# Flows into de Sitter space from anisotropic initial conditions:
An effective field theory approach

Feraz Azhar<EMAIL_ADDRESS>Department of Philosophy, University of Notre Dame,
South Bend, IN 46556 USA  David I. Kaiser<EMAIL_ADDRESS>Department of
Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA

(August 27, 2024)

###### Abstract

For decades, physicists have analyzed various versions of a “cosmic no-hair”
conjecture, to understand under what conditions a spacetime that is initially
spatially anisotropic and/or inhomogeneous will flow into an isotropic and
homogeneous state. Wald’s theorem, in particular, established that homogeneous
but anisotropic spacetimes, if filled with a positive cosmological constant
plus additional matter sources that satisfy specific energy conditions, will
necessarily flow toward an (isotropic) de Sitter state at late times. In this
paper we study the flow of homogeneous but anisotropic spacetimes toward
isotropic states under conditions more general than those to which Wald’s
theorem applies. We construct an effective field theory (EFT) treatment for
generic “single-clock” systems in anisotropic spacetimes—which are not limited
to realizations compatible with scalar-field constructions—and identify fixed
points in the resulting phase space. We identify regions of this phase space
that flow to isotropic fixed points—including a de Sitter fixed point—even in
the absence of a bare cosmological constant, and for matter sources that do
not obey the energy conditions required for Wald’s theorem. Such flows into de
Sitter reveal the emergence of an effective cosmological constant.

##  I Introduction

Cosmic inflation provides the leading account of the dynamics of the very
early universe. (See Refs. Guth and Kaiser (2005); Bassett  _et al._ (2006);
Lyth and Liddle (2009); Martin  _et al._ (2014); Guth  _et al._ (2014);
Baumann and McAllister (2015) for reviews.) Models of inflation describe a
brief period of accelerated expansion of space and can account for the highly
symmetric spacetime we observe today—namely, one that is homogeneous and
isotropic to high accuracy on large scales, with relatively small differences
in energy density from place to place.

A natural question that arises is: how inhomogeneous and anisotropic can the
universe be prior to inflation such that (i) inflation initiates and (ii)
assuming that it does initiate, the inhomogeneities and anisotropies
dissipate? If highly symmetric initial states are required for (i) and (ii),
then models of inflation would face an “initial conditions problem,” such that
the conditions that inflation looks to explain would be required as starting
assumptions. This potential problem was recognized early in the history of the
development of cosmic inflation and has received renewed interest of late. For
recent discussion, see Refs. East  _et al._ (2016); Kleban and Senatore
(2016); Clough  _et al._ (2017, 2018); Brandenberger (2016); Linde (2018);
Bloomfield  _et al._ (2019); Chowdhury  _et al._ (2019); Aurrekoetxea  _et
al._ (2020); Creminelli  _et al._ (2020a, b); Azhar (2020); Joana and Clesse
(2021); Tenkanen and Tomberg (2020); Wang and Senatore (2021); Corman and East
(2022).

The question of whether initially inhomogeneous and anisotropic backgrounds
can evolve via inflation into homogeneous and isotropic backgrounds has
crystallized into a “cosmic no-hair” conjecture. There are various versions of
this conjecture (see Ref. (Schmidt, 2004, §9.5)) but it primarily amounts to
the claim that—given certain constraints placed on the matter content of the
universe, such as the presence of a cosmological constant, and assuming a
metric theory of gravity, such as general relativity—initially inhomogeneous
and/or anisotropic degrees of freedom dissipate as the universe expands. The
presence of a cosmological constant in certain formulations of the conjecture
serves as a proxy for a broad class of inflationary scenarios that employ
potential-energy domination as a means of driving accelerated expansion.

Probing spacetime dynamics assuming generic inhomogeneities and anisotropies
of space is a formidable challenge. A more tractable approach is to relax the
assumption of isotropy of the background space while preserving homogeneity,
and investigate any ensuing “isotropization.” This approach was influentially
addressed by Wald (1983). In particular, Wald demonstrated that initially
expanding homogeneous backgrounds (generically for all Bianchi spacetimes
except for Bianchi type IX)—with a positive cosmological constant and matter
degrees of freedom that satisfy the dominant and strong energy conditions—will
isotropize. Moreover, such conditions generically yield de Sitter-like
expansion at late times, in which the cosmological constant eventually
dominates the energy density of the background, leading to accelerated
expansion. (See Refs. Watanabe  _et al._ (2009); Maleknejad and Sheikh-Jabbari
(2012); Cannone  _et al._ (2015); East  _et al._ (2016); Kleban and Senatore
(2016); Andréasson and Ringström (2016); Clough  _et al._ (2017, 2018);
Brandenberger (2016); Linde (2018); Carroll and Chatwin-Davies (2018);
Bloomfield  _et al._ (2019); Chowdhury  _et al._ (2019); Aurrekoetxea  _et
al._ (2020); Creminelli  _et al._ (2020a, b); Azhar (2020); Joana and Clesse
(2021); Tenkanen and Tomberg (2020); Wang and Senatore (2021); Moss and Sahni
(1986); Kitada and Maeda (1993); Rendall (2004); Kleban and Senatore (2016);
Schmidt (2004); Corman and East (2022) for more recent, related work.)

Wald’s powerful theorem Wald (1983) requires that the matter sources (other
than the bare, positive cosmological constant) satisfy both the strong and the
dominant energy conditions. Yet since Wald’s original work, several groups
have demonstrated that fairly simple models can yield well-behaved
cosmological dynamics despite failing to satisfy any of the energy conditions
on which Wald’s theorem depends Kandrup (1992); Barcelo and Visser (1999);
Visser and Barcelo (2000); Barcelo and Visser (2000); Bellucci and Faraoni
(2002); Barcelo and Visser (2002); Dubovsky  _et al._ (2006); Nicolis  _et
al._ (2010); Rubakov (2014); Martin-Moruno and Visser (2017). Such examples
point to physically relevant scenarios that fall entirely outside the domain
to which Wald’s theorem applies. To assess the dynamics of anisotropic
spacetimes in such scenarios, one must develop distinct methods, complementary
to those with which Wald’s theorem has been explored to date.

We pursue such a new approach in this work. In our analysis, we do not assume
an explicit cosmological constant, nor do we place a priori conditions (such
as energy conditions) on the matter degrees of freedom. Rather than imposing
various energy conditions—the limitations of which have been scrutinized in
detail Kandrup (1992); Barcelo and Visser (1999); Visser and Barcelo (2000);
Barcelo and Visser (2000); Bellucci and Faraoni (2002); Barcelo and Visser
(2002); Dubovsky  _et al._ (2006); Nicolis  _et al._ (2010); Rubakov (2014);
Martin-Moruno and Visser (2017)—we adapt the powerful effective field theory
(EFT) approach to inflationary dynamics Cheung  _et al._ (2008); Weinberg
(2008). The EFT approach enables us to analyze the evolution of matter
sources, including anisotropic pressures and other features associated with
imperfect fluids, in quantitative detail yet also in a model-independent way.
In particular, within this framework, the accelerated expansion of space need
not be driven by the potential energy $V(\phi^{I})$ associated with one or
more scalar fields $\phi^{I}$. More generally, our “single-clock” formalism,
akin to the one developed in Ref. Azhar and Kaiser (2018), can accommodate
scenarios that are not limited to those compatible with canonical scalar-field
constructions.

Within this general framework, we identify regions within the effective phase
space that flow into (isotropic) de Sitter-like expansion at late times,
without having included a bare cosmological constant, and for matter sources
that need not satisfy either the strong or dominant energy conditions.111By
“late times,” we mean times $t\gg\tau$, where $\tau$ is the relevant dynamical
time-scale set by the inverse of the trace of the extrinsic curvature. Such
dynamics fall entirely outside the conditions under which Wald’s theorem Wald
(1983) holds. These examples do not contradict Wald’s theorem. Rather, they
indicate a new physical phenomenon—the emergence of an effective cosmological
constant—that could not be identified, let alone analyzed, in physical systems
to which Wald’s theorem applies.

In more detail, we construct an EFT of anisotropic backgrounds that describes
the simplest anisotropic spacetime, a Bianchi I cosmological model. Our
formalism consists of an action with effective matter degrees of freedom that
(i) reflect certain symmetries of the background space at early times and (ii)
are minimally coupled to (Einstein) gravity. We construct a dynamical system
from this EFT, without imposing energy conditions nor inserting an explicit
cosmological constant. We focus on the flow of such systems within the
“principal phase space,” the lowest-dimensional phase space of the class of
dynamical models developed here.

We investigate the stability of initially isotropic backgrounds in the
presence of anisotropic matter sources, identifying conditions under which the
space flows back to isotropy. (We do not treat more general questions about
the stability of de Sitter spacetime, related to possible infrared divergences
associated with light scalar fields Burgess  _et al._ (2016); Gorbenko and
Senatore (2019); Baumgart and Sundrum (2020).) We also investigate the
evolution of initially anisotropic backgrounds, identifying conditions under
which such spacetimes isotropize over time. For each of these scenarios, we
find nontrivial regions of parameter space that flow into isotropic, de
Sitter-like expansion at late times, despite the presence of significant
anisotropic matter sources at early times.

Our plan for this paper is as follows. In Sec. II we develop our EFT formalism
for treating anisotropic backgrounds and translate this EFT into a dynamical
system. We describe fixed points and stability properties for these fixed
points for the principal phase space. In Sec. III we describe our results
where, in Sec. III.1, we probe stability properties of isotropic backgrounds,
and in Sec. III.2 we demonstrate cases that isotropize at late times, with
space undergoing accelerated expansion due to the emergence of an effective
(rather than a bare) cosmological constant. Concluding remarks follow in Sec.
IV, and we collect some relevant results in Appendixes A–D.

##  II  EFT formalism

We begin by introducing, for the class of models in which we are interested:
the relevant effective degrees of freedom, a characterization of energy
conditions (to facilitate a comparison with previous work), and dynamical
equations. We build upon EFT techniques first introduced in Refs. Cheung  _et
al._ (2008); Weinberg (2008); techniques for handling dynamical systems
introduced in Ref. Frusciante  _et al._ (2014) (further developed in Ref.
Azhar and Kaiser (2018)); and a parameterization for anisotropic backgrounds
introduced in Ref. van den Hoogen and Coley (1995). We work in units where
$\hbar=c=1$, so that the reduced Planck mass may be written as
$M_{\textrm{pl}}=(8\pi G)^{-1/2}\simeq 2.4\times 10^{18}\,\textrm{GeV}$. We
restrict attention to four spacetime dimensions and adopt the metric signature
$(-,+,+,+)$. Lower-case Greek letters $\alpha,\beta,\dots=0,1,2,3$, label
spacetime indices and lower-case Latin letters $i,j,\dots=1,2,3$, label
spatial indices.

###  II.1 An EFT of anisotropic backgrounds

Wald derived his influential cosmic no-hair theorem Wald (1983) for the case
of a universe filled with a bare, positive cosmological constant $\Lambda$
plus additional matter sources described by an energy-momentum tensor
$\tilde{T}_{\mu\nu}$, such that the total energy-momentum tensor for the
system took the form

$T_{\mu\nu}=-\Lambda g_{\mu\nu}+\tilde{T}_{\mu\nu}.$ (1)

Wald’s proof relied upon several postulates. First, that the dynamics of the
system were governed by Einstein’s field equations of ordinary general
relativity, with $T_{\mu\nu}$ of the form in Eq. (1). Second, that the trace
of the extrinsic curvature of the spacetime was positive at an initial time,
corresponding to an initial volume-averaged expansion of space. Third—and
crucially—that $\tilde{T}_{\mu\nu}$ satisfied both the dominant energy
condition,

$\tilde{T}_{\mu\nu}u^{\mu}u^{\nu}\geq 0,$ (2)

and the strong energy condition,

$\left(\tilde{T}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\tilde{T}\right)u^{\mu}u^{\nu}\geq
0,$ (3)

where $u^{\mu}$ is a future-directed time-like unit vector orthogonal to the
spatial hypersurfaces. Under those conditions, Wald elegantly demonstrated
that the square of the shear tensor
$\sigma^{2}\equiv\sigma_{\alpha\beta}\sigma^{\alpha\beta}/2$ must vanish
exponentially quickly over time, on a time-scale set by $M_{\rm
pl}/\Lambda^{1/2}$. The shear tensor is defined as Hawking and Ellis (1973);
Wald (1984); Visser (1995)

$\sigma_{\alpha\beta}\equiv\frac{1}{2}\left(\nabla_{\nu}u_{\mu}+\nabla_{\mu}u_{\nu}\right)h^{\mu}_{\>\>\alpha}h^{\nu}_{\>\>\beta}-\frac{1}{3}\nabla_{\mu}u^{\mu}\,h_{\alpha\beta},$
(4)

where

$h_{\mu\nu}\equiv g_{\mu\nu}+u_{\mu}u_{\nu}$ (5)

is the metric on spatial hypersurfaces. The quantity $\sigma^{2}$ vanishes
identically for spacetimes that are homogeneous and isotropic, but in general
is nonvanishing for anisotropic spacetimes. Therefore the condition
$\sigma^{2}\rightarrow 0$ indicates isotropization.

Critical to Wald’s theorem are both the presence of a bare, positive
cosmological constant, $\Lambda>0$, and the constraint that the other sources
of matter obey both the dominant and strong energy conditions of Eqs. (2)–(3).
Given the discussion in Refs. Barcelo and Visser (1999); Visser and Barcelo
(2000); Barcelo and Visser (2000, 2002); Bellucci and Faraoni (2002); Dubovsky
_et al._ (2006); Nicolis  _et al._ (2010); Rubakov (2014); Martin-Moruno and
Visser (2017) of how readily these (and related) energy conditions can be
violated by simple, well-behaved systems, on the other hand, we study the
dynamics of effective systems in which we set $\Lambda=0$, and we do not
require that $\tilde{T}_{\mu\nu}$ obey either the dominant or strong energy
conditions.

In order to study whether anisotropic spacetimes can isotropize over time
under conditions more general than those to which Wald’s theorem applies, we
must therefore develop a complementary approach to the one deployed so
powerfully in Ref. Wald (1983). We develop the key ingredients of our
alternative approach in this section.

We study dynamical flows for a homogeneous spacetime that is generally
anisotropic, with a line element corresponding to a Bianchi I background given
by

$ds^{2}=-dt^{2}+a^{2}_{i}(t)(dx^{i})^{2}.$ (6)

The three functions $a_{i}(t)$ (for $i=1,2,3$) encode (generally different)
scale factors for expansion in each of three independent spatial directions.
One may define an average scale factor, $a(t)$, as the geometric mean of these
three scale factors:

$a(t)\equiv\left[a_{1}(t)a_{2}(t)a_{3}(t)\right]^{1/3}.$ (7)

One can then rewrite Eq. (6) as

$ds^{2}=-dt^{2}+a^{2}(t)\,\delta_{ij}\,e^{2\beta_{i}(t)}\,dx^{i}\,dx^{j}\,,$
(8)

so that the functions $\beta_{i}(t)$ correspond to the anisotropic parts of
the scale factors $a_{i}(t)$. Equating coefficients in Eqs. (6) and (8) and
then using Eq. (7) yields a consistency condition:

$\beta_{1}(t)+\beta_{2}(t)+\beta_{3}(t)=0\,.$ (9)

Although the line element in Eq. (8) represents a particularly simple form of
a homogeneous yet anisotropic spacetime, for our purposes this ansatz is
actually quite general. As demonstrated in Ref. Cropp and Visser (2011), the
immediate vicinity of any timelike geodesic in an arbitrary spacetime can be
described by a line element corresponding to a Bianchi I spacetime in the so-
called “ultra-local limit.” Eq. (8) therefore provides a fairly general basis
for constructing an effective field theory treatment.

To lowest order in our effective description, we consider fluid quantities
that are spatially homogeneous (hence they depend only on cosmic time, $t$),
but that are not necessarily spatially isotropic (hence they need not remain
spherically symmetric at any given time $t$). In addition to the average scale
factor $a(t)$ and its change over time, we also need to consider a nonzero
shear tensor $\sigma_{\alpha\beta}$. For the line element of Eq. (8), the
nonvanishing components of $\sigma_{\alpha\beta}$ are given by

$\sigma_{ij}=a^{2}\dot{\beta}_{i}e^{2\beta_{i}}\,\delta_{ij}\quad{\rm(no\>sum\>on\>}i)\,,$
(10)

where overdots denote a derivative with respect to cosmic time, $t$, and hence

$\sigma^{2}\equiv\frac{1}{2}\sigma_{\alpha\beta}\,\sigma^{\alpha\beta}=\frac{1}{2}\left(\dot{\beta}_{1}^{2}+\dot{\beta}_{2}^{2}+\dot{\beta}_{3}^{2}\right)\,.$
(11)

For a line element of the form in Eq. (8), an isotropic spacetime corresponds
to the case $\dot{\beta}_{i}\rightarrow 0$ for each $i$, and hence
$\sigma^{2}\rightarrow 0$.

Much as in Ref. Azhar and Kaiser (2018), we are interested in dynamics
associated with an action that is linear in perturbations about the background
(and, as usual, is described in unitary gauge). We are interested in dynamics
of the background, so we ignore terms that are quadratic (and higher) in
perturbations and that contain terms suppressed in the EFT expansion by an
assumed large energy scale. (See Ref. Pereira  _et al._ (2007) for a gauge-
invariant treatment of spatially inhomogeneous perturbations about a Bianchi I
spacetime.) We thus consider an effective action for matter degrees of freedom
of the form:

$\displaystyle S^{({\cal M})}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\left[-L(t)-c(t)g^{00}-X(t)\left(g^{11}-g^{22}\right)-Y(t)\left(g^{11}-g^{33}\right)\right]$
(12) $\displaystyle\equiv\int
d^{4}x\sqrt{-g}\left[-L(t)-c(t)g^{00}-X_{11}(t)g^{11}-X_{22}(t)g^{22}-X_{33}(t)g^{33}\right]\,,$
(13)

where ‘${\cal M}$’ stands for ‘matter.’ In Eq. (12), we have introduced four
free functions of time, $L(t),c(t),X(t)$, and $Y(t)$. Whereas only two of
these functions, $L(t)$ and $c(t)$, are required (at background order) to
account for the dynamics of a homogeneous and isotropic Friedmann-Lemaître-
Robertson-Walker (FLRW) spacetime Cheung  _et al._ (2008); Weinberg (2008);
Azhar and Kaiser (2018), we require two more functions, $X(t)$ and $Y(t)$, for
the anisotropic case, given the metric functions $\beta_{i}$ of Eq. (8)
subject to the constraint of Eq. (9). In Eq. (13), to more easily keep track
of indices, we have introduced the identifications: $X_{11}(t)\equiv
X(t)+Y(t)$, $X_{22}(t)\equiv-X(t)$, $X_{33}(t)\equiv-Y(t)$. As a result of
these identifications, we have

$\displaystyle X_{11}(t)+X_{22}(t)+X_{33}(t)=0.$ (14)

The authors of Ref. Cannone  _et al._ (2015), in considering backgrounds that
perturb away from a FLRW background, include comparable terms proportional to
the inverse metric in the effective action, and then work to lowest order in
the terms that multiply the inverse metric. In our case, we need not consider
the anisotropic source terms $X_{ii}$ to be perturbatively small. On the other
hand, the authors of Ref. Cannone  _et al._ (2015) include a term in their
effective action proportional to $g^{0i}$ that would be sourced by a
nonvanishing momentum flux $T^{0i}$, but, as we will describe shortly, such a
term must vanish in general in a spatially homogeneous spacetime, so we
neglect it here.

Varying $S^{({\cal M})}$ with respect to $g^{\mu\nu}$ yields an effective
energy-momentum tensor:

$\begin{split}T_{\mu\nu}&=g_{\mu\nu}\left[-L(t)-c(t)g^{00}-\tilde{X}(t)\right]-2\frac{\delta}{\delta
g^{\mu\nu}}\left[-L(t)-c(t)g^{00}-\tilde{X}(t)\right]\\\
&=g_{\mu\nu}\left[c(t)-L(t)-\tilde{X}(t)\right]+2c(t)\,\delta^{0}_{\>\>\mu}\delta^{0}_{\>\>\nu}+2\sum_{i}X_{ii}(t)\,\delta^{i}_{\>\>\mu}\delta^{i}_{\>\>\nu}\,.\end{split}$
(15)

For convenience, we have introduced the notation

$\tilde{X}(t)\equiv X_{11}(t)g^{11}+X_{22}(t)g^{22}+X_{33}(t)g^{33}\,.$ (16)

We may write Eq. (15) in the canonical form,

$T_{\mu\nu}=\rho\,u_{\mu}u_{\nu}+P(g_{\mu\nu}+u_{\mu}u_{\nu})+\pi_{\mu\nu}\,,$
(17)

where $\rho$ is the isotropic energy density, $P$ is the isotropic pressure,
$\pi_{\mu\nu}$ is the anisotropic pressure, and $u^{\mu}$ is the effective
fluid 4-velocity, which (to background order) takes the form
$u^{\mu}=\delta^{\mu}_{\>\>0}$. In this reference frame, the anisotropic
pressure is orthogonal to the fluid velocity, $u^{\mu}\pi_{\mu\nu}=0$, and
hence $\pi_{0\nu}=0$ for all $\nu$ (again, to background order, that is, in
the absence of spatial inhomogeneities). Likewise, $\pi_{\mu\nu}$ is
symmetrical and traceless: $\pi_{\mu\nu}=\pi_{\nu\mu}$ and
$\pi^{\mu}_{\>\>\mu}=0$, respectively.

Equivalently, we may define the fluid components in terms of the mixed-index
energy-momentum tensor (see, for example, Refs. Bassett  _et al._ (2006); Lyth
and Liddle (2009)):

$\begin{split}T^{0}_{\>\>0}&=-\rho\,,\\\ T^{0}_{\>\>i}&=\partial_{i}\delta
q\,,\\\ T^{i}_{\>\>j}&=\delta^{i}_{\>\>j}P+\pi^{i}_{\>\>j}\,.\end{split}$ (18)

In a homogeneous universe, the spatial gradient of the momentum flux
$\partial_{i}\delta q$ vanishes. This form also makes manifest that the
anisotropic pressure $\pi_{ij}$ cannot be proportional to $\delta_{ij}$; if it
were, then it would simply contribute to the “ordinary” (isotropic) pressure
$P$. Given that the Einstein tensor associated with the line element in Eq.
(8) has vanishing off-diagonal terms, $G_{ij}=0$ for $i\neq j$, we expect the
off-diagonal components within $\pi_{ij}$ to vanish as well. We therefore
expect the anisotropic pressure for the case of interest to include up to
three nonvanishing components, $\pi_{11}$, $\pi_{22}$, and $\pi_{33}$, which
are not all equal to each other.

In Appendix A, we demonstrate that if the universe were filled only with
scalar fields, each with a canonical coupling to gravity and with gravity
described by the usual Einstein-Hilbert action, then the anisotropic pressure
would vanish (at background order, neglecting spatial inhomogeneities). In
that case, $\pi_{ij}=0$ for all $i,j$, and then $T_{ij}\propto g_{ij}$, even
for multifield models with nontrivial field-space manifolds. On the other
hand, if we consider generalizations beyond canonical general relativity, such
as the broader class of Horndeski actions, then there do arise terms in the
effective action that behave like an anisotropic pressure, even for spatially
homogeneous spacetimes filled only with scalar fields. (See also Ref. Rubakov
(2014).) More generally, anisotropic pressures can arise even with canonical
general relativity if the universe contains sources of matter with nonzero
spin, such as vector fields Cropp and Visser (2011).

Dropping explicit dependences on time, we may write Eq. (15) as

$T_{\mu\nu}=\left(c+L+\tilde{X}\right)u_{\mu}u_{\nu}+\left(c-L-\frac{1}{3}\tilde{X}\right)\left(g_{\mu\nu}+u_{\mu}u_{\nu}\right)+\pi_{\mu\nu}\,,$
(19)

with

$\pi_{\mu\nu}\equiv
2\sum_{i=1}^{3}X_{ii}\left(-\frac{1}{3}h_{\mu\nu}\,g^{ii}+\delta^{i}_{\>\mu}\,\delta^{i}_{\>\nu}\right)$
(20)

and $h_{\mu\nu}$ defined in Eq. (5). Comparing Eqs. (17) and (19), we may
define

$\rho\equiv c+L+\tilde{X}\,,\>\>P\equiv c-L-\frac{1}{3}\tilde{X}\,,$ (21)

and then, as expected,

$\pi_{\mu\nu}=\pi_{\nu\mu}\,,\>\>u^{\mu}\,\pi_{\mu\nu}=0\,,\>\>g^{\mu\nu}\,\pi_{\mu\nu}=0\,.$
(22)

We may also introduce the three nonvanishing components of the anisotropic
pressure that source the components of the shear tensor:

$p_{i}\equiv(g_{ii})^{-1}\,\pi_{ii}=-\frac{2}{3}\tilde{X}+2(g_{ii})^{-1}\,X_{ii}\>\>\>\>({\rm
no\>\>sum\>\>on\>\>}i)\,.$ (23)

The tracelessness of $\pi_{\mu\nu}$ translates to the constraint that

$p_{1}+p_{2}+p_{3}=0\,.$ (24)

Note that in the isotropic limit, with $X_{ii}=0$ (for each $i$), $c$
contributes to $\rho$ and $P$ like the kinetic energy of a canonically
normalized scalar field, and $L$ contributes like the scalar field’s potential
$V$ Azhar and Kaiser (2018).

In order to compare properties of our system with previous investigations of
cosmic no-hair theorems, including Wald’s work Wald (1983), we next identify
salient pointwise energy conditions for the effective matter degrees of
freedom (see Refs. Hawking and Ellis (1973); Wald (1984); Visser (1995);
Visser and Barcelo (2000); Barcelo and Visser (2000, 2002); Rubakov (2014);
Martin-Moruno and Visser (2017)). The null energy condition (NEC) may be
written (where there is no sum on $i$ in any term)

$\textrm{NEC: }\>\rho+P+p_{i}\geqslant 0\equiv c+g^{ii}X_{ii}\geqslant 0.$
(25)

The weak energy condition (WEC) becomes

WEC: $\displaystyle\rho\geqslant\;0\;\;\&\;\;\rho+P+p_{i}\geq 0$
$\displaystyle\equiv c+L+\tilde{X}\geqslant 0\;\;\&\;\;c+g^{ii}X_{ii}\geqslant
0.$ (26)

The dominant energy condition (DEC) may be written

$\displaystyle\textrm{DEC: }\>\rho\geqslant 0\;\;\&\;\;\lvert
P+p_{i}\rvert\leqslant\rho$ $\displaystyle\;\equiv c+L+\tilde{X}\geqslant
0\;\;\&\;\;0\leqslant c+g^{ii}X_{ii}\leqslant c+L+\tilde{X}.$ (27)

And finally, the strong energy condition (SEC) becomes

SEC: $\displaystyle\rho+3P\geqslant\;0\;\;\&\;\;\rho+P+p_{i}\geq 0$
$\displaystyle\equiv 2c-L\geqslant 0\;\;\&\;\;c+g^{ii}X_{ii}\geqslant 0.$ (28)

As noted in Refs. Barcelo and Visser (1999); Visser and Barcelo (2000);
Barcelo and Visser (2000, 2002); Bellucci and Faraoni (2002); Dubovsky  _et
al._ (2006); Nicolis  _et al._ (2010); Rubakov (2014); Martin-Moruno and
Visser (2017), although many powerful theorems in general relativity rely on
the assumption of various energy conditions, the pointwise energy conditions
in Eqs. (25)–(28) can be violated by scalar fields in curved spacetimes (even
at the classical level), without necessarily inducing unphysical
instabilities. Therefore we do not impose constraints such as the DEC or SEC
on our system a priori, in contrast to previous studies of isotropization.

The Einstein field equations yield coupled equations of motion for various
dynamical quantities. Given the form of $T_{\mu\nu}$ in Eq. (19), the
generalized Friedmann equations for $H\equiv\dot{a}/a$ take the form

$H^{2}=\frac{1}{3M_{\rm
pl}^{2}}\left(c+L+\tilde{X}\right)+\frac{1}{3}\sigma^{2}\,,$ (29)
$\dot{H}+H^{2}=-\frac{1}{3M_{\rm
pl}^{2}}\left(2c-L\right)-\frac{2}{3}\sigma^{2}\,.$ (30)

Each anisotropic scale factor evolves as

$\ddot{\beta}_{i}+3H\dot{\beta}_{i}=\frac{1}{M_{\rm pl}^{2}}p_{i}\,,$ (31)

from which one may evaluate the evolution of the shear scalar

$\frac{d}{dt}\sigma^{2}+6H\sigma^{2}=\frac{1}{M_{\rm
pl}^{2}}\left(\dot{\beta}_{1}p_{1}+\dot{\beta}_{2}p_{2}+\dot{\beta}_{3}p_{3}\right)\,.$
(32)

Finally, the continuity equation $\nabla_{\nu}\,T^{\nu}_{\>\>\mu}=0$ yields

$\dot{\rho}+3H(\rho+P)+\sigma_{\mu\nu}\,\pi^{\mu\nu}=0\,.$ (33)

Upon using Eqs. (20) and (21), this expression may be written as

$\dot{c}+\dot{L}+\dot{\tilde{X}}+3H\left(2c+\frac{2}{3}\tilde{X}\right)+\dot{\beta}_{1}p_{1}+\dot{\beta}_{2}p_{2}+\dot{\beta}_{3}p_{3}=0\,.$
(34)

One may readily check that the two Friedmann equations in Eqs. (29) and (30)
are mutually consistent upon using Eq. (34). Upon subtracting Eq. (29) from
Eq. (30), we can solve for $\dot{H}$, and from that expression we can solve
for the Hubble slow-roll parameter $\epsilon$:

$\epsilon\equiv-\frac{\dot{H}}{H^{2}}=\frac{1}{3M_{\rm
pl}^{2}H^{2}}\left(3c+\tilde{X}\right)+\frac{\sigma^{2}}{H^{2}}\,.$ (35)

As expected, this expression reduces to the usual case, in which
$\epsilon\propto c$ Azhar and Kaiser (2018), when anisotropic degrees of
freedom vanish: $\tilde{X},\sigma^{2}\rightarrow 0$.

Much as in Wald’s analysis Wald (1983), we restrict attention to scenarios in
which the initial extrinsic curvature is positive, corresponding to volume-
averaged expansion of space. For the line element of Eq. (8), the lapse
function $N(x^{\mu})=1$ and, since $g_{0i}=0$, the shift vector vanishes, so
the extrinsic curvature takes the simple form Wald (1984); Cropp and Visser
(2011)

$K_{ij}=\frac{1}{2}\partial_{t}h_{ij}=(H+\dot{\beta}_{i})h_{ij},$ (36)

where $h_{ij}=a^{2}(t)\delta_{ij}e^{2\beta_{i}(t)}$ is the metric on equal-
time spatial slices. Then we find

$K=h^{ij}K_{ij}=3H+\sum_{i}\dot{\beta}_{i}=3H,$ (37)

where the final expression follows upon using the time derivative of the
constraint in Eq. (9). Hence we see that, much as for an FLRW spacetime, $K>0$
corresponds to the case of (volume-averaged) spatial expansion, $H>0$, upon
which we will focus in our analysis.

###  II.2 New variables

We now introduce a change of variables that will allow us to rewrite the
Einstein equations expressed in Eqs. (29), (30), and (31), as well as the
continuity equation in Eq. (34), in a form that makes them amenable to a
useful dynamical analysis. (See also Ref. van den Hoogen and Coley (1995) for
a complementary analysis.)

First note that the shear tensor, $\sigma_{\alpha\beta}$, is traceless:
$\sigma^{\mu}_{\;\;\mu}=0$. This reduces the three non-zero (purely spatial)
components of $\sigma_{\alpha\beta}$ to two independent components. We can
represent these two independent components with the following new variables:

$\displaystyle\sigma_{1}$
$\displaystyle\equiv\sigma^{1}_{\;\;1}-\sigma^{2}_{\;\;2},$ (38)
$\displaystyle\sigma_{2}$
$\displaystyle\equiv\sigma^{1}_{\;\;1}-\sigma^{3}_{\;\;3}.$ (39)

For the Bianchi I background in Eq. (8), these are given by

$\displaystyle\sigma_{1}$ $\displaystyle=\dot{\beta}_{1}-\dot{\beta}_{2},$
(40) $\displaystyle\sigma_{2}$
$\displaystyle=\dot{\beta}_{1}-\dot{\beta}_{3}.$ (41)

One may now show—using the (derivative of the) constraint on the sum of the
$\beta_{i}$’s in Eq. (9)—that the quantity we had previously used to signal
the degree of anisotropy of the background space, namely $\sigma^{2}$, can be
written as

$\displaystyle\sigma^{2}=\frac{1}{3}(\sigma_{1}+\sigma_{2})^{2}-\sigma_{1}\sigma_{2}.$
(42)

A similar procedure can be implemented for the anisotropic pressure
$\pi_{\mu\nu}$, since it is also traceless, symmetric, and purely
spatial—which means it has just two independent components. We thus define

$\displaystyle\pi_{1}$ $\displaystyle\equiv\pi^{1}_{\;\;1}-\pi^{2}_{\;\;2},$
(43) $\displaystyle\pi_{2}$
$\displaystyle\equiv\pi^{1}_{\;\;1}-\pi^{3}_{\;\;3}.$ (44)

One finds, using Eqs. (20) and (23), that $\pi^{i}_{\;\;i}=p_{i}$ (no sum on
$i$). These new variables thus each encode differences in the anisotropic
pressures:

$\displaystyle\pi_{1}$ $\displaystyle=p_{1}-p_{2},$ (45)
$\displaystyle\pi_{2}$ $\displaystyle=p_{1}-p_{3}.$ (46)

In describing the Einstein field equations with the variables
$\\{\sigma_{1},\sigma_{2},\pi_{1},\pi_{2}\\}$, one may replace the evolution
equations for the $\beta_{i}$’s in Eq. (31) with equations of evolution for
the $\sigma_{A}$’s, with $A=1,2$. Indeed, employing Eq. (31), one finds

$\displaystyle\dot{\sigma}_{A}$
$\displaystyle=-3H\sigma_{A}+\frac{1}{M_{\textrm{pl}}^{2}}\pi_{A}.$ (47)

Finally, the continuity equation of Eq. (34) can be rewritten in terms of
these new variables by using the constraint on the sum of the $p_{i}$’s in Eq.
(24), and the derivative of the constraint on the sum of the $\beta_{i}$’s in
Eq. (9). We find

$\displaystyle\dot{\beta}_{1}p_{1}+\dot{\beta}_{2}p_{2}+\dot{\beta}_{3}p_{3}=\frac{1}{3}\left[\sigma_{1}\left(2\pi_{1}-\pi_{2}\right)+\sigma_{2}\left(2\pi_{2}-\pi_{1}\right)\right].$
(48)

Using Eqs. (42), (47), and (48), we may then rewrite the coupled dynamical
equations as

$\displaystyle H^{2}$
$\displaystyle=\frac{1}{3M_{\textrm{pl}}^{2}}\left(c+L+\tilde{X}\right)+\frac{1}{9}(\sigma_{1}+\sigma_{2})^{2}-\frac{1}{3}\sigma_{1}\sigma_{2},$
(49) $\displaystyle\dot{H}+H^{2}$
$\displaystyle=-\frac{1}{3M_{\textrm{pl}}^{2}}(2c-L)-\frac{2}{9}(\sigma_{1}+\sigma_{2})^{2}+\frac{2}{3}\sigma_{1}\sigma_{2},$
(50) $\displaystyle\dot{\sigma}_{1}$
$\displaystyle=-3H\sigma_{1}+\frac{1}{M_{\textrm{pl}}^{2}}\pi_{1},$ (51)
$\displaystyle\dot{\sigma}_{2}$
$\displaystyle=-3H\sigma_{2}+\frac{1}{M_{\textrm{pl}}^{2}}\pi_{2},$ (52)
$\displaystyle 0$
$\displaystyle=\dot{c}+\dot{L}+\dot{\tilde{X}}+2H\left(3c+\tilde{X}\right)+\frac{1}{3}\left[\sigma_{1}\left(2\pi_{1}-\pi_{2}\right)+\sigma_{2}\left(2\pi_{2}-\pi_{1}\right)\right].$
(53)

###  II.3 A dynamical analysis

Equations (49)–(53) suggest expansion-normalized, dimensionless dynamical
variables, which yield a dynamical system with some useful features for our
analysis of isotropization of homogeneous spaces. In particular, one may
extract a closed dynamical system in which one can identify (i) de Sitter
evolution of the background space and (ii) necessary and sufficient conditions
for the satisfaction of each energy condition, purely in terms of the
dynamical variables. This latter feature helps to facilitate a comparison with
previous work, including Wald’s theorem Wald (1983).

Following Ref. Azhar and Kaiser (2018) (see also Ref. Frusciante  _et al._
(2014)) we define the dimensionless variables

$\displaystyle x$
$\displaystyle\equiv\frac{c+L+\tilde{X}}{3M_{\textrm{pl}}^{2}H^{2}},$ (54)
$\displaystyle y$
$\displaystyle\equiv\frac{3c+\tilde{X}}{3M_{\textrm{pl}}^{2}H^{2}}\,,\qquad\lambda_{l}\equiv-\frac{(3c+\tilde{X})^{(l+1)}}{H(3c+\tilde{X})^{(l)}}\,,$
(55) $\displaystyle\omega_{1}$
$\displaystyle\equiv\frac{\pi_{1}}{3M_{\textrm{pl}}^{2}H^{2}}\,,\qquad\mu_{m}\equiv-\frac{\pi_{1}^{(m+1)}}{H\pi_{1}^{(m)}}\,,$
(56) $\displaystyle\omega_{2}$
$\displaystyle\equiv\frac{\pi_{2}}{3M_{\textrm{pl}}^{2}H^{2}}\,,\qquad\nu_{n}\equiv-\frac{\pi_{2}^{(n+1)}}{H\pi_{2}^{(n)}}\,,$
(57) $\displaystyle z_{1}$ $\displaystyle\equiv\frac{\sigma_{1}}{3H},$ (58)
$\displaystyle z_{2}$ $\displaystyle\equiv\frac{\sigma_{2}}{3H}.$ (59)

Here $(m)$ denotes the $m$th derivative with respect to cosmic time $t$, and
$l,m,n\geq 0$. The expressions for $\lambda_{l}$, $\mu_{m}$, and $\nu_{n}$
introduce infinite towers of dimensionless variables that encode implicit
choices for the functional dependences of $3c+\tilde{X}$, $\pi_{1}$, and
$\pi_{2}$ (respectively) on $t$—though in practice one need only consider a
finite number of such variables for a given phase-space analysis. Note further
that in light of Eq. (21), we may write

$\displaystyle x$ $\displaystyle=\frac{\rho}{3M_{\textrm{pl}}^{2}H^{2}},$ (60)
$\displaystyle y$ $\displaystyle=\frac{\rho+P}{2M_{\textrm{pl}}^{2}H^{2}}.$
(61)

Thus we can interpret $x$ as an expansion-normalized isotropic (effective)
energy density, and $y$ as a normalized sum of the isotropic energy density
and the isotropic pressure. As such, in a purely isotropic spacetime, the
vanishing of $y$ corresponds to a background with a constant energy
density—that is, effectively, to a background with a cosmological constant.

One may also derive a set of coupled, ordinary differential equations with
respect to which we may perform a dynamical-systems analysis. We find

$\displaystyle\frac{dx}{d\ln a}=2\epsilon x-2y$ $\displaystyle-
z_{1}(2\omega_{1}-\omega_{2})-z_{2}(2\omega_{2}-\omega_{1}),$ (62)
$\displaystyle\frac{dy}{d\ln a}$ $\displaystyle=2\epsilon y-\lambda_{0}y,$
(63) $\displaystyle\frac{d\omega_{1}}{d\ln a}$
$\displaystyle=2\epsilon\omega_{1}-\mu_{0}\omega_{1},$ (64)
$\displaystyle\frac{d\omega_{2}}{d\ln a}$
$\displaystyle=2\epsilon\omega_{2}-\nu_{0}\omega_{2},$ (65)
$\displaystyle\frac{dz_{1}}{d\ln a}$
$\displaystyle=z_{1}(\epsilon-3)+\omega_{1},$ (66)
$\displaystyle\frac{dz_{2}}{d\ln a}$
$\displaystyle=z_{2}(\epsilon-3)+\omega_{2},$ (67)
$\displaystyle\frac{d\lambda_{l}}{d\ln a}$
$\displaystyle=\lambda_{l}\left(-\lambda_{l+1}+\lambda_{l}+\epsilon\right),$
(68) $\displaystyle\frac{d\mu_{m}}{d\ln a}$
$\displaystyle=\mu_{m}\left(-\mu_{m+1}+\mu_{m}+\epsilon\right),$ (69)
$\displaystyle\frac{d\nu_{n}}{d\ln a}$
$\displaystyle=\nu_{n}\left(-\nu_{n+1}+\nu_{n}+\epsilon\right),$ (70)

for $l,m,n\geq 0$ and where the Hubble slow-roll parameter, $\epsilon$,
described in Eq. (35), is given by

$\displaystyle\epsilon\equiv-\frac{\dot{H}}{H^{2}}=y+3(z_{1}+z_{2})^{2}-9z_{1}z_{2}.$
(71)

Furthermore, there is a constraint, derived from the first Friedmann equation,
Eq. (49):

$\displaystyle 1=x+(z_{1}+z_{2})^{2}-3z_{1}z_{2}.$ (72)

Equations (62)–(72) do not form a closed system due to the three infinite
towers in Eqs. (68)–(70), but we can construct a closed system by fixing
$\lambda_{L},\mu_{M},\nu_{N}$ to be constants for some fixed $L,M,N\geq 0$. In
such a case, the effective phase space will naturally be coordinatized by the
variables $x,y,\omega_{1},\omega_{2},z_{1},z_{2}$, together with
$\lambda_{0},\lambda_{1},\dots,\lambda_{L-1},\mu_{0},\mu_{1},\dots,\mu_{M-1},\nu_{0},\nu_{1},\dots,\nu_{N-1}$
(when $L,M,N>0$). Taking into account the constraint in Eq. (72), the dynamics
will thus be $(6+L+M+N-1)$-dimensional.

The dynamics can be further simplified because there exist invariant manifolds
for any choice of $(L,M,N)$. For example, it is straightforward to show that
the constraint surface is an invariant manifold because
$d\left[x+(z_{1}+z_{2})^{2}-3z_{1}z_{2}-1\right]/d\ln
a=2\epsilon\left[x+(z_{1}+z_{2})^{2}-3z_{1}z_{2}-1\right]=0$, given Eq. (72).
Hence the system does not move off the constraint surface if it begins on it.
Similarly, the following surfaces are also invariant manifolds: $y=0$;
$\omega_{A}=0$, for $A=1,2$; $\lambda_{l}=0$, for $l=0,1,\dots,L-1$;
$\mu_{m}=0$, for $m=0,1,\dots,M-1$; and $\nu_{n}=0$, for $n=0,1,\dots,N-1$.

For any particular phase space—defined by fixing, for some $(L,M,N)$,
$\lambda_{L},\mu_{M}$, and $\nu_{N}$ to be constants—the dynamics in that
phase space are consistent with

$\displaystyle{(3c+\tilde{X})}^{(L)}(t)$
$\displaystyle={(3c+\tilde{X})}^{(L)}(t_{i})\left[\frac{a(t_{i})}{a(t)}\right]^{\lambda_{L}},$
(73) $\displaystyle\pi_{1}^{(M)}(t)$
$\displaystyle=\pi_{1}^{(M)}(t_{i})\left[\frac{a(t_{i})}{a(t)}\right]^{\mu_{M}},$
(74) $\displaystyle\pi_{2}^{(N)}(t)$
$\displaystyle=\pi_{2}^{(N)}(t_{i})\left[\frac{a(t_{i})}{a(t)}\right]^{\nu_{N}}.$
(75)

Here $t_{i}$ is some fixed initial time. Thus, for example, setting $\mu_{M}$
to be a constant corresponds to assuming that the $M$th time derivative of
$\pi_{1}(t)$ scales as $[a(t)]^{-\mu_{M}}$.

Note also that for any phase space one can uniquely identify when the
background is isotropic. In particular (as shown in Appendix B) assuming a
finite Hubble expansion rate $H$:

$\textrm{Background is isotropic}\iff z_{1}=0=z_{2},$ (76)

since the variables $z_{A}$ (for $A=1,2$) are proportional to the independent
components of the shear tensor $\sigma_{\alpha\beta}$. Furthermore, we can
identify when such a background undergoes de Sitter expansion: for $z_{A}=0$,
the Hubble slow-roll parameter becomes $\epsilon=y$, which vanishes whenever
$y=0$. Thus the background undergoes de Sitter evolution whenever
$y=z_{1}=z_{2}=0$.

One may also now express the energy conditions described in Sec. II.1 purely
in terms of the dynamical variables introduced in Eqs. (54)–(59). The NEC
amounts to the condition that, for each $i$,

$\displaystyle c+g^{ii}X_{ii}\geqslant
0\iff\frac{c+g^{ii}X_{ii}}{3M_{\textrm{pl}}^{2}H^{2}}\geqslant 0.$ (77)

The three conditions on the right-hand side of Eq. (77) can be expressed in
terms of the dynamical variables described above, so that the NEC is satisfied
iff each of the following three conditions is satisfied:

$\textrm{NEC}\begin{cases}2y+\omega_{1}+\omega_{2}&\geqslant 0,\\\
2y-2\omega_{1}+\omega_{2}&\geqslant 0,\\\ 2y+\omega_{1}-2\omega_{2}&\geqslant
0.\end{cases}$ (78)

Similarly, the WEC is satisfied iff each condition for the NEC is satisfied,
together with one extra condition:

$\textrm{WEC: NEC \&}\;x\geqslant 0.$ (79)

The DEC is satisfied iff each of the following four conditions is satisfied:

$\textrm{DEC}\begin{cases}6x&\geqslant 2y+\omega_{1}+\omega_{2}\geqslant 0,\\\
6x&\geqslant 2y-2\omega_{1}+\omega_{2}\geqslant 0,\\\ 6x&\geqslant
2y+\omega_{1}-2\omega_{2}\geqslant 0,\\\ x&\geqslant 0.\end{cases}$ (80)

Finally, the SEC is satisfied iff each condition for the NEC is satisfied,
together with one extra condition:

$\textrm{SEC: NEC \&}\;y\geqslant x.$ (81)

###  II.4 The principal phase space

The dynamical system described in the previous subsection contains a countably
infinite number of possible phase spaces, depending on choices for the non-
negative integers $(L,M,N)$. In this paper we will focus on the simplest
(lowest-dimensional) phase space, which we will refer to as the principal
phase space. This phase space is found by choosing $(L,M,N)=(0,0,0)$ and
thereby arises by fixing $\lambda_{0},\mu_{0}$, and $\nu_{0}$ to be constants.
Of course, if one chooses different nonnegative values for $(L,M,N)$, one will
obtain phase spaces of higher dimensionality, and we expect that such phase
spaces will contain richer dynamical flows. There is a trade-off, however, in
that nonzero values for $(L,M,N)$ lead to larger phase spaces and hence
expanded parameter spaces to try to characterize. Moreover, we found in our
previous analysis Azhar and Kaiser (2018) (for the case of homogeneous and
isotropic backgrounds) that each of the fixed points within the principal
phase space also appeared as fixed points in the higher-dimensional
generalizations, even as the total number of fixed points grew with the
dimensionality of the phase space. Hence we focus on an analysis of the fixed
points within the principal phase space in the present work and defer analysis
of higher-dimensional generalizations to future work.

In addition to assuming $H>0$, we also restrict attention in what follows to
fixed points at which $\epsilon>0$. (In FLRW spacetimes, $\epsilon<0$ can
yield a so-called “big rip” singularity Starobinsky (2000); Caldwell (2002);
Caldwell  _et al._ (2003).) As we will see below, once we fix
$(L,M,N)=(0,0,0)$ to construct our principal phase space, the slow-roll
parameter $\epsilon$ when evaluated at various fixed points is proportional to
$\lambda_{0},\mu_{0}$, or $\nu_{0}$. Furthermore, setting
$\lambda_{0},\mu_{0}$, or $\nu_{0}$ equal to zero would collapse various fixed
points to the (isotropic) de Sitter fixed point, which would artificially
favor flows to de Sitter at late times. Hence we consider
$\lambda_{0},\mu_{0}$, and $\nu_{0}$ to be positive.

For the principal phase space, the values of $\lambda_{0},\mu_{0}$, and
$\nu_{0}$ determine the rate at which various combinations of parameters decay
with time, in accord with Eqs. (73), (74), and (75). In particular
$|3c+\tilde{X}|\sim a(t)^{-\lambda_{0}}$, $|\pi_{1}|\sim a(t)^{-\mu_{0}}$, and
$|\pi_{2}|\sim a(t)^{-\nu_{0}}$, respectively. The latter two equations
dictate the rate at which anisotropic pressures decay as the background
expands; cf. Eqs. (45) and (46) with Eq. (24). If, in addition to such a decay
of anisotropic pressures, one has that $|3c+\tilde{X}|\sim
a(t)^{-\lambda_{0}}$, we see from Eq. (53) that
$\dot{\rho}=\dot{c}+\dot{L}+\dot{\tilde{X}}\to 0$. That is, the system is
driven towards an energy-density dominated state, where this energy density is
constant in time. Such a scenario corresponds to the emergence of an effective
cosmological constant. So we can think of $\mu_{0}$ and $\nu_{0}$ as
determining the rate of decay of relevant anisotropic degrees of freedom; a
sufficiently large (positive) value of $\lambda_{0}$, in turn, can create
conditions for an effective cosmological constant.

Equations of evolution for trajectories in this phase space are given by

$\displaystyle\frac{dx}{d\ln a}=2\epsilon x-2y$ $\displaystyle-
z_{1}(2\omega_{1}-\omega_{2})-z_{2}(2\omega_{2}-\omega_{1}),$ (82)
$\displaystyle\frac{dy}{d\ln a}$ $\displaystyle=2\epsilon y-\lambda_{0}y,$
(83) $\displaystyle\frac{d\omega_{1}}{d\ln a}$
$\displaystyle=2\epsilon\omega_{1}-\mu_{0}\omega_{1},$ (84)
$\displaystyle\frac{d\omega_{2}}{d\ln a}$
$\displaystyle=2\epsilon\omega_{2}-\nu_{0}\omega_{2},$ (85)
$\displaystyle\frac{dz_{1}}{d\ln a}$
$\displaystyle=z_{1}(\epsilon-3)+\omega_{1},$ (86)
$\displaystyle\frac{dz_{2}}{d\ln a}$
$\displaystyle=z_{2}(\epsilon-3)+\omega_{2}$ (87)

where the equation for the Hubble slow-roll parameter, Eq. (71), continues to
hold, as does the constraint in Eq. (72).

One may picture the five-dimensional principal phase space—formed from the six
directions $\\{x,y,\omega_{1},\omega_{2},z_{1},z_{2}\\}$ with the constraint
of Eq. (72)—as a product of two spaces (essentially
$\mathbb{R}^{2}\times\mathbb{R}^{3}$): one (constrained) space coordinatized
by $(z_{1},z_{2},x)$ with the other space coordinatized by
$(\omega_{1},\omega_{2},y)$. Note that the constraint surface is described
purely in terms of $(z_{1},z_{2},x)$ whereas the energy conditions largely
reference $(\omega_{1},\omega_{2},y)$. We will refer to the space
coordinatized by $(\omega_{1},\omega_{2},y)$ as the “internal space.” The
constraint surface is depicted in Fig. 1 (left) and the projection of this
surface onto the $z_{1}$-$z_{2}$ plane is depicted in Fig. 1 (right). At each
point of the constraint surface, one may envision three further (“internal”)
directions coordinatized by $(\omega_{1},\omega_{2},y)$.

Figure 1: (Left) The constraint surface (upside down elliptic paraboloid in
gray) for $0\leq x\leq 1$. (Right) The projection of the constraint surface
onto the $z_{1}$-$z_{2}$ plane. The outermost ellipse (in black) corresponds
to the contour where $x=0$. The contours in gray correspond to
$x=0.2,0.4,0.6$, and $0.8$ (moving inwards, towards the origin, from the black
ellipse).

####  II.4.1 Fixed points

Dynamical flows in the principal phase space are naturally studied by
considering fixed points in this space. One finds these fixed points by
setting the left-hand sides of Eqs. (82)–(87) to zero, subject to the
constraint of Eq. (72).

One can more efficiently calculate the fixed points by considering two
distinct cases, namely (i) $x=0$ and (ii) $x\neq 0$. The former case leads to
a one-parameter set of fixed points that is consistent with the Kasner circle,
a one-parameter set of anisotropic vacuum solutions of the Einstein field
equations first described by Kasner (1921).222These are solutions for which
the energy density vanishes and yet the background has anisotropies that are
“frozen in.” We do not believe these anisotropies can be removed by a
coordinate transformation, though $\sigma^{2}$ does decrease quadratically
with time, as we show in Appendix D. (See also Ref. (Misner  _et al._ , 1973,
§30.2).) In what follows we will compute the fixed points explicitly for the
case $x=0$, and then simply quote results for the case with $x\neq 0$.

When $x=0$, the constraint of Eq. (72) becomes

$\displaystyle 1=(z_{1}+z_{2})^{2}-3z_{1}z_{2},$ (88)

and this implies that the Hubble slow-roll parameter, $\epsilon$, is given by

$\displaystyle\epsilon=y+3(z_{1}+z_{2})^{2}-9z_{1}z_{2}=y+3.$ (89)

To find the fixed points we need to solve:

$\displaystyle 0$
$\displaystyle=-2y-z_{1}(2\omega_{1}-\omega_{2})-z_{2}(2\omega_{2}-\omega_{1}),$
(90) $\displaystyle 0$ $\displaystyle=2\epsilon y-\lambda_{0}y,$ (91)
$\displaystyle 0$ $\displaystyle=2\epsilon\omega_{1}-\mu_{0}\omega_{1},$ (92)
$\displaystyle 0$ $\displaystyle=2\epsilon\omega_{2}-\nu_{0}\omega_{2},$ (93)
$\displaystyle 0$ $\displaystyle=z_{1}(\epsilon-3)+\omega_{1},$ (94)
$\displaystyle 0$ $\displaystyle=z_{2}(\epsilon-3)+\omega_{2}.$ (95)

Equation (91) together with Eq. (89) yields

$\displaystyle 0$ $\displaystyle=2(y+3)y-\lambda_{0}y=y(2y+6-\lambda_{0})$
$\displaystyle\implies y=0\;\;{\rm or}\;\;y=\frac{\lambda_{0}-6}{2}.$ (96)

One can show numerically that there is no solution to the system of equations
of Eqs. (90)–(95) when $y\neq 0$, so we proceed under the assumption that
$y=0$. When $y=0$ we have, from Eq. (89), that $\epsilon=3$. This means that
Eqs. (94) and (95) are satisfied (indeed, for any value of $z_{1}$ and
$z_{2}$) iff $\omega_{1}=0=\omega_{2}$. Equations (90), (92), and (93) are
then automatically satisfied. Thus, when $x=0$ we find that the fixed points
(which we will refer to as FPK for “fixed point Kasner”) are given by

$\displaystyle{\bf FPK:}\;\;x$
$\displaystyle=y=\omega_{1}=\omega_{2}=0\;\;\&\;\;$ $\displaystyle 1$
$\displaystyle=(z_{1}+z_{2})^{2}-3z_{1}z_{2}.$ (97)

In Appendix D we derive the metric that results from assuming these values for
the dynamical variables, showing that they are consistent with Einstein field
equations whose solution is given by the Kasner metric.

Below, we will separate the fixed points FPK into two distinct sets of fixed
points, FPK- and FPK+. The former set corresponds to the lower portion of the
ellipse traced out in the $z_{1}$-$z_{2}$ plane by
$1=(z_{1}+z_{2})^{2}-3z_{1}z_{2}$, while the latter corresponds to the upper
portion of the ellipse. We consider each such set to be given via a function,
$z_{2}(z_{1})$, where $z_{1}$ takes values in a prescribed domain. We collect
all fixed points (including those for which $x\neq 0$) in Table 1.

Label | $z_{1}$ | $z_{2}$ | $x$ | $\omega_{1}$ | $\omega_{2}$ | $y$ | $\epsilon$
---|---|---|---|---|---|---|---
dS | $0$ | $0$ | $1$ | $0$ | $0$ | $0$ |  $0$
FPb | $0$ | $0$ | $1$ | $0$ | $0$ | $\frac{\lambda_{0}}{2}$ | $\frac{\lambda_{0}}{2}$
FPc | $0$ | $-\sqrt{\frac{\nu_{0}}{6}}$ | $1-\frac{\nu_{0}}{6}$ | $0$ | $\frac{1}{2}\sqrt{\frac{\nu_{0}}{6}}(\nu_{0}-6)$ | $0$ | $\frac{\nu_{0}}{2}$
FPd | $0$ | $\sqrt{\frac{\nu_{0}}{6}}$ | $1-\frac{\nu_{0}}{6}$ | $0$ | $-\frac{1}{2}\sqrt{\frac{\nu_{0}}{6}}(\nu_{0}-6)$ | $0$ | $\frac{\nu_{0}}{2}$
FPe | $-\sqrt{\frac{\mu_{0}}{6}}$ | $0$ | $1-\frac{\mu_{0}}{6}$ | $\frac{1}{2}\sqrt{\frac{\mu_{0}}{6}}(\mu_{0}-6)$ | $0$ | $0$ | $\frac{\mu_{0}}{2}$
FPf | $\sqrt{\frac{\mu_{0}}{6}}$ | $0$ | $1-\frac{\mu_{0}}{6}$ | $-\frac{1}{2}\sqrt{\frac{\mu_{0}}{6}}(\mu_{0}-6)$ | $0$ | $0$ | $\frac{\mu_{0}}{2}$
FPK- | $[-\frac{2}{\sqrt{3}},\frac{2}{\sqrt{3}})$ | $\frac{1}{2}(z_{1}-\sqrt{4-3z_{1}^{2}})$ | $0$ | $0$ | $0$ | $0$ |  $3$
FPK+ | $(-\frac{2}{\sqrt{3}},\frac{2}{\sqrt{3}}]$ | $\frac{1}{2}(z_{1}+\sqrt{4-3z_{1}^{2}})$ | $0$ | $0$ | $0$ | $0$ |  $3$
Table 1: Fixed points in the principal phase space. There is a single
(isotropic) de Sitter fixed point, dS, as well as another isotropic fixed
point FPb. The fixed points corresponding to the Kasner circle, FPK, have been
separated into two one-parameter families of fixed points, labeled FPK- and
FPK+.

Note that there is a single fixed point that corresponds to de Sitter
expansion of the background, which we label dS. There is a second fixed point
that corresponds to isotropic expansion, FPb, which (for $\lambda_{0}\neq 0$)
does not correspond to de Sitter expansion of the background. The isotropic
fixed point FPb has the same properties as those identified as fixed point
FP0b in Ref. Azhar and Kaiser (2018), for the case of dynamical flows within
homogeneous and isotropic background spacetimes. In particular, for constant
$\lambda_{0}>0$, the (volume-averaged) scale factor $a(t)$ grows as
$a(t)=a(t_{i})(t/t_{i})^{p}$ with $p=2/\lambda_{0}$, which yields accelerated
expansion for $0<\lambda_{0}<2$. Such dynamics can be realized in canonical
single-scalar-field models with an effective potential of the form Azhar and
Kaiser (2018)

$V(\phi)=V_{0}\exp\left[-\alpha\left(\frac{\phi}{M_{\rm
pl}}\right)^{\beta}\right]$ (98)

with $\alpha=\sqrt{\lambda_{0}}$ and $\beta=1$.

The remaining fixed points (for arbitrary positive values of $\mu_{0}$ and
$\nu_{0}$) each correspond to anisotropic expansion. In Appendix C we consider
the spacetime line elements associated with each of these anisotropic fixed
points.

####  II.4.2 Stability properties of fixed points

One can probe the stability properties of the fixed points by considering the
entire (unconstrained) phase space, or by focusing only on the region of the
phase space in which the constraint of Eq. (72) is satisfied. In what follows,
we will focus on the latter case, as the constraint surface is an invariant
manifold and all trajectories we will study will satisfy the constraint for
all time. In this case, there is a straightforward argument (for any dynamical
system derived from our framework) to the effect that we can simply ignore the
dynamical variable $x$ and focus on the other variables. More specifically, on
the constraint surface, we can rearrange the constraint of Eq. (72) such that

$\displaystyle x=1-\left[(z_{1}+z_{2})^{2}-3z_{1}z_{2}\right].$ (99)

The dynamical equations for the remaining variables
$\\{y,\omega_{1},\omega_{2},z_{1},z_{2}\\}$ are indeed closed:

$\displaystyle\frac{dy}{d\ln a}$ $\displaystyle=2\epsilon y-\lambda_{0}y,$
(100) $\displaystyle\frac{d\omega_{1}}{d\ln a}$
$\displaystyle=2\epsilon\omega_{1}-\mu_{0}\omega_{1},$ (101)
$\displaystyle\frac{d\omega_{2}}{d\ln a}$
$\displaystyle=2\epsilon\omega_{2}-\nu_{0}\omega_{2},$ (102)
$\displaystyle\frac{dz_{1}}{d\ln a}$
$\displaystyle=z_{1}(\epsilon-3)+\omega_{1},$ (103)
$\displaystyle\frac{dz_{2}}{d\ln a}$
$\displaystyle=z_{2}(\epsilon-3)+\omega_{2},$ (104)

where the Hubble slow-roll parameter, $\epsilon$, is given by Eq. (71).

Results for the case in which we focus on stability properties for the
constrained phase space are shown in Table 2, for select values of
$\lambda_{0},\mu_{0}$, and $\nu_{0}$.

Label | Eigenvalues of Jacobian | Sample condition | Stability
---|---|---|---
dS | $\\{-3,-3,-\lambda_{0},-\mu_{0},-\nu_{0}\\}$ |  $\lambda_{0}>0$, $\mu_{0}>0$, $\nu_{0}>0$ | Attractor
FPb | $\left\\{\frac{1}{2}(-6+\lambda_{0}),\frac{1}{2}(-6+\lambda_{0}),\lambda_{0},\lambda_{0}-\mu_{0},\ \lambda_{0}-\nu_{0}\right\\}$ |  $\lambda_{0}=2$, $\mu_{0}=1/3$, $\nu_{0}=1$ | Saddle
FPc | $\left\\{-\lambda_{0}+\nu_{0},-\mu_{0}+\nu_{0},\frac{1}{2}(-6+\nu_{0}),c^{-},c^{+}\right\\}$ |  $\lambda_{0}=2$, $\mu_{0}=1/3$, $\nu_{0}=1$ | Saddle
FPd | $\left\\{-\lambda_{0}+\nu_{0},-\mu_{0}+\nu_{0},\frac{1}{2}(-6+\nu_{0}),c^{-},c^{+}\right\\}$ |  $\lambda_{0}=2$, $\mu_{0}=1/3$, $\nu_{0}=1$ | Saddle
FPe | $\left\\{-\lambda_{0}+\mu_{0},\frac{1}{2}(-6+\mu_{0}),d^{-},d^{+},\mu_{0}-\nu_{0}\right\\}$ |  $\lambda_{0}=2$, $\mu_{0}=1/3$, $\nu_{0}=1$ | Saddle
FPf | $\left\\{-\lambda_{0}+\mu_{0},\frac{1}{2}(-6+\mu_{0}),d^{-},d^{+},\mu_{0}-\nu_{0}\right\\}$ |  $\lambda_{0}=2$, $\mu_{0}=1/3$, $\nu_{0}=1$ | Saddle
FPK- | $\left\\{6,0,6-\lambda_{0},6-\mu_{0},6-\nu_{0}\right\\}$ | $0<\lambda_{0},\mu_{0},\nu_{0}<6$ | Unstable∗
FPK+ | $\left\\{6,0,6-\lambda_{0},6-\mu_{0},6-\nu_{0}\right\\}$ | $0<\lambda_{0},\mu_{0},\nu_{0}<6$ | Unstable∗
Table 2: Stability properties of fixed points for the principal phase space.
Sample conditions for FPb–FPf yield the stability properties shown in the
rightmost column (in each case, the fixed point is a saddle point for the
sample condition). The parameters $c^{\pm}$ and $d^{\pm}$ are defined in Eqs.
(105) and (106). The superscript ∗ denotes that the fixed points are normally
hyperbolic. That is, one has a set of non-isolated fixed points for which the
only eigenvalues with zero real part are those whose eigenvectors are tangent
to the set of points. In such a case, one may read off the stability
properties of the fixed points from the remaining eigenvalues. (See Ref. Coley
(1999).)

In Table 2, the quantities $c^{\pm}$ and $d^{\pm}$ are given by

$\displaystyle c^{\pm}$
$\displaystyle=\frac{1}{4}\left(3\nu_{0}-6\pm\sqrt{-7\nu_{0}^{2}+60\nu_{0}+36}\right),$
(105) $\displaystyle d^{\pm}$
$\displaystyle=\frac{1}{4}\left(3\mu_{0}-6\pm\sqrt{-7\mu_{0}^{2}+60\mu_{0}+36}\right).$
(106)

Generally, the de Sitter fixed point is an attractor, the remaining non-Kasner
fixed points are saddle points, and the Kasner fixed points are unstable.

##  III An emergent cosmological constant

We now analyze flows through the principal phase space. We begin, in Sec.
III.1, with an analysis that probes the stability of isotropic backgrounds. In
particular, we start dynamical trajectories at $z_{1}=0=z_{2}$ (and therefore
at $x=1$)—so that the background is initially isotropic—and identify regions
of the internal space coordinatized by $(\omega_{1},\omega_{2},y)$ that flow
to dS (the de Sitter fixed point) or to FPb (the non-de Sitter isotropic fixed
point), even in the presence of nonvanishing anisotropic pressure. In Sec.
III.2 we analyze states that begin with an anisotropic background
characterized by nonvanishing shear, which then subsequently flow to dS or
FPb.

For such analyses there are various choices for parameters that need to be
made. First, recall that the principal phase space is defined by fixing
$\lambda_{0},\mu_{0}$, and $\nu_{0}$ to be constants, which we take to be
real-valued and positive. In order to probe a regime in which anisotropic
degrees of freedom do not dissipate arbitrarily quickly, we select relatively
small values for the parameters $\mu_{0}$ and $\nu_{0}$, such as
$\\{\mu_{0},\nu_{0}\\}\in\\{1/10,2/3\\}$. (We also analyze a scenario in which
the average of $\mu_{0}$ and $\nu_{0}$ is $2/3$, with $\mu_{0}=1/3$ and
$\nu_{0}=1$.) Furthermore, to explore various possibilities for the emergence
of a quasi-constant energy density for the background, we consider a broader
range of values for $\lambda_{0}$, with $\lambda_{0}\in\\{1/10,1,2\\}$. Recall
from Eqs. (55) and (61) that $y\sim\rho+P$ and
$\lambda_{0}=-\partial_{t}(\rho+P)/[H(\rho+P)]$. De Sitter expansion
corresponds to $y=0$ together with each $X_{ii}=0$. Hence a large value of
$\lambda_{0}$ would seem to favor the emergence of an effective cosmological
constant (especially if combined with large values of $\mu_{0}$ and $\nu_{0}$,
which drive rapid dissipation of the anisotropic pressure contributions),
whereas, prima facie, smaller values of $\lambda_{0}$ would not favor such
asymptotic behavior. (See also the discussion near the beginning of Sec.
II.4.)

Second, our EFT framework (like any EFT framework) has a limited range of
validity. The dimensionless variable $x\sim\rho$ is constrained via Eq. (72)
to be $x\leq 1$. We therefore analyze initial conditions such that the
magnitude of each of our dimensionless dynamical variables is less than or
equal to 1. We consider our EFT no longer valid if, during subsequent
evolution of the system, the magnitude of at least one dimensionless dynamical
variable grows to be greater than 10.

Third, we distinguish between cases in which the initial value $y(t_{i})<0$
from those in which $y(t_{i})\geq 0$. Note from Eq. (61) that $y<0$
corresponds to $P<-\rho$. Scenarios with $y(t_{i})<0$ cannot be realized with
single scalar field models for which the scalar field has a minimal coupling
to Einstein gravity and a canonical kinetic term. Nonetheless, such scenarios
can be studied within our more general EFT framework. In an isotropic
spacetime, an equation of state corresponding to $y<0$ could yield a “big rip”
singularity Starobinsky (2000); Caldwell (2002); Caldwell  _et al._ (2003). As
we discuss further at the end of Sec. III.1, however, scenarios exist within
our effective system for which $y<0$ does not trigger such singular behavior.

###  III.1 Flows back to isotropy

We first study the stability of isotropic backgrounds in the presence of
nonvanishing anisotropic matter sources. In particular, we consider spacetimes
that are initially isotropic, with $z_{1}=0=z_{2}$ and hence $x=1$, and
establish the range of initial conditions in the internal space
$\\{\omega_{1},\omega_{2},y\\}$ that yield an isotropic background at late
times. Recall from Eqs. (56)–(59) that $\omega_{A}\sim\pi_{A}$ and
$z_{A}\sim\sigma_{A}$ (for $A=1,2$), and from Eqs. (66)–(67) that nonzero
anisotropic pressures $\omega_{A}$ will induce nonvanishing components of the
shear $z_{A}$.

In Fig. 2, we present plots of a slice through initial conditions in the
internal space $\\{\omega_{1},\omega_{2},y\\}$.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 2: Slices (corresponding to $\omega_{2}=0$) through the internal space
for six dynamical systems. The light blue region denotes initial conditions
that flow to the dS fixed point, dS. The orange region denotes initial
conditions that exit the EFT region with a background that is anisotropic.
Fractions of the internal space that flow to different states are displayed in
Table 3. The dominant energy condition (DEC) is satisfied for initial
conditions that lie in the shaded region.

The six plots correspond to six different choices for $\lambda_{0}$ and
$\mu_{0}=\nu_{0}$. For each plot, the light blue region describes initial
conditions that flow to dS (the isotropic dS fixed point) while the orange
region describes initial conditions that exit the regime of validity of our
EFT dynamical system with a background that is anisotropic. In other words,
points within the orange region represent evolution such that the magnitude of
at least one of the dynamical variables grows to be greater than 10 at some
finite time, and at that time at least one of the $z_{A}$ remains nonzero.
There are also some cases (that appear in white) in which initial conditions
yield flows that exit the regime of validity of our EFT dynamical system with
a background that is isotropic, even though they did not reach either
isotropic fixed point, dS or FPb. We set the threshold to consider a late-time
state sufficiently isotropic to be $|z_{A}|\leq 10^{-5}$ for $A=1,2$,
comparable to the observed anisotropy in our own universe on long length-
scales as measured in the cosmic microwave background radiation.

These plots exhibit evolution from initial conditions beyond the domain to
which Wald’s theorem Wald (1983) applies. As noted around Eq. (1), Wald
derived his cosmic no-hair theorem for systems described by an energy-momentum
tensor $T_{\mu\nu}$ that included a bare, positive cosmological constant
$\Lambda$ (which violates the SEC), as well as additional matter sources
$\tilde{T}_{\mu\nu}$ that obey both the SEC and the DEC. Hence Wald’s theorem
applies to systems in which $T_{\mu\nu}$ always satisfies the DEC and may or
may not satisfy the SEC (depending on the magnitude of $\Lambda$ compared to
the components of $\tilde{T}_{\mu\nu}$). The plots in Fig. 2, on the other
hand, include scenarios (outside of the shaded regions) in which the initial
conditions violate the DEC, but for which the system nonetheless flows to the
de Sitter fixed point. Indeed, the fraction of points within the (full)
internal space that exhibit such behavior can be significant, as suggested by
the slice shown in Fig. 2a.

For each case shown in Fig. 2, we initialize the system with
$-1\leq\\{\omega_{1}(t_{i}),\omega_{2}(t_{i}),y(t_{i})\\}\leq 1$. (This range
includes sets of initial conditions that lie exactly along the $y$-axis, for
which the initial anisotropic pressures vanish identically.) The fraction of
the internal space $\\{\omega_{1},\omega_{2},y\\}$ that subsequently flows to
various late-time states is shown in Table 3. Within the table, the row that
yields the highest fraction of the space that flows to the fixed point dS, for
$y(t_{i})\geq 0$, is highlighted in gray. For $y(t_{i})\geq 0$, we find that
decreasing $\lambda_{0}$ (while keeping $\mu_{0}=\nu_{0}$ fixed) leads to a
decrease in the fraction of points that flow to dS; whereas decreasing
$\mu_{0}=\nu_{0}$ (while keeping $\lambda_{0}$ fixed) leads to a pronounced
decrease in the fraction of points that flow to dS. This pattern makes sense,
given that smaller values of $\mu_{0}$ and $\nu_{0}$ correspond to scenarios
in which the components of the anisotropic pressure dissipate more slowly over
time, whereas larger values of $\lambda_{0}$ more effectively drive
$(\rho+P)\rightarrow 0$. On the other hand, if we include cases where
$y(t_{i})<0$ we find an interesting pattern: decreasing $\lambda_{0}$ while
holding $\mu_{0}=\nu_{0}=1/10$ fixed yields a larger fraction of points that
flow to dS. Trajectories with $y(t_{i})<0$ begin with $P<-\rho$ (see Eq.
[61]), which yields super-accelerated expansion at early times. Such early
growth quickly dilutes the anisotropic pressures, driving the system toward
the dS fixed point.

| Initial $y$-values |  |  | Fraction of the full 3D internal space that flows to …
---|---|---|---|---
Fig. | considered | $\lambda_{0}$ | $\mu_{0}=\nu_{0}$ | dS | FPb | isotropic EXIT | anisotropic EXIT
2a | $|y(t_{i})|\leq 1$ | 2 | 2/3 | 0.7030 | – | 0 | 0.2970
2a | $0\leq y(t_{i})\leq 1$ | 2 | 2/3 | 0.4336 | – | 0 | 0.5664
2b | $|y(t_{i})|\leq 1$ | 2 | 1/10 | 0.1841 | – | 0 | 0.8159
2b | $0\leq y(t_{i})\leq 1$ | 2 | 1/10 | 0.0487 | – | 0 | 0.9513
2c | $|y(t_{i})|\leq 1$ | 1 | 2/3 | 0.6122 | – | 0.0001 | 0.3877
2c | $0\leq y(t_{i})\leq 1$ | 1 | 2/3 | 0.2469 | – | 0.0001 | 0.7530
2d | $|y(t_{i})|\leq 1$ | 1 | 1/10 | 0.2710 | – | 0.0001 | 0.7289
2d | $0\leq y(t_{i})\leq 1$ | 1 | 1/10 | 0.0261 | – | 0.0001 | 0.9738
2e | $|y(t_{i})|\leq 1$ | 1/10 | 2/3 | 0.5138 | 0 | 0.0009 | 0.4853
2e | $0\leq y(t_{i})\leq 1$ | 1/10 | 2/3 | 0.0465 | 0 | 0.0017 | 0.9518
2f | $|y(t_{i})|\leq 1$ | 1/10 | 1/10 | 0.4350 | 0 | 0.0001 | 0.5649
2f | $0\leq y(t_{i})\leq 1$ | 1/10 | 1/10 | 0.0058 | 0 | 0.0002 | 0.9940
Table 3: For certain values of $\lambda_{0},\mu_{0}$, and $\nu_{0}$, we
display the fraction of the full internal space—coordinatized by
$\\{\omega_{1},\omega_{2},y\\}$—that either flows to the isotropic de Sitter
fixed point (dS) or the isotropic (non-de Sitter) fixed point (FPb).
Trajectories that reach neither fixed point and that leave the regime of
validity of the EFT in an (an)isotropic state undergo ‘(an)isotropic EXIT ’. A
dash ‘–’ indicates a value $<0.0001$.

We next consider three trajectories of interest that begin with isotropic
initial conditions, $(z_{1},z_{2},x)=(0,0,1)$, and for which the anisotropic
pressures vanish, $\omega_{A}=0$ for $A=1,2$. For finite $H$, the case
$\omega_{A}=0$ corresponds to $\pi_{A}=0$, which in turn means that
$p_{1}=p_{2}=p_{3}$. The components $p_{i}$ also satisfy $p_{1}+p_{2}+p_{3}=0$
from Eq. (24), so we find

$p_{i}=0\>\>\>{\rm for}\>\>i=1,2,3\quad({\rm for}\>\>\omega_{A}=0).$ (107)

Given that both $\omega_{A}=0$ (for $A=1,2$) are invariant manifolds,
$p_{i}=0$ therefore persists under the system’s evolution.

To explore such scenarios, we choose
$(\lambda_{0},\mu_{0},\nu_{0})=(1,1/10,1/10)$ so that FPb, with coordinates
$(\omega_{1},\omega_{2},y)=(0,0,1/2)$, lies within the range of initial
conditions under consideration. We then vary the initial value of $y$ and
consider three cases.

Case 1: $y(t_{i})=0.49$. The trajectory flows straight down, along the
$y$-axis, to the de Sitter fixed point dS, while maintaining $x=1$, $z_{A}=0$.
In this scenario the background is driven to the de Sitter fixed point and not
to FPb. This is consistent with dS being an attractor for
$(\lambda_{0},\mu_{0},\nu_{0})=(1,1/10,1/10)$.

Case 2: $y(t_{i})=0.51$. The trajectory flows straight up, along the $y$-axis,
while maintaining $x=1$, $z_{A}=0$. In this case, initially inflating initial
conditions (as diagnosed by initial conditions for which $\epsilon=y<1$) are
blocked. Some insight can be gained into the nature of this trajectory by
noting that if $x=1$ then $3M_{\textrm{pl}}^{2}H^{2}=c+L+\tilde{X}$, in which
case

$\displaystyle
y\equiv\frac{3c+\tilde{X}}{3M_{\textrm{pl}}^{2}H^{2}}=\frac{3c+\tilde{X}}{c+L+\tilde{X}}.$
(108)

In this case, a growing $y$ means that $3c$ grows relative to $c+L$ (or $2c$
grows relative to $L$)—and this corresponds, as described in Ref. Azhar and
Kaiser (2018), to a kinetic-energy-density dominated regime, in which one
would expect inflation to come to an end.

This observation about evolution under conditions dominated by kinetic energy
can be made more precise by considering the behavior of $\tilde{X}$. As noted
in Eq. (107), $p_{i}=0$ (for $i=1,2,3$) when $\omega_{A}=0$ (for $A=1,2$). But
$p_{i}\equiv-\frac{2}{3}\tilde{X}+2g^{ii}X_{ii}$ (no sum on $i$). So
$p_{i}=0\iff\tilde{X}=3g^{ii}X_{ii}$ (no sum on $i$). Thus we have that

$\displaystyle g^{11}X_{11}$ $\displaystyle=g^{22}X_{22}=g^{33}X_{33}$
$\displaystyle\implies\frac{X_{11}}{a^{2}e^{2\beta_{1}}}=\frac{X_{22}}{a^{2}e^{2\beta_{2}}}=\frac{X_{33}}{a^{2}e^{2\beta_{3}}}.$
(109)

These equations yield

$\displaystyle X_{22}$ $\displaystyle=e^{2(\beta_{2}-\beta_{1})}X_{11},$ (110)
$\displaystyle X_{33}$ $\displaystyle=e^{2(\beta_{3}-\beta_{1})}X_{11}.$ (111)

Note further that the $z_{A}$’s remain zero along the trajectory, in which
case, for finite $H$, we have that
$\sigma_{1}=0\iff\dot{\beta}_{1}=\dot{\beta}_{2}$ and
$\sigma_{2}=0\iff\dot{\beta}_{1}=\dot{\beta}_{3}$, and hence
$\dot{\beta}_{1}=\dot{\beta}_{2}=\dot{\beta}_{3}$. From the derivative of the
constraint on the $\beta_{i}$’s in Eq. (9), we therefore find that
$\dot{\beta}_{i}=0$ for $i=1,2,3$. Upon substituting Eqs. (110) and (111) into
the constraint on the components $X_{ii}$ in Eq. (14), we find

$\displaystyle
X_{11}\left[1+e^{2(\beta_{2}-\beta_{1})}+e^{2(\beta_{3}-\beta_{1})}\right]=0.$
(112)

The term in brackets is a positive constant. Therefore we find that $X_{11}=0$
and, from Eqs. (110) and (111), $X_{22}=0=X_{33}$. Therefore $\tilde{X}=0$,
and our system reduces to precisely the FLRW case studied in Ref. Azhar and
Kaiser (2018). As analyzed there, trajectories in which $2c$ grows relative to
$L$ do indeed correspond to the case of kinetic-energy density growing
relative to the potential-energy density, and hence a flow away from inflating
conditions.

Case 3: $y(t_{i})=-10$. Given $\omega_{A}=0$ for $A=1,2$, this trajectory
begins with large and negative $y$ and flows straight up the $y$-axis,
maintaining $x=1,z_{A}=0,\omega_{A}=0$, to dS, the de Sitter fixed point. Just
as in Case 2, this trajectory satisfies $\tilde{X}=0$.

We may define an equation of state for this trajectory, $w\equiv{P}/{\rho}$.
Using Eq. (21) with $\tilde{X}=0$, we find

$\displaystyle w=\frac{c-L}{c+L}.$ (113)

Given that

$\displaystyle c-L$
$\displaystyle=3M_{\textrm{pl}}^{2}H^{2}\left(-x+\frac{2}{3}y\right),$ (114)
$\displaystyle c+L$ $\displaystyle=3M_{\textrm{pl}}^{2}H^{2}x,$ (115)

we find

$\displaystyle w=-1+\frac{2y}{3x}=-1+\frac{2y}{3},$ (116)

since $x=1$ along this trajectory. So $y<0\iff w<-1$ and

$\displaystyle y\to 0^{-}\iff w\to-1^{-}.$ (117)

That is, the trajectory of Case 3 evolves with an equation of state that
varies with time and approaches that of a cosmological constant, $w=-1$, from
below. Such trajectories represent EFT realizations of the “pseudo-rip”
scenarios identified in Ref. Frampton  _et al._ (2012a), in which $H$ flows to
a finite constant at late times, unlike the more familiar “big rip”
singularities usually associated with $w<-1$ Starobinsky (2000); Caldwell
(2002); Caldwell  _et al._ (2003). As in the scenarios described in Refs.
Stefancic (2005); Nojiri  _et al._ (2005); Nojiri and Odintsov (2005);
Frampton  _et al._ (2011, 2012b, 2012a), our Case 3 presumably avoids the big-
rip singularity by violating various energy conditions.

###  III.2 Flows to isotropy starting from anisotropic backgrounds

We now consider scenarios in which the background is initially anisotropic. In
these cases, the $z_{A}$’s (proportional to components of the shear) are
nonzero at early times, and hence, from the constraint of Eq. (72), $x\neq 1$.

To study such scenarios, we first select values for $\lambda_{0},\mu_{0}$, and
$\nu_{0}$. For each chosen value of $\lambda_{0},\mu_{0}$, and $\nu_{0}$, we
consider six different starting points on the constraint surface described by
Eq. (72) and depicted in Fig. 3.

(a)  Figure 3: Constraint surface projected onto the $z_{1}$-$z_{2}$ plane.
Starting points of trajectories analyzed in this section are presented in
green. For each chosen set of values of $\\{\lambda_{0},\mu_{0},\nu_{0}\\}$,
we consider six different starting points, labeled 1a, 1b, 2a, 2b, 3a, and 3b.

Those starting points correspond to three different values of $x$:
$x=0.1,0.5,0.7$. Increasing initial values of $x$ correspond to decreasing
degrees of initial anisotropy. (Recall that $x=1$ corresponds to an isotropic
state.) For each value of $x$, we select two different points, denoted by the
letters ‘a’ and ‘b’ in Fig. 3. Points marked with an ‘a’ satisfy
$z_{1}=z_{2}=\sqrt{1-x}$, whereas points marked with a ‘b’ satisfy
$z_{1}=-z_{2}=-\sqrt{(1-x)/{3}}$. We first describe two representative cases,
then consider more general behavior of such systems.

We first consider the selection $\lambda_{0}=2$, $\mu_{0}=\nu_{0}=2/3$. Slices
through the internal space $\\{\omega_{1},\omega_{2},y\\}$ for the six
different starting points on the constraint surface are displayed in Fig. 4.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 4: Slices (corresponding to $\omega_{2}=0$) through the internal space
for six different starting points on the constraint surface. In all cases,
$\lambda_{0}=2,\mu_{0}=\nu_{0}=2/3$. Each figure corresponds to one of the
initial conditions identified by green dots in Fig. 3: (a) corresponds to
point 1a; (b) corresponds to point 1b; (c) corresponds to point 2a; (d)
corresponds to point 2b; (e) corresponds to point 3a; (f) corresponds to point
3b. Trajectories that begin within the light blue region flow to the de Sitter
fixed point, dS. Trajectories that begin in the orange region exit the EFT
regime of validity with a background that is anisotropic. Fractions of the
internal space that flow to various final states are displayed in Table 4.

The fraction of the internal space that flows either to dS or leaves the EFT
regime of validity in an anisotropic state is presented in Table 4. Note that
in this case, there are significant portions of the internal space (in light
blue in Fig. 4) that both violate DEC and that flow to dS. All of these
initial conditions lie outside the domain to which Wald’s theorem Wald (1983)
applies.

|  | Initial $y$-values |  |  | Fraction of the internal space that flows to …
---|---|---|---|---|---
Fig. | $x(t_{i})$ | considered | $\lambda_{0}$ | $\mu_{0}=\nu_{0}$ | dS | anisotropic EXIT
4a | 0.1 | $|y(t_{i})|\leq 1$ | 2 | 2/3 | 0.2261 | 0.7739
4a | 0.1 | $0\leq y(t_{i})\leq 1$ | 2 | 2/3 | 0.0065 | 0.9935
4b | 0.1 | $|y(t_{i})|\leq 1$ | 2 | 2/3 | 0.2583 | 0.7417
4b | 0.1 | $0\leq y(t_{i})\leq 1$ | 2 | 2/3 | 0.0065 | 0.9935
4c | 0.5 | $|y(t_{i})|\leq 1$ | 2 | 2/3 | 0.4303 | 0.5697
4c | 0.5 | $0\leq y(t_{i})\leq 1$ | 2 | 2/3 | 0.0996 | 0.9004
4d | 0.5 | $|y(t_{i})|\leq 1$ | 2 | 2/3 | 0.4397 | 0.5603
4d | 0.5 | $0\leq y(t_{i})\leq 1$ | 2 | 2/3 | 0.1030 | 0.8970
4e | 0.7 | $|y(t_{i})|\leq 1$ | 2 | 2/3 | 0.5401 | 0.4599
4e | 0.7 | $0\leq y(t_{i})\leq 1$ | 2 | 2/3 | 0.1965 | 0.8035
4f | 0.7 | $|y(t_{i})|\leq 1$ | 2 | 2/3 | 0.5419 | 0.4581
4f | 0.7 | $0\leq y(t_{i})\leq 1$ | 2 | 2/3 | 0.2072 | 0.7928
Table 4: The fraction of the internal space—coordinatized by
$(\omega_{1},\omega_{2},y)$—that either flows to the isotropic de Sitter fixed
point (dS) or leaves the regime of validity of the EFT in an anisotropic
state, denoted: ‘anisotropic EXIT ’. The row that gives the highest fraction
of the space that flows to dS, for $y(t_{i})\geq 0$, is highlighted in gray.

Next we consider $\lambda_{0}=1/10$, $\mu_{0}=\nu_{0}=1/10$. Compared to the
scenarios featured in Fig. 4 and Table 4, for these selections of $\mu_{0}$
and $\nu_{0}$, the anisotropic pressures dissipate considerably more slowly.
Slices through the internal space of $\\{\omega_{1},\omega_{2},y\\}$ for the
six different starting points on the constraint surface are displayed in Fig.
5.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 5: Slices (corresponding to $\omega_{2}=0$) through the internal space
for six different starting points on the constraint surface. In all cases,
$\lambda_{0}=1/10,\mu_{0}=\nu_{0}=1/10$. Each figure corresponds to one of the
initial conditions identified by green dots in Fig. 3: (a) corresponds to
point 1a; (b) corresponds to point 1b; (c) corresponds to point 2a; (d)
corresponds to point 2b; (e) corresponds to point 3a; (f) corresponds to point
3b. Trajectories that begin within the light blue region flow to the de Sitter
fixed point, dS. Trajectories that begin in the orange region exit the EFT
regime of validity with a background that is anisotropic. Fractions of the
internal space that flow to various final states are displayed in Table 5.

The fraction of the internal space that flows either to dS or leaves the EFT
regime of validity in an anisotropic state is presented in Table 5. Compared
to Fig. 4, there are fewer initial conditions that yield flows to dS for non-
negative $y(t_{i})$, though trajectories with $y(t_{i})<0$ still generically
flow to dS. As we noted in Section III.1, trajectories with $y(t_{i})<0$ begin
with $P<-\rho$; hence they accelerate very efficiently at early times,
diluting the anisotropic pressures and driving the system toward the dS fixed
point. Again, all parts of the internal space that are shaded light blue
correspond to initial conditions that lie outside the domain to which Wald’s
theorem Wald (1983) applies.

|  | Initial $y$-values |  |  | Fraction of the internal space that flows to …
---|---|---|---|---|---
Fig. | $x(t_{i})$ | considered | $\lambda_{0}$ | $\mu_{0}=\nu_{0}$ | dS | isotropic EXIT | anisotropic EXIT
5a | 0.1 | $|y(t_{i})|\leq 1$ | 1/10 | 1/10 | 0.2996 | 0 | 0.7004
5a | 0.1 | $0\leq y(t_{i})\leq 1$ | 1/10 | 1/10 | 0.0001 | 0 | 0.9999
5b | 0.1 | $|y(t_{i})|\leq 1$ | 1/10 | 1/10 | 0.2991 | 0 | 0.7009
5b | 0.1 | $0\leq y(t_{i})\leq 1$ | 1/10 | 1/10 | 0.0001 | 0 | 0.9999
5c | 0.5 | $|y(t_{i})|\leq 1$ | 1/10 | 1/10 | 0.3776 | – | 0.6224
5c | 0.5 | $0\leq y(t_{i})\leq 1$ | 1/10 | 1/10 | 0.0012 | – | 0.9988
5d | 0.5 | $|y(t_{i})|\leq 1$ | 1/10 | 1/10 | 0.3741 | – | 0.6259
5d | 0.5 | $0\leq y(t_{i})\leq 1$ | 1/10 | 1/10 | 0.0011 | – | 0.9988
5e | 0.7 | $|y(t_{i})|\leq 1$ | 1/10 | 1/10 | 0.4044 | – | 0.5956
5e | 0.7 | $0\leq y(t_{i})\leq 1$ | 1/10 | 1/10 | 0.0024 | – | 0.9976
5f | 0.7 | $|y(t_{i})|\leq 1$ | 1/10 | 1/10 | 0.4024 | – | 0.5975
5f | 0.7 | $0\leq y(t_{i})\leq 1$ | 1/10 | 1/10 | 0.0024 | – | 0.9976
Table 5: The fraction of the internal space—coordinatized by
$(\omega_{1},\omega_{2},y)$—that either flows to the isotropic de Sitter fixed
point (dS) or that leaves the regime of validity of the EFT in an
(an)isotropic state, denoted: ‘(an)isotropic EXIT ’. The rows that give the
highest fraction of the space that flows to dS, for $y(t_{i})\geq 0$, are
highlighted in gray. A dash ‘–’ indicates a value $<0.0001$.

We also studied scenarios intermediate between these two sets of cases,
including scenarios in which $\lambda_{0}\in\\{1/10,1,2\\}$ and
$\mu_{0}=\nu_{0}\in\\{1/10,2/3\\}$. See Table 6 for a summary.

Initial $y$-values | $\lambda_{0}$ | $\mu_{0}=\nu_{0}$ | Max. fraction to dS
---|---|---|---
$|y(t_{i})|\leq 1$ | 2 | 2/3 |  $\sim 54$%
$0\leq y(t_{i})\leq 1$ | 2 | 2/3 |  $\sim 21$%
$|y(t_{i})|\leq 1$ | 2 | 1/10 |  $\sim 11$%
$0\leq y(t_{i})\leq 1$ | 2 | 1/10 |  $\sim 2$%
$|y(t_{i})|\leq 1$ | 1 | 2/3 |  $\sim 51$%
$0\leq y(t_{i})\leq 1$ | 1 | 2/3 |  $\sim 13$%
$|y(t_{i})|\leq 1$ | 1 | 1/10 |  $\sim 20$%
$0\leq y(t_{i})\leq 1$ | 1 | 1/10 |  $\sim 1$%
$|y(t_{i})|\leq 1$ | 1/10 | 2/3 |  $\sim 50$%
$0\leq y(t_{i})\leq 1$ | 1/10 | 2/3 |  $\sim 7$%
$|y(t_{i})|\leq 1$ | 1/10 | 1/10 |  $\sim 40$%
$0\leq y(t_{i})\leq 1$ | 1/10 | 1/10 |  $<1$%
Table 6: A summary of results for the maximum fraction of the internal space
that flows to dS for each of the values of $\lambda_{0}$ and $\mu_{0}=\nu_{0}$
considered. For each value of $\lambda_{0}$ and $\mu_{0}=\nu_{0}$, the points
on the constraint surface with respect to which the maximum fraction has been
computed are described in Fig. 3.

The results obtained collectively yield the following conclusions.

  * –

An initial $x(t_{i})=0.7$ always yields a larger fraction of the internal
space that flows to dS than scenarios with smaller $x(t_{i})$, for all values
of the parameters $\lambda_{0}$ and $\mu_{0}=\nu_{0}$ probed; this fraction
gets progressively smaller as $x(t_{i})$ decreases. This trend makes sense,
since larger values of $x(t_{i})$ correspond to initial states that are closer
to initially being isotropic.

  * –

There were no flows to FPb (the non-de Sitter isotropic fixed point): that
fixed point is generally a saddle point (at least for values where
$\lambda_{0}\neq\mu_{0}$ and $\lambda_{0}\neq\nu_{0}$), and one therefore
expects that only trajectories with very specific initial conditions would
reach it.

  * –

For $0\leq y(t_{i})\leq 1$ and for a fixed $\mu_{0}=\nu_{0}$, increasing
$\lambda_{0}$ generally increases the fraction of the internal space that
flows to dS. This effect is far more appreciable for $\mu_{0}=\nu_{0}=2/3$
than for $\mu_{0}=\nu_{0}=1/10$.

  * –

If we include trajectories with $y(t_{i})<0$, a larger fraction of the
internal space generally flows to dS for smaller values of $\lambda_{0}$
(holding $\mu_{0}=\nu_{0}$ fixed). The super-accelerated expansion driven by
$y(t_{i})<0$ (corresponding to $P(t_{i})<-\rho(t_{i})$) lingers longer for
smaller $\lambda_{0}$. For $\mu_{0}=\nu_{0}=2/3$, the maximum value of the
fraction that flows to dS is rather stable, at around 50%.

  * –

Much as we discussed at the beginning of Sec. II.4, we again find that larger
values of $\lambda_{0},\mu_{0}$, and $\nu_{0}$ generally create conditions
that mimic evolution of a system with a cosmological constant.
Correspondingly, a larger fraction of the internal space flows to dS.

Finally, we considered scenarios in which $\mu_{0}\neq\nu_{0}$, such as the
case $\mu_{0}=1/3$ and $\nu_{0}=1$, to keep their average the same as the
cases with $\mu_{0}=\nu_{0}=2/3$. In Fig. 6 we display results for
$\lambda_{0}=2$.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 6: Slices (corresponding to $\omega_{2}=0$) through the internal space
for six different starting points on the constraint surface. In all cases,
$\lambda_{0}=2$, $\mu_{0}=1/3$, and $\nu_{0}=1$. Each figure corresponds to
one of the initial conditions identified by green dots in Fig. 3: (a)
corresponds to point 1a; (b) corresponds to point 1b; (c) corresponds to point
2a; (d) corresponds to point 2b; (e) corresponds to point 3a; (f) corresponds
to point 3b. Trajectories that begin within the light blue region flow to the
de Sitter fixed point, dS. Trajectories that begin in the orange region exit
the EFT regime of validity with a background that is anisotropic. Fractions of
the internal space that flow to various final states are displayed in Table 7.

The fraction of the internal space that flows either to dS or leaves the EFT
regime of validity in an anisotropic state is presented in Table 7.

|  | Initial $y$-values |  |  |  | Fraction of the internal space that flows to …
---|---|---|---|---|---|---
Fig. | $x(t_{i})$ | considered | $\lambda_{0}$ | $\mu_{0}$ | $\nu_{0}$ | dS | anisotropic EXIT
6a | 0.1 | $|y(t_{i})|\leq 1$ | 2 | 1/3 | 1 | 0.1876 | 0.8124
6a | 0.1 | $0\leq y(t_{i})\leq 1$ | 2 | 1/3 | 1 | 0.0037 | 0.9963
6b | 0.1 | $|y(t_{i})|\leq 1$ | 2 | 1/3 | 1 | 0.2110 | 0.7890
6b | 0.1 | $0\leq y(t_{i})\leq 1$ | 2 | 1/3 | 1 | 0.0063 | 0.9937
6c | 0.5 | $|y(t_{i})|\leq 1$ | 2 | 1/3 | 1 | 0.3623 | 0.6377
6c | 0.5 | $0\leq y(t_{i})\leq 1$ | 2 | 1/3 | 1 | 0.0711 | 0.9289
6d | 0.5 | $|y(t_{i})|\leq 1$ | 2 | 1/3 | 1 | 0.3759 | 0.6241
6d | 0.5 | $0\leq y(t_{i})\leq 1$ | 2 | 1/3 | 1 | 0.0783 | 0.9217
6e | 0.7 | $|y(t_{i})|\leq 1$ | 2 | 1/3 | 1 | 0.4668 | 0.5332
6e | 0.7 | $0\leq y(t_{i})\leq 1$ | 2 | 1/3 | 1 | 0.1470 | 0.8530
6f | 0.7 | $|y(t_{i})|\leq 1$ | 2 | 1/3 | 1 | 0.4716 | 0.5284
6f | 0.7 | $0\leq y(t_{i})\leq 1$ | 2 | 1/3 | 1 | 0.1535 | 0.8465
Table 7: Fraction of the internal space—coordinatized by
$\\{\omega_{1},\omega_{2},y\\}$—that either flows to the isotropic de Sitter
fixed point (dS) or that leaves the regime of validity of the EFT in an
anisotropic, denoted: ‘anisotropic EXIT ’. The row that gives the highest
fraction of the space that flows to dS, for $y(t_{i})\geq 0$, is highlighted
in gray.

Again we find regions (in light blue) where the DEC is not satisfied that
nonetheless flow to the de Sitter fixed point.

##  IV Discussion

By developing an effective field theory (EFT) treatment for model-independent,
“single-clock” systems in homogeneous yet anisotropic spacetimes, we have
analyzed dynamical flows of initially expanding spaces into isotropic states
at late times—in particular, into de Sitter space. Our approach applies to all
such single-clock scenarios and is not limited to realizations that are
compatible with the usual single-scalar-field constructions. Our analysis can
accommodate scenarios that fall outside the domain to which Wald’s influential
work Wald (1983) on the isotropization of homogeneous spacetimes applies.

Wald’s cosmic no-hair theorem Wald (1983) applies to systems in which the
energy-momentum tensor includes a bare cosmological constant plus additional
matter degrees of freedom that satisfy the dominant and strong energy
conditions. Our EFT treatment, on the other hand, does not include a bare
cosmological constant. In addition, in light of work that describes how
readily various energy conditions may be violated in otherwise well-behaved
classical and semi-classical settings Kandrup (1992); Barcelo and Visser
(1999); Visser and Barcelo (2000); Barcelo and Visser (2000, 2002); Bellucci
and Faraoni (2002); Dubovsky  _et al._ (2006); Nicolis  _et al._ (2010);
Rubakov (2014); Martin-Moruno and Visser (2017), we drop the requirement that
the effective matter degrees of freedom initially satisfy such energy
conditions.

The EFT we develop includes an action for matter degrees of freedom that is
linear in perturbations about a suitably general anisotropic (Bianchi I)
background. This action yields an energy-momentum tensor that may be
reorganized into canonical form, including terms that describe anisotropic
pressures. We identify salient variables that allow us to define expansion-
normalized dimensionless dynamical variables, akin to the analysis in Refs.
van den Hoogen and Coley (1995); Frusciante  _et al._ (2014); Azhar and Kaiser
(2018). One may then construct a closed dynamical system in which one can
uniquely identify isotropic states, including de Sitter evolution.

Our analysis of the “principal phase space” (the lowest-dimensional phase
space) in this setting reveals rich dynamical structure. We identify two fixed
points that correspond to isotropic backgrounds, including a de Sitter
attractor (labeled dS) and a non-de Sitter fixed point (that is generally a
saddle point). In addition to four further fixed points that correspond to
anisotropic backgrounds, we identify a one-parameter set of (unstable) fixed
points that are consistent with the Kasner circle—a one-parameter set of
anisotropic vacuum solutions in which anisotropies are “frozen-in” (though
they dissipate over time).

Within this framework, we perform two different analyses. In Sec. III.1 we
investigate the stability of initially isotropic backgrounds in the presence
of anisotropic pressures, identifying regions of phase space that flow to
various final states, including flows to the isotropic de Sitter fixed point,
dS. (See Fig. 2 and Table 3.) For certain reasonable choices of parameters, we
find that as much as $70\%$ of the phase space can flow to dS, including a
significant portion that corresponds to initial conditions that do not satisfy
the dominant or strong energy conditions.

In Sec. III.2 we investigate scenarios in which the background is initially
anisotropic, and identify regions of phase space that flow to various final
states, including dS. (See Figs. 4, 5, 6 and Tables 4, 5, and 7.) Again, for
reasonable choices of parameters that describe the dynamical systems, we find
that as much as about $50\%$ of the phase space flows to dS, including
significant swaths of initial conditions that do not satisfy the dominant or
strong energy conditions. (If we restrict attention to those regions of
parameter space for which $P\geq-\rho$ at the initial time, then we find as
much as $10\%-20\%$ of the initial conditions flow to dS.) Such flows lie
outside the domain to which Wald’s no-hair theorem Wald (1983) applies.

Using our general EFT-based dynamical framework, we thus describe evolution of
homogeneous spacetimes consistent with the emergence of an effective
cosmological constant, for which the equation of state evolves toward
$P/\rho\rightarrow-1$ while the anisotropic pressures redshift away. Our work
reveals a broader context in which “cosmic no-hair conjectures” may apply,
beyond the powerful results first identified by Wald Wald (1983). Such
generalized isotropization can further shed light on the range of conditions
under which cosmic inflation may begin.

The present work can be extended in several interesting directions. One topic
for future research is to identify fixed points and explore dynamical flows
for these systems beyond the principal phase space, that is, upon setting at
least one of $L,M,N>0$. Another is to construct an EFT consistent with more
general anisotropic backgrounds than the Bianchi I line element of Eq. (8).
Using tools like those developed here, such analyses would further help
clarify just how general various cosmic no-hair conjectures may ultimately be.

Finally, an open question regarding any cosmic no-hair analysis is how to
account for the second law of thermodynamics for gravitational systems.
Penrose, for example, has suggested that entropy might scale with the Weyl
curvature tensor Penrose (New York: Cambridge University Press, 1979, New
York: Cambridge University Press, 2008, 2018), in which case any system that
flows from anisotropic to isotropic conditions would appear to violate the
second law of thermodynamics. (For a recent review, see Ref. Hu (2021).)
Whether the Weyl tensor is an appropriate measure of entropy, or some distinct
method of understanding the second law of thermodynamics for cosmological
systems would be more appropriate, remains a fascinating question for further
research.

###### Acknowledgements.

We gratefully acknowledge helpful discussions with Jérôme Martin. F. A.
acknowledges support from the Faculty Research Support Program (FY2019) at the
University of Notre Dame. Portions of this work were conducted in MIT’s Center
for Theoretical Physics and supported in part by the U. S. Department of
Energy under Contract No. DE-SC0012567.

##  Appendix A Anisotropic pressure in scalar field models

We first consider a universe filled only with scalar fields, each with a
canonical coupling to gravity, and further assume that the gravitational
degrees of freedom are described by the canonical Einstein-Hilbert action. The
action for such models can be written in the form

$\begin{split}S&=\int d^{4}x\sqrt{-g}\left[\frac{M_{\rm pl}^{2}}{2}R+{\cal
L}^{({\cal M})}\right]\\\ &=\int d^{4}x\sqrt{-g}\left[\frac{M_{\rm
pl}^{2}}{2}R-\frac{1}{2}g^{\mu\nu}\,{\cal
G}_{IJ}(\phi^{K})\partial_{\mu}\phi^{I}\partial_{\nu}\phi^{J}-V(\phi^{K})\right]\,.\end{split}$
(118)

The energy-momentum tensor then takes the form (see, for example, Ref. Kaiser
_et al._ (2013))

$\begin{split}T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S^{({\cal
M})}}{\delta g^{\mu\nu}}&=g_{\mu\nu}{\cal L}^{({\cal
M})}-2\left(\frac{\delta{\cal L}^{({\cal M})}}{\delta g^{\mu\nu}}\right)\\\
&={\cal
G}_{IJ}\partial_{\mu}\phi^{I}\partial_{\nu}\phi^{J}-g_{\mu\nu}\left[\frac{1}{2}g^{\alpha\beta}\,{\cal
G}_{IJ}\,\partial_{\alpha}\phi^{I}\partial_{\beta}\phi^{J}+V(\phi^{K})\right]\,,\end{split}$
(119)

where $S^{({\cal M})}\equiv\int d^{4}x\sqrt{-g}\,{\cal L}^{({\cal M})}$. Even
if the field-space manifold has nondiagonal terms, such that ${\cal
G}_{IJ}(\phi^{K})\neq 0$ for $I\neq J$, and each of the scalar fields couples
directly to the others in the interaction potential $V(\phi^{K})$, we still
find that $T_{ij}\propto g_{ij}$ for a spatially homogeneous system, such that
each $\phi^{I}\rightarrow\phi^{I}(t)$. In that case,

${\cal G}_{IJ}\,\partial_{\mu}\phi^{I}\partial_{\nu}\phi^{J}={\cal
G}_{IJ}\,\dot{\phi}^{I}\dot{\phi}^{J}\delta^{0}_{\>\>\mu}\delta^{0}_{\>\>\nu}\,,$
(120)

so that

$T_{ij}=g_{ij}\left[\frac{1}{2}{\cal
G}_{IJ}\,\dot{\phi}^{I}\dot{\phi}^{J}-V(\phi^{K})\right]\,.$ (121)

Given that $g_{0i}=0$ for our background spacetime, we then readily find

$T^{i}_{\>\>j}=g^{ik}\,T_{kj}=g^{ik}\,g_{kj}\left[\frac{1}{2}{\cal
G}_{IJ}\dot{\phi}^{I}\dot{\phi}^{J}-V(\phi^{K})\right]\propto\delta^{i}_{\>\>j}\,.$
(122)

Comparing with Eq. (18), we see that $T^{i}_{\>\>j}\propto\delta^{i}_{\>\>j}$
and hence $\pi_{ij}=0$ for all such models, even those that include curved
field-space manifolds characterized by nontrivial metrics ${\cal
G}_{IJ}(\phi^{K})$.

We may consider more general scenarios that move beyond the canonical
Einstein-Hilbert gravitational action. In particular, we can examine the
Horndeski action, which consists of the most general set of terms in
$(3+1)$-dimensional spacetime that yields a second-order equation of motion
for a scalar field $\phi(x^{\mu})$ coupled to gravity. The action may be
parameterized as Kobayashi (2019); Rubakov (2014)

$\begin{split}S=\int
d^{4}x\sqrt{-g}&\Bigg{\\{}G_{2}(\phi,Z)-G_{3}(\phi,Z)\,\Box\phi+G_{4}(\phi,Z)R+G_{4,Z}\Big{[}(\Box\phi)^{2}-\nabla^{\mu}\nabla^{\nu}\phi\,\nabla_{\mu}\nabla_{\nu}\phi\Big{]}\\\
&+G_{5}(\phi,Z)G^{\mu\nu}\,\nabla_{\mu}\nabla_{\nu}\phi-\frac{1}{6}G_{5,Z}\Big{[}(\Box\phi)^{3}-3\Box\phi\,\nabla^{\mu}\nabla^{\nu}\phi\,\nabla_{\mu}\nabla_{\nu}\phi+2\nabla_{\mu}\nabla_{\nu}\phi\,\nabla^{\nu}\nabla^{\lambda}\phi\,\nabla^{\mu}\nabla_{\lambda}\phi\Big{]}\Bigg{\\}}\,,\end{split}$
(123)

with

$Z\equiv-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\,\partial_{\nu}\phi\,.$ (124)

(To avoid confusion with $X_{ij}$, we will use $Z$ to denote the kinetic term
of the scalar field, as in Eq. (124); in the literature this term is usually
denoted by $X$.) In Eq. (123), each term $G_{i}(\phi,Z)$ is an arbitrary
function of $\phi$ and $Z$, and $G_{i,Z}\equiv\partial G_{i}(\phi,Z)/\partial
Z$. One candidate term of the form $g^{ij}\,X_{ij}$ appears within the term
proportional to $G_{5}(\phi,Z)$, which involves the Einstein tensor
$G^{\mu\nu}$. In particular, we have, as usual,

$G^{\mu\nu}=R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R\,,$ (125)

and hence the action in Eq. (123) includes the term

$g^{ij}X_{ij}=g^{ij}\left(-\frac{1}{2}G_{5}(\phi,Z)\,R\,\nabla_{i}\nabla_{j}\phi\right)\,,$
(126)

keeping in mind that even in a homogeneous background, for which
$\phi\rightarrow\phi(t)$,
$\nabla_{i}\nabla_{j}\phi=-\Gamma^{0}_{\>ij}\dot{\phi}\neq 0$. Given that the
dynamical coefficients $G_{i}(\phi,Z)$ are arbitrary, the most general
statement we can make is that in a spatially homogeneous yet anisotropic
spacetime, terms of the form $g^{ij}X_{ij}$ are generic in the action for
single-field Horndeski models.

##  Appendix B A unique isotropic state

Here we establish that in any (constrained) effective phase space described in
Sec. II.3, the background space is isotropic if and only if $z_{1}=0=z_{2}$.
This allows us to easily identify the isotropic background state in any phase
space.

For a finite (and nonzero) value of the Hubble expansion rate $H$,
$z_{1}=0=z_{2}\iff\sigma_{1}=0=\sigma_{2}\implies\sigma^{2}\equiv\frac{1}{3}\left(\sigma_{1}+\sigma_{2}\right)^{2}-\sigma_{1}\sigma_{2}=0$.
Note however that it is also the case that the final implication sign in the
previous sentence can be reversed:
$\sigma^{2}=0\implies\sigma_{1}=0=\sigma_{2}$. This is because

$\displaystyle\sigma^{2}=0$
$\displaystyle\equiv\frac{1}{3}\left(\sigma_{1}+\sigma_{2}\right)^{2}-\sigma_{1}\sigma_{2}=0$
(127)
$\displaystyle\iff\sigma_{2}=\left(\frac{1\pm\sqrt{3}i}{2}\right)\sigma_{1},$
(128)

so that $\sigma_{1}$ and $\sigma_{2}$ are both real if and only if
$\sigma_{1}=0=\sigma_{2}$. Thus we have that

$z_{1}=0=z_{2}\iff\sigma_{1}=0=\sigma_{2}\iff\sigma^{2}=0.$ (129)

Now, also by definition, we have that

$\displaystyle\sigma^{2}$
$\displaystyle=\frac{1}{2}\left(\dot{\beta}_{1}^{2}+\dot{\beta}_{2}^{2}+\dot{\beta}_{3}^{2}\right)$
(130)
$\displaystyle=\dot{\beta}_{1}^{2}+\dot{\beta}_{2}^{2}+\dot{\beta}_{1}\dot{\beta}_{2},$
(131)

where we have used the (derivative of the) constraint over the $\beta_{i}$’s
of Eq. (31) to establish the second line above. Thus we have

$\displaystyle\sigma^{2}=0$
$\displaystyle\implies\dot{\beta}_{2}^{2}+\dot{\beta}_{1}\dot{\beta}_{2}+\dot{\beta}_{1}^{2}=0$
(132)
$\displaystyle\iff\dot{\beta}_{2}=\left(\frac{-1\pm\sqrt{3}i}{2}\right)\dot{\beta}_{1},$
(133)

so that $\dot{\beta}_{1}$ and $\dot{\beta}_{2}$ are both real if and only if
$\dot{\beta}_{1}=0=\dot{\beta}_{2}$. The (derivative of the) constraint on the
$\beta_{i}$’s then ensures that $\dot{\beta}_{3}=0$, so that the $\beta_{i}$’s
are all constant and the background space is isotropic. Thus we have shown
that

$\sigma^{2}=0\implies\textrm{Background is isotropic}.$ (134)

Note also that if the background is isotropic, then it has an FLRW form
wherein the shear scalar also vanishes. Thus we have that

$\textrm{Background is isotropic}\implies\sigma^{2}=0.$ (135)

Combining Eqs. (129), (134), and (135) therefore gives us the required result:
that (assuming $H$ is finite and nonzero)

$\displaystyle z_{1}=0=z_{2}$
$\displaystyle\iff\sigma_{1}=0=\sigma_{2}\iff\sigma^{2}=0$
$\displaystyle\iff\textrm{Background is isotropic}.$ (136)

##  Appendix C Spacetime structure associated with anisotropic fixed points
FPc–FPf

Fixed points FPc–FPf in Table 1 are anisotropic for arbitrary (positive)
values of $\mu_{0}$ and $\nu_{0}$. Here we describe the spacetime metric
corresponding to each of these fixed points. Generally, we find that the
metric has rotational symmetry about an axis (the $x^{3}$-axis for FPc and
FPd, and the $x^{2}$-axis for FPe and FPf) and corresponds to a background
that is expanding in each independent direction.

Consider FPc, for which $z_{1}=0$ and thus $\sigma_{1}=0$. (We will focus, in
what follows, on expanding solutions where the Hubble expansion rate is
finite, so that $0<H<\infty$.) This implies

$\displaystyle\dot{\beta}_{1}=\dot{\beta}_{2}.$ (137)

This result, together with the derivative of the constraint on the
$\beta_{i}$’s ($\sum_{i=1}^{3}\dot{\beta}_{i}=0$) yields

$\displaystyle\dot{\beta}_{3}=-2\dot{\beta}_{1}.$ (138)

Furthermore, $z_{2}=-\sqrt{\nu_{0}/6}$ and thus from the definition of $z_{2}$
and $\sigma_{2}$:

$\displaystyle\dot{\beta}_{1}-\dot{\beta}_{3}=-3H\sqrt{\frac{\nu_{0}}{6}}.$
(139)

Substituting Eq. (138) into Eq. (139) we find

$\displaystyle\dot{\beta}_{1}=-\sqrt{\frac{\nu_{0}}{6}}\frac{\dot{a}}{a},$
(140)

so that

$\displaystyle\beta_{1}=-\sqrt{\frac{\nu_{0}}{6}}\ln a+C_{1},$ (141)

where $C_{1}$ is a constant of integration. Combining Eqs. (137) and (140) we
find

$\displaystyle\beta_{2}=-\sqrt{\frac{\nu_{0}}{6}}\ln a+C_{2},$ (142)

where $C_{2}$ is a constant. Similarly, combining Eqs. (138) and (140) we find

$\displaystyle\beta_{3}=\sqrt{\frac{2\nu_{0}}{3}}\ln a+C_{3},$ (143)

where $C_{3}$ is a constant.

The scale factor in each direction, as described in Eq. (6), can be written as
$a_{i}=ae^{\beta_{i}}$ so that, using Eqs. (141), (142), and (143), we find

$\displaystyle a_{1}$ $\displaystyle=A_{1}a^{1-\sqrt{\nu_{0}/6}},$ (144)
$\displaystyle a_{2}$ $\displaystyle=A_{2}a^{1-\sqrt{\nu_{0}/6}},$ (145)
$\displaystyle a_{3}$ $\displaystyle=A_{3}a^{1+\sqrt{2\nu_{0}/3}},$ (146)

where $A_{i}\equiv e^{C_{i}}$ (for $i=1,2,3$) is a positive constant.
Substituting Eqs. (144)–(146) into the original form of the metric [Eq. (6)]
and implementing a (straightforward) rescaling of the spatial coordinates
(which we now refer to using $\\{x,y,z\\}$), we find

$\displaystyle ds^{2}=-dt^{2}$
$\displaystyle+a^{2\left(1-\sqrt{\nu_{0}/6}\right)}\left(dx^{2}+dy^{2}\right)$
$\displaystyle+a^{2\left(1+\sqrt{2\nu_{0}/3}\right)}dz^{2}.$ (147)

Note further that the Hubble slow-roll parameter is given by
$\epsilon\equiv-\dot{H}/H^{2}=\nu_{0}/2$, so that

$\displaystyle a\sim t^{2/\nu_{0}}.$ (148)

This metric has rotational symmetry about the $z$-axis. For $0<\nu_{0}<6$,
scale factors in the $x$\- and $y$-directions are growing with time though the
expansion in these directions is slower than that in the $z$-direction.

A similar calculation for FPd (where the only difference between FPd and FPc,
for the purposes of computing the metric, is in the sign of $z_{2}$) yields

$\displaystyle ds^{2}=-dt^{2}$
$\displaystyle+a^{2\left(1+\sqrt{\nu_{0}/6}\right)}\left(dx^{2}+dy^{2}\right)$
$\displaystyle+a^{2\left(1-\sqrt{2\nu_{0}/3}\right)}dz^{2},$ (149)

where $a$ is given in Eq. (148). This metric also has rotational symmetry
about the $z$-axis. For $0<\nu_{0}<3/2$ the scale factor in the $z$-direction
is growing with time though the expansion in this direction is slower than in
the $x$\- and $y$-directions.

Spacetime metrics for fixed points FPe and FPf can be obtained by symmetry
from the metrics computed for FPc and FPd (respectively). For FPe and FPf,
$z_{2}=0$, so that $\dot{\beta_{1}}=\dot{\beta_{3}}$. Thus the scale factor in
the $x^{2}$-direction will differ from the scale factors in the other two
directions (as opposed to FPc and FPd, where the scale factor in the
$x^{3}$-direction was different from that in the other two directions). Note,
in addition, that FPe and FPf are defined in terms of $\mu_{0}$ instead of
$\nu_{0}$.

The metric for FPe can thus be obtained from that of FPc by swapping
$z\leftrightarrow y$ and setting $\nu_{0}\to\mu_{0}$. We find that

$\displaystyle ds^{2}=-dt^{2}$
$\displaystyle+a^{2\left(1-\sqrt{\mu_{0}/6}\right)}\left(dx^{2}+dz^{2}\right)$
$\displaystyle+a^{2\left(1+\sqrt{2\mu_{0}/3}\right)}dy^{2},$ (150)

with

$\displaystyle a\sim t^{2/\mu_{0}}.$ (151)

This metric has rotational symmetry about the $y$-axis. For $0<\mu_{0}<6$,
scale factors in the $x$\- and $z$-directions are growing with time though the
expansion in these directions is slower than that in the $y$-direction.

The metric for FPf can similarly be obtained from that of FPd by swapping
$z\leftrightarrow y$ and setting $\nu_{0}\to\mu_{0}$. We find that

$\displaystyle ds^{2}=-dt^{2}$
$\displaystyle+a^{2\left(1+\sqrt{\nu_{0}/6}\right)}\left(dx^{2}+dz^{2}\right)$
$\displaystyle+a^{2\left(1-\sqrt{2\mu_{0}/3}\right)}dy^{2},$ (152)

where $a$ is given by Eq. (151). This metric has rotational symmetry about the
$y$-axis. For $0<\mu_{0}<3/2$ the scale factor in the $y$-direction is growing
with time though the expansion in this direction is slower than in the $x$\-
and $z$-directions.

##  Appendix D A Kasner circle of fixed points

Here we demonstrate that fixed points in the principal phase space when $x=0$
correspond to the Kasner circle. Recall that those fixed points are given by

$x=y=\omega_{1}=\omega_{2}=0\;\;\&\;\;1=(z_{1}+z_{2})^{2}-3z_{1}z_{2}.$ (153)

Assume at the outset that $0<H<\infty$ (so that we only ever consider
expanding solutions where the Hubble expansion rate is finite). Then, for
these fixed points, the definitions of the dynamical variables yield:

$\displaystyle x=0$ $\displaystyle\iff c+L+\tilde{X}=0,$ (154) $\displaystyle
y=0$ $\displaystyle\iff 3c+\tilde{X}=0,$ (155) $\displaystyle\omega_{1}=0$
$\displaystyle\iff\pi_{1}=0\iff p_{1}=p_{2},$ (156)
$\displaystyle\omega_{2}=0$ $\displaystyle\iff\pi_{2}=0\iff p_{1}=p_{3}.$
(157)

From the last two equations above, and the constraint that
$p_{1}+p_{2}+p_{3}=0$ [Eq. (24)], we find

$\displaystyle p_{1}=p_{2}=p_{3}=0.$ (158)

Substituting Eqs. (154)–(157) and Eq. (158) into the first Friedmann equation
[Eq. (29)], the equation of evolution for the $\beta_{i}$’s [Eq. (31)], and
the equation of evolution of the shear scalar yields:

$\displaystyle H^{2}$ $\displaystyle=\frac{1}{3}\sigma^{2},$ (159)
$\displaystyle\ddot{\beta_{i}}+3H\dot{\beta_{i}}$ $\displaystyle=0,$ (160)
$\displaystyle\frac{d}{dt}\sigma^{2}+6H\sigma^{2}$ $\displaystyle=0.$ (161)

Note that the constraint on the $z_{A}$’s in Eq. (153) (which we have not yet
explicitly used) just yields the first Friedmann equation [viz. Eq. (159)].
Integrating Eq. (161), we find

$\displaystyle\sigma^{2}=e^{{C}}a^{-6},$ (162)

where ${C}$ is a constant. Substituting this into Eq. (159) and then
integrating, we obtain

$\displaystyle\frac{a^{3}}{3}=\sqrt{\frac{e^{{C}}}{3}}t+{D},$ (163)

where ${D}$ is a constant. Assuming the initial condition $a(t=0)=0$, yields

$\displaystyle a=\gamma
t^{1/3},\;\;\textrm{where}\;\;\gamma\equiv\left(3e^{{C}}\right)^{1/6}.$ (164)

Note that therefore, we have

$\displaystyle H=\frac{1}{3t}.$ (165)

Integrating Eq. (160), and using the expression for $H$ in Eq. (165), we find
(for each $i=1,2,3$) that

$\displaystyle|\dot{\beta}_{i}|=\frac{e^{B_{i}}}{t},$ (166)

where $B_{i}$ is a constant. Another way to write this, which will make things
easier in what follows is:

$\displaystyle\dot{\beta}_{i}=\pm\frac{e^{B_{i}}}{t}\equiv\frac{{A}_{i}}{t},$
(167)

where ${A}_{i}\equiv\pm e^{B_{i}}$ is a constant that can take either sign.
Constraints on these constants arise due to the derivative of the constraint
on the sum of the $\beta_{i}$’s. In particular:

$\displaystyle\dot{\beta}_{1}+\dot{\beta}_{2}+\dot{\beta}_{3}=0\implies\frac{\left({A}_{1}+{A}_{2}+{A}_{3}\right)}{t}=0,$
(168)

so that

$\displaystyle{A}_{1}+{A}_{2}+{A}_{3}=0.$ (169)

Also, note that we have

$\displaystyle\sigma^{2}$
$\displaystyle=\frac{1}{2}\left(\dot{\beta}_{1}^{\,2}+\dot{\beta}_{2}^{\,2}+\dot{\beta}_{3}^{\,2}\right)$
(170)
$\displaystyle=\frac{1}{2t^{2}}\left({A}_{1}^{2}+{A}_{2}^{2}+{A}_{3}^{2}\right),$
(171)

where the second line above uses Eq. (167). However, from Eqs. (162) and
(164), we also have

$\displaystyle\sigma^{2}=\frac{e^{{C}}}{a^{6}}=\frac{e^{{C}}}{\gamma^{6}t^{2}}=\frac{1}{3t^{2}}.$
(172)

Combining Eqs. (171) and (172) gives

$\displaystyle{A}_{1}^{2}+{A}_{2}^{2}+{A}_{3}^{2}=\frac{2}{3}.$ (173)

To complete the derivation of the metric, we may integrate Eq. (167) (for
$i=1,2,3$), to find

$\displaystyle\beta_{i}={A}_{i}\ln{t}+\mu_{i},$ (174)

where $\mu_{i}$ is a constant. Combining the expression for $a$ [in Eq. (164)]
with the expressions for the $\beta_{i}$’s [in Eq. (174)] we can write the
metric in the following way:

$\displaystyle ds^{2}$
$\displaystyle=-dt^{2}+{\gamma^{2}\,e^{2\mu_{1}}}\,t^{2/3}\,e^{2{A}_{1}\ln{t}}\,dx^{2}+{\gamma^{2}\,e^{2\mu_{2}}}\,t^{2/3}\,e^{2{A}_{2}\ln{t}}\,dy^{2}+{\gamma^{2}\,e^{2\mu_{3}}}\,t^{2/3}\,e^{2{A}_{3}\ln{t}}\,dz^{2}$
(175) $\displaystyle\equiv-
dt^{2}+{\gamma^{2}\,e^{2\mu_{1}}}\,t^{2q_{1}}\,dx^{2}+{\gamma^{2}\,e^{2\mu_{2}}}\,t^{2q_{2}}\,dy^{2}+{\gamma^{2}\,e^{2\mu_{3}}}\,t^{2q_{3}}\,dz^{2},$
(176)

where the final line uses the following definition (for each $i=1,2,3$):

$\displaystyle q_{i}$ $\displaystyle\equiv\frac{1}{3}+{A}_{i}.$ (177)

A change of spatial coordinates allows us to bring the metric into a standard
form (as quoted in, for example, Ref. (Misner  _et al._ , 1973, §30.2)):

$\displaystyle
ds^{2}=-dt^{2}+t^{2q_{1}}\,d\bar{x}^{2}+t^{2q_{2}}\,d\bar{y}^{2}+t^{2q_{3}}\,d\bar{z}^{2}.$
(178)

Given the constraints on the ${A}_{i}$’s, this solution is not a function of
three free parameters (the $q_{i}$’s)—there are (as we will describe) two
constraints that these parameters must satisfy and hence we have a one-
parameter family of solutions. From the definitions of the $q_{i}$’s and the
constraints on the ${A}_{i}$’s, it is straightforward to show that

$\displaystyle q_{1}+q_{2}+q_{3}$ $\displaystyle=1,$ (179) $\displaystyle
q_{1}^{2}+q_{2}^{2}+q_{3}^{2}$ $\displaystyle=1.$ (180)

The metric is more precisely specified by those values of the $q_{i}$’s that
are given by the intersection of the plane (in the first constraint above) and
the sphere (in the second constraint). This circle (in general) is known as
the Kasner circle.

We have thus shown that the fixed points of interest [in Eq. (153)] are
consistent with Einstein field equations whose solution is given by the Kasner
metric in Eqs. (176) or (178).

Note that our fixed-point analysis for the principal phase space yields a one-
parameter family of solutions—as expressed by the constraint on the $z_{A}$’s,
in Eq. (153). One can expressly relate this one-parameter family of fixed
points to the Kasner circle as follows.

The Kasner circle can be characterized by eliminating $q_{1}$ from Eq. (180),
using Eq. (179). In particular, one obtains:

$\displaystyle q_{2}^{2}+q_{3}^{2}+q_{2}q_{3}-q_{2}-q_{3}=0.$ (181)

Now, let the $z_{A}$’s satisfy Eq. (153), and let us use the expressions for
$H$ and $\dot{\beta}_{i}$ found earlier [in Eqs. (165) and (167)]. That is, we
can write

$\displaystyle
z_{1}\equiv\frac{\sigma_{1}}{3H}=\frac{\dot{\beta_{1}}-\dot{\beta_{2}}}{3H}$
$\displaystyle=\frac{{{A}_{1}}/{t}-{{A}_{2}}/{t}}{{1}/{t}}$
$\displaystyle={A}_{1}-{A}_{2}\equiv q_{1}-q_{2}.$ (182)

Similarly, we can write

$\displaystyle z_{2}=q_{1}-q_{3}.$ (183)

It is now straightforward to show, using Eq. (182), Eq. (183), and the
constraints on the $q_{i}$’s, that

$\displaystyle(z_{1}+z_{2})^{2}-3z_{1}z_{2}-1$
$\displaystyle=z_{1}^{2}-z_{1}z_{2}+z_{2}^{2}-1$ (184)
$\displaystyle=q_{2}^{2}+q_{3}^{2}+q_{2}q_{3}-q_{2}-q_{3}.$ (185)

Thus the set of points traced out by the constraint on the $z_{A}$’s is
consistent with the set of points traced out by the constraints on the
$q_{i}$’s.

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