only if, this formula can be
reduced to

$\mathbf{g}=-\,e^{2\phi}\,dt\otimes dt+a(t)^{2}\,\delta_{ij}\,dx^{i}\otimes
dx^{j}\,.$ (2)

In both cases, we can set $\phi=0$ without loss of generality if, and only if,
$\mathbf{u}$ is geodesic, that is, $\nabla_{\mathbf{u}}\mathbf{u}=0$.

###### Proof.

Since $\mathbf{u}$ is vorticity-free, there are coordinates adapted to the
splitting of space + time generated by it with [7]

$\mathbf{u}=e^{-\phi}\,\frac{\partial}{\partial
t}\quad\textrm{and}\quad\mathbf{g}=-\,e^{2\phi}\,dt\otimes
dt+\tilde{h}_{ij}(t,y)\,dy^{i}\otimes dy^{j}\,,$

where the flat space section $\Sigma_{t_{0}}$ passing through the point
$(t_{0},y_{0})$ is locally characterized by $t=t_{0}$. Denoting the spatial
projection of the gradient of $\phi$ by $\bar{\nabla}\phi$, $\phi$ is a
function that can be set to zero without loss of generality if, and only if,
it is a function of $t$ only, that is,
$\nabla_{\mathbf{u}}\mathbf{u}=\bar{\nabla}\phi=0$.

The intrinsically flat coordinate system is built in the following way: Let
$F_{0}^{t}$ denote the flux of the vector field $\partial_{t}$ and fix
$p\in\mathcal{M}$ at the space section $t=0$ with coordinates $(0,y_{0})$.
Define the spatial frame along the curve $F^{t}_{0}(0,y_{0})=(t,y_{0})$ as

$\mathbf{e}_{1}(t,y_{0}):=\frac{\partial}{\partial
y^{1}}\quad,\,\ldots\mathbf{e}_{m-1}(t,y_{0}):=\frac{\partial}{\partial
y^{m-1}}\,.$

Along each flat hypersurface $t=t_{0}$ we can locally propagate this frame
such that the $\mathbf{e}_{i}$’s are all Killing fields of $\mathbf{h}$
representing local translations, that is, $[\mathbf{e}_{i},\mathbf{e}_{j}]=0$.
As we take $F_{i}^{s}(p)$ to represent the flux of $\mathbf{e}_{i}$, the map

$\psi(t,x^{1},\ldots,x^{m-1}):=F_{1}^{x^{1}}\circ\ldots\circ
F_{m-1}^{x^{m-1}}\,(t,y_{0})$ (3)

gives us the desired coordinate transformation. In fact, $\psi$ leaves any
space section invariant, since the flux of each field $\mathbf{e}_{k}$ do it.
Furthermore, as they all commute with each other, the fluxes also commute, so
that $\mathbf{e}_{k}$ is represented by $\partial/\partial x^{k}$ in the new
coordinates. But they are Killing fields of $h_{ij}(t,x)\,dx^{i}\otimes
dx^{j}$ for each $t$, that is, $\partial h_{ij}/\partial x^{k}=0$. ∎

In the dimension $m=4$, the theorem 1 states that any space-flat spacetime is
locally a Bianchi-I model if, and only if, $\mathbf{u}$ is geodesic. In
particular, the flat RW case follows as we take $\mathbf{u}$ to be both
geodesic and shear-free. This shows that space-flatness allows suitable
generalizations of the models often considered in Cosmology by including both
inhomogeneity, as $\phi$ depends on the space coordinates, and anisotropy,
presented in the shear tensor of $h_{ij}(t)$. In this manuscript, we shall
focus mainly in the inhomogeneity property, so that in the most parts of it we
will assume $\mathbf{u}$ to be shear-free.

There are non-orthogonal versions of the proposition 1 in the literature (see
[6], section 2). We shall emphasize that our results are valid in those
contexts as well as in any spacetime which is foliated by conformally flat
spatial sections. To see that, it is enough to take $\mathbf{u}$ to be the
vector field normal to the conformally flat foliation, which is clearly
vorticity-free, and apply the proposition 1 to $e^{2\xi}\,\mathbf{g}$, for an
appropriate function $\xi$. Since the theorem is local, this function need not
to be globally defined. Therefore, that result can be readily extended to any
spacetime foliated by conformally flat slices:

###### Corollary 1

If $(\mathcal{M},\mathbf{g})$ is foliated by conformally flat spatial
sections, then around each point there are coordinates where the metric is
represented as

$\mathbf{g}=-\,e^{2\phi(t,x)}\,dt\otimes
dt+e^{2\xi(t,x)}\,h_{ij}(t)\,dx^{i}\otimes dx^{j}\,.$ (4)

In particular, $h_{ij}(t)$ can be reduced to $a(t)^{2}\,\delta_{ij}$ only in
the shear-free case, that is, when the expansion tensor
$\theta_{i}^{j}=e^{\phi}\,(2\dot{\xi}\,\delta_{i}^{j}+\,h^{jk}\dot{h}_{ki})/2$
is everywhere proportional to the identity.

###  2.2 The Einstein’s Field Equations

We now describe the Einstein’s field equations (EFE) for space-flat spacetimes
in the adapted coordinates used in equation (2). We assume that the vector
field $\mathbf{u}$ is shear-free and the matter content is represented as a
fluid idealization co-moving along its flux, in which case the energy-momentum
tensor turns into [7, 16]:

$\mathbf{T}=\rho\,\mathbf{u}^{\flat}\otimes\mathbf{u}^{\flat}+p\,\mathbf{h}+\mathbf{u}^{\flat}\otimes\mathbf{q}^{\flat}+\mathbf{q}^{\flat}\otimes\mathbf{u}^{\flat}+\Pi,$
(5)

where $\rho$ is the energy density, $p$ is isotropic thermodynamic pressure,
$\mathbf{q}$ is the relativistic momentum density and $\Pi$ is the anisotropic
pressure.

The $0-0$ component of the EFE provides the generalized Friedmann equation

$\frac{1}{2}\scalebox{0.7}[0.8]{$(m-1)(m-2)$}\mathrm{H}^{2}=e^{2\phi}\rho\,,$
(6)

while the $0-i$ components set the thermodynamic phenomenology relating the
relativistic momentum density and the acceleration of $\mathbf{u}$ through

$q_{i}=\scalebox{0.7}[0.8]{$(m-2)$}\mathrm{H}\,\partial_{i}e^{-\phi},$ (7)

where we introduce the Hubble parameter $\mathrm{H}(t)$ in the usual way

$\mathrm{H}(t)=\frac{1}{a}\,\frac{da}{dt}\,.$ (8)

These equations can be used to express the conservation of energy
$\nabla_{\mu}T^{\mu}_{0}=0$ as

$\frac{\partial}{\partial
t}\left(\,e^{2\phi}\rho\right)=\scalebox{0.7}[0.8]{$(m-1)$}\mathrm{H}\,\left[\,-\,e^{2\phi}\left(\,\rho+p\right)+\scalebox{0.7}[0.8]{$(m-2)$}\left(\frac{e^{\phi}\,\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}\,e^{\phi}}{\scalebox{0.7}[0.8]{$(m-1)$}\,a^{2}}+\mathrm{H}\dot{\phi}\right)\right],$
(9)

where  ${\scriptstyle\Delta}$ stands for the Euclidean Laplacian
$\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}\phi:=\delta^{ij}\partial_{i}\partial_{j}\phi$.
Written in terms of $\mathrm{H}$, it reads

$\dot{\mathrm{H}}=-\,\frac{e^{2\phi}}{\scalebox{0.7}[0.8]{$m-2$}}\left(\rho+p\right)+\frac{e^{\phi}\,\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}\,e^{\phi}}{\scalebox{0.7}[0.8]{$(m-1)$}\,a^{2}}+\mathrm{H}\dot{\phi}\,.$
(10)

It is clear that for $\phi=0$ the formulas (6), (9) and (10) reduce to the
standard equations in a flat RW spacetime.

The last part of the Einstein’s equations is the traceless spatial one given
explicitly as

$\frac{1}{m-1}\,\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}e^{\phi}\,\delta_{ij}-\partial_{i}\partial_{j}e^{\phi}=e^{\phi}\,\pi_{ij},$
(11)

which is in fact an equation of state, in the terms discussed in [17, 18, 9]

$\pi_{ij}=-\frac{(m-2)}{(m-3)}\,E_{ij}\,,$ (12)

where $E_{ij}$ is the electric part of the Weyl tensor. Note that it is a
consequence of the non-geodesic and shear-free comoving flow, besides the
space-flatness.

It is worth noticing that the generalized Friedmann equation (6) enforces the
positivity of $\rho$, except for the static universe $\mathrm{H}=0$, of
course. Hence, we can rewrite the conservation law (9) as an equation
involving only $\rho$ and $p$ with the scale factor as a “time” variable

$\frac{\partial\rho}{\partial
a}=\frac{\scalebox{0.7}[0.8]{$(m-2)$}}{a^{3}}\,\rho^{1/2}\,\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}\,\rho^{-1/2}\,\,-\,\frac{\scalebox{0.7}[0.8]{$(m-1)$}}{a}\,\left(\,\rho+p\right)\,.$
(13)

As long as the pressure is a known function $p=p(a,x,\rho)$, in principle, we
can solve this equation for $\rho(a,x)$. Therefore, the components of energy-
momentum tensor and the metric are now functions of the new coordinates
$(a,x)$ as we use the identity
$e^{2\phi}=\frac{1}{2}\scalebox{0.7}[0.8]{$(m-1)(m-2)$}\mathrm{H}^{2}\,\rho$
in Eqs. (2), (7) and (11). So, the relativistic momentum density becomes

$q_{i}={\rm sgn}(\mathrm{H})\,\sqrt{\frac{2(m-2)}{m-1}}\,\partial_{i}\rho,$
(14)

the anisotropic pressure tensor turns to

$\pi_{ij}=P_{ij}-\frac{\delta_{ij}}{m-1}\,\delta^{k\ell}\,P_{k\ell}\,,$ (15)

where
$P_{ij}=(2\,\rho\,\partial_{i}\partial_{j}\,\rho-3\,\partial_{i}\rho\,\partial_{j}\rho)/4\,\rho^{2}$
and the metric is

$\mathbf{g}=-\,\frac{2(m-1)(m-2)}{a^{2}\,\rho(a,x)}\,da\otimes
da+a^{2}\,\delta_{ij}\,dx^{i}\otimes dx^{j}\,.$ (16)

Note that $\mathrm{H}=\mathrm{H}(a)$ is a free function independent from the
distribution of matter throughout the space and the equation of state used for
$p$. Therefore, the problem here is no longer to determine $\mathrm{H}(a)$, as
in the standard model [7], but instead, we have to understand the significance
of the time variable $t$ that arises when we take $H(a)$ to fit the data. This
argumentation will fail only if we return to the homogeneous and isotropic
case ($\phi=0$). Indeed, the problem of connecting such solutions with the
observations is much more intricate, since the inhomogeneity plays a non-
trivial role in it [9].

There are some known solutions of Einstein’s equations in the space-flat
context. In [5, 6], the authors find many characterizations for the solutions
in the perfect fluid case. In [19], the co-moving dust solutions are well
studied, while in [20], an extensive analysis of the vacuum and general
perfect fluid solutions is given. In this manuscript, we keep ourselves as
close as possible to the cosmological applications of such spacetimes. Hence,
we look at them in a different perspective and take advantage of the natural
structure such models have that is suitable to accommodate an inhomogeneous
pattern of matter distribution repeated throughout the universe, as we
describe in the next section.

###  2.3 The cosmological cells

Our aim in this manuscript is to set suitable conditions to ensure that our
model will be useful as a cosmological one. We shall accomplish it by
mimicking the process of observation and analysis often encountered in the
description of our universe: there is a scale of length, $L_{0}$, typically of
some hundreds of megaparsecs (Mpc), so that the universe is assumed to be
homogeneous from that point on. This means that at each instant of time, by
which we mean that one measured by some special free-falling observers, as
those in the critical values of $\phi$ [9], we could divide the universe in
boxes of length $L_{0}$ and assume that virtually the same distribution of
mass and energy is encountered in each of them. This is essentially what is
meant by homogeneity in the standard $\Lambda$-CDM model. Here, we want to
move one step further and consider this internal structure from the beginning.
As we are going to show, we have a natural mathematical framework for
describing it.

Let us assume each flat space section of $\mathcal{M}$ to be a copy of the
Euclidean space, $\Sigma_{p}=\mathbb{R}^{m-1}$. To any discrete subgroup
$\Gamma$ of the group of the Euclidean symmetries,
$O(m-1)\ltimes\mathbb{R}^{m-1}$, it can be associated a closed set
$\mathrm{K}\subset\mathbb{R}^{m-1}$ that is repeated throughout the space in
similar copies as we apply the elements of $\Gamma$ to it. Such a set is
called a fundamental domain of $\Gamma$ [21]. If the action is free, the
quotient $\Sigma/\Gamma$ is a flat Riemannian manifold. We are interested in
the case when $\mathrm{K}$ is compact of volume $V_{0}=L_{0}^{m-1}$. It plays
the role of the boxes described in the last paragraph, so that we will refer
to it as a _cosmological cell_ , that is, a compact fundamental domain such
that the quotient $\mathbb{R}^{m-1}/\Gamma$ is a compact flat Riemann
manifold.

To illustrate this scheme, in two space dimensions ($m=3$), any compact
orientable flat Riemannian manifold is isometric to a torus
$\mathbb{R}^{2}/\Gamma\cong\mathbb{T}^{2}$, where the discrete subgroup of
translations is generated by two linearly independent vectors
$\vec{v}_{1},\vec{v}_{2}\in\mathbb{R}^{2}$,
$\Gamma=\\{n_{1}\,\vec{v}_{1}+n_{2}\,\vec{v}_{2}\,|\,n_{1},n_{2}\in\mathbb{Z}\\}$.
In this case, the vectors $\vec{v}_{1}$ and $\vec{v}_{2}$ generate a grid in
$\mathbb{R}^{2}$, each of its compact parts being a cosmological cell of area
$L_{0}^{2}=||\vec{v}_{1}\times\vec{v}_{2}||$. In three space dimensions,
besides the torus generated by three linearly independent vectors, we have
five more non-diffeomorphic classes of grids in $\mathbb{R}^{3}$ generating a
compact orientable flat Riemannian manifold of volume $V_{0}$. In higher
dimensions these classes increase in number [22].

For the sake of simplicity, one might prefer to think of the cosmological cell
as the hypercube of side $L_{0}$, as follows

$\mathrm{K}=\left\\{(x_{1},\ldots,x_{m-1})\,|\,-\frac{L_{0}}{2}\leq
x_{i}\leq\frac{L_{0}}{2}\right\\}\,.$ (17)

Although this is enlightening, and in fact it is essentially all we can have
in dimension $m=3$, we should keep in mind that there are many other non-
equivalent possibilities in more than two space dimensions.

In our space-flat model, the universe should be considered homogeneous at
large scales $L\gg L_{0}$, where each cosmological cell can be viewed as a
point, but completely inhomogeneous as we go to scales of the order or smaller
than $L_{0}$. Therefore, we must look to the field equations with this in
mind, adding to them boundary conditions that allow this interpretation.

###  2.4 The boundary and regularity conditions

The problem of determining the solution of Einstein’s field equations is well
defined as far as we set the right boundary and regularity conditions. For
this, we assume that any spatial function, as $\phi(t,\cdot)$, $\rho(t,\cdot)$
or $p(t,\cdot)$, for each time $t$, to be defined in an open and dense subset
$U\subset\mathbb{R}^{m-1}$ and to be $\Gamma$-periodic. This means that, as
"$\cdot$" denotes the action of $\Gamma$ in $\mathbb{R}^{m-1}$ and $f$ is such
a function, we demand $U$ to be invariant under $\Gamma$, that is,
$\Gamma\cdot U=U$, and for every $x\in U$ and $g\in\Gamma$, $f(g\cdot
x)=f(x)$. As an example, if $\Gamma$ has as its cosmological cell the
hypercube defined in equation (17), then we demand that for any $x\in U$,
$i=1,\ldots,m-1$, we have $(x^{1},\ldots,x^{i}+L_{0},\ldots,x^{m-1})\in U$ and

$f(x^{1},\ldots,x^{i}+L_{0},\ldots,x^{m-1})=f(x^{1},\ldots,x^{m-1})\,.$ (18)

In order to obtain general results concerning existence and uniqueness of
solutions to the Einstein’s equations, we need to specify the regularity
assumptions for the spatial functions. We will accomplish it by working in the
Sobolev spaces $\mathcal{H}^{N}(\mathbb{R}^{m-1})$ and
$\mathcal{H}^{N}(\mathbb{R}^{m-1}/\Gamma)$, with $\mathbb{R}^{m-1}/\Gamma$ a
compact manifold (see [23], Chapter 2). For instance, any spatial function
which is $N$ times continuously differentiable in $\mathbb{R}^{m-1}$
($\mathcal{C}^{N}$) and $\Gamma$-periodic has all their derivatives up to
order $N$ bounded by a constant $C>0$, that is, $|\partial^{\alpha}f(x)|\leq
C$ for every $x\in U$ and $|\alpha|\leq N$, where we have used the usual
multi-index notation for partial derivatives, so that it is in the space
$\mathcal{H}^{N}(\mathbb{R}^{m-1})$. Furthermore, any function in the quotient
manifold differentiable up to the order $N$, say
$\tilde{f}\in\mathcal{C}^{N}(\mathbb{R}^{m-1}/\Gamma)$, gives rise to a unique
$\Gamma$-periodic functions $f\in\mathcal{C}^{N}(\mathbb{R}^{m-1})$, a fact
that can be straightforwardly verified from the relation
$f(x)=\tilde{f}(\Gamma\cdot x)$. These aspects will be of great importance in
the proof of the theorem 1.

##  3 On the solutions for simple fluids

###  3.1 Existence, uniqueness and early homogeneity

Let us now turn to the problem of the existence and uniqueness for the
solutions of Einstein’s equations. In order to settle our hypothesis on the
behavior of the cosmological fluid, we must keep in mind that the purpose in
this work is to emphasize the role of local inhomogeneity in the global
cosmological dynamics. So, if we define the function $\gamma(t,x)$ as
$p(t,x)=(\gamma(t,x)-1)\,\rho(t,x)$, we shall expect to find its averaged
behavior on large scales ($L\gg L_{0}$) in the form $\overline{\gamma}_{\rm
eff}=\overline{\gamma}_{\rm eff}(t)$. We will also assume it at small scales,
that is: the ratio $p/\rho$ is a known function of the scale factor ‘$a$’
only, that is, $\gamma=\gamma(a)$. In this picture, the continuity equation
(9) in terms of $a$ and $\rho$ turns out to be

$a\,\frac{\partial\rho}{\partial
a}=-(m-1)\,\gamma(a)\,\rho+\frac{(m-2)}{2\,a^{2}}\,\left(\frac{3}{2}\left(\frac{|\nabla\rho|}{\rho}\right)^{2}-\frac{\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}\rho}{\rho}\right)\,.$
(19)

In terms of the function

$\zeta(a)=-(m-1)\,\int_{a_{0}}^{a}\,\frac{\gamma(a^{\prime})}{a^{\prime}}\,da^{\prime}\,$
(20)

and the new “time” variable

$s=-\,\frac{m-2}{2}\,\int_{a_{0}}^{a}\,\frac{e^{-\zeta(a^{\prime})}}{a^{\prime
3}}\,da^{\prime}$ (21)

it turns into

$\frac{\partial}{\partial
s}\left(\rho\,e^{-\zeta(s)}\right)=\frac{\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}\rho}{\rho}-\frac{3}{2}\left(\frac{|\nabla\rho|}{\rho}\right)^{2}\,.$
(22)

###### Theorem 1

Let $\Gamma$ be a discrete subgroup of isometries of the Euclidean space
$\mathbb{R}^{m-1}$, such that $\mathbb{R}^{m-1}/\Gamma$ is a compact manifold,
and $\mathrm{K}$ the associated cosmological cell. Fix $a_{0}>0$, $a_{\rm
min}=\inf\,\\{a\,|\,0<a<a_{0}\,,\,0<s(a)<\infty\\}$ and $\rho_{0}(x)$ a
positive and $\Gamma$-periodic function which is $N$ times continuously
differentiable on $\mathbb{R}^{m-1}$, $N>(m+3)/2$. There is a solution
$\rho(a,x)$ of the equation (19) satisfying the following conditions:

  1. (i)

$\rho$ is the unique solution which is continuous in $(a_{\rm
min},a_{0}]\,{\scriptstyle\times}\,\mathbb{R}^{m-1}$, smooth
($\mathcal{C}^{\infty}$) in $(a_{\rm
min},a_{0})\,{\scriptstyle\times}\,\mathbb{R}^{m-1}$, $\rho(a,\cdot)$ is
$\Gamma$-periodic in $\mathbb{R}^{m-1}$ for every $a\in(a_{\rm min},a_{0}]$
and $\rho(a_{0},x)=\rho_{0}(x)$ for every $x\in\mathbb{R}^{m-1}$;

  2. (ii)

For every $x\in\mathbb{R}^{m-1}$, we have

$a_{min}<a\leq a_{0}\quad\Rightarrow\quad\rho_{0}^{\rm
min}\,e^{\zeta(a)}\leq\rho(a,x)\leq\rho_{0}^{\rm max}\,e^{\zeta(a)}\,,$ (23)

where $\rho_{0}^{\rm min}$ and $\rho_{0}^{\rm max}$ are the maximum and
minimum values of $\rho_{0}(x)$, respectively, and $\zeta$ is defined in (20).
In particular, $\rho$ is everywhere positive: $\rho(a,x)>0$.

###### Proof.

It is enough if we prove our theorem of existence and uniqueness of solutions
for the equation (19) in the compact Riemannian manifold (without boundary)
$\mathbb{R}^{m-1}/\Gamma$. The unique extension of the main theorem to the
entire spacetime follows just as explained in the section 2.4.

In the first place, if there is a solution which is continuous in the interval
$(a_{\rm min},a_{0}]$, then, by the positivity of $\rho_{0}$, the energy
density can be described by the function $v(a,x)$ through

$\rho=v^{2}\,e^{\zeta(a)}\geq 0\,,$ (24)

in an interval $(a,a_{0}]$. Moreover, uniqueness of $\rho$ follows from the
uniqueness of $v$, at least along the maximal interval $(a_{1},a_{0}]$ where
$v(a)$ has no zeros in the cell $\mathrm{K}$, for
$v=\sqrt{\rho\,e^{-\zeta(a)}\,}$. Hence, we rewrite the continuity equation
with the “time” variable $s$ given by the equation (22) as the quasi-linear
parabolic one:

$\frac{\partial v}{\partial s}=\nabla\left(\frac{1}{v^{2}}\,\nabla
v\right),\,\qquad v(0)\equiv v(0,x)=\,\sqrt{\rho_{0}(x)}\,.$ (25)

As a direct consequence of the propositions 8.3 and 9.8 in chapter 15 of the
reference [24], where we use the hypothesis $N>(m+3)/2$, there is only one
solution $\tilde{v}$ of this initial value problem in
$\mathbb{R}^{m-1}/\Gamma$ which is continuous on
$[0,\infty)\,{\scriptstyle\times}\,\mathbb{R}^{m-1}/\Gamma$ and
$\mathcal{C}^{\infty}$ on
$(0,\infty)\,{\scriptstyle\times}\,\mathbb{R}^{m-1}/\Gamma$. Therefore, as we
have argued in the section 2.4, there is only one extension $v(s,x)$ to
$[0,\infty)\,{\scriptstyle\times}\,\mathbb{R}^{m-1}\subset\mathcal{M}$, which
is continuous on $[0,\infty)\,{\scriptstyle\times}\,\mathbb{R}^{m-1}/\Gamma$
and $\mathcal{C}^{\infty}$ on
$(0,\infty)\,{\scriptstyle\times}\,\mathbb{R}^{m-1}/\Gamma$. Hence, The
existence and uniqueness of the solution $\rho(a,x)$, continuous on
$(a_{1},a_{0}]\,{\scriptstyle\times}\,\mathbb{R}^{m-1}$ and
$\mathcal{C}^{\infty}$ on
$(a_{1},a_{0}]\,{\scriptstyle\times}\,\mathbb{R}^{m-1}$, follows
straightforwardly, as well as the positiveness property, that is, $\rho\geq 0$
everywhere. It remains to show that $a_{1}=a_{\textrm{min}}$.

We now proceed to show that the maximal interval $[0,s^{*})$ where $v(s)$ has
no zeros in the cell is in fact $[0,\infty)$, which ensures the uniqueness and
positivity of $\rho$ in the interval $(a_{min},a_{0}]$. First of all, take a
point $x_{1}\in\mathrm{K}$ that is a minimum of $v(s_{1})$ with
$0<s_{1}<s^{*}$. If
$\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}v(s_{1},x_{1})\neq
0$ then it is positive, since the point in question is a minimum. By
continuity, there must be an open set
$I\,{\scriptstyle\times}\,U\subset(0,s^{*})\,{\scriptstyle\times}\,\mathrm{K}$
containing $(s_{1},x_{1})$ where
$v\,\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}v-\scalebox{0.7}[0.8]{$2$}|\nabla
v|^{2}>0$, since $v(s_{1},x_{1})>0$,
$\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}v(s_{1},x_{1})>0$
and $\nabla v(s_{1},x_{1})=0$. Using the equation (25), we conclude that
$\frac{\partial v}{\partial s}>0$ in this interval, that is,

$s\in I\,:\quad s>s_{1}\quad\Rightarrow\quad v(s)>v(s_{1})\quad\mbox{in $U$}.$
(26)

If
$\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}v(s_{1},x_{1})=0$,
let $\Sigma_{1}$ be the connected component containing $(s_{1},x_{1})$ of the
set in $(0,s^{*})\,{\scriptstyle\times}\,\mathrm{K}$ defined by the $m$
equations $\nabla v=0$,
$\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}v=0$. If
$\Sigma_{1}$ is a point then there will be $(s_{2},x_{2})$ with $x_{2}$ a
(local) minimum point of $v$ such that $s_{2}>s_{1}$ is arbitrarily close to
$s_{1}$ and
$\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}v(s_{2},x_{2})>0$,
so that the argument above still works. If not, such a point will be in
$\Sigma_{1}$ and there will be a piecewise differentiable curve
$(s,x(s))\in\Sigma_{1}$ connecting it to $(s_{1},x_{1})$. For this curve we
have, using the equation (25) and the fact that $\nabla v=0$ and
$\,\raisebox{0.09995pt}{\scalebox{1.2}[1.3]{${\scriptstyle\Delta}$}}v=0$ along
it,

$\frac{d}{ds}v(s,x(s))=\frac{\partial v}{\partial s}(s,x(s))=0\,.$ (27)

Therefore we conclude that

$0<s_{1}<s_{2}<s^{*}\quad\Rightarrow\quad v^{\rm min}(s_{1})\leq v^{\rm
min}(s_{2})\,,$ (28)

where $v^{\rm min}(s)$ stands for the minimum value of $v(s)$ in $\mathrm{K}$.
Changing $v\to-v$, we readily see that the analogous relation is satisfied by
the maximum $v^{\rm max}(s)$ of $v(s)$ in $\mathrm{K}$:

$0<s_{1}<s_{2}<s^{*}\quad\Rightarrow\quad v^{\rm max}(s_{2})\leq v^{\rm
max}(s_{1})\,.$ (29)

So, by continuity of $v$, we conclude that

$0<v^{\rm min}(0)\leq v(s)\leq v^{\rm max}(0)\,,$ (30)

proving the formula (23) and showing that $s^{*}=\infty$, that is, $\rho$ is
unique. ∎

Theorem 1 presents a different perspective for the Einstein’s equations.
Instead of determining the future behavior of the spacetime from its initial
conditions, we have found its past history from the “final” or today’s
configuration. This feature is a consequence of picking up the scalar factor
as the time variable together with the intrinsic flatness hypothesis. It does
not only tells us that our inhomogeneous picture is mathematically consistent,
for past existence and uniqueness are granted as we set our current state, but
it also allows us to infer some general aspects of such spacetimes, as for
instance, the fact that the spatial homogeneity never decreases as the
universe expands.

In order to make sense of our last statement, let us define the maximum, the
minimum and the mean energy density in each cell, respectively, as

$\rho^{\rm max}(a):=\max_{x\in\mathrm{K}}\,\rho(a,x)\quad,\quad\rho^{\rm
min}(a):=\min_{x\in\mathrm{K}}\,\rho(a,x)$ (31)

and

$\overline{\rho}(a):=\frac{1}{Vol_{m-1}(\mathrm{K})}\,\int_{\mathrm{K}}\rho(a,x)\,d^{m-1}x\,.$
(32)

We define the inhomogeneity modulus $\Delta(a)$ as

$\Delta(a):=\frac{\rho^{\rm max}(a)-\rho^{\rm min}(a)}{\rho^{\rm max}(a)}\,,$
(33)

while the density contrast, as we interpret $\overline{\rho}(a)$ to be the
mean background mass density [1], is

$\delta(a,x)=\frac{\rho(a,x)-\overline{\rho}(a)}{\overline{\rho}(a)}\,.$ (34)

Both $\Delta$ and $\delta$ are measures of the “degree of homogeneity” of the
spacetime. In the period when $\Delta\ll 1$ or $|\delta|\ll 1$, the spacetime
can be considered spatially homogeneous. From the standard model of cosmology,
we should expect a highly homogeneous early universe, with the homogeneity
growing as the universe expands. Theorem 1 tells us that our model is
consistent with this picture, for it demands the following corollary:

###### Corollary 2

Under the hypothesis of Theorem 1, setting $\Delta_{0}=\Delta(a_{0})$, we have

$\quad a\leq
a_{0}\quad\Rightarrow\quad\cases{0}\leq\Delta(a)\leq\Delta_{0}\leq 1\\\
-\Delta_{0}\leq-\Delta(a)\leq\delta(a,x)\leq\frac{\Delta(a)}{1-\Delta(a)}\leq\frac{\Delta_{0}}{1-\Delta_{0}}.$
(35)

In particular, $\Delta(a)$ do not decrease in an expanding universe.

###### Proof.

This comes from the inequality (23). First of all, we note that the same
inequality holds for $\rho^{\rm max}(a)$, $\rho^{\rm min}(a)$ and
$\overline{\rho}(a)$. Hence, we conclude that

$\quad a\leq a_{0}\quad\Rightarrow\quad\frac{\rho^{\rm min}_{0}}{\rho^{\rm
max}_{0}}\leq\frac{\rho^{\rm min}}{\rho^{\rm
max}}\leq\frac{\rho}{\overline{\rho}}\leq\frac{\rho^{\rm max}}{\rho^{\rm
min}}\leq\frac{\rho^{\rm max}_{0}}{\rho^{\rm min}_{0}}\quad{\rm
and}\quad\frac{\rho^{\rm min}_{0}}{\rho^{\rm max}_{0}}\leq\frac{\rho^{\rm
min}(a)}{\rho^{\rm max}(a)}\,.$ (36)

The inequalities (35) follow from the fact that $\rho^{\rm
min}=(1-\Delta)\,\rho^{\rm max}$ and $\rho=(1+\delta)\,\overline{\rho}$. If
$\Delta(a)$ decreases at $a=a_{1}$, then we could change the "initial"
conditions of the theorem 1 to that of the instant $a=a_{1}$, thus
contradicting the inequalities just obtained. Hence, $\Delta(a)$ cannot
decrease with the increasing of the scale factor. ∎

The reader should notice that the choice of the equation of state does not
determine a priori the Hubble parameter, except when we return to the RW
spacetime. However, there are some ways to overcome this situation. For a
better understanding of this fact, let us exemplify it by taking any free-
falling observer with its proper time, which it imprints to the whole universe
as being the cosmological time function $t$. Let us also assume it is
initially at rest with respect to the flat space sections, so that, by direct
inspection of the geodesic equations and the uniqueness of their solutions, we
readily conclude that this observer is placed along the points $(t,x_{0})$
where the spatial gradient of $\phi$, and of $\rho$ as well, vanishes. From
this point of view, the metric in the formula (2) can be set with
$\phi(t,x_{0})=0$, which looks like a plane RW metric around these points, and
the “Friedmann” equation for $m=4$ would appear as
$3\mathrm{H}(t)^{2}=\rho(t,x_{0})$, just like in the flat standard model. We
could do that along the points where the energy density has its maximum, thus
obtaining a Hubble parameter $\mathrm{H}=\mathrm{H}^{max}$, or its minimum,
with $\mathrm{H}=\mathrm{H}^{min}$. If we think that “the Hubble” parameter
satisfies a phenomenological relation $\mathrm{H}=\mathrm{H}(a)$, just as in
the standard $\Lambda$CDM model, we could be neglecting a multiplicative term
of order $\mathrm{H}^{min}/\mathrm{H}^{max}=\sqrt{1-\Delta}$ in the
determination of the density parameters $\Omega$’s. This shows us that the
choice of the “cosmological” observers, the cosmic time $t$ and any comparison
with the $\Lambda$CDM model should be taken with care. We leave this task to
another investigation, which is beyond the scope of this manuscript.

###  3.2 Some exact solutions

According to the fourth case discussed in section 6.1.2 of the reference [25],
it is straightforward the comparison with our equation (25). Thus, one can
find implicit solutions in a traveling-wave form by taking the ansatz

$|\vec{k}|^{2}\int_{v_{{0}}}^{v\left(s,\vec{x}\right)}\\!\frac{{\rm
d}u}{u^{2}(\lambda u+c_{1})}=\lambda s+\vec{k}\cdot\vec{x},$ (37)

where $\vec{k}\in\mathbb{R}^{m-1}$, $\lambda\in\mathbb{R}$, $v_{0}$ and
$c_{1}\neq 0$ are arbitrary constants. It is direct to verify that this
expression satisfies the equation (25). Besides, this integral can be
particularly solved, yielding

$\frac{1}{v_{0}}-\frac{1}{v}+\frac{\lambda}{c_{1}}\ln\left(\frac{\frac{\lambda}{c_{1}}+\frac{1}{v}}{\frac{\lambda}{c_{1}}+\frac{1}{v_{0}}}\right)=\frac{c_{1}(\lambda
s+\vec{k}\cdot\vec{x})}{|\vec{k}|^{2}}.$ (38)

We can isolate $v(s,\vec{x})$ in this equation and find the energy density, as
follows

$\quad\rho(s,x,y)=e^{\zeta(s)}\left\\{\frac{\lambda}{c_{1}}+\frac{\lambda}{c_{1}}\,W\left[-\left(\frac{c_{1}}{\lambda
v_{0}}+1\right)\exp\left(\frac{c_{1}^{2}(\vec{k}\cdot\vec{x}+\lambda
s)}{\lambda|\vec{k}|^{2}}-\frac{c_{1}}{\lambda
v_{0}}-1\right)\right]\right\\}^{-2},$ (39)

being $W(X)$ the Lambert W-function (for details see the reference [26]). As
we are dealing only with real arguments for the Lambert W-function, there are
only two branches to be considered: $W_{0}(X)$ for $X>-1/e$ and $W_{-1}(X)$
for $-1/e\leq X<0$. Note that the equation (39) is invariant under the
transformations $\vec{k}\rightarrow\beta\vec{k}$,
$\lambda\rightarrow\beta\lambda$ and $c_{1}\rightarrow\beta c_{1}$, where
$\beta\neq 0$ is a constant, letting $\nu_{0}$ unchanged. It means that the
profile responsible for the matter distribution inside the cosmological cell
may have the same shape for different scales in terms of the wave parameters
$\vec{k}$ and $\lambda$ by choosing $\beta$ adequately.

For the sake of illustration, we study a dust-dominated ($\gamma(a)=1$)
cosmological scenario for $m=3$. In this case, the auxiliary functions
$\zeta(a)$ and $s(a)$ can be found straightforwardly, reducing to

$\zeta(a)=2\ln\left(\frac{a_{0}}{a}\right),\quad\mbox{and}\quad
s(a)=\frac{1}{2a_{0}^{2}}\ln\left(\frac{a_{0}}{a}\right),$ (40)

(a) Cosmological cell

(b) Cosmological lattice

Figure 1:  Plot of $\rho(1,x,y)$. The cosmological cell (1(a)) is designed by
taking four sub-cells of unit side, where each of them is given from the
equation (41) with different values for the parameters: $k_{x}=k_{y}=1$ for
$x\in[0,1]$ and $y\in[0,1]$, $-k_{x}=k_{y}=1$ for $x\in[-1,0]$ and $y\in[0,1]$
and so on, respecting the reflection symmetry of the cell. The other constants
remain the same for the sub-cells and they are chosen as $\lambda=1$,
$c_{1}=-2$ and $v_{0}=1$. At the end, they are all joined continuously forming
a cosmological cell of side $L_{0}=2$. The cosmological lattice (1(b)) is
constructed by gluing continuously the boundaries of the cosmological cells.

where $a_{0}$ is interpreted as the scale factor today and it is set to unit
without loss of generality. Then, we can explicitly write $\rho(a,x,y)$ with
the help of equations (20), (21), (24) and (39), as follows

$\quad\rho(a,x,y)=\frac{\lambda}{a^{2}\,c_{1}}\left\\{1+W_{0}\left[-\left(\frac{c_{1}}{\lambda
v_{0}}+1\right)a^{-\frac{c_{1}^{2}|\,\vec{k}|^{2}}{2}}\exp\left(\frac{c_{1}^{2}(k_{1}x+k_{2}y)}{\lambda|\vec{k}|^{2}}-\frac{c_{1}}{\lambda
v_{0}}-1\right)\right]\right\\}^{-2}.$ (41)

We take the cosmological cell to be a square of side $L_{0}$ centered at the
origin of the coordinate system and we divide it in four equal smaller squares
of sides $L_{0}/2$. Thus, we define the solution in
$[0,L_{0}/2]\times[0,L_{0}/2]$ with the use of formula (41) and then reflect
it in the remaining three parts, say, $[0,L_{0}/2]\times[-L_{0}/2,0]$,
$[-L_{0}/2,0]\times[-L_{0}/2,0]$ and $[-L_{0}/2,0]\times[0,L_{0}/2]$. The
whole periodic solution is obtained by gluing them throughout the other
cosmological cells. The profile of $\rho(1,x,y)$ is depicted by the figure
(1(a)). Note that we combine four solutions provided by the equation (41)
using different wave parameters in order to get the peaked profile for the
energy density. In the figure (1(b)), we depicted the cosmological lattice in
the studied case where one can see the succession of matter clumps and voids
all over the space. Finally, with the help of the figures (2(a)) and (2(b)),
the former illustrates the density contrast for a spatial section of the
cosmological cell and the latter depicts the evolution of the inhomogeneity
modulus according to the scale factor. Both present complementary results
predicted by the Corollary (2), namely, the growing of the inhomogeneity
inside the cell and its time dependence with respect to the scale factor. As
$a$ increases the relative inhomogeneities also increase and they vanish as
$a$ goes to zero.

(a) The density contrast as function of $x$ (with $y=0$) for different values
of $a$.

(b) The inhomogeneity modulus as function of $a$.

Figure 2: The spatial distribution and time evolution of the inhomogeneity
parameters of our model.

For the particular case in which $c_{1}=0$, the integral in the equation (37)
is rather trivial and the energy density can be written in terms of elementary
functions, as follows

$\rho(a,x,y)=\frac{\nu_{0}^{2}\,|\vec{k}|^{2}}{a^{2}\left|\,|\vec{k}|^{2}+\lambda\nu_{0}^{2}[\lambda\ln
a-2(k_{1}x+k_{2}y)]\right|}.$ (42)

Figure 3: For better illustration, we take the slice $y=1$ of the energy
density inside the cosmological cell for different values of $a$, with the
parameters $k_{x}=k_{y}=\lambda=\nu_{0}=1$.

Note that now $\rho(a,x,y)$ diverges when the denominator of this equation
vanishes. Indeed, from the figure (3), it is clear to see the traveling-wave
character of this particular solution and the process of concentration of
matter as the scale factor evolves. It is worth mentioning that both cases do
not satisfy the hypothesis of Theorem (1), since the energy density is not
differentiable all over the cosmological cell.

##  4 Concluding Remarks

From the observations, we know that the universe has inhomogeneites
responsible for the emergence of the large scale structures. From the
homogeneous and isotropic viewpoint, with a RW background metric, this issue
is approached perturbatively. They are obtained from small quantum
fluctuations that start growing when the scale factor is small, evolving to
larger scales. According to our results, it is reasonable to expect that
models such as ours have ingredients enough to explain qualitatively the same
cosmological data in a different way, by taking small perturbations of the
fully nonlinear inhomogeneous background, where the matter content is
periodically distributed throughout the space. One can find in the literature
arguments that these approaches are not equivalent and can lead to difference
explanations of the same set of data [27, 28, 29]. Furthermore, it is
important to mention that the current standard picture of cosmology is passing
through revision, as in the case of the Hubble tension [30] or in the
explanation of the accelerated expansion [31]. Hence, it is fair to exploit
other alternatives the General Theory of Relativity offers to us, specially
after the period when the inhomogeneities begin to grow significantly. This
manuscript is a starting point for one of them, which we think is worth
pursuing.

Starting with an intrinsically flat spacetime filled in with a dissipative
fluid, we have established the general mathematical setting for dealing with a
cosmological model under periodic boundary conditions, ending up with an
inhomogeneous cosmological model defined in a lattice, in which the matter
distribution is nontrivial inside the fundamental cells. As we have adapted
important theorems from the theory of partial differential equations to our
cosmological scenario, we have been able to prove existence and uniqueness
theorems of solutions for the Einstein’s equations. We also obtained the
Corollary 2, which ensures their compatibility with a highly homogeneous phase
of the universe for small values of the scale factor.

We have also found a class of exact solutions for the Einstein’s equations,
given in terms of the expression (39) of the energy density, that carries
interesting properties, as: (i) regularity of $\rho$ over the cell for all
values of the scale factor; (ii) Spatial homogeneity as $a$ goes to zero
($\Delta\to 0$ as $a\to 0$ for suitable choice of the parameters); (iii) the
global picture representing regions of high concentration of matter alternated
by voids. Finally, we have studied the particular case of a dust-dominated
universe in order to better illustrate the properties of our model, with the
cosmological cells and lattice they originate, as well as the asymptotic
homogeneous phase in the early universe.

There is a long road to be traced in understanding the space-flat spacetimes
with periodic distribution of matter. Some effort to theoretically underpin
their observational characteristics have already been made, as the study of
the relation luminosity distance $\times$ redshift [9], although a rigorous
connection to the observational data is still missing. The many other
observational features should be probed in the future, as for instance, the
link of such geometric structure with the cosmic microwave background.

## References

## References

  * [1] Peebles P J E 1993 Principles of Physical Cosmology (Princeton University Press)
  * [2] Wald R M 1984 General Relativity (The University of Chicago Press)
  * [3] Ringström H 2013 On the Topology and Future Stability of the Universe (Oxford University Press)
  * [4] Collins C B and Szafron D A 1979 Journal of Mathematical Physics 20 2347
  * [5] Szafron D A and Collins C B 1979 Journal of Mathematical Physics 20 2354
  * [6] Collins C B and Szafron D A 1979 Journal of Mathematical Physics 20 2362
  * [7] Ellis G F R, Maartens R and MacCallum M A H 2012 Relativistic Cosmology (Cambridge University Press)
  * [8] Abbott T, Aguena M, Alarcon A, Allam S, Alves O, Amon A, Andrade-Oliveira F, Annis J, Avila S, Bacon D et al. 2022 Physical Review D 105 023520
  * [9] Bittencourt E, Gomes L and Santos G 2021 International Journal of Modern Physics D 30 2150033
  * [10] Clifton T, Ferreira P G and Land K 2008 Phys. Rev. Lett. 101(13) 131302
  * [11] February S, Larena J, Smith M and Clarkson C 2010 Monthly Notices of the Royal Astronomical Society 405 2231–2242
  * [12] Kolb E W 2011 Classical and Quantum Gravity 28 164009
  * [13] Buchert T 2000 General Relativity and Gravitation 32 105
  * [14] Wiltshire D L 2011 Classical and Quantum Gravity 28 164006
  * [15] Bentivegna E, Clifton T, Durk J, Korzyński M and Rosquist K 2018 Classical and Quantum Gravity 35 175004
  * [16] Ellis G F R and Hawking S W 1975 The Large Scale Structure of Space-Time (Cambridge University Press)
  * [17] Mimoso J P and Crawford P 1993 Classical and Quantum Gravity 10 315
  * [18] Bittencourt E, Salim J M and Dos Santos G B 2014 General Relativity and Gravitation 46 1790
  * [19] Sopuerta C F 2000 Classical and Quantum Gravity 17 4685–4695
  * [20] Wolf T 1986 Journal of Mathematical Physics 27 2340–2353
  * [21] Vinberg E 1993 Geometry II (Springer, Berlin, Heidelberg)
  * [22] Wolf J A 2011 Spaces of Constant Curvature (AMS Chelsea Publishing)
  * [23] Aubin T 1982 Nonlinear Analysis on Manifolds. Monge-Ampère Equations (Springer-Verlag)
  * [24] Taylor M E 2011 Partial Differential Equations III: Nonlinear Equations (Springer, New York, NY)
  * [25] Polyanin A D and Zaitsev V F 2012 Handbook of nonlinear partial differential equations second edition ed (Taylor & Francis Group, LLC)
  * [26] Roy R and Olver F W J 2010 Lambert w-function NIST Handbook of Mathematical Functions ed Olver F W J, Lozier D M, Boisvert R F and Clark C W (Cambridge University Press)
  * [27] Green S R and Wald R M 2014 Classical and Quantum Gravity 31 234003
  * [28] Buchert T, Carfora M, Ellis G F R, Kolb E W, MacCallum M A H, Ostrowski J J, Räsänen S, Roukema B F, Andersson L, Coley A A and Wiltshire D L 2015 Classical and Quantum Gravity 32 215021
  * [29] Green S R and Wald R M 2015 Comments on backreaction (Preprint 1506.06452)
  * [30] Riess A G, Macri L M, Hoffmann S L, Scolnic D, Casertano S, Filippenko A V, Tucker B E, Reid M J, Jones D O, Silverman J M, Chornock R, Challis P, Yuan W, Brown P J and Foley R J 2016 The Astrophysical Journal 826 56
  * [31] Rácz G, Dobos L, Beck R, Szapudi I and Csabai I 2017 Monthly Notices of the Royal Astronomical Society: Letters 469 L1–L5