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# Nonzero Spatial Curvature in Symmetric Teleparallel Cosmology

Andronikos Paliathanasis<EMAIL_ADDRESS>Institute of Systems Science,
Durban University of Technology, Durban 4000, South Africa  Departamento de
Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280
Antofagasta, Chile

###### Abstract

We consider the symmetric teleparallel $f\left(Q\right)$-gravity in
Friedmann–Lemaître–Robertson–Walker cosmology with nonzero spatial curvature.
For a nonlinear $f\left(Q\right)$ model there exist always the limit of
General Relativity with or without the cosmological constant term. The de
Sitter solution is always provided by the theory and for the specific models
$f_{A}\left(Q\right)\simeq
Q^{\frac{\alpha}{\alpha-1}}~{},~{}f_{B}\left(Q\right)\simeq
Q+f_{1}Q^{\frac{\alpha}{\alpha-1}}$ and $f_{C}\left(Q\right)\simeq Q+f_{1}Q\ln
Q$ it was found to be the unique attractor. Consequently small deviations from
STGR can solve the flatness problem and lead to a de Sitter expansion without
introduce a cosmological constant term. This result is different from that
given by the power-law theories for the other two scalar of the trinity of
General Relativity. What makes the nonlinear symmetric teleparallel theory to
stand out are the new degree of freedom provided by the connection defined in
the non-coincidence frame which describes the nonzero spatial curvature.

Cosmology; Curvature; $f\left(Q\right)$-gravity; Symmetric teleparallel
gravity

##  I Introduction

Cosmological observations reveals the universe’s homogeneity and isotropy on
large scales, described by the Friedmann–Lemaître–Robertson–Walker (FLRW)
geometry. The value of the spatial curvature for the three-dimensional
hypersurface is a debate and recent studies indicate that it is not zero cr1 ;
cr2 ; cr3 ; cr4 ; cr5 . As a solution to the observational data, it has been
proposed that the universe underwent a rapid expansion known as inflation guth
. During the inflationary era, the universe was dominated by a matter source
with negative pressure.

There is a category of gravitational models where the Einstein-Hilbert Action
Integral is modified in an effort to include new geometrodynamical terms in
the field equations and explain cosmic acceleration. In this category of
models, the modification of the Action Integral for the gravitational theory
is a direct consequence for the introduction of geometric invariants. For a
four-dimensional manifold $V^{4}$, with a metric tensor $g_{\mu\nu}$ and a
generic connection $\hat{\Gamma}_{\mu\nu}^{\kappa}$, we can define the Riemann
tensor $R_{\;\lambda\mu\nu}^{\kappa}$, the torsion tensor
$\mathrm{T}_{\;\mu\nu}^{\lambda}$ and the non-metricity tensor
$Q_{\lambda\mu\nu}~{}$eisn . In General Relativity the connection
$\hat{\Gamma}_{\mu\nu}^{\kappa}$ takes the form of the Levi-Civita connection,
where $\mathrm{T}_{\;\mu\nu}^{\lambda}$ and $Q_{\lambda\mu\nu}$ vanishes, i.e.
$\mathrm{T}_{\;\mu\nu}^{\lambda}=0$ and $Q_{\lambda\mu\nu}=0$, while the Ricci
scalar is the fundamental scalar for the gravitational Action Integral.
Nevertheless, in the context of Teleparallel Equivalent of General Relativity
(TEGR) it holds $R_{\;\lambda\mu\nu}^{\kappa}=0$ and $Q_{\lambda\mu\nu}=0,$
and the gravitational the torsion scalar $T$ constructed by the tensor
$\mathrm{T}_{\;\mu\nu}^{\lambda}$ plays the role of the gravitational
Lagrangian function ein28 ; Tsamp . However, in Symmetric Teleparallel General
Relativity (STGR) the non-metricity scalar $Q$ is the fundamental
gravitational scalar; while at the same time $R_{\;\lambda\mu\nu}^{\kappa}=0$
and $\mathrm{T}_{\;\mu\nu}^{\lambda}=0~{}$Hohmann . These three gravitational
theories provide the same gravitational field equations, which is why the
Ricci scalar $R$, the torsion scalar $T$, and the non-metricity scalar $Q$ are
known as the trinity of gravity tr1 . However, this equivalence of the three
theories is lost when nonlinear components are included in the gravitational
Action Integral.

General Relativity is a second-order theory of gravity; however
$f\left(R\right)$-theory Buda  is a higher-order theory; specifically
$f\left(R\right)$-theory is fourth-order theory and equivalent with a special
case of the Brans-Dicke gravitational model. The power-law
$f\left(R\right)=R^{\lambda}$, $\lambda\neq 0,1$ gravitational model has been
widely studied in cosmological studies cl1 ; cl2 ; cl3 . Indeed, for
$\lambda\neq 2$ in a spatially flat FLRW the exact solution of the Friedmann’s
equations is the scaling solution which describes an perfect fluid with
constant equation of state parameter anbas1 . Nevertheless, for $\lambda=2$,
the exact solution of the cosmological field equations is the de Sitter
universe sd3 . In the presence of the spatial curvature, or in the presence of
anisotropies, the power-law $f\left(R\right)=R^{\lambda}$ model can solve the
flatness problem but the de Sitter universe is recovered only for
$\lambda=2~{}$gl1 ; gl2 . In teleparallelism and in $f\left(T\right)$-gravity,
the power-law model $f\left(T\right)=T^{\lambda}$ leads only to scaling
cosmological solutions sd4 , while for nonzero spatially curvature the
exponential expansion is provided only for other $f\left(T\right)$ functions
sd2 .

In this study, we are interested in the evolution of cosmological dynamics in
symmetric teleparallel $f(Q)-$theory of gravity for the FLRW universe with
nonzero spatial curvature. $f(Q)$-theory has been widely studied by
cosmologists because it can provide an alternative geometric mechanism for
describing cosmological phenomena f6 . For various applications of symmetric
teleparallel theory we refer the reader to ff1 ; ff2 ; ff3 ; ff4 ; ff4a ; ff4b
; ff5 ; ff6 ; ff7 ; nn5 ; nn7 ; ww2 ; qq2 ; qq3 ; ffq1 ; ffq2 ; ffq3 ; ffq4 ;
ffq5  and references therein. Recently, in qq4  a detailed analysis for the
evolution of cosmological parameters performed within the framework of
$f(Q)$-gravity for a spatially flat FLRW universe. Similar studies have been
presented before qq4a ; qq5 ; qq6  in the case of the coincidence gauge for
the connection. The novelty of qq4  lies in assuming the case of the non-
coincidence connection, where new dynamic degrees of freedom are introduced in
the field equation from the connection. This leads to a higher-order theory of
gravity, as also analyzed recently in qq6 . Consequently, the additional
degrees of freedom provide a more complex cosmological evolution compared to
that given by the coincidence connection. This means that various epochs of
cosmological history can be recovered. An analysis to use cosmological
observations to understand the effects of the different connections performed
recently in ds00 . See also the recent review rev10 .

The purpose of this study is to investigate if the nonlinear
$f\left(Q\right)$-theory can solve the flatness problem and if there exist a
mechanism to provide a de Sitter expansion without a cosmological constant
term. The rapid expansion of the universe, that is, the inflation, it has been
proposed the flatness problem guth . During inflation the universe expands
such that the energy density of the curvature term to be neglected. With this
study and the analysis of the dynamics we investigate if in
$f\left(Q\right)$-gravity the existence of curvature in the initial conditions
leads to trajectories which describe spatially flat universes. The structure
of the paper follows.

In Section II we introduce the cosmological model of our consideration which
that of a FLRW spacetime with nonzero spatial curvature in symmetric
teleparallel $f\left(Q\right)$-gravity. We present the modified field
equations for arbitrary $f\left(Q\right)$ function where we observe that
dynamical degrees of freedom are introduced by the non-coincidence gauge for
the connection. In Section III we study the evolution of the physical
parameters by studying the phase-space. We compute the stationary points for a
generic function $f\left(Q\right)$. However in order to investigate the
stability properties we specify function $f\left(Q\right)$. We found that
there exist always a future attractor which correspond to the spatially flat
de Sitter universe. Furthermore, the limit of General Relativity is always
recovered by the nonlinear function $f\left(Q\right)$. Consequently, the
nonlinear $f\left(Q\right)$-theory can solve the flatness problem and provide
an inflationary mechanism without to introduce a cosmological constant term.
Finally, in Section V we draw our conclusions.

##  II $f\left(Q\right)$-cosmology with nonzero curvature

In cosmological scales the universe is considered to be isotropic and
homogeneous described by the FLRW geometry. In spherical coordinates the
latter spacetime is described by the line element

$ds^{2}=-dt^{2}+a\left(t\right)^{2}\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\right).$ (1)

Parameter $k$ is the spatial curvature of the three-dimensional hypersurface,
$k=1$ describes a closed universe, $k=0$ a flat universe and $k=-1$ describes
an open universe. Function $a\left(t\right)$ is the scalar factor which
corresponds the radius of the universe. The Hubble function is defined as
$H=\frac{\dot{a}}{a}$.

The compatible connections $\Gamma_{\mu\nu}^{\kappa}$ for the FLRW geometry
which inherits the spacetime isometries and they describe flat geometry, that
is, $R_{\;\lambda\mu\nu}^{\kappa}\left(\Gamma\right)=0$, they have been
derived before in Hohmann ; Heis2 . For zero spatial curvature, $k=0$, it has
been found that there exist three different families of connections, however
for $k\neq 0$, there exist a unique connection with nonzero components in the
spherical coordinates ds1

$\begin{split}&\Gamma_{\;tt}^{t}=-\frac{k+\dot{\gamma}(t)}{\gamma(t)},\quad\Gamma_{\;rr}^{t}=\frac{\gamma(t)}{1-kr^{2}}\quad\Gamma_{\;\theta\theta}^{t}=\gamma(t)r^{2},\quad\Gamma_{\;\phi\phi}^{t}=\gamma(t)r^{2}\sin^{2}(\theta)\\\
&\Gamma_{\;tr}^{r}=\Gamma_{\;rt}^{r}=\Gamma_{\;t\theta}^{\theta}=\Gamma_{\;\theta
t}^{\theta}=\Gamma_{\;t\phi}^{\phi}=\Gamma_{\;\phi
t}^{\phi}=-\frac{k}{\gamma(t)},\quad\Gamma_{\;rr}^{r}=\frac{kr}{1-kr^{2}},\quad\Gamma_{\;\theta\theta}^{r}=-r\left(1-kr^{2}\right),\\\
&\Gamma_{\;\phi\phi}^{r}=-r\sin^{2}(\theta)\left(1-kr^{2}\right),\Gamma_{\;r\theta}^{\theta}=\Gamma_{\;\theta
r}^{\theta}=\Gamma_{\;r\phi}^{\phi}=\Gamma_{\;\phi
r}^{\phi}=\frac{1}{r},\Gamma_{\;\phi\phi}^{\theta}=-\sin\theta\cos\theta,\Gamma_{\;\theta\phi}^{\phi}=\Gamma_{\;\phi\theta}^{\phi}=\cot\theta.\end{split}$
(2)

For the latter connection the non-metricity scalar reads qq3

$Q=-6H^{2}+\frac{3\gamma}{a^{2}}H+\frac{3\dot{\gamma}}{a^{2}}+k\left(\frac{6}{a^{2}}+\frac{3}{\gamma}\left(\frac{\dot{\gamma}}{\gamma}-3H\right)\right).$
(3)

while the resulting modified Friedmann’s equations in $f\left(Q\right)$
gravity are qq3

$3H^{2}f^{\prime}(Q)+\frac{1}{2}\left(f(Q)-Qf^{\prime}(Q)\right)-\frac{3\gamma\dot{Q}f^{\prime\prime}(Q)}{2a^{2}}+3k\left(\frac{f^{\prime}(Q)}{a^{2}}-\frac{\dot{Q}f^{\prime\prime}(Q)}{2\gamma}\right)=0,$
(4)
$-2\left(f^{\prime}(Q)H\right)^{\cdot}-3H^{2}f^{\prime}(Q)-\frac{1}{2}\left(f(Q)-Qf^{\prime}(Q)\right)+\frac{\gamma\dot{Q}f^{\prime\prime}(Q)}{2a^{2}}-k\left(\frac{f^{\prime}(Q)}{a^{2}}+\frac{3\dot{Q}f^{\prime\prime}(Q)}{2\gamma}\right)=0.$
(5)

Furthermore, the connection satisfies the equation of motion qq3

$\dot{Q}^{2}f^{\prime\prime\prime}(Q)\left(1+\frac{ka^{2}}{\gamma^{2}}\right)+\left[\ddot{Q}\left(1+\frac{ka^{2}}{\gamma^{2}}\right)+\dot{Q}\left(\left(1+\frac{3ka^{2}}{\gamma^{2}}\right)H+\frac{2\dot{\gamma}}{\gamma}\right)\right]f^{\prime\prime}(Q)=0.$
(6)

We remark that connection (2) depends on an function $\gamma\left(t\right)$,
and that for nonlinear $f\left(Q\right)$ function there is not any coordinate
transformation in which the connection can be written in the coincidence gauge
$\Gamma_{\kappa\lambda}^{\mu}=0~{}$ds1 . For the flat space, $k=0$, there are
three different connections but only one is recovered from (2) by replacing
$k=0.$ For the spatially flat spacetime the coincidence universe exist, and
then modified Friedmann equations are similar with that of the
$f\left(T\right)$-gravity. Because for $k\neq 0$ there coincidence gauge does
not exist, function $\gamma\left(t\right)$ can not neglected, which means that
the field equations considered in ref1  do not describe
$f\left(Q\right)$-cosmology with nonzero spatial curvature.

For a linear function $f\left(Q\right)$, that is,
$f\left(Q\right)=Q-2\Lambda$, the field equations are that of General
Relativity, where $\Lambda$ is the cosmological constant term. In this
analysis we assume the power-law function $f\left(Q\right)=Q^{\lambda}$, with
$\lambda\neq 0,1$.

For the power-law function $f\left(Q\right)$, the gravitational field
equations (4)-(6) read

$3\lambda
H^{2}+\frac{1-\lambda}{2}Q-\frac{3\gamma\dot{Q}\lambda\left(\lambda-1\right)}{2a^{2}Q}+3k\left(\frac{\lambda}{a^{2}}-\frac{\lambda\left(\lambda-1\right)\dot{Q}}{2\gamma
Q}\right)=0,$ (7)
$-2\left(\lambda\dot{H}+\lambda\left(\lambda-1\right)H\frac{\dot{Q}}{Q}\right)-3\lambda
H^{2}-\frac{1-\lambda}{2}Q+\frac{\gamma\dot{Q}\lambda\left(\lambda-1\right)}{2a^{2}Q}-k\left(\frac{\lambda}{a^{2}}+\frac{3\lambda\left(\lambda-1\right)\dot{Q}}{2\gamma
Q}\right)=0,$ (8)
$\left(\lambda-2\right)\dot{Q}^{2}\left(1+\frac{ka^{2}}{\gamma^{2}}\right)+Q\left[\ddot{Q}\left(1+\frac{ka^{2}}{\gamma^{2}}\right)+\dot{Q}\left(\left(1+\frac{3ka^{2}}{\gamma^{2}}\right)H+\frac{2\dot{\gamma}}{\gamma}\right)\right]=0.$
(9)

where we have assumed that $Q$ is nonzero.

The modified Friedmann’s equations can be written equivalently as follows

$3\lambda H^{2}=\rho_{Q}-\rho_{k}+\rho_{\gamma}$ (10)
$-2\dot{H}-3H^{2}=p_{Q}+\frac{1}{3}\rho_{k}+p_{\gamma}$ (11)

where we have defined the effective fluid components

$\rho_{Q}=\frac{\lambda-1}{2\lambda}Q~{},~{}\rho_{k}=\frac{3k}{a^{2}}~{},~{}$
(12)
$\rho_{\gamma}=\frac{3}{2}\left(\frac{\gamma}{a^{2}}+\frac{k}{\gamma}\right)\left(\lambda-1\right)\left(\ln
Q\right)^{\cdot},$ (13) $p_{Q}=2\left(\lambda-1\right)H\left(\ln
Q\right)^{\cdot}-\frac{\lambda-1}{2\lambda}Q$
$p_{\gamma}=\frac{1}{2}\left(-\frac{\gamma}{a^{2}}+\frac{3k}{\gamma}\right)\left(\lambda-1\right)\left(\ln
Q\right)^{\cdot}.$ (14)

The effective fluid, characterized by components $\rho_{\gamma}$ and
$p_{\gamma}$, incorporates the dynamical terms associated with the function
$\gamma$, which arises from the non-coincidence connection (2). In the
asymptotic limit where $\left(\ln Q\right)^{\cdot}\rightarrow 0$, we find that
$\rho_{\gamma}=0$ and $p_{\gamma}=0$, while $\rho_{Q}=\rho_{Q_{0}}$ and
$p_{Q}=-\rho_{Q_{0}}$. Consequently, the dynamical behavior aligns with that
of General Relativity, including a cosmological constant term, where
$\rho_{Q0}$ plays the role of the cosmological constant.

To gain deeper insights into the cosmological dynamics and evolution of the
cosmological parameters in the presence of curvature for
$f\left(Q\right)$-gravity, we will conduct a comprehensive analysis of the
asymptotic dynamics for the field equations (4)-(6). This detailed examination
will enable us to determine the general behavior of the physical parameters
within the framework of nonlinear symmetric teleparallel theory, while also
shedding light on the influence of non-zero spatial curvature on the
cosmological dynamics.

##  III Phase-space description

We introduce the scalar field $\psi=f^{\prime}\left(Q\right)$, such that to
write the field equations (4)-(6) in the following form

$3\psi
H^{2}+V\left(\psi\right)-\frac{3\gamma}{2a^{2}}\dot{\psi}+3k\left(\frac{\psi}{a^{2}}-\frac{\dot{\psi}}{2\gamma}\right)=0,$
(15) $-2\left(\psi H\right)^{\cdot}-3\psi
H^{2}-V\left(\psi\right)+\frac{\gamma\dot{\psi}}{2a^{2}}-k\left(\frac{\psi}{a^{2}}+\frac{3\dot{\psi}}{2\gamma}\right)=0,$
(16)
$\left(1+\frac{ka^{2}}{\gamma^{2}}\right)\ddot{\psi}+\dot{\psi}\left(\left(1+\frac{3ka^{2}}{\gamma^{2}}\right)H+\frac{2\dot{\gamma}}{\gamma}\right)=0,$
(17)

in which $V\left(\psi\right)=\frac{1}{2}\left(f(Q)-Qf^{\prime}(Q)\right)$
plays the role of the scalar field potential, and
$Q=-2V\left(\psi\right)_{,\psi}$

Therefore the effective energy density and pressure components related to the
nonlinear $f\left(Q\right)$ are $\rho_{Q}^{eff}=\rho_{Q}+\rho_{\gamma}$,
$p_{eff}=p_{Q}+p_{\gamma}$ with

$\rho_{Q}=-\frac{V\left(\psi\right)}{\psi}~{},~{}p_{Q}=2H\left(\ln\psi\right)^{\cdot}~{},~{}$
(18)
$~{}\rho_{\gamma}=\frac{3}{2}\left(\frac{\gamma}{a^{2}}+\frac{k}{\gamma}\right)\left(\ln\psi\right)^{\cdot}+\frac{V\left(\psi\right)}{\psi}~{},$
(19)
$p_{\gamma}=\frac{1}{2}\left(-\frac{\gamma}{a^{2}}+\frac{3k}{\gamma}\right)\left(\ln\psi\right)^{\cdot}~{}.$
(20)

Finally, in the scalar field description, equation (3) reads

$-6H^{2}+\frac{3\gamma}{a^{2}}H+\frac{3\dot{\gamma}}{a^{2}}+k\left(\frac{6}{a^{2}}+\frac{3}{\gamma}\left(\frac{\dot{\gamma}}{\gamma}-3H\right)\right)+2V\left(\psi\right)_{,\psi}=0.$
(21)

###  III.1 Dimensionless variables

To analyze the dynamics effectively, we incorporate the use of new dependent
variables

$x=\frac{\dot{\psi}}{2\psi H}~{},~{}y=\frac{V\left(\psi\right)}{3\psi
H^{2}}~{},~{}z=\frac{\gamma}{a^{2}H}~{},~{}\Omega_{k}=\frac{k}{a^{2}H^{2}}~{},~{}\mu=\psi\frac{V_{,\psi}}{V},$

and the new independent variable $\tau=\ln a$. Parameter $\tau$ is introduced
in order the trajectories to depend on the size of the universe and not the
time-parameter.

Hence, the field equations (15), (16), (17) and (21) are written in the
equivalent form of the first-order dynamical system

$\displaystyle\frac{2z\left(\Omega_{k}+z^{2}\right)^{2}}{x}\frac{dx}{d\tau}$
$\displaystyle=\Omega_{k}^{2}z\left(\Omega_{k}-3+3y\right)+2\Omega_{k}\left(\Omega_{k}-7+3y\right)z^{3}$
$\displaystyle+8\left(\Omega_{k}-1+\mu
y\right)z^{4}+\left(5+\Omega_{k}+3y\right)z^{5}-x\left(\Omega_{k}+z^{2}\right)^{2}\left(3\Omega_{k}-z^{2}\right),$
(22)
$\frac{dy}{d\tau}=y\left(3\left(1+y\right)+\Omega_{k}+x\left(2\left(1+\mu\right)+\frac{3\Omega_{k}}{z}-z\right)\right),$
(23) $\displaystyle 2\frac{dz}{d\tau}$
$\displaystyle=x\left(3\Omega_{k}-z\left(z-4\right)\right)$
$\displaystyle+\frac{z}{\Omega_{k}+z^{2}}\left(\Omega_{k}\left(5+\Omega_{k}+3y\right)-4\left(\Omega_{k}-1+\mu
y\right)z+\left(\Omega_{k}-3\left(1-y\right)\right)z^{2}\right)$ (24)
$\frac{d\Omega_{k}}{d\tau}=\Omega_{k}\left(1+\Omega_{k}+3y+x\left(4+\frac{3\Omega_{k}}{z}-z\right)\right),$
(25) $\frac{d\mu}{d\tau}=2\mu x\left(1-\mu+\mu\Gamma\left(\mu\right)\right),$
(26)

with algebraic constraint

$1+\Omega_{k}+y-\frac{x}{z}\left(\Omega_{k}+z^{2}\right)=0,$ (27)

and function $\Gamma\left(\mu\right)$ is defined as

$\Gamma\left(\mu\right)=\frac{V_{,\psi\psi}V}{\left(V_{,\psi}\right)^{2}}.$
(28)

Therefore, the deceleration parameter is expressed as

$q\left(x,y,z,\Omega_{k}\right)=\frac{1}{2}\left(1+\Omega_{k}+3y+z\left(4+\frac{3\Omega_{k}}{z}-z\right)\right).$
(29)

With the application of the constraint equation (27) the dynamical system lies
on a four dimensional manifold. Hence, by substituting
$y=-1-\Omega_{k}+\frac{x}{z}\left(\Omega_{k}+z^{2}\right)$ in (22), (24), (25)
and (26) we end with the system

$\displaystyle\frac{2z\left(\Omega_{k}+z^{2}\right)^{2}}{x}\frac{dx}{d\tau}$
$\displaystyle=x\left(\Omega_{k}+z^{2}\right)\left(3\Omega_{k}^{2}+4\Omega_{k}z^{2}+4\mu
z^{3}+z^{4}\right)$
$\displaystyle-z\left(\Omega_{k}^{2}\left(3+\Omega_{k}\right)+2\Omega_{k}\left(\Omega_{k}+5\right)z^{2}+4\left(\mu+1+\left(\mu-1\right)\Omega_{k}\right)z^{3}+\left(\Omega_{k}-1\right)z^{4}\right),$
(30) $\displaystyle\frac{dz}{d\tau}$
$\displaystyle=-\frac{z}{\Omega_{k}+z^{2}}\left(\left(\Omega_{k}-1\right)\Omega_{k}-2\left(1+\mu+\left(\mu-1\right)\Omega_{k}\right)z+\left(3+\Omega_{k}\right)z^{2}\right)$
$\displaystyle+x\left(3\Omega_{k}+z\left(2\left(1-\mu\right)+z\right)\right),$
(31)
$\frac{d\Omega_{k}}{d\tau}=2\Omega_{k}\left(-\left(1+\Omega_{k}\right)+2x\left(2+\frac{3\Omega_{k}}{z}+z\right)\right),$
(32) $\frac{d\mu}{d\tau}=2x\mu\left(1-\mu+\mu\Gamma\left(\mu\right)\right).$
(33)

There are two special functional forms of the scalar field potential of
special interest. The power-law potential
$V\left(\psi\right)=V_{0}\psi^{\mu_{0}}$, where equation (33) becomes
$\frac{d\mu}{d\tau}=0$, that is, $\mu=\mu_{0}$, and the exponential potential
$V\left(\psi\right)=V_{0}\psi^{\alpha}e^{\frac{1}{2}\psi_{0}\left(\ln\psi\right)^{2}}$,
in which $\mu\left(1-\mu+\mu\Gamma\left(\mu\right)\right)=\psi_{0}$. For the
power-law potential the dimension of the dynamical system is reduced by one;
while for the second potential function stationary points exist only when
$x=0$.

That is an important observation because for an arbitrary function
$f\left(Q\right)$, the existence of a stationary point $\mu=\mu_{0}$ it means
that at the asymptotic limit $f\left(Q\right)$ function is described by the
power-law function $Q^{\frac{\mu_{0}}{\mu_{0}-1}}$, which provides the power-
law potential $V\left(\psi\right)=V_{0}\psi^{\mu_{0}}$.

That is similar to the quintessence scalar field cop1  where the two
exponential potential is related to the scaling solution, while the power-law
potential attributes the de Sitter limit cop2 . Thus, any other potential
function consists the scaling solution at the limit in which the potential is
described by the exponential function, or the de Sitter limit when the
potential is described by the power-law function. As an example we recall the
hyperbolic potential cop3  which has the power-law and the exponential limits
cop4 , that is, why the hyperbolic potential provides de Sitter and scaling
solutions for the background equations.

###  III.2 Asymptotic solutions at the finite regime

Now, let us proceed to determine the stationary/equilibrium points of the
system (30)-(33). Each stationary point corresponds to an asymptotic solution
for the field equations. Let
$P=\left(x\left(P\right),z\left(P\right),\Omega_{k}\left(P\right),\mu\left(P\right)\right)$
be one such stationary point of the dynamical system, and as a stationary
point, the right-hand sides of equations (30)-(33) become zero.

For an arbitrary potential $V\left(\psi\right)$, that is, for an arbitrary
potential function $\Gamma\left(\mu\right)$, then if $\mu_{0}$, is a real root
of equation $\mu\left(1-\mu+\mu\Gamma\left(\mu\right)\right)=0,$ then the
equilibrium points of the dynamical system (30)-(33) are

$\displaystyle P_{1}$
$\displaystyle=\left(-1,z_{1},-z_{1},\mu_{0}\right),~{}P_{2}=\left(\frac{1}{2},2,0,\mu_{0}\right),$
$\displaystyle P_{3}$
$\displaystyle=\left(\frac{5}{1-3\mu_{0}},-1-\mu_{0},0,\mu_{0}\right)\text{
and }P_{4}=\left(0,\frac{2\left(1+\mu_{0}\right)}{3},0,\mu_{0}\right).$

In the special case where $\mu_{0}=2$, there exist the additional equilibrium
point

$P_{5}=\left(-1,-3,\Omega_{k5},2\right).$

For the asymptotic solutions described at the letter points we derive the
deceleration parameters

$q\left(P_{1}\right)=0~{},~{}q\left(P_{2}\right)=1~{},~{}q\left(P_{3}\right)=\frac{2\left(2-\mu_{0}\right)}{1-3\mu_{0}}~{},~{}q\left(P_{5}\right)=-1~{},$

and for $\mu_{0}=2,~{}$

$q\left(P_{5}\right)=0.$

The stationary points have distinct characteristics: Stationary point $P_{1}$
represents a family of Milne-like universes with a scale factor
$a\left(t\right)=a_{0}t$. Point $P_{2}$ corresponds to a spatially flat FLRW
asymptotic solution, where the effective fluid behaves as a stiff fluid. At
stationary point $P_{3}$, we find a spatially flat self-similar universe with
a deceleration parameter
$q\left(P_{3}\right)=\frac{2\left(2-\mu_{0}\right)}{1-3\mu_{0}}$. This point
exists for $\mu_{0}\neq\frac{1}{3}$ and describes an accelerated universe when
$\frac{1}{3}<\mu<2$. Point $P_{4}$ holds special significance as it
corresponds to the de Sitter solution. Lastly, stationary point $P_{5}$
describes Milne-like universes with a scale factor $a\left(t\right)=a_{0}t$.
Notably, this result aligns with the findings in qq3 , which analyzed the
existence of self-similar solutions.

In addition to the equilibrium points mentioned above, there always exist
other stationary points in the system.

$P_{6}=\left(0,0,-1,0\right)~{},~{}P_{7}=\left(0,\frac{2\left(1+\mu\right)}{3},0,\mu\right)$

where in $P_{7}$, $\mu$ is arbitrary parameter such that the algebraic
expression $\mu\left(1-\mu+\mu\Gamma\left(\mu\right)\right)$ to be non-
singular. Points $P_{6}$ and the family of points $P_{7}$ are the only points
which exist in the case where
$\mu\left(1-\mu+\mu\Gamma\left(\mu\right)\right)=0$ has no real solutions. We
remark that $P_{4}$ is a special case of point $P_{7}$. These two points,
describe the asymptotic limit of the theory in the context of General
Relativity with or without the cosmological constant term.

We derive the deceleration parameters

$q\left(P_{6}\right)=0\text{ and }q\left(P_{7}\right)=-1\text{\thinspace}.$

Therefore, point $P_{6}$ is the Milne solution of General Relativity, and
$P_{7}$ are family of de Sitter points.

We note that for arbitrary function $f\left(Q\right)$, these are the possible
families of asymptotic solutions described by the theory. For a specific
definition for the function $f\left(Q\right)$, the existence conditions are
defined and the stability properties can be examined.

The significance of the General Relativity limit lies in the existence of
points $P_{6}$ and $P_{7}$ for any arbitrary function $f\left(Q\right)$. The
novelty introduced by the non-coincidence gauge it follows from the the
inclusion of the gauge function $\gamma$ in the connection. It is important to
note that for the non-flat FLRW spacetime, this represents the only possible
connection, as referred to in ds1 .

###  III.3 Stability properties

Until now we have discussed only the existence conditions and the physical
properties of the stationary points for the field equations. However, in order
to reconstruct the cosmological history the stability properties of the points
should be determined. Hence, we should specify the potential function
$V\left(\psi\right)$. In order to demonstrate the stability properties we
consider the following cases for the scalar field potential,
$V_{A}\left(\psi\right)=V_{0}\psi^{\alpha}$,
$V_{B}\left(\psi\right)=V_{0}\left(\psi-1\right)^{\alpha}$ and
$V_{C}=V_{0}e^{\alpha x}$. The corresponding $f\left(Q\right)$-theories
related to these potentials are

$f_{A}\left(Q\right)\simeq
Q^{\frac{\alpha}{\alpha-1}}~{},~{}f_{B}\left(Q\right)\simeq
Q+f_{1}Q^{\frac{\alpha}{\alpha-1}}\text{ and }f_{C}\left(Q\right)\simeq
Q+f_{1}Q\ln Q.$

####  III.3.1 Power law potential $V_{A}\left(\psi\right)=V_{0}\psi^{\alpha}$

For the power-law potential we calculate $\mu=\alpha$ and
$\Gamma\left(\mu\right)=\frac{\alpha-1}{\alpha}$, that is, the dimension of
the dynamical system is reduced by one (30)-(33). Consequently, we study the
stability of the stationary points on the constant surface $\mu=\alpha$.

The eigenvalues of the linearized system (30)-(32) around the stationary point
$P_{2}$ are, $\left\\{-2,2,3+\mu\right\\}$, which means that $P_{2}$ is a
saddle point. For the point $P_{3}$ the three-dimensional system provides the
eigenvalues
$\left\\{\frac{4\left(2-\mu\right)}{1-3\mu},\frac{2\left(\mu-2+\sqrt{4\mu\left(9+4\mu\right)-11}\right)}{1-3\mu},\frac{2\left(\mu-2-\sqrt{4\mu\left(9+4\mu\right)-11}\right)}{3\mu-1}\right\\}$,
from where we can infer that $P_{3}$ is always a saddle point. For the de
Sitter point $P_{4}$ we derive the eigenvalues $\left\\{-2,-3,-5\right\\}$,
that is, $P_{4}$ is a future attractor. For the limit of General Relativity,
i.e. point $P_{6}$, the eigenvalues are $\left\\{-2,2,2\right\\}$, thus, the
Milne solution is always unstable and $P_{6}$ is a saddle point. Finally, for
the family of points $P_{1}$ the eigenvalues are
$\left\\{0,0,2\left(2-\mu\right)\right\\}$. Because of the zero eigenvalues in
order to infer about the stability properties of the point we shall study the
trajectories numerically.

In Fig. 1 we present two-dimensional phase-portraits for the dynamical system
(30)-(32) for different values of the free parameters $\alpha$. The surfaces
are where the family of points $P_{1}$ are defined. Thus, from the Fig. we
observe that $P_{1}$ are always saddle points.

Figure 1: Two-dimensional phase-space portraits for the dynaical system
(30)-(32) in the surfaces $\left(x,z\right),~{}\left(x,\Omega_{k}\right)$ and
$\left(z,\Omega_{k}\right)$ where points $P_{1}$ are defined. Figures are for
various values of the free parameter $\alpha$, where we remark that points
$P_{1}$ describe unstable solutions.

Furthermore, for the special case where $\alpha=2$, that is,
$V_{A}\left(\psi\right)=V_{0}\psi^{2}$, for the linearized system (30)-(33)
around the equilibrium point $P_{5}$ we find the eigenvalues
$\left\\{-1,2,7\right\\}$; we conclude that $P_{5}$ is always a saddle point.

####  III.3.2 Power law potential
$V_{B}\left(\psi\right)=V_{0}\left(\psi-1\right)^{\alpha}$

For the second potential function
$V_{B}\left(\psi\right)=V_{0}\left(\psi-1\right)^{\alpha}$ it follows,
$\psi=\frac{\mu}{\mu-\alpha}$ and
$\Gamma\left(\mu\right)=\frac{\alpha}{\alpha-1}$. Consequently, equation (33)
becomes

$\frac{d\mu}{d\tau}=\frac{2}{\alpha}x\mu\left(\mu-\alpha\right).$ (34)

Indeed, there are two roots of the algebraic equation
$\mu\left(\mu-\alpha\right)=0$, they are, $\mu_{1}=0$ and $\mu_{2}=\alpha$.

The dynamical system now exists in four dimensions, and accordingly, there are
four eigenvalues associated with it. In the case of General Relativity, with
or without the cosmological constant, represented by points $P_{6}$ and
$P_{7}$ respectively, the eigenvalues of the linearized system are
${-2,2,2,0}$ for $P_{6}$ and ${-2,-3,-5,0}$ for $P_{7}$.

As a result, point $P_{6}$ always functions as a saddle point, while our focus
shifts towards the de Sitter solutions at point $P_{7}$. The existence of the
zero eigenvalue, as derived from equation (34), suggests that the trajectories
lie along the $\mu$-direction. Despite this, the physical properties of the
asymptotic solution, such as the deceleration parameter, remain unchanged.
Although it may appear that $P_{7}$ also behaves as a saddle point, it is
essential to recognize that this family of points defines a space where the
trajectories reside on a three-dimensional surface, specifically corresponding
to the de Sitter universe.

For $\mu_{1}=0$, and points $P_{1}$ we calculate the eigenvalues
$\left\\{0,0,2,4\right\\}$, hence $P_{1}$ are saddle points. Moreover, the
eigenvalues related to point $P_{2}$ are ${-1,-2,2,3}$, while the eigenvalues
related to $P_{3}$ are ${8,-10,-4+2\sqrt{11}i,-4+2\sqrt{11}i}$, which means
that $P_{2}$ and $P_{3}$ are saddle points. Finally, for $P_{4}$ it holds the
same discussion with $P_{7}$.

On the surface for $\mu_{1}=\alpha$, with omit the presentation of the
eigenvalues, however we find that the stability properties of the stationary
points $P_{2}$, $P_{3}$ and $P_{4}$ are the same as before. Specifically
$P_{2}$, $P_{3}$ are saddle points and $P_{4}$ has the same behaviour as
points $P_{7}$. However, for points $P_{1}$ we derive the eigenvalues
$\left\\{0,0,4-2\alpha,-2\right\\}$, where the phase-space portraits 1 are
valid, and the same conclusion holds.

####  III.3.3 Exponential potential
$V_{C}\left(\psi\right)=V_{0}e^{\alpha\psi}$

Consider now the exponential potential
$V_{C}\left(\psi\right)=V_{0}e^{\alpha\psi}$, from where we calculate
$\mu=\alpha x$ and $\Gamma\left(\mu\right)=1$. Hence, equation (33) reads

$\frac{d\mu}{d\tau}=2\mu x.$ (35)

The stationary points consist of those with $\mu_{0}=0$ and the ones
corresponding to the limit of General Relativity, namely $P_{6}$ and $P_{7}$.
As observed previously, the dynamical system reveals that the de Sitter
universe stands as the only attractor, while all the other asymptotic
solutions are unstable and represented by saddle points.

##  IV Poincare variables

The variables $\left\\{x,z,\Omega_{k},\mu\right\\}$ of the dynamical system
(30)-(33) are not constraint and they can take values in all the range of real
numbers; that is, they can reach infinity. To examine the existence of
stationary points at the infinity we introduce Poincare variables.

For simplicity on our calculations we select $\mu=\alpha$ which corresponds to
the power-law function $f\left(Q\right)=Q^{\alpha}$.

###  IV.1 Closed universe $\Omega_{k}>0$

For $\Omega_{k}>0$ we define the variables

$x=\frac{x}{\sqrt{1-X^{2}-W^{2}-Z^{2}}}~{},~{}z=\frac{Z}{\sqrt{1-X^{2}-W^{2}-Z^{2}}}~{},~{}~{}\Omega_{k}=\frac{W^{2}}{1-X^{2}-W^{2}-Z^{2}},$

where now the deceleration parameter reads

$q\left(X,W,Z\right)=\frac{Z\left(Z^{2}+X^{2}-1\right)+X\left(3W^{2}+Z\left(Z+2\sqrt{1-X^{2}-W^{2}-Z^{2}}\right)\right)}{Z\left(1-X^{2}-W^{2}-Z^{2}\right)},$

and the field equations read

$\frac{dX}{dT}=\chi\left(X,W,Z,\alpha\right)~{},~{}\frac{dW}{dT}=\omega\left(X,W,Z,\alpha\right)~{},~{}\frac{dZ}{dT}=\zeta\left(X,W,Z,\alpha\right)$

where $Z\neq 0$ and $dT=\sqrt{1-X^{2}-W^{2}-Z^{2}}d\tau.$

At the infinity limit, $1-X^{2}-W^{2}-Z^{2}$, the stationary points
$A=A\left(X\left(A\right),W\left(A\right),Z\left(A\right)\right)$ of the field
equations are

$A^{\pm}=\left(0,0,\pm 1\right).$

The deceleration parameters of the asymptotic solutions at these stationary
points are

$q\left(A^{\pm}\right)=0\text{.}$

Consequently, the points $A^{\pm}$ correspond to Milne-like solutions.
Although the eigenvalues of the linearized points are all zero, numerical
solutions indicate that these points are consistently unstable.

It is worth noting that a family of transition points exists on the surface
$Z=0$, comprising the four points $C_{1}^{\pm}=\left(\pm 1,0,0\right)$ and
$C_{2}^{\pm}=\left(0,\pm 1,0\right)$. These transition points represent Big
Rip singularities, with points $C_{1}^{-}$ and $C_{2}^{+}$ corresponding to
such scenarios, while points $C_{1}^{+}$ and $C_{2}^{-}$ describe a super
collapse universe.

Figure 2: Two-dimensional phase-space portraits for the dynaical system
(30)-(32) at the infinity for a closed FLRW spacetime, on diferent surfaces
from where it is clear that there are not attractors at the finite regime.
Figures of the left column are for $\mu=4$, and figures of the right column
are for $\mu=-4$.

In Fig. 2, we provide two-dimensional phase-space portraits for the dynamical
system (30)-(32) and the power-law theory at infinity, observed on different
surfaces. It is evident from the figures that there are no attractors within
the finite regime.

###  IV.2 Open universe $\Omega_{k}<0$

In the case of an open universe, that is, $\Omega_{k}<0$, the Poincare
variables are

$x=\frac{x}{\sqrt{1-X^{2}-W^{2}-Z^{2}}}~{},~{}z=\frac{Z}{\sqrt{1-X^{2}-W^{2}-Z^{2}}}~{},~{}~{}\Omega_{k}=-\frac{W^{2}}{1-X^{2}-W^{2}-Z^{2}}.$

We follow the same procedure as before and we find that the there is not any
attractor at the infinity. Milne universes are described by the saddle points
$A_{3}^{\pm}$. In Fig. 3 we present phase-space portraits of the dynamical
system for different values of the free parameters.

Figure 3: Two-dimensional phase-space portraits for the dynaical system
(30)-(32) at the infinity for an open FLRW spacetime, on diferent surfaces
from where it is clear that there are not attractors at the finite regime.
Figures of the left column are for $\mu=4$, and figures of the right column
are for $\mu=-4$.

It is important to note that the results presented above hold true for a
general $f\left(Q\right)$ function. However, the existence of the stationary
points is contingent upon the additional equation (33). While this aspect is
of particular interest from a mathematical analysis perspective, it does not
significantly contribute to the physical discussion regarding the viability of
$f\left(Q\right)$-theory.

##  V Conclusions

We studied the dynamics of the cosmological field equations in the symmetric
teleparallel $f\left(Q\right)\,$-theory, for a FLRW geometry with nonzero
spatial curvature. We study the phase-space and we determined the stationary
points such that to construct all the asymptotic solutions and the
cosmological history.

Through our analysis, we determined the equilibrium points for the general
nonlinear function $f\left(Q\right)$, and notably, we found that the theory
always yields the Milne spacetime as a limit, which is consistent with General
Relativity. Additionally, we recovered the de Sitter spacetime, representing
the limit of General Relativity when a cosmological constant term is present.
Furthermore, our investigation yielded novel asymptotic solutions originating
from the geometric terms of the theory. These solutions are associated with
Milne-like scenarios and spatially flat scaling spacetimes.

We examined the stability characteristics of the stationary points for three
distinct functions in our study: $f_{A}\left(Q\right)\simeq
Q^{\frac{\alpha}{\alpha-1}}$, $f_{B}\left(Q\right)\simeq
Q+f_{1}Q^{\frac{\alpha}{\alpha-1}}$, and $f_{C}\left(Q\right)\simeq
Q+f_{1}Q\ln Q$. Through our investigation, we made a significant observation
that the spatially flat de Sitter universe serves as the exclusive attractor
within the theoretical framework.

In addition to the above analysis, we delved into the stability properties of
the stationary points at infinity specifically for the power-law function
$f_{A}\left(Q\right)$. To achieve this, we utilized Poincare variables in our
study.

Therefore, nonlinear symmetric teleparallel theory can solve the flatness
problem and lead to de Sitter expansion without introduce a cosmological
constant term. This conclusion for the $f\left(Q\right)$-gravity is different
from that of the nonlinear theories constructed by the other two scalar of the
trinity of General Relativity, that is, of the $f\left(R\right)=R^{\lambda}$
and of the $f\left(T\right)=T^{\lambda}$ theories.

In power-law $f\left(R\right)$-gravity the de Sitter universe is recovered
when $\lambda=2~{}$sd3 , while in the power-law $f\left(T\right)$-gravity only
scaling cosmological solutions are provided sd4 . However, what makes the
nonlinear symmetric teleparallel theory to stand out is the introduction of
the additional dynamical variables provided by the non-coincidence connection
and described by the function $\gamma$. While someone will expect that the
power-law theory $f_{A}\left(Q\right)\simeq Q^{\frac{\alpha}{\alpha-1}}$ will
not recover General Relativity, as in the case of spatially flat FLRW in the
coincidence gauge, that is not true, because of the contribution by the non-
coincidence gauge terms.

While this research was still under consideration by the journal, a preprint
with a similar study was made available online arx1 . In the mentioned work,
the authors adopted the Ansatz $\gamma\simeq a\left(t\right)$, restricting
their analysis to a specific surface. As a result, their findings may not be
applicable to arbitrary initial conditions or the original problem in its
entirety. Moreover, in their specification of the function $f\left(Q\right)$,
the authors ended up with a system that has more equations than the original
degrees of freedom. Consequently, the analysis presented in arx1  is deemed
incomplete. In contrast, in our work, we considered a generic function
$\gamma$, allowing for a more comprehensive exploration of the topic, and our
results remain valid for any set of initial conditions.

Before we conclude we would like to discuss what will be evolution of the
perturbations in the case of the non-coincidence gauge. Assume the spatially
flat case, $k=0$, then for this connections, the field equations (4)-(5)
becomes

$3H^{2}f^{\prime}(Q)+\frac{1}{2}\left(f(Q)-Qf^{\prime}(Q)\right)=\rho_{\gamma},$
(36)
$-2\left(f^{\prime}(Q)H\right)^{\cdot}-3H^{2}f^{\prime}(Q)-\frac{1}{2}\left(f(Q)-Qf^{\prime}(Q)\right)=p_{\gamma}.$
(37)

with

$\rho_{\gamma}=\frac{3\gamma\dot{Q}f^{\prime\prime}(Q)}{2a^{2}}~{}~{},~{}p_{\gamma}=-\frac{\gamma\dot{Q}f^{\prime\prime}(Q)}{2a^{2}}~{}.$
(38)

From (38) we derive the equation of state parameter
$p_{\gamma}=-\frac{1}{3}\rho_{\gamma}$.

The lhs parts of equations (36), (37) form the modified field Friedmann’s
equations of $f\left(Q\right)$-theory in the coincidence gauge. Therefore the
field equations for the non-coincidence gauge introduce an matter component
which is coupled to $\psi=f^{\prime}\left(Q\right)$. Because of that analogue
it is expected the evolution of the perturbations to be similar with that of
the coincidence gauge with the extra matter term per1 ; per2 . Where
$\gamma\rightarrow 0$, that is, on the surface with $~{}z=0$, the limit of the
coincidence gauge is recovered.

###### Acknowledgements.

This work was financially supported in part by the National Research
Foundation of South Africa (Grant Numbers 131604). The author thanks the
support of Vicerrectoría de Investigación y Desarrollo Tecnológico (Vridt) at
Universidad Católica del Norte through Núcleo de Investigación Geometría
Diferencial y Aplicaciones, Resolución Vridt No - 098/2022.

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