# Insights into Gravitinos Abundance, Cosmic Strings and Stochastic
Gravitational Wave Background

K. El Bourakadi1,2<EMAIL_ADDRESS>G. Otalora1
<EMAIL_ADDRESS>A. Burton-Villalobos1
<EMAIL_ADDRESS>H. Chakir2<EMAIL_ADDRESS>M.
Ferricha-Alami3<EMAIL_ADDRESS>M. Bennai3<EMAIL_ADDRESS>1Departamento de Física, Facultad de Ciencias, Universidad de Tarapacá,
Casilla 7-D, Arica, Chile.
2Subatomic Research and Applications Team, Faculty of Science Ben M’sik,
Casablanca Hassan II University, Morocco
3Quantum Physics and Magnetism Team, LPMC, Faculty of Science Ben M’sik,
Casablanca Hassan II University, Morocco.

###### Abstract

In this paper, we investigate D-term inflation within the framework of
supergravity, employing the minimal Kähler potential. Our study reveals that
this model can overcome the $\eta$-problem found in F-term models.
Additionally, we explore reheating dynamics and gravitino production,
emphasizing the interplay between reheating temperature, spectral index, and
gravitino abundance. Our analysis indicates that gravitino production is
sensitive to the equation of state during reheating, affecting the reheating
temperature and subsequent dark matter relic density. Perturbation theory
reveals that Primordial Gravitational Waves (PGWs) evolve according to second-
order effects arising from first-order curvature perturbations. In fact, the
spectral energy density of these waves is particularly relevant to Pulsar
Timing Array (PTA) observations. Furthermore, we analyze gravitational waves
generated by cosmic strings, providing critical constraints on early Universe
dynamics and cosmic string properties. The dimensionless string tension
significantly influences the stochastic gravitational wave background (SGWB)
produced by cosmic strings post-inflation. Finally, the SGWB produced by
cosmic strings is influenced by the energy scales of inflation and string
formation.

##  I Introduction

Supersymmetric (SUSY) inflation [1, 2, 3, 4] provides an appealing framework
for connecting inflation with particle physics models rooted in grand unified
theories (GUTs) [5]. In the simplest models, the soft supersymmetry breaking
terms are crucial for aligning predictions of the scalar spectral index with
cosmic microwave background (CMB) observations [6],[7]. SUSY provides an
appealing solution to the analogous hierarchy problem in the Standard Model
(SM) of particle physics and also facilitates the unification of the three
gauge couplings. Notably, the local version of supersymmetry — supergravity
(SUGRA) — would dominate the dynamics of the early Universe when high-energy
physics played a crucial role. Therefore, considering inflation within the
framework of supergravity is quite natural. However, integrating inflation
into supergravity is a challenging task. This difficulty arises mainly because
the SUSY breaking potential term, essential for inflation, typically imparts
an additional mass to the would-be inflaton, thereby disrupting the flatness
of the inflaton potential [8, 9]. To achieve successful inflation that aligns
with observational data from large-scale structures and the anisotropy of CMB
radiation, the potential of the scalar field, known as the inflaton, must be
very flat. This required flatness of the potential can be achieved through the
mechanisms provided by supersymmetry or supergravity. In supersymmetric
models, the scalar potential is composed of contributions from both the F-term
and D-term.

Specifically, in F-term inflation models, where the vacuum energy that drives
the inflationary expansion predominantly comes from the F-term, the inflaton
mass is generally on the same order as the Hubble parameter $H$ during
inflation [10]. Consequently, achieving a sufficiently long expansion to
address the horizon and flatness problems is challenging. This issue is
referred to as the eta problem in inflation models within the framework of
supergravity. Conversely, D-term inflation, in which the vacuum energy is
supplied by the D-term, does not encounter this problem [11, 12]. Therefore,
from this perspective, D-term inflation is more appealing than F-term
inflation. However, it has been discovered that the D-term inflation model
also has its own set of issues [13]. For instance, from an observational
standpoint, cosmic strings generated after inflation can significantly impact
the spectrum of CMB anisotropy [14], as this model is a type of hybrid
inflation. Additionally, there are concerns about the validity of the D-term
inflation potential, since the inflaton requires a large initial value on the
order of the (sub-)Planck scale for a natural model parameter[15]. Therefore,
D-term inflation appears to be under considerable scrutiny. Additionally,
within the framework of (heterotic) string models, there are two further
issues: the runaway behavior of the dilaton and an excessively large magnitude
of the Fayet-Iliopoulos (FI) term [16].

Given the current and anticipated future constraints on inflaton decay via CMB
limits on $N_{re}$ and the competition with constraints from gravitino
abundance in supersymmetric models, we revisit the issue of gravitino
production following inflation. A well-understood source of gravitinos is
their production through particle collisions in the thermal plasma that fills
the Universe after reheating [17, 18, 19]. However, gravitinos could also have
been produced by particle collisions before the reheating process was
complete, either from collisions of relativistic inflaton decay products
before thermalization or within any dilute thermal plasma formed by these
collisions while inflaton decay was still ongoing [20]

Cosmic strings are intriguing messengers from the early Universe due to their
distinctive signatures in the stochastic gravitational wave background (SGWB).
Recent NANOGrav 12.5-year data has provided evidence of a stochastic process
at nanohertz frequencies, which has been interpreted as SGWB in numerous
recent studies [21, 22, 23, 24]. Relic gravitational waves (GWs) offer a
fascinating window into the exploration of early Universe cosmology [25].
Cosmic strings generate powerful bursts of gravitational radiation that could
be detected by interferometric gravitational wave detectors such as LIGO,
Virgo, and LISA [26, 27]. Additionally, the SGWB can be detected or
constrained through various observations, including big bang nucleosynthesis
(BBN), pulsar timing experiments, and interferometric gravitational wave
detectors [28].

The paper is organized as follows, in Sec. II we examine supergravity
formalism, in Sec. III we focus on the D-term model for the case of the
minimal Kähler potential. In Sec. IV, we apply the previous findings in the
study of the reheating process and its relation to gravitinos production. In
Sec. V, we link the primordial gravitational waves production from different
epochs in the light of the chosen supergravity formalism, which is constrained
by PTA observations. In Sec. VI, we study the gravitational waves production
due to networks of cosmic strings. In Sec. VII, we conclude.

##  II Supergravity Formalism

The supergravity lagrangian scalar part can be defined by the superpotential
$W\left(\Theta_{i}\right)$, Kähler potential
$K\left(\Theta_{i},\Theta_{i}^{\ast}\right)$, and gauge kinetic function
$f\left(\Theta_{i}\right)$ [29, 30, 31]. $W$ and $f$ are holomorphic functions
of complex scalar fields, whereas the initial function $K$ is non-holomorphic
and operates as a real function of the scalar fields $\Theta_{i}$ and their
conjugates $\Theta_{i}^{\ast}$. The three functions mentioned above are
initially defined in terms of (chiral) superfields. Since our primary focus is
on the scalar component of a superfield, we equate a superfield with its
complex scalar counterpart, denoting both by the same symbol. The action of
the complex scalar fields, when minimally coupled to gravity, comprises both
kinetic and potential components.

$S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{\sqrt{-g}}{\mathcal{L}}_{kin}-V\left(\Theta_{i},\Theta_{i}^{\ast}\right)\right].$
(1)

The kinetic terms of the scalar fields are dictated by the Kähler potential,
denoted as K, and expressed as follows:

$\frac{1}{\sqrt{-g}}{\mathcal{L}}_{kin}=-K_{ij\ast}D_{\mu}\Theta_{i}D_{\nu}\Theta_{j}^{\ast}g^{\mu\nu},$
(2)

with $K_{ij\ast}=\partial^{2}K/\partial\Theta_{i}\partial\Theta_{j}^{\ast},$
and $D_{\mu}$ denotes the covariant derivative in the gauge field context. The
potential $V$ for scalar fields $\Theta_{i}$ consists of two components,
namely, the F-term $V_{F}$ and the D-term $V_{D}$. Here the F-term component
which is determined by both the superpotential $W$ and the Kähler potential
$K$ is given by,

$V_{F}=e^{K}\left[D_{\Theta_{i}}WK_{ij\ast}^{-1}D_{\Theta_{j}^{\ast}}W^{\ast}-\frac{3\left|W\right|^{2}}{M_{p}^{2}}\right],$
(3)

where

$D_{\Theta_{i}}W=\frac{\partial
W}{\partial\Theta_{i}}+\frac{1}{M_{p}^{2}}\frac{\partial
K}{\partial\Theta_{i}}W.$ (4)

Conversely, the D-term $V_{D}$ is associated with gauge symmetry and is
defined by the gauge kinetic function and the Kähler potential,

$V_{D}=\frac{1}{2}\sum\left[Ref_{ab}\left(\Theta_{i}\right)\right]^{-1}g_{a}g_{b}D_{a}D_{b},$
(5)

where

$D_{a}=\Theta_{i}\left(T_{a}\right)_{j}^{i}\frac{\partial
K}{\partial\Theta_{i}}+\xi_{a}.$ (6)

In this context, the subscripts $\left(a\right)$ and $\left(b\right)$ denote
gauge symmetries, where $g_{a}$ represents the gauge coupling constant, and
$T_{a}$ stands for the corresponding generator. The term $\xi_{a}$ is referred
to as the Fayet–Iliopoulos $\left(FI\right)$ term, which is non-zero
exclusively when the gauge symmetry is Abelian, specifically $U(1)$ symmetry.
Keep in mind that within a supersymmetric framework, the tree-level potential
during inflation comprises both an F-term and a D-term. These two components
exhibit distinct characteristics, and in the context of all inflationary
models, it is noteworthy that dominance is typically attributed to only one of
these terms.

In certain models rooted in Supergravity, such as the one under consideration
here with a nonminimal Kähler potential, the exponential term $e^{K}$
contributes a term $V$ to the effective mass squared, roughly of the order of
$H^{2}$ (the Hubble scale squared), affecting all scalar fields. Consequently,
inflation experiences an effective mass squared
$V_{g}=\sum\left|\frac{\partial W}{\partial\Theta_{i}}\right|^{2}=3H^{2}$
[32], introducing a contribution to the slow-roll parameter $\eta$. This
contribution results in a violation of the slow-roll conditions
($\left|\eta\right|\ll 1$) and leads to the so-called $\eta$-problem

$\eta=M_{p}^{2}\frac{V^{\prime\prime}}{V}\simeq\frac{1}{3}\left(\frac{m}{H}\right)^{2}\simeq
1.$

Addressing this challenge has prompted the proposal of various approaches.
Here, we are only interested in the use of D-term hybrid inflation, which can
provide positive energy in D-term potential. This allows for the realization
of inflation while avoiding the $\eta$-problem.

##  III D-term Hybrid Model

Constructing an inflation model in supergravity faces a significant challenge
arising from the F-term, particularly the exponential factor associated with
it. Hence, achieving a positive potential energy in the D-term could pave the
way for successful inflation without encountering the $\eta$ problem. This
insight was initially highlighted in Ref. [33].

Let us consider a D-term model of hybrid inflation as introduced in [34, 35],

$W=\lambda S\Theta_{+}\Theta_{-},$ (7)

where $\lambda$ is the coupling constant, and $S,$ $\Theta_{+},$ and
$\Theta_{-}$ are three (chiral) superfields. The superpotential remains
unchanged under a $U(1)$ gauge symmetry, with assigned charges of $0$, $+1$,
and $-1$ for the fields $S$, $\Theta_{+}$, and $\Theta_{-}$, respectively.
Additionally, it exhibits an R symmetry governing the transformation of these
fields as follows:

$S\left(\theta\right)\rightarrow e^{2i\alpha}S\left(\theta
e^{i\alpha}\right),~{}~{}~{}~{}~{}~{}\Theta_{+}\Theta_{-}(\theta)\rightarrow\Theta_{+}\Theta_{-}(\theta
e^{i\alpha}),$ (8)

and considering the minimal Kähler potential,

$K=\left|S\right|^{2}+\left|\Theta_{+}\right|^{2}+\left|\Theta_{-}\right|^{2}.$
(9)

From Eqs. (3) and (5) the tree level scalar potential is given by,

$\displaystyle V\left(S,\Theta_{+},\Theta_{-}\right)$ $\displaystyle=$
$\displaystyle\lambda^{2}e^{\left|S\right|^{2}+\left|\Theta_{+}\right|^{2}+\left|\Theta_{-}\right|^{2}}\left[\left|\Theta_{+}\Theta_{-}\right|^{2}+\left|S\Theta_{-}\right|^{2}+\left|S\Theta_{+}\right|^{2}+\left(\left|S\right|^{2}+\left|\Theta_{+}\right|^{2}+\left|\Theta_{-}\right|^{2}+3\right)\left|S\Theta_{+}\Theta_{-}\right|^{2}\right]$
(10)
$\displaystyle+\frac{g^{2}}{2}\left(\left|\Theta_{+}\right|^{2}-\left|\Theta_{-}\right|^{2}+\xi\right).$

Here, $g$ represents the gauge coupling constant, $\xi$ is a non-zero FI term,
and we have adopted a minimal gauge kinetic function. This potential exhibits
a distinctive global minimum $V=0$ at $S=\Theta_{+}=0,\Theta_{-}=\sqrt{\xi}.$
Yet, when $\left|S\right|$ is significantly large, the potential displays a
local minimum with a positive energy density at $\Theta_{+}=\Theta_{-}=0$. To
determine the critical value $S_{c}$ of $\left|S\right|$, we compute the mass
matrix of $\Theta_{+}$ and $\Theta_{-}$ along the inflationary trajectory
where $\Theta_{+}=\Theta_{+}=0$. This is expressed as:

$V_{mass}=m_{+}^{2}\left|\Theta_{+}\right|^{2}+m_{-}^{2}\left|\Theta_{-}\right|^{2},$
(11)

with

$m_{+}^{2}=\lambda^{2}\left|S\right|^{2}e^{\left|S\right|^{2}}+g^{2}\xi,~{}~{}~{}~{}~{}~{}~{}~{}~{}m_{-}^{2}=\lambda^{2}\left|S\right|^{2}e^{\left|S\right|^{2}}-g^{2}\xi.$
(12)

Hence, provided that $m_{-}^{2}\geq 0$, corresponding to $\left|S\right|\geq
S_{c}\simeq g\sqrt{\xi}/\lambda$ for $S_{c}\lesssim 1$, the local minimum at
$\Theta_{+}=\Theta_{+}=0$ remains stable, leading to inflation driven by the
positive potential energy density of $g^{2}\xi^{2}/2$. Moreover, this mass
split induces quantum corrections computed using the standard formula [32,
36],

$V_{1L}=\frac{1}{32\pi^{2}}\left[m_{+}^{4}\ln\left(\frac{\lambda^{2}\left|S\right|^{2}e^{\left|S\right|^{2}}+g^{2}\xi}{\Lambda^{2}}\right)+m_{-}^{4}\ln\left(\frac{\lambda^{2}\left|S\right|^{2}e^{\left|S\right|^{2}}+g^{2}\xi}{\Lambda^{2}}\right)-2\lambda^{4}\left|S\right|^{4}e^{\left|2S\right|^{2}}\ln\left(\frac{\lambda^{2}\left|S\right|^{2}e^{\left|S\right|^{2}}}{\Lambda^{2}}\right)\right],$
(13)

and when $\left|S\right|\gg S_{c},$ the effective potential of S $S$ during
inflation is approximated as,

$V\left(S\right)\simeq\frac{g^{2}\xi^{2}}{2}\left[1+\frac{g^{2}}{8\pi^{2}}\ln\left(\frac{\lambda^{2}\left|S\right|^{2}e^{\left|S\right|^{2}}}{\Lambda^{2}}\right)\right].$
(14)

Given that the potential is independent of the phase of the complex scalar
field $S$, we can propose its real part as $\sigma\equiv\sqrt{2}ReS$.
Subsequently, for $\sigma_{c}\ll\sigma\lesssim 1$, the effective potential of
the inflaton field $\sigma$ is expressed as:

$V\left(\sigma\right)\simeq\frac{g^{2}\xi^{2}}{2}\left[1+\frac{g^{2}}{8\pi^{2}}\ln\left(\frac{\lambda^{2}\sigma^{2}}{2\Lambda^{2}}\right)\right].$
(15)

During the inflationary period, the slow roll slow-roll parameters take the
form

$\displaystyle\epsilon$ $\displaystyle\simeq$
$\displaystyle\frac{g^{4}}{32\pi^{4}\sigma^{2}},$ (16) $\displaystyle\eta$
$\displaystyle\simeq$ $\displaystyle-\frac{g^{2}}{4\pi^{2}\sigma^{2}},$ (17)

and the e-folding number N is estimated as

$N\simeq\frac{2\pi^{2}}{g^{2}}\left(\sigma_{k}^{2}-\sigma_{end}^{2}\right).$
(18)

Throughout the inflationary regime, curvature perturbations arose from
inflaton fluctuations. The amplitude of these perturbations in the comoving
gauge ${\mathcal{R}}$ [37, 38, 39, 40, 41], measured on the comoving scale of
$2\pi/k$, is determined as,

${\mathcal{R}}^{2}\simeq\frac{1}{24\pi^{2}}\frac{V}{\epsilon}=\frac{N}{3}\xi^{2}.$
(19)

Here, $k$ represents the epoch when the $k$ mode exited the Hubble radius in
the course of inflation, as indicated in [42, 43, 46]. Conversely,
gravitational wave tensor perturbations $\mathit{h}$ are also generated, and
their amplitude on the comoving scale of $2\pi/k$ is specified in Ref. [47],
and calculated in this model as,

$h_{k}^{2}\simeq\frac{2}{3\pi^{2}}V\left(\sigma_{k}\right)=\frac{g^{2}\xi^{2}}{3\pi^{2}}.$
(20)

Then, the tensor-to-scalar ratio $r$ is given by

$r\equiv\frac{h_{k}^{2}}{{\mathcal{R}}^{2}}\simeq 16\epsilon$ (21)

where $h_{k}^{2}=2H_{k}^{2}/\pi^{2}$, and for a chosen pivot scale, the power
spectrum of scalar perturbation can be considered as ${\mathcal{R}}^{2}\simeq
A_{s}$.

##  IV  Reheating Dynamics and Gravitinos Production

In the initial stages, the quasi-de-Sitter phase is driven by the inflaton
field, resulting in $N_{k}$ e-folds of expansion. The comoving horizon scale
decreases proportionally to $\sim a^{-1}$ during this period. Following the
conclusion of accelerated expansion and the subsequent expansion of the
comoving horizon, the reheating phase begins [44]. After an additional
$N_{re}$ e-folds of expansion, all the energy stored in the inflaton field is
completely dissipated, leading to the formation of a hot plasma with a
reheating temperature of $T_{re}$. Following this phase, the Universe
undergoes $N_{RD}$ e-folds of expansion in a state of radiation domination
before transitioning into a state of matter domination.

In cosmology, we observe perturbation modes that exhibit magnitudes comparable
to the size of the cosmic horizon. For instance, Planck identifies the pivot
scale at $k=0.05Mpc^{-1}$ [45]. The comoving Hubble scale, denoted as
$a_{k}H_{k}=k$ is associated with the current timescale in relation to the
moment when this particular mode crossed the horizon [48, 49],

$\frac{k}{a_{0}H_{0}}=\frac{a_{k}}{a_{end}}\frac{a_{end}}{a_{re}}\frac{a_{re}}{a_{eq}}\frac{a_{eq}H_{eq}}{a_{0}H_{0}}\frac{H_{k}}{H_{eq}}.$
(22)

Quantities represented by a subscript $\left(k\right)$ are calculated at the
point of horizon exit. Other subscripts denote various epochs, including the
end of inflation $(_{end})$, reheating $(_{re})$, radiation-matter equality
$(_{eq})$, and the present time $(_{0})$. It is noteworthy that
$\ln\left(\frac{a_{k}}{a_{end}}\right)=N_{k},$
$\ln\left(\frac{a_{end}}{a_{re}}\right)=N_{re}$ and
$\ln\left(\frac{a_{re}}{a_{eq}}\right)=N_{RD}.$

A connection between the temperature at the end of reheating $T_{re}$ and the
CMB temperature $T_{0}$ can be established by considering factors is given by,

$T_{re}=\left(\frac{43}{11g_{re}}\right)^{\frac{1}{3}}\left(\frac{a_{0}T_{0}}{k}\right)H_{k}e^{-N_{k}}e^{-N_{re}}.$
(23)

On the other hand, the reheating duration can be expressed as,

$N_{re}=\frac{1}{3\left(1+\omega_{re}\right)}\ln\left(\frac{30\cdot\frac{3}{2}V_{end}}{\pi^{2}g_{re}T_{re}^{4}}\right),$
(24)

and thus, the final form of reheating temperature is described as a function
of the number of inflationary e-foldings $N_{k}$, the Hubble parameter $H_{k}$
and the potential value at the end of inflation $V_{end}$ as [43, 46, 50],

$T_{re}=\left[\left(\frac{43}{11g_{re}}\right)^{\frac{1}{3}}\frac{a_{0}T_{0}}{k}H_{k}e^{-N_{k}}\left[\frac{3^{2}\cdot
5V_{end}}{\pi^{2}g_{re}}\right]^{-\frac{1}{3\left(1+\omega_{re}\right)}}\right]^{\frac{3\left(1+\omega_{re}\right)}{3\omega_{re}-1}}.$
(25)

To determine the reheating temperature, $T_{re}$, for a specific model, one
must calculate $N_{k}$, $H_{k}$, $V_{end}$, and the potential at the end of
inflation by using the formula $V_{end}=V\left(\sigma_{end}\right)$, and
knowing that $\sigma_{end}$ is determined considering $\left|\eta\right|=1.$

The SUGRA effects result in the coupling of the inflaton field $\phi$ to all
matter fields, provided there is a non-zero vacuum expectation value (VEV).
The interactions with fermions are appropriately expressed in the context of
the total Kähler potential, denoted as $G=K+\ln|W|^{2}$, and such that:

${\mathcal{L}}=-\frac{1}{2}e^{G/2}G_{\phi
ijk}\phi\psi^{i}\psi^{j}\varphi^{k}+h.c.$ (26)

In the given context, $\varphi^{i}$ represents a scalar field, and $\psi^{i}$
represents a fermion in a 2-spinor representation. We make the assumption that
$G_{i}$ is much smaller than ${\mathcal{O}}(1)$ for all fields, excluding the
field responsible for SUSY breaking. The presence of the SUSY breaking field
may lead to the suppression of the contribution proportional to $G_{\phi}$
during the inflaton decay due to interference [51, 52]. In this section, we
simplify our analysis by assuming the minimal Kähler potential. Consequently,
the Kähler potential lacks non-renormalizable terms. Despite isolating the
inflaton field from other fields in the global SUSY Lagrangian, the presence
of (SUGRA) corrections enables its decay. The coupling constants are expanded
at the vacuum state [53].

$G_{\phi ijk}=-\frac{W_{\phi}}{W}\frac{W_{ijk}}{W}+\frac{W_{\phi
ijk}}{W}\simeq K_{\phi}\frac{W_{ijk}}{W}+\frac{W_{\phi ijk}}{W}.$ (27)

In previous analysis [53], the assumption was made that the Vacuum Expectation
Values (VEVs) are effectively negligible for all fields except the inflaton.
We employed the condition $G_{\phi}\ll\left\langle\phi\right\rangle$ in the
final equation. Notably, we observed that the outcome remains invariant under
Kähler transformation, and these constants tend to zero in the global SUSY
limit. Subsequently, the decay rates are computed

$\Gamma_{3/2}\left(\phi\longrightarrow\psi^{i}\psi^{j}\varphi^{k}\right)\simeq\frac{N_{f}}{1536\pi^{3}}\left|Y_{ijk}\right|^{2}\left(\frac{\left\langle\phi\right\rangle}{M_{p}}\right)^{2}\frac{m_{\phi}^{3}}{M_{p}^{2}},$
(28)

Here, $N_{f}$ represents the count of final states, and the Yukawa coupling
$Y_{ijk}$ is denoted as $W_{ijk}$. In this context, we have disregarded the
masses of the particles in the final state and employed
$K=\varphi^{\dagger}\varphi$ for the inflaton. Additionally, it is assumed
here that the particles $\psi^{i}$ and $\psi^{j}$ are non-identical . The
decay rates of the inflaton into scalar particles align with the previously
mentioned results. Indeed, considering the scalar potential,
$V=e^{G}\left(G^{j}G_{j}-3\right),$ the estimated decay amplitude of
$\phi^{\ast}\Longrightarrow\varphi^{i}\varphi^{j}\varphi^{k}$ is denoted as
$V_{\bar{\phi}ijk}$. Given the sizable SUSY mass of the inflaton, this
amplitude is approximately proportional to $e^{G/2}G_{\phi ijk}$, multiplied
by the inflaton mass, $m_{\phi}\simeq
e^{G/2}\left|G_{\bar{\phi}}^{\phi}\right|$. The inflaton’s decay into a pair
of gravitinos occurs at the rate $\Gamma$. The resulting gravitino abundance
$Y_{3/2}$ is then calculated, with $Y_{3/2}$ equal to the ratio of the final
gravitino abundance to the entropy density $\frac{n_{3/2}}{s}$ [54, 55, 56].

$Y_{3/2}\simeq 2\times
10^{-11}\left(\frac{10^{6}GeV}{T_{re}}\right)\left(\frac{m_{\phi}\times\left(\left\langle\phi\right\rangle\right)}{10^{27}GeV^{2}}\right)^{2}.$
(29)

Figure 1:  The gravitinos abundance $Y_{3/2}$ as functions of the spectral
index $n_{s}$ for different values of the equation of state $\omega.$

The plot in Fig. 1 displays the gravitino abundance $Y_{3/2}$ as a function of
the spectral index $n_{s}$ for different values of the inflaton mass
$m_{\phi}$. Each subplot corresponds to a specific $m_{\phi}$ values, ranging
from $10^{8}GeV$ to $10^{14}GeV$Ḋifferent curves representing various equation
of state parameter values during reheating $\omega$, with a shaded region
highlighting the Planck bound on $n_{s}=0.9649\pm 0.0042$ according to recent
observations [57].

From the given formalism, the abundance of gravitinos produced during the
reheating phase, denoted as $Y_{3/2}$ is significantly dependent on the
reheating temperature $T_{re}$,which is affected by $\omega$. The curves of
the gravitinos abundance tend towards the central value where all the lines
with the different equation of state values converge. This central value of
the abundance curves corresponds to the highest reheating temperature, and all
the lines coincide within the observed bound on the spectral index $n_{s}$.
Higher $m_{\phi}$ values generally lead to increased gravitino production when
considering the maximum reheating temperature, while different $\omega$ values
impact the reheating temperature and thus the gravitino abundance, causing
$Y_{3/2}$ to decrease away from the observed bound of $n_{s}$.

In fact, lower equation of state values tend to fall outside the $n_{s}$ bound
faster than higher values of $\omega$. Consequently, this plot underscores the
need to reconcile reheating scenarios with observational constraints on
gravitino abundance.

##  V Gravitational Waves Background

###  V.1 Scalar induced gravitational waves

Detecting the background of primordial gravitational waves would provide
strong evidence for the inflation paradigm and offer insights into the
fundamental physics of the early Universe [45]. Recent studies have focused on
Primordial Gravitational Waves (PGW) from the period immediately following
inflation [58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69]. Within the
inflationary scenario, the presence of a kinetic epoch preceding inflation
leads to a distinct blue tilt in the spectra of primordial gravitational waves
at higher frequencies [70]. Gravitational waves are identified as the
transverse-traceless component of metric perturbations. According to linear
perturbation theory, scalar, vector, and tensor modes do not interact. Now,
let us calculate the gravitational waves generated by second-order
gravitational interactions resulting from first-order curvature perturbations
[71, 72, 73, 74]. Using the Newtonian gauge, the perturbed metric is expressed
as [75]:

$ds^{2}=a^{2}\left(\tau\right)\left[-\left(1+2\Phi\right)d\tau^{2}+\left(\delta_{ij}\left(1-2\Phi\right)+\frac{h_{ij}}{2}\right)dx^{i}dx^{j}\right],$
(30)

where $\tau$ is the conformal time, $\Phi$ represents the first-order Bardeen
gravitational potential, and $h_{ij}$ denotes the second-order tensor
perturbation. We can reformulate the spectral abundance of gravitational
waves, defined as the energy density of gravitational waves per logarithmic
comoving scale, as [76, 77],

$\Omega_{GW}(k)=\frac{k^{2}}{12H_{0}}{\mathcal{T}}(\mathbf{\tau}_{0},\mathbf{k}){\mathcal{P}}_{t}(k).$
(31)

The time evolution of a gravitational wave field, denoted as
$h_{\mathbf{k}}(\tau_{i})$ at an initial conformal time $\tau_{i}$ and
characterized by its tensor spectrum, can be determined by computing the GW
transfer function
${\mathcal{T}}\mathbf{(\tau,k)=}h_{\mathbf{k}}(\tau)/h_{\mathbf{k}}(\tau_{i})$
[78]. Here, $h_{\mathbf{k}}(\tau)$ is evaluated at a conformal time
$\eta\gg\eta_{i}$ [79, 80]. The current conformal time is denoted by
$\mathbf{\tau}_{0}$, and the Hubble constant is represented by $H_{0}$ [85].
Our focus is on the spectral energy density parameter $\Omega_{GW}(k)$ at
Pulsar Timing Array (PTA) scales, where $f\sim{\mathcal{O}}(10-9)Hz$
corresponds to wavenumbers $k\sim{\mathcal{O}}(10^{6})Mpc^{-1}$ which are
significantly larger than the wavenumber linked to a mode crossing the horizon
at matter-radiation equality. Stated differently, the modes observed at PTA
scales crossed the horizon deep within the radiation era, well before matter-
radiation equality. In the regime where $k\gg k_{eq},$ the gravitational wave
spectral energy density associated with PTA signals can be expressed as [86,
87, 88, 89, 90, 91]

$\Omega_{GW}(f)=\frac{2\pi^{2}f_{yr}^{2}}{3H_{0}^{2}}A^{2}\left(\frac{f}{f_{yr}}\right)^{\alpha}.$
(32)

The NANOGrav measurements [92, 93], estimate ranges of several parameters
which are given as follows, $\alpha=\left(5-\gamma\right)$ with $\gamma=13/3$,
$f_{yr}$ is best estimated to $f_{yr}\simeq 3.1\times 10^{-8}$ and $H_{0}$ is
given by $H_{0}\equiv 100h~{}km/s/Mpc$. The relationship linking the amplitude
of the Pulsar Timing Array (PTA) signal, with the cosmological parameters were
found to be [94, 95]

$A=\sqrt{\frac{45\Omega_{m}^{2}A_{s}}{32\pi^{2}(\eta_{0}k_{eq})^{2}}}c\frac{f_{yr}}{\eta_{0}}\left(\frac{f_{yr}^{-1}}{f_{\star}}\right)^{\frac{n_{T}}{2}}\sqrt{r}.$
(33)

The dependence on $n_{T}$ in this context arises from the substantial ”lever
arm” between the Cosmic Microwave Background (CMB) pivot frequency, where
$A_{s}$ is constrained, and the frequency of the Pulsar Timing Array (PTA)
signal $\left[1yr^{-1}\right]$.

$g$ | $\sigma$ | $r$ | $A$ | $f(Hz)$ | $h^{2}\Omega_{gw}(f)$
---|---|---|---|---|---
$3\times 10^{-3}$ | $2\times 10^{-5}$ | $0.001$ | $1.44\times 10^{-15}$ | $4\times 10^{-8}$ | $2.91\times 10^{-9}$
$2\times 10^{-3}$ | $3\times 10^{-6}$ | $0.009$ | $4.33\times 10^{-15}$ | $2\times 10^{-8}$ | $1.85\times 10^{-8}$
$4\times 10^{-3}$ | $7\times 10^{-6}$ | $0.026$ | $9.38\times 10^{-15}$ | $1\times 10^{-8}$ | $3.78\times 10^{-8}$
$5\times 10^{-3}$ | $9\times 10^{-6}$ | $0.039$ | $1.03\times 10^{-14}$ | $8\times 10^{-7}$ | $5.07\times 10^{-7}$
$4\times 10^{-3}$ | $5\times 10^{-6}$ | $0.052$ | $1.04\times 10^{-14}$ | $6\times 10^{-7}$ | $5.86\times 10^{-7}$
$5\times 10^{-3}$ | $5\times 10^{-6}$ | $0.128$ | $1.63\times 10^{-14}$ | $4\times 10^{-7}$ | $1.17\times 10^{-6}$
Table 1: Testing the density of gravitational waves predicted by PTA as
functions of several parameters related to the D-term hybrid model. Here, we
have incorporated the best-fit values for the cosmological parameters
according to the Planck results [57], $\Omega_{m}=0.315,$ $\eta_{0}\simeq
1.38,$ $A_{s}=2,1\times 10^{-9},k_{eq}\simeq 0.01,f_{\star}\approx 7.7\times
10^{-17}Hz,$ $f_{yr}\simeq 3.1\times 10^{-8}Hz$ and $H_{0}\equiv
100~{}h~{}km/s/Mpc.$

Table 1 presents the NanoGrav proposed model for the density
$h^{2}\Omega_{gw}(f)$ as a function of frequency $f,$ scalar-to-tensor ratio
$r$, and the amplitude of the PTA signal related to the potential parameters
of the D-term hybrid model. The model for $\Omega_{gw}(f)$ depends on various
cosmological parameters and constants defined earlier. The analytical
expression for, $h^{2}\Omega_{gw}(f)$ is derived from a recent theoretical
model for primordial gravitational waves. $h^{2}\Omega_{gw}(f)$ represents the
fractional energy density of gravitational waves in the universe, indicating
the energy carried by the gravitational wave background relative to the
critical energy density needed for a flat universe. By examining the table 1,
one can identify the parameters that predict higher or lower densities of
gravitational waves $h^{2}\Omega_{gw}(f)$ in line with NANOGrav predictions
[81, 82, 83, 84]. The amplitude of PTA scales $A\lesssim 10^{-20}$ provides
accurate predictions for $h^{2}\Omega_{gw}(f)$ values. Furthermore, the Table
1 offers insights into the behavior of gravitational wave background (GWB)
density concerning frequency, scalar-to-tensor ratio, and the D-term model
parameters, demonstrating good consistency with predicted bounds on density
and frequency when fine-tuning the inflationary parameters associated with PTA
signals.

###  V.2 Primordial Gravitational Waves

For primordial gravitational waves in a spatially flat FLRW background, the
metric element can be expressed as follows:

$ds^{2}=a^{2}(\tau)\left[-d\tau^{2}+\left(\delta_{ij}+h_{ij}\right)dx^{i}dx^{j}\right].$
(34)

The perturbation $h_{ij}$ satisfies the transverse-traceless (TT) conditions:
$h^{i}_{~{}i}=0$ and $\partial^{i}h_{ij}=0$. The tensor perturbation
$h_{ij}\left(\tau,\vec{x}\right)$ can be decomposed into its Fourier modes,
which are associated with two polarization tensors, satisfying the equation of
motion

$h_{\mathbf{k}}^{\lambda\prime\prime}(\tau)+2{\mathcal{H}}h_{\mathbf{k}}^{\lambda\prime}(\tau)+k^{2}h_{\mathbf{k}}^{\lambda}(\tau)=0.$
(35)

Here $\left({}^{\prime}\right)$ denotes the derivative with respect to
conformal time $\tau$, where $d\tau=$ $\frac{dt}{a}$ and
${\mathcal{H}}=\frac{a^{\prime}}{a}$. The normalized gravitational wave energy
density spectrum is defined as the energy density per logarithmic frequency
interval

$\Omega_{gw}(k)=\frac{1}{\rho_{c}}\frac{d\rho_{gw}}{d\ln k},$ (36)

here $\rho_{c}$ represents the energy density. Additionally,

$\Omega_{gw,0}(k)=\frac{1}{12}\left(\frac{k^{2}}{a_{0}^{2}H_{0}^{2}}\right){\mathcal{P}}_{h}(k),$
(37)

knowing that
${\mathcal{P}}_{h}(k)\equiv\frac{k^{3}}{\pi^{2}}\sum_{\lambda}\left|h_{\mathbf{k}}^{\lambda}\right|^{2}.$
By considering the scale at which horizon re-entry occurs
$\left(k=a_{hc}H_{hc}\right)\ $and analyzing the horizon re-entry scale
alongside the Hubble parameter across different epochs, we can derive the
present-day primordial gravitational wave spectrum for mode re-entry during
the matter-dominated (M), radiation-dominated (R), and kinetic (K) eras,
respectively, as follows:

$\displaystyle\Omega_{gw,0}^{\left(\mathit{M}\right)}$ $\displaystyle=$
$\displaystyle\frac{1}{24}\Omega_{m,0}^{2}\frac{a_{0}^{2}H_{0}^{2}}{k^{2}}{\mathcal{P}}_{t}~{}~{}~{}~{}~{}~{}\left(k_{0}<k\leq
k_{eq}\right),$ (38) $\displaystyle\Omega_{gw,0}^{\left(\mathit{R}\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{24}\Omega_{r,0}^{2}\left(\frac{g_{\ast}}{g_{\ast
0}}\right)\left(\frac{g_{\ast s}}{g_{\ast
s0}}\right){\mathcal{P}}_{t}~{}~{}~{}~{}~{}~{}\left(k_{eq}<k\leq
k_{r}\right),$ (39) $\displaystyle\Omega_{gw,0}^{\left(\mathit{K}\right)}$
$\displaystyle=$
$\displaystyle\Omega_{gw,0}^{\left(\mathit{R}\right)}\left(\frac{k}{k_{r}}\right)~{}~{}~{}~{}~{}\left(k_{r}<k\leq
k_{\max}\right),$ (40)

Figure 2: D-term $\xi$ parameter as a function of the comoving wavenumber $k$
across the primordial gravitational wave spectrum $\Omega_{gw,0}$.

Fig. 2 provides a detailed visualization of the primordial tensor power
spectrum and the corresponding energy density parameters $\Omega_{gw,0}$
across different cosmological eras: matter-dominated, radiation-dominated, and
kinetic eras. In the matter-dominated era, $\Omega_{gw,0}$ decreases with
increasing comoving wavenumber $k$, indicating larger scales are more
influenced by primordial tensor perturbations. During the radiation-dominated
era, the energy density of gravitational waves is only affected directly by
the D-term hybrid parameter $\xi$, while the scale $k$ has a constant effect
on $\Omega_{gw,0}$. In the kinetic era, the energy density of gravitational
waves is increasing with respect to the comoving scale to reach its maximum
values at superhorizon scales. This comprehensive depiction illustrates how
primordial gravitational waves evolve through different epochs, influenced by
the universe’s thermal history and expansion dynamics, and underscores the
significance of these factors on the energy densities observed.

##  VI Constraints on Gravitational Waves from Cosmic Strings

Cosmic strings form at the end of inflation, impacting the anisotropies
observed in CMB and contributing to the creation of stochastic gravitational
waves [100]. The dimensionless string tension, $G\mu_{cs},$ is key to
understanding these phenomena, where $G=\frac{1}{8\pi M_{p}^{2}}=6.7\times
10^{-39}GeV^{-2},$ and $\mu_{cs}$ is the string’s mass per unit length.
Current CMB constraints place limits on this tension as $G\mu_{cs}\lesssim
1.3\times 10^{-7}$ [100]. The SGWB arises from a mix of sources including
inflation, cosmic strings, and phase transitions [101]. Specifically,
inflationary tensor perturbations re-entering the horizon generate an SGWB
[102, 103, 104, 105], leaving a unique imprint on the CMB B-mode polarization.
The amplitude and scale dependence of this background are described by the
tensor-to-scalar ratio $r$ and the tensor spectral index $n_{T},$ adhering to
the inflationary consistency relation $r=-8n_{T}$ [106]. Given that $r\geq 0,$
this implies $n_{T}\leq 0$ indicating a red spectrum [107]. With current
limits on the tensor-to-scalar ratio, the amplitude of the inflationary SGWB
at pulsar timing array (PTA) and interferometer scales remains too low for
detection by these instruments, necessitating a primordial tensor power
spectrum with a strong blue tilt $\left(n_{T}\geq 0\right)$ for detection
[108, 109].

The detection of gravitational waves from cosmic strings is primarily
influenced by two key scales: the energy scale of inflation $\Lambda_{\inf}$
and the scale at which cosmic strings generate the GW spectrum
$\Lambda_{cs}\equiv\sqrt{G\mu_{cs}}$. The amplitude of the tensor mode
anisotropy in the cosmic microwave background fixes the energy scale of
inflation to approximately $\Lambda_{\inf}\sim V^{1/4}\sim 3.3\times
10^{16}r^{1/4}$ [110]. By applying the Planck $2\sigma$ bounds on the tensor-
to-scalar ratio $r$, we derive an upper limit on the inflation scale,
$\Lambda_{\inf}<1.6\times 10^{16}GeV$ [111]. In our model, cosmic strings form
post-inflation, indicating $\Lambda_{\inf}>\Lambda_{cs},$ which results in a
SGWB generated from undiluted strings. The SGWB from metastable cosmic string
networks is expressed relative to the critical density as described in the
folowing form [112],

$\displaystyle\Omega_{GW}\left(f\right)$ $\displaystyle=$
$\displaystyle\frac{8\pi
G}{3H_{0}^{2}}f\left(G\mu_{cs}\right)^{2}\Sigma_{n=1}^{\infty}C_{n}\left(f\right)P_{n},$
(41) $\displaystyle=$ $\displaystyle\frac{8\pi
G}{3H_{0}^{2}}f{\mathcal{G}}_{cs}(C_{n},P_{n}).$

Here, ${\mathcal{G}}_{cs}(C_{n},P_{n})$ represent the term of cosmic strings
contribution to GWs production, which contain the power spectrum
$P_{n}\simeq\frac{50}{\zeta\left(4/3\right)n^{4/3}}$ that represent the
gravitational waves (GWs) emitted by the n-th harmonic of a cosmic string
loop, and $C_{n}$ which denotes the number of loops emitting GWs observed at a
specific frequency $f$ . The number of loops emitting GWs, observed at a given
frequency $f$ is defined as [112],

$C_{n}\left(f\right)=\frac{2n}{f^{2}}\int_{z_{\min}}^{z_{\max}}\frac{dz}{H(z)\left(1+z\right)^{6}}{\mathcal{N}}(\mathit{l},t).$
(42)

The integration range spans the lifetime of the cosmic string network,
starting from its formation at $z_{\max}\simeq\frac{T_{R}}{2.7K},$ with
$T_{R}$ being approximately $10^{9}GeV$ to its decay at
$z_{\min}=\left(\frac{70}{H_{0}}\right)^{1/2}\left(\frac{\Gamma\left(G\mu_{cs}\right)^{2}}{2\pi\times
6.7\times 10^{-39}}\exp\left(-\pi\kappa_{cs}\right)\right)^{1/4}$ [113], where
$\Gamma\simeq 50$ is a numerical factor specifying the cosmic strings decay
rate. Here, ${\mathcal{N}}(\mathit{l},t)$ represents the number density of CS
loops of length $\mathit{l}=\frac{2n}{\left(1+z\right)f}.$ The loop density is
defined by considering their formation and decay across different epochs. For
the region of interest, the dominant contribution arises from the loops
generated during the radiation-dominated era which is given by [113],

${\mathcal{N}}_{r}(\mathit{l},t)=\frac{0.18}{t^{2/3}\left(\mathit{l}+\Gamma
G\mu_{cs}t\right)^{5/2}}.$ (43)

The cosmological time and the Hubble rate with the current values of matter,
radiation and dark energy densities, are respectively expressed as a function
of the redshift $z$ as
$t(z)=\int_{z_{\min}}^{z_{\max}}\frac{dz}{H(z)\left(1+z\right)},$
$H(z)=H_{0}\sqrt{\Omega_{\Lambda}+\Omega_{m}\left(1+z\right)^{3}+\Omega_{r}\left(1+z\right)^{4}}.$

Figure 3: Gravitational waves density $\Omega_{GW}$ as a function of the
frequency $f$ and the cosmic string parameter $G\mu_{cs}.$

Fig. 3 illustrates the density of gravitational waves $\Omega_{GW}$ as a
function of frequency $f$ and the cosmic string parameter $G\mu_{cs}$. The
contour plot uses a logarithmic scale for both the frequency and $G\mu_{cs}$,
highlighting how gravitational wave densities vary across different scales and
parameters. The gravitational wave energy density is computed by integrating
over the redshift $z$, taking into account the contributions from matter,
radiation, and dark energy to the Hubble parameter $H(z)$ The function
${\mathcal{N}}_{r}(\mathit{l},t)$ reflects the number density of gravitational
waves, which is influenced by the redshift and other cosmological parameters.
$C_{n}$ and $P_{n}$ are functions of $n$ , frequency $f$, and the cosmic
string parameter ${\mathcal{G}}_{cs},$ representing cosmic strings
contribution to the power spectrum of the gravitational waves. The resulting
plot shows that at higher frequencies, the gravitational wave density varies
more significantly with changes in $G\mu_{cs},$ indicating a strong dependence
on the string tension parameter. This comprehensive depiction underscores the
intricate relationship between cosmic string dynamics and gravitational wave
emissions, offering insights into how different frequencies and cosmic string
tensions contribute to the observable gravitational wave background. The use
of a logarithmic scale for both axes ensures that a wide range of values is
represented, making it easier to visualize the detailed structure and
variation of $\Omega_{GW}$ across different cosmological scenarios.

Figure 4: Cosmic strings factor ${\mathcal{G}}_{cs}$ as functions of the
frequency $f$ and the tensor-to-scalar ratio $r$.

Fig. 4 presents the cosmic strings factor ${\mathcal{G}}_{cs}$ as a function
of frequency and the tensor-to-scalar ratio $r.$ The contour plot uses a
logarithmic scale for frequency, ranging from $10^{-10}Hz$ to $10^{-6}Hz$ ,
and spans multiple values of $r=\left[0.026,0.128\right]$, reflecting
different possible strengths of primordial tensor perturbations. The frequency
dependence of ${\mathcal{G}}_{cs}$ incorporates a power-law behavior with an
exponent $\alpha$, representing the spectral shape of the cosmic string
signal. The resulting plot visually demonstrates that ${\mathcal{G}}_{cs}$
increases with higher values of $r$ and decreases with increasing frequency.
The contour levels indicate the logarithmic values of, with color gradients
representing the intensity. This visualization helps to elucidate the
relationship between cosmic string dynamics and gravitational wave signals
across a range of frequencies and tensor-to-scalar ratios. Specifically, it
shows that stronger primordial tensor perturbations lead to higher
${\mathcal{G}}_{cs}$ values, particularly at lower frequencies, which is
crucial for understanding the potential observational signatures of cosmic
strings in gravitational wave experiments. This comprehensive depiction
provides insights into the scale and strength of cosmic string contributions
to the gravitational wave background, highlighting key dependencies and aiding
in the interpretation of potential observational data in the context of early
universe cosmology.

##  VII Conclusion

In this paper, we investigated the framework of supergravity model, employing
the minimal Kähler potential. Our study revealed that D-term potential can
circumvent the $\eta$-problem inherent in F-term models, offering a viable
path to successful inflation. Mathematically, we derived the scalar potential,
incorporating both F-term and D-term contributions, and demonstrated the
stability conditions for the inflaton field $S$. The effective potential
during inflation was approximated, highlighting the importance of the gauge
coupling constant $g$ and the Fayet-Iliopoulos term $\xi$. The slow-roll
parameters and the resulting e-folding number were calculated, providing
insights into the inflationary dynamics. Furthermore, we explored reheating
dynamics and gravitino production, emphasizing the interplay between reheating
temperature $T_{re}$, spectral index $n_{s}$, and gravitino abundance
$Y_{3/2}$. Our analysis indicated that gravitino production is sensitive to
the equation of state during reheating, impacting the reheating temperature
and the subsequent dark matter relic density.

The study of scalar induced gravitational waves offers crucial insights into
the early Universe, particularly within the inflationary paradigm. Detecting
the primordial gravitational wave background can substantiate inflationary
models by providing evidence for a kinetic epoch preceding inflation,
characterized by a distinct blue tilt at higher frequencies. The perturbation
theory framework reveals that the transverse-traceless components of these
waves evolve according to second-order gravitational interactions induced by
first-order curvature perturbations. The spectral energy density of these
waves at various scales, particularly those relevant to Pulsar Timing Array
(PTA) observations, can be quantified through analytical models, which align
well with empirical data, such as that from NANOGrav. These models illustrate
the dependence of gravitational wave amplitudes on key cosmological
parameters, thus highlighting the interplay between inflationary dynamics and
observable gravitational wave spectra.

The analysis of gravitational waves generated by cosmic strings provides
critical constraints on the dynamics of the early Universe and the properties
of cosmic strings. The dimensionless string tension, $G\mu_{cs}$,
significantly influences the stochastic gravitational wave background (SGWB)
produced by cosmic strings post-inflation. Current constraints from CMB
observations limit $G\mu_{cs}\leq 1.3\times 10^{-7}.$ The SGWB from cosmic
strings is shaped by the energy scales of inflation and the string formation,
with the amplitude of tensor mode anisotropy fixing the inflation scale around
$\Lambda_{\inf}\sim 3.3\times 10^{16}r^{1/4}$. Cosmic string contributions to
the gravitational wave spectrum, represented by $\Omega_{GW}(f)$, depend on
the number density and dynamics of cosmic string loops across different
cosmological epochs. These contributions exhibit a strong frequency dependence
and vary significantly with the string tension parameter, highlighting the
complex relationship between cosmic string properties and observable
gravitational wave signals. The comprehensive models and visualizations
underscore the importance of cosmic string dynamics in shaping the SGWB and
provide insights into potential observational signatures in gravitational wave
experiments, aiding in the broader understanding of early universe cosmology.

## Acknowledgments

G.O. acknowledges the financial support of Fondecyt Grant 1220065.

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