# Breaking the Cosmological Principle into pieces: a prelude to the
intrinsically homogeneous and isotropic spacetimes

L. G. Gomes  Federal University of Itajubá (UNIFEI), Av. BPS, 1303, Itajubá-
MG, 37500-903, Brazil<EMAIL_ADDRESS>
###### Abstract

In this manuscript, we show that there are three fundamental building blocks
supporting the Cosmological Principle. The first of them states that there is
a special frame in the universe where the spatial geometry is intrinsically
homogeneous and isotropic. The second demands the existence of a fiducial
observer to whom the Hubble parameter is isotropic. The last piece states that
matter and radiation behave as a perfect fluid. We show that these three
hypotheses give us the Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes,
the central pillar of the standard model of Cosmology. We keep with the first
of them and start to investigate the so-called intrinsically homogeneous and
isotropic spacetimes. They emerge after the decoupling of the CMB with the
geometric frame of reference. Furthermore, a “$\Lambda$CDM-like" effective
theory arises naturally in those backgrounds, together with some new density
parameters relating to the local inhomogeneities, the internal energy density,
and the local and global magnitudes of the Hubble anisotropy. All those
properties make this class of inhomogeneous models, which roughly speaking,
keeps "1/3" of the Cosmological Principle, worth investigating in applications
to Cosmology, for it can accommodate the same ingredients of the standard
model, as a geometric frame and a free-falling isotropic cosmic background
radiation, and reduce to the latter when some observable parameters vanish.

##  1  Introduction

Cosmology has been established on the belief that the Universe is homogeneous
and isotropic on large scales. Although this is a quite vague notion, it is
largely accepted that, when put on sound mathematical grounds, the
Cosmological Principle is equivalent to demanding the spacetime $\mathsf{M}$
to be of the FLRW type, named after their precursors Friedmann-Lemaître-
Robertson-Walker. It stands as a framework around which the standard model of
Cosmology is built [1, 2]. Nonetheless, the recent observational data have
been testing the limits of this theory, and their tenets were put under
intense scrutiny after the discovery of many “tensions" [3]. In particular,
the observational basis for the Cosmological principle has been investigated
in many aspects [4], and the doubts on whether it is indeed suitable to
describe our large-scale universe have been growing in the last few years[5,
6]. Regardless of the outcome of this cosmic conundrum but still inspired by
this current debate, in this manuscript, we shall pay a visit to the
mathematical hypotheses that lead us to the FLRW spacetimes, keeping those of
more fundamental relevance, while re-considering the role of the others, for
which the validity is a matter of observational verification.

The FLRW spacetimes are characterized by their highly symmetric spatial
sections: there is a Lie group acting on $\mathsf{M}$ by isometries whose
orbits are space-like hypersurfaces, the homogeneity condition, with a
transitive induced action of the isotropy group on their tangent spaces, the
isotropic character of the action. A well-known theorem ensures the existence
of a special coordinate representation where the metric is completely
characterized by the scale factor $a(t)$ and the constant $K_{0}$ representing
the spatial curvature [7]. As we open up the many restrictions imposed by the
large symmetry group of the FLRW spacetimes, we come to the conclusion that
such models are characterized by the vanishing of many spatial anisotropies.
In fact, the spatial geometry, the Hubble expansion, the energy flux, and the
pressure coming from the matter content, impose no spatial asymmetry along the
many allowed directions in such spacetimes.

As we move towards the foundations of the Cosmological Principle, we first
realize that it demands space to have the simplest of the geometries, those
which do not distinguish two different directions nor distinct points. These
homogeneous and isotropic environments are the simply connected Riemannian
manifolds with constant sectional curvature $K_{0}$, which are restricted to
three types of space: The Euclidean ($K_{0}=0$), the hyperbolic ($K_{0}<0$),
and the spheric ($K_{0}>0$) [8, 9]. We can set this condition as the existence
of special observers in the Universe, represented by a time-like unitary
vector field $\mathrm{u}$, to whom the notions of space and time are naturally
distinguished, that is, $\mathrm{u}$ has vanishing vorticity 111If
$\mathrm{u}$ has non-vanishing vorticity, then its orthogonal space
distribution is no longer integrable ([10], chapter 4). Hence, the notion of
“space” to those observers cannot be given by a unique natural choice., such
that the equal-time spatial sections have one of those simplest
configurations. They are formally defined in hypothesis H1, and the resulting
structure is called intrinsically homogeneous and isotropic spacetime, the
subject of investigation in this manuscript. They constitute a subclass of the
broader family of spacetimes admitting intrinsic symmetries [11, 12, 13].

The second fundamental notion behind the Cosmological Principle is the
isotropy of the Hubble sky, that is, the property of expanding equally in all
directions. It has a strong observational appeal, so much so that its validity
has been questioned in some recent data analyses (see Ref.[4] and references
therein). Regardless of the constraints imposed by observational Cosmology,
this piece of the Cosmological Principle has its own subtleties to be
understood and dealt with even on a theoretical basis. Nonetheless, the Hubble
isotropy dwells in the core of the standard model of Cosmology and is largely
accepted in the scientific community. The intrinsically homogeneous and
isotropic spacetimes readily recover this feature under the assumption of the
existence of a fiducial observer to whom the Hubble sky is the same in all
directions. In addition, we also assume that it is able to distinguish a scale
factor $\ell$ such that the local volume grows with $\ell^{3}$, if we are in a
four-dimensional spacetime, and the curvature decreases as $\ell^{-2}$. This
is the second piece of the Cosmological Principle, written down in hypothesis
H2.

The third and last piece is the isotropy of the physical constituents of
matter and radiation. It has a strong physical appeal by stating that there is
no net energy flux nor anisotropic stress in the universe. In fact, as we
settle the cosmological observers as those to whom the cosmic background
radiation (CMB) is nearly isotropic, for instance, it represents the idea that
there would be no giant magnetic field, nor any constituent of matter moving
in a relativistic speed, nor any phenomena that could account for significant
breaking of the perfect fluid form of the energy-momentum tensor. It is set in
the hypothesis H3.

Those are the three building blocks of the Cosmological Principle, put down in
section2. The first, mathematical, ensures the homogeneity and isotropy of the
spatial geometry to describe the universe. The second, observational,
guarantees isotropic expansion. The third, physical, assumes a perfect fluid
form for the overall combination of matter and radiation. As we put all of
them together and assume Einstein’s equations to hold, we recover the FLRW
spacetime, with each one of these assumptions being absolutely necessary for
it. We discuss that in section 4.

The main purpose of this manuscript is to start investigating the
intrinsically homogeneous and isotropic spacetimes, whose mathematical
foundations are the subject of section 3. This is a framework where just the
first “1/3" of the Cosmological Principle is kept. It turns out that this
structure naturally appears when we allow the CMB observers to decouple from
these geometric ones, as discussed in section 6. It corresponds to that
counterpart less accessible by our direct observations or our physical
intuition, and therefore, less likely to be ruled out by physical
considerations or surveys probing the cosmological sky. In this sense, the
hypothesis of homogeneity and isotropy of spatial geometry seems to be the one
to keep with, while leaving the rest "2/3" to observational scrutiny. Even
large-scale homogeneity and a $\Lambda$CDM-like approach to Cosmology
naturally arise from this rich framework, as we show in sections 5 and 7,
respectively. The notations and sign conventions follow Ref. [10].

##  2 On the hypotheses laying in the foundations of the Cosmological
Principle

Here we follow a methodology closely related to the co-moving approach of
General Relativity, very often used in Cosmology [14, 10], differing from it
only in some choices of variables and in the interpretation that our reference
system will not be necessarily co-moving with a fluid. We start with a
$m$-dimensional spacetime $\mathsf{M}$ endowed with the Lorentzian metric
$\mathrm{g}$ and establish our cosmological observers by choosing a time-like
vector field $\mathrm{u}$, oriented to the future, unitary,
$\mathrm{u}^{\,2}=-1$, and vorticity-free, that is, its orthogonal
distribution $\mathrm{u}^{\bot}$ is integrable. This last condition ensures
that we are not breaking the isotropy condition from the beginning, for
otherwise, we would have the vorticity vector pointing to some preferred
direction, at least if $\dim\mathsf{M}=4$. It stands as the intuitive notion
of a “rigid" system of observables, with no relative rotation among its
constituents. Furthermore, the curvature of the spatial sections orthogonal to
$\mathrm{u}$ should not distinguish between two different points or
directions, meaning that our space sections have constant sectional curvature
[8, 9].

Looking for a conceptual understanding of the mathematical foundations behind
the Cosmological Principle, we set our first hypothesis as follows:

###### Hypothesis 1

In the spacetime $\mathsf{M}$ with metric $\mathrm{g}$ there is a time-like
vector field $\mathrm{u}$, oriented to the future, unitary,
$\mathrm{u}^{\,2}=-1$, and vorticity-free. For each point $p\in\mathsf{M}$,
the maximum integrable manifold of $\mathrm{u}^{\bot}$ passing through $p$,
$\mathsf{S}_{p}$, the spatial section at $p$, is a space of constant curvature
$K(p)$ with the inherited Riemannian metric

$\mathrm{h}=\mathrm{g}+\mathrm{u}^{\flat}\mathrm{u}^{\flat}\,,$ (1)

where $\mathrm{u}^{\flat}:=\mathrm{g}(\mathrm{u},\cdot)$ is the $1$-form
naturally associated to $\mathrm{u}$ through $\mathrm{g}$, the lowering of its
index. Any two spatial sections are diffeomorphic. Such spacetimes are called
intrinsically homogeneous and isotropic.

Throughout the manuscript, we assume hypothesis 1 to hold, so that our
investigation is concentrated on the general intrinsically homogeneous and
isotropic spacetimes. In this context, there is a natural splitting in our
spacetime settling our choices for time and space, respectively, notions of
which can be incorporated into any tensor field. For instance, we will denote
$\varphi:\mathsf{M}\to\mathbb{R}$ by a time function whenever
$d\varphi(p)\cdot v=0$ for every $p\in\mathsf{M}$ and
$v\in\mathrm{u}(p)^{\bot}$. Analogously, it is a space function if
$d\varphi\cdot\mathrm{u}=0$. A vector field $X$ is a time vector field if it
is everywhere proportional to $\mathrm{u}$ and a space vector field if it is
everywhere orthogonal to $\mathrm{u}$. Any covariant tensor field in
$\mathsf{M}$ is also referred to as a space tensor field if it vanishes
whenever contracted with $\mathrm{u}$, and so on. Moreover, the spatial
gradient is the spatial projection of the gradient, which coincides with the
gradient defined in the intrinsic Riemannian geometry of each spatial section
(denote $\mathrm{u}\cdot\phi:=\mathrm{u}^{\nu}\partial_{\nu}\phi$) :

$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\phi=\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}\phi+(\mathrm{u}\cdot\phi)\,\mathrm{u}\,.$
(2)

An important role is played by the acceleration vector field
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\mathrm{u}$.
Besides being orthogonal to $\mathrm{u}$, its associated one form
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\mathrm{u}^{\flat}:=\mathrm{g}(\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\mathrm{u},\cdot)$,
which in coordinate representation is simply
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\mathrm{u}_{\nu}$,
is closed when restricted to each space section $\mathsf{S}_{p}$, a fact that
is equivalent to the vanishing of the vorticity of $\mathrm{u}$ (Compare with
formula (4.36) in [10]). From that intrinsic viewpoint, it is locally the
spatial gradient of a “potential". In a more precise manner, in the following
lemma, we establish this well-known result, for the sake of completeness:

###### Proposition 1

Let $\mathrm{u}$ be a time-like and vorticity-free vector field and $U$ an
open set in $\mathsf{M}$ such that $\mathsf{S}_{p}\cap U$ is connected and
simply connected along each spatial section $\mathsf{S}_{p}$, $p\in U$. There
are smooth functions, a time $t:U\to\mathbb{R}$ and a “potential"
$\phi:U\to\mathbb{R}$, such that each space section $\mathsf{S}_{p}$ is
locally characterized by $t=t(p)$, that is, $\mathsf{S}_{p}\cap
U=t^{-1}(t(p))$, and the spatial gradient of $\phi$ gives the acceleration
vector,

$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\phi=\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\mathrm{u}\,.$
(3)

Furthermore, any two “potentials" in $U$ differ by a time function in $U$:

$\phi_{2}-\phi_{1}=\varphi(t)\,.$ (4)

###### Proof.

With no loss of generality, put $U=\mathsf{M}$ connected and simply connected,
and set a curve of reference points in each space section by starting from a
point $p_{0}\in\mathsf{M}$ flowing along $\mathrm{u}$,
$c(t)=F_{\mathrm{u}}^{t}(p_{0})$, $F_{\mathrm{u}}^{t}$ being its flux. If we
assume connected space sections, each $p\in\mathsf{M}$ would correspond to a
time $t_{p}$ for which $p\in\Sigma_{c(t_{p})}$. Now we take the integral of
$\left(\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\mathrm{u}\right)^{\flat}$
along a path lying entirely in $\Sigma_{c(t_{p})}$, from $c(t_{p})$ to $p$,

$\phi(p)=\int_{c(t_{p})}^{p}\,\left(\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\mathrm{u}\right)^{\flat}\,.$
(5)

This defines a function as long as the path in the integral is taken to lie
entirely in a simply connected region of $\Sigma_{p}$, since
$\left(\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\mathrm{u}\right)^{\flat}$
is closed along this space section. By definition, it is clear that the space
component of
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}\phi$ is the
acceleration vector field. Assuming that both $\phi_{1}$ and $\phi_{2}$
satisfy equation (3), uniqueness follows from the fact that for
$\varphi=\phi_{2}-\phi_{1}$ and $X$ a space vector field, we have
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}\phi_{i}=\lambda_{i}\mathrm{u}+\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\mathrm{u}$
and

$d\varphi\cdot
X=\mathrm{g}(\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}\varphi,X)=(\lambda_{2}-\lambda_{1})\,\mathrm{g}(\mathrm{u},X)=0\,.$

∎

In local coordinates $(x^{\mu})=(t,x^{i})$ adapted to the observers
$\mathrm{u}$, guaranteed by the vanishing of the vorticity, any “potential"
$\phi$ appears as $\mathrm{u}=e^{-\phi}\frac{\partial}{\partial t}$ and in the
metric as $\ln\sqrt{-\mathrm{g}_{00}}$, that is,

$\mathrm{g}=-\,e^{2\phi(t,x)}\,dt^{2}+\mathrm{h}_{ij}(t,x)\,dx^{i}dx^{j}\,.$
(6)

Note that $\mathrm{g}_{00}\approx-(1+2\phi)$ in the nearly Newtonian regime
[15]. Therefore, any such function has a simple interpretation: it is the
gravitational potential in the usual textbook-like Newtonian limit of General
Relativity [15, 16]. Needless to say, this special status of a true potential
loses its meaning in General Relativity. Throughout the text, our coordinate
system is assumed to be adapted to $\mathrm{u}$, just like in the formula (6)
above.

An $\mathrm{u}$-observer is a curve $c(t)$ which is an integral line of
$\mathrm{u}$, that is, $\dot{c}(t)=\mathrm{u}(c(t))$. Due to the gauge freedom
of proposition 1, we can always assume $\phi$ to satisfy $\phi(c(t))=0$, so
that in $\mathrm{u}$-adapted coordinates, $c(t)=(t,x_{0})$ for a fixed
$x_{0}$. Moreover,
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\dot{c}(t)}\dot{c}(t)=\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\mathrm{u}(c(t))=\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\phi(c(t))$,
that is, $c(t)$ is a geodesic if, and only if,
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\phi(c(t))=0$.
This is the relativistic way to express that there is no net gravitational
force acting on this observer.

Another important geometric structure induced by $\mathrm{u}$ is the expansion
tensor [10], $\theta^{\mu}_{\nu}$, which is the symmetric component of the
spatial projection of
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}^{\mu}\mathrm{u}_{\nu}$.
Since its skew-symmetric counterpart, the vorticity, vanishes, it satisfies
$\theta_{\mu 0}=\theta_{0\mu}=0$, and for the spatial components,
$\theta^{i}_{k}=\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}^{i}\mathrm{u}_{k}$,

$\theta^{i}_{k}=\frac{e^{-\phi}}{2}\mathrm{h}^{i\ell}\frac{\partial}{\partial
t}\mathrm{h}_{\ell
k}=\mathrm{H}\,\left(\delta^{i}_{k}+\scalebox{0.9}[1.1]{$\mathtt{C}$}\scalebox{1.1}[1.0]{$\Sigma$}^{i}_{k}\right)\quad,\quad\scalebox{1.1}[1.0]{$\Sigma$}_{k}^{k}=0\,,$
(7)

with $\mathrm{H}$ the Hubble parameter and
$\scalebox{1.1}[1.0]{$\Sigma$}^{i}_{k}$ the Hubble anisotropy tensor,
measuring the mean expansion rate of the spatial length and the anisotropies
in it, respectively. The constant in the formula (7) appears in order to set
$\scalebox{1.1}[1.0]{$\Sigma$}=1$ as a turning point in the generalized
Friedmann equation (26), that is,

$\scalebox{0.9}[1.1]{$\mathtt{C}$}=\sqrt{(m-1)(m-2)}\quad,\quad
m=\dim\mathsf{M}\,.$ (8)

The reason to opt for the dimensionless Hubble anisotropy
$\scalebox{1.1}[1.0]{$\Sigma$}^{i}_{k}$ instead of the usual shear
$\scalebox{0.9}[1.1]{$\mathtt{C}$}\mathrm{H}\scalebox{1.1}[1.0]{$\Sigma$}^{i}_{k}$
is justified by its straightforward representation in the Kasner plane and
natural interpretation of their magnitudes as regular
($\scalebox{1.1}[1.0]{$\Sigma$}<1/2$), intermediate
($1/2<\scalebox{1.1}[1.0]{$\Sigma$}<1$) and extreme
($\scalebox{1.1}[1.0]{$\Sigma$}>1$) [17], where the Hubble anisotropy
magnitude is

$\scalebox{1.1}[1.0]{$\Sigma$}=\sqrt{\scalebox{1.1}[1.0]{$\Sigma$}_{i}^{k}\scalebox{1.1}[1.0]{$\Sigma$}_{k}^{i}}\,.$
(9)

At each point $p\in\mathsf{M}$, $\mathrm{H}(p)$ is the value of the expansion
rate averaged over all possible spatial directions $\hat{\bf n}$ at $p$,
$\mathrm{H}(p)=\langle\theta_{ik}(p)n^{i}n^{k}\rangle_{S^{m-2}}$, while
$\Sigma$ is proportional to the standard deviation of the Hubble anisotropy at
$p$,
$\sigma_{A}(p)^{2}=\left\langle\left(\theta_{ik}(p)n^{i}n^{k}\right)^{2}\right\rangle_{S^{m-2}}-\mathrm{H}(p)^{2}$.222If
we represent $\hat{\bf n}=n^{i}e_{i}(p)$ in an orthonormal basis of
$T_{p}\mathsf{S}_{p}$ diagonalizing
$\scalebox{1.1}[1.0]{$\Sigma$}^{i}_{k}(p)$, whose eigenvalues are
$\scalebox{1.1}[1.0]{$\Sigma$}_{1},\ldots,\scalebox{1.1}[1.0]{$\Sigma$}_{m-1}$,
we have $(n^{1})^{2}+\ldots+(n^{m-1})^{2}=1$,
$\scalebox{1.1}[1.0]{$\Sigma$}_{1}+\ldots+\scalebox{1.1}[1.0]{$\Sigma$}_{m-1}=0$,
and
$\scalebox{1.1}[1.0]{$\Sigma$}^{2}=(\scalebox{1.1}[1.0]{$\Sigma$}_{1})^{2}+\ldots+(\scalebox{1.1}[1.0]{$\Sigma$}_{m-1})^{2}$.
By the symmetries of the sphere, we also have
$\langle(n^{i})^{2}\rangle_{S^{m-2}}=1/(m-1)$,
$\langle(n^{i})^{2}(n^{k})^{2}\rangle_{S^{m-2}}$ equal to $A_{m}$, for $i=k$,
and $B_{m}$, for $i\neq k$. Applying these relations, we get
$\sigma_{A}^{2}/\mathrm{H}^{2}=\scalebox{0.9}[1.1]{$\mathtt{C}$}^{2}\left\langle\left(\scalebox{1.1}[1.0]{$\Sigma$}_{ik}n^{i}n^{k}\right)^{2}\right\rangle_{S^{m-2}}=(m-1)(m-2)(A_{m}-B_{m})\scalebox{1.1}[1.0]{$\Sigma$}^{2}$.
For $m=4$, $A_{4}=1/5$ and $B_{4}=1/15$. For instance,

$m=4\quad\Rightarrow\quad\scalebox{1.1}[1.0]{$\Sigma$}=\frac{\sqrt{5}}{2}\frac{\sigma_{A}}{|\mathrm{H}|}\,.$
(10)

The Hubble parameter has another well-known interpretation: the
$\mathrm{u}$-observer passing at $p=(t_{1},x_{1})$, which in the adapted
coordinates (6) is just the curve $c(t)=(t,x_{1})$, measures the infinitesimal
spatial volume $dV(t)$ at each instant of time, giving rise to the volumetric
scale factor at $p$, $\ell_{V}(t)$, with

$dV(t)=\ell_{V}(t)^{m-1}dV(t_{1})\quad\textrm{and}\quad\mathrm{H}(t,x_{1})=\frac{1}{\ell_{V}(t)}\frac{d\ell_{V}(t)}{d\tau_{1}}\,,$
(11)

where $d\tau_{1}=e^{\phi(t,x_{1})}dt$ is its proper-time. This means that
$\mathrm{H}$ measures the expansion rate of the universe with respect to the
volumetric scale factor. In a similar manner, if the scalar curvature of the
spatial sections does not vanish along the observer passing at $p$,
$\overline{\mathrm{R}}(t,x_{1})\neq 0$, we can define the curvature scale
factor and its ratio by, respectively,

$\overline{\mathrm{R}}(t,x_{1})=\frac{\overline{\mathrm{R}}(t_{1},x_{1})}{\ell_{K}(t)^{2}}\quad\textrm{and}\quad\mathrm{H}_{K}(t,x_{1})=\frac{1}{\ell_{K}(t)}\frac{d\ell_{K}(t)}{d\tau_{1}}\,.$
(12)

Our general intuition involving volume, area, and curvature would demand the
factors $\ell^{m-1}$, $\ell^{2}$, and $\ell^{-2}$, respectively, as we re-
define the length scale by a factor $\ell$. In that case, we would have
$\ell_{V}=\ell_{K}$. In General Relativity, things can get much more
complicated, in part due to the freedom of choice of the observers
$\mathrm{u}$, and consequently to the evolution of the induced spatial
geometry. Nonetheless, in the late-time universe, we could imagine that some
special observers would be free-falling in a "calm" region of the universe,
where our naive basic intuition on re-scaling parameters holds true. For this
reason, we define:

###### Definition 1

An $\mathrm{u}$-observer $c(t)$ is called fiducial if the following conditions
hold:

  1. 1.

It is free-falling, that is, $c(t)$ is a geodesic;

  2. 2.

If $\overline{\mathrm{R}}\neq 0$, then the curvature and volumetric scale
factors coincide along $c(t)$;

After setting our considerations on the Hubble parameter and its anisotropy,
we are ready to formulate what we mean by the second piece of the Cosmological
Principle:

###### Hypothesis 2

There is a fiducial $\mathrm{u}$-observer $c(t)$ to whom the Hubble parameter
is isotropic, that is, $\scalebox{1.1}[1.0]{$\Sigma$}(c(t))=0$.

The splitting of space and time imposed by the $\mathrm{u}$-observers
naturally and uniquely decomposes the energy-momentum tensor as [10]

$T_{\mu\nu}=\mathrm{\rho}\mathrm{u}_{\mu}\mathrm{u}_{\nu}+q_{\mu}\mathrm{u}_{\nu}+q_{\nu}\mathrm{u}_{\mu}+\scalebox{0.9}[1.0]{$\mathsf{P}$}\mathrm{h}_{\mu\nu}+\pi_{\mu\nu}\,,$
(13)

where $\mathrm{\rho}=T_{\mu\nu}\mathrm{u}^{\mu}\mathrm{u}^{\nu}$ is the energy
density,
$q_{\mu}=-\,\left(T_{\mu\nu}\mathrm{u}^{\nu}+\mathrm{\rho}\mathrm{u}_{\mu}\right)$
is the spatial vector field ($q_{\mu}\mathrm{u}^{\mu}=0$) representing the
energy flux,
$\scalebox{0.9}[1.0]{$\mathsf{P}$}=(T^{\mu}_{\mu}+\mathrm{\rho})/(m-1)$ is the
relativistic pressure, and $\pi_{\mu\nu}$ is the anisotropic stress, which is
symmetric, spatial ($\pi^{\mu}_{\nu}\mathrm{u}^{\nu}=0$), and traceless
($\pi^{\mu}_{\mu}=0$). We recall that this decomposition does not presuppose a
single fluid in our spacetime. In an environment with many fluids,
$T_{\mu\nu}=T_{\mu\nu}^{(a)}+T_{\mu\nu}^{(b)}+\ldots$ and the energy density
is $\mathrm{\rho}=\mathrm{\rho}^{(a)}+\mathrm{\rho}^{(b)}+\ldots$, where
$\mathrm{\rho}^{(a)}=T_{\mu\nu}^{(a)}\mathrm{u}^{\mu}\mathrm{u}^{\nu}$, and so
on. Hence, the physical parameters of the energy-momentum tensor described
above are effective in the sense that they sum all the contributions coming
from every component of matter and radiation present in our model.
Furthermore, $\mathrm{u}$ is not necessarily assumed to be co-moving with any
of them.

The last ingredient of the Cosmological Principle is the absence of
anisotropies in the energy and momentum as it is seen by the
$\mathrm{u}$-observers:

###### Hypothesis 3

The total energy-momentum tensor has a perfect fluid form
$T_{\mu\nu}=\mathrm{\rho}\mathrm{u}_{\mu}\mathrm{u}_{\nu}+\scalebox{0.9}[1.0]{$\mathsf{P}$}\mathrm{h}_{\mu\nu}$
in the referential frame of the $\mathrm{u}$-observers.

These three hypotheses put together with the Einstein’s equations give us the
Cosmological Principle (Theorem 3). The first of them, H1, states that there
are special observers in the universe to whom the spatial geometry is
intrinsically homogeneous and isotropic, a matter of mathematical aesthetics
and simplicity while establishing the framework where Cosmology is built onto.
The second, H2, has a stronger observational appeal, as it states that there
is a preferred observer to whom the Hubble parameter is isotropic in a region
where the curvature and the volume scale properly. The third, H3, concerns the
physical aspects of the universe by demanding matter and radiation to behave
as an effective perfect fluid. Each one is necessary to obtain the Friedmann-
Lemaître-Robertson-Walker (FLRW) spacetime, and therefore, to set the central
pillar of the standard model of Cosmology, as we will see in section 4. In
other words, they are the fundamental building blocks of the Cosmological
Principle.

##  3 Intrinsically homogeneous and isotropic spacetimes

###  3.1 Local coordinate representation

We begin by analyzing the local form of the intrinsically homogeneous and
isotropic spacetimes. We recall that among the complete and simply connected
Riemannian spaces of constant curvature, there are only three possibilities:
the Euclidean space $\mathbb{E}^{m-1}$, the Sphere $\mathbb{S}^{m-1}$ and the
Hyperbolic Space $\mathbb{H}^{m-1}$. Any other, as far as it is complete, is
the quotient of one of them with a discrete subgroup of isometries. Despite
the different topologies they can have, thus differing globally, locally they
are all isometric as far as they have the same curvature, $K$, putting them in
three distinct homothetic classes, depending on whether $K<0$, $K=0$ or $K>0$
[8, 9]. In our general relativistic context, each leaf of the foliation
defined along the hypothesis H1 behaves just as equal. The difference between
the theory of the spaces of constant curvature and our relativistic framework
is that we have to cope with the geometry varying with time. In order to avoid
topological problems appearing in the aforementioned global context, we assume
the curvature to keep its sign along $\mathsf{M}$, that is, $K=0$ throughout
$\mathsf{M}$ or $K(p)K(q)>0$ for any two points $p,q\in\mathsf{M}$. The result
is laid down in the following theorem:

###### Theorem 1

Let $(\mathsf{M},\mathrm{g})$ be an intrinsically homogeneous and isotropic
spacetime with respect to the set of observers described by the time-like and
vorticity-free vector field $\mathrm{u}$, $\mathrm{u}^{2}=-1$. For each
spatial section orthogonal to $\mathrm{u}$, denote its spatially constant
curvature by the time function $K$, which does not change sign along
$\mathsf{M}$. Hence, around any point $p\in\mathsf{M}$, there is a chart
$\varphi=(t,x)$ defining coordinates where the metric is represented as

$\mathrm{g}=-\,e^{2\phi(t,x)}\,dt^{2}+\frac{a(t)^{2}\gamma_{ik}(t)\,dx^{i}dx^{k}}{\left(\,1+\frac{K(t)a(t)^{2}}{4}\,\gamma_{ik}(t)(x^{i}-x^{i}_{0}(t))(x^{k}-x^{k}_{0}(t))\,\right)^{2}}\,\,,$
(14)

with $\det(\gamma_{ij}(t))=1$, $\varphi(p)=(t_{0},0)$, and
$x^{i}_{0}(t_{0})=0$. We call $\varphi$ a canonical coordinate system centered
at $p$.

###### Proof.

The case $K=0$ has been done in Ref.[18]. According to the hypothesis that $K$
does not change sign, we will assume that $K\neq 0$ for any point in
$\mathsf{M}$. Since $\mathrm{u}$ is vorticity-free, we can take an open
neighborhood $\widetilde{U}\subset\mathsf{M}$ of a given point $p_{0}$ in
$\mathsf{M}$ where there is an adapted coordinate system
$(\tilde{t},\tilde{x})$, as in formula (6), for which
$\widetilde{\mathrm{h}}_{ik}(\tilde{t},\tilde{x})$ has constant curvature
$K(\tilde{t})$. For each time $\tilde{t}$, since spaces of constant curvature
are conformally flat, we can define a function $\psi(\tilde{t},\tilde{x})$
such that
$\psi(\tilde{t},\tilde{x})^{2}\widetilde{\mathrm{h}}_{ik}(\tilde{t},\tilde{x})$
is a family of matrices with unitary determinant representing flat metrics
parametrized by $\tilde{t}$. In this case, $\psi(\tilde{t},\tilde{x})$ is
defined by
$\det(\widetilde{\mathrm{h}}_{ik}(\tilde{t},\tilde{x}))^{-1/2(m-1)}$, that is,
it is smooth. Hence, $\psi^{2}\mathrm{g}$ is a space-flat metric in
$\widetilde{U}$, implying that we can use the proposition 1 of Ref.[18] in
order to obtain a new coordinate system $\varphi=(t,x)$ around $p$, also
adapted to the observer $\mathrm{u}$, such that

$\mathrm{h}_{ik}(t,x)=\frac{1}{\psi(t,x)^{2}}\gamma_{ik}(t)\,.$ (15)

As we fix $t_{0}$ with $p$ in the spatial section $t=t_{0}$ and apply a well-
known theorem [8] for this Riemannian manifold with metric
$\mathrm{h}_{ij}(t_{0},x)$ of constant curvature $K_{0}=K(t_{0})$, we can
assume, without loss of generality, that the spatial coordinates $x^{i}$ are
centered at $p$ with

$\varphi(p)=(t_{0},0)\quad,\quad\gamma_{ik}(t_{0})=\delta_{ik}\quad,\quad\textrm{and}\quad\psi(t_{0},x)=1+(K_{0}/4)\delta_{ik}x^{i}x^{k}\,.$
(16)

Since the curvature tensor of $\mathrm{h}_{ik}(t,x)$, as a metric with
constant curvature, is

$\overline{\mathrm{R}}_{ijk\ell}=K(t)\psi(t,x)^{-4}\left(\gamma_{ik}(t)\gamma_{j\ell}(t)-\gamma_{i\ell}(t)\gamma_{jk}(t)\right)\,,$

a straightforward calculation of $\overline{\mathrm{R}}_{ijk\ell}$ using the
relation (15) and posterior comparison with the formula above lead us to

$\chi_{ik}\gamma_{j\ell}(t)+\chi_{j\ell}\gamma_{ik}(t)-\chi_{i\ell}\gamma_{jk}(t)-\chi_{jk}\gamma_{i\ell}(t)=0\,,$
(17)

where

$\chi_{ik}=\left(K(t)-\gamma^{j\ell}(t)\partial_{j}\psi\partial_{\ell}\psi\right)\gamma_{ik}(t)-2\psi\partial_{i}\partial_{k}\psi\,.$
(18)

If we first contract the above expression with $\mathrm{h}^{j\ell}(t)$, we get
$(m-3)\chi_{ik}=-\chi\,\gamma_{ik}(t)$ , $\chi=\chi_{j\ell}\gamma^{j\ell}(t)$,
and then with $\gamma^{ik}(t)$, we obtain that $\chi=0$, and therefore that
$\chi_{ik}=0$.

For each $t$, define $y^{k}=A^{k}_{i}(t)x^{i}$, where
$\gamma_{j\ell}(t)A^{j}_{i}(t)A^{\ell}_{k}(t)=\delta_{ik}$. The condition
$\chi_{ik}=0$ for $i\neq k$ implies $\partial^{2}\psi/\partial y^{i}\partial
y^{k}=0$, that is, $\psi=\psi_{1}(t,y^{1})+\ldots+\psi_{m-1}(t,y^{m-1})$.
Denoting the derivative with respect to $y^{k}$ by $\psi_{k}^{\prime}$, the
condition $\chi_{ik}=0$ implies

$\psi_{k}^{\prime\prime}(t,y^{k})=\frac{K(t)-[\psi_{1}^{\prime}(t,y^{1})]^{2}-\ldots-[\psi_{m-1}^{\prime}(t,y^{m-1})]^{2}}{2\left(\psi_{1}(t,y^{1})+\ldots+\psi_{m-1}(t,y^{m-1})\right)}=\frac{A(t)}{2}\,.$

This over-determined system can be solved only for

$\psi_{k}(t,y^{k})=\frac{A(t)}{4}(y^{k}-y^{k}_{0}(t))^{2}+b_{k}(t)\,,$

with $A(t),y^{k}_{0}(t),b_{k}(t)$ smooth functions and $A(t)B(t)=K(t)$ for
$B=\sum b_{k}$. This implies that
$\psi(t,y)=A(t)\delta_{ik}(y^{i}-y^{i}_{0}(t))(y^{k}-y^{k}_{0}(t))/4+B(t)$.
Hence, returning to the $x$-coordinates, we obtain

$\psi(t,x)=\frac{A(t)}{4}\gamma_{ik}(t)(x^{i}-x^{i}_{0}(t))(x^{k}-x^{k}_{0}(t))+B(t)\,.$

A simple comparison with the initial condition for $\psi$ in equation (16)
tells us that $A(t_{0})=K_{0}$, $x_{0}^{k}(t_{0})=0$ and $B(t_{0})=1$. Hence,
we set $a(t)=\det(\gamma_{ij}(t))^{1/2(m-1)}/B(t)$ and change
$\gamma_{ij}(t)=B(t)^{2}a(t)^{2}\widehat{\gamma}_{ij}(t)$, with
$\det(\widehat{\gamma}_{ij})=1$. As we ommit the hat in
$\widehat{\gamma}_{ij}$, we get the representation given in the formula (14),
thus proving the theorem. ∎

The formula (14) is our reference point to investigate the local properties of
the intrinsically homogeneous and isotropic spacetimes. Note that in those
coordinates, the Hubble parameter and anisotropy tensor of the
$\mathrm{u}$-observers look like as, respectively,

$\mathrm{H}=e^{-\phi}\left(\mathrm{H}_{S}-\mathrm{H}_{\psi}\right)\quad\textrm{and}\quad\scalebox{1.1}[1.0]{$\Sigma$}^{i}_{k}=\frac{\mathsf{S}^{i}_{k}}{\scalebox{0.9}[1.1]{$\mathtt{C}$}\left(\mathrm{H}_{S}-\mathrm{H}_{\psi}\right)}\,,$
(19)

where we have defined

$\mathrm{H}_{S}=\frac{1}{a}\frac{da}{dt}\quad,\quad\mathrm{H}_{\psi}=\frac{1}{\psi}\frac{\partial\psi}{\partial
t}\quad\textrm{and}\quad\mathsf{S}^{i}_{k}=\frac{1}{2}\gamma^{i\ell}\frac{d}{dt}\gamma_{\ell
k}\,,$ (20)

with $\mathsf{S}^{k}_{k}=0$, and

$\psi=1+\frac{K(t)a(t)^{2}}{4}\gamma_{ik}(t)(x^{i}-x^{i}_{0}(t))(x^{k}-x^{k}_{0}(t))\,.$
(21)

Hence, we get:

###### Corollary 1

An intrinsically homogeneous and isotropic spacetime is shear-free
($\scalebox{1.1}[1.0]{$\Sigma$}=0$) if, and only if, we can set
$\gamma_{ik}=\delta_{ik}$ in the local representation (14):

$\mathrm{g}=-\,e^{2\phi(t,x)}\,dt^{2}+\frac{a(t)^{2}\,\left((dx^{1})^{2}+\ldots+(dx^{m-1})^{2}\right)}{\left(\,1+\frac{K(t)a(t)^{2}}{4}\,\left((x^{1}-x^{1}_{0}(t))^{2}+\ldots+(x^{m-1}-x^{m-1}_{0}(t))^{2}\right)\,\right)^{2}}\,.$
(22)

###### Proof.

It is clear that the representation (22) follows from (14) if, and only if,
$S_{ik}\sim\dot{\gamma}_{ik}=0$. ∎

Formulas as (22) are usual to the so-called Stephani-Barnes spacetimes [19,
20, 21, 22, 23], that is, those Hubble-isotropic
($\scalebox{1.1}[1.0]{$\Sigma$}=0$), or shear-free, where H1 and H3 also hold.
In that case, the curve $x_{0}(t)$ is often called the “wandering center of
symmetry". We can eliminate it in the vicinity of the free-falling observers
to whom the Hubble parameter looks homogeneous in the first-order
approximation. Rigorously, we have:

###### Theorem 2

In an intrinsically homogeneous and isotropic spacetime where the curvature
does not change sign, around each free-falling $\mathrm{u}$-observer $c(t)$
for which the Hubble parameter is spatially homogeneous up to the first order,
that is,
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\mathrm{H}(c(t))=0$,
there is a coordinate system adapted to $\mathrm{u}$ where $c(t)=(t,0)$ and
the metric is

$\mathrm{g}=-\,e^{2\phi(t,x)}\,dt^{2}+\frac{a(t)^{2}\gamma_{ik}(t)\,dx^{i}dx^{k}}{\left(\,1+\frac{K(t)a(t)^{2}}{4}\,\gamma_{ik}(t)x^{i}x^{k}\,\right)^{2}}\,\,,$
(23)

with $\det(\gamma_{ij}(t))=1$. In particular, if it is a fiducial
$\mathrm{u}$-observer, then $K(t)=K_{0}/a(t)^{2}$ with $a(t)$ standing for
both curvature and volumetric scale factors along $c(t)$.

###### Proof.

This theorem is needless in the flat case, hence we assume $K\neq 0$. Let us
take canonical coordinates centered at $c(t_{0})=(t_{0},0)$ as in theorem 1
with $\phi(t,0)=0$, with no loss of generality. Since $c(t)$ is an integral
line of $\mathrm{u}$, $c(t)=(t,0)$. The condition of being free-falling
implies
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\dot{c}}\dot{c}=\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\mathrm{u}=\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\phi=0$
along $c(t)$, that is,
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\phi(t,0)=0$.
In order for the Hubble parameter to be spatially homogeneous up to the first
order, we must have

$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}_{i}\mathrm{H}(t,0)=-\partial_{i}\mathrm{H}_{\psi}(t,0)=\frac{\partial}{\partial
t}\left(\frac{Ka^{2}\gamma_{ik}x_{0}^{k}}{2\psi(t,0)}\right)=0\,,$ (24)

where we used
$\partial_{i}\mathrm{H}_{\psi}(t,0)=\partial_{t}(\partial_{i}\ln\psi)_{x=0}$.
Hence, $Ka^{2}\gamma_{ik}x_{0}^{k}=2\psi(t,0)A_{i}$, for some constants
$A_{i}$. However, since the coordinates are centered at $(t_{0},0)$,
$x^{i}_{0}(t_{0})=0$, that is, $A_{i}=0$ for every $i=1,\ldots,m-1$. This is
possible only if $x_{0}^{i}(t)=0$ for any $t$, which proves the first part of
the theorem. The rest follows from the fact that, if $c(t)$ is fiducial, then
the volume scale factor along it is $a(t)$ so that $K(t)=K_{0}/a(t)^{2}$. ∎

###  3.2 Einstein’s equations

The spatial sections of the intrinsic homogeneous and isotropic spacetimes
have constant sectional curvature, which means that the Riemann tensor, the
Ricci tensor, and the curvature scalar are, respectively

$\overline{\mathrm{R}}^{ik}_{j\ell}=K(\delta^{i}_{j}\delta^{k}_{\ell}-\delta^{i}_{\ell}\delta^{k}_{j})\quad,\quad\overline{\mathrm{R}}^{i}_{j}=(m-2)K\delta^{i}_{j}\quad,\textrm{and}\quad\overline{\mathrm{R}}=\scalebox{0.9}[1.1]{$\mathtt{C}$}^{2}K\,.$
(25)

Using those relations in Einstein’s equations
$R_{\mu}^{\nu}-R/2\delta_{\mu}^{\nu}=T_{\mu}^{\nu}$, where we have adopted the
convention of inserting the cosmological constant in the energy density and
relativistic pressure through the substitutions
$\mathrm{\rho}\to\mathrm{\rho}+\Lambda$ and
$\scalebox{0.9}[1.0]{$\mathsf{P}$}\to\scalebox{0.9}[1.0]{$\mathsf{P}$}-\Lambda$,
we obtain:

  1. (i)

Generalized Friedmann equation:

$\frac{1}{2}\scalebox{0.9}[1.1]{$\mathtt{C}$}^{2}\left((1-\scalebox{1.1}[1.0]{$\Sigma$}^{2})\mathrm{H}^{2}+K\right)=\rho$
(26)

  2. (ii)

Energy flux equation:

$q_{i}=(m-2)\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}_{i}\,\mathrm{H}-\scalebox{0.9}[1.1]{$\mathtt{C}$}\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}_{k}\left(\mathrm{H}\scalebox{1.1}[1.0]{$\Sigma$}^{k}_{i}\right)$
(27)

  3. (iii)

Raychaudhuri’s equation:

$\displaystyle
e^{-\phi}\,\dot{\mathrm{H}}+\left(\,1+(m-2)\scalebox{1.1}[1.0]{$\Sigma$}^{2}\right)\,\mathrm{H}^{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{\scalebox{0.9}[1.1]{$\mathtt{C}$}^{2}}\left((m-3)\mathrm{\rho}+(m-1)\scalebox{0.9}[1.0]{$\mathsf{P}$}\right)$
(28) $\displaystyle+$
$\displaystyle\frac{1}{m-1}\,e^{-\phi}\,\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}^{2}\,e^{\phi}$

  4. (iv)

Hubble anisotropy equation:

$e^{-\phi}\frac{\partial}{\partial
t}\left(\mathrm{H}\scalebox{1.1}[1.0]{$\Sigma$}^{k}_{i}\right)+(m-1)\mathrm{H}^{2}\scalebox{1.1}[1.0]{$\Sigma$}^{k}_{i}=\frac{1}{\scalebox{0.9}[1.1]{$\mathtt{C}$}}\left(\pi_{i}^{k}+\widehat{\Phi}_{i}^{k}\right)\,,$
(29)

where

$\widehat{\Phi}_{i}^{k}=e^{-\phi}\,\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}_{i}\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}^{\,k}e^{\phi}-\frac{1}{m-1}\,e^{-\phi}\,\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}^{2}e^{\phi}\,\delta_{i}^{k}\,.$
(30)

Note that $\widehat{\Phi}_{i}^{k}$ resembles the Newtonian tidal tensor in the
weak potential regime $\phi<<1$. For $m=4$, as we make the substitutions
$T_{\mu\nu}\to 8\pi GT_{\mu\nu}$, $\mathrm{H}\to\theta/3$,
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}_{i}\phi=\dot{\mathrm{u}}_{i}$,
and $\scalebox{1.1}[1.0]{$\Sigma$}^{k}_{i}\to 3\sigma^{k}_{i}/\sqrt{6}\theta$,
with the notations $\sigma^{2}=(\sigma^{k}_{i}\sigma^{i}_{k})/2$ and
“$\dot{x}=\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}x$",
we can verify that
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mu}\mathrm{u}^{\mu}=e^{-\phi}\,\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}^{2}\,e^{\phi}$,
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{\mathrm{u}}\sigma^{k}_{i}=e^{-\phi}\partial\sigma^{k}_{i}/\partial
t$, and the above equations are the vorticity-free ($\omega^{\nu}=0$)
Einstein’s equations in Ref. [10], as in the case of eq. (6.23), for the
generalized Friedmann, eq. (6.20), for the energy flux, eq. (6.4), for
Raychaudhuri’s, and eq. (6.25), for the Hubble anisotropy equation.

The intrinsically homogeneous and isotropic spacetimes have been considered
earlier in the literature, but not in its most general aspects. There are a
lot of quantitative and qualitative results concerning the homogeneous Bianchi
models [24, 25], which encompasses our framework of the free-falling situation
($\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\phi=0$).
When $K=0$, it is equivalent to the general Bianchi I class, and when $K\neq
0$, it means we are in special cases of Bianchi V ($K<0$) or Bianchi IX
($K>0$). In the inhomogeneous context
($\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\phi\neq
0$), Einstein’s equations for $K=0$ seem to have been most studied, and a
series of exact solutions in different contexts can be found [26, 27, 28, 17,
18]. For the inhomogeneous but isotropic ($\scalebox{1.1}[1.0]{$\Sigma$}=0$)
case with general curvature, see Chapter 4 in Ref. [23] and the references
therein.

##  4 The three pieces of the the Cosmological Principle

So far, we have described the three pieces composing the Cosmological
Principle. As we put them together two-by-two without the third party, the
spacetime cannot be FLRW, even when we admit Einstein’s equations to hold. In
fact, we have:

  * •

H2 \+ H3 \+ Einstein’s Equations do not imply H1: In any generalized
Robertson-Walker spacetime [29], obtained by putting $\phi=0$ and
$\mathrm{h}_{ik}=a(t)^{2}\gamma_{ik}(x)$ in formula (6) for the metric
$\mathrm{g}$, every $\mathrm{u}$-observer is fiducial and shear-free
($\scalebox{1.1}[1.0]{$\Sigma$}=0$). The Hubble parameter is homogeneous
($\mathrm{H}=\mathrm{H}(t)$) and $\mathrm{g}$ is a perfect fluid solution of
Einstein’s equations. However, except when $\gamma$ has constant curvature, it
does not satisfy H1.

  * •

H1 \+ H3 \+ Einstein’s Equations do not imply H2: Any perfect fluid Bianchi I
solution of Einstein’s equations with $\scalebox{1.1}[1.0]{$\Sigma$}\neq 0$
can be used as example (see the general barotropic solution in section 4.1 of
Ref. [30], for instance).

  * •

H1 \+ H2 +Einstein’s Equations do not imply H3: for the metric as in corollary
1 with $K=0$, any free-falling $\mathrm{u}$-observer is fiducial with
$\scalebox{1.1}[1.0]{$\Sigma$}=0$. In Ref. [17] there is such a solution for
an imperfect fluid.

However, when we assume all those hypotheses and Einstein’s equations to hold,
we recover the mathematical formulation of the Cosmological Principle:

###### Theorem 3

An intrinsically homogeneous and isotropic spacetime filled with a perfect
fluid where a fiducial observer detects an isotropic Hubble parameter is FLRW
around it provided the curvature does not change sign. In this sense,

$H\ref{Hyp:IsotropyCurvature}+H\ref{Hyp:IsotropyHubble}+H\ref{Hyp:IsotropyEMTensor}+\textrm{Einstein's
equations}\qquad\Leftrightarrow\qquad\textrm{FLRW.}$ (31)

If one of the hypotheses H1, H2, or H3 is lacking, the implication cannot hold
true.

###### Proof.

We use canonical coordinates, as in theorem 1, centered at any point of the
fiducial $\mathrm{u}$-observer of hypothesis H2. According to formula (19),
$\scalebox{1.1}[1.0]{$\Sigma$}=0$ for at least one observer means that
$\dot{\gamma}_{ik}=0$, that is, $\gamma_{ik}(t)=\delta_{ik}$, whith no loss of
generality. In particular, $\scalebox{1.1}[1.0]{$\Sigma$}=0$ everywhere. Since
the fluid is perfect, the energy flux equation (27) implies
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\mathrm{H}=0$.
This condition and the hypothesis H2 allow us to apply theorem 2, and
therefore obtain $x^{i}_{0}(t)=0$ and $K=K_{0}/a^{2}$. In particular, we have
$\mathrm{H}_{\psi}=0$ and
$(\dot{a}/{a})e^{\phi}\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\phi=\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\mathrm{H}=0$,
that is, $\phi=0$ with no loss of generality. In short, we have canonical
coordinates as in theorem 1, centered at the fiducial $\mathrm{u}$-observer of
hypothesis H2, with $\phi=0$, $x^{i}_{0}(t)=0$, $K=K_{0}/a^{2}$, and
$\gamma_{ik}(t)=\delta_{ik}$. In other words, a FLRW spacetime. ∎

The characterization of FLRW spacetimes is an old topic in the literature. The
most famous result is the seminal Ehlers-Geren-Sachs theorem [31], stating
that in regions where an isotropic CMB is the unique source of energy-
momentum, the spacetime must be FLRW. Another very interesting
characterization relies upon the redshift against the angular diameter
distance up to third order [32], which has a beautiful observational appeal.
In Geometry, this kind of problem is often referred to as rigidity conditions
for the FLRW spacetimes (see [33] and references therein).

The Stephani-Barnes universes [19, 20, 21, 22, 23] are "almost FLRW"
spacetimes, characterized by

$\textrm{Stephani-
Barnes}=\left\\{H\ref{Hyp:IsotropyCurvature}+(\scalebox{1.1}[1.0]{$\Sigma$}=0)+H\ref{Hyp:IsotropyEMTensor}+\textrm{Einstein's
equations}\right\\}\,.$ (32)

Here we have unveiled the two subtle characteristics differentiating both,
which are the existence of a free-falling $\mathrm{u}$-observer and the fact
that along it the curvature and the volume can be properly scaled. According
to the fact that
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\mathrm{H}=0$
in such environments and using theorem 2, these mean that the "wandering
center of symmetry" can be settled as $x^{i}_{0}(t)=0$ and that $K\sim
a^{-2}$, respectively.

##  5 Naturalness for homogeneity on large scales: the periodic boundary
conditions

###  5.1 Periodic boundary conditions

One distinguished property of the intrinsically homogeneous and isotropic
spacetimes is the natural geometric structure they have for building effective
homogeneous models. In fact, we can transcribe in our spacetime the concept
that the universe, after taking averages on a large scale, is described by an
effective model that behaves like an homogeneous one. In such a framework, we
can smooth the problems arising in the process of “averaging" in Cosmology
[34]. This idea of space being divided into a lattice formed by identical
regions, with matter behaving equally in each one of its cells, and therefore
giving rise to a large-scale homogeneous structure, has appeared in the
literature since the 1950s, after the seminal work of Lindquist and Wheeler
[35], and continued in lines like “Archipelagian Cosmology" [36, 37, 38],
“black hole lattice spacetimes" [39], and others[40, 41, 42, 43, 44]. Here we
extend the ideas first developed in Ref. [18].

The notion of tilling space into identical cells is natural to spaces of
constant curvature, which can be accomplished in a quite straightforward way.
Hence, let us assume that each one of our spatial sections, identified with
$\mathsf{S}$, is one of the following geometric forms: the Euclidean space
$\mathbb{R}^{m-1}$, for $K_{0}=0$, the hyperbolic space $\mathbb{H}^{m-1}$,
for $K_{0}<0$, or the sphere $\mathbb{S}^{m-1}$, for $K_{0}>0$. Take a
discrete subgroup $\Gamma$ of the group of the isometries of the space
$\mathsf{S}$ of constant curvature $K_{0}$, and assume its action on
$\mathsf{S}$ is free and properly discontinuous so that the quotient
$\mathsf{K}=\mathsf{S}/\Gamma$ is a complete manifold of constant curvature
$K_{0}$ (corollary 2.3.17 in [8]), which we denote by $\mathsf{K}$. For our
purposes here, we shall also assume that $\mathsf{K}$ is compact. Hence, we
have the following picture: the space $\mathsf{S}$ is divided into identical
regions, called fundamental domains of $\Gamma$ [9], so that any
$\Gamma$-periodic object, like a function invariant by $\Gamma$, for example,
turns out to be equivalent to the corresponding object in $\mathsf{K}$.

Let us translate this scheme to Cosmology by assuming our spacetime to be
composed of maximally symmetric spatial sections, that is, one of the standard
models above of constant curvature $K_{0}$. At large scales, the universe is
quite homogeneous. That means patterns of galaxy distribution and other
physical parameters are distributed virtually in an equal manner along
identical regions of typical length $L>>L_{0}$, with $L_{0}$ of the order of
$100$ Mpc or greater (see Ref. [4] for an up-to-date discussion). These are
the cosmological cells, which we identify with the fundamental domains of
$\Gamma$, as described above. Therefore, the “Cosmological Principle in
practice" tells us that the inhomogeneities are $\Gamma$-periodically
distributed throughout space such that dealing with these local irregularities
in each cosmological cell turns out to be equivalent as if we were in the
compact manifold $\mathsf{K}$. In order to illustrate it, consider the
following two cases:

  * •

If $K_{0}=0$ and $m=4$, we could proceed as in Ref.[18] and take our three-
dimensional space $\mathbb{R}^{3}$ to be divided into an infinite number of
cubic boxes of length $L_{0}$, our cosmological cells. In this case, the
discrete subgroup of symmetry is formed by the translations
$(x,y,z)\mapsto(x+n_{x}L_{0},y+n_{y}L_{0},z+n_{z}L_{0})$,
$n_{x},n_{y},n_{z}\in\mathbb{Z}$. Hence, the periodicity of galaxy
distribution along those boxes would reduce the mathematical approach to the
inhomogeneities to the three-dimensional torus
$\mathbb{T}^{3}=\mathbb{R}^{3}/\Gamma$. In particular, this periodic boundary
condition would demand Einstein’s equation to be formulated in the torus
$\mathbb{T}^{3}$, the compact manifold that we could identify with any one of
the cosmological cells.

  * •

If $K_{0}=-1$, we could first set $m=3$, for the sake of simplicity. Hence our
spatial sections would be copies of the hyperbolic plane $H^{2}$, and the
discrete subgroup of symmetries $\Gamma$ preserving orientation, called a
Fuchsian group, would lead us to the $g$-connected sum
$\mathsf{K}=H^{2}/\Gamma=\mathbb{T}^{2}\\#\ldots\\#\mathbb{T}^{2}$, classified
by the genus $g\geq 2$. For instance, if $g=2$, we would formulate our
equations in the bi-torus $\mathbb{T}^{2}\\#\mathbb{T}^{2}$, which could also
be identified with any one of the cosmological cells. In this case, the
galactic distribution would resemble one of Escher’s “Circle Limit"
masterpieces. For the dimension of interest in Cosmology, $m=4$, the richness
of the hyperbolic geometry allows us to decompose the space $H^{3}$ in an
immense amount of possibilities.

The preceding argumentation allows us to formulate the periodic boundary
condition by demanding the spatial functions as $\mathrm{H}(t,*)$,
$\mathrm{\rho}(t,*)$, and $\phi(t,*)$ to be $\Gamma$-periodic for all possible
time $t$, that is, they are such that $f(t,g\cdot x)=f(t,x)$ for all
$x\in\mathsf{S}$, as "$\cdot$" denotes the action of $\Gamma$ on $\mathsf{S}$
(See [18] for examples with $K_{0}=0$). Formally, we have:

######  Definition 2 (Periodic Boundary Condition)

The spacetime is $\mathsf{M}=\mathsf{I}\times\mathsf{S}$,
$\mathsf{I}\subset\mathbb{R}$ an open interval, and the restriction of its
metric $\mathrm{g}$ to the space section $\\{t\\}\times\mathsf{S}$ is
$a(t)^{2}\gamma(t)$, where $(\mathsf{S},\gamma(t))$ is a maximally symmetric
Riemannian space of constant curvature $K=K(t)$. Furthermore, for each
$t\in\mathsf{I}$, there is a subgroup $\Gamma(t)$ of the isometries of the
space $(\mathsf{S},\gamma(t))$ such that the quotient $\mathsf{S}/\Gamma(t)$
endowed with the inherited Riemannian metric induces an isometry onto a
compact Riemannian manifold $\mathsf{K}$ with constant curvature $K=K(t)$ and
fixed volume $L_{0}^{m-1}$. Any function or spatial tensor field involved in
the dynamics of Einstein’s equations is $\Gamma(t)$-periodic in the sense that
they are well defined in the quotient $\mathsf{S}/\Gamma(t)$.

Following the periodic condition above, we shall often identify any
cosmological cell of the tilling imposed by $\Gamma(t)$ at each time $t$ with
the compact manifold $\mathsf{K}\cong\mathsf{S}/\Gamma(t)$. $\mathsf{K}$ does
not depend on time, even though $\Gamma(t)$ does. For instance, in the case
$m=2$ and $K=0$, the only compact and orientable possibility for the manifold
$\mathsf{K}$ is the torus $\mathbb{T}^{2}$, while $\Gamma(t)$ could be, for
instance, the time-dependent subgroup of translations in the Euclidean plane
given by $(x,y)\mapsto(x+e^{t}L_{0}n,y+e^{-3t}L_{0}m)$, with
$n,m\in\mathbb{Z}$. In this case, we identify any rectangle defined by $0\leq
x<e^{t}L_{0}$ and $0\leq y<e^{-3t}L_{0}$ and their $\Gamma(t)$-translations as
a cosmological cell, with
$\mathsf{K}=\mathbb{T}^{2}\cong\mathbb{R}^{2}/\Gamma(t)$, and $\gamma(t)$ as a
flat metric on the torus given by
$\gamma_{11}(t)dx^{2}+\gamma_{12}(t)dxdy+\gamma_{22}(t)dy^{2}$ for which the
canonical infinitesimal area,
$\sqrt{\gamma_{11}\gamma_{22}-\gamma_{12}^{2}}dxdy$, turns out to be
$e^{2t}dxdy$, such that $\mathbb{T}^{2}$ has a constant area $L_{0}^{2}$.

###  5.2 Naturalness for the averaging process

The separation of the spatial metric into a scale factor $a(t)$ and a
Riemannian metric $\gamma(t)$, that keeps the constant volume $L_{0}^{m-1}$
for $\mathsf{K}$, is strategic for the cosmological applications. Note that
following the canonical local coordinates of theorem 1 we can put
$\gamma(t)=\gamma_{ik}(t)dx^{i}dx^{k}/\psi(t,x)^{2}$, with $\psi$ given in
equation (21), only if $K(t)=K_{0}/a(t)^{2}$. In fact, if we take two instants
$t_{0}$ and $t_{1}$, with $\gamma_{ik}(t_{0})=\delta_{ik}$ and
$x_{0}^{i}(t_{0})=0$, with no loss of generality, and define the affine
oriented coordinates change $x^{i}=x_{0}^{i}(t_{1})+a^{i}_{k}y^{k}$, with
$\gamma_{ik}(t_{1})a_{j}^{i}a_{\ell}^{k}=\delta_{j\ell}$, since
$\det(\gamma_{ij})=1$, we get $\det(a^{k}_{i})=1$ as well, that is,
$dx^{1}\wedge\ldots\wedge dx^{m-1}=dy^{1}\wedge\ldots\wedge dy^{m-1}$.
Therefore, the volume of $\mathsf{K}$ at the instant $t_{1}$ is

$\frac{V(t_{1})}{a(t_{1})^{m-1}}=\int_{\mathsf{K}}\frac{d^{m-1}x}{\psi(t_{1},x)^{m-1}}=\int_{\mathsf{K}}\frac{d^{m-1}y}{\left(1+\frac{K_{0}}{4}\delta_{ik}y^{i}y^{k}\right)^{m-1}}=\frac{V(t_{0})}{a(t_{0})^{m-1}}\,.$
(33)

Choosing an instant $t_{0}$ with $a(t_{0})=1$ and $V(t_{0})=L_{0}^{m-1}$, we
obtain

$K(t)=\frac{K_{0}}{a(t)^{2}}\qquad\Rightarrow\qquad
V(t)=a(t)^{m-1}\,L_{0}^{m-1}\,.$ (34)

In particular, this occurs if there is a fiducial $\mathrm{u}$-observer
perceiving a Hubble parameter homogeneous to the first order, as in theorem 2.
Moreover, the average value of any $\Gamma(t)$-periodic function does not
depend on $a(t)$, for

$\langle
F\rangle(t)=\frac{1}{L_{0}^{m-1}}\int_{\mathsf{K}}F(t,x)\frac{d^{m-1}x}{\psi(t,x)^{m-1}}\,.$
(35)

Using the definitions in equation (20), and always denoting
$\langle\cdot\rangle$ for the mean value along the compact manifold
$\mathsf{K}$, we have

$\frac{d}{dt}\langle F\rangle=\langle\frac{\partial F}{\partial
t}\rangle-(m-1)\langle F\,\mathrm{H}_{\psi}\rangle\,.$ (36)

In the resume, we have a natural geometric structure for the averaging process
induced by the discrete subgroup of symmetries of the periodic condition, such
that the problem of non-commutativity of the time derivatives is smoothed out,
and even disappears if $\mathrm{H}_{\psi}=0$.

##  6 Decoupling the geometric and the CMB frames: the need for intrinsically
homogeneous and isotropic spacetimes

Let us now identify where the $\mathrm{u}$-observers are, the ones to whom the
spatial geometry is homogeneous and isotropic. We call it the geometric frame
of reference. Besides that, we admit the existence of free-falling, vorticity-
free, homogeneous, and isotropic background radiation whose co-moving frame is
defined by the time-like vector field $\tilde{\mathrm{u}}$, with
$v=\tilde{\mathrm{u}}-\mathrm{u}$ the spatial velocity relative to the
$\mathrm{u}$-observers. The CMB in dimension $m=4$ is given by a perfect fluid
energy-momentum tensor with respect to the $\tilde{\mathrm{u}}$-observers
whose pressure has three times the magnitude of the spatially homogeneous
energy density $\widetilde{\mathrm{\rho}}_{r}$, that is,

$T^{CMB}_{\mu\nu}=\frac{\widetilde{\mathrm{\rho}}_{r}}{3}\left(4\,\tilde{\mathrm{u}}_{\mu}\tilde{\mathrm{u}}_{\nu}+\mathrm{g}_{\mu\nu}\right)\quad,\quad\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\nabla}$}}_{X}\widetilde{\mathrm{\rho}}_{r}=0\quad\textrm{whenever}\quad\mathrm{g}(X,\tilde{\mathrm{u}})=0\,.$
(37)

From the $\mathrm{u}$ viewpoint, this gives an energy-momentum tensor with
energy density $\mathrm{\rho}_{r}=\widetilde{\mathrm{\rho}}_{r}$, pressure
$\scalebox{0.9}[1.0]{$\mathsf{P}$}_{r}=(1+v^{2})\mathrm{\rho}_{r}/3$, energy
flux $q_{r}^{i}=\mathrm{\rho}_{r}v^{i}$ and stress $\pi_{r}^{ik}$ given by the
trace-less part of $\mathrm{\rho}_{r}v^{i}v^{k}$. Note that at any point
$p\in\mathsf{M}$ where $v=0$ these different descriptions of the CMB coincide.

The fact that the CMB is free-falling means that the unique integral curve of
$\tilde{\mathrm{u}}$ passing at $p$ in the instant $t_{0}$, that is, the curve
$c(\tau)\in\mathsf{M}$ such that $c(\tau(t_{0}))=p$ and
$x^{\prime}(\tau)=\tilde{\mathrm{u}}(c(\tau))$, is in fact a geodesic. As long
as we allow the relative velocity of the CMB frame to vanish along $c(\tau)$,
$v(c(\tau))=0$, we realize that $\tau=t$ and $c(t)$ is a free-faling
$\mathrm{u}$-observer. In particular,
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\phi(c(t))=0$
for every $t$. In other words, $c(t)$ is a special observer to whom the CMB
and constant curvature frames coincide if, and only if, it is placed exactly
where the net "gravitational force" vanishes, as in the case of the fiducial
observers of definition 1.

Due to the considerations above, if we assume that our geometric reference
frame does not coincide with the CMB’s, except for some fiducial observers
symmetrically distributed along the geometric frame, our spacetime is no
longer FLRW, but instead, an intrinsically homogeneous and isotropic one. The
discrete subgroup of symmetries defining the periodic condition of section 5
would be characterized by the disposition of these fiducial
$\mathrm{u}$-observers along space, at least in principle. Our current
measurements based on the dipole anisotropy of the CMB can tell us that our
peculiar velocity relative to these special observers is something near $300$
km/s in a specific direction [45], but cannot distinguish this picture from
the canonical FLRW counterpart. Therefore, probing the validity of the
hypothesis stating that the CMB and geometric frames do not coincide
everywhere could be achieved only through indirect means. For this reason, we
must build effective cosmological models over the framework of intrinsically
homogeneous and isotropic spacetimes for later scrutinizing their
observational validity. In the next section, we comment on how we could start
pushing forward this procedure.

##  7 Towards an effective homogeneous model from 1/3 of the Cosmological
Principle

Among the intrinsically homogeneous and isotropic spacetimes, those satisfying
the periodic boundary condition of section 5 have a natural structure to build
average values of the physical quantities. In particular, we can mount
effective cosmological models similar to the standard $\Lambda$CDM one, but
still quite different due to the characteristics of the geometric background.
However, there is a warning in advance to the reader: our intention here is to
furnish a glimpse of the potential such models can have in Cosmology. A
complete development of such a scheme will appear soon elsewhere.

We start by setting the dimension to be $m=4$ and the scale of homogeneity as
$L_{0}$. We assume the existence of a set of fiducial observers symmetrically
spread in space based on whom we define the geometric frame and the discrete
subgroup of symmetries, $\Gamma(t)$. Hence, they define the periodic boundary
condition set in section 5, and thus the typical cosmological cell denoted by
$\mathsf{K}$ and identified with a compact manifold endowed with a time-
dependent family of metrics $\mathrm{h}(t)=a(t)^{2}\gamma(t)$. In canonical
coordinates, they are represented as in theorem 2, with constant curvature
$K(t)=K_{0}/a(t)^{2}$ such that $\mathsf{K}$ has volume $a(t)^{3}L_{0}^{3}$.
We define the mean value of any continuous function as in the formula (35). We
calibrate the $\mathrm{u}$-observers’ clocks to show the same time $t_{0}$ for
"today", that is, redshift $z=0$, when we set $a(t_{0})=1$. Our fundamental
assumption is that those fiducial observers are simultaneously at rest in the
CMB and geometric frames.

The energy density is split into two parts. The first one emulates the
homogeneous $\Lambda$CDM model. For this, we borrow the notation from the
standard picture and denote the critical density by $\mathrm{\rho}_{c}$, which
we are going to use as a reference for the energy density magnitude. As usual,
we define the density parameters for the dark energy, radiation, and matter
by, respectively, [2]

$\mathrm{\rho}_{c}=3\langle\mathrm{H}\rangle^{2}(t_{0})\quad,\quad\Omega_{\Lambda}=\frac{\Lambda}{\mathrm{\rho}_{c}}\quad,\quad\Omega_{r}=\frac{\langle\mathrm{\rho}_{r}\rangle(t_{0})}{\mathrm{\rho}_{c}}\quad,\quad\Omega_{m}=\frac{\langle\mathrm{\rho}_{m}\rangle(t_{0})}{\mathrm{\rho}_{c}}\,,$
(38)

where $\Lambda$ is the cosmological constant, $\mathrm{\rho}_{r}(t,x)$ is the
energy density due to radiation, and $\mathrm{\rho}_{m}(t,x)$ the part coming
from baryonic and dark matter inside the cosmological cells. The first one is
composed mainly of free-falling background radiation whose frame is moving
with the spatial velocity $v$ described in section 6. The total energy density
becomes

$\frac{\mathrm{\rho}(t,x)}{\mathrm{\rho}_{c}}=\Omega_{\Lambda}+\frac{\Omega_{m}}{a^{3}}+\frac{\Omega_{r}}{a^{4}}+\mathrm{\mu}(t,x)\,.$
(39)

The last term in the equation above is the internal energy density (in units
of $\mathrm{\rho}_{c}$), which should be understood as the contribution of
anything that could make the energy density depart from the $\Lambda$CDM
proposal. Its physical interpretation would vary with the background
assumptions on the matter/radiation content, which could stand for things as
the internal motion and interactions within each cosmological cell, the energy
of some fields inside them, corrections due to the fact that the geometric and
CMB’s frames are not the same, and so on. In general, the real aspect of it
will only show up when we further investigate Einstein’s equations in a
desired given situation. The density parameter associated with it is defined
as

$\widehat{\Omega}_{\mu}(a)=\langle\mathrm{\mu}\rangle\quad,\quad\widehat{\Omega}_{\mu}(1):=\Omega_{\mu}\,.$
(40)

Its dependence on the scale factor, interpretation, and relevance to the
cosmological picture must be analyzed case by case, which should vary with
different configurations for the energy-momentum tensor.

The inhomogeneities inside each cosmological cell $\mathsf{K}$ also contribute
with a density parameter, the one representing the standard deviation of the
true Hubble parameter $\mathrm{H}$ from its homogeneous substitute
$\langle\mathrm{H}\rangle$. Formally, it is defined as

$\widehat{\Omega}_{I}(a)\,\mathrm{\rho}_{c}=\langle\mathrm{H}^{2}\rangle-\langle\mathrm{H}\rangle^{2}=\langle\left(\mathrm{H}-\langle\mathrm{H}\rangle\right)^{2}\rangle\quad,\quad\widehat{\Omega}_{I}(1):=\Omega_{I}\,.$
(41)

Hence, $\widehat{\Omega}_{I}(a(t))=0$ if, and only if, the Hubble parameter is
homogeneous at time $t$, that is,
$\mathrm{H}(t,\cdot)=\langle\mathrm{H}\rangle(t)$. It is important to note
here that, while the cosmic expansion occurs apparently in a homogeneous way
driven by $a(t)$, its perception through the $\mathrm{u}$-observers must take
into account their different proper times, $d\tau=e^{\phi}dt=da/\mathrm{H}$,
since the geometric frame is not free-falling. Hence,
$\widehat{\Omega}_{I}(a)$ would also be a measure of the dispersion of the
$\mathrm{u}$-proper times along space.

The average value of the standard deviation of the anisotropy along the
cosmological cells gives us another density parameter, which is defined as

$\widehat{\Omega}_{A}(a)=\frac{\langle\sigma_{A}^{2}\rangle}{\mathrm{\rho}_{c}}\quad,\quad\widehat{\Omega}_{A}(1):=\Omega_{A}\,,$
(42)

where $\sigma_{A}$ has been introduced in equation (10). Its magnitude has
both local and global contributions. The first one would come from the
irregular distribution of matter inside each cosmological cell, even for
vacuum, since the Newtonian-like tidal term
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}^{i}\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}_{k}\phi$
would behave as a source for the anisotropy equation (29). On the other hand,
the global counterpart would be characteristic of a large-scale effect. In
that case, it seems reasonable to expect those contributions to be quite
similar to their counterpart in the homogeneous Bianchi models. It is common
to attribute to this Hubble anisotropy the behavior $\Omega_{A}/a^{6}$, which
is characteristic of a flat and perfect fluid Bianchi I universe [46].
Notwithstanding, the most expected situation is quite different, for the FLRW
universes are in general unstable equilibrium points in the Kasner disc [47],
and there is a plethora of possibilities for the asymptotic behavior even in
the simplest case of vanishing curvature [48, 49, 50, 51, 17, 52].
Furthermore, kind of behaviors such as the BKL picture of the early universe
are not excluded here, at least not a priori, for they would be a possibility
in the positive curvature case [53, 54, 55].

As we put all the contributions from the internal energy density,
inhomogeneities, and anisotropies, we recover an average version of the
Friedmann equation (26), which turns out to be

$3\frac{\langle\mathrm{H}\rangle^{2}}{\mathrm{\rho}_{c}}=\Omega_{\Lambda}+\frac{\Omega_{r}}{a^{4}}+\frac{\Omega_{m}}{a^{3}}+\frac{\Omega_{K}}{a^{2}}+\widehat{\Omega}_{\mu}(a)+\frac{15}{4}\widehat{\Omega}_{A}(a)-3\widehat{\Omega}_{I}(a)\,,$
(43)

where $\Omega_{K}=-\,3K_{0}/\mathrm{\rho}_{c}$ is the density parameter for
the curvature, as usual. Clearly, depending on the values of the constants
$\Omega_{\mu}$, $\Omega_{I}$, and $\Omega_{A}$, a new interpretation of the
constituents of the Universe might emerge. In particular, $\mathrm{\rho}_{c}$
is no longer a “critical" value that determines the sign of the curvature, for

$\langle\mathrm{\rho}\rangle(t_{0})-\mathrm{\rho}_{c}=3\Omega_{I}-\frac{15}{4}\Omega_{A}-\Omega_{K}\,.$
(44)

The $\Lambda$CDM model is recovered as we set
$\Omega_{\mu}\approx\Omega_{A}\approx\Omega_{I}\approx 0$. Nonetheless, the
vanishing of these new constants does not imply the background geometry is
FLRW. Instead, it imposes some constraints on the average values of these
inhomogeneous parameters. That is, even the $\Lambda$CDM model might surge
from an inhomogeneous background in this approach.

As we remarked before, we shall not extend the development of this effective
averaged model, for it requires much more effort and would take us out of the
scope of this manuscript. However, the arguments put above show us that it is
possible to have an environment where a $\Lambda$CDM-like model naturally
coexists with non-linear inhomogeneities and anisotropies, and therefore turns
out to be an interesting arena to investigate how one affects the others in
different phases of the universe. Hence, it is natural to ask here whether a
highly isotropic CMB would imply $\Omega_{A}\approx 0$ or how intense would be
the natural Hubble tension of these models. We finish by commenting on these
two topics below.

Since the CMB radiation seems to be very isotropic as it is seen by the
fiducial observers to whom the geometric and CMB frames coincide, as mentioned
in section 6, there is a highly spread common sense that this fact must imply
$\Omega_{A}$ to vanish, and observational analysis very often take this
implication for granted [56]. Furthermore, the "cosmic no-hair" theorem [57],
which excludes the positive curvature situation, states that a positive
cosmological constant would exponentially kill any anisotropy in the universe.
However, recent probes suggest that the Hubble sky might not be isotropic on
cosmic scales [5, 6, 4] and the no-hair argument does not necessarily imply a
collapse of the Hubble anisotropy in the sky we observe today, for the time
required for that mechanism to start is greater than the age of our universe
(see sec. 6 in Ref.[16]). In order to show how we can shed some light on this
conundrum, let us permit ourselves to interpret $\phi$ as a Newtonian
gravitational potential so that CMB/fiducial $\mathrm{u}$-observers are in
preferred locations of the late-time universe where the net gravitational
force on them vanishes, as for instance, in the center of mass of their local
galactic cluster. In this context, it is not clear the constraints a small
anisotropy of the CMB would impinge in the tidal tensor $\Phi^{i}_{k}$ nor in
the Hubble anisotropy, neither it seems clear the dynamical behavior they
would develop through the equation (29). Even if  $\Sigma$ is small, the
simple fact that they could have different orders of magnitude would leave
observable marks in the deceleration parameter [58]. Therefore, our background
spacetime provides a natural theoretical arena for further investigations on
the behavior of those anisotropies and how they would leave their
observational signals in the late-time sky.

We finally call attention to the fact that a natural Hubble tension must arise
in our framework in the late-time universe. The measurement of the Hubble
constant from the CMB analysis would presumably come together with their
cosmological partners from the $\Lambda$CDM model, since they are meaningful
in the time of last scattering, as it is very often considered [59]. However,
there is no reason to believe, at least a priory, that we could keep the same
interpretation for the new parameters composing the average Friedmann equation
(43). In fact, it is the other way around. For the sake of simplicity, let us
consider only the isotropic context, that is, assume $\widehat{\Omega}_{A}=0$,
which implies $\mathrm{H}_{\psi}=0$ (see theorem 2). Under this condition, for
the flat spatial sections, it has been verified that the density contrast in
our model is compatible with our expectations: it allows a very homogeneous
early state to become more and more inhomogeneous as time goes by in the
expansion scenario [18]. Therefore, if we think that from the time of the last
scattering through the era where the large-scale structures were formed and
until today, $\widehat{\Omega}_{I}$ might have significantly changed. Hence,
it seems improbable to conceive it in the determination of the Hubble constant
from the CMB measurements, even knowing that some effects due to the late-time
inhomogeneities leave their imprints there, as the integrated Sachs-Wolf[2],
for instance. On the other hand, the behavior of $\widehat{\Omega}_{\mu}$ is
more difficult to assert in this general context, but it might as well give
rise to some new parameters invisible to the CMB and/or some possible
contributions to the $\Lambda$CDM existing ones. In the surroundings of the
observers to whom the geometric and CMB frames coincide the universe is
virtually FLRW, for
$\raisebox{0.09995pt}{\scalebox{1.1}[1.4]{${\scriptstyle\overline{\nabla}}$}}\phi=0$
along their paths. Hence, the redshift measurements and the luminosity
distance on large scales $L>>L_{0}$ must indeed be dealt with as in the
homogeneous and isotropic standard model, except for the fact that we use the
mean Hubble parameter in the averaged Friedmann equation (43). From this, we
readily conclude that the usual FLRW formula for the luminosity distance would
be a good approximation, that is, the formula (Sec. 7.4.5 in Ref. [10])

$d_{L}(z)=(1+z)\,S_{K_{0}}\left(\int_{0}^{z}\frac{dz^{\prime}}{\langle\mathrm{H}\rangle(z^{\prime})}\right)\quad,\quad
1+z=\frac{1}{a}\,,$ (45)

with $S_{K_{0}}(x)=\sin x$, for $K_{0}>0$, $S_{K_{0}}(x)=\sinh x$, for
$K_{0}<0$, and $S_{K_{0}}(x)=x$, for $K_{0}=0$. Therefore, the discrepancy in
the determination of the Hubble constant from the CMB and from the Luminous
distance observations is inexorably present in our effective model.

In the literature, one can find different tentatives to solve the Hubble
tension through the inhomogeneous and anisotropic context (see section 3.1. of
Ref. [60] and the references therein). Whether the scheme above will shed some
light on or, on the contrary, intensify the empiric Hubble tension in
Cosmology, is a matter of further analysis, that can only be known as we probe
inside the parameters $\widehat{\Omega}_{I}$ and $\widehat{\Omega}_{\mu}$
through Einstein’s equations in specific situations and examine them against
real observations. Regardless of the outcome of such investigations, all the
general considerations we have made so far are in order to show that the
intrinsically homogeneous and isotropic spacetimes are worth investigating,
and the effective $\Lambda$CDM-like models arising naturally from them can
shed light on some of the most elusive problems encountered in Cosmology
today.

##  8 Final remarks

In this manuscript, we have investigated the foundations of the Cosmological
Principle, which has been broken into three parts. The first, mathematical,
ensures the homogeneity and isotropy of the spatial geometry to describe the
universe, thus defining the geometric frame. The second, observational,
guarantees isotropic expansion along fiducial observers. The third, physical,
assumes a perfect fluid form for the overall combination of matter and
radiation. As we permit ourselves to decouple the CMB from the geometric
frame, we remain with the more general structure where just the first of those
pieces is kept, the so-called intrinsically homogeneous and isotropic
spacetimes. In this context, we have an interesting geometric framework where
taking spatial averages and mounting $\Lambda$CDM-like effective models, with
a Hubble tension inexorably attached, are naturally conceived. All these
properties make them a prolific arena for further theoretical investigations
and astrophysical applications.

## Acknowledgments

The author is thankful for the support from FAPEMIG, project number
RED-00133-21.

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