# Teleparallel Robertson-Walker geometries and applications

A. A. Coley<EMAIL_ADDRESS>Department of Mathematics and Statistics,
Dalhousie University, Halifax, Nova Scotia, Canada, B3H 3J5  A. Landry
<EMAIL_ADDRESS>Department of Mathematics and Statistics, Dalhousie
University, Halifax, Nova Scotia, Canada, B3H 3J5  F Gholami<EMAIL_ADDRESS>Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova
Scotia, Canada, B3H 3J5

###### Abstract

In teleparallel geometries the coframe and corresponding spin-connection are
the principal geometric objects and consequently the appropriate definition of
a symmetry is that of an affine symmetry. The set of invariant coframes and
their corresponding spin connections that respect the full six dimensional Lie
algebra of Robertson-Walker affine symmetries are displayed and discussed. We
will refer to such geometries as teleparallel Robertson-Walker (TRW)
geometries, where the corresponding derived metric is of Robertson-Walker form
and is characterized by the parameter $k=(-1,0,1)$. The field equations are
explicitly presented for the $F(T)$ class of teleparallel TRW spacetimes. We
are primarily interested in investigating the $k\neq 0$ TRW models. After
first studying the $k=0$ models and, in particular, writing their governing
field equations in an appropriate form, we then study their late time
stability with respect to perturbations in $k$ in both the cases of a
vanishing and non-vanishing effective cosmological constant term. As an
illustration we consider both quadratic $F(T)$ theories and power-law
solutions.

##  I Introduction

In the covariant approach to teleparallel theories of gravity [2, 1], the
(co)frame and corresponding spin-connection are the main geometric objects of
study (and the metric is a derived quantity). The appropriate definition of a
symmetry in a teleparallel geometry is that of an affine symmetry. In a
teleparallel manifold an affine frame symmetry on the frame bundle is a
diffeomorphism from the manifold to itself that leaves the spin-connection
invariant and affects the invariant frame in a particular way, and is defined
by the existence of a vector field, ${\bf X}$, which satisfies [3]:

$\mathcal{L}_{{\bf X}}\mbox{{\bf h}}_{a}=\lambda_{a}^{~{}b}\,\mbox{{\bf
h}}_{b}\mbox{ and }\mathcal{L}_{{\bf X}}\omega^{a}_{~{}bc}=0,$ (1)

where $\omega^{a}_{~{}bc}$ represents the spin-connection defined with respect
to the geometrically preferred invariant frame $\mbox{{\bf h}}_{a}$ determined
by the Cartan-Karlhede algorithm and $\lambda_{a}^{~{}b}$ is an element of the
linear isotropy group thereby obtained. This definition is a frame-dependent
analogue of the definition of a symmetry presented in [4, 5].111In the Cartan-
Karlhede algorithm adopted for teleparallel geometry [3], the parameters of
the Lorentz frame transformations are fixed by the normalization of the
components of the torsion tensor and its covariant derivatives in an invariant
manner. This algorithm consequently leads to an invariantly defined frame up
to linear isotropy (defined as the (sub)group of Lorentz transformations that
leave the torsion tensor and its covariant derivatives invariant). In
particular, the Cartan-Karlhede algorithm consequently provides a set of
Cartan invariants which can then be used to uniquely characterize a geometry
locally and determine the dimension of the affine symmetry group.

Using this algorithm, the invariant coframes and the corresponding spin
connection that respects the imposed affine frame symmetries were constructed.
In particular, the teleparallel geometries which are invariant under the full
$G_{6}$ Lie algebra of affine symmetries were displayed [6]. In addition, the
proper coframe has also been obtained in each case and the field equations
(FE) displayed. We denote such geometries as teleparallel Robertson-Walker
(TRW) geometries [6], where the associated (derived) metric is of Robertson-
Walker form characterized by a constant parameter $k=(-1,0,1)$. We note that
this parameter $k$, which is usually interpreted as the constant spatial
curvature in the RW pseudo-Riemannian metric, cannot be so related here since
the Riemann tensor is identically zero in teleparallel spacetimes; while “k”
may be considered as the curvature of the $3$-space, in $4$-space it actually
occurs as part of the torsion scalar. We also comment that in the TRW
geometries, an appropriate spin connection/coframe pair results in a situation
in which the antisymmetric part of the FEs are identically zero.

Most of the research up until now has been concerned with the analysis of the
flat ($k=0$) TRW cosmological models [7, 8] (and references within). In
particular, specific forms for $F(T)$ have been investigated (using a priori
ansatz such as, e.g., polynomial functions), and reconstruction methods have
been explored extensively (in which the function $F(T)$ is reconstructed from
the underlying assumptions of the models; e.g., a functional form for the
solution such as the scale factor is a simple power-law). Dynamical systems
methods (e.g., fixed point and stability analysis) in flat TRW models have
been widely utilized [7, 9] (also see [10, 11, 12, 13]), which include the
study of the stability of the standard de Sitter fixed point.

There has been much confusion in the analysis of TRW geometries since it is
often unclear whether the frame and the corresponding spin connection admits
the full $G_{6}$ Lie algebra of spatially homogeneous and isotropic affine
symmetries (which are then necessarily isometries of the metric). In
particular, for non-zero $k$ the geometries investigated often do not have a
$G_{6}$ of affine frame symmetries (usually only $3$ of the Killing vectors
(KVs) considered are affine frame symmetries). Indeed, for the $k=\pm 1$
cases, there have been a number of incorrect attempts to find solutions, often
involving the use of complex tetrads or inappropriate spin connections.
However, recently frame/connection pairs for geometries satisfying a $G_{6}$
group of symmetries have been constructed [15, 4, 5, 14] in a similar (but not
equivalent) fashion to that presented in [3] and utilized here. In addition,
when a proper frame is assumed additional confusion can occur.

The earliest attempts to include a non zero $k$ properly were done in [16, 17,
4, 5, 14], although the derivation was questionable in one of the first times
the FE were written down [18]. However, it is still not clear within this
alternative approach (see also [15, 19, 20, 21]) which functions are essential
in finding a solution and which depend on the coordinates chosen. In addition,
subclasses of geometries with an additional symmetry can’t be determined
explicitly using this approach; larger symmetry groups must be assumed from
the outset and it must then be determined whether they exist or not by trial
and error. Nevertheless, the resulting eqns. are the same as those obtained in
[6] and displayed below.

We are primarily interested in studying the $k\neq 0$ TRW models here.
Geometries with non-zero $k$ have been recently studied in bounce models [22]
and in inflation [23] (although the analysis may only apply for $k=1$, due to
the use of a complex tetrad). Also, perturbations have been studied in non-
flat cosmology [24].

##  II  Teleparallel Robertson-Walker (TRW) spacetimes

We shall work in coordinates $(t,r,\theta,\phi)$, where the $G_{6}$ Lie
algebra of affine symmetries are [3]:

$\displaystyle
X_{z}=\partial_{\phi},~{}X_{y}=-\cos\phi\partial_{\theta}+\frac{\sin\phi}{\tan\theta}\partial_{\phi},X_{x}=\sin\phi\partial_{\theta}+\frac{\cos\phi}{\tan\theta}\partial_{\phi},$
$\displaystyle
X_{1}=\chi\sin\theta\cos\phi\partial_{r}+\frac{\chi}{r}\cos\theta\cos\phi\partial_{\theta}-\frac{\chi\sin\phi}{r\sin\theta}\partial_{\phi},$
$\displaystyle
X_{2}=\chi\sin\theta\sin\phi\partial_{r}+\frac{\chi}{r}\cos\theta\sin\phi\partial_{\theta}+\frac{\chi\cos\phi}{r\sin\theta}\partial_{\phi},$
(2) $\displaystyle
X_{3}=\chi\cos\theta\partial_{r}-\frac{\chi}{r}\sin\theta\partial_{\theta},$

where $\chi\equiv\sqrt{1-kr^{2}}$. We write
$\\{X_{I}\\}_{I=1}^{6}=\\{X_{1},X_{2},X_{3},X_{x},X_{y},X_{z}\\}$. The largest
linear isotropy group allowed by a spatially homogeneous geometry is $SO(3)$
and a matrix basis for its Lie algebra is of the form:

$\lambda_{\hat{1}}=\left[\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\end{array}\right],~{}\lambda_{\hat{2}}=-\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&1&0\\\ 0&-1&0&0\\\
0&0&0&0\end{array}\right],~{}\lambda_{\hat{3}}=-\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&1\\\ 0&0&0&0\\\ 0&-1&0&0\end{array}\right]$ (3)

We also work with the following coframe:

$h^{a}_{~{}\mu}=\left[\begin{array}[]{cccc}1&0&0&0\\\
0&\frac{a(t)}{\sqrt{1-kr^{2}}}&0&0\\\ 0&0&a(t)r&0\\\
0&0&0&a(t)r\sin(\theta)\end{array}\right].$ (4)

The derived metric is the Robertson-Walker (RW) metric:

$\displaystyle
ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-k\,r^{2}}+r^{2}\,\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right)\right],$
(5)

where the parameter $k=(-1,0,1)$ but it is not interpreted as a constant
spatial curvature here. It is this metric that leads to the Friedmann-
Lemaitre-RW models in GR.

Assuming that the connection is metric compatible, we necessarily have that
$\omega_{abc}=-\omega_{bac}$. The solution to eqns (1) for each of the affine
symmetry vector fields in (2) gives the non-trivial components [6]:

$\displaystyle\omega_{122}=\omega_{133}=\omega_{144}=W_{1}(t),$
$\displaystyle\omega_{234}=-\omega_{243}=\omega_{342}=W_{2}(t),$
$\displaystyle\omega_{233}=\omega_{244}=-\frac{\sqrt{1-kr^{2}}}{a(t)r},$ (6)
$\displaystyle\omega_{344}=\frac{\cos(\theta)}{a(t)r\sin(\theta)},$

where $W_{1}$ and $W_{2}$ are arbitrary functions. This is the general
connection for any Riemann-Cartan geometry admitting the symmetry group with
generators given by eqn (2). The tensor-part of the torsion tensor is then
identically zero and the torsion tensor can be further decomposed into a
vector-part, $V_{a}=T^{b}_{~{}ba}$, and a axial-part,
$A_{a}=\epsilon_{abcd}T^{bcd}$.

In order to determine the connections that represent a teleparallel geometry
we must then impose the Riemmann flatness conditions, which have the distinct
solutions [6]:

  * •

$W_{1}(t)=0$, $\displaystyle W_{2}(t)=\pm\frac{\sqrt{k}}{a(t)}$ where
$\displaystyle{\bf V}=-\frac{3\dot{a}(t)}{a(t)}\mbox{{\bf h}}^{1},\mbox{\ \rm
and\ }{\bf A}=\mp\frac{2\sqrt{k}}{a(t)}\mbox{{\bf h}}^{1}$,

  * •

$\displaystyle W_{1}(t)=\pm\frac{\sqrt{-k}}{a(t)}$, $W_{2}(t)=0$ where
$\displaystyle{\bf V}=\frac{3(\pm\sqrt{-k}+\dot{a}(t))}{a(t)}\mbox{{\bf
h}}^{1},\mbox{\ \rm and\ }{\bf A}=0$,

where $a=a(t)$ is the frame function and $\dot{a}=\dot{a}(t)$ is its time
derivative. Each case above contains the subcase $k=0$. In the first case the
$\pm$ solutions are equivalent under a local Lorentz transformation and in the
second case the $\pm$ solutions are equivalent under a discrete coordinate
transformation $t\,\rightarrow\,-t$.

In conclusion, each of the TRW geometries given by the coframe (4) with the
connection (6) (and such that the two arbitrary functions $W_{1}$ and $W_{2}$
satisfy one of the forms above) is a teleparallel geometry admitting the
required symmetry group with generators given by eqn (2).

The torsion scalar for each of $k=(-1,0,1)$ is of the form

$\displaystyle T(t)$ $\displaystyle=$ $\displaystyle
6\left(\frac{\dot{a}}{a}\right)^{2}-12W_{1}\frac{\dot{a}}{a}-6W_{1}^{\,2}+6W_{2}^{\,2}$
(7) $\displaystyle=$ $\displaystyle
6\left(\frac{\dot{a}}{a}+W_{1}+W_{2}\right)\left(\frac{\dot{a}}{a}+W_{1}-W_{2}\right)$
$\displaystyle=$ $\displaystyle\frac{2}{3}V^{2}-\frac{3}{2}A^{2},$

where the vectorial and axial term magnitudes are given by:

$V^{2}=-9\left(\frac{\dot{a}}{a}+W_{1}\right)^{2},\qquad A^{2}=-4W_{2}^{2}.$
(8)

We note that a non-trivial axial part of the torsion scalar only occurs in the
$k=+1$ case. Also note that there are $4$ independent (Cartan) invariant
scalars in the positive $k$ case, while there are only $3$ in the $k=-1$ case
[3].

###  II.1 TRW Proper Frames

In teleparallel geometry, there always exists a frame in which the spin-
connection is zero. This frame is called the proper frame (or Weitzenbock
frame). If we assume an orthonormal diagonal coframe in the spherical
coordinate system (4) with corresponding spin connection computed above, we
can investigate each of the 3 values for $k$ independently. For $k=-1$ we have
$[W_{1}(t),W_{2}(t)]=[\delta\sqrt{-k}/a(t),0]$, for $k=0$ we have
$[W_{1}(t),W_{2}(t)]=[0,0]$, and for $k=+1$ we have that
$[W_{1}(t),W_{2}(t)]=[0,\delta\sqrt{k}/a(t)]$, where $\delta=\pm 1$ is a
discrete parameter. The spin connection can then be determined, for a matrix
$\Lambda^{a}_{~{}b}\in SO(1,3)$, from the differential equation

$\omega^{a}_{~{}b}=(\Lambda^{-1})^{a}_{~{}c}d\Lambda^{c}_{~{}b};$ (9)

that is, we now simply need to solve this differential equation for
$\Lambda^{a}_{~{}b}$ for each of $k=-1$ and $k=+1$ (where $k=0$ constitutes a
subcase of both), and hence easily determine a proper coframe
$\widetilde{h}{}^{a}=\Lambda^{a}_{~{}b}h^{b}$, where $h^{b}$ is given by eqn.
(4).

####  II.1.1 Negative $k$-parameter TRW Case:

When $k=-1$, a Lorentz transformation that satisfies the differential eqn. (9)
is

$\Lambda^{a}_{~{}b}=\left[\begin{array}[]{cccc}\sqrt{1-kr^{2}}&-\delta\sqrt{-k}r&0&0\\\
-\delta\sqrt{-k}r\sin(\theta)\sin(\phi)&\sqrt{1-kr^{2}}\sin(\theta)\sin(\phi)&\cos(\theta)\sin(\phi)&\cos(\phi)\\\
-\delta\sqrt{-k}r\sin(\theta)\cos(\phi)&\sqrt{1-kr^{2}}\sin(\theta)\cos(\phi)&\cos(\theta)\cos(\phi)&-\sin(\phi)\\\
\delta\sqrt{-k}r\cos(\theta)&-\sqrt{1-kr^{2}}\cos(\theta)&\sin(\theta)&0\end{array}\right].$
(10)

Clearly any global Lorentz transformation multiplying this transformation also
gives rise to a solution. Therefore, we can either formulate the $k=-1$
spacetime geometry by using the proper frame
$\widetilde{h}{}^{a}=\Lambda^{a}_{~{}b}h^{b}$ with this Lorentz transformation
and with $h^{b}$ given by eqn. (4), which necessarily has a trivial spin
connection, or by using the diagonal coframe (4) and the corresponding spin
connection one-form [6]:

$\omega^{a}_{~{}b}=\left[\begin{array}[]{cccc}0&-\frac{\delta\sqrt{-k}}{\sqrt{1-kr^{2}}}dr&-\delta\sqrt{-k}rd\theta&-\delta\sqrt{-k}r\sin(\theta)d\phi\\\
-\frac{\delta\sqrt{-k}}{\sqrt{1-kr^{2}}}dr&0&-\sqrt{1-kr^{2}}d\theta&-\sqrt{1-kr^{2}}\sin(\theta)d\phi\\\
-\delta\sqrt{-k}rd\theta&\sqrt{1-kr^{2}}d\theta&0&-\cos(\theta)d\phi\\\
-\delta\sqrt{-k}r\sin(\theta)d\phi&\sqrt{1-kr^{2}}\sin(\theta)d\phi&\cos(\theta)d\phi&0\end{array}\right].$
(11)

####  II.1.2 Positive $k$-parameter TRW Case:

When $k=+1$, a Lorentz transformation that satisfies the differential eqn. (9)
is

$\displaystyle\Lambda^{a}_{~{}b}=$
$\displaystyle\left[\begin{array}[]{cccc}1&0&0&0\\\
0&\cos(\theta)&-\sqrt{1-kr^{2}}\sin(\theta)&-\delta\sqrt{k}r\sin(\theta)\\\
0&\sin(\theta)\cos(\phi)&\sqrt{1-kr^{2}}\cos(\theta)\cos(\phi)-\delta\sqrt{k}r\sin(\phi)&\delta\sqrt{k}r\cos(\theta)\cos(\phi)-\sqrt{1-kr^{2}}\sin(\phi)\\\
0&\sin(\theta)\sin(\phi)&\sqrt{1-kr^{2}}\cos(\theta)\sin(\phi)+\delta\sqrt{k}r\cos(\phi)&\delta\sqrt{k}r\cos(\theta)\sin(\phi)+\sqrt{1-kr^{2}}\cos(\phi)\end{array}\right].$
(16)

Any global Lorentz transformation times this transformation is again a
solution. We can properly formulate the $k=+1$ geometry by constructing the
proper frame $\widetilde{h}{}^{a}=\Lambda^{a}_{~{}b}h^{b}$ using this Lorentz
transformation and with $h^{b}$ given by eqn. (4), which necessarily has a
trivial spin connection, or by using the diagonal coframe (4) and
corresponding spin connection one-form [6]:

$\displaystyle\omega^{a}_{~{}b}=$
$\displaystyle\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&-\sqrt{1-kr^{2}}d\theta+\delta\sqrt{k}r\sin(\theta)d\phi&-\delta\sqrt{k}rd\theta-\sqrt{1-kr^{2}}\sin(\theta)d\phi\\\
0&\sqrt{1-kr^{2}}d\theta-\delta\sqrt{k}r\sin(\theta)d\phi&0&\frac{\delta\sqrt{k}}{\sqrt{1-kr^{2}}}dr-\cos(\theta)d\phi\\\
0&\delta\sqrt{k}rd\theta+\sqrt{1-kr^{2}}\sin(\theta)d\phi&-\frac{\delta\sqrt{k}}{\sqrt{1-kr^{2}}}dr+\cos(\theta)d\phi&0\end{array}\right].$
(22)

###  II.2 Field equations

Using either the proper coframe (and trivial connection) or the diagonal
coframe/ connection pair (4) and (11), the torsion scalar for $k=-1$ is:

$T=6\left(\frac{\dot{a}}{a}+\frac{\delta\sqrt{-k}}{a}\right)^{2}.$ (24)

We shall assume an energy momentum tensor of the form
$T_{ab}=\rho(t)u_{a}u_{b}+(u_{a}u_{b}+g_{ab})p(t)$, which formally represents
a perfect fluid source with energy density and pressure denoted by $\rho(t)$
and $p(t)$, respectively. The antisymmetric part of the FEs are then
automatically satisfied (and hence all of the frames defined above are ”good”
frames in the terminology of [2]) in the $F(T)$ teleparallel geometries. The
remaining linearly independent eqns. from the symmetric part of the FEs in the
$F(T)$ teleparallel geometries are then given by:

$\displaystyle-\frac{F(T)}{2}+6F^{\prime}(T)\left(\frac{\dot{a}}{a}\right)\left(\frac{\dot{a}}{a}+\frac{\delta\sqrt{-k}}{a}\right)$
$\displaystyle=$ $\displaystyle\kappa\rho,$ (25a) $\displaystyle
F(T)-6F^{\prime}(T)\left(\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}+\frac{\delta\sqrt{-k}}{a}\right)^{2}\right)-6F^{\prime\prime}(T)\dot{T}\left(\frac{\dot{a}}{a}+\frac{\delta\sqrt{-k}}{a}\right)$
$\displaystyle=$ $\displaystyle\kappa(\rho+3p).$

Obviously, these coframe/connection pairs also apply in the subcase $k=0$.
Care must be taken in the case $k=+1$ since complex valued coframes or spin
connections may occur.

Using either the proper coframe (and trivial spin connection) or the diagonal
coframe/ connection pair (4) and (22), the torsion scalar for $k=+1$ is:

$T=6\left[\left(\frac{\dot{a}}{a}\right)^{2}-\frac{k}{a^{2}}\right],$ (26)

which is independent of $\delta$. Assuming a perfect fluid as in the $k=-1$
case, the antisymmetric part of the FEs are identically satisfied and the
symmetric part of the FEs are

$\displaystyle-\frac{F(T)}{2}+6F^{\prime}(T)\left(\frac{\dot{a}}{a}\right)^{2}$
$\displaystyle=$ $\displaystyle\kappa\rho,$ (27a) $\displaystyle
F(T)-6F^{\prime}(T)\left(\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}-\frac{k}{a^{2}}\right)-6F^{\prime\prime}(T)\dot{T}\left(\frac{\dot{a}}{a}\right)$
$\displaystyle=$ $\displaystyle\kappa(\rho+3p).$ (27b)

In this $k=+1$ case, there is no dependence on the discrete parameter
$\delta$. Further, the equations once again reduce to the $k=0$ FEs by setting
$k=0$.

The energy-momentum conservation equation

${\dot{\rho}}+3\frac{\dot{a}}{a}(\rho+p)=0,$ (28)

follows from the two FEs above.

Notice that the FEs reduce to those of FLRW models when $F(T)=T$.

###  II.3 The function $F(T)$

A number of particular examples of $F(T)$ theories studied in the literature
include polynomial functions in $T$, and especially quadratic $T^{2}$ theory
[7]. Recently, it has been shown that the theory

$F(T)=-\Lambda+T+\gamma\,T^{2}$ (29)

can alleviate a variety of cosmological tensions [25]. It is also of interest
to study the theory with $F(T)=T+\gamma T^{\beta}$ or an exponential function.

##  III Analysis and Equations

  * •

We recall that if $T=$ const., then the theory reduces to a renormalized GR
and we obtain teleparallel analogues of special solutions in GR [1]. Such
solutions with a non-vanishing ”effective” cosmological constant will be
asymptotic to ($k=0$) teleparallel analogues of de Sitter solutions [6]. We
are only interested here in non-GR type solutions.

  * •

We shall assume an eqn. of state of the form: $p=\alpha\rho$, with
$-1<\alpha\leq 1$, so that $\rho$ is monotonically decreasing for
$H\equiv\frac{\dot{a}}{a}>0$. We are particularly interested in the case
$k\neq 0$.

  * •

A detailed analysis of the theory $F(T)=-\Lambda+T+\gamma T^{\beta}$ or for
power-laws solutions is possible.

###  III.1 Equation for $k=0$

In the case $k=0$, the non constant torsion scalar is given by $T=6H^{2}$ and
is positive (and we assume that $H\geq 0$). The conservation eqn. is given by
(28), and so for $p=\alpha\rho$, with $\alpha>-1$, $\rho$ is monotonically
decreasing to zero at late times. Note that in the case $k=0$ at the present
time $\rho_{c}\,=\,\frac{3H^{2}}{8\pi G}\,\sim\,10^{-26}$ $kg/m^{3}$. For
positive $\rho$ we have that $-\frac{F}{2}+T\,F^{\prime}>0$. From the two FEs
we then find that (compare with [12]):

$\frac{dT}{d\tau}=\frac{3(1+\alpha)(\frac{F}{2}-T\,F^{\prime})}{(\frac{F^{\prime}}{2}+T\,F^{\prime\prime})}\equiv\mathcal{F}(T),$
(30)

where the new time coordinate $\tau$ is defined by

$\frac{d\tau}{dt}\equiv H\equiv\sqrt{\frac{T}{6}},$

which is well defined since $H>0$. Note that for positive $\rho$,
$(\frac{F}{2}-T\,F^{\prime})<0$ and so $\mathcal{F}(T)<0$ and
$\frac{dT}{d\tau}<0$ if $(\frac{F^{\prime}}{2}+T\,F^{\prime\prime})>0$. From
(30) we have that:

$\frac{1}{\mathcal{F}}=-\frac{1}{3(1+\alpha)}\frac{d}{dT}\left[\ln\left(\frac{F}{2}-T\,F^{\prime}\right)\right],$
(31)

so that eqn. (30) can be integrated to obtain

$\displaystyle\frac{F}{2}-T\,F^{\prime}$ $\displaystyle=$
$\displaystyle-\tilde{A}_{0}e^{-3(1+\alpha)\tau},$ (32)

where $\tilde{A}_{0}$ is a positive constant.

###  III.2 Equation for $k=\pm 1$

When $k=-1$, $T$ is given by (24), and we obtain (where here
$\frac{d\tau}{dt}\equiv\sqrt{\frac{T}{6}}$):

$\displaystyle\frac{dT}{d\tau}$ $\displaystyle=$
$\displaystyle\,\mathcal{F}(T)\,\sqrt{1-\frac{6\,K}{T}}\,\left[\frac{1+\left(\frac{2\left(1+3\alpha\right)\,T\,F^{\prime}(T)}{\left(1+\alpha\right)\left(\frac{F(T)}{2}-T\,F^{\prime}(T)\right)}\right)\,\frac{K}{T}}{1-\left(\frac{6\,T\,F^{\prime\prime}(T)}{\left(\frac{F^{\prime}(T)}{2}+T\,F^{\prime\prime}(T)\right)}\right)\,\frac{K}{T}}\right],$
(33)

where $K=K(T)\equiv\frac{\sqrt{-k}}{a^{2}}$. For $\frac{K}{T}\ll 1$, we can
approximate this as:

$\frac{dT}{d\tau}\approx\,\mathcal{F}(T)\,+\,\mathcal{G}(T)\,K(T),$ (34)

where

$\mathcal{G}(T)=\frac{6\left(1+3\alpha\right)\,F^{\prime}(T)}{\left[\frac{F^{\prime}(T)}{2}\,+T\,F^{\prime\prime}(T)\right]}+\frac{9\left(1+\alpha\right)\left[\frac{F(T)}{2}-T\,F^{\prime}(T)\right]\left[T\,F^{\prime\prime}(T)-\frac{F^{\prime}(T)}{2}\right]}{T\,\left[\frac{F^{\prime}(T)}{2}\,+T\,F^{\prime\prime}(T)\right]^{2}}.$
(35)

When $k=+1$, $T$ is given by (26) (where $\frac{d\tau}{dt}\equiv H$), we
obtain:

$\displaystyle\frac{dT}{d\tau}$ $\displaystyle=$
$\displaystyle\,\mathcal{F}(T)\,\sqrt{1+\frac{6\,K}{T}}\,\left[\frac{1-\left[\frac{2\left(1+3\alpha\right)\,T\,F^{\prime}(T)}{\left(1+\alpha\right)\,\left(\frac{F(T)}{2}-T\,F^{\prime}(T)\right)}\right]\,\frac{K}{T}}{1+\left[\frac{6\,T\,F^{\prime\prime}(T)}{\left(\frac{F^{\prime}(T)}{2}+T\,F^{\prime\prime}(T)\right)}\right]\,\frac{K}{T}}\right]$

where $K=K(T)=\frac{\left|k\right|}{a^{2}}$. If $\frac{K}{T}\ll 1$, we obtain:

$\displaystyle\frac{dT}{d\tau}\approx$
$\displaystyle\,\mathcal{F}(T)\,-\,\mathcal{G}(T)\,K(T)$ (37)

####  III.2.1 Comment

Note that in both cases, we have that:

$\displaystyle\frac{dT}{d\tau}=\mathcal{F}(T)\,-k\,\mathcal{G}(T)\,K(T)$ (38)

where $k=-1$ gives exactly the eqn. (34) with $K(T)=\frac{\sqrt{-k}}{a^{2}}$
and $k=+1$ gives exactly the eqn. (37) with
$K(T)=\frac{\left|k\right|}{a^{2}}$. The relationship is exact when $k=0$.
Clearly late time stability of the $k=0$ solutions to non-zero $k$
perturbations will occur when $\mathcal{F}(T)\gg\mathcal{G}(T)\,K(T)$ for
$\tau\,\rightarrow\,\infty$.

##  IV Quadratic $F(T)$ Solutions

For example, if we have the theory $F(T)=-\Lambda+T+\gamma\,T^{\beta}$, the
functions $\mathcal{F}(T)$ and $\mathcal{G}(T)$ become, respectively:

$\displaystyle\mathcal{F}(T)$ $\displaystyle=$
$\displaystyle-\frac{3\,\left(1+\alpha\right)\,\left[\Lambda+T+\gamma\,\left(2\beta-1\right)\,T^{\beta}\right]}{\left[1+\gamma\,\beta\,(2\beta-1)\,T^{\beta-1}\right]},$
(39) $\displaystyle\mathcal{G}(T)$ $\displaystyle=$
$\displaystyle\frac{12\left(1+3\alpha\right)\,\left(1+\gamma\,\beta\,T^{\beta-1}\right)}{\left[1+\gamma\,\beta\,(2\beta-1)\,T^{\beta-1}\right]}$
(40)
$\displaystyle\quad\quad+\frac{9\,\left(1+\alpha\right)\left[1-\gamma\,\beta\,(2\beta-3)\,T^{\beta-1}\right]\,\left[\Lambda+T+\gamma\,(2\,\beta-1)\,T^{\beta}\right]}{T\,\left[1+\gamma\,\beta\,(2\beta-1)\,T^{\beta-1}\right]^{2}}.$

For $F(T)=-\Lambda+T+\gamma\,T^{\beta}$, the $k=0$ solution (of eqn. (32)) is
given by:

$\displaystyle
A_{0}\,\exp\left(-3\,\left(1+\alpha\right)\tau\right)=\Lambda+T(\tau)+\gamma\,\left(2\beta-1\right)\,T^{\beta}(\tau),$
(41)

where $A_{0}=2\tilde{A}_{0}$. In the quadratic case $\beta=2$ (explicit
solutions in the cubic case $\beta=3$ are also possible), we then obtain from
(29) the function $T(\tau)$ explicitly as:

$\displaystyle T(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{6\,\gamma}\,\left[-1+\sqrt{1-12\gamma\,\left(\Lambda-
A_{0}\,\exp\left(-3\,\left(1+\alpha\right)\tau\right)\right)}\right],$ (42)

since $T>0$ (and $1-12\gamma\,\Lambda>0$) and where

$\displaystyle H(\tau)=$ $\displaystyle\sqrt{\frac{T(\tau)}{6}}.$ (43)

Note that as $\tau\rightarrow\infty$, $H(\tau)\rightarrow H_{c}$ and
$T(\tau)\rightarrow\,T_{c}$, both positive constants for $\Lambda\neq 0$. From
$\frac{d\tau}{dt}=H(\tau)$, we get by integration $\tau=\tau(t)$, or using
$H=\frac{\dot{a}}{a}$, $\tau=\tau_{0}+\ln\left(a(t)\right)$. From the
conservation eqns. $\rho(t)=\rho_{0}\,a^{-3\left(1+\alpha\right)\,t}$, so that
$\rho\rightarrow 0$ as $t\rightarrow\infty$.

In the quadratic case eqns. (39) and (40) explicitly become:

$\displaystyle{\mathcal{F}(\tau)}$ $\displaystyle=$
$\displaystyle{-\frac{3\,\left(1+\alpha\right)\,A_{0}\,\exp\left(-3\,\left(1+\alpha\right)\tau\right)}{\sqrt{1-12\gamma\,\left(\Lambda-
A_{0}\,\exp\left(-3\,\left(1+\alpha\right)\tau\right)\right)}},}$ (44a)
$\displaystyle{\mathcal{G}(\tau)}$ $\displaystyle=$
$\displaystyle{\frac{4\left(1+3\alpha\right)\,\left(2+\sqrt{1-12\gamma\,\left(\Lambda-
A_{0}\,\exp\left(-3\,\left(1+\alpha\right)\tau\right)\right)}\right)}{\sqrt{1-12\gamma\,\left(\Lambda-
A_{0}\,\exp\left(-3\,\left(1+\alpha\right)\tau\right)\right)}}}$
$\displaystyle\quad{+\frac{18\gamma\,\left(1+\alpha\right)\left[4-\sqrt{1-12\gamma\,\left(\Lambda-
A_{0}\,\exp\left(-3\,\left(1+\alpha\right)\tau\right)\right)}\right]\,A_{0}\,\exp\left(-3\,\left(1+\alpha\right)\tau\right)}{\left[-1+\sqrt{1-12\gamma\,\left(\Lambda-
A_{0}\,\exp\left(-3\,\left(1+\alpha\right)\tau\right)\right)}\right]\,\left[1-12\gamma\,\left(\Lambda-
A_{0}\,\exp\left(-3\,\left(1+\alpha\right)\tau\right)\right)\right]}.}$

Alternatively, the eqns. (39) and (40) can be written for $\beta=2$ as:

$\displaystyle{\mathcal{F}(T)}$ $\displaystyle=$
$\displaystyle{-\frac{3\,\left(1+\alpha\right)\,\left(\Lambda+T+3\gamma\,T^{2}\right)}{\left(1+6\gamma\,T\right)},}$
(45a) $\displaystyle{\mathcal{G}(T)}$ $\displaystyle=$
$\displaystyle{\frac{12\left(1+3\alpha\right)\,\left(1+2\,\gamma\,T\right)}{\left(1+6\gamma\,T\right)}}+{\frac{9\,\left(1+\alpha\right)\left(1-2\,\gamma\,T\right)\,\left(\Lambda+T+3\gamma\,T^{2}\right)}{T\,\left(1+6\,\gamma\,T\right)^{2}},}$
(45b)

where $T$ is given exactly by eq (42).

If $K=K(\tau)=\frac{|k|}{a^{2}}=|k|\,e^{-2\tau}$, then
$K(T)=K_{0}\,\left(\Lambda+T+3\gamma\,T^{2}\right)^{\frac{2}{3\,\left(1+\alpha\right)}}$
where $K_{0}$ is a constant, and the product of eqn. (45b) with $K(T)$ is:

$\displaystyle{K(T)\,\mathcal{G}(T)}$ $\displaystyle=$
$\displaystyle{\frac{12\,K_{0}\,\left(1+3\alpha\right)\,\left(1+2\,\gamma\,T\right)}{\left(1+6\gamma\,T\right)}\left(\Lambda+T+3\gamma\,T^{2}\right)^{\frac{2}{3\,\left(1+\alpha\right)}}}$
(46)
$\displaystyle\quad+{\frac{9\,K_{0}\,\left(1+\alpha\right)\left(1-2\,\gamma\,T\right)}{T\,\left(1+6\,\gamma\,T\right)^{2}}\,\left(\Lambda+T+3\gamma\,T^{2}\right)^{1+\frac{2}{3\,\left(1+\alpha\right)}}.}$

Due to the $T^{-1}$ term in the denominator of the second term,
$K(T)\,\mathcal{G}(T)$ always diverges when $T_{c}=0$. Thus, in the case
$\Lambda=0$ all zero curvature solutions are unstable to the inclusion of non-
zero spatial curvature.

In the case $\Lambda\,\neq\,0$, eqns. (45a) and (46) become for $T_{c}$
sufficiently less that unity:

$\displaystyle{\mathcal{F}(T)}$ $\displaystyle\approx$
$\displaystyle{-3\,\left(1+\alpha\right)\,T\,\left[1-3\gamma\,T\right]}$ (47a)
$\displaystyle{K(T)\,\mathcal{G}(T)}$ $\displaystyle\approx$
$\displaystyle{12\,K_{0}\,\left(1+3\alpha\right)\,T^{\frac{2}{3(1+\alpha)}}\left[1+2\,\gamma\,T\left(\frac{1}{(1+\alpha)}-2\right)+\gamma^{2}\,T^{2}\left(\frac{2}{(1+\alpha)^{2}}-\frac{11}{(1+\alpha)}+24\right)\right]}$
(47b)
$\displaystyle+{9\,K_{0}\,\left(1+\alpha\right)\,T^{\frac{2}{3(1+\alpha)}}\left[1+\gamma\,T\,\left(\frac{2}{(1+\alpha)}-11\right)+\gamma^{2}\,T^{2}\left(\frac{2}{(1+\alpha)^{2}}-\frac{25}{(1+\alpha)}+90\right)\right]}$
$\displaystyle\,\equiv\,$ $\displaystyle
T^{\frac{2}{3(1+\alpha)}}\left[1+A_{1}\,T+A_{2}\,T^{2}\right]$

For local stability, we consequently need $T^{\frac{2}{3(1+\alpha)}}\ll T$ for
$T\,\rightarrow\,0$, so that $\frac{2}{3(1+\alpha)}>1$ and finally
$-1<\alpha<-\frac{1}{3}$, corresponding to an inflating perfect fluid eqn. of
state.

##  V Power-Law solutions

Let us work in terms of the coordinate $t$ instead of $\tau$ (i.e.,
$a=a(\tau)=e^{\tau}$). By assuming $a(t)=a_{n}\,t^{n}$, so that

$\displaystyle H(t)=\frac{n}{t},\quad\quad
T(t)=\frac{6\,n^{2}}{t^{2}},\quad\quad\frac{dT(t)}{dt}=-\frac{12\,n^{2}}{t^{3}}=-\frac{2\,T(t)}{t},$
(48)

where we include negative values of $n$ (corresponding to contracting models
for $t>0$, but negative values of $t$ are also possible). We then get from
eqn. (32) for the total energy density $\rho_{sum}$ by adding the matter
density ($\rho(t)$) and a cosmological constant source ($\Lambda$):

$\displaystyle\frac{F}{2}-T\,F^{\prime}$ $\displaystyle=$
$\displaystyle-\tilde{A}_{0}\,\left[a(t)\right]^{-3(1+\alpha)}-\frac{\Lambda}{2},$
(49)

where $\tilde{A}_{0}$ is a positive constant. We recall that $-1<\alpha\leq
1$.

By substituting eqns. (48) into eqn. (49) and using
$t=\frac{\sqrt{6}\,n}{\sqrt{T}}$, we obtain:

$\displaystyle F(T)$ $\displaystyle=$
$\displaystyle\frac{C}{1-3(1+\alpha)n}\,T^{\left[\frac{3(1+\alpha)n}{2}\right]}+B\,\sqrt{T}-\Lambda.$
(50)

where $B$ and $C$ are non-zero constants and $n\neq\frac{1}{3(1+\alpha)}$. By
substituting eqn. (50) into eqns. (30) and (35), the functions
$\mathcal{F}(T)$ and $\mathcal{G}(T)$ are given by:

$\displaystyle\mathcal{F}(T)=-\frac{2}{n}\,T+\frac{2\,\Lambda}{n\,C}\,T^{1-\frac{3(1+\alpha)\,n}{2}}$
(51a) $\displaystyle\mathcal{G}(T)=$
$\displaystyle\frac{6\left[(1-3\alpha)\,n-3\right]}{n\,\left[1-3(1+\alpha)\,n\right]}-\frac{4B\,\left[(1+3\alpha)n+1\right]}{(1+\alpha)\,C\,n^{2}}\,T^{\left[1-3(1+\alpha)\,n\right]/2}$
$\displaystyle-4\Lambda\left[\frac{9\,\left[(1+\alpha)\,n-1\right]}{2\,C\,n\,\left[1-3(1+\alpha)\,n\right]}\,T^{-3(1+\alpha)\,n/2}-\frac{B}{C^{2}\,n^{2}\,(1+\alpha)}\,T^{\left[1-6(1+\alpha)\,n\right]/2}\right].$
(51b)

But we need to compare $\mathcal{F}(T)$ with $K(T)\,\mathcal{G}(T)$ where
$K(T)=\frac{\left|k\right|}{a^{2}(t)}=K_{0}\,T^{n}$ according to eqn. (48).
The product of eqn. (51) with $K(T)$ is:

$\displaystyle K(T)\,\mathcal{G}(T)=$
$\displaystyle\frac{6\,K_{0}\left[(1-3\alpha)\,n-3\right]}{n\,\left[1-3(1+\alpha)\,n\right]}\,T^{n}-\frac{4B\,K_{0}\,\left[(1+3\alpha)n+1\right]}{(1+\alpha)\,C\,n^{2}}\,T^{\left[1-(1+3\alpha)\,n\right]/2}$
$\displaystyle-4\Lambda\,K_{0}\left[\frac{9\,\left[(1+\alpha)\,n-1\right]}{2\,C\,n\,\left[1-3(1+\alpha)\,n\right]}\,T^{-(1+3\alpha)\,n/2}-\frac{B}{C^{2}\,n^{2}\,(1+\alpha)}\,T^{\left[1-2(2+3\alpha)\,n\right]/2}\right].$
(52)

###  V.1 Stability Conditions

We wish to consider the stability of the $k=0$ solutions to curvature
perturbations (see comments earlier). The limit of eqn. (51a) for
$T\rightarrow 0$ (i.e., $t\rightarrow\,\infty$) is:

$\displaystyle\mathcal{F}(T)\,\sim\,-\frac{2}{n}\,T+\frac{2\,\Lambda}{n\,C}\,T^{1-\frac{3(1+\alpha)\,n}{2}},$
(53)

where $n\neq 0$. For $n<0$ (contracting model), eqn. (53) becomes for
$T\,\rightarrow 0$:

$\displaystyle\mathcal{F}(T)\,\sim\,T.$ (54)

For $n>0$ (expanding model), eqn. (53) becomes for $T\,\rightarrow 0$:

$\displaystyle\mathcal{F}(T)\,\sim\,T^{1-\frac{3(1+\alpha)\,n}{2}},$ (55)

where $n\leq\frac{2}{3(1+\alpha)}$ for a converging $\mathcal{F}(T)$ when
$T\,\rightarrow 0$.

####  V.1.1 General $n<0$ Case

We assume that $\mathcal{F}(T)\,\sim\,T$ and that $B\neq 0$ and $\Lambda\neq
0$. For stability, we obtain the following criteria for the first term of eqn.
(V):

$\displaystyle(1-3\alpha)\,n-3=0\quad\Rightarrow\quad
n=\frac{3}{(1-3\alpha)}<0,$ (56)

so that $\frac{1}{3}<\alpha\leq 1$. The eqn. (V) becomes:

$\displaystyle K(T)\,\mathcal{G}(T)=$
$\displaystyle-\frac{8B\,K_{0}\,(1-3\alpha)\left(2+3\alpha\right)}{9(1+\alpha)\,C}\,T^{\frac{1}{2}-\frac{3(1+3\alpha)}{2(1-3\alpha)}}$
$\displaystyle-4\Lambda\,K_{0}\left[-\frac{3\,(1-3\alpha)\left(1+3\alpha\right)}{4C\,\left(2+3\alpha\right)}\,T^{-\frac{3(1+3\alpha)}{2(1-3\alpha)}}-\frac{B(1-3\alpha)^{2}}{9C^{2}\,(1+\alpha)}\,T^{\frac{1}{2}-\frac{3(2+3\alpha)}{(1-3\alpha)}}\right],$
(57)

where $\alpha\neq-\frac{2}{3}$ (and $\alpha>-1$).

For stability, we need to compare all the powers of terms in eqn. (V.1.1) with
unity (the power of $T$). Therefore, we require that simultaneously:

$\displaystyle\frac{1}{2}-\frac{3(1+3\alpha)}{2(1-3\alpha)}$ $\displaystyle>$
$\displaystyle 1,$ (58a) $\displaystyle-\frac{3(1+3\alpha)}{2(1-3\alpha)}$
$\displaystyle>$ $\displaystyle 1,$ (58b)
$\displaystyle\frac{1}{2}-\frac{3(2+3\alpha)}{(1-3\alpha)}$ $\displaystyle>$
$\displaystyle 1,$ (58c)

and we obtain the condition $\frac{1}{3}<\alpha\leq 1$ (for
$K(T)\,\mathcal{G}(T)\ll\mathcal{F}(T)$).

####  V.1.2 General $n>0$ Case

In the more physical expanding case for $B\neq 0$ and $\Lambda\neq 0$,
$\mathcal{F}(T)\,\sim\,T^{1-\frac{3(1+\alpha)\,n}{2}}$ when $T\,\rightarrow
0$, and so we compare all powers of the terms in eqn. (V) with the power of
$T^{1-\frac{3(1+\alpha)\,n}{2}}$:

$\displaystyle n$ $\displaystyle>$ $\displaystyle 1-\frac{3(1+\alpha)\,n}{2},$
(59a) $\displaystyle\frac{1}{2}-\frac{(1+3\alpha)\,n}{2}$ $\displaystyle>$
$\displaystyle 1-\frac{3(1+\alpha)\,n}{2},$ (59b)
$\displaystyle-\frac{(1+3\alpha)\,n}{2}$ $\displaystyle>$ $\displaystyle
1-\frac{3(1+\alpha)\,n}{2},$ (59c) $\displaystyle\frac{1}{2}-(2+3\alpha)\,n$
$\displaystyle>$ $\displaystyle 1-\frac{3(1+\alpha)\,n}{2}.$ (59d)

From eqns. (59a) to (59d), we obtain that $-1<\alpha<-\frac{1}{3}$ as
stability conditions (with $n>1$ for $-1<\alpha\leq-\frac{2}{3}$ and
$n>-\frac{1}{1+3\alpha}$ for $-\frac{2}{3}<\alpha<-\frac{1}{3}$).

####  V.1.3 Special cases

For $B=0$ ($\Lambda\neq 0$), we have as stability conditions:

  * •

For $n<0$: we only need to satisfy eqn. (58b) and we still obtain
$\frac{1}{3}<\alpha\leq 1$ for stable solutions
($K(T)\,\mathcal{G}(T)\,\sim\,T^{-\frac{3(1+3\alpha)}{2(1-3\alpha)}}$).

  * •

For $n>0$: we get from eqns. (59a) and (59c) for
$\mathcal{F}(T)\,\sim\,T^{1-\frac{3(1+\alpha)\,n}{2}}$:

$\displaystyle n$ $\displaystyle>$ $\displaystyle 1-\frac{3(1+\alpha)\,n}{2},$
(60a) $\displaystyle-\frac{(1+3\alpha)\,n}{2}$ $\displaystyle>$ $\displaystyle
1-\frac{3(1+\alpha)\,n}{2}.$ (60b)

From eqns. (60a) and (60b), we obtain that $-1<\alpha\leq 1$ and $n>1$ for
stable solutions where $K(T)\,\mathcal{G}(T)\,\sim T^{-(1+3\alpha)\,n/2}$.

For $\Lambda=0$ ($B\neq 0$), we have as stability conditions:

  * •

For $n<0$: we only need to satisfy eqn. (58a) and we still find that
$\frac{1}{3}<\alpha\leq 1$ and then
$K(T)\,\mathcal{G}(T)\,\sim\,T^{\frac{1}{2}-\frac{3(1+3\alpha)}{2(1-3\alpha)}}$.

  * •

For $n>0$: we have that $\mathcal{F}(T)\,\sim\,T$ and from eqn. (V) we obtain
the stability conditions:

$\displaystyle n$ $\displaystyle>$ $\displaystyle 1,$ (61a)
$\displaystyle\frac{1}{2}-\frac{(1+3\alpha)\,n}{2}$ $\displaystyle>$
$\displaystyle 1.$ (61b)

From eqns. (61a) and (61b), we have that $-1<\alpha<-\frac{1}{3}$ for stable
solutions (with $n>1$ for $-1<\alpha\leq-\frac{2}{3}$ and
$n>-\frac{1}{1+3\alpha}$ for $-\frac{2}{3}<\alpha<-\frac{1}{3}$). These exotic
fields with negative pressure are as appear in GR. Thus we get that
$K(T)\,\mathcal{G}(T)\,\equiv\,A_{1}\,T^{n}+B_{1}\,T^{\left[1-(1+3\alpha)\,n\right]/2}$,
and then:

$\displaystyle K(T)\,\mathcal{G}(T)\,$ $\displaystyle\sim$
$\displaystyle\,T^{n}\quad\quad\quad\quad\quad\quad\;\text{if}\quad
n<\frac{1}{3(1+\alpha)},$ (62a) $\displaystyle\sim$
$\displaystyle\,T^{\left[1-(1+3\alpha)\,n\right]/2}\quad\;\text{if}\quad
n>\frac{1}{3(1+\alpha)}.$ (62b)

In the case $\Lambda=0$ and $B=0$, we obtain the GR solutions.

####  V.1.4 Generalizations and physical consequences

Physically we are primarily interested here in the stability of the $k=0$ TRW
models to non-zero $k$ perturbations and the question of late time
acceleration. The assumed eqn. of state is $p=\alpha\,\rho$, where
$\alpha=\frac{1}{3}$ and $\alpha=0$ correspond to the physically important
cases of radiation and dust (matter domination) ($\alpha=-1$ formally
corresponds to a cosmological constant). However, more general perfect fluid
sources with non-constant $\alpha$ are possible, as are even more general
sources such as, for example, a non-interacting mixture of dust and radiation
with $\rho=\rho_{m}+\rho_{r}$ and $p=\frac{\rho_{r}}{3}$, which also admits
power-laws solutions. However, the important asymptotic behaviour of these
models (e.g., the non-interacting two-fluid models alluded to above) will
often be described by the $\alpha$ constant power-laws solutions above (hence
illustrating the role of such constant $\alpha$ power law solutions for more
physical cosmological models). Studying more physical models with a scalar
field will need to be done using different techniques. A scalar field can be
included with an effective $\rho_{\phi}$ and $p_{\phi}$, defined by:

$\displaystyle\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\dot{\phi}^{2}}{2}+V(\phi),$ (63a) $\displaystyle
p_{\phi}$ $\displaystyle=$ $\displaystyle\frac{\dot{\phi}^{2}}{2}-V(\phi),$
(63b)

subject to the corresponding conservation law which can be written in the form
of the Klein-Gordon eqn:

$\displaystyle 0=\ddot{\phi}+V^{\prime}(\phi)+3\,H\,\dot{\phi},$ (64)

where $\phi$ is the scalar field, $V(\phi)$ is the scalar field potential and
then $\dot{\phi}$ and $\ddot{\phi}$ are, respectively, the first and second
time-derivatives of $\phi$ (all of these eqns are for a constant coupling
function in $\phi$ [26]). The form of the potential $V(\phi)$ must have a
physical motivation, and not all potentials will admit exact power-law
solutions although solutions will often be power-law asymptotically.

##  VI Discussion

We have presented all teleparallel geometries that are invariant under the
full $G_{6}$ Lie algebra of RW affine symmetries. We have discussed and
clarified their properties, especially in the cases of non-zero $k$ (i.e.,
$k=-1$ and $k=+1$). In particular, we have explicitly computed the geometries
in the proper coframes. We have displayed the correct FE for the $F(T)$ class
of teleparallel TRW spacetimes.

In order to analyse the resulting cosmological models, we first considered the
$k=0$ model and wrote the governing FE as an ordinary differential eqn., and
discussed its properties. Subsequently, we formulated the $k\neq 0$ TRW models
as pertubations of the $k=0$ model, which enabled us to carry out a late time
stability analysis. In particular, we presented a detailed stability analysis
for the quadratic $F(T)=-\Lambda+T+\gamma T^{2}$ models and the class of
power-laws solutions, in both the case of a vanishing and non-vanishing
$\Lambda$.

We explicitly considered models with a constant eqn. of state parameter
$\alpha$, but made some comments on more general models with non-constant
$\alpha$, such as models with a mixture of two fluids and scalar field models.
It is the latter models, which include a scalar field, that are of more
interest from a physical point of view (e.g., inflation). But different
techniques are necessary for their investigation. We shall return to this in
future work. Ultimately, of course, the models need to be confronted with
observations [7, 27] (see for example [25]).

## Acknowledgments

AAC is funded by the Natural Sciences and Engineering Research Council of
Canada. AL is funded by an AARMS fellowship. We would like to thank Robert van
den Hoogen and David McNutt for helpful comments.

## Abbreviations

The following abbreviations are used in this manuscript:

DE | Differential Equation
---|---
FE | Field Equation
GR | General Relativity
KV | Killing Vectors
RW | Robertson-Walker
TRW | Teleparallel Robertson-Walker

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