# Spatially flat universes with isotropic tidal forces

Fabio Scalco Dias<EMAIL_ADDRESS>Leandro Gustavo Gomes
<EMAIL_ADDRESS>Luis Fernando Mello<EMAIL_ADDRESS>Universidade
Federal de Itajubá, Av. BPS, 1303, Itajubá-MG, Brazil. ZIP: 37500-903 .

###### Abstract

We investigate the dynamics of the spatially flat universes submitted to
isotropic tidal forces and adiabatic expansion under Einstein’s equations.
Surprisingly, the tendency to a high Hubble anisotropy at late times starts to
appear as far as we assume the strong energy condition, a characteristic which
becomes dominant in the radiation era and even more stringent under a stiff
matter regime. We introduce the parameter $b$ which measures the relative
change in the magnitudes of the Hubble anisotropy and the scale factor and use
it to show that the anisotropies must be kept small as long as we assume the
Universe has passed through an inflationary period, in accordance with the
cosmic no-hair theorem. Hence, we have a class of models that shows us in a
simple and straightforward way the instability of the FLRW universes, the
furtiveness of the isotropy concept, and how they can still be consistent with
the standard model of Cosmology as far as we assume the occurrence of an
inflationary early period.

Anisotropy; Bianchi I; spatially homogeneous cosmology

##  I Introduction

Isotropy is a quite elusive property in Cosmology. The Hubble ratio of
expansion vary no more than $1\%$ along the different directions of the sky
[1, 2], the CMB temperature is known to be independent of direction with a
high degree of precision [3, 4], and the number count of radio sources seems
to be consistent with the isotropy hypothesis [5]. Therefore, as we apply the
Copernican principle, meaning that we are not privileged observers of the
cosmic drift, we would be in the comfortable position to set our spacetime as
a small perturbation of a homogeneous and isotropic FLRW universe, the current
standard model of Cosmology [6]. That would be just as perfect as one could
expect, except for one important point: the FLRW universes are not stable
under small perturbations.

Historically, just after the discovery of the CMB in the ’60s, which by the
time was showing its first traces of isotropy, the picture of a homogeneous,
chaotic, and highly anisotropic early epoch had been proposed [7], often
referred to as the BKL scenario. The current state of the universe would be
achieved as the anisotropy dies out during the expansion, which could be
caused by neutrino viscosity, for instance [8, 9]. Soon after that, Collins
and Hawking showed that the spatially homogeneous universes do not, in
general, isotropize [10]. That was a distinguished point in the conceptual
evidence for the instability of the FLRW models, which in turn became a
barrier for the chaotic BKL picture [11]. Since then, the BKL approach
survived as a general framework for approaching the big bang singularity [12],
the dynamical aspects of the anisotropies have been further studied [13, 14,
15, 16], and the inflationary theory has taken into the scene, leaving no
cosmological hair to account neither for homogeneity nor isotropy [17].

Today, some decades after the results of Collins and Hawking, the behavior of
the anisotropy under Einstein’s dynamics can still baffle our intuition, even
in the simplest of the models, under the most reasonable physical conditions.
In order to show that, we start with a general spatially flat model, a Bianchi
type I spacetime, which is put in an adiabatic expansion with only one
restriction: the gravitational forces are everywhere isotropic, so that the
gravitational pull (or push) felt by the components of our free-falling cosmic
fluid has no distinguished direction. As we are going to show, even under this
framework favoring an isotropic behavior, the tendency to anisotropy still
persists, and when the thermodynamic pressure is as high as in the case of
pure radiation, it dominates completely. They form a class of simple and
rather counter-intuitive examples, which has a three-fold implication for
Cosmology: (i) It enhances our conceptual understanding of the cosmic
anisotropies. In particular, it gives us a reminder of how the FLRW universes
are unstable under small perturbations, even when the overall situation is
favoring stability; (ii) It displays a complete nonlinear account for the
anisotropy dynamics in a quite simple and physically reliable manner. This
could be useful in case the observations start pointing to an anisotropic
Hubble sky [18, 19], which would demand models beyond the small perturbations
of an isotropic background; (iii) Even if the observed universe comes to be
plainly isotropic, it also furnishes a good class of examples where the cosmic
no-hair theorem can be appreciated and clearly understood. As we are going to
see, during an inflationary period, the Hubble anisotropy in our model decays
twice as fast as the scale factor. Hence, we are able to follow
straightforwardly how inflation would drive the leftovers of the cosmic
anisotropies to an insignificant position in the cosmic evolution.

The manuscript is divided as follows: in the section II, we define what we
mean by isotropy in the gravitational tidal forces and adiabatic expansion in
the spatially flat models. In the following section, we describe the
Einstein’s equations in this setting, showing the first traces of the growth
of the anisotropies. In the section IV, we specialize the dynamics to those
cases with a linear equation of state between the energy density and the
thermodynamic pressure. This simplification gives us a whole class of examples
where the global anisotropy dynamics can be seen in the Kasner disc,
enlightening our understanding of such behaviors. In particular, we introduce
a parameter $b$ which measures the relative change in the magnitudes of the
Hubble anisotropy and the scale factor. This has an important role in
justifying why the anisotropies must be kept small after an inflationary
period. At the final section, we make our concluding considerations. The
notations and sign conventions follow the reference [20].

##  II Isotropy of the tidal forces and adiabatic expansion in spatially flat
universes

Our main assumption is that the universe allows a class of free-falling
observers to whom the space sections are flat and the tidal forces are
isotropic. This means that the gravitational forces should balance out in a
way that the pull (or push) felt by any of these observers is the same in any
direction. As we put this forward in the mathematical language, the observers
are represented by a unitary, geodesic, and vorticity-free fundamental
velocity, $\mathrm{u}$, whose space sections are flat, that is, a Bianchi type
I model, in a way the tidal force operator,
$-\,\mathrm{R}^{\mu}_{\kappa\nu\lambda}\mathrm{u}^{\kappa}\mathrm{u}^{\lambda}$,
has no preferred spatial directions. This means that it is multiple of the
identity along the spatial directions, which is equivalent to the equation of
state [21]

$E^{\mu}_{\nu}=\tfrac{1}{2}\,\Pi^{\mu}_{\nu}\,,$ (1)

where $E^{\mu}_{\nu}$ is the electric part of the Weyl tensor and
$\Pi^{\mu}_{\nu}$ the anisotropic stress tensor. In this case, as we use
adapted coordinates $(t,x^{i})$ for which $\mathrm{u}=\partial_{t}$ and the
metric is

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

the geodesic deviation equations along the spatial directions turn into

$\frac{d^{2}\xi^{i}}{d\tau^{2}}=-\,\frac{1}{6}\,\left(\mathrm{\rho}+3\,\mathrm{P}\right)\,\xi^{i}\,,$
(3)

where $\mathrm{\rho}=T_{\mu\nu}\mathrm{u}^{\mu}\mathrm{u}^{\nu}$ and
$\mathrm{P}=T_{i}^{i}$ stand for the energy density and total pressure,
respectively. The Hubble and the matter-radiation anisotropies will be
identified with the dimensionless spatial tensors [16]
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{i}^{j}=\sigma_{i}^{j}/(\sqrt{6}\,\mathrm{H})$
and
$\raisebox{0.09995pt}{\scalebox{0.8}[0.8]{${\mathcal{N}}$}}_{i}^{j}=\Pi_{i}^{j}/(\sqrt{6}\,\mathrm{P})$,
respectively, where $\sigma_{i}^{j}$ is the shear tensor. Hence, the electric
Weyl tensor becomes

$\frac{1}{\sqrt{6}}\,E_{k}^{i}=\mathrm{H}^{2}\,\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{k}^{i}-\,\frac{1}{2}\,\mathrm{P}\,\raisebox{0.09995pt}{\scalebox{0.8}[0.8]{${\mathcal{N}}$}}_{k}^{i}-\,\sqrt{6}\,\mathrm{H}^{2}\,\left(\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{i}_{\ell}\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{\ell}_{k}-\,\frac{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{2}}{3}\,\delta^{i}_{k}\right)\,.$
(4)

Finally, we can re-write the equation of state (1) as

$\mathrm{P}\,\raisebox{0.09995pt}{\scalebox{0.8}[0.8]{${\mathcal{N}}$}}^{i}_{k}=\mathrm{H}^{2}\,\left(\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{k}^{i}-\,\sqrt{6}\,\left(\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{i}_{\ell}\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{\ell}_{k}-\,\tfrac{\raisebox{0.06996pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{2}}{3}\,\delta^{i}_{k}\right)\right)\,.$
(5)

We assume an adiabatic expansion, with no energy flux, $q^{i}=0$, nor any
spatial heat diffusion, $\partial_{i}T=0$, so that the entropy is conserved:
$\dot{S}=0$. As we split the pressure in its bulk and thermodynamic
counterparts, $\mathrm{P}_{B}$ and $\mathrm{P}_{T}$, respectively, we obtain
[21]

$\frac{dS}{dt}=0\quad\Rightarrow\quad\mathrm{P}_{B}=\mathrm{P}-\mathrm{P}_{T}=-\,2\,\mathrm{P}\,\raisebox{0.09995pt}{\scalebox{0.8}[0.8]{${\mathcal{N}}$}}_{ik}\,\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{ik}\,.$
(6)

This means that the bulk pressure is composed by the Hubble and matter-
radiation components of the anisotropy. On the other hand, the energy
conservation,
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\nabla}$}}_{\mu}\,T^{\mu}_{0}=0$,
turns out to be dependent only on the thermodynamical pressure, since it is
written as

$\frac{d\mathrm{\rho}}{dt}=-\,3\mathrm{H}\left(\mathrm{\rho}+\mathrm{P}_{T}\right)\,.$
(7)

It is worth mentioning that this equation emulates the conservation of energy
in the FLRW spacetimes. Therefore, $\mathrm{P}_{T}$ should be interpreted as
the “isotropic” part of the total pressure, which is responsible for the net
force the cosmic fluid exerts upon the fabric of the spacetime. On the other
hand, $\mathrm{P}_{B}$ accommodates that part remaining in the process of
taking thermodynamical averages, no less important, since they are connected
to the entropy increase.

##  III The cosmic dynamics from Einstein’s equations

The Einstein’s equations in the variables
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{k}^{i}$
and $\raisebox{0.09995pt}{\scalebox{0.8}[0.8]{${\mathcal{N}}$}}_{k}^{i}$ have
been put forward in Ref. [16]. They are equivalent to the Generalized
Friedmann equation,

$\mathrm{\rho}=3\,\left(\,1-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{2}\,\right)\,\mathrm{H}^{2}\,,$
(8)

where
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=\sqrt{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{k}^{i}\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{k}_{i}}$
is the Hubble anisotropy magnitude, the conservation of energy (7) and the
anisotropy equation, which after applying the condition (5) for isotropic
tides, becomes

$\frac{1}{\mathrm{H}}\frac{d}{dt}\,\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{k}^{i}=\left(1-\frac{\mathrm{\rho}-\mathrm{P}}{2\,\mathrm{H}^{2}}\right)\,\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{k}^{i}-\,\sqrt{6}\,\left(\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{i}_{\ell}\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{\ell}_{k}-\,\frac{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{2}}{3}\,\delta^{i}_{k}\right)\,.$
(9)

The first consequence of the tidal anisotropy is that the relation (5) allows
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{i}_{k}$
and $\raisebox{0.09995pt}{\scalebox{0.8}[0.8]{${\mathcal{N}}$}}^{i}_{k}$ to be
simultaneously diagonalizable, and hence, as we put
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{i}_{k}(t_{0})$
in the diagonal form, the Einstein’s equations tell us that it will continue
to be like that along the entire expansion. In other words, our Bianchi I
spacetime is diagonalizable. Hence, the system (9) is completely determined by
the equations for the Hubble anisotropy magnitude  ${\scriptstyle\Sigma}$ and
the Kasner angle $\alpha$, as

$(\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{k}_{i})=\sqrt{\frac{2}{3}}\,\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\,\left[\begin{array}[]{ccc}\sin\alpha&0&0\\\
0&\sin\left(\alpha+\frac{2}{3}\pi\right)&0\\\
0&0&\sin\left(\alpha-\frac{2}{3}\pi\right)\\\ \end{array}\right]\,.$ (10)

From this and the relation (5), we can write the bulk pressure as

$\mathrm{P}_{B}=-\,2\,\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{2}\,\mathrm{H}^{2}\,\left(1+\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\,\sin(3\alpha)\,\right)\,.$
(11)

By using the new time parameter $ds=\mathrm{H}dt$ and the “equation-of-state”
variable w, that is,

$s(t)=\ln\left(\frac{a(t)}{a_{0}}\right)\quad\textrm{and}\quad\textrm{w}(t)=\frac{\mathrm{P}_{T}(t)}{\mathrm{\rho}(t)}\,,$
(12)

the energy conservation becomes

$\mathrm{\rho}^{\prime}=-3(\textrm{w}+1)\,\mathrm{\rho}\,,$ (13)

while the anisotropy equation turns into

$\left\\{\begin{aligned}
\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{\prime}&=\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\,\left(1-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{2}\right)\,\left(\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\,\sin(3\alpha)+\,\frac{3\textrm{w}-1}{2}\right),\\\
\alpha^{\prime}&=\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\,\cos(3\alpha),\end{aligned}\right.$
(14)

where we have used the abreviation $z^{\prime}=dz/ds$. We will consider only
the inner part of the Kasner disc
($\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\leq 1$),
since this is equivalent of keeping the energy density non-negative, according
to the generalized Friedmann equation (8).

##  IV The cosmic dynamics for fluids with a linear equation of state

In order to have a glimpse of the different features of the dynamical behavior
of our cosmic system, we will consider the expansion with w constant, that is,
with the energy density and the thermodynamical pressure satisfying the linear
equation of state $\mathrm{\rho}=\textrm{w}\,\mathrm{P}_{T}$, with
$\textrm{w}^{\prime}=0$.

###  IV.1 General properties of the solutions

Equations (14) define a smooth and autonomous system in the Kasner disc
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\leq 1$. In
the “Cartesian” coordinates, $x=\scalebox{0.7}[0.8]{$\Sigma$}\,\cos(3\alpha)$
and $y=\scalebox{0.7}[0.8]{$\Sigma$}\,\sin(3\alpha)$, it turns out to be
polynomial, as

$\left\\{\begin{aligned}
x^{\prime}&=x\left(1-x^{2}-y^{2}\right)\left(y+\frac{3\textrm{w}-1}{2}\right)-3xy,\\\
y^{\prime}&=y\left(1-x^{2}-y^{2}\right)\left(y+\frac{3\textrm{w}-1}{2}\right)+3x^{2}.\\\
\end{aligned}\right.$ (15)

Since all the solutions are kept inside the compact disc, they are defined for
every real value of $s$, that is, for every $a>0$. Hence, for all of them, as
we assume expansion ($\dot{a}>0$), we have two distinct epochs, just as in the
FLRW case: the early ($a\ll a_{0}$) and the late-time ($a\gg a_{0}$)
universes. Furthermore, the conservation (13) is also analogous to its
counterpart in the isotropic universes, so that the energy density turns out
to be

$\mathrm{\rho}(t)=\mathrm{\rho}_{0}\,\left(\frac{a_{0}}{a(t)}\right)^{3(\textrm{w}+1)}\,.$
(16)

The anisotropy magnitude  ${\scriptstyle\Sigma}$ and the Kasner angle
$\alpha$, in general, cannot be fully integrated from the equations in (14).
Notwithstanding, as we observe that

$\frac{d\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}}{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}(1-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{2})}=\tan(3\alpha)d\alpha+\frac{3\textrm{w}-1}{2}\frac{da}{a}$
(17)

whenever $\cos(3\alpha_{0})\neq 0$, we obtain the constraint

$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}(t)=\frac{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{0}}{\sqrt{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{0}^{2}+(1-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{0}^{2})\xi(t)}}$
(18)

with

$\xi(t)=\left(\frac{a_{0}}{a(t)}\right)^{3\textrm{w}-1}\left(\frac{\cos(3\alpha(t))}{\cos(3\alpha_{0})}\right)^{\frac{2}{3}}\,.$
(19)

On the other hand, the solutions with $\sin(3\alpha_{0})=\pm 1$ satisfy
$\alpha^{\prime}=0$. Hence, by a direct integration of the first of the
equations in (14), we obtain

$\int_{\raisebox{0.06996pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{0}}^{\raisebox{0.06996pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}(t)}\frac{d\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}}{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\,\left(1-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{2}\right)\,\left(\frac{3\textrm{w}-1}{2}\pm\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\right)}=\ln\left(\frac{a(t)}{a_{0}}\right)\,.$
(20)

Note that in the generic case (18), when $\textrm{w}>1/3$, as we set
$a\to\infty$ we get $\xi\to 0$ and
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\to 1$. This
implies that these solutions get more and more anisotropic as the universe
expands. This rather counter-intuitive behavior, since the tides are kept
isotropic, persists even when w attains smaller values, up to the breaking
point of the strong energy condition, $\textrm{w}=-1/3$. This important fact
will be addressed throughout the text.

It is interesting to analyze the relation of the orders of magnitude both the
scale factor and the anisotropy went through between two specific moments of
the expansion of the universe, say from $t_{1}$ to $t_{2}$. This is
characterized by the parameter

$b=\left|\frac{\ln(\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}/\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1})}{\ln(a_{2}/a_{1})}\right|\,.$
(21)

As we analyze it along the solutions $\sin(3\alpha_{0})=\pm 1$, we have,
according to (20),

$\frac{1}{b}=\left|\frac{1}{\ln(\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}/\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1})}\int_{\raisebox{0.06996pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}}^{\raisebox{0.06996pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}}\frac{d\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}}{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\,\left(1-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{2}\right)\,\left(\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\pm\frac{3\textrm{w}-1}{2}\right)}\right|\,.$
(22)

As we take
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}/\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}\to
0$ or
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}/\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}\to
0$ in (22), we conclude that

$b\approx\frac{|3\textrm{w}-1|}{2}$ (23)

whenever one of the variables
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}$ or
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}$
overcomes the other in many orders of magnitude. These formulas will be of
suitable usage in order to estimate the variation of the anisotropy magnitudes
during the different epochs of the universe.

###  IV.2 The qualitative aspects of the dynamics

The equilibrium points of the system (14), with w constant, are the origin,
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=0$,
representing the flat FLRW universe, the Taub points in the Kasner circle
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=1$, and the
LRS points inside the disc 111For the Taub points, see [15]. Any point with
Kasner angular coordinates $n\pi/6$, $n$ odd, is Locally Rotationally
Symmetric (LRS), which refers to the more symmetric configuration of the
spacetime. A good analogy is to compare the ellipsoids of revolution in the
Euclidean spaces (LRS) with their less symmetric partners., with
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=|3\textrm{w}-1|/2$.
They come in two categories: the $T$’s and the $Q$’s. The Taub points
$T_{1},T_{2},T_{3}$, with
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=1$, and the
LRS ones $\widetilde{T}_{1},\widetilde{T}_{2},\widetilde{T}_{3}$, with
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=(3\textrm{w}-1)/2$,
these last ones existing only in the case $1/3\leq\textrm{w}\leq 1$, have the
following Kasner angles coordinates, respectively,

$\alpha_{1}=\frac{\pi}{2},\,\alpha_{2}=\frac{11\pi}{6},\,\alpha_{3}=\frac{7\pi}{6}.\\\
$

The Taub points $Q_{1},Q_{2},Q_{3}$, with
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=1$, and the
LRS ones $\widetilde{Q}_{1},\widetilde{Q}_{2},\widetilde{Q}_{3}$, with
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=(1-3\textrm{w})/2$,
these last ones existing only in the case $-1\leq\textrm{w}\leq 1/3$, have the
following Kasner angles coordinates, respectively,

$\alpha_{1}=\frac{3\pi}{2},\,\alpha_{2}=\frac{5\pi}{6},\,\alpha_{3}=\frac{\pi}{6}.$

Note that when $\textrm{w}=-1/3$, the points $\widetilde{Q}_{i}$’s coincide
with the $Q_{i}$’s, and when $\textrm{w}=1$, the points $\widetilde{T}_{i}$’s
coincide with the $T_{i}$’s. Moreover, as $\textrm{w}\to 1/3^{+}$,
$\widetilde{T}_{i}$ approach the origin, as well as $\widetilde{Q}_{i}$, when
$\textrm{w}\to 1/3^{-}$.

The linear part of system (15) at the origin is

$\left[\begin{array}[]{cc}(3\textrm{w}-1)/2&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr 0&(3\textrm{w}-1)/2\end{array}\right].$ (24)

At the Taub points, the linear part of system (14) is

$\left[\begin{array}[]{cc}3-3\textrm{w}&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr 0&3\end{array}\right]\quad\\\
\quad\text{or}\quad\left[\begin{array}[]{cc}-(3\textrm{w}+1)&0\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr 0&-3\end{array}\right]\,,$ (25)

whether the point is $T_{i}$ or $Q_{i}$ $(i=1,2,3)$, respectively. Finally,
for the points $\widetilde{T}$’s or $\widetilde{Q}$’s, we obtain

$\left[\begin{array}[]{cc}\frac{3}{8}(9\textrm{w}^{2}-1)(\textrm{w}-1)&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&\frac{3}{2}(3\textrm{w}-1))\end{array}\right].$ (26)

In short, the stability of each of these points is given in Table 1 where the
following notation is used: S (saddle), UN (unstable node), SN (stable node)
HS (hyperbolic sectors) and S-N (saddle-node) [23].

w / Point | Origin | $Q_{i}$ | $T_{i}$ | $\widetilde{Q}_{i}$ | $\widetilde{T}_{i}$
---|---|---|---|---|---
$\textrm{w}\in[-1,-1/3)$ | SN | S | UN | $\nexists$ | $\nexists$
$\textrm{w}=-1/3$ | SN | S-N | UN | $Q_{i}$ | $\nexists$
$\textrm{w}\in(-1/3,1/3)$ | SN | SN | UN | S | $\nexists$
$\textrm{w}=1/3$ | 6 HS | SN | UN | Origin | Origin
$\textrm{w}\in(1/3,1)$ | UN | SN | UN | $\nexists$ | S
$\textrm{w}=1$ | UN | SN | S-N | $\nexists$ | $T_{i}$
Table 1: Topological type of each equilibrium point of system (14) for
$-1\leq\textrm{w}\leq 1$.

In what follows, we analyze the dynamics in the Kasner disc
($\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\leq 1$)
with the variables  ${\scriptstyle\Sigma}$ and $\alpha$ working as “polar
coordinates” in the plane [15, 16]. We are interested in the interval
$-1\leq\textrm{w}\leq 1$. We could abuse of our intuition and refer to the
condition $\textrm{w}=0$ as “dust”, $\textrm{w}=1/3$ as “radiation”,
$\textrm{w}=1$ as “stiff-matter”, and so on. This would be justified as far as
the anisotropies are kept small, so that we could interpret those situations
as small perturbations of the proposed physical situation. In the case the
anisotropies grow large, those proposed names could be quite misleading. For
this reason, we will adopt the names “dust-like”, “radiation-like”, “stiff-
matter-like”, and so on.

####  IV.2.1 From the inflation-like scenario to the breaking point of the
strong energy condition

According to the standard picture of Cosmology, the interval
$-1\leq\textrm{w}<-1/3$ encompass the very early inflationary era as well as
the late-time dark energy period, both satisfying
$\mathrm{P}_{T}\approx-\mathrm{\rho}$. The state $\textrm{w}=-1/3$ will be
referred to as the breaking point where the strong energy condition 222Indeed,
the strong energy condition demands $\mathrm{\rho}+3\mathrm{P}>0$ and
$\mathrm{\rho}+\mathrm{P}+\Pi_{i}>0$, $i=1,2,3$, $\Pi_{i}$ the eigenvalues of
$\Pi_{ij}$. The name refers to the state at which it begins to be violated, at
least for small anisotropy
($\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\approx
0$).. Here, the $T$-Taub points are unstable nodes while the $Q$’s are
saddles, with all the solutions inside the Kasner disc converging to the late-
time isotropic cosmology, that is,
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\to 0$ as
$a\to\infty$ (see Fig. 1).

Figure 1: The phase portrait in the Kasner disc
($\raisebox{0.08995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\leq 1$)
for $-1\leq\textrm{w}\leq-1/3$. Any solution isotropizes towards the FLRW
universe at the center. The $T$-Taub points are unstable nodes. The $Q$-Taub
points are saddles up to the value $\textrm{w}=-1/3$, when they coincide with
the $\widetilde{Q}$’s and begin the transition to stable nodes. In this case,
they are saddle-nodes.

Since the anisotropy decreases as the universe expands, let us pick an initial
and a final states,
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}$ and
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}$,
respectively, with
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}\ll\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}$.
According to the formula (23), we have $b\approx(1-3\textrm{w})/2$. This means
that the anisotropy diminishes twice as fast as the universe expands, if
$\textrm{w}=-1$, or at the same ratio, if $\textrm{w}=-1/3$. Let us analyze
the first case separately, due to its conceptual importance.

The parameter $b$ in the case $\textrm{w}=-1$ can be straightforwardly
calculated through the integral (22) along the solution $\alpha=\pi/2$. If we
take
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}=\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}\times
10^{-n}$, $n>1$, we obtain

$\frac{2}{b}-1\approx\frac{-\ln(1-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1})+\ln(1+\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}/2)-2\ln(1+\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1})}{3n\ln
10}\,,$ (27)

where we have used
$\ln(1+\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}\times
10^{-n})\approx 0$. For
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}$ not
too close to $1$, we have $b\approx 2$, just as aforementioned. On the other
hand, if the initial condition was extremely anisotropic, as
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}=1-10^{-n_{1}}$,
we would have $b\approx 2-2n_{1}/(3n+n_{1})$, meaning that $b$ could attain
smaller values, but we would still have $b\geq 4/3$. Hence, when
$\mathrm{P}_{T}=-\mathrm{\rho}$, the anisotropy vanishes faster than the
universe expands. Furthermore, if the universe have passed through $N$ e-folds
during this period, the anisotropy would diminish something near to $2N$
e-orders of magnitude. This is in agreement with the no-hair picture of the
cosmic evolution [17]. As we apply it to the inflationary period, where the
universe is believed to stay long enough as $N\gtrsim 60$ [21], we would have
the anisotropy at the beginning of the reheating period probably as tiny as
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{rh}\lesssim
e^{-120}$. If not that, at least it would not be greater than $e^{-80}$, as we
put $b=4/3$. An analogous situation would occur during the late-time dominance
of the dark energy, but now with an inferior value for $N$.

####  IV.2.2 From the breaking point of the strong energy condition to the
radiation-like condition

From the dynamical viewpoint, the interval $-1/3\leq\textrm{w}<1/3$ is
characterized by the origin still being a stable node, but at this time its
basin of attraction is not the inner Kasner disc anymore. In fact, new saddle-
type equilibrium points appear along the straight segments connecting the
$Q$-Taub points to the origin, those LRS ones labeled as $\widetilde{Q}_{1}$,
$\widetilde{Q}_{2}$ and $\widetilde{Q}_{3}$. To each of them, there correspond
two separatrices splitting the disc into two parts: the inner one, which
contains the basin of attraction for the equilibrium at the origin, where the
universe tends to a late-time isotropic state, and the outermost one, where
the anisotropization takes place, and the solutions tend to the highly
anisotropic LRS universes at the $Q$-Taub points.

Figure 2: The phase portrait in the Kasner disc
($\raisebox{0.08995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\leq 1$)
for $-1/3<\textrm{w}<1/3$. There are separatrices sppliting the dynamics in
two main global behaviors. In the innermost part, the solutions isotropize
towards the FLRW universe at late-times. In the outermost one, the universes
tend to the highly anisotropic LRS models at the $Q$-Taub points. As we get
closer to the radiation epoch ($\textrm{w}\to 1/3^{-}$), anisotropization
starts to dominate over isotropization.

The $\widetilde{Q}$’s equilibrium points at
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=(1-3\textrm{w})/2$
split the line $\alpha=(1+4n)\pi/6$ in two: the isotropization and
anisotropization segments, where
$0<\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}<(1-3\textrm{w})/2$
and
$(1-3\textrm{w})/2<\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}<1$,
respectively (see Fig. 2). As we estimate the order of magnitude parameter $b$
with the aid of the formula (23), we obtain

$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}\ll\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}<\frac{1-3\textrm{w}}{2}\quad\Rightarrow\quad
b=\frac{1-3\textrm{w}}{2}\,.$ (28)

Therefore, as
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}<\frac{1-3\textrm{w}}{2}$,
the anisotropy decays slower than the universe expands, by a factor $b$, with
$0<b<1$, such that $b\to 1$ as $\textrm{w}\to-1/3^{+}$ and $b\to 0$ as
$\textrm{w}\to 1/3^{-}$. In the case of dust-like solutions ($\textrm{w}=0$),
we have that each two e-folds of the universe corresponds to one of the
anisotropy, such that $b=1/2$, as long as
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}<1/2$.
On the other hand, when
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}>\frac{1-3\textrm{w}}{2}$,
the parameter $b$ loses its general character, so that the estimate it is
intended for should be directly calculated from the integral (22), since it
can give any positive number.

####  IV.2.3 The radiation-like era

The radiation-like era is distinguished as being the transition point to pure
anisotropization. In fact, the $\widetilde{Q}$-type equilibrium points
coalesce with the origin, so that the anisotropization sector becomes
virtually the only one available inside the Kasner disc, except for the
segments where $\sin(3\alpha_{0})=-1$, the remains of the former
isotropization region. Therefore, the isotropic universe is no longer stable
for small perturbations, for
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\to 1$ as
$a\to\infty$, no matter the initial values as far as we keep
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{0}\neq 0$
and $\sin(3\alpha_{0})\neq-1$. The special cases where $\sin(3\alpha_{0})=-1$
are still tending to the late-time FLRW model. The dynamics in the Kasner disc
for this epoch is depicted in Fig. 3.

Figure 3: The phase portrait in the Kasner disc
($\raisebox{0.08995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\leq 1$)
for the radiation era ($\textrm{w}=1/3$). The solutions anisotropize towards
the LRS models at the $Q$-Taub points, where
$\raisebox{0.08995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=1$, except
the FLRW universe at the origin and those at the straight segment with
$\alpha=\pi/2+2n\pi/3$.

In the case $\cos(3\alpha_{0})\neq 0$, we obtain from the formula (18) the
first integral

$\frac{(1-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{2})^{3/2}}{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}^{3}\cos(3\alpha)}=c_{0}.$
(29)

This implies that system (14) is integrable for $\textrm{w}=1/3$.

Let us assume that our model describes the hot and dense epoch dominated by
radiation, when the input anisotropy was
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}$, and
by the end of this period, when the CMB was released from the initial plasma
and started to propagate freely, the anisotropy became
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}$.
Indeed, this is very plausible if these anisotropies are kept small. Since we
get $b\approx 0$ from the limiting case (23), we conclude that the anisotropy
left that epoch almost with the same magnitude it entered there.

Since we have got little information on the vanishing of $b$, we might go
deeper into its analysis and make it from the scratch. So, setting
$\alpha=\pi/6$, for the sake of simplicity, and using the integral (22), we
get

$b=\frac{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}\ln(\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}/\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1})}{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}+\Delta}\,,$
(30)

where

$\Delta=\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}\ln\sqrt{\left(\frac{1+\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}}{1+\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}}\right)\left(\frac{1-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}}{1-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}}\right)}\,.$
(31)

As we let the universe to expand $N$ e-folds during this epoch,
$\ln(a_{2}/a_{1})=N$, we get
$\ln(\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}/\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1})=Nb$.
Hence, as we note that
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}<\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}$
implies $\Delta>0$, we obtain

$\frac{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}}{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}}\leq\frac{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}-\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}}{\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}}<N\,,$
(32)

that is,

$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}<\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}<(N+1)\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}\,,$
(33)

which means that $b<[\ln(N+1)]/N$. In other words, the change in the
anisotropy magnitude along the radiation era, as the universe expands $N$
e-folds, is not greater than
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{2}\approx(N+1)\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}_{1}$.

####  IV.2.4 From radiation-like to stiff-matter-like periods

In the isotropic model, the interval $1/3<\textrm{w}\leq 1$ is a candidate for
the epoch between the inflationary and the radiation eras at “ultrahigh”
densities. It contains the limiting stiff matter condition ($\textrm{w}=1$),
for which the sound propagation attains the speed of light [25, 26].
Interesting enough, our model has naturally separated this regime from the
others. Indeed, it appears as the physically relevant interval for which the
origin is an unstable equilibrium. Hence, the universe tends to anisotropize
even for arbitrarily small and non-vanishing values of  ${\scriptstyle\Sigma}$
. This is quite unexpected, for the tidal forces are still isotropic.

Figure 4: The phase portrait in the Kasner disc
($\raisebox{0.08995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\leq 1$)
for $1/3<\textrm{w}<1$. Any solution but the FLRW universe at the center
anisotropizes. In the late-time regime they approach the LRS universes at the
$Q$-Taub points, with
$\raisebox{0.08995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=1$. The
only exceptions are in the lines $\alpha=\pi/2+2n\pi/3$, where
$\raisebox{0.08995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\to(3\textrm{w}-1)/2$
as $a\to\infty$, the $\widetilde{T}$’s saddle-type equilibrium points.

Figure 5: The phase portrait in the Kasner disc
($\raisebox{0.08995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\leq 1$)
for $\textrm{w}=1$. Any solution but the FLRW universe at the center
anisotropizes to the borders of the Kasner disc
($\raisebox{0.08995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\to 1$ as
$a\to\infty$). The saddle-type equilibrium points ($\widetilde{T}$’s) have
coalesced with the $T$-Taub points at
$\raisebox{0.08995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=1$.

As we allow the equation of state to run from $\textrm{w}=1/3$ to
$\textrm{w}=1$, the saddle-type equilibrium points $\widetilde{T}$’s appear at
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=(3\textrm{w}-1)/2$
and $\alpha=\pi/2+2(1-k)\pi/3$. In the late-time regime, the solutions
approach the LRS universes at the $Q$-Taub points, with
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=1$, except
the isotropic model at the origin and those on the straight line connecting it
to $\widetilde{T}$’s, for which
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\to(3\textrm{w}-1)/2$
as $a\to\infty$ (see Fig. 4).

The phase portrait in the Kasner disc for the stiff-matter-like situation is
plotted in Fig. 5. In this case, the inner equilibrium points
$\widetilde{T}$’s have coalesced with the $T$-Taub points, so that any
solution but the isotropic FLRW model at
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}=0$ tends to
the maximum anisotropy
$\raisebox{0.09995pt}{\scalebox{0.9}[1.3]{${\scriptstyle\Sigma}$}}\to 1$ at
late-times.

From the formula (23), we get $b\approx(3\textrm{w}-1)/2$. This means that in
the stiff-matter-like era, if there has been one, with $\textrm{w}=1$ and
$b\approx 1$, have the universe passed through $N$ e-folds during this high-
density period, the anisotropy would increase just as well.

##  V Final remarks

In this manuscript, we have analyzed the spatially flat spacetimes under
adiabatic expansion and isotropic tidal forces. The total pressure has
naturally been divided into the thermodynamical ($P_{T}$) and bulk ($P_{B}$)
components, the first emulating its isotropic counterpart while the other
exists due to anisotropic effects. We analyzed the dynamics of the Hubble
anisotropy in the Kasner disc during the different epochs when the ratio
$\textrm{w}=\mathrm{P}_{T}/\mathrm{\rho}$ could be held constant. The final
framework is a simple and physically relevant scenario where the anisotropy
can be understood in its fully non-linear aspects, which was dully depicted in
the Kasner disc. For instance, the reader can have a glimpse of the overall
aspects of the system and how it changes with w just by passing from the
figures 1 to 5, from the isotropizing environment with $\textrm{w}=-1$ (Fig.
1) to the completely anisotropizing dynamics with $\textrm{w}=1$ (Fig. 5).

For the very early universe, our model has much more to tell about the
inflationary than the BKL scenario. The reason for this is simple: while the
first is highly compatible with the hypothesis of isotropic tidal forces, the
chaotic behavior of the second would hardly let this characteristic to be
attained. Hence, as we assume inflation took place, the isotropic components
of the energy-momentum tensor for the scalar field driving it would overcome
the energy density and thermodynamical pressure, so that
$\mathrm{P}_{T}\approx-\mathrm{\rho}$. The universe would commence a strong
isotropization process, with  ${\scriptstyle\Sigma}$ decreasing twice as fast
as its expansion rate ($b\approx 2$). At the end of this era, when
$\textrm{w}\sim-1/3$ [27], the (pre)reheating epoch would have begun, with
$-1/3\leq\textrm{w}\leq 1$. As we put the e-fold duration of inflation and
reheating as $N_{inf}$ and $N_{rh}$, with $\textrm{w}=-1$ and $\textrm{w}=1$,
respectively, we would have a estimate for the net decrease of the anisotropy
as $e^{-N}$, where $N\sim 2N_{inf}-N_{rh}$, according to our analysis
following the formula (23). Recent estimates point to $N_{inf}\sim N_{rh}\sim
60$ [21, 27]. This would led the anisotropy still insignificant to be detected
in the late-time sky, for the periods coming afterwards which are dominated by
radiation (anisotropization with $b\approx 0$), dust (anisotropization with
$b\approx 1/2$) and dark energy (isotropization with $b\approx 2$), would not
last long enough to significantly change this tiny scale. Notwithstanding, we
see that a possible detection of anisotropy in the Hubble sky [18, 19] is not
necessarily in contradiction with inflation, but instead, might shed some
light in the period just following it. In principle, our model could favor a
quiescent era [28] lasting about twice the preceding inflationary one.
Needless to say, this statement is of speculative character. Despite that,
this is an example of how our findings can drive us to a qualitative scenario
that could hardly be achieved from the perturbative approach, at least in
their completeness.

Regardless of the estimates for the very early universe, our model was
designed to attend a gap in the cosmology literature. Whether or not the small
perturbations around isotropic models are enough to describe our cosmic
observations [29, 30] is of less importance here. Our aim resides in the
belief that a comprehensive knowledge of our universe could be achieved only
as we understand Einstein’s equations and the concepts brought by them as a
whole. In that direction, we have presented a model that could be considered
of physical interest just as a perfect fluid can 333Anisotropy is often
considered in conjunction to perfect fluids, for which $\Pi_{ij}=0$.
Typically, such models have a simple dynamics, with $\dot{\alpha}=0$ and
$\sigma_{ij}\sim a^{-3}$., but containing much more of the complex aspects of
the anisotropies than the latter. In particular, we have a broader view of
their global dynamics with the aid of the Kasner disc as well as we can follow
their tracks along the different epochs of the cosmic expansion, just as we
did with the Hubble anisotropy by using the parameter $b$, which enhanced our
view of this furtive concept. Last but not least important, the constraints
our model imposes on the observational data, in the lines of Ref. [32], is
something we have not touched upon in this manuscript. We leave this issue to
be accomplished elsewhere.

## Acknowledgments

FSD is partially supported by Fundação de Amparo à Pesquisa do Estado de Minas
Gerais – FAPEMIG [grant number APQ–01158–17]. LFM is partially supported by
Fundação de Amparo à Pesquisa do Estado de Minas Gerais – FAPEMIG [grant
number APQ–01105–18] and by Conselho Nacional de Desenvolvimento Científico e
Tecnológico – CNPq [grant number 311921/2020–5].

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