# Exploring the impact of Tree Level Higgs Potential on Reheating in $f(Q)$
Gravity

K. El Bourakadi1,2<EMAIL_ADDRESS>1Quantum Physics and Magnetism
Team, LPMC, Faculty of Science Ben M’sik,
Casablanca Hassan II University, Morocco
2LPHE-MS Laboratory department of Physics, Faculty of Science, Mohammed V
University in Rabat, Morocco.

###### Abstract

This article investigates the effects of Tree Level Higgs Potential on
reheating in $f(Q)$ gravity. We examine the formalism of tree level inflation
in a modified $f(Q)$ model, finding that the slow roll and e-folds number
during inflation depend on the $f(Q)$ model $\alpha$ parameter. Additionally,
we calculate the reheating as a function of the decay rate, constraining the
inflationary parameters and reheating temperature. Our study reveals that the
Tree Level Higgs Potential influences the reheating temperature and decay
rate. We conclude that increasing the decay rate energy leads to higher
reheating temperature, and the chosen $\alpha$ parameter affects the reheating
temperature curve. This research advances our understanding of the interplay
between Tree Level Higgs inflation, matter production, and modified gravity in
$f(Q)$ gravity.

##  I Introduction

The most likely successful inflation scenario is the single-field chaotic
model [1, 2]. However, polynomial and monomial potentials are also viable
options. For example, many monomial potentials have quadratic or quartic
forms, like the simple double-well potential used to explain the production of
Primordial Black Holes [3, 4, 5]. Additionally, Ref. [6] employed the Higgs
potential at the tree level to develop inflation in non-supersymmetric grand
unified theories $\left(GUT\right)$ and calculate proton lifetime. Similarly,
Refs. [7] and [8] explored the effects of inflation couplings with other
particles, such as $\left(GUT\right)$ scalars, to investigate quantum
smearing. They found that the Higgs potential can be divided into three
branches. Inflationary models were created to resolve several problems
associated with Big Bang cosmology, including the flatness problem, horizon
problem, homogeneity, and the numerical density of monopoles [9].
Additionally, inflation provides a causal explanation for the observed
anisotropy of the cosmic microwave background radiation $\left(CMB\right)$
[10]. Recently, braneworld models have emerged as a novel alternative approach
to test certain predictions and incorporate corrections to general relativity
[11, 12].

Within the framework of GR, the motion of spacetime is determined by the Ricci
curvature scalar $R$, which is derived from the Levi-Civita connection.
Gravity is considered a metric theory in this context, meaning it adheres to
established assumptions such as Lorentz invariance and causality. However,
there are other geometric quantities such as torsion and non-metricity that
can also be used to describe the gravitational field. This has been explored
in previous studies [13, 14]. As part of this approach, a version of GR called
the symmetric teleparallel equivalent of GR $\left(STEGR\right)$ was suggested
[15, 16, 17, 18, 19, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. This model
utilizes a non-metric connection for spacetime geometry, but with the
constraint that the total curvature and torsion are both zero. STEGR can also
be extended to $f(Q)$ gravity [28], which does not require the Equivalence
Principle as an initial assumption. This theory can be approached using gauge
theories and has several benefits, including no significant coupling issues
caused by extra scalar modes [29]. Recently, $f(Q)$ gravity has gained
attention [30, 31], particularly in regards to explaining problems with late-
time acceleration and dark energy [32, 33, 34]. However, its effects on the
early universe have not been fully explored.

Following inflation, there is a period during which the energy density stored
in the inflaton converts into a thermal bath consisting of relativistic
particles [35, 36, 37, 38, 39, 40, 41, 42]. This process is known as reheating
and takes place between the end of inflation and the start of the radiation-
dominated epoch. During the reheating process, ordinary matter is produced as
a result of the inflaton field losing energy. In the simplest case, reheating
happens through the inflaton’s perturbative decay into particles of standard
model matter as it oscillates around the minimum of its potential [43, 44].
However, this scenario has been challenged due to its inability to explain the
cohesive nature of the inflaton field [45, 46]. In some instances, a
preheating stage takes place before reheating, during which particles are
created through nonperturbative mechanisms like parametric resonance decay
[47]. The potential for both preheating and reheating to occur between the end
of inflation and the start of the radiation-dominated epoch was explored in
[48].

The structure of our paper is outlined as follows: Section II provides an
overview of the fundamental principles of $f(Q)$ gravity. Section III
introduces the tree level inflation model, while Section IV describes the tree
level inflationary scenario within $f(Q)$ gravity. In Section V, we summarize
our findings and present our conclusions.

##  II Inflationary cosmology in $f(Q)$ gravity

In Weyl-Cartan geometry, gravitational effects are caused not only by a change
in the direction of a vector in parallel transport, but also by a change in
its length. Non-metricity represents the geometric variation in the length of
a vector and is theoretically defined as the covariant derivative of the
metric tensor. In this paper, we will look at the modified Einstein-Hilbert
action in f(Q) symmetric teleparallel gravity, which is written as,

$S=\int\frac{1}{2\kappa}f(Q)\sqrt{-g}d^{4}x+S_{\phi},$ (1)

where $f(Q)$ is a generic function of the non-metricity scalar $Q$, $g$ is the
determinant of the metric tensor $g_{\mu\nu}$ i.e.
$g=\det\left(g_{\mu\nu}\right)$, $\kappa=1/M_{p}^{2}$ and $S_{\phi}$ is the
inflaton action defined in the next section. In addition, the non-metricity
scalar $Q$ is given as,

$Q\equiv-g^{\mu\nu}(L_{\,\,\,\alpha\mu}^{\beta}L_{\,\,\,\nu\beta}^{\alpha}-L_{\,\,\,\alpha\beta}^{\beta}L_{\,\,\,\mu\nu}^{\alpha}).$
(2)

We obtain the field equations of $f(Q)$ gravity by varying the gravitational
action (1) with regard to the metric tensor $g_{\mu\nu}$,

$-\frac{2}{\sqrt{-g}}\nabla_{\beta}\left(f_{Q}\sqrt{-g}P_{\,\,\,\,\mu\nu}^{\beta}\right)-\frac{1}{2}fg_{\mu\nu}-f_{Q}(P_{\mu\beta\alpha}Q_{\nu}^{\,\,\,\beta\alpha}-2Q_{\,\,\,\mu}^{\beta\alpha}P_{\beta\alpha\nu})=\kappa
T_{\mu\nu}.$ (3)

Here, $f_{Q}=\frac{df\left(Q\right)}{dQ}$, and $\nabla_{\beta}$ is the
covariant derivative is shown. We now investigate a homogeneous and spatially
flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric. As a result, the
analogous non-metricity scalar is $Q=6H^{2}$. In our current investigation, we
suppose that the Universe is a perfect non-viscous fluid with the energy-
momentum tensor given by

$T_{\nu}^{\mu}=diag\left(-\rho,p,p,p\right),$ (4)

The updated Friedmann equations are stated as where $p$ is the ideal non-
viscosity fluid pressure and the Universe’s energy density is $\rho$.

$\kappa\rho=\frac{f}{2}-6FH^{2}$ (5)

and

$\kappa p=-\frac{f}{2}+6FH^{2}+2\left(\dot{F}H+F\dot{H}\right).$ (6)

In this case, $\left(\text{\textperiodcentered}\right)$ denotes a derivative
with regard to cosmic time $\left(t\right)$, while $F\equiv f_{Q}$ denotes
differentiation with respect to $Q$. Combining Eqs (5) and (6) yields the
evolution equation for the Hubble parameter $H$ as,

$\overset{.}{H}+\frac{\overset{.}{F}}{F}H=\frac{\kappa}{2F}\left(\rho+p\right).$
(7)

The field equations of Einstein (5) and (6) can be thought of as extended
symmetric teleparallel counterparts to conventional Friedmann’s equations,
with added terms from space-time non-metricity and the trace of the energy-
momentum tensor $T$ acting as an effective component. As a result, the
effective energy density $\rho_{eff}$ and effective pressure $p_{eff}$ are
calculated.

$3H^{2}=\kappa\rho_{eff}=\frac{f}{4F}-\frac{\kappa}{2F}\rho,$ (8)

with

$2\dot{H}+3H^{2}=-\kappa
p_{eff}=\frac{f}{4F}-\frac{2\dot{F}H}{F}+\frac{\kappa}{2F}\left(\rho+2p\right).$
(9)

Taking into account Eqs. (5) and (6) one gets

$\rho=\frac{f-12H^{2}F}{2\kappa}.$ (10)

The inflationary universe model was developed to address a number of
cosmological concerns, including the horizon, flatness, and monopole issues.
The Klein-Gordon equation $\ddot{\phi}+3H\dot{\phi}+V^{\prime}=0$, which
governs the dynamics of the scalar field $\phi$, connects the scalar field
with the scalar potential $V(\phi)$ in inflation theory. The slow-roll
approximation involves ignoring the kinetic components and the second temporal
derivatives of the field $\phi$, i.e. $\dot{\phi}^{2}\ll V$ and
$\ddot{\phi}\ll V$ respectively, thus,

$\displaystyle H^{2}$ $\displaystyle\sim$
$\displaystyle\frac{V(\phi)}{3M_{p}^{2}},$ (11) $\displaystyle\dot{\phi}$
$\displaystyle\sim$ $\displaystyle-\frac{V^{\prime}\left(\phi\right)}{3H}.$
(12)

The Hubble parameter may be used to generate the slow-roll parameters as [53]

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle-\frac{\dot{H}}{H^{2}},$ (13) $\displaystyle\eta$
$\displaystyle=$
$\displaystyle\frac{\dot{\epsilon}}{H\epsilon}\approx-\frac{\ddot{H}}{2\dot{H}H}.$
(14)

we define the number of e-folds associated with inflation from the beginning
$t$ to $t_{end}$ as follows

$N\equiv\ln\left(\frac{a_{end}}{a}\right)=\int_{t}^{t_{end}}Hdt,$ (15)

The spectral index $n_{s}$ describes the relationship between curvature
disturbances and slow roll parameters. The e-folding number $N$, which
reflects the pace of the Univers expansion during this era, the tensor to
scalar perturbations ratio $r$, and the spectral index $n_{s}$ are also used
to investigate the period of inflation.

$\displaystyle n_{s}-1$ $\displaystyle=$ $\displaystyle-6\epsilon+2\eta,$ (16)
$\displaystyle r$ $\displaystyle=$ $\displaystyle 16\epsilon.$ (17)

Next, we will discuss how the tree level potential inflationary model can be
described in the context of $f(Q)$ gravity.

##  III The tree level potential in $f(Q)$ gravity

###  III.1 The inflationary parameters on $f(Q)$ gravity

In the context of cosmic inflation, we consider the model $f(Q)=\alpha Q,$
with $\alpha=F\neq 0$, by taking Eqs. (7), (8), (9) and (10) we obtain the
following equations

$\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle\frac{\kappa\rho\left(1+\omega\right)}{2\alpha},$ (18)
$\displaystyle 3H^{2}$ $\displaystyle=$
$\displaystyle\kappa\rho_{eff}=-\frac{\kappa\rho}{\alpha},$ (19)
$\displaystyle 2\dot{H}+3H^{2}$ $\displaystyle=$ $\displaystyle-\kappa
p_{eff}=\frac{\kappa\omega\rho}{\alpha},$ (20) $\displaystyle\rho$
$\displaystyle=$ $\displaystyle-\frac{3\alpha H^{2}}{\kappa}.$ (21)

In of $f(Q)$ gravity, we consider the tachyonic field leads from Eqs. (31),
(32) along with Eqs.(17), (29) gives

$\displaystyle 3H^{2}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{\alpha}V(\phi),$ (22) $\displaystyle
2\dot{H}+3H^{2}$ $\displaystyle=$ $\displaystyle\frac{\kappa}{\alpha}\omega
V(\phi).$ (23)

Taking into account the previous two equations one gets

$\dot{H}=\frac{\kappa}{2\alpha}\dot{\phi}^{2},$ (24)

after the derivation of Eq.(19) and using Eq. (20), we obtain the modified
equation of motion

$\ddot{\phi}+3H\dot{\phi}+V^{\prime}=0.$ (25)

From Eqs. (LABEL:eps2) we know that the condition $\eta\ll 1$ leads to
$\ddot{\phi}\ll 3H\dot{\phi},$ which means that the time derivative of the
tachyonic field is given as $\dot{\phi}\sim-V^{\prime}\left(\phi\right)/3H$.
Now, let us use the slow-roll parameters that will be calculated in the
context of the $f(Q)$ gravity. For this purpose, we should insert Eqs. (22)
and (24) into the slow roll parameter, knowing that we need to consider the
case $\dot{\phi}^{2}<2/3$ which gives

$\displaystyle\epsilon$ $\displaystyle\approx$
$\displaystyle\frac{3}{2}\frac{\dot{\phi}^{2}}{V}\simeq-\frac{\alpha}{2\kappa}\left(\frac{V^{\prime}}{V}\right)^{2},$
(26) $\displaystyle\eta$ $\displaystyle\approx$
$\displaystyle-\frac{\ddot{\phi}}{H\dot{\phi}}\simeq-\frac{\alpha}{\kappa}\frac{V^{\prime\prime}}{V},$
(27)

the e-folds number for this chosen model can be written using Eq.(22) and
Eq.(12) in the following way

$N\equiv\int_{\phi_{k}}^{\phi_{end}}\frac{H}{\dot{\phi}}d\phi\approx-\int_{\phi_{end}}^{\phi_{k}}\frac{\kappa}{\alpha}\frac{V}{V^{\prime}}d\phi.$
(28)

Our goal next is to calculate the inflationary parameters for tree level
potential in the chosen model of $f(Q)$ gravity.

###  III.2  Constraints on the tree level potential in $f(Q)$ gravity

The proposed tree level Higgs potential [6], [7], [54] is given by

$V(\phi)=V_{0}\left[1-\left(\frac{\phi}{\mu}\right)^{2}\right]^{2},$ (29)

here $V_{0}$ is the vacuum energy density at the origin and $\mu$ is the
vacuum expectation value of the inflation, this potential will be discussed
for three branches: $\phi^{2}\gg\mu^{2}$ (AV branch), $\phi\sim\mu$ (Near
branch), and $\phi^{2}\ll\mu^{2}$ (BV branch).

####  AV branch: $\phi^{2}\gg\mu^{2}$

In the case of an AV branch, the potential becomes

$V(\phi)=V_{0}\left(\frac{\phi}{\mu}\right)^{4},$ (30)

the slow-roll parameters in this case are determined by the values of the
field at the beginning of inflation $\phi_{k}$ and is given by

$\displaystyle\epsilon$ $\displaystyle\simeq$
$\displaystyle-\frac{8\alpha}{\kappa}\frac{1}{\phi_{k}^{2}},$ (31)
$\displaystyle\eta$ $\displaystyle\simeq$
$\displaystyle-\frac{12\alpha}{\kappa}\frac{1}{\phi_{k}^{2}}.$ (32)

Equation (28), on the other hand, offers the integral growth from $\phi_{k}$
to $\phi_{end}$ as

$N\simeq-\frac{\kappa}{8\alpha}\left(\phi_{k}^{2}-\phi_{end}^{2}\right)\simeq-\frac{\kappa}{8\alpha}\phi_{k}^{2}.$
(33)

Now since the inflationary parameters are determined by the value of the
inflaton field $\phi_{k}$, we can link these parameters to observation using
Eq.(16) to obtain $\left(n_{s}-1\right)/3=8\alpha/\kappa\phi_{k}^{2}.$

####  Near branch: $\phi\sim\mu$

We will analyze the physical consequences of the solutions in the situation of
$\phi\sim\mu$ when the tree level potential becomes

$V(\phi)=V_{0}\left[1-\left(\frac{\phi}{\mu}\right)^{2}\right]^{2}=\frac{V_{0}}{\mu^{4}}\left(\phi+\mu\right)^{2}\left(\mu-\phi\right)^{2}$
(34)

in the interests of clarity, we take the approximation: $\phi+\mu=2\mu$ where
$\phi\sim\mu$, whereas the potential becomes

$V(\phi)\simeq\frac{4V_{0}}{\mu^{2}}\left(\mu-\phi\right)^{2}.$ (35)

In this context, one can also consider the slow-roll parameters for this
potential by

$\displaystyle\epsilon$ $\displaystyle\simeq$
$\displaystyle-\frac{2\alpha}{\kappa}\frac{1}{\left(\phi_{k}-\mu\right)^{2}},$
(36) $\displaystyle\eta$ $\displaystyle\simeq$
$\displaystyle-\frac{2\alpha}{\kappa}\frac{1}{\left(\phi_{k}-\mu\right)^{2}}.$
(37)

On the other hand, for the Near branch _,_ the number of e-folds gives

$N\simeq-\frac{\kappa}{2\alpha}\left[\left(\phi_{k}-\mu\right)^{2}-\left(\phi_{end}-\mu\right)^{2}\right]\simeq-\frac{\kappa}{2\alpha}\left(\phi_{k}-\mu\right)^{2}$
(38)

Because the inflationary parameters are defined by the value of the inflaton
field $\phi_{k}$, we can now relate these parameters to observations to obtain
$\left(n_{s}-1\right)=8\alpha/\kappa\left(\phi_{k}-\mu\right)^{2}.$

####  BV branch: $\phi^{2}\ll\mu^{2}$

In this section, we will focus on inflationary potential, which may be written
in the BV branch as [7]

$V(\phi)=V_{0}\left(1-2\frac{\phi^{2}}{\mu^{2}}\right).$ (39)

In this situation, the slow-roll settings can also be considered. knowing that
$\left(\phi/\mu\right)^{2}\ll 1$. The two first parameters for this branch are
given by

$\displaystyle\epsilon$ $\displaystyle\simeq$
$\displaystyle-\frac{8\alpha}{\kappa}\frac{\phi_{k}^{2}}{\mu^{4}},$ (40)
$\displaystyle\eta$ $\displaystyle\simeq$
$\displaystyle\frac{4\alpha}{\kappa}\frac{1}{\mu^{2}}.$ (41)

On the other hand, the number of e-folding $N$ can be expressed as

$N\simeq-\frac{\kappa}{4\alpha}\mu^{2}\left(\ln\left(\phi_{end}\right)-\ln\left(\phi_{k}\right)\right),$
(42)

here we can simply calculate $\phi_{end}$ considering $\epsilon=1$ that can be
applied at the end of inflation when $\phi\rightarrow\phi_{end}.$

We can now link the inflationary parameters to observations. However, this
time we have to specify precise values of $\phi_{k},$ $\mu,$ and $\alpha$
parameters since the spectral index parameter is not specified by the value of
the inflaton field $\phi_{k}$ and taking into account the approximation
$\left(\phi/\mu\right)^{2}\ll 1,$ $n_{s}$ is given by the following equation$\
\left(n_{s}-1\right)=8\alpha/\kappa\mu^{2}.$

##  IV Reheating mechanism from the inflaton decay

The process of reheating plays a crucial role in the evolution of the
universe. It marks the emergence of matter towards the end of the rapid
expansion phase when the inflaton field breaks down, and standard model
particles are formed, resulting in a rise in temperature that brings about a
restoration of the standard Big Bang cosmology model. [55]. As the universe
expands, the amplitude of the oscillations gradually decreases, and the
oscillating field transfers its energy to the newly formed particles. In
addition, we take into account the crucial interaction Lagrangian [56].

$L_{int}=\frac{1}{2}g^{2}\phi^{2}\chi^{2}+h\psi\bar{\psi}\phi,$ (43)

where $h$ and $\sigma$ are dimensionless couplings and $\sigma$ has the
dimensions of mass. The decay rate of the inflaton field is defined in the
following way
$\Gamma=\Gamma_{\left(\phi\rightarrow\chi\chi\right)}+\Gamma_{\left(\phi\rightarrow\psi\psi\right)}$
with $\Gamma_{\left(\phi\rightarrow\chi\chi\right)}=g^{4}\sigma^{2}/8\pi
m_{\phi},$ $\Gamma_{\left(\phi\rightarrow\psi\psi\right)}=h^{2}m_{\phi}/8\pi$
and $m_{\phi}$ is the mass of the inflaton field.

Next, we demonstrate the correlation between the decay rate and reheating
parameters. We start with the Friedmann equation in the $f(Q)$ gravity
scenario, which is $H{{}^{2}}=-\kappa\rho/3\alpha$. Subsequently, we express
the energy density at the onset of the reheating phase as a function of the
reheating temperature.

$\rho_{re}=\frac{\pi^{2}}{30}g_{\ast}T_{re}^{4},$ (44)

The reheating relativistic degree of freedom is denoted as $g_{\ast}$. In the
reheating phase, we can substitute $\rho$ with $\rho_{re}$ to obtain the
Friedmann equation expressed as follows:

$H^{2}=-\frac{\kappa}{3\alpha}\frac{\pi^{2}}{30}g_{\ast}T_{re}^{4}$ (45)

After inflation ends, the energy densities of the scalar field and radiation
become comparable when $H\simeq\Gamma$. From this point onwards, the Universe
is governed by radiation, which determines the reheating temperature. The
temperature can be expressed as:

$T_{re}=\left(-\frac{3\cdot
30\alpha}{\pi^{2}g_{\ast}}\frac{\Gamma^{2}}{\kappa}\right)^{\frac{1}{4}}.$
(46)

Figure 1: The evolution of the reheating temperature as a function of the
total decay rate for different values of $\alpha.$

In Fig. (1) we plot the variation of the reheating temperature $T_{re}$ as a
function of the $\alpha$ parameter related to the $f(Q)$ gravity model, for
mathematical consistency we observe that our model is consistent only for
negative values of $\alpha.$ To study the behavior of $T_{re}$ we chose
different values of $\alpha$ bounded in the following way $\alpha\leq-1$ which
makes the increasing curve of reheating temperature moves towards lower values
of reheating temperature for $\Gamma\propto 10^{21}GeV$ , our results
illustrate the predictions of Eq.(46) which lead to the conclusion that higher
values of the decay rate energy will lead to higher reheating temperature.

##  V Conclusion

In recent years, despite the undeniable success of ’the standard model of
cosmology’ governed by Einstein’s general theory of relativity (GTR), its
limitations have become apparent in various aspects. These include
cosmological singularity and black holes, tensions in $H_{0}$ and $\sigma_{8}$
on observational grounds, and most importantly, the need for yet undetected
dark sectors to explain the late-time cosmic acceleration. These limitations
have prompted researchers to explore alternative theories of gravity.
Teleparallel gravity has emerged as a significant area of research in this
regard. Specifically, researchers have considered replacing the Levi-Civita
connection, which forms the foundation of GTR, with an affine connection on
spatially flat spacetime with vanishing torsion. This allows non-metricity to
entirely define gravity, giving rise to the theory known as ’symmetric
teleparallel gravity’. The focus of our paper is to examine tree level Higgs
inflation and reheating in $f(Q)$ gravity. Our study involves limiting the
inflationary parameters and the reheating temperature. We begin by outlining
the formalism for tree level inflation within the chosen modified $f(Q)$
model. We discovered that the slow roll and e-folds number related to
inflation relies on the $f(Q)$ model $\alpha$ parameter. Additionally, we have
computed the reheating in relation to the decay rate. We selected various
values of $\alpha$ that are bounded by $\alpha\leq-1$. This constraint causes
the upward trend of the reheating temperature curve to shift towards lower
values of reheating temperature. This leads to the conclusion that increasing
the energy of the decay rate results in a corresponding increase in the
reheating temperature.

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