# Probing symmetric teleparallel gravity in the early universe

Avik De1, Dalia Saha2,3, Ganesh Subramaniam4 and Abhik Kumar Sanyal2

###### Abstract

General theory of relativity can be equivalently formulated on a flat
spacetime that associates with a torsion-free affine connection of non-
vanishing non-metricity scalar $Q$. In this work, we present an extension of
this, the $f(Q)$ theory of gravity, and explore the early evolution of the
universe in the background of anisotropic Bianchi-I model. The $f(Q)$ theory
in the current setting through its geometric modification is quite successful
in explaining the late time accelerated expansion. Here we notice that it also
fits inflationary parameters with excellent precession, but fails to produce a
decelerated expansion in the radiation dominated era.

1,4 Department of Mathematical and Actuarial Sciences, Universiti Tunku Abdul
Rahman, Jalan Sungai Long, 43000 Cheras, Malaysia.
2 Department of Physics, Jangipur College, Murshidabad, West Bengal, India -
742213.
3 Department of Physics, University of Kalyani, Nadia, India - 741235.

11footnotetext:  Electronic address:
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<EMAIL_ADDRESS>
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2sanyal_<EMAIL_ADDRESS>

##  1 Introduction

Despite the undeniable success of ‘the standard model of cosmology’, governed
by Einstein’s general theory of relativity (GTR), limitations in several
different aspects are evident in recent years. Cosmological singularity and
black holes, $H_{0}$ and $\sigma_{8}$ tensions in the observational grounds,
and more importantly, the dependencies on the yet undetected dark sectors to
explain the late time cosmic acceleration motivated researchers to venture out
for alternate theories of gravity. Teleparallel gravity has been extensively
researched in this regard, in recent years. To be specific, the Levi-Civita
connection, which serves as the foundation of GTR, may be replaced by an
affine connection on spatially flat spacetime with vanishing torsion, allowing
its non-metricity to assume complete responsibility for defining gravity.
‘Symmetric teleparallel gravity’ is the name given to this particular theory.
Long back, in order to unify gravity with electromagnetism, Einstein [1]
attributed gravity to the torsion of spacetime in a more developed ‘metric
teleparallel theory’, which is based on an affine connection with vanishing
curvature and non-metricity. In teleparallel theories, one can construct
either the so-called torsion scalar $\mathbb{T}$ from this torsion tensor in
the metric teleparallel theory or the non-metricity scalar $Q$ in the
symmetric teleparallel theory. Thereafter, much like the Lagrangian density
$\mathcal{L}=\sqrt{-g}R$ ($R$ being the Ricci scalar) in the Einstein-Hilbert
action $S=\int{d^{4}x}\mathcal{L}$ in GTR, it is also possible to consider the
Lagrangian density $\mathcal{L}=\sqrt{-g}\mathbb{T}$ in the metric
teleparallel theory and also the Lagrangian density $\mathcal{L}=\sqrt{-g}Q$
in the symmetric teleparallel theory to obtain the respective field equations.
However, the latter two theories are equivalent to GTR upto a boundary term
and thus also rely on dark components of the universe as GTR does. To address
this dark sector issue of the universe, without inviting a scalar field,
extensions in terms of $f(\mathbb{T})$ and $f(Q)$-theories were formulated in
the metric teleparallel and symmetric teleparallel gravity respectively, the
same way as the modified $f(R)$-theory was developed from the Einstein-Hilbert
action in the realm of GTR. Indeed, there is an instant benefit in these two
theories over GTR. First, both $f(\mathbb{T})$ and $f(Q)$ theories are
featured in second order field equations instead of the fourth order ones in
$f(R)$, and next, these theories can also present an alternative explanation
for the accelerating universe [2, 3, 5, 4, 6, 7, 8, 9, 10]. These are in
particular the main reasons behind their popularity and extensive research in
recent time. However, models based on $f(\mathbb{T})$ suffer from substantial
coupling issues which is absent in $f(Q)$ theories, in general. In a further
closer look at the field equations of $f(\mathbb{T})$ theories we can observe
an eerie presence of skew-symmetric elements, which is not the case in $f(Q)$
or $f(R)$ or most other theories of gravity. As a result, $f(\mathbb{T})$
theories (except $f(\mathbb{T})=\mathbb{T}$) are also not locally Lorentz
invariant and possess extra degrees of freedom not present in GTR [11]. This
is a huge loss, as now the lack of Lorentz invariance calls for a system of
$16$ equations in $f(\mathbb{T})$ theories instead of $10$ equations in GTR
[12]. On a side by side comparison, the $f(\mathbb{T})$ connections must
comply with $Q_{\mu\nu\gamma}=0$, which are $40$ independent equations due to
the symmetry of the non-metricity tensor $Q_{\mu\nu\gamma}$ in the second and
third indices. Whereas, the $f(Q)$ connections must obey the vanishing torsion
tensor criteria, $\mathbb{T}^{\gamma}_{\,\,\,\mu\nu}=0$; only $24$ independent
equations since the $\mu=\nu$ case is trivially satisfied, making
$f(\mathbb{T})$ connections much more restrictive than $f(Q)$. For further
reading in this interesting topic of geometric trinity of gravity, one can
refer to [13, 14, 15, 16, 17] and the references therein. One can also look
into an array of publications [18, 19, 11] for other issues of $f(\mathbb{T})$
theory.

By all means, theories of gravity based on curvature and torsion are almost in
their adult age, whereas theories based on the non-metricity are still in
their infancy. Therefore, symmetric teleparallel gravity theory requires a lot
of scrutiny and detailed analysis, in its formulation, theoretical viability
and observational support. After the first brief introductory paper [20], a
lot of publications are emerging in the $f(Q)$-gravity theory in this past few
years, such as, its covariant formulation [21], interior and exterior
spherically symmetric solution [22], cosmography [23], effective pressure and
energy [24], signature of $f(Q)$-gravity in cosmology [25], as an alternative
to $\Lambda$CDM theory [7], an investigation from the reshift [26], the growth
index in matter perturbation [27], its dynamical system analysis [28, 6],
coupled with (pseudo-) scalar fields to the nonmetricity [29, 30], among
others. The geodesic deviation equation in $f(Q)$-gravity was also studied and
some fundamental results were obtained [31]. However, this relatively new
$f(Q)$ gravity theory is not examined in great depth (if not at all) in the
early stages of the universe, and this motivates the present study.

The current universe is modelled as isotropic and homogeneous at a large
enough scale, and so it is considered to be spatially maximally symmetric
Friedmann-Lemaître-Robertson-Walker (FLRW) type. This is quite a handy model
to work with, however, there are enough observational evidences to picture a
not so symmetrical looking universe, atleast in its beginning near the initial
singularity [32, 33, 34]. Nor should we take it for granted to be so in the
finite future. The Wilkinson Microwave Anisotropy Probe (WMAP) dataset clearly
call for some additional morphology than the standard FLRW model of the
universe. According to a significant finding in the field of theoretical
cosmology, a subset of anisotropic Bianchi models (models I, VII0, V, VIIh,
and IX) can be interpreted as the (appropriately defined) homogeneous limit of
(standard) linear cosmological perturbations of the FLRW spacetime [35, 36].
This significant finding closes the gap between the FLRW model and the Bianchi
model, compelling us to view the latter as a valid illustration of the
conventional cosmological framework. These types of models are specifically
those, which under the limit of zero anisotropy, result in Friedmann
spacetimes. Even though the “great event” of the inflationary paradigm is
successful in isotropizing the early cosmos into the homogeneous and isotropic
state that it is in today, it naturally cannot provide a complete account of
evolution. One must still relax the original assumption of FLRW geometry and
investigate the transition from an anisotropic and inhomogeneous state to the
observed level of homogeneity and isotropy. Therefore, as a first step, one
may think about homogeneous cosmological models of the Bianchi type. These
models make for a sizable and nearly exhaustive class of cosmological models;
they are homogeneous, but they aren’t necessarily isotropic. Further, in the
isotropic and homogeneous space-time, the teleparallel $f(\mathbb{T})$ gravity
is identical to the symmetric teleparallel $f(Q)$ gravity in coincidence
gauge. This is another important reason to study anisotropic model. In the
current study, we take into account such an anisotropy (axially symmetric
Bianchi-1 metric) and couple a scalar field to drive inflation that occurred
at the very early stage of the cosmological evolution of the universe, between
$10^{-42}-10^{-26}$ sec. If we find a viable inflation followed by graceful
exit, and an oscillatory behaviour of the scalar field, then scalar field
decays to produce particles. We shall next study radiation era, and see if a
viable deceleration is admissible. Finally, if the remnant of the scalar field
is red-shifted away, then a pressureless dust era, with early deceleration and
late-time acceleration follows, without dark energy. We consider that the
shear is not too large at the advent of radiation era (particularly less than
the expansion scalar), to find the scenarios under which the energy conditions
should be satisfied [37, 38].

The present paper is organised as follows: After the brief introductory
discussion presented above, comparing GTR, $f(\mathbb{T})$ and $f(Q)$ theories
of gravity, in the following section 2 we present the mathematical formulation
of $f(Q)$ theory along with the general form for the field equations. In
section 3, the field equations in the axially symmetric Bianchi-I spacetime
are presented and the possible existence of a vacuum de-Sitter type universe
have been explored. Next, in section 4, we study scalar field inflation in two
different $f(Q)$ models and an excellent fit with the recent dataset is
observed. Finally in section 5 we investigate the radiation-dominated era and
search for a decelerated expansion of the cosmos. We finish the present study
with concluding remarks in section 6. Throughout the text, we use the
notations $\dot{()}$, $()^{\prime}$ and $()_{Q}$ to denote the partial
differentiation with respect to the time variable $t$, scalar field $\phi$ and
the non-metricity scalar $Q$, respectively.

##  2 $f(Q)$ formulation

We discuss the detailed construction of the $f(Q)$ theory in symmetric
teleparallelism in this section. Basically, we begin with a $4$-dimensional
Lorentzian manifold $M^{4}$, a line element governed by the metric tensor
$g_{\mu\nu}$ in certain coordinate system $\\{x^{0},x^{1},x^{2},x^{3}\\}$ and
a non-tensor term, the connection coefficient
$\Gamma^{\alpha}_{\,\,\,\mu\nu}$, defining the covariant derivative $\nabla$
and also taking care of the curvature of the spacetime. In general, the
triplet $(M,g,\Gamma)$ regulates the spacetime geometry until we restrict
ourself to specifically consider both the metric-comptaibility and torsion-
free conditions on the connection coefficient. In such case there is only a
unique connection available, the Levi-Civita connection $\mathring{\Gamma}$
and it has a well-known relation with the metric $g$ given by

$\mathring{\Gamma}^{\alpha}_{\,\,\,\mu\nu}=\frac{1}{2}g^{\alpha\beta}\left(\partial_{\nu}g_{\beta\mu}+\partial_{\mu}g_{\beta\nu}-\partial_{\beta}g_{\mu\nu}\right).$
(1)

So, the Levi-Civita connection is basically a function of the metric $g$ and
not an independent contributor in the spacetime geometry. There is a drastic
development once we slightly relax these conditions and consider a torsion-
free affine connection $\Gamma$ on a flat spacetime which is not metric-
compatible, the incompatibility is characterised by the non-metricity tensor

$Q_{\lambda\mu\nu}:=\nabla_{\lambda}g_{\mu\nu}=\partial_{\lambda}g_{\mu\nu}-\Gamma^{\beta}_{\,\,\,\lambda\mu}g_{\beta\nu}-\Gamma^{\beta}_{\,\,\,\lambda\nu}g_{\beta\mu}\neq
0\,,$ (2)

We can always express

$\Gamma^{\lambda}{}_{\mu\nu}:=\mathring{\Gamma}^{\lambda}{}_{\mu\nu}+L^{\lambda}{}_{\mu\nu}$
(3)

where $L^{\lambda}{}_{\mu\nu}$ is the disformation tensor. It follows that,

$L^{\lambda}{}_{\mu\nu}=\frac{1}{2}(Q^{\lambda}{}_{\mu\nu}-Q_{\mu}{}^{\lambda}{}_{\nu}-Q_{\nu}{}^{\lambda}{}_{\mu})\,.$
(4)

We can construct two different types of non-metricity vectors,

$Q_{\mu}:=g^{\nu\lambda}Q_{\mu\nu\lambda}=Q_{\mu}{}^{\nu}{}_{\nu}\,,\qquad\tilde{Q}_{\mu}:=g^{\nu\lambda}Q_{\nu\mu\lambda}=Q_{\nu\mu}{}^{\nu}\,.$

The superpotential (or the non-metricity conjugate) tensor
$P^{\lambda}{}_{\mu\nu}$ is given by

$P^{\lambda}{}_{\mu\nu}=\frac{1}{4}\left(-2L^{\lambda}{}_{\mu\nu}+Q^{\lambda}g_{\mu\nu}-\tilde{Q}^{\lambda}g_{\mu\nu}-\frac{1}{2}\delta^{\lambda}_{\mu}Q_{\nu}-\frac{1}{2}\delta^{\lambda}_{\nu}Q_{\mu}\right)\,.$
(5)

Finally, the non-metricity scalar $Q$ is defined as

$Q=Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}.$ (6)

It should be mentioned that $R^{\rho}_{\sigma\mu\nu}=0$ is one of the
restrictions that was utilised when developing the $f(Q)$-theory. This
indicates that there is a unique coordinate system that can be chosen to make
the affine connection disappears, denoted by the expression
$\Gamma^{\lambda}_{\mu\nu}=0$. The term “coincident gauge” refers to this kind
of circumstance. In this particular setting, there is only one dynamical
variable, and that is the metric itself. By altering the action phrase

$S=\int\left[\frac{1}{2}f(Q)+\mathcal{L}_{M}\right]\sqrt{-g}\,d^{4}x$ (7)

with respect to the metric, we obtain the field equations

$\frac{2}{\sqrt{-g}}\nabla_{\lambda}(\sqrt{-g}f_{Q}P^{\lambda}{}_{\mu\nu})-\frac{1}{2}fg_{\mu\nu}+f_{Q}(P_{\nu\rho\sigma}Q_{\mu}{}^{\rho\sigma}-2P_{\rho\sigma\mu}Q^{\rho\sigma}{}_{\nu})=T^{m}_{\mu\nu}$
(8)

where $T^{m}_{\mu\nu}$ is the energy-momentum tensor generated from the matter
Lagrangian $\mathcal{L}_{M}$. We assume a barotropic perfect fluid including
dark matter $T^{m}_{\mu\nu}=(p+\rho)u_{\mu}u_{\nu}+pg_{\mu\nu}$ with isotropic
pressure $p$, energy-density $\rho$ and the four-velocity vector $u^{\mu}$.
Thereafter, using the relation (3), after some calculations (see the Appendix
of [31]) we can rewrite the field equations (8) in its covariant formulation
as

$f_{Q}\mathring{G}_{\mu\nu}+\frac{1}{2}g_{\mu\nu}(Qf_{Q}-f)+2f_{QQ}\mathring{\nabla}_{\lambda}QP^{\lambda}{}_{\mu\nu}=T^{m}_{\mu\nu}$
(9)

where

$\mathring{G}_{\mu\nu}=\mathring{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathring{R}.$

All the expressions with a $\mathring{()}$ is calculated with respect to the
Levi-Civita Connection $\mathring{\Gamma}$. Hence in its GTR equivalent
effective form we can express it as

$\displaystyle\mathring{G}_{\mu\nu}=\frac{1}{f_{Q}}T^{eff}_{\mu\nu}=\frac{1}{f_{Q}}T^{m}_{\mu\nu}+T^{DE}_{\mu\nu},$
(10)

where

$\displaystyle
T^{DE}_{\mu\nu}=\frac{1}{f_{Q}}\left[\frac{1}{2}g_{\mu\nu}(f-Qf_{Q})-2f_{QQ}\mathring{\nabla}_{\lambda}QP^{\lambda}_{\mu\nu}\right].$

##  3 Field equations and some consequences:

Starting from the most general linearly perturbed spatially flat FLRW metric

$\displaystyle
ds^{2}=-(1+2\Phi(\bar{x},t))dt^{2}-2B_{i}(\bar{x},t)dtdx^{i}+a^{2}(t)[(1-2\Psi(\bar{x},t))\delta_{ij}+2h_{ij}(\bar{x},t)]dx^{i}dx^{j},$
(11)

under suitable gauge choice and the homogeneity bound [39], in the current
discussion, we consider the axially symmetric Bianchi universe with line
element given in terms of Cartesian coordinates

$g=-\mathrm{d}t\otimes\mathrm{d}t+e^{\left(-4\,\chi\left(t\right)+2\,\xi\left(t\right)\right)}\mathrm{d}x\otimes\mathrm{d}x+e^{\left(2\,\chi\left(t\right)+2\,\xi\left(t\right)\right)}\mathrm{d}y\otimes\mathrm{d}y+e^{\left(2\,\chi\left(t\right)+2\,\xi\left(t\right)\right)}\mathrm{d}z\otimes\mathrm{d}z.$
(12)

It is straightforward to check that this particular coordinate system is
automatically a coincident gauge. The Ricci scalar $\mathring{R}$ and the non-
metricity scalar $Q$ are calculated to be

$\mathring{R}=6\dot{\chi}^{2}+12\dot{\xi}^{2}+6\ddot{\xi};~{}~{}~{}~{}~{}Q(t)=6\dot{\chi}^{2}-6\dot{\xi}^{2}.$
(13)

The equations of motion are:

$\begin{split}&{f\over 2}-Qf_{Q}=\rho,\\\ &{f\over
2}+2f_{Q}(\ddot{\xi}+\ddot{\chi}+3\dot{\xi}^{2}+3\dot{\xi}\dot{\chi})+2\dot{Q}f_{QQ}(\dot{\xi}+\dot{\chi})=-p,\\\
&{f\over 2}+2f_{Q}(\ddot{\xi}-{1\over 2}\ddot{\chi}+3\dot{\xi}^{2}-{3\over
2}\dot{\xi}\dot{\chi})+2\dot{Q}f_{QQ}(\dot{\xi}-{1\over
2}\dot{\chi})=-p.\end{split}$ (14)

Here $\rho$ and $p$ stand for the energy density and the thermodynamic
pressure of a perfect barotropic fluid. The non-metricity scalar, the
effective average Hubble parameter and the effective average scale factors may
be expressed as,

$\begin{split}&Q=-6(\dot{\xi}^{2}-\dot{\chi}^{2})={2\over
3}(3\sigma^{2}-\theta^{2})=2(\sigma^{2}-3H^{2}),\\\
&\theta=3H=3\dot{\xi},~{}~{}a=e^{\xi},~{}~{}\sigma^{2}=3\dot{\chi}^{2}.\end{split}$
(15)

where $\theta$ is the expansion scalar, $\sigma^{2}$ the shear scalar, while
the proper volume is $\mathcal{V}=e^{3\xi}$.

From the pair of pressure equations one immediately arrives at,

$f_{Q}\dot{\chi}=c_{1}e^{-3\xi}$ (16)

and it produces

$\dot{Q}f_{QQ}=-f_{Q}\left[3\dot{\xi}+{\ddot{\chi}\over\dot{\chi}}\right].$
(17)

In view of equations (14) and (17), one can also construct the following
equation,

$\rho-3p=2f+6f_{Q}\left(\ddot{\xi}+\dot{\xi}^{2}-\dot{\chi}^{2}-\dot{\xi}{\ddot{\chi}\over\dot{\chi}}\right),$
(18)

which we shall use shortly. The covariant conservation of the energy momentum
in the present scenario follows from [17]. This checkpoint is not trivial as
in the case of GTR or $f(R)$ theories. Since in certain scenarios, for
example, in a static spherically symmetric spacetime, this imposes one
additional constraint over the model parameters. One can show that the
covariant conservation of energy momentum is equivalent to the field equation
obtained by varying the same $f(Q)$ action term with respect to the affine
connection (recall that the symmetric teleparallel theory is necessarily a
metric-affine theory), but the situation does not improve any further, the
model parameters has to satisfy this second field equation now. In the present
scenario, however, we do not have any additional burden arising from this. The
result can also be shown directly from the equations of motion (14).
Differentiating the energy equation of (14) with respect to time variable, we
obtain

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle-\frac{\dot{Q}}{2}f_{Q}-Q\dot{Q}f_{QQ}$ (19)
$\displaystyle=f_{Q}\left[3\dot{\xi}Q+\frac{\ddot{\chi}}{\dot{\chi}}Q-\frac{\dot{Q}}{2}\right],\text{
due to }(\ref{fQ})$
$\displaystyle=6f_{Q}\left[3(\dot{\chi}^{2}-\dot{\xi}^{2})+\ddot{\xi}-\dot{\xi}\frac{\ddot{\chi}}{\dot{\chi}}\right]\dot{\xi}.$

On the other hand, using (17) in (14) we also have

$\displaystyle
p+\rho=-2f_{Q}\left[3(\dot{\chi}^{2}-\dot{\xi}^{2})+\ddot{\xi}-\dot{\xi}\frac{\ddot{\chi}}{\dot{\chi}}\right]$
(20)

which combined to yield the usual continuity relation
$\dot{\rho}=-3\dot{\xi}(p+\rho)$. Hence, one can instead use the set of three
independent equations (the density equation, one of the pressure equations and
the relation (16)) with five ($f(Q),\rho,p,\xi,\chi$) unknowns. Clearly we
need two physically legitimate assumptions to obtain exact solutions. Other
than the barotropic equation of state $p=\omega\rho$, the most legitimate
assumption is $\sigma^{2}\propto\theta^{2}$, which has been widely used
earlier in different Bianchi models [40, 41, 42, 43, 44, 45, 46].
Additionally, it has been found that if a relation between the metric
coefficients are assumed in orthogonal, locally rotationally-symmetric and
spatially-homogeneous Bianchi II, VIII and IX metrics, the relation
$\sigma^{2}\propto\theta^{2}$ automatically follows [47]. Finally, de-Sitter
solution in modified theory of gravity associated with higher order curvature
invariant terms, also leads to the same relation $\sigma^{2}\propto\theta^{2}$
[48, 49]. However under such assumption, here one obtains,

$\theta^{2}\propto\sigma^{2},~{}~{}\mathrm{i.e.,}~{}~{}~{}\dot{\xi}=k\dot{\chi}.$
(21)

As a result,

$\xi=k\chi+k_{1},~{}~{}\mathrm{and}~{}~{}Q=-6(k^{2}-1)\dot{\chi}^{2}=-6\dot{\xi}^{2}\left({{k^{2}-1}\over
k^{2}}\right).$ (22)

Now, in view of equation (18), one finds,

$\rho-3p=2f+6\dot{\xi}^{2}\left({{k^{2}-1}\over k^{2}}\right)f_{Q}=2f-Qf_{Q}.$
(23)

Hence in the radiation era $p={1\over 3}\rho$, $f(Q)=f_{0}Q^{2}$ is the
allowed form for $f(Q)$, and ‘General theory of Relativity’ can not be
retrieved. Further, substituting (21) and (22) in the pressure equation, one
obtains $f(Q)=0$, both in the pressure-less dust era ($p=0$) as well as in the
early vacuum dominated era ($p=\rho=0$). Furthermore, in the case of pure
vacuum ($\rho=p=0$), the first equation of the set (14) yields
$f(Q)\propto\sqrt{Q}$. Note that, a non-negative $Q$ warrants the expansion
rate is smaller than the shear scalar ($\theta^{2}<\sigma^{2}$) throughout the
cosmic evolution, and the universe never isotropizes. In this context, we
recall that such uncanny behaviour is absent from $f(R)$ theories of gravity,
which admits de-Sitter solution in pure vacuum, and therefore has much rich
structure than $f(Q)$ gravity. However, in order to avert such pathologies, we
need either to withdraw the assumption (21) or to modify the right hand side
of the field equations. As already mentioned, the assumption (21) has largely
been used in the literature earlier, and also appears automatically in some
cases, so the simplest modifications are either to associate viscosity and/or
a scalar field. In the present work, we consider a scalar field $\phi$, which
is required to drive inflation in the very early universe. The independent set
of field equations are therefore,

$\begin{split}&{f\over 2}-Qf_{Q}=\rho+{1\over 2}\dot{\phi}^{2}+V(\phi),\\\
&{f\over
2}+2f_{Q}(\ddot{\xi}+\ddot{\chi}+3\dot{\xi}^{2}+3\dot{\xi}\dot{\chi})+2\dot{Q}f_{QQ}(\dot{\xi}+\dot{\chi})=-p-{1\over
2}\dot{\phi}^{2}+V,\\\ &f_{Q}\dot{\chi}=c_{1}e^{-3\xi}.\end{split}$ (24)

Additionally, we have the Bianchi identity and the Klein-Gordon equation,
which are,

$\begin{split}&\dot{\rho}+3\dot{\xi}(\rho+p)=0,\\\
&\ddot{\phi}+3\dot{\xi}\dot{\phi}+V^{\prime}(\phi)=0.\end{split}$ (25)

Combining, we also find

$\rho-3p-\dot{\phi}^{2}+4V=2f+6f_{Q}\left[\ddot{\xi}+\dot{\xi}^{2}-\dot{\xi}{\ddot{\chi}\over\dot{\chi}}-\dot{\chi}^{2}\right].$
(26)

###  3.1 Energy condition:

Before we proceed further, it is necessary to fix the signature of the
coefficients $(\alpha~{}\mathrm{and}~{}\beta)$ associated with two different
models $f(Q)=\beta Q^{2}$ and $f(Q)=\alpha Q+\beta Q^{2}$, that we shall
consider to study inflation and else. This is possible in view of energy
conditions. We remind that the essence of all the modified theories of gravity
including the symmetric telleparallel geometry under present consideration,
developed so far, is to resolve the cosmic puzzle without the need of dark
energy. Thus, although scalar field has been considered to study inflation, it
must not remain present at the end. Hence much of the scalar field should be
redshifted during radiation era, and all we shall left with is the barotropic
fluid in the pressureless dust era. In addition, the universe should turn out
to be isotropic, as par current observation, hence contribution from shear
should be considerably low. For perfect fluid
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}$, the energy conditions are:
1\. Null energy condition: $\rho+p\geq 0$.
2\. Weak energy condition: $\rho+p\geq 0$ and $\rho\geq 0$.
3\. Dominant energy condition:$\rho\geq|p|$.
4\. Strong energy condition:$\rho+p\geq 0$ and $\rho+3p\geq 0$.
It is important to mention that $p<0$ is allowed, and strong energy condition
does not automatically satisfy the weak energy condition. However, if both
$\rho>0$, and $p>0$, all energy conditions are satisfied simultaneously.

For the present purpose, the field equations (24) may be cast in the following
manner:

$\begin{split}&\rho={f\over 2}-Qf_{Q}-\rho_{\phi},\\\
&\rho+p=-Qf_{Q}-2c_{1}e^{-3\xi}{d\over
dt}\left(H\over\sigma\right)-(\rho_{\phi}+p_{\phi}),\\\
&\rho+3p=-f(Q)-Qf_{Q}-6c_{1}e^{-3\xi}{d\over
dt}\left(H\over\sigma\right)-(\rho_{\phi}+3p_{\phi}).\end{split}$ (27)

In the above, $\rho_{\phi}={1\over 2}\dot{\phi}^{2}+V(\phi)$, and
$p_{\phi}={1\over 2}\dot{\phi}^{2}-V(\phi)$. Since in the matter dominated
era, $\dot{\xi}=H>0$ while, $\dot{\chi}=\sigma$ may be positive or negative,
if not zero, the sign of the second term appearing on the right hand side of
both the second and the third equations of (LABEL:EC) remains arbitrary.
However, as the universe is large enough ($e^{3\xi}=a^{3}$), this term
contributes little, and as long as the assumption (21) holds, the term
vanishes.

1\. Now for ‘General theory of Relativity’, $f(Q)=\alpha
Q=\alpha(2\sigma^{2}-6H^{2})$. Hence, $\rho=-{\alpha Q\over 2}-\rho_{\phi}$,
$\rho+p=-\alpha Q-2c_{1}e^{-3\xi}{d\over
dt}\left(H\over\sigma\right)-(\rho_{\phi}+p_{\phi})$ and $\rho+3p=-2\alpha
Q-6c_{1}e^{-3\xi}{d\over
dt}\left(H\over\sigma\right)-(\rho_{\phi}+3p_{\phi})$. Hence, only if
anisotropy is small, $\sigma^{2}<3H^{2}$, then $Q<0$, and all the energy
conditions may be satisfied for $\alpha>0$ ($\alpha={1\over 16\pi G}$, say in
the unit $\hbar=c=1$), and contribution from the scalar field, in particular,
is small.

2\. Next let us consider, $f(Q)=\beta Q^{2}$. Clearly, $\rho=-{3\over 2}\beta
Q^{2}-\rho_{\phi}$ indicates $\beta<0$. Further, $\rho+p=-2\beta
Q^{2}-2c_{1}e^{-3\xi}{d\over
dt}\left(H\over\sigma\right)-(\rho_{\phi}+p_{\phi})>0$, and $\rho+3p=-3\beta
Q^{2}-6c_{1}e^{-3\xi}{d\over
dt}\left(H\over\sigma\right)-(\rho_{\phi}+3p_{\phi})>0$ also require
$\beta<0$, and little contribution from the scalar field.

3\. Finally, for $f(Q)=\alpha Q+\beta Q^{2}$, $\rho=-{\alpha\over 2}Q-{3\over
2}\beta Q^{2}-\rho_{\phi}$, $\rho+p=-\alpha Q-2\beta
Q^{2}-2c_{1}e^{-3\xi}{d\over
dt}\left(H\over\sigma\right)-(\rho_{\phi}+p_{\phi})$ and $\rho+3p=-2\alpha
Q-3\beta Q^{2}-6c_{1}e^{-3\xi}{d\over
dt}\left(H\over\sigma\right)-(\rho_{\phi}+3p_{\phi})$. Clearly again, our
earlier choice, $\alpha>0$ and $\beta<0$ ensure $\rho>0$, throughout cosmic
evolution, provided contribution from the scalar field remains subdominant.

In a nutshell, energy conditions in all the three cases are satisfied
provided, $\alpha>0$, $\beta<0$ and $|\alpha Q|>2\rho_{\phi}$, $|\alpha
Q|>\rho_{\phi}+p_{\phi}$ hold. It is worth mentioning that the sign of $c_{1}$
may be fixed from the solution of the field equations.

Let us emphasize the fact that, if most of the scalar is used up in creating
particles, at the end of inflation and during re-heating, while the rest is
redshifted in the following radiation dominated era, by some means, then
matter-dominated era, would be left without the scalar field. Field equations
for $f(Q)=\alpha Q-\beta Q^{2}$, then can be formulated in terms of the
effective energy density and the effective pressure in the following forms:

$\begin{split}&-{\alpha\over 2}Q=\rho-{3\over 2}\beta Q^{2}=\rho_{e},\\\
&{\alpha\over 2}Q+2\alpha\dot{\chi}{d\over
dt}\left({\dot{\xi}\over\dot{\chi}}\right)=-{1\over 2}\beta Q^{2}-2\beta
Q\dot{\chi}{d\over
dt}\left({\dot{\xi}\over\dot{\chi}}\right)=-p_{e},\end{split}$ (28)

so that, the effective equation of state $\omega_{e}$ reads as:

$\omega_{e}={p_{e}\over\rho_{e}}={{1\over 2}\beta Q^{2}+2\beta
Q\dot{\chi}{d\over dt}\left({\dot{\xi}\over\dot{\chi}}\right)\over\rho-{3\over
2}\beta Q^{2}}.$ (29)

Since, the second term in the numerator may possibly be negative, so early
deceleration followed by acceleration in the matter dominated era may be
envisaged.

###  3.2 In search of vacuum de-Sitter solution:

Let us now seek de-Sitter solution in the form $a=e^{\xi}=a_{0}e^{\Lambda
t},~{}\mathrm{i.e.},~{}\dot{\xi}=\Lambda$, under the assumption,

$\sigma^{2}\propto\theta^{2},~{}\mathrm{for~{}which},~{}\dot{\xi}=k\dot{\chi}.$
(30)

Note that in the case of $f(R)$ gravity, de-Sitter solution enforces a
constant Ricci scalar. Likewise, under the above choice (30),
$Q=-{6(k^{2}-1)\over k^{2}}\Lambda^{2}$ also becomes a constant. In the vacuum
dominated era ($p=0=\rho$), the set of field equations (24) and (25) simplify
to:

$\begin{split}&{f\over 2}-Qf_{Q}={1\over 2}\dot{\phi}^{2}+V(\phi),\\\ &{f\over
2}=-{1\over 2}\dot{\phi}^{2}+V(\phi),\\\ &f_{Q}\dot{\chi}=c_{1}e^{-3\xi},\\\
&\ddot{\phi}+3\dot{\xi}\dot{\phi}+V^{\prime}(\phi)=0.\end{split}$ (31)

Note that the Klein-Gordon equation is not an independent one, and so we now
have three independent equations with ($f(Q),\chi,\phi,V(\phi)$) four
unknowns. From the first pair we find,

$(Qf_{Q})\dot{\chi}=-\dot{\phi}^{2}\dot{\chi},~{}~{}\mathrm{or,}~{}~{}\dot{\phi}^{2}=6c_{1}(k^{2}-1)\dot{\chi}e^{-3\xi},~{}~{}\mathrm{or,}~{}~{}\dot{\phi}^{2}=6c_{1}\left({k^{2}-1\over
k}\right)\dot{\xi}e^{-3\xi},$ (32)

in view of the assumption (30). Now as de-Sitter solution is imposed, we find,

$\dot{\phi}^{2}=6c_{1}\Lambda\left({{k^{2}-1}\over
ka_{0}^{3}}\right)e^{-3\Lambda t}.$ (33)

At this point, we recall that the convention adopted in the present study
leads to $Q=-6H^{2}<0$ in homogeneous and isotropic Robertson-Walker space-
time. Hence, if shear is not too large from the beginning,
$3\sigma^{2}<\theta^{2}$, implying $k>1$, and so $c_{1}>0$. Now, (33) may be
integrated to yield

$\phi=\left[\phi_{0}-{2\over 3\Lambda}\sqrt{{6\Lambda c_{1}(k^{2}-1)\over
ka_{0}^{3}}}e^{-{3\over 2}\Lambda t}\right].$ (34)

Now under repeated differentiation of (34) we find,

$\dot{\phi}={3\Lambda\over
2}(\phi_{0}-\phi),~{}~{}~{}\mathrm{and}~{}~{}~{}\ddot{\phi}=-{9\over
4}\Lambda^{2}(\phi_{0}-\phi).$ (35)

Therefore we can find the form of the potential from the Klein-Gordon equation
as:

$V^{\prime}=-{9\over
4}\Lambda^{2}(\phi-\phi_{0}),~{}\mathrm{and~{}so}~{}~{}V(\phi)=-{9\over
8}\Lambda^{2}\phi^{2}+{9\over 4}\Lambda^{2}\phi_{0}\phi+V_{0}.$ (36)

Unfortunately,

$f(Q)=-{9\over 4}\Lambda^{2}\phi_{0}^{2}-{9\over
2}\Lambda^{2}\phi^{2}+9\Lambda^{2}\phi_{0}\phi+2V_{0},$ (37)

cannot be expressed in terms of $Q$ alone. In any case there appears a
contradiction, since for $Q=$constant, $f(Q)$ must also be a constant, while
here $f(Q)$ evolves with time along with the scalar field $\phi$. In a host of
modified gravity models, de-Sitter solution is admissible under the choice
$\phi=\phi_{0}e^{-\Lambda t}$ in isotropic metric [50, 51, 52, 53, 54, 55, 56,
57], as well as in anisotropic Bianchi metrics [48, 49]. However, such
assumption here leads to imaginary $\chi(t)$. In a nutshell, a legitimate de-
Sitter solution is not admissible in $f(Q)$ theory of gravity. This is
definitely yet another serious pathology of the theory.

Nonetheless, we can still study inflation with exponential expansion, since
while studying inflation, our assumption of de-Sitter solution is relaxed, and
$\dot{\xi}$ is assumed to vary slowly, instead of being a constant
($\Lambda$). Thus we can choose arbitrary form of the potential and $f(Q)$.
Further slow roll inflation requires a flat potential, which we shall choose
for the purpose.

##  4 Scalar field inflation

Despite some alternative suggestions [64, 65, 66], Inflation has been
contemplated as a scenario, since it is the most viable model that can resolve
the issues like flatness, horizon, monopole and structure formation problems
singlehandedly, which were left unanswered by FLRW model. Recent analysis of
the combined data (TT, TE, EE + lowE +lensing + BK15 + BAO+ Bicep2) puts up
tight constrains on inflationary parameters, viz., the scalar to tensor ratio
$(r<0.055)$ and (scalar) spectral index $(0.9631<n_{s}<0.9705)$, along with a
very small tensor spectral index $n_{t}$ [58],[59]. It is already known that
to solve the horizon and the flatness problems, the number of e-folds $N$
should preferably lie within the range $45<N<65$. In fact, a more recent
analysis [60] suggested even tighter constraint on scalar to tensor ratio
$(r<0.032)$. In this section, we consider a vacuum-dominated
$(p_{m}=0;~{}~{}\rho_{m}=0)$ scalar field driven inflation in the very early
universe, find the inflationary parameters and compare with data sets, to test
viability of the action (7). The effective Friedmann equation, can be written
as,

$\displaystyle\left[{\dot{\phi}^{2}\over
2}+V(\phi)-\frac{f}{2}+Qf_{Q}\right]=0,$ (38)

and the Klein Gordon equation can be written in its standard form as,

$\displaystyle\ddot{\phi}+3H{\dot{\phi}}+V^{\prime}(\phi)=0.$ (39)

###  4.1 $f(Q)=-\beta Q^{2},~{}\beta>0$:

Since we have found $f(Q)\propto Q^{2}$ in radiation era, so we first check
its outcome in inflationary regime. Under the above quadratic form of $f(Q)$,
the effective Friedmann equation (38) reads as,

$\displaystyle-54\beta\frac{(k^{2}-1)^{2}}{k^{4}}H^{4}+\left[{\dot{\phi}^{2}\over
2}+V(\phi)\right]=0.$ (40)

Now, enforcing the standard slow-roll conditions $\dot{\phi}^{2}\ll V(\phi)$
and $|\ddot{\phi}|\ll 3{H}|\dot{\phi}|$, equations (40) and (39), reduce to,

$\displaystyle-54\beta\frac{(k^{2}-1)^{2}}{k^{4}}H^{4}+V(\phi)=0,~{}~{}~{}~{}~{}~{}3{H}\dot{\phi}\simeq-V^{\prime}.$
(41)

Solving $H^{2}$ from (41) one obtains,

$\displaystyle
H^{2}={\frac{k^{2}}{\sqrt{54}(k^{2}-1)}\sqrt{V(\phi)\over\beta}}.$ (42)

Now, combining the pair of equations (41), one can show that the potential
slow roll parameter $\epsilon$ equals the Hubble slow roll parameter
($\epsilon_{1}$) under the condition,

$\displaystyle\begin{split}\epsilon\equiv-{\dot{H}\over{H}^{2}}=\frac{\sqrt{3\beta}(k^{2}-1)V^{\prime
2}}{2{\sqrt{2}}k^{2}V^{3\over 2}};\hskip
21.68121pt\eta=2\alpha\left({V^{\prime\prime}\over V}\right);\end{split}$ (43)

where $\epsilon\ll 1$, during inflation. The primordial spectral index of
scalar perturbation $(n_{s})$, the tensor-to-scalar ratio $(r)$, and the
tensor spectral index $(n_{t})$ may now be expressed as,

$\displaystyle
n_{s}=1-6\epsilon+2\eta,~{}~{}~{}~{}r=16\epsilon,~{}~{}~{}n_{t}=-{r\over
8}=-2\epsilon,$ (44)

where, $|n_{s}|<1$ ensures deviation from scale invariance. Further, since
$\frac{H}{\dot{\phi}}=-\frac{k^{2}\sqrt{V(\phi)}}{\sqrt{6\beta}(k^{2}-1)V^{\prime}}$,
therefore, the number of e-folds may be computed as follows:

$\displaystyle{N}(\phi)\simeq\int_{t_{i}}^{t_{f}}{H}dt=\int_{\phi_{i}}^{\phi_{f}}\frac{{H}}{\dot{\phi}}d\phi\simeq\int_{\phi_{f}}^{\phi_{i}}\Big{(}\frac{k^{2}\sqrt{V(\phi)}}{\sqrt{6\beta}(k^{2}-1)V^{\prime}}\Big{)}d\phi,$
(45)

where, $\phi_{i}$ and $\phi_{f}$ denote the scalar fields at the beginning
$(t_{i})$ and at the end $(t_{f})$ of inflation. Let us choose the potential
in the following form, $V(\phi)=V_{0}-{V_{1}\over\phi}$, so that the potential
remains almost flat as $\phi$ is large, and slow roll is admissible. Now, as
$V(\phi)=V_{0}-{V_{1}\over\phi}$, $V^{\prime}(\phi)={V_{1}\over\phi^{2}}$ and
$V^{\prime\prime}(\phi)=-{2V_{1}\over\phi^{3}}$, then the equations (43) and
(45) reduces to,

$\displaystyle\begin{split}\epsilon=\frac{\sqrt{3\beta
V_{1}}(k^{2}-1)}{2{\sqrt{2}}k^{2}\phi^{5\over
2}(\frac{V_{0}}{V_{1}}\phi-1)^{3\over 2}},\hskip
36.135pt\eta=-\frac{2M_{P}^{2}}{\phi^{2}(\frac{V_{0}}{V_{1}}\phi-1)},\hskip
36.135ptN=\int_{\phi_{f}}^{\phi_{i}}\Big{[}\frac{k^{2}\phi^{3\over
2}(\frac{V_{0}}{V_{1}}\phi-1)^{1\over 2}}{\sqrt{6\beta
V_{1}}(k^{2}-1)}\Big{]}d\phi.\end{split}$ (46)

$\phi_{i}$ in $M_{P}$ | $n_{s}$ | $r$ | ${N}$
---|---|---|---
4.00 | 0.96277 | 0.00403 | 45
4.05 | 0.96422 | 0.00383 | 47
4.10 | 0.96559 | 0.00363 | 49
4.15 | 0.96889 | 0.00345 | 51
4.20 | 0.96813 | 0.00328 | 53
4.25 | 0.96930 | 0.00312 | 55
4.30 | 0.97042 | 0.00297 | 57
4.35 | 0.97148 | 0.00283 | 59

Table 1: Data set for the inflationary parameters with
$\phi_{f}=0.702211~{}M_{P}$, $\beta V_{1}=1~{}M_{P}^{5}$, ${V_{0}\over
V_{1}}=2~{}M_{P}^{-1}$ and $k=1.15$, varying $\phi_{i}$.

In Table 1, we present a data set for the inflationary parameters, varying
$\phi_{i}$ between $4.00M_{P}\leq\phi_{i}\leq 4.35M_{P}$, so that $r$ and
$n_{s}$ lie within the experimental limit. Note that slow rollover terminates
$(\epsilon=1)$ at $\phi_{f}=0.702211M_{P}$ in every case. Further, the number
of e-folds $45\leq N\leq 59$ found is sufficient to solve the horizon and
flatness problems. Since data fits wonderfully with current observation, let
us now proceed to find the energy scale of inflation in view of the relation
(42). Taking into consideration the data set of Table 1, associated with
$N=57$, for which $\phi_{i}=4.30M_{P}$, we find

$\displaystyle{H_{*}}^{2}=\frac{1.3225{\sqrt{\beta
V_{1}\left(\frac{V_{0}}{V_{1}}-\frac{1}{\phi_{i}}\right)}}}{{\sqrt{5}4}(1.3225-1)\beta}=\frac{1.75826}{2.3699\beta}=\frac{0.74191}{\beta},~{}~{}~{}~{}\text{or}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}H_{*}=\sqrt{\frac{0.77131}{\beta}},$
(47)

where we have considered $k=1.15$, ${V_{0}\over V_{1}}=2M_{P}^{-1}$ and $\beta
V_{1}=1M_{P}^{5}$. Note that $\beta$ still remains arbitrary. Now, the energy
scale of inflation in a single scalar field model corresponding to GTR is
given by the expression [63] as,

$\displaystyle{H_{*}}=8\times 10^{13}\sqrt{\frac{r}{0.2}}GeV=9.76\times
10^{12}GeV\approx 3.93\times 10^{-6}M_{P}.$ (48)

In the above computation, we have used the data $r=0.00297$ of Table 1. Thus,
in order to match the scale of inflation (47) with the single field scale of
inflation (48), $\beta$ has to be of the order, $\beta\approx 4.80\times
10^{10}$. Note that, we have fixed only two parameters $\beta V_{1}$ and
$V_{0}\over V_{1}$, for the present inflationary model. However, with the
above consideration (sub-Planckian scale of inflation), the parameter $\beta$
is fixed. Further, since individually each term has to be of the same order of
magnitude, one can compare the two terms of (42) to find that:

$\displaystyle V(\phi)=V_{1}\left({V_{0}\over
V_{1}}-{1\over\phi}\right)=1.76V_{1}=3.678\times 10^{-11}M_{P}^{4},$ (49)

implying

$\displaystyle V_{1}\approx 2.09\times
10^{-11}M_{P}^{5},~{}~{}~{}~{}~{}~{}V_{0}\approx 4.18\times
10^{-11}M_{P}^{4}.$ (50)

Thus, all the parameters are fixed once and forever.

Finally, we need to test whether the model gracefully exits from inflation.
Thus, with the form of the potential, $V(\phi)=(V_{0}-{V_{1}\over\phi})$, with
$\beta$ negative, equation (40) may be expressed as,

$\displaystyle{54(k^{2}-1)^{2}H^{4}\over
k^{4}V_{1}}=\frac{1}{\beta}\left[{\dot{\phi}^{2}\over 2V_{1}}+{V_{0}\over
V_{1}}-{1\over\phi}\right].$ (51)

During inflation, $H^{2}$ and $V_{1}$ are of the same order of magnitude,
while Hubble parameter varies slowly. However, at the end, Hubble rate usually
decreases sharply, and $H^{4}$ falls much below $V_{1}$. Hence, $H^{4}\over
V_{1}$ may be neglected without loss of generality. Thus, one can write

$\displaystyle{\dot{\phi}}^{2}=-2\left[V_{0}-{V_{1}\over\phi}\right].$ (52)

Let us choose the following oscillatory form of $\phi$ to check if the
consequence is physically admissible.

$\displaystyle\phi=\exp({i\omega t}).$ (53)

Therefore, equation (52) reduces to,

$\displaystyle\omega^{2}\phi^{2}=2\left(V_{0}-{V_{1}\over\phi}\right).$ (54)

Hence, with $V_{0}=4.18\times 10^{-11}M_{P}^{4},~{}~{}V_{1}=2.09\times
10^{-11}M_{P}^{5},~{}\phi_{f}=0.702211M_{P}$, one finds $\omega=6.98\times
10^{-6}M_{P}$. Clearly, $\phi$ exhibits oscillatory behaviour, provided
$\omega=6.98\times 10^{-6}M_{P}$ .

###  4.2 $f=\alpha Q-\beta Q^{2},\alpha>0,~{}\beta>0:$

In this case, for vacuum era $(p_{m}=0;~{}~{}\rho_{m}=0)$ the Friedmann
equation may be expressed as,

$\displaystyle{\dot{\phi}^{2}\over 2}+V(\phi)-\left(\frac{\alpha Q-\beta
Q^{2}}{2}\right)+(\alpha-2\beta Q)Q=0.$ (55)

Again using the equation (21) the above equation reduces to,

$-54\beta H^{4}\left({{k^{2}-1}\over k^{2}}\right)^{2}-3\alpha
H^{2}\left({{k^{2}-1}\over k^{2}}\right)+\left[{\dot{\phi}^{2}\over
2}+V(\phi)\right]=0,$ (56)

and the Klein Gordon equation can be written in its regular form as,

$\displaystyle\ddot{\phi}+3H{\dot{\phi}}+V^{\prime}(\phi)=0.$ (57)

We now enforce the standard slow-roll conditions $\dot{\phi}^{2}\ll V(\phi)$
and $|\ddot{\phi}|\ll 3{H}|\dot{\phi}|$, on equations (56) and (57), which
thus finally reduce to,

$\displaystyle-54\beta H^{4}\left({{k^{2}-1}\over k^{2}}\right)^{2}-3\alpha
H^{2}\left({{k^{2}-1}\over
k^{2}}\right)+V(\phi)=0,~{}~{}~{}~{}~{}~{}3{H}\dot{\phi}\simeq-V^{\prime}.$
(58)

Solving for $H^{2}$ in view of (58), we readily obtain,

$\displaystyle H^{2}=\frac{k^{2}\left[-\alpha\pm\sqrt{\alpha^{2}+24\beta
V(\phi)}\right]}{36\beta(k^{2}-1)}.$ (59)

Further, combining equations (58), one can show that the ‘potential slow roll
parameter’ $\epsilon$ equals the ‘Hubble slow roll parameter’ ($\epsilon_{1}$)
under the condition,

$\displaystyle\begin{split}\epsilon\equiv-{\dot{H}\over{H}^{2}}=\frac{72\beta^{2}(k^{2}-1)V^{\prime
2}}{k^{2}\left(\sqrt{\alpha^{2}+24\beta
V(\phi)}\right)\left[-\alpha\pm\sqrt{\alpha^{2}+24\beta
V(\phi)}\right]^{2}};\hskip
21.68121pt\eta=2\alpha\left({V^{\prime\prime}(\phi)\over
V(\phi)}\right).\end{split}$ (60)

Further, since
$\frac{H}{\dot{\phi}}=-\frac{k^{2}\left[-\alpha\pm\sqrt{\alpha^{2}+24\beta
V(\phi)}\right]}{12\beta(k^{2}-1)V^{\prime}}$, one can compute the number of
e-folds as,

$\displaystyle{N}(\phi)\simeq\int_{t_{i}}^{t_{f}}{H}dt=\int_{\phi_{i}}^{\phi_{f}}{{H}\over{\dot{\phi}}}d\phi=\int_{\phi_{f}}^{\phi_{i}}\frac{k^{2}\left[-\alpha\pm\sqrt{\alpha^{2}+24\beta
V(\phi)}\right]}{12\beta(k^{2}-1)V^{\prime}}d\phi.$ (61)

Let us now make the same choice of the potential as in the case-1, viz.,
$V(\phi)=V_{0}-{V_{1}\over\phi}$, so that it remains almost flat as $\phi$ is
large, and slow roll is admissible. Therefore, following two sets for the
expressions of $\epsilon,~{}\eta$ (60) and $N$ (61) are found,

$\displaystyle\begin{split}&\epsilon=\frac{72\beta^{2}(k^{2}-1){V_{1}}^{2}}{k^{2}\phi^{4}\left(\sqrt{\alpha^{2}+24\beta
V_{1}\left(\frac{V_{0}}{V_{1}}-\frac{1}{\phi}\right)}\right)\left[-\alpha+\sqrt{\alpha^{2}+24\beta
V_{1}\left(\frac{V_{0}}{V_{1}}-\frac{1}{\phi}\right)}\right]^{2}};\hskip
21.68121pt\eta=-\frac{2M_{P}^{2}}{\phi^{2}(\frac{V_{0}}{V_{1}}\phi-1)},\\\
&N=\int_{\phi_{f}}^{\phi_{i}}\frac{k^{2}\phi^{2}\left[-\alpha+\sqrt{\alpha^{2}+24\beta
V_{1}\left(\frac{V_{0}}{V_{1}}-\frac{1}{\phi}\right)}\right]}{12\beta(k^{2}-1)V_{1}}d\phi.\end{split}$
(62)
$\displaystyle\begin{split}&\epsilon=\frac{72\beta^{2}(k^{2}-1){V_{1}}^{2}}{k^{2}\phi^{4}\left(\sqrt{\alpha^{2}+24\beta
V_{1}\left(\frac{V_{0}}{V_{1}}-\frac{1}{\phi}\right)}\right)\left[-\alpha-\sqrt{\alpha^{2}+24\beta
V_{1}\left(\frac{V_{0}}{V_{1}}-\frac{1}{\phi}\right)}\right]^{2}};\hskip
21.68121pt\eta=-\frac{2M_{P}^{2}}{\phi^{2}(\frac{V_{0}}{V_{1}}\phi-1)},\\\
&N=\int_{\phi_{f}}^{\phi_{i}}\frac{k^{2}\phi^{2}\left[-\alpha-\sqrt{\alpha^{2}+24\beta
V_{1}\left(\frac{V_{0}}{V_{1}}-\frac{1}{\phi}\right)}\right]}{12\beta(k^{2}-1)V_{1}}d\phi.\end{split}$
(63)

corresponding to a plus (62) and minus (63) sign appearing before the square
root. In Table-2, we present a data set for the set of expressions (62), while
Table-3 is presented for (63). In both the tables, we have varied ${V_{0}\over
V_{1}}$ between $4.4M_{P}^{-1}\leq\phi_{i}\leq 5.6M_{P}^{-1}$, so that $r$ and
$n_{s}$ lie within the experimental limit. Further, the number of e-folds for
Table-2 and Table-3 are found to vary with in the range $48\leq N\leq 56$ and
$53\leq N\leq 61$ respectively, which are sufficient to solve the horizon and
flatness problems. Clearly, the fit is outstanding, since the scalar to tensor
ratio is able to sustain further constraints, which might appear from future
analysis.

$\phi_{f}$ in $M_{P}$ |  ${V_{0}\over V_{1}}$ in $M_{P}^{-1}$ | $n_{s}$ | $r$ | ${N}$
---|---|---|---|---
0.37422 | 5.6 | 0.97128 | 0.00156 | 56
0.38181 | 5.4 | 0.97014 | 0.00166 | 54
0.38990 | 5.2 | 0.96889 | 0.00176 | 53
0.39856 | 5.0 | 0.96755 | 0.00188 | 52
0.40787 | 4.8 | 0.96607 | 0.00201 | 51
0.41788 | 4.6 | 0.96447 | 0.00216 | 50
0.42872 | 4.4 | 0.96270 | 0.00233 | 48

Table 2: Data set (62) for the inflationary parameters with
$\phi_{i}=3.0M_{P}$, $\beta V_{1}=1.0M_{P}^{5}$, $\alpha=0.5M_{P}^{2}$ and
$k=1.08$ under the variation of ${V_{0}\over V_{1}}$.

$\phi_{f}$ in $M_{P}$ |  ${V_{0}\over V_{1}}$ in $M_{P}^{-1}$ | $n_{s}$ | $r$ | ${|N|}$
---|---|---|---|---
0.35797 | 5.6 | 0.97138 | 0.00130 | 61
0.36494 | 5.4 | 0.97024 | 0.00138 | 60
0.37237 | 5.2 | 0.96901 | 0.00146 | 58
0.38031 | 5.0 | 0.96767 | 0.00156 | 57
0.38883 | 4.8 | 0.96621 | 0.00166 | 56
0.39780 | 4.6 | 0.96461 | 0.00177 | 55
0.40790 | 4.4 | 0.96286 | 0.00190 | 53

Table 3: Data set (63) for the inflationary parameters with
$\phi_{i}=3.0M_{P}$, $\beta V_{1}=1.0M_{P}^{5}$, $\alpha=0.5M_{P}^{2}$ and
$k=1.08$ under the variation of ${V_{0}\over V_{1}}$.

Let us now compute the energy scale of inflation in view of the relation (59),
considering the last but one data, ($N=50$, for which
$\phi_{i}=3.0M_{P},~{}\alpha=0.5,~{}k=1.08,~{}{{V_{0}\over
V_{1}}=4.6M_{P}^{-1}},~{}\beta V_{1}=1.0M_{P}^{5}$), appearing in Table 2.
Correspondingly, we compute,

$\displaystyle{H_{*}}^{2}=\frac{1.1664\left[-\alpha\pm\sqrt{\alpha^{2}+24\beta
V_{1}\left({V_{0}\over
V_{1}}-{1\over\phi_{i}}\right)}\right]}{36\beta(1.1664-1)}=\frac{1.1664(-0.5\pm
10.13)}{-5.990\beta}=\frac{1.875}{\beta},$ (64)

where positive value of ${H_{*}}^{2}$ is taken. Note that $\beta$ still
remains arbitrary. Now, the energy scale of inflation in a single scalar field
model corresponding to GTR [63] is given by the following expression,

$\displaystyle{H_{*}}=8\times 10^{13}\sqrt{\frac{r}{0.2}}GeV=8.314\times
10^{12}GeV\approx 3.34\times 10^{-6}M_{P}.,$ (65)

whose numerical value is computed taking into account the value of the tensor-
to-scalar ratio $r=0.00216$ from the last but one data set of Table 2. Thus,
in order to match the scale of inflation (64) with the single field scale of
inflation (65), $\beta$ has to be, $\beta\approx 1.681\times 10^{11}$. We have
therefore been able to fix the parameter $\beta$, from physical ground (sub-
Planckian inflation), and hence the values of $V_{1}$ and $V_{0}$ may be found
as well, which are,

$\displaystyle V_{1}=\frac{1.0}{1.68\times 10^{11}}\approx 5.95\times
10^{-12}M_{P}^{5},\hskip 36.135ptV_{0}=4.6V_{1}\approx 2.74\times
10^{-11}M_{P}^{4}.$ (66)

Finally, to handle the issue of gracefully exit from inflation, we recall
equation (56), which in view of the above form of the potential,
$V(\phi)=V_{0}-{V_{1}\over\phi}$, is expressed as,

$\displaystyle-\frac{54\beta H^{4}(k^{2}-1)^{2}}{V_{1}k^{4}}-\frac{3\alpha
H^{2}(k^{2}-1)}{V_{1}k^{2}}+\left[{\dot{\phi}^{2}\over
2V_{1}}+\left({V_{0}\over V_{1}}-{1\over{\phi}}\right)\right]=0.$ (67)

During inflation, $H^{2}$ and $V_{1}$ are of the same order of magnitude,
while the Hubble parameter varies slowly. But, at the end of inflation, the
Hubble rate usually decreases sharply, and $\beta H^{4}$ falls much below
$V_{1}$. Hence, one can neglect both the terms ${\beta H^{4}\over V_{1}}$ and
${\alpha H^{2}\over V_{1}}$ without any loss of generality. Thus, one can
write

$\displaystyle{\dot{\phi}}^{2}=-2\left[V_{0}-{V_{1}\over\phi}\right].$ (68)

Taking into account, $\phi_{f}=0.41788M_{P}$, $V_{0}=2.74\times
10^{-11}M_{P}^{4}$ and $V_{1}=5.95\times 10^{-12}M_{P}^{5}$ from Table-2, it
is possible to show that the above equation exhibits oscillatory behavior

$\displaystyle\phi=\exp({i\omega t}),$ (69)

provided, $\omega\approx 1.23\times 10^{-5}M_{P}$. It may be mentioned that
such astounding fit was also found for
$f(\mathbb{T})=\alpha\mathbb{T}+\beta{\mathbb{T}}^{2}$ teleparallel gravity
theory in isotropic and homogeneous model [67]. As mentioned in the
introduction, $f(\mathbb{T})$ and $f(Q)$ in coincidence gauge (adopted here)
are indistinguishable in homogeneous and isotropic model, so the result holds
for $f(Q)$ gravity, as well.

##  5 Radiation era

No doubt, the $f(Q)$ theory fits the latest released inflationary data with
extremely good precession. It is therefore required to scrutinize its validity
in the subsequent epoch. After graceful exit from inflation, the universe re-
heats and reaches the temperature ($\sim 10^{32}~{}K$) of hot big-bang, whence
the universe enters the radiation dominated era. In this era, The seeds of
perturbation which enter the horizon at the end of inflation, evolve to form
structure. Further, CMB is also formed in this epoch. These are possible only
if the universe evolves with decelerated expansion. So let us seek a solution
in the form,

$a(t)=e^{\xi}=a_{0}t^{n},~{}\mathrm{i.e.}~{}\dot{\xi}={n\over t}.$ (70)

Next, let us inspect under what condition the Klein-Gordon equation (25) is
validated, which currently takes the following form, for the same choice of
the potential $V(\phi)=V_{0}-{V_{1}\over\phi}$, as

$\ddot{\phi}+3{n\over t}\dot{\phi}+{V_{1}\over\phi^{2}}=0.$ (71)

The above equation is satisfied for $l={2\over 3}$, if we seek a solution in
the form $\phi=\phi_{0}t^{l}$. This means $\phi\propto t^{2\over 3}$, and
$\phi$ increases with time. Nothing to worry about it, since $V_{1}\sim
10^{-11}~{}M_{P}^{5}$ (50) , and $V_{1}\sim 10^{-12}~{}M_{P}^{5}$ (66), and
therefore, soon the last term $V_{1}\over\phi^{2}$ becomes insignificantly
small and may be neglected, leaving only $\ddot{\phi}+3{n\over
t}\dot{\phi}=0$, behind. This admits a solution in the form

$\phi=\phi_{0}t^{1-3n},$ (72)

and so for $3n<1$, the remnant of $\phi$ is red-shifted away and there might
not remain any trace of a scalar field in the low energy limit. In this manner
the cosmic puzzle may be solved through $f(Q)$ gravity, avoiding dark energy
issue.

###  5.1 $f(Q)\propto Q^{m}$:

We have already noticed that $f(Q)=-\beta Q^{2}$ fits with inflationary
parameter with excellence. So, in this subsection, we make our way to test its
behaviour in the radiation dominated era, starting from an arbitrary power
$m$. So, let us take up equation (16), viz., $f_{Q}\dot{\chi}=c_{1}e^{-3\xi}$,
and consider the earlier assumption $\dot{\xi}=k\dot{\chi}$ (21) to obtain:

$f_{Q}\dot{\xi}=kc_{1}e^{-3\xi},$ (73)

which in view of (70), implies

$f_{Q}={kc_{1}\over
na_{0}^{3}}t^{1-3n},~{}~{}\mathrm{i.e.},~{}f_{Q}\propto{1\over t^{3n-1}}.$
(74)

Further considering, $f(Q)=f_{0}Q^{m}$, we find

$f_{Q}=mf_{0}Q^{m-1}=mf_{0}[-6(k^{2}-1)]^{m-1}\dot{\chi}^{2(m-1)}=mf_{0}\left[-{6(k^{2}-1)\over
k^{2}}\right]^{m-1}\dot{\xi}^{2(m-1)};~{}~{}\mathrm{i.e.},~{}f_{Q}\propto{1\over
t^{2(m-1)}},$ (75)

using solution (70). Thus, equations (74) and (75) simultaneously hold
provided, $3n-1=2m-2$ or $n={1\over 3}(2m-1)$. Now for $m=1,~{}n={1\over 3}$,
and $f(Q)=f_{0}Q$, which is simply ‘General Theory of Relativity’. The
effective scale factor therefore evolves as $a(t)=a_{0}t^{1\over 3}$, while
the scalar field seizes to evolve in view of (72). Note that in reference to
equation (26), the potential vanishes. This is the solution obtained earlier
with viscous fluid in the B-I space-time, in the absence of a scalar field
[42]. This gives slower expansion rate than the standard Friedmann model
($a\propto\sqrt{t}$). On the contrary, if we seek $a\propto\sqrt{t}$, as in
the Friedmann model, then $m={5\over 4}$, i.e. $f=f_{0}Q^{5\over 4}$, which is
also not much promising. Particularly for $m=2$, which we have considered to
exhibit outstanding fit with observed inflationary parameters, leads to
coasting solution $a\propto t$ and for $m>2$ accelerating expansion is
revealed. Therefore despite wonderful fit with inflationary parameters,
$f(Q)\propto Q^{2}$ is not a good choice to explain radiation dominated era.

###  5.2 $f(Q)=\alpha Q-\beta Q^{2}$:

From the above analysis, it is clear that the assumption
$\sigma^{2}\propto\theta^{2}$ does not allow a combination of two terms
($f(Q)=\alpha Q-\beta Q^{2}$) as considered in the present subsection. We
therefore withdraw this assumption as a result of which $\xi(t)$ and $\chi(t)$
are no longer related. Hence, we are required to solve the set of equations
(24). Again we start with our master equation (16) as before, viz,
$f_{Q}\dot{\chi}=c_{1}e^{-3\xi}$, and use solution (70) to find,

$\alpha\dot{\chi}+12\beta\left({n^{2}\over
t^{2}}-\dot{\chi}^{2}\right)\dot{\chi}-{c_{1}\over a_{0}^{3}t^{3n}}=0.$ (76)

Equation (76) is satisfied under the condition:

$\dot{\chi}={\chi_{0}\over t};~{}\chi_{0}=n={1\over 3};~{}\alpha={c_{1}\over
a_{0}^{3}\chi_{0}}={3c_{1}\over a_{0}^{3}};~{}a=a_{0}t^{1\over 3}.$ (77)

Note that in the process we arrive at $\dot{\chi}^{2}=\dot{\xi}^{2}$, leading
to $Q=0$. Further, for $n={1\over 3}$, the scalar field (72) seizes to evolve,
and so it would appear as dark energy in the matter dominated era. In this
process we get tangled up, since all the development in regard of symmetric
telleparallel theory to resolve cosmic puzzle without dark energy ended in the
requirement of dark energy, yet again. In a nutshell, $f(Q)$ theory of gravity
fails to produce a viable cosmological evolution in the radiation era.

##  6 Concluding remarks

Resolving the cosmic puzzle without dark energy, requires to modify GTR.
Several such modified theories of gravity have been developed in the last two
decades. Recent hearsay being the ‘teleparallel gravity theory’. The primary
advantage of ‘teleparallel gravity theory’ is that the field equations are
second order unlike $f(R)$ theory of gravity. This avoids Ostrogradski
instability and associated ghost degrees of freedom. However, Lovelock gravity
also has the same characteristics, but, it is applicable either in higher
dimensions ($D>4$), or requires to associate a dilatonic coupled scalar field
in $4$ dimension. Two types of teleparallism have been investigated in recent
years, the metric $f(\mathbb{T})$ and the symmetric $f(Q)$ teleparallel
gravity theories. Models that are based on $f(\mathbb{T})$ have significant
coupling problems, which are generically absent in $f(Q)$ theories. There is
an unsettling presence of skew-symmetric elements in the field equations of
$f(\mathbb{T})$ theories, which is not the case in $f(Q)$ or $f(R)$ or in the
majority of other theories of gravity. This may be seen when one takes an even
deeper look at the $f(\mathbb{T})$ theories. As a consequence of this,
$f(\mathbb{T})$ theories (with the exception of those in which
$f(\mathbb{T})=T$) are not locally Lorentz invariant and have more degrees of
freedom in comparison to GTR. This is a significant setback when one proposes
to establish $f(\mathbb{T})$ theory as an alternative of GTR. On a side by
side comparison, the $f(\mathbb{T})$ connections are much more restrictive
than $f(Q)$, as we see from the symmetries of the torsion and non-metricity
tensor components. We have made the following observations after a thorough
investigation of the comparatively new $f(Q)$ gravity:

  * •

In the present article we have studied $f(Q)$ theory of gravity in the
anisotropic yet homogeneous axially symmetric Bianchi-I model, under the
coincident gauge coordinates. Although its remarkable fit with recently
released inflationary parameters is encouraging; unfortunately, it miserably
fails to envisage a decelerated expansion in the radiation dominated era,
which follows inflationary regime and is responsible to produce CMBR.

  * •

In the case of $f(\mathbb{T})$ theories of gravity, it has been shown that its
second order field equations is rather a disadvantage as it inherits the
unresolved issue of ‘Branched Hamiltonian’ [68]. The authors attempted to
bypass this issue by adding a curvature squared term $(R^{2})$ in the action
and thus extending the theory to essentially a $f(R,\mathbb{T})$ class of
theories. To note that, the $R$ in the mentioned paper [68] is the Ricci
scalar corresponding to the Levi-Civita connection which we denote here by
$\mathring{R}$. This is a delicate and very risky technique as it involves
working with two separate connections at every step. The modified symmetric
teleparallel gravity seems to suffer from the same unresolved disease, which
has no definite cure yet and may only be bypassed by considering an extension
like $f(\mathring{R},Q)$ theory.

  * •

There is a continuing dispute on whether geodesics of the Levi-Civita
connection or autoparallel of the flat affine connection, should be utilised
as the worldline of a test particle in the neighbourhood of a gravitating
source. Although an experimental confirmation is still pending, an
investigation of the most natural trajectory of a test particle has been
conducted from a theoretical perspective in [69].

  * •

Another contentious topic that, at least for the time being, appears to have
been resolved is the question of how the second clock effect in parallel
transported objects in teleparallel geometry (both metric and symmetric) works
[70, 71, 72].

However, it is too early to discard the ‘symmetric teleparallel theory’, since
the coincident gauge connections that we have worked with, and most other
literature also adapted to, is not the unique connection to formulate the
$f(Q)$ theory [73]. In future works we hope to derive all the connections in
the Bianchi type-I spacetime which are compatible with symmetric
teleparallelism, and associate shear and bulk viscosities, to further
investigate the effect in the early (inflationary epoch) and mid-stage
evolution (radiation dominated epoch) of the universe.

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