# Tensor Perturbations from Bounce Inflation Scenario in $f(Q)$ Gravity

Kun Hu<EMAIL_ADDRESS>School of Physics, Huazhong University of
Science and Technology, Wuhan, 430074, China  Institute of Astrophysics,
Central China Normal University, Wuhan 430079, China  Tanmoy Paul
<EMAIL_ADDRESS>National Institute of Technology Jamshedpur, Department
of Physics, Jamshedpur - 831 014, India  Taotao Qiu<EMAIL_ADDRESS>(corresponding author) School of Physics, Huazhong University of Science and
Technology, Wuhan, 430074, China

###### Abstract

In this paper, we construct a bounce inflation cosmological scenario in the
framework of the modified symmetric teleparallel gravity, namely $f(Q)$
theory, and investigate the tensor perturbations therein. As is well-known,
the tensor perturbations generated in the very early Universe (inflation and
pre-inflation regions) can account for the primordial gravitational waves
(PGWs) that are to be detected by the next generation of GW experiments. We
discuss the stability condition of the tensor perturbations in the bounce
inflation process and investigate in detail the evolution of the perturbation
variable. The general form of the tensor power spectrum is obtained both for
large as well as small scale modes. As a result, we show for both kinds of
modes (short or long wavelength modes), the tensor spectrum may get a positive
tilt in the parametric range where the tensor perturbation proves to be stable
— this interestingly hints an enhancement of gravitational waves’ amplitude in
the background of the $f(Q)$ bounce-inflation scenario. Moreover, we study the
LQC-like scenario as a specific case of our model, in which, the primordial
tensor power spectrum turns out to be nearly scale-invariant on both small and
large scales.

##  I Introduction

It is well-known that our Universe, described by the Standard Big Bang Theory,
suffered from the notorious problem of Cosmic Singularity [1] at the time of
its birth. More precisely, as has been proved by S. Hawking et al. in the
1970s, the Universe will inevitably meet the singularity if it is 1) described
by 4D classical General Relativity, 2) satisfying the Null Energy Condition
(NEC) [2, 3]. In the vicinity of the singularity, physical quantities such as
energy density blow up, therefore failing to present the real nature.
Nonetheless, one of the simple and feasible ways of avoiding such a
singularity is to have the universe start with a contracting phase, and bounce
to the expanding one as we observed [4, 5]. Such a scenario is usually dubbed
the “cosmic bounce”.

To realize a cosmic bounce, at least one of the two conditions mentioned above
needs to be broken. It might be more straightforward to break the NEC by
including some exotic matter such as ghost field [6], but in order to avoid
releasing the ghost instability, the models should be delicately designed.
There are examples made of ghost condensate mechanism [7] or Horndeski
theories [8, 9, 10, 11, 12]. However, one can also obtain the bounce by
modifying Einsein’s General Relativity, namely referring to modified gravity
theories, where the effective NEC is also broken. This includes non-minimal
coupling bounce [13, 14], $f(R)$ bounce [15, 16], $f(R,\mathcal{G})$ bounce
[17, 18], $f(T)$ bounce [19, 20, 21], $f(Q)$ bounce [22, 23, 24, 25, 26],
Loop-quantum bounce [27] and so on.

Compared to other modified gravity theories, the bounce scenario in $f(Q)$
theory has been less studied, especially, the perturbation analysis of $f(Q)$
bounce scenario is far from sufficient. This motivates the work in the current
paper. The $f(Q)$ theory is one of the interesting alternative theories of
gravity, which extends from a metric theory to a metric-affine theory [28]. It
is well-known that in GR, the affine connection
$\varGamma^{\lambda}_{~{}\mu\nu}$, and therefore the Ricci scalar $R$ of the
gravity action, can be written in terms of metric only, making GR a pure
metric theory. (In such case, $\varGamma$ is known the Christoffel symbol).
This is because of two conditions: 1) torsionlessness, namely the connection
is symmetric,
$\varGamma^{\lambda}_{~{}\mu\nu}=\varGamma^{\lambda}_{~{}\nu\mu}$. 2) metric
compatibility, namely the covariant derivative on metric vanishes,
$\nabla_{\lambda}g_{\mu\nu}=0$. However, if one breaks one of the two
conditions, the metric and connection become independent of each other, and
the theory becomes metric-affine theory, in which the metric and the affine
connection are both independent fundamental variables. To be precise, if one
breaks condition 1) while keeping condition 2), the theory will become a
theory with torsion, with asymmetric connection; In contrast, if one breaks
condition 2) while maintaining condition 1), one will get another gravity
theory with symmetric connection and no torsion but instead have non-
metricity.

Nonetheless, both the two kinds of gravity theories can be made equivalent to
GR, which has obtained great success at least in the range of low energy. For
torsion theory, the action can be made of the torsion scalar $T$:

$S=\int
d^{4}x\,eT~{},~{}T\equiv\frac{1}{4}T_{\lambda\mu\nu}T^{\lambda\mu\nu}+\frac{1}{2}T_{\lambda\mu\nu}T^{\mu\lambda\nu}-T_{\lambda}T^{\lambda}~{},$
(1)

where we define the torsion tensor:
$T^{\lambda}_{~{}\mu\nu}\equiv\varGamma^{\lambda}_{~{}\mu\nu}-\varGamma^{\lambda}_{~{}\nu\mu}$,
which is to describe how the hypersurfaces twist about a vector after a
parallel transportation, and $T_{\mu}\equiv T^{\lambda}_{~{}\mu\lambda}$, $e$
is the volume element given by the tetrad. This is dubbed as Teleparrallel
Equivalent of General Relativity (TEGR) [29]. Similarly, for non-metricity
theory, the action can be made of the nonmetricity scalar $Q$:

$S=\int
d^{4}x\,\sqrt{-g}Q~{},~{}Q\equiv\frac{1}{2}Q_{\mu\nu\lambda}Q^{\lambda\mu\nu}-\frac{1}{4}Q_{\mu\nu\lambda}Q^{\mu\nu\lambda}+\frac{1}{4}Q_{\lambda}Q^{\lambda}-\frac{1}{2}Q_{\lambda}\tilde{Q}^{\lambda}~{},~{}$
(2)

where we define the non-metricity tensor:
$Q_{\lambda\mu\nu}\equiv\nabla_{\lambda}g_{\mu\nu}$, which is to describe the
variation of the norm of vector after parallel transportation, and its traces
$Q_{\lambda}\equiv g^{\mu\nu}Q_{\lambda\mu\nu}$, $\tilde{Q}_{\lambda}\equiv
g^{\mu\nu}Q_{\nu\mu\lambda}$. This is known to be Symmetric Teleparrallel
Equivalent of General Relativity (STEGR) [30]. It has been proved in [31, 32]
that these two actions, as well as the Einstein-Hilbert action of GR, only
differ by a total derivative from each other, therefore they can share many of
the physical properties and phenomenological predictions. However, if the
actions are furtherly modified to be any functions of these scalars, this is
no longer the case because the total derivatives inside the functions can no
longer be thrown out. Therefore, the modified gravity theories $f(R)$, $f(T)$
and $f(Q)$ are completely different theories, thus may give rise to very
different (and interesting) phenomenons.

There has already been a lot of discussion on $f(Q)$ theory and its various
applications. For the development of the theory itself, we see [33] for the
construction of a conformal invariant theory, and [34, 35] for a
generalization to the so-called general teleparallel quadratic tensor
theories. Also see [36] for a discussion on the covariant formulation. For
cosmological applications, [37, 38, 39, 40, 41, 42, 43, 44, 45, 46]
investigated various cosmological evolutions, while [47, 48] studied the
cosmological perturbations. Especially, the gravitational waves generated by
the theory are studied in [49, 50, 51, 52, 53]. Moreover, there are a lot of
constraints on this theory from observations, see [54, 55, 56, 57]. See also
[58, 59] for comprehensive reviews.

Recently, in [60], some of our authors studied the physical degrees of freedom
of the perturbative $f(Q)$ theory in coincident gauge, by performing the
Hamiltonian analysis approach. Our result shows that different from the usual
GR theory, the $f(Q)$ theory in such a gauge can have many more additional
degrees of freedom (up to 8), which might be attributed to the violation of
the general covariance. The additional degrees of freedom are basically scalar
and vector ones, which are still under investigation. Nevertheless, it is much
more evident that only two tensor degrees of freedom remain, since there is no
room for redundant ones, which can be seen just from the Scalar-Vector-Tensor
(SVT) decomposition. Therefore the tensor perturbations of this theory can be
easily determined. Moreover, with the emergence of more and more Primordial
Gravitational Waves (PGW) experiments such as NANOGrav [61], AliCPT [62, 63],
CMB-S4 [64] and LiteBIRD [65], the investigation of tensor perturbations of
the early universe models becomes more and more striking. Recently released
data of Pulsar Timing Array from NANOGrav 15yr [66], EPTA [67], PPTA [68] and
CPTA [69] shows the hint of stochastic GW background at 2 to $4.6\sigma$
level. Therefore, in the following, we will investigate the construction of a
bounce (inflation) scenario in the framework of $f(Q)$ theory, as well as the
stability of the tensor perturbation and its evolution process through the
bounce.

The contents are arranged as follows. In Sec. II we briefly review the basic
formulation of $f(Q)$ theory. In Sec. III we construct a model from $f(Q)$
theory with bouncing behavior at the background level. Sec. IV devotes itself
to the perturbations generated in such a bounce model. Sec. V comes to the
conclusions and outlook. Throughout the paper, we work in units with
$c=\hbar=1$ and for simplicity, we set the gravitational constant
$\kappa^{2}=8\pi G_{N}=1$.

##  II the $f(Q)$ theory

The most general action of $f(Q)$ gravity is given by the following expression
[47, 37, 70]

$S=\int d^{4}x\,\sqrt{-g}\left[\frac{1}{2}f(Q)+\mathcal{L}_{m}\right]~{},$ (3)

where $g$ is the determinant of the metric $g_{\mu\nu}$, $f(Q)$ is an
arbitrary function of the nonmetricity scalar $Q$ appeared in (2) and
$\mathcal{L}_{m}$ stands for the matter lagrangian density. We consider the
general linear affine connection as

$\varGamma^{\alpha}_{\,\mu\nu}=\left\\{{}^{\,\alpha}_{\mu\nu}\right\\}+K_{\,\mu\nu}^{\alpha}+L_{\,\mu\nu}^{\alpha}~{}.$
(4)

Here, the first term of the right-hand side of the above equation is the well-
known Levi-Civita connection

$\left\\{{}^{\,\alpha}_{\mu\nu}\right\\}=\frac{1}{2}g^{\alpha\lambda}\left(\partial_{\mu}g_{\lambda\nu}+\partial_{\nu}g_{\mu\lambda}-\partial_{\lambda}g_{\mu\nu}\right)~{},~{}$
(5)

the second term is the contortion tensor $K_{\,\mu\nu}^{\alpha}$, which is a
function of the torsion tensor $T^{\lambda}_{\,\mu\nu}$

$K^{\alpha}_{\,\mu\nu}=\frac{1}{2}g^{\alpha\lambda}\left(T_{\lambda\mu\nu}+T_{\mu\nu\lambda}-T_{\nu\lambda\mu}\right)~{},$
(6)

and the last term $L_{\mu\nu}^{\alpha}$ is disformation tensor, and it is
composed of nonmetricity tensor $Q_{\lambda\mu\nu}$

$\displaystyle
L^{\alpha}_{\,\mu\nu}=-\frac{1}{2}g^{\alpha\lambda}\left(Q_{\mu\lambda\nu}+Q_{\nu\lambda\mu}-Q_{\lambda\mu\nu}\right)~{}.$
(7)

The nonmetricity tensor is defined by

$Q_{\alpha\mu\nu}\equiv\hat{\nabla}_{\alpha}g_{\mu\nu}=\frac{\partial
g_{\mu\nu}}{\partial
x^{\alpha}}-g_{\nu\sigma}{\varGamma}_{\mu\alpha}^{\sigma}-g_{\sigma\mu}{\varGamma}_{\nu\alpha}^{\sigma}~{},$
(8)

where $\hat{\nabla}_{\alpha}$ denotes the covariant derivative associated with
the general affine connection. Meanwhile, the space-time curvature is
described by the usual Riemann tensor

$R_{\beta\mu\nu}^{\alpha}(\varGamma)=\partial_{\mu}\varGamma_{\nu\beta}^{\alpha}-\partial_{\nu}\varGamma_{\mu\beta}^{\alpha}+\varGamma^{\alpha}_{\mu\lambda}\varGamma_{\nu\beta}^{\lambda}-\varGamma_{\nu\lambda}^{\alpha}\varGamma_{\mu\beta}^{\lambda}~{}.$
(9)

Symmetric teleparallelism, by definition, requires free curvature and torsion,
namely $R_{\beta\mu\nu}^{\alpha}(\varGamma)\stackrel{{\scriptstyle!}}{{=}}0$
and $T_{\mu\nu}^{\alpha}=K_{\mu\nu}^{\alpha}\stackrel{{\scriptstyle!}}{{=}}0$
respectively. Thereby, the form of general affine connection has been
restricted as following

$\varGamma_{\,\mu\nu}^{\alpha}=\frac{\partial
x^{\alpha}}{\partial\xi^{\lambda}}\partial_{\mu}\partial_{\nu}\xi^{\lambda}~{}.$
(10)

Equation (10) turns out to be a pure-gauge connection and hence it is always
possible to make it vanish by utilizing general coordinate transformation to
set $\xi^{\lambda}=x^{\lambda}$. This gauge choice was named the coincident
gauge.

Substituting Eq. (4) (with $K_{\mu\nu}^{\alpha}=0$) into Eq. (9) one can find
that the Ricci scalar of the Einstein-Hilbert action $\mathring{R}$, composed
of the Levi-Civita connection (5), can be expressed by the nonmetricity scalar
plus a total divergence of two independent traces of the nonmetricity tensor,
which become the boundary term after integration

$\mathring{R}=Q-{\nabla}_{\alpha}\left(Q^{\alpha}-\tilde{Q}^{\alpha}\right)~{}.~{}$
(11)

It is also useful to introduce the so-called nonmetricity conjugate or
superpotential tensor

$P^{\alpha}{}_{\mu\nu}\equiv\frac{1}{2}\frac{\partial Q}{\partial
Q_{\alpha}{}^{\mu\nu}}=-\frac{1}{4}Q^{\alpha}{}_{\mu\nu}+\frac{1}{2}Q_{\left(\mu\nu\right)}^{\alpha}+\frac{1}{4}g_{\mu\nu}Q^{\alpha}-\frac{1}{4}\left(g_{\mu\nu}\tilde{Q}^{\alpha}+\delta^{\alpha}_{(\mu}Q_{\nu)}\right)~{},$
(12)

and the equation of motion can directly be obtained by taking the variation of
the action towards metric $g_{\mu\nu}$, namely

$\frac{2}{\sqrt{-g}}\nabla_{\alpha}\left(\sqrt{-g}f_{Q}P_{\
\mu\nu}^{\alpha}\right)+\frac{1}{2}g_{\mu\nu}f+f_{Q}\left(P_{\mu\alpha\beta}Q_{\nu}^{\
\alpha\beta}-2Q_{\alpha\beta\mu}P_{\ \ \
\nu}^{\alpha\beta}\right)=T_{\mu\nu}~{},$ (13)

where the energy-momentum tensor of the matter is defined by

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\sqrt{-g}\mathcal{L}_{m}}{\delta
g^{\mu\nu}}~{}.$ (14)

Here and thereafter, $f_{Q}$ stands for the derivative of a function $f(Q)$
with respect to nonmetricity scalar $Q$. In particular, with the help of
$P^{\alpha\lambda}{}_{\lambda}=(Q^{\alpha}-\tilde{Q}^{\alpha})/2$, the
relation (11) and the identity

$\nabla_{\alpha}\sqrt{-g}=\left(2\sqrt{-g}\right)^{-1}\frac{\partial
g}{\partial
g_{\mu\nu}}\nabla_{\alpha}g_{\mu\nu}=\frac{\sqrt{-g}}{2}Q_{\alpha}~{},$ (15)

we can rewrite the Einstein tensor in GR in terms of superpotential and
nonmetricity tensor [71],

$\stackrel{{\scriptstyle\circ}}{{G}}_{\mu\nu}=2\nabla_{\lambda}P^{\lambda}{}_{\mu\nu}-\frac{1}{2}Qg_{\mu\nu}+P_{\rho\mu\nu}Q^{\rho\sigma}{}_{\sigma}+P_{\nu\rho\sigma}Q_{\mu}{}^{\rho\sigma}-2P_{\rho\sigma\mu}Q^{\rho\sigma}{}_{\nu}~{},~{}$
(16)

Plugging the relation (16) and into Eq. (13), the field equation can be
written into a more compact form [36]:

$2f_{QQ}P_{\mu\nu}^{\alpha}\partial_{\alpha}Q+\frac{1}{2}g_{\mu\nu}\left(f-f_{Q}Q\right)+f_{Q}\stackrel{{\scriptstyle\circ}}{{G}}_{\mu\nu}=T_{\mu\nu}~{}.~{}$
(17)

In the case of STEGR, i.e., $f(Q)=Q$, the above equation reduces to Einstein’s
equation in GR, thus the degrees of freedom from the affine connection do not
enter into the field equation. But in a more general case, the $f(Q)$ is not
the linear function of nonmetricity scalar, which gives a totally different
evolution of the physical system from GR.

##  III Bounce Model within $f(Q)$ gravity

###  III.1 Background equations

In bouncing cosmology, the Universe begins from an epoch in the contracting
phase in the far past, when it contracts to a very small but non-zero scale,
and enters the expansion phase through a bounce, bridging with the thermal
history of the expansion of the Universe that we observe today. Throughout the
process, the scale factor of the Universe never reaches zero, and thus
provides a possible way to avoid the singularity problem of the Universe.

To begin with, we would have a brief analysis of the cosmological background
evolution, considering the flat Friedmann-Lemaitre-Robertson-Walker (FLRW)
background with the metric

$ds^{2}=-dt^{2}+a^{2}\left(t\right)\delta_{ij}dx^{i}dx^{j}~{},~{}$ (18)

where $a(t)$ is the scale factor. On the other hand, if we regard the
background of the Universe as a perfect fluid, then the stress-energy tensor
takes the form of

$T_{\,\,\nu}^{(m)\mu}=\left(\rho_{m}+p_{m}\right)u^{\mu}u_{\nu}+p_{m}\,\delta_{\,\,\nu}^{\mu}~{},$
(19)

where $\rho_{m}$ and $p_{m}$ are the energy density and pressure of the fluid,
while $u^{\mu}=(1,0,0,0)$ is the fluid’s four-velocity. More explicitly, we
can write stress-energy tensor into a matrix form

$T^{(m)}_{\mu\nu}=\left[\begin{array}[]{cccc}\rho_{m}&0&0&0\\\
0&a^{2}(t)p_{m}&0&0\\\ 0&0&a^{2}(t)p_{m}&0\\\
0&0&0&a^{2}(t)p_{m}\end{array}\right]~{}.$ (20)

In the contracting epoch, the “velocity” of the scale factor $\dot{a}(t)$ is
negative, which causes the Hubble parameter $H\equiv\dot{a}/{a}<0$. However,
it turns positive in the expanding phase. That means we must have $H=0$,
$\dot{H}>0$ for the cosmological turnaround, which is exactly the bouncing
point. In the case of Einstein’s gravity with some matter fields, we have

$\frac{\dot{H}}{H^{2}}=-\frac{3}{2}\left(\omega_{m}+1\right)~{},~{}$ (21)

where $\omega_{m}\equiv p_{m}/\rho_{m}$ is the equation of state (EoS) of the
matter field under consideration, it shows that only with $\omega_{m}=-\infty$
has the power to produce the bounce, which clearly violates all the energy
conditions. This clearly reflects that in the case of GR, the Universe can not
achieve bounce without the presence of some extremely exotic matter. However,
it is not the case in the framework of $f(Q)$ gravity, since the geometric
effects of space-time can be interpreted as effective matter that violates the
NEC.

For the $f(Q)$ gravity, we take the action form as in Eq. (3) and work in the
coincident gauge $\varGamma_{\ \mu\nu}^{\lambda}\equiv 0$, in which we have
$Q_{\alpha\mu\nu}=\partial_{\alpha}g_{\mu\nu}$. Using metric (18) the
equations of motion (17) becomes

$\displaystyle 6f_{Q}H^{2}-\frac{1}{2}f$ $\displaystyle=\rho_{m},$ (22)
$\displaystyle\left(12H^{2}f_{QQ}+f_{Q}\right)\dot{H}$
$\displaystyle=-\frac{1}{2}(\rho_{m}+p_{m})~{}.~{}$

Comparing Eq. (22) to the Friedmann equation in GR, we can rewrite the
effective Friedmann equations as $3H^{2}=\rho_{\text{eff}}$ and
$2\dot{H}+3H^{2}=-p_{\text{eff}}$, where $\rho_{\text{eff}}$ and
$p_{\text{eff}}$ are effective density and pressure of the total fluid:

$\displaystyle\rho_{\text{eff}}$
$\displaystyle=\frac{1}{2f_{Q}}\left(\rho_{m}+\frac{1}{2}f\right)~{},$ (23)
$\displaystyle p_{\text{eff}}$
$\displaystyle=\frac{1}{f_{Q}}\left[12H^{2}f_{QQ}\dot{H}-\frac{f}{4}+\frac{p_{m}}{2}-\dot{H}f_{Q}\right]~{}.$

Moreover, the EoS parameter of matter is $\omega_{m}=p_{m}/\rho_{m}$, which
satisfies the continuity equation

$\overset{\cdot}{\rho_{m}}+3H\rho_{m}\left(1+\omega_{m}\right)=0~{}.~{}$ (24)

It is also worth mentioning that, the relation between the nonmetricity scalar
and Hubble parameter is simply

$Q=6H^{2}(t)~{},~{}$ (25)

which can be explicitly deduced from the definition of nonmetricity scalar (2)
for the background metric (18). Therefore, one can make use of the Friedmann
equations (22) to solve $f(Q)$ in terms of $H$. Define $f_{H}\equiv
f_{Q}dQ/dH$ to replace $f_{Q}$ in the first equation in (22), one gets

$f_{H}=\frac{1}{H}f(H)+\frac{2\rho_{m}(H)}{H}~{},$ (26)

the general solution of which is given by,

$f(H)=H\left[\sqrt{6}A+2\int\frac{\rho_{m}(H)}{H^{2}}\text{d}H\right]~{}.~{}$
(27)

Here $A$ is an integration constant. However, in order to get the exact
expression of $f$ in terms of $Q$ or $H$, the dependence of $\rho_{m}$ on $H$
must be known as well.

###  III.2 Reconstructing the bounce action

In order to realize a bounce scenario in this $f(Q)$ gravity, we apply the
reconstruction approach and start with a specific, simple but quite natural
scale factor with the form

$a\left(t\right)=a_{B}\left(1+\alpha\,t^{2}\right)^{\frac{1}{3(\gamma+1)}}~{},~{}$
(28)

where $a_{B}$ is the scale factor at the bouncing point, $\alpha$ is a
positive parameter that describes how fast the bounce takes place, and
$\gamma$ is a constant parameter. The corresponding Hubble parameter is given
by

$H(t)=\frac{\overset{\cdot}{a}}{a}=\frac{2\alpha
t}{3\left(1+\gamma\right)\left(1+\alpha t^{2}\right)}~{}.~{}$ (29)

We plot the Hubble parameter as well as the nonmetricity scalar in terms of
$t$ in Fig. 1.

Figure 1: The evolution plots of the Hubble parameter $H$ (left panel) and
nonmetricity scalar $Q$ (right panel) in terms of the cosmic time $t$ for the
scale fator (28) (the parameters are chosen as $\alpha=9$ and $\gamma=0$ for
both plots). The Hubble parameter and nonmetricity scalar are both zero at the
bounce point and meet their critical value $H_{\max}/H_{\min}$ and $Q_{\max}$
at $t=\pm 1/\sqrt{\alpha}$ respectively. When $t\rightarrow\infty$ the Hubble
parameter and nonmetricity scalar both asymptotically go to zero.

Now, we can define the effective equation of state (EoS) as
$\omega_{\text{eff}}=p_{\text{eff}}/\rho_{\text{eff}}$, so that Eq. (21) still
holds with $\omega_{m}\rightarrow\omega_{eff}$. Plugging the expression of
Hubble parameter (29) into Eq. (21), we have

$\omega_{\text{eff}}\left(t\right)=\left(1+\gamma\right)\left(1-\frac{1}{\alpha
t^{2}}\right)-1~{}.$ (30)

We immediately see that for the late time Universe, i.e., $|t|\gg
1/\sqrt{\alpha}$, we have the relation $\omega_{\text{eff}}\approx\gamma$,
while when the Universe is near the bounce point, i.e., $t\rightarrow 0$,
$\omega_{\text{eff}}$ goes to $-\infty$.

With the help of parametrization form (29), one can firstly express the
inverse relation of cosmic time $t$ in terms of $H$ as:

$\displaystyle t_{1}$ $\displaystyle=\frac{\alpha-\sqrt{\alpha^{2}-9\alpha
H^{2}-18\alpha H^{2}\gamma-9\alpha H^{2}\gamma^{2}}}{3(\alpha H+\alpha
H\gamma)}~{},$ (31) $\displaystyle t_{2}$
$\displaystyle=\frac{\alpha+\sqrt{\alpha^{2}-9\alpha H^{2}-18\alpha
H^{2}\gamma-9\alpha H^{2}\gamma^{2}}}{3(\alpha H+\alpha H\gamma)}~{}.$

The solutions have been separated into two branches. In the first branch
$t_{1}$ increases (or decreases) with increasing (or decreasing) Hubble
parameter, while the scenario is reversed in the second branch. From the Fig.
1 one can see that, for $-1\leqslant\sqrt{\alpha}\,t\leqslant 1$, the relation
$t=t(H)$ is given by the first branch; while for
$-\infty<\sqrt{\alpha}\,t\leqslant-1$ and $1\leqslant\sqrt{\alpha}\,t<\infty$
the relation $t=t(H)$ is given by the second branch.

Note that the straightforward solution of energy conservation equation (24)
gives the evolution of matter

$\rho_{m}(H)=\rho_{B}\left(\frac{a_{B}}{a(t)}\right)^{3(\omega_{m}+1)}~{},~{}$
(32)

with $\rho_{B}$ its value at the bouncing point. Plugging the scale factor
(28) and the expression (31) into Eq. (32), for the time branches $t_{1}$ and
$t_{2}$, the corresponding energy densities evolve as

$\displaystyle\rho_{m1}\left(H\right)$
$\displaystyle=\rho_{B}\left(\frac{Q_{m}-Q_{m}\sqrt{1-6H^{2}/Q_{m}}}{3H^{2}}\right)^{n}~{},$
(33) $\displaystyle\rho_{m2}\left(H\right)$
$\displaystyle=\rho_{B}\left(\frac{Q_{m}+Q_{m}\sqrt{1-6H^{2}/Q_{m}}}{3H^{2}}\right)^{n}~{}.~{}$

Here, for simplicity, we introduce factors
$n\equiv-(1+\omega_{m})/(1+\gamma)$, and $Q_{m}\equiv
2\alpha/[3\left(1+\gamma\right)^{2}]$ which denotes the maximum value of $Q$
(see Fig. 1).

The interesting fact is that if we choose $\gamma=\omega_{m}=0$ and
$\rho_{B}=4\alpha/3$, the evolution Eq. (33) takes the form

$\rho_{m}\left(H\right)=\frac{6H^{2}}{1+\sqrt{1-9H^{2}/\alpha}}~{},$ (34)

which has the same form as the effective Friedmann equation in the context of
Loop Quantum Cosmology (LQC) [72]

$3H^{2}=\rho_{m}\left(1-\frac{\rho_{m}}{\rho_{B}}\right)~{}.~{}$ (35)

For $\alpha\rightarrow\infty$, it leads to $\rho_{B}\rightarrow\infty$ and
thus Eq. (35) reduces to the Friedmann equation in GR. The above analysis
illustrates our model reduces to LQC when we take the special case $n=-1$. For
the sake of generality, we keep the parameter $n$ arbitrary in the subsequent
calculation.

Inserting Eq. (33) and Eq. (29) into expression (27), we can get the form of
$f=f(H)$ in the present context. Finally, utilizing the relation (25) again,
the reconstructed Lagrangian of $f(Q)$ is obtained:

For $-1\leqslant\sqrt{\alpha}\,t\leqslant 1$:

$\displaystyle
f\left(Q\right)=A\sqrt{Q}-\rho_{B}2^{n+1}\left(1+\sqrt{1-\hat{Q}}\right)^{-n}$
$\displaystyle+\sqrt{2}\rho_{B}n\hat{Q}\left(1+\sqrt{1-\hat{Q}}\right)^{-\frac{1}{2}}$
(36)
$\displaystyle\times\text{2F1}\left[\frac{1}{2},\frac{3}{2}+n,\frac{3}{2},\frac{1}{2}-\frac{1}{2}\sqrt{1-\hat{Q}}\,\right]~{},~{}$

For $-\infty<\sqrt{\alpha}\,t\leqslant-1$ and
$1\leqslant\sqrt{\alpha}\,t<\infty$:

$\displaystyle
f\left(Q\right)=A\sqrt{Q}-\rho_{B}\left(\frac{1+\sqrt{1-\hat{Q}}}{\hat{Q}}\right)^{n}$
$\displaystyle\left\\{2^{n+1}+n\left(\sqrt{1-\hat{Q}}\right)^{n}\sqrt{\hat{Q}}\right.$
(37)
$\displaystyle\left.\times\text{Beta}\left[\frac{1}{2}-\frac{1}{2}\sqrt{1-\hat{Q}},-\frac{1}{2}-n,\frac{1}{2}\right]\right\\}~{}.~{}$

In above expressions, the 2F1 is the Hypergeometric function, and the notation
$\hat{Q}\equiv Q/Q_{m}$ in which $Q_{m}=2\alpha/[3\left(1+\gamma\right)^{2}]$.
In addition, for the case of LQC, we have $\omega_{\text{eff}}=\gamma=0$, and
the Lagrangian (36) and (37) gets reduced to the following forms respectively

For $-1\leqslant\sqrt{\alpha}\,t\leqslant 1$:

$f_{\text{LQC}}\left(Q\right)=A\sqrt{Q}-\rho_{B}\left[1+\sqrt{1-\frac{3Q}{2\alpha}}+\sqrt{\frac{3Q}{2\alpha}}\arcsin\left(\sqrt{\frac{1}{2}-\frac{1}{2}\sqrt{1-\frac{3Q}{2\alpha}}}\right)\right]~{},$
(38)

For $-\infty<\sqrt{\alpha}\,t\leqslant-1$ and
$1\leqslant\sqrt{\alpha}\,t<\infty$:

$f_{\text{LQC}}\left(Q\right)=A\sqrt{Q}+\rho_{B}\left[-1+\sqrt{1-\frac{3Q}{2\alpha}}+\sqrt{\frac{3Q}{2\alpha}}\arcsin\left(\sqrt{\frac{1}{2}-\frac{1}{2}\sqrt{1-\frac{3Q}{2\alpha}}}\right)\right]~{}.$
(39)

Next, we are interested in studying the evolution of the Hubble horizon of our
model. In the scenario of (28), the co-moving Hubble radius is given by

$r_{H}\left(t\right)=\frac{1}{aH}=\frac{3\left(\gamma+1\right)}{2a_{B}\alpha
t}\left(1+\alpha t^{2}\right)^{1-\frac{1}{3\left(\gamma+1\right)}}~{}.$ (40)

We can distinguish it into three different cases

Figure 2: The plot of the co-moving horizon of our model (orange) and the
fluctuation mode that cross the horizon before and after the bounce (blue). We
choose the parameters $\alpha=1,\gamma=0,a_{B}=1$ for the left panel, and
$\alpha=1,\gamma=-2/3,a_{B}=1$ for the right panel.

For the case of $\gamma>-1/3$, as shown in the left panel of FIG. 2., the
primordial perturbation modes generate far away from the bounce in the deep
contracting era. Along with the contraction of the Universe, the quantum
fluctuations exit the Hubble radius because the horizon decreases fast near
the bounce. After passing the bounce point, it will stay outside of the radius
until re-enter the horizon at some point in the late time expansion era.

For $-1<\gamma\leqslant-1/3$, as shown in the right panel of FIG. 2., when
$t>0$ the Hubble radius decreases monotonically over time and eventually goes
to zero in the far future. Thus the primordial perturbation modes generate
near the bounce area where $r_{H}\rightarrow\infty$, and it will stay outside
of the horizon till today. As a result, the perturbation modes in the distant
past lie in the super-Hubble regime, and thus the resolution of the horizon
problem is problematic in this case. Thereby this situation should be
excluded.

For $\gamma<-1$, the horizon evolves similarly to the case of $\gamma>-1/3$.
However, the scale factor would permanently decrease after the bounce and
equal to zero today, this case should also be ruled out.

Therefore in order to have a viable bounce scenario, here we consider the
parameter $\gamma$ (present in Eq. (28)) to be restricted by $\gamma>-1/3$ in
the subsequent calculations.

##  IV  Tensor Perturbations in $f(Q)$ bounce

###  IV.1 Perturbative action

In this section, we study the cosmological linear tensorial perturbations
generated from the $f(Q)$ bounce model. In order to do this, we work with the
comoving time $\eta\equiv\int{dt/a(t)}$ and consider the tensorial
perturbation around the background metric (18), namely:

$ds^{2}=a^{2}(\eta)[-d\eta^{2}+(\delta_{ij}+h_{ij})dx^{i}dx^{j}]~{},~{}$ (41)

where $h_{ij}$ is traceless and divergent-free. Subsituting this perturbed
metric (41) into the action (3) one can straightforwardly get the second order
action as [47]

$\mathcal{S}_{\text{tensor }}^{(2)}=\frac{1}{2}\sum_{\lambda}\int
a^{2}(\eta)f_{Q}\left[\left(h_{(\lambda)}^{\prime}\right)^{2}-k^{2}h_{(\lambda)}^{2}\right]\mathrm{d}^{3}k\mathrm{~{}d}\eta~{}.~{}$
(42)

The $h_{(\lambda)}(\eta)$ are eigenmode functions of $h_{ij}$ in Fourier modes
while $\lambda=1,2$ represent its two helicity modes, and $k$ is the comoving
wavenumber of planewave perturbations. Moreover, the overprime ′ denotes
differentiation with respect to comoving time $\eta$. It can be observed that
both the gravity polarization modes evolve in an equal footing. This indicates
that $f(Q)$ cosmology does not introduce any parity-violating terms in the
gravity wave action. By taking the variation of Eq. (42), one can obtain the
equation of motion of ${h}_{(\lambda)}$:

$h^{\prime\prime}_{\left(\lambda\right)}+\left(\frac{2a^{\prime}}{a}+\frac{1}{f_{Q}}\frac{\partial
f_{Q}}{\partial\eta}\right)h^{\prime}_{\left(\lambda\right)}+k^{2}h_{\left(\lambda\right)}=0~{}.~{}$
(43)

With respect to Mukhanov-Sasaki variable
$v_{\left(\lambda\right)}\left(\eta\right)=z(\eta)h_{(\lambda)}$ with
$z^{2}(\eta)=a^{2}(\eta)f_{Q}$, the above equation can be recast as

$v^{\prime\prime}_{\left(\lambda\right)}\left(\eta\right)+\left(k^{2}-\frac{1}{z}\frac{d^{2}z}{d\eta^{2}}\right)v_{\left(\lambda\right)}\left(\eta\right)=0~{}.~{}$
(44)

###  IV.2 Stability analysis

The present subsection is reserved for investigating the stability of the
tensor perturbation, which is ensured if the kinetic term of the perturbed
action in (43) remains positive during the entire cosmic era, i.e.,

$\frac{\partial f\left(Q\right)}{\partial
Q}>0~{},~{}\text{for}~{}-\infty<t<\infty~{}.$ (45)

Clearly, the stability condition can be checked with the form of $f(Q)$ that
we have determined in Eq. (36) and (37). Since the value of $Q$ is much
smaller than $Q_{m}$ either when the Universe is close to bounce or far from
bounce, for simplicity, we expand our action up to the first order for the
$Q/Q_{m}$.

For $-1\leqslant\sqrt{\alpha}\,t\leqslant 1$, i.e. near the bounce,

$\displaystyle f\left(Q\right)$ $\displaystyle\simeq
A\sqrt{Q}-2\rho_{B}+\frac{nQ\rho_{B}}{2Q_{m}}+\mathcal{O}\left(Q^{2}\right)~{},$
(46) $\displaystyle f_{Q}\left(Q\right)$
$\displaystyle\simeq\frac{A}{2\sqrt{Q}}+\frac{n\rho_{B}}{2Q_{m}}+\mathcal{O}\left(Q^{2}\right)~{}.~{}$

Since the $Q_{m},\rho_{B}$ and $Q(t)$ are all positive factors, the
requirement for $f_{Q}>0$ near the bounce seems to be equivalent to the
condition $A>-n\rho_{B}\sqrt{Q}/Q_{m}$. Since $Q$ varies from $0$ to $Q_{m}$,
thus for the case $n>0$, the constraint on $A$ immediately reduces to $A>0$.
However, for $n\leq 0$, the condition becomes $A>|n|\rho_{B}/\sqrt{Q_{m}}$.

For $-\infty<\sqrt{\alpha}\,t\leqslant-1$ and
$1\leqslant\sqrt{\alpha}\,t<\infty$, i. e. far from the bounce,

$\displaystyle f\left(Q\right)$ $\displaystyle\simeq
A\sqrt{Q}-4^{n}\rho_{B}\left(\frac{Q_{m}}{Q}\right)^{n}\left(\frac{2}{1+2n}\right)+\mathcal{O}\left(Q^{2}\right)~{},$
(47) $\displaystyle f_{Q}\left(Q\right)$
$\displaystyle\simeq\frac{A}{2\sqrt{Q}}+4^{n}\frac{2n\rho_{B}}{\left(1+2n\right)Q}\left(\frac{Q_{m}}{Q}\right)^{n}+\mathcal{O}\left(Q^{2}\right)~{}.~{}$

In order to avoid the tensor perturbation instability far from the bounce, the
condition for the integration constant
$A>-4^{n+1}n\rho_{B}Q^{-1/2}(Q_{m}/Q)^{n}/(1+2n)$ must be satisfied. Thus from
the perspective of the stability condition of the tensor perturbation during
$-\infty<\sqrt{\alpha}\,t\leqslant-1$ and $1\leqslant\sqrt{\alpha}\,t<\infty$,
we get three different cases depending on the value of $n$ as: for the cases
$n>0$ or $n<-1/2$, we get
$A>-\frac{4^{n+1}n\rho_{B}}{\left(1+2n\right)\sqrt{Q_{m}}}$; however, for
$-1/2<n<0$, the tensor perturbation is not stable for any value of $A$. We
summarize our results in Table. 1.

Table 1: Value Ranges of the model parameters from stability analysis. | $n<-1/2$ | $-1/2\leqslant n\leqslant 0$ | $n>0$
---|---|---|---
Near the bounce | $A>\frac{\left|n\right|\rho_{B}}{\sqrt{Q_{m}}}$ | $A>\frac{\left|n\right|\rho_{B}}{\sqrt{Q_{m}}}$ | $A>0$
Far from the bounce | $A>-\frac{4^{n+1}n\rho_{B}}{\left(1+2n\right)\sqrt{Q_{m}}}$ | Unstable | $A>-\frac{4^{n+1}n\rho_{B}}{\left(1+2n\right)\sqrt{Q_{m}}}$

###  IV.3 Evolution of tensor perturbations and power spectrum

In this subsection, we study the evolution process of the tensor perturbations
in our model. To be more precise, in order to solve the early Universe issues
like flatness, horizon, and monopole problems as well, we consider the
universe enters a period of inflation phase after the bounce, and this
scenario is commonly known as bounce inflation [73, 12]. The inflationary
phase may be realized by considering a scalar field (or inflaton) with non-
zero potential in the current $f(Q)$ scenario over the normal matter fields.
For instance, one may consider that the EoS parameter of the scalar field
remains less than that of the normal matter field during cosmic evolution. As
a result, the scalar field energy density remains suppressed in the
contraction phase; however, after the bounce, the matter field energy gets
diluted with the expansion of the Universe and the scalar field (or the
inflaton) becomes the dominant component. Such inflaton field, along with the
$f(Q)$ given in Eq. (47), triggers an inflationary phase after the bounce in
the Universe’s evolution. We briefly sketch the evolution plot of the Universe
in Fig. 3. Having set the stage, we calculate how the tensor perturbations
evolve during the whole process.

Figure 3: The schematic diagram of the evolutionary behavior of the co-moving
Hubble horizon in cosmic time $t$. The yellow wavy line represents
perturbation mode with long wavelength, while the blue wavy line represents
perturbation mode with short wavelength.

####  IV.3.1 In contracting phase

At the deep contracting phase when all the perturbation modes are initially
submerged inside the Hubble radius, i.e when $k^{2}\gg z^{\prime\prime}/z$,
the last term in the left-hand side of Eq. (44) can be neglected. As a result,
the tensor perturbation variable starts from the Bunch-Davies vacuum which
corresponds to

$v_{k}(\eta\rightarrow-\infty)=\frac{1}{\sqrt{2k}}e^{-ik\eta}~{}.~{}$ (48)

Here we omit the subscript $(\lambda)$ for $v_{(\lambda)}$ as both the
polarization modes have similar evolution. By using such Bunch-Davies vacuum
as the initial condition for $v_{k}(\eta)$, we now try to solve the full
Mukhanov-Sasaki Eq. (44). For this purpose, we need to determine
$z(\eta)=a(\eta)f_{Q}$ (and hence $z^{\prime\prime}/z$) in the contracting
phase where the form of $f(Q)$ is given by (47) and $a\left(t\right)\propto
t^{\frac{2}{3(\gamma+1)}}$. With such form of the scale factor along with
$d\eta=a^{-1}(t)dt$, we get $\eta-\tilde{\eta}_{B-}\propto
t^{\frac{1+3\gamma}{3\left(\gamma+1\right)}}$, where
$\tilde{\eta}_{B-}\equiv\eta_{B-}-[(1+3\gamma){\cal H}_{B-}/2]^{-1}$, and
$\eta_{B-}$ and ${\cal H}_{B-}$ are the conformal time and conformal Hubble
parameter at the transition point from contraction to bounce era. Such
expressions of $f_{Q}$ and $a(\eta)$ lead to the following $z(\eta)$ in the
contracting phase:

$\displaystyle z\left(\eta\right)$
$\displaystyle=a\left(\eta\right)\sqrt{f_{Q}}$ (49) $\displaystyle\simeq{\cal
M}\cdot\left|\eta-\tilde{\eta}_{B-}\right|^{s}~{},~{}$

where the exponent $s=\frac{2\left(2+n\right)}{1+3\gamma}+n+1$, and ${\cal M}$
is a complicated function, see the footnote 222The coefficient $M$ takes the
form of ${\cal
M}=\alpha^{\frac{1+s}{3\left(1+\gamma\right)}}\frac{3^{-s}\left(1+\gamma\right)^{2+n}}{2\left(1+3\gamma\right)^{-s}}\left(\frac{a_{B}}{1+\gamma}\right)^{1+s}\sqrt{\frac{2^{-n}3^{1+n}n\rho_{B}}{1+2n}(Q_{m})^{n}}$.
By choosing $\omega_{m}=\gamma=0$, it reduces to ${\cal M}=\frac{\alpha
a_{B}^{3}}{18}\sqrt{\frac{2\rho_{B}}{Q_{m}}}$.. Here we set the integral
constant $A=0$ for simplicity, according to the stability condition in Table.
1.

Thanks to Eq. (49), the Mukhanov-Sasaki Eq. (44) turns out to be

$v_{k}^{\prime\prime}+\left[k^{2}-\frac{s\left(s-1\right)}{(\eta-\tilde{\eta}_{B-})^{2}}\right]v_{k}=0~{},~{}$
(50)

by solving which we get $v_{k}(\eta)$ in terms of Hankel functions as follows:

$v_{k}\left(\eta\right)=\sqrt{\left|\eta-\tilde{\eta}_{B-}\right|}\left[c_{1}(k)\cdot
H^{(1)}_{\nu_{1}}\left(k\left|\eta-\tilde{\eta}_{B-}\right|\right)+c_{2}(k)\cdot
H^{(1)}_{-\nu_{1}}\left(k\left|\eta-\tilde{\eta}_{B-}\right|\right)\right],\quad\nu_{1}=\frac{1}{2}\left(1-2s\right)~{},~{}$
(51)

where $c_{1}(k)$, $c_{2}(k)$ are integration constants. From Fig. 3 we can see
that, in the very far past where the fluctuation modes are deep inside the
horizon, one gets $k^{2}\gg|z^{\prime\prime}/z|$, and $v_{k}$ becomes:

$v_{k}(\eta)\simeq\sqrt{\frac{2}{\pi
k}}\left[c_{1}\cos(k|\eta-\tilde{\eta}_{B-}|)+c_{2}\sin(k|\eta-\tilde{\eta}_{B-}|)\right]~{},$
(52)

On the other hand, the horizon shrinks rapidly to a minimum value as the
bounce point is approached. In this case, some perturbation modes with small
$k$, i.e., $k^{2}\ll\left|z^{\prime\prime}/z\right|$, will exit the horizon.
For this kind of superhorizon modes, $v_{k}$ becomes:

$\displaystyle
v_{k}\left(\eta\right)\simeq\frac{c_{1}\left(k\left|\eta-\tilde{\eta}_{B-}\right|\right)^{\frac{1}{2}+\nu_{1}}}{2^{\nu_{1}}\Gamma(1+\nu_{1})\sqrt{k}}+\frac{c_{2}\left(k\left|\eta-\tilde{\eta}_{B-}\right|\right)^{\frac{1}{2}-\nu_{1}}}{2^{-\nu_{1}}\Gamma(1-\nu_{1})\sqrt{k}}~{}.$
(53)

As a whole, tensor perturbation in the contraction phase is obtained as

(Superhorizon): $\displaystyle\quad
h_{k}\left(\eta\right)=\frac{v_{k}\left(\eta\right)}{z\left(\eta\right)}\simeq\frac{c_{1}\left(k\left|\eta-\tilde{\eta}_{B-}\right|\right)^{2\nu_{1}}}{2^{\nu_{1}}\Gamma(1+\nu_{1})k^{\nu_{1}}{\cal
M}}+\frac{c_{2}}{2^{-\nu_{1}}\Gamma(1-\nu_{1})k^{\nu_{1}}{\cal M}}~{},$ (54)
(Subhorizon): $\displaystyle\quad
h_{k}\left(\eta\right)\simeq\frac{\left|\eta-\tilde{\eta}_{B-}\right|^{\nu_{1}-1/2}}{\cal
M}\sqrt{\frac{2}{\pi
k}}\left[c_{1}\cos(k|\eta-\tilde{\eta}_{B-}|)+c_{2}\sin(k|\eta-\tilde{\eta}_{B-}|)\right]~{}.~{}$

It may be observed that the subhorizon modes oscillate with a time-dependent
amplitude. For the superhorizon modes, since $|\eta|$ is decreasing, for
positive $\nu_{1}$ the second term in the expression of
$h_{k}\left(\eta\right)$ decreases with the Universe’s evolution and remains
sub-dominant, hence the tensor perturbation gets frozen in the superhubble
scale. However, in the LQC-like case where $n=-1$, or equivalently
$\gamma=\omega_{m}=0$, $\nu_{1}=-3/2$ turns out to be negative, hence the
first term in the superhorizon mode is increasing and dominant. Moreover,
matching the subhorizon solution (54) with the Bunch-Davis vacuum (coming from
Eq. (48)) leads one to get the exact value of the coefficients:

$c_{1}=\frac{\sqrt{\pi}}{2}~{},~{}~{}~{}c_{2}=-i\frac{\sqrt{\pi}}{2}~{}.$ (55)

####  IV.3.2 Through the bounce

When the co-moving Hubble horizon $r_{H}$ passes through its minimum value, it
will rapidly increase to infinity at the bounce point. During this process,
all the perturbation modes lie within the subhorizon regime until cross the
bouncing phase. Considering the $f(Q)$ (near the bounce) determined in Eq.
(46) and the scale factor (28), we obtain

$z\left(\eta\right)=\sqrt{\frac{n\rho_{B}}{2Q_{m}}+\frac{\sqrt{3}\left(1+\gamma\right)A}{4\sqrt{2}\alpha
a_{B}\left(\eta-\eta_{B}\right)}}\left(1+\alpha
a_{B}^{2}(\eta-\eta_{B})^{2}\right)^{\frac{1}{3\left(1+\gamma\right)}}\simeq\sqrt{\frac{\sqrt{3}\left(1+\gamma\right)A}{4\sqrt{2}\alpha
a_{B}\left(\eta-\eta_{B}\right)}}~{},~{}$ (56)

and consequently, the Mukhanov-Sasaki Eq. (44) becomes (in the leading order
of $\eta\rightarrow\eta_{B}$)

$v_{k}^{\prime\prime}+\left[k^{2}-\frac{3}{4\left(\eta-\eta_{B}\right)^{2}}\right]v_{k}=0~{}.~{}$
(57)

From the above equation we can see that, in the bounce phase, what is to be
compared with $k^{2}$ (representing the wavelengths of the fluctuation modes)
is not horizon, but the effective potential. This is different from the
contracting or expanding phase. Moreover, since $A>0$ in the function of
$f(Q)$ from the analysis in the previous subsection, $f(Q)$ is dominated by
$\sqrt{Q}\sim|H|$ rather than $H^{2}$ in GR, the effective potential is no
longer constant as in [74, 13], but proportional to $(\eta-\eta_{B})^{2}$.
This means for $k|\eta-\eta_{B}|\ll 1$ is also a case to be considered, even
if the fluctuation modes is inside the horizon.

One can solve Eq. (57) to get $v_{k}$ for large and small $k$ modes. Eq. (57)
gives:

$v_{k}\left(\eta\right)=\sqrt{\left|\eta-\eta_{B}\right|}\left[d_{1}(k)\cdot
H^{(1)}_{1}\left(k\left|\eta-\eta_{B}\right|\right)+d_{2}(k)\cdot
H^{(1)}_{-1}\left(k\left|\eta-\eta_{B}\right|\right)\right]~{},~{}$ (58)

and for small $k$ modes ($k|\eta-\eta_{B}|\ll 1$),

$v_{k}\simeq\frac{d^{(s)}_{1}(k|\eta-\eta_{B}|)^{\frac{3}{2}}}{2\Gamma(2)\sqrt{k}}+\frac{2d^{(s)}_{2}(k|\eta-\eta_{B}|)^{-\frac{1}{2}}}{\Gamma(0)\sqrt{k}}~{},$
(59)

for large $k$ modes ($k|\eta-\eta_{B}|\gg 1$),

$v_{k}\simeq\sqrt{\frac{2}{\pi
k}}[d^{(l)}_{1}(k)\cos(k|\eta-\eta_{B}|)+d^{(l)}_{2}(k)\sin(k|\eta-\eta_{B}|)]~{}.$
(60)

Here we use the superscript “$s$” and “$l$” to distinguish the coefficients
associated with small $k$ and large $k$, respectively. The corresponding
curvature perturbation in the bouncing phase can be calculated as

(Small $k$): $\displaystyle\quad
h_{k}\left(\eta\right)\simeq\left(\frac{32}{3}\right)^{\frac{1}{4}}\sqrt{\frac{\alpha
a_{B}}{\left(1+\gamma\right)A}}\left[\frac{d^{(s)}_{1}|\eta-\eta_{B}|^{2}}{2\Gamma(2)}k+\frac{2d^{(s)}_{2}}{\Gamma(0)k}\right]~{},$
(61) (Large $k$): $\displaystyle\quad
h_{k}\left(\eta\right)\simeq\left(\frac{128}{3}\right)^{\frac{1}{4}}\sqrt{\frac{\alpha
a_{B}\left|\eta-\eta_{B}\right|}{\left(1+\gamma\right)Ak\pi}}[d^{(l)}_{1}(k)\cos(k|\eta-\eta_{B}|)+d^{(l)}_{2}(k)\sin(k|\eta-\eta_{B}|)]~{}.~{}$

The solutions for the bouncing phase should be matched to that during the
contracting phase. For this purpose, we will use the Hwang-Vishniac (Deruelle-
Mukhanov) matching condition [75, 76], which requires that the $h_{k}$ and its
derivative $h_{k}^{\prime}$ should be continuous at the transition point of
contraction-to-bounce phase, namely $t_{B-}=-1/\sqrt{\alpha}$ in the present
model. Utilizing the solutions (54) and (61) we can determine the coefficients
as follows
i) For small $k$:

$\displaystyle d_{1}^{\left(s\right)}\simeq$
$\displaystyle\left(\frac{3}{2}\right)^{1/4}c_{1}\left(\frac{k}{|{\cal
H}_{B-}|}\right)^{-1+\nu_{1}}\left(\frac{2}{|{\cal
H}_{B-}|}\right)^{\nu_{1}}\Gamma(1-\nu_{1})\sin(\pi\nu_{1})\sqrt{\frac{A(1+\gamma)a_{B}}{{\cal
M}^{2}\pi^{2}(\Delta\eta)^{2}\alpha}}~{},$ (62) $\displaystyle
d_{2}^{\left(s\right)}\simeq$ $\displaystyle
2^{\nu_{1}}\left(\frac{3}{2^{17}}\right)^{1/4}c_{1}k^{1+\nu_{1}}|{\cal
H}_{B-}|^{-2\nu_{1}}({\cal
H}_{B-}\Delta\eta-4)\Gamma(-\nu_{1})\Gamma(0)\sin(\pi\nu_{1})\sqrt{\frac{A(1+\gamma)a_{B}}{{\cal
M}^{2}\pi^{2}\alpha}}~{},~{}(\nu_{1}<0)$ $\displaystyle\simeq$ $\displaystyle
2^{\nu_{1}}\left(\frac{3}{2^{9}}\right)^{1/4}c_{2}k^{1-\nu_{1}}\Gamma(\nu_{1})\Gamma(0)\sin(\pi\nu_{1})\sqrt{\frac{A(1+\gamma)a_{B}}{{\cal
M}^{2}\pi^{2}\alpha}}~{}.~{}(\nu_{1}>0)$

ii) For large $k$:

$\displaystyle d_{1}^{\left(l\right)}\simeq$
$\displaystyle\left(\frac{3}{8}\right)^{1/4}\left(\frac{2}{|{\cal
H}_{B-}|}\right)^{-1/2+\nu_{1}}\sqrt{\frac{A(1+\gamma)a_{B}}{\alpha\Delta\eta{\cal
M}^{2}}}\left(c_{1}\cos{\left[\frac{k}{2}\left(\Delta\eta-\frac{4}{{\cal
H}_{B-}}\right)\right]}+c_{2}\sin{\left[\frac{k}{2}\left(\Delta\eta-\frac{4}{{\cal
H}_{B-}}\right)\right]}\right)~{},$ (63) $\displaystyle
d_{2}^{\left(l\right)}\simeq$
$\displaystyle\left(\frac{3}{8}\right)^{1/4}\left(\frac{2}{|{\cal
H}_{B-}|}\right)^{-1/2+\nu_{1}}\sqrt{\frac{A(1+\gamma)a_{B}}{\alpha\Delta\eta{\cal
M}^{2}}}\left(c_{2}\cos{\left[\frac{k}{2}\left(\Delta\eta-\frac{4}{{\cal
H}_{B-}}\right)\right]}-c_{1}\sin{\left[\frac{k}{2}\left(\Delta\eta-\frac{4}{{\cal
H}_{B-}}\right)\right]}\right)~{},$

where $\Delta\eta\equiv\eta_{B-}-\eta_{B+}\simeq 2(\eta_{B-}-\eta_{B})$ is the
duration of bounce phase in conformal time.

####  IV.3.3 In inflation phase

As mentioned earlier that after the bounce i.e. during
$\sqrt{\alpha}t\geqslant 1$, the inflaton field becomes the dominating energy
component of the Universe, which, along with the $f(Q)$ given in Eq. (47),
triggers the inflation process for a suitable choice of inflaton potential.
During the inflation era, the Universe expands almost exponentially, while the
Hubble parameter $H$ is nearly constant. Furthermore, the transition point
from bounce-to-inflation locates at $t_{B+}=1/\sqrt{\alpha}$, because the
Hubble horizon $1/H(t)$ given by Eq. (28) has a minimum value when
$t_{B+}=1/\sqrt{\alpha}$, which makes smooth conjunction between the bounce
and the inflation.

For convenience, we assume that the scale factor during inflation takes the
following form

$a_{\text{inf}}\left(t\right)\simeq\kappa\left(1+\epsilon\cdot\beta\cdot
t\right)^{1/\epsilon}~{},~{}0<\epsilon\ll 1~{},~{}$ (64)

where $\beta$ and $\kappa$ are the parameters used to adjust the speed and the
amplitude of the inflation, and the slow-roll parameter $\epsilon$ is defined
by $\epsilon=-\dot{H}/H^{2}$. In order to satisfy the continuity from bounce-
to-inflation, i.e. the matching of the Hubble parameters and scale factors
given by Eq. (64) and Eq. (28) at $t_{B+}$, there should be intrinsic
relationships between the parameters as the following:

$\displaystyle\beta\simeq\frac{\sqrt{\alpha}}{3\left(\gamma+1\right)}~{},~{}~{}~{}~{}~{}~{}\kappa=2^{\frac{1}{3\left(1+\gamma\right)}}a_{B}\left(1+\frac{\epsilon}{3(1+\gamma)}\right)^{-1/\epsilon}~{}~{},$

where we consider $1+\epsilon\approx 1$. Furthermore, the conformal time
during the inflation is given by

$\eta-\tilde{\eta}_{B+}=\frac{(1+\epsilon\beta
t)^{\frac{\epsilon-1}{\epsilon}}}{\kappa\beta(\epsilon-1)}$ (65)

where $\tilde{\eta}_{B+}\equiv\eta_{B+}-[(\epsilon-1){\cal H}_{B+}]^{-1}$, and
$\eta_{B+}$ and ${\cal H}_{B+}$ are the conformal time and conformal Hubble
parameter at the transition point from bounce to inflation era. Having this,
we can determine the inflationary scale factor (64) in terms of the conformal
time:

$a_{\text{inf}}\left(\eta\right)\simeq\kappa[\kappa\beta(\epsilon-1)(\eta-\tilde{\eta}_{B+})]^{\frac{1}{\epsilon-1}}~{},$
(66)

The above expression of $a(\eta)$ and the $f(Q)$ (given by Eq. (47))
immediately leads to

$z\left(\eta\right)\simeq{\cal
N}\cdot|\eta-\tilde{\eta}_{B+}|^{\frac{\epsilon(n+1)+1}{\epsilon-1}}$ (67)

while ${\cal
N}=\kappa^{\frac{\epsilon(n+2)}{\epsilon-1}}\beta^{\frac{n+2}{\epsilon-1}}\sqrt{\frac{2^{n}n\left(Q_{m}\right)^{n}\rho_{B}}{3^{n+1}(1+2n)}}$.
For $\omega_{m}=\gamma=0$, it reduces to ${\cal N}=3\sqrt{\rho_{B}/(2\alpha
Q_{m})}$. The Mukhanov-Sasaki Eq. (44) in inflation phase becomes

$v_{k}^{\prime\prime}+\left[k^{2}-\frac{2+3\left(n+2\right)\epsilon}{(\eta-\tilde{\eta}_{B+})^{2}}\right]v_{k}=0~{},~{}$
(68)

by solving which we get

$v_{k}\left(\eta\right)=\sqrt{\left|\eta-\tilde{\eta}_{B+}\right|}\left[f_{1}H^{(1)}_{\nu_{2}}\left(k\left|\eta-\tilde{\eta}_{B+}\right|\right)+f_{2}H^{(1)}_{-\nu_{2}}\left(k\left|\eta-\tilde{\eta}_{B+}\right|\right)\right],\quad\nu_{2}\simeq\frac{3}{2}+\left(n+2\right)\epsilon~{}.~{}$
(69)

Note that the above solution of $v_{k}(\eta)$ during the inflation is of the
same form compared to that of in the contraction phase — this is expected,
because in both phases, the term $z^{\prime\prime}/z$ turns out to be
proportional to $1/(\eta-\tilde{\eta}_{B+})^{2}$.

What we are observing are the superhorizon modes at the end of the inflation
phase, and they come in two different ways. As shown in Fig. 3, there are two
branches of this solution: The first branch is the small $k$ modes, which have
already exited the horizon at the contracting phase. The second branch comes
from the large $k$ modes, which are always subhorizon modes until it exits the
Hubble horizon during the inflation phase. The solution for the first branch
is

$\displaystyle v_{k}\left(\eta\right)\simeq$
$\displaystyle\frac{f^{(s)}_{1}\left(k\left|\eta-\tilde{\eta}_{B+}\right|\right)^{\frac{1}{2}+\nu_{2}}}{2^{\nu_{2}}\Gamma(1+\nu_{2})\sqrt{k}}+\frac{f^{(s)}_{2}\left(k\left|\eta-\tilde{\eta}_{B+}\right|\right)^{\frac{1}{2}-\nu_{2}}}{2^{-\nu_{2}}\Gamma(1-\nu_{2})\sqrt{k}}~{},$
(70) $\displaystyle h_{k}\left(\eta\right)\simeq$
$\displaystyle\frac{f^{(s)}_{1}\left(k\left|\eta-\tilde{\eta}_{B-}\right|\right)^{2\nu_{2}}}{2^{\nu_{2}}\Gamma(1+\nu_{2})k^{\nu_{2}}{\cal
N}}+\frac{f^{(s)}_{2}}{2^{-\nu_{2}}\Gamma(1-\nu_{2})k^{\nu_{2}}{\cal
N}}~{},~{}$

and the solution for the second branch is

$\displaystyle v_{k}\left(\eta\right)\simeq$ $\displaystyle\sqrt{\frac{2}{\pi
k}}\left[f^{(l)}_{1}\cos(k|\eta-\tilde{\eta}_{B+}|)+f^{(l)}_{2}\sin(k|\eta-\tilde{\eta}_{B+}|)\right]~{},$
(71) $\displaystyle h_{k}\left(\eta\right)\simeq$
$\displaystyle\frac{|\eta-\tilde{\eta}_{B+}|}{\cal N}\sqrt{\frac{2}{\pi
k}}\left[f^{(l)}_{1}\cos(k|\eta-\tilde{\eta}_{B+}|)+f^{(l)}_{2}\sin(k|\eta-\tilde{\eta}_{B+}|)\right]~{},~{}$

The coefficients $f_{1}$, $f_{2}$ can be determined by the same method in the
previous discussion. For the first branch, the solution (70) should be matched
with the small $k$ modes in bouncing phase (61), by which we get:

$\displaystyle f_{1}^{\left(s\right)}\simeq$
$\displaystyle-\left(\frac{1}{6}\right)^{1/4}d^{(s)}_{1}\left(\frac{1}{2{\cal
H}_{B+}(1-\epsilon)}\right)^{\frac{1}{2}-\nu_{2}}\left(\frac{k}{{\cal
H}_{B+}(1-\epsilon)}\right)^{\frac{1}{2}-\nu_{2}}\Gamma(\nu_{2})\sqrt{\frac{\alpha
k(\Delta\eta)^{2}{\cal N}^{2}}{A(1+\gamma)a_{B})}}~{},$ (72)
$\displaystyle\simeq$ $\displaystyle 2^{\nu_{1}+\nu_{2}-1}c_{1}\frac{{\cal
N}}{{\cal M}\pi}|{\cal
H}_{B-}|^{1-2\nu_{1}}k^{-\frac{1}{2}+\nu_{1}}\left(\frac{1}{{\cal
H}_{B+}(1-\epsilon)}\right)^{\frac{1}{2}-\nu_{2}}\left(\frac{k}{{\cal
H}_{B+}(1-\epsilon)}\right)^{\frac{1}{2}-\nu_{2}}\Gamma(1-\nu_{1})\Gamma(\nu_{2})\sin(\pi\nu_{1})~{},$
$\displaystyle f_{2}^{\left(s\right)}\simeq$
$\displaystyle-\left(\frac{1}{3\times
2^{9}}\right)^{1/4}(-k\Delta\eta)^{-\frac{3}{2}}\left(\frac{2}{{\cal
H}_{B+}(1-\epsilon)}\right)^{\frac{1}{2}-\nu_{2}}\left(\frac{k}{{\cal
H}_{B+}(1-\epsilon)}\right)^{\frac{1}{2}+\nu_{2}}\Gamma(-\nu_{2})\sqrt{\frac{\alpha\Delta\eta{\cal
N}^{2}}{A(1+\gamma)a_{B}}}$ $\displaystyle\times$
$\displaystyle\left(d^{(s)}_{1}(k\Delta\eta)^{2}[2+\nu_{2}(1-\epsilon){\cal
H}_{B+}\Delta\eta]+16\frac{d^{(s)}_{2}}{\Gamma(0)}[\nu_{2}(1-\epsilon){\cal
H}_{B+}\Delta\eta]\right)~{},$ $\displaystyle\simeq$ $\displaystyle
2^{\nu_{1}-\nu_{2}-1}c_{1}\frac{{\cal N}}{{\cal
M}\pi}k^{\nu_{2}-\nu_{1}}\left(\frac{k}{|{\cal
H}_{B-}|}\right)^{2\nu_{1}}\Gamma(1-\nu_{1})\Gamma(-\nu_{2})\sin(\pi\nu_{1})\left(\frac{{\cal
H}_{B-}}{{\cal
H}_{B+}(1-\epsilon)}+\frac{2\nu_{2}}{\nu_{1}}\right)~{},~{}(\nu_{1}<0)$
$\displaystyle\simeq$ $\displaystyle 2^{\nu_{1}-\nu_{2}}c_{2}\frac{{\cal
N}}{{\cal
M}\pi}k^{\nu_{2}-\nu_{1}}\Gamma(1-\nu_{2})\Gamma(\nu_{1})\sin(\pi\nu_{1})~{},~{}(\nu_{1}>0)~{}$

while the second branch of solution (71) should be matched with the large $k$
modes in bouncing phase (61), from which we have:

$\displaystyle f_{1}^{\left(l\right)}\simeq$
$\displaystyle\left(\frac{8}{3}\right)^{1/4}[{\cal
H}_{B+}(1-\epsilon)]^{\nu_{2}-\frac{1}{2}}\sqrt{\frac{\alpha\Delta\eta{\cal
N}^{2}}{A(1+\gamma)a_{B}}}\left(d^{(l)}_{1}\cos\left[\frac{k\Delta\eta}{2}+\frac{k}{{\cal
H}_{B+}(1-\epsilon)}\right]-d^{(l)}_{2}\sin\left[\frac{k\Delta\eta}{2}+\frac{k}{{\cal
H}_{B+}(1-\epsilon)}\right]\right)~{},$ (73) $\displaystyle\simeq$
$\displaystyle\left(\frac{2}{|{\cal H}_{B-}|}\right)^{-1/2+\nu_{1}}\frac{\cal
N}{\cal M}[{\cal
H}_{B+}(1-\epsilon)]^{\nu_{2}-\frac{1}{2}}\left(c_{1}\cos\left[k\left(\frac{2}{{\cal
H}_{B-}}+\frac{1}{{\cal
H}_{B+}(1-\epsilon)}\right)\right]-c_{2}\sin\left[k\left(\frac{2}{{\cal
H}_{B-}}+\frac{1}{{\cal H}_{B+}(1-\epsilon)}\right)\right]\right)~{},$
$\displaystyle f_{2}^{\left(l\right)}\simeq$
$\displaystyle\left(\frac{8}{3}\right)^{1/4}[{\cal
H}_{B+}(1-\epsilon)]^{\nu_{2}-\frac{1}{2}}\sqrt{\frac{\alpha\Delta\eta{\cal
N}^{2}}{A(1+\gamma)a_{B}}}\left(d^{(l)}_{2}\cos\left[\frac{k\Delta\eta}{2}+\frac{k}{{\cal
H}_{B+}(1-\epsilon)}\right]+d^{(l)}_{1}\sin\left[\frac{k\Delta\eta}{2}+\frac{k}{{\cal
H}_{B+}(1-\epsilon)}\right]\right)~{},$ $\displaystyle\simeq$
$\displaystyle\left(\frac{2}{|{\cal H}_{B-}|}\right)^{-1/2+\nu_{1}}\frac{\cal
N}{\cal M}[{\cal
H}_{B+}(1-\epsilon)]^{\nu_{2}-\frac{1}{2}}\left(c_{2}\cos\left[k\left(\frac{2}{{\cal
H}_{B-}}+\frac{1}{{\cal
H}_{B+}(1-\epsilon)}\right)\right]+c_{1}\sin\left[k\left(\frac{2}{{\cal
H}_{B-}}+\frac{1}{{\cal H}_{B+}(1-\epsilon)}\right)\right]\right)~{}.~{}$

We notice that Eq. (70) contains two terms, of which the first term in the
bracket depend on $\left|\eta\right|$ while the last term is a constant. Since
the $\left|\eta\right|$ is decreasing during inflation, thus the first term
would decay along the evolution of time for positive $\nu_{2}$. For instance,
in the LQC-like case where $n=-1$, $\nu_{2}=(3/2+\epsilon)$ turns out to be
positive, hence the last term becomes dominant.

####  IV.3.4 the tensor power spectrum

With the above solutions in hand, we’re ready to obtain the final tensor power
spectrum. The tensor spectrum is defined as:

$P_{T}(k)\equiv
2\frac{k^{3}}{\pi^{2}}\left|h_{k}\right|^{2}=2\frac{k^{3}}{\pi^{2}}\left|\frac{v_{k}}{z}\right|^{2}~{},$
(74)

where the factor 2 corresponds to the two independent polarization modes of
the gravitational waves. Note that for both the first and the second branch,
we are observing the perturbation modes when they are in the superhubble
regime in the inflationary phase. Using Eq.(70), we have

$P_{T}(k)\simeq
2^{1+2\nu_{2}}\frac{k^{3-2\nu_{2}}|f_{2}|^{2}}{\Gamma^{2}(1-\nu_{2}){\cal
N}^{2}\pi^{2}}~{},~{}$ (75)

where $\nu_{2}\simeq 3/2+\left(n+2\right)\epsilon$. So we get:
i) For small $k$:

$\displaystyle P_{T}^{(s)}=$ $\displaystyle\frac{2^{2\nu_{1}-3}|{\cal
H}_{B-}|^{-4\nu_{1}}[{\cal H}_{B-}\nu_{1}+2{\cal
H}_{B+}(1-\epsilon)\nu_{2}]^{2}\Gamma^{2}(-\nu_{1})\sin^{2}(\pi\nu_{1})}{{\cal
H}_{B+}^{2}{\cal
M}^{2}\pi^{3}(1-\epsilon)^{2}\nu_{2}^{2}}k^{3+2\nu_{1}}~{},~{}(\nu_{1}<0)$
(76) $\displaystyle=$
$\displaystyle\frac{2^{2\nu_{1}-1}\Gamma^{2}(\nu_{1})\sin^{2}(\pi\nu_{1})}{{\cal
M}^{2}\pi^{3}}k^{3-2\nu_{1}}~{},~{}(\nu_{1}>0)~{}$

ii) For large $k$:

$P_{T}^{(l)}=\frac{4^{\nu_{1}+\nu_{2}-1}|{\cal H}_{B-}|^{1-2\nu_{1}}[{\cal
H}_{B+}(1-\epsilon)]^{2\nu_{2}-1}}{\pi{\cal
M}^{2}\Gamma^{2}(1-\nu_{2})}k^{3-2\nu_{2}}.~{}$ (77)

Clearly, the power spectrum gets frozen in the superhorizon scale. Here we
need to recall that both $f_{1}$ and $f_{2}$ depend on $k$, and they have
different forms for small $k$ and large $k$ (see Eq. (72) and Eq. (73)
respectively). This indicates that the tilt of the tensor perturbation
($n_{T}$) for small $k$ is different compared to that of large $k$. It is
worth mentioning that the integral constant $A$ does not appear in the final
power spectrum; this is due to the fact that $A$ appears as a global constant
in the action (46) and does not change the structure of the action, thus it is
not surprising that the final power spectrum does not contain $A$. In
particular, $n_{T}$ is obtained as:

$\displaystyle\text{(Small $k$):}\quad n_{T}$
$\displaystyle\equiv\frac{d\text{ln}P_{T}}{d\text{ln}k}\simeq 3+2\nu_{1}\simeq
4-2s~{},~{}\text{for}~{}s\geqslant\frac{1}{2}~{},$ (78) $\displaystyle n_{T}$
$\displaystyle\equiv\frac{d\text{ln}P_{T}}{d\text{ln}k}\simeq 3-2\nu_{1}\simeq
2+2s~{},~{}\text{for}~{}s\leqslant\frac{1}{2}~{},$ $\displaystyle\text{(Large
$k$):}\quad n_{T}$
$\displaystyle=3-2\nu_{2}\simeq-2\left(2+n\right)\epsilon~{}.$

To derive the expression of $n_{T}$ for small $k$, we used
$\nu_{1}=\left(1-2s\right)/2$. An important point to be noted from the above
expression is that for $n<-2$ (for which the perturbation remains stable, see
Sec.[IV.2]), the tensor perturbation at large $k$ gets a positive tilt — this
interestingly hints an enhancement of gravitational waves’ amplitude at large
$k$ in the background of a bounce-inflation scenario. Moreover Eq.(78) also
hints at a consistency of the present $f(Q)$ bounce model with the recent PTA
data [66, 67, 68, 69], as the tensor tilt for small $k$ may acquire positive
value for suitable parameters [77, 78, 79]. Such possibilities clearly
indicate that primordial gravitational waves’ spectrum at the present Universe
(and its possibility to cross various sensitivity curves of GWs observatories)
in the context of $f(Q)$ bounce-inflation is worthwhile to examine, and this
is expected to study in the near future.

For the special case related to LQC, after substituting all the required
coefficients into the Eq. (75) and by choosing $n=-1$ (or equivalently
$\gamma=\omega_{m}=0$) and $\rho_{B}=4\alpha/3$, we acquire the following
expression for the corresponding tensor power spectrum:

i) For small $k$:

$P_{T}^{(s)}=\frac{{\cal H}_{B-}^{6}[3{\cal H}_{B-}+2{\cal
H}_{B+}((2\epsilon^{2}+\epsilon-3)]^{2}}{256{\cal H}_{B+}^{2}{\cal
M}^{2}\pi^{2}(2\epsilon^{2}+\epsilon-3)^{2}}~{},~{}n_{T}^{(s)}=0~{}.~{}$ (79)

ii) For large $k$:

$P_{T}^{(l)}=\frac{4^{\epsilon-1}|{\cal H}_{B-}|^{4}[{\cal
H}_{B+}(1-\epsilon)]^{2+2\epsilon}}{\pi{\cal
M}^{2}\Gamma^{2}\left(-\frac{1}{2}-\epsilon\right)}k^{-2\epsilon}~{},~{}n_{T}^{(l)}\simeq-2\epsilon~{}.~{}$
(80)

We can see that regardless of whether the wavelength is small or large, the
tensor spectrum index is nearly zero, implying a nearly scale-invariant power
spectrum. This is consistent with the well-known results of matter bounce
inflation models [80, 81]. Another interesting feature is that the power
spectrum (79) (80) is proportional to $\alpha$ (the $\cal M\sim\alpha$,
$\cal{H}_{B+}$ and $\cal{H}_{B-}$ are proportional to $\sqrt{\alpha}$), hence
those for different $k$ differs only in the amplitude of the power spectrum.
From the above analysis, one can see that, $\alpha$ has triple identities: it
can measure the duration of the bounce process of the Universe; indicate the
peak value of Hubble parameter/nonmetricity scalar at the transition point of
contraction-to-bounce/bounce-to-inflation; and is also related to the matter
density at the bounce point, $\rho_{B}$. If we get the constraints on $\alpha$
from the detection of primordial gravitational waves, we can constrain on
these three quantities at the same time. We expect the future observations
such as AliCPT [62, 63] can help us in this direction. 333Actually, what
AliCPT (and other PGW experiments) detects is the tensor/scalar ratio of the
perturbations. This means that to get constraints on $\alpha$, we should know
the scalar perturbation as well. However, the scalar perturbation is more
difficult to deal with, due to the redundant degrees of freedom [47, 60]. This
issue is under investigation now.

##  V discussion and conclusions

In this work, we propose a bounce inflation cosmological scenario in a $f(Q)$
modified gravity theory, where the Universe initially contracts with a
bouncing-like behavior, and smoothly transits to an inflationary era which can
be further connected to the standard Big-Bang cosmology. In such a scenario,
the presence of the inflation (after the bounce) ensures the natural
resolution of the flatness problem, while the horizon problem gets resolved
due to the initial Bunch-Davies vacuum at distant past where all the
perturbation modes lie within the sub-Hubble regime. Using a suitable
reconstruction technique, we obtain the form of $f(Q)$ corresponding to this
scenario.

First of all, we presented a parametrized form of bounce solution preceded by
an inflationary era, and reconstructed the $f(Q)$ action using the inverse-
function method. We found that for the case of matter bounce where
$\gamma=\omega_{m}=0$, our model behaves similar as the Loop Quantum Cosmology
scenario, where the Friedmann equation receives a correction term in
proportional to $\rho_{m}^{2}$.

Next, we analyze the cosmological perturbation. However, some of our authors
in [60] show that it may exist 8 degrees of freedom of the perturbative $f(Q)$
theory in the case of coincident gauge. Due to the large number of possible
propagation modes, the number of the rest degrees of freedom are unknown
except for the two tensor degrees of freedom. For this reason, we studied the
tensor perturbation, starting from the contracting phase to the inflation
phase.

We investigated the stability issue of the tensor perturbations and showed the
range of viable parameters in order to have stable perturbations. Our results
show that there exist suitable parameter ranges which ensure the ghost-free
nature of the present model during the entire cosmic era. We then calculate
the tensor modes of primordial perturbations generated from the Bunch-Davies
vacuum in the contracting phase, both for the large and small $k$ modes. We
present our result for general unspecified parameters as well as the specific
case of matter bounce, namely $\gamma=\omega_{m}=0$. As a result, the tensor
power spectrum in the current context shows the following characteristics —
(a) for the case of matter bounce, the spectrum is found to be frozen in the
superhubble regime, presenting nearly scale-invariance in both large and small
scales. This is consistent with the general result of matter bounce inflation
scenario. Moreover, the amplitude of the tensor perturbation is proportional
to the parameter $\alpha$, which represents the bouncing duration, the Hubble
parameter and nonmetricity scalar at the transition points and the matter
density at the bouncing point. For future detection of primordial
gravitational waves such as AliCPT, CMB-S4 and LiteBIRD, this parameter can be
furtherly constrained along with the scalar perturbations. (b) the tensor
spectrum for the small modes $k$ as well as the large $k$ modes may also get a
positive tilt depending on the model parameters, which can be consistent with
recent stochastic GW data from recent PTA observations. This also hints at a
rich possibility of GWs’ enhancement and its detection in the future.

As a side remark, let’s briefly comment on the similarity of the matter bounce
case in our model and the LQC model. There are many cosmological models
studied in the context of LQC, including the Matter Bounce Scenario in LQC
[82], the Slow-roll Inflation [83, 84] and so on. People also found the
relationship between LQC and $f(T)$ modified gravity theory [85]. Here, our
result may also help us to deepen our understanding of the connection between
LQC and modified gravity; especially the symmetric teleparallel version of
LQC. Although there may be a certain amount of accidents, it might hint that
there can be some quantum gravity nature hidden inside the phenomenological
models of these non-Riemannian gravity theories, such as torsion or
nonmetricity theory.

Although we mainly focus on the tensor perturbations (Gravitational waves) of
the $f(Q)$ bounce scenario, it is definitely interesting and necessary to
investigate the more complicated scalar perturbations of this scenario as
well, which is undergoing now. We will present our progress and results on
this subject in the near future.

###### Acknowledgements.

We thank Taishi Katsuragawa for helpful discussions. T.Q. and K.H. are
supported by the National Key Research and Development Program of China under
Grant No. 2021YFC2203100, and the National Science Foundation of China under
Grant No. 11875141.

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