Following the method developed in zs  (see also Section $9.2$ of gmmbook ), we
will write the modes as follows,

$\displaystyle\chi_{k}(\tau)=\alpha_{k}(\tau)\frac{e^{-i\int^{\tau}\omega_{k}(\bar{\eta})d\bar{\eta}}}{\sqrt{2\omega_{k}(\tau)}}+\beta_{k}(\tau)\frac{e^{i\int^{\tau}\omega_{k}(\bar{\eta})d\bar{\eta}}}{\sqrt{2\omega_{k}(\tau)}},$
(240)

where $\alpha_{k}(\tau)$ and $\beta_{k}(\tau)$ are the time-dependent
Bogoliubov coefficients. Now, imposing that the modes satisfy the conditions

$\displaystyle\chi_{k}^{\prime}(\tau)=-i\omega_{k}(\tau)\left(\alpha_{k}(\tau)\frac{e^{-i\int^{\tau}\omega_{k}(\bar{\eta})d\bar{\eta}}}{\sqrt{2\omega_{k}(\tau)}}-\beta_{k}(\tau)\frac{e^{i\int^{\tau}\omega_{k}(\bar{\eta})d\bar{\eta}}}{\sqrt{2\omega_{k}(\tau)}}\right),$
(241)

one easily shows that the Bogoliubov coefficients must satisfy the fundamental
system

$\displaystyle\left\\{\begin{array}[]{ccc}\alpha_{k}^{\prime}(\tau)&=&\frac{\omega_{k}^{\prime}(\tau)}{2\omega_{k}(\tau)}e^{2i\int^{\tau}\omega_{k}(\bar{\eta})d\bar{\eta}}\beta_{k}(\tau)\\\
\beta_{k}^{\prime}(\tau)&=&\frac{\omega_{k}^{\prime}(\tau)}{2\omega_{k}(\tau)}e^{-2i\int^{\tau}\omega_{k}(\bar{\eta})d\bar{\eta}}\alpha_{k}(\tau)\end{array}\right.$
(244)

in order for the expression (240) to be a solution of the equation (220).

Finally, inserting (240) into the expression for vacuum energy density (239),
and taking into account that the Bogoliubov coefficients satisfy the equation
$|\alpha_{k}(\tau)|^{2}-|\beta_{k}(\tau)|^{2}=1$, one finds that

$\displaystyle\rho_{\chi}(\tau)=\frac{1}{2\pi^{2}a^{4}(\tau)}\int_{0}^{\infty}k^{2}\omega_{k}(\tau)|\beta_{k}(\tau)|^{2}dk.$
(245)

At this point, it is very important to notice that $|\beta_{k}(\tau)|^{2}$
encodes the vacuum polarization effects and also the particle creation, which
only happens when the adiabatic evolution breaks. In fact, the quantity

$\displaystyle
N_{\chi}(\tau)=\frac{1}{2\pi^{2}a^{3}(\tau)}\int_{0}^{\infty}k^{2}|\beta_{k}(\tau)|^{2}dk$
(246)

was named in the Russian literature as the number density of quasi-particles,
which is very different from the number density of the produced particles
because, as we will see immediately, it also contains the vacuum polarization
effects.

As an application, we will consider our toy model (217) once again. Then, in
order to obtain the value of the $\beta$-Bogoliubov coefficient we come back
to the equation (244) and in the first approximation we take
$\alpha_{k}(\tau)=1$, getting

$\displaystyle\beta_{k}(\tau)=\int^{\tau}\frac{\omega_{k}^{\prime}(\eta)}{2\omega_{k}(\eta)}e^{-2i\int^{\eta}\omega_{k}(\bar{\eta})d\bar{\eta}}d\eta.$
(247)

After integrating by parts before the beginning of kination, it yields

$\displaystyle\beta_{k}(\tau)=\left(-\frac{\omega^{\prime}_{k}(\tau)}{4i\omega_{k}^{2}(\tau)}+\frac{1}{8\omega_{k}(\tau)}\left(\frac{\omega^{\prime}_{k}(\tau)}{\omega_{k}^{2}(\tau)}\right)^{\prime}\right.$
$\displaystyle\left.+\frac{1}{16i\omega_{k}(\tau)}\left(\frac{1}{\omega_{k}(\tau)}\left(\frac{\omega^{\prime}_{k}(\tau)}{\omega_{k}^{2}(\tau)}\right)^{\prime}\right)^{\prime}+....\right)e^{-2i\int^{\tau}\omega_{k}(\bar{\eta})d\bar{\eta}}.$
(248)

However, after kination the $\beta$-Bogoliubov coefficient must be given by

$\displaystyle\beta_{k}(\tau)=\left(-\frac{\omega^{\prime}_{k}(\tau)}{4i\omega_{k}^{2}(\tau)}+\frac{1}{8\omega_{k}(\tau)}\left(\frac{\omega^{\prime}_{k}(\tau)}{\omega_{k}^{2}(\tau)}\right)^{\prime}\right.$
$\displaystyle\left.+\frac{1}{16i\omega_{k}(\tau)}\left(\frac{1}{\omega_{k}(\tau)}\left(\frac{\omega^{\prime}_{k}(\tau)}{\omega_{k}^{2}(\tau)}\right)^{\prime}\right)^{\prime}+....\right)e^{-2i\int^{\tau}\omega_{k}(\bar{\eta})d\bar{\eta}}+C,$
(249)

where the constant $C$ has to be chosen in order that the $\beta$-Bogoliubov
coefficient becomes continuous at $\tau_{kin}$ because the equation (244) is a
first order differential equation, and then one has to demand continuity to
the solution. Thus, after some cumbersome calculation hpa  one has

$\displaystyle
C=\left(\frac{m_{\chi}^{2}m^{3}a^{5}_{kin}}{16i\omega^{5}_{k}(\tau_{kin})}+....\right)e^{-2i\int^{\tau_{kin}}\omega_{k}(\bar{\eta})d\bar{\eta}},$
(250)

where we have denoted $a_{kin}\equiv a(\tau_{kin})$.

Notice that the terms that do not contain $C$ lead to sub-leading geometric
quantities in the energy density. Effectively, the term
$-\frac{\omega^{\prime}_{k}(\tau)}{4i\omega_{k}^{2}(\tau)}$ leads to the
following contribution to the energy density:
$\frac{m_{\chi}^{2}H^{2}}{96\pi}\ll 3M_{pl}^{2}H^{2}$. The same happens with
$\frac{1}{8\omega_{k}(\tau)}\left(\omega^{\prime}_{k}(\tau)/\omega_{k}^{2}(\tau)\right)^{\prime}$,
which leads to a term of order $H^{4}$, meaning that
$\frac{H^{4}}{M_{pl}^{2}}\ll H^{2}$. The product of the first and second term
generates in the right-hand side of the modified semi-classical Friedmann
equation a term of the order $\frac{H^{3}m_{\chi}}{M_{pl}^{2}}$, which is also
sub-leading compared with $H^{2}$. Finally, the third term of (VI.1.3) leads
in the right-hand side of the semi-classical Friedmann equation to the sub-
leading term $\frac{H^{6}}{m_{\chi}^{2}M_{pl}^{2}}$.

Fortunately, this does not happen with $C$, whose leading term gives the main
contribution of the vacuum energy density due to the gravitational particle
production. In fact, the time dependent terms, which as we have already seen
are always sub-leading, are vacuum polarization effects, and they rapidly
disappear in the adiabatic regime. That is, soon after the beginning of
kination $|\beta_{k}(\tau)|^{2}$ approaches to $|C|^{2}$, whose value
coincides with the one obtained using the WKB approach (see formula (225)),
getting once again

$\displaystyle\rho_{\chi}(\tau)\cong\left\\{\begin{array}[]{ccc}0&\mbox{
when}&\tau<\tau_{kin}\\\
10^{-5}\left(\frac{m}{m_{\chi}}\right)^{2}m^{4}\left(\frac{a_{kin}}{a(\tau)}\right)^{3}&\mbox{
when}&\tau\geq\tau_{kin},\end{array}\right.$ (253)

which at the beginning of kination is sub-dominant with respect to the energy
density of the background but it will eventually dominate because the one of
the background decreases during kination as $a^{-6}(\tau)$.

In order to understand better these results it is useful to recall, as we have
already explained, that the authors of the diagonalization method assume that
during the whole evolution of the universe quanta named quasi-particles are
created and annihilated due to the interaction of the quantum field with
gravity gmmbook , i.e., this is a vacuum polarization effect. And, following
this interpretation, the number density of the created quasi-particles at time
$\tau$ is given by
$N_{\chi}(\tau)=\frac{1}{2\pi^{2}a^{3}(\tau)}\int_{0}^{\infty}k^{2}|\beta_{k}(\tau)|^{2}dk$.
However, one has to be very careful with this interpretation and especially
keep in mind that, as we have shown in our toy model (217), real particles are
only created when the adiabatic regime breaks. Effectively, before the
beginning of kination the main term of the $\beta$-Bogoliubov coefficient is
given by $-\frac{\omega^{\prime}_{k}(\tau)}{4i\omega_{k}^{2}(\tau)}$, whose
contribution to the energy density is $\frac{m_{\chi}^{2}H^{2}}{96\pi}$, and
to the number density of quasi-particles is
$\frac{m_{\chi}^{2}H^{2}}{512\pi}$, and thus, at time $\tau$ before the
beginning of kination $\rho_{\chi}(\tau)\not=m_{\chi}N_{\chi}(\tau)$, meaning
that the quasi-particles do not evolve as real massive particles. On the
contrary, during kination the leading term of $N_{\chi}(\tau)$ is given by
$10^{-5}\left(\frac{m}{m_{\chi}}\right)^{3}m^{3}\left(\frac{a_{kin}}{a(\tau)}\right)^{3}$,
so we have $\rho_{\chi}(\tau)=m_{\chi}N_{\chi}(\tau)$ and the decay follows
$a^{-3}(\tau)$, which justifies the interpretation of massive particle
production.

So, once we have understood the gravitational particle production, we review
our last work partlorentzian  about the creation of particles in the LQI
scenario. Coming back to our potential (118), first of all one has to
integrate numerically the conservation equation for the inflaton field, namely

$\displaystyle\ddot{\varphi}+3H\dot{\varphi}+V_{\varphi}=0,$ (254)

where
$H=\frac{1}{\sqrt{3}M_{pl}}\sqrt{\frac{\dot{\varphi}^{2}}{2}+V(\varphi)}$,
with initial conditions at the horizon crossing (when the pivot scale leaves
the Hubble radius). Recalling that in that moment the system is in the slow-
roll phase and, since this regime is an attractor, one only has to take
initial conditions in the basin of attraction of the slow-roll solution, for
example, $\varphi_{*}=-0.154M_{pl}$ and $\dot{\varphi}_{*}=0$, where the
“star” denotes, as usual, that the quantities are evaluated at the horizon
crossing.

Once the evolution of the background is obtained, and in particular the
evolution of the Hubble rate, one can compute the evolution of the scale
factor, which is given by

$\displaystyle a(t)=a_{*}e^{\int_{t_{*}}^{t}H(s)ds},$ (255)

where one can choose $a_{*}=1$ as the value of the scale factor at the horizon
crossing.

From the evolution of the scale factor, one can see in Figure 9 that a spike
appears during the phase transition from the end of inflation to the beginning
of kination, i.e., when the adiabatic evolution is broken and particles are
gravitationally produced.

Figure 10: Plot of the adiabatic evolution for a heavy field with mass
$m_{\chi}\cong 10^{15}$ GeV, when the background is given by the Lorentzian
Quintessential Inflation potential. The value $k=a_{kin}H_{kin}$ has been used
for the quantities $a_{kin}$, $H_{kin}$ of this model.

Next, taking into account what we have learned about our toy model (217), one
can numerically solve the equation (244), with initial conditions
$\alpha_{k}(\tau_{*})=1$ and $\beta_{k}(\tau_{*})=0$ at the horizon crossing
(there were neither particles nor polarization effects at that moment because
during the slow-roll regime the derivatives of the Hubble rate are negligible
compared with the powers of $H$, i.e., the system is in the adiabatic regime).

In order to get rid of complex exponentials it is useful to transform the
equation (244) into a second order differential equation, namely

$\displaystyle\left\\{\begin{array}[]{cc}&\alpha_{k}^{\prime\prime}(\tau)=\alpha_{k}^{\prime}(\tau)\left(\frac{\omega_{k}^{\prime\prime}(\tau)}{\omega_{k}^{\prime}(\tau)}-\frac{\omega_{k}^{\prime}(\tau)}{\omega_{k}(\tau)}+2i\omega_{k}(\tau)\right)+\left(\frac{\omega_{k}^{\prime}(\tau)}{2\omega_{k}(\tau)}\right)^{2}\alpha_{k}(\tau)\\\
&\beta_{k}^{\prime\prime}(\tau)=\beta_{k}^{\prime}(\tau)\left(\frac{\omega_{k}^{\prime\prime}(\tau)}{\omega_{k}^{\prime}(\tau)}-\frac{\omega_{k}^{\prime}(\tau)}{\omega_{k}(\tau)}-2i\omega_{k}(\tau)\right)+\left(\frac{\omega_{k}^{\prime}(\tau)}{2\omega_{k}(\tau)}\right)^{2}\beta_{k}(\tau)\end{array}\right..$
(258)

Given that $\alpha_{k}(\tau_{*})=1$ and $\beta_{k}(\tau_{*})=0$ leads to
$\alpha_{k}^{\prime}(\tau_{*})=0$, one is interested in solving the equation
for $\alpha_{k}(\tau)$, which can be split into the real and imaginary form in
the following way,

$\displaystyle\left\\{\begin{array}[]{cc}&\alpha_{k,Re}^{\prime\prime}(\tau)=\alpha_{k,Re}^{\prime}(\tau)\left(\frac{\omega_{k}^{\prime\prime}(\tau)}{\omega_{k}^{\prime}(\tau)}-\frac{\omega_{k}^{\prime}(\tau)}{\omega_{k}(\tau)}\right)-2\omega_{k}(\tau)\alpha_{k,Im}^{\prime\prime}(\tau)+\left(\frac{\omega_{k}^{\prime}(\tau)}{2\omega_{k}(\tau)}\right)^{2}\alpha_{k,Re}(\tau)\\\
&\alpha_{k,Im}^{\prime\prime}(\tau)=\alpha_{k,Im}^{\prime}(\tau)\left(\frac{\omega_{k}^{\prime\prime}(\tau)}{\omega_{k}^{\prime}(\tau)}-\frac{\omega_{k}^{\prime}(\tau)}{\omega_{k}(\tau)}\right)-2\omega_{k}(\tau)\alpha_{k,Re}^{\prime\prime}(\tau)+\left(\frac{\omega_{k}^{\prime}(\tau)}{2\omega_{k}(\tau)}\right)^{2}\alpha_{k,Im}(\tau)\end{array}\right.,$
(261)

and then $|\beta_{k}(\tau)|^{2}=|\alpha_{k}(\tau)|^{2}-1$ because of the well-
known conservation property of the Wronskian. For the value
$k=a_{kin}H_{kin}$, one can see in Figure 11 that $|\beta_{k}(\tau)|^{2}$
stabilizes soon to a non-zero value after the beginning of kination,
containing only particle production effects. One can numerically check that
this happens for the range $0.05\lesssim\frac{k}{a_{kin}H_{kin}}\lesssim
7\times 10^{4}$, which leads to values of $|\beta_{k}|^{2}$ of the order of
$10^{-10}$ and $10^{-11}$.

Figure 11: Evolution of $|\beta_{k}(\tau)|^{2}$

Then, introducing these values of the $\beta$-Bogoliubov coefficient in the
equation (245), one obtains a vacuum energy density of the order of
$10^{44}\mbox{GeV}^{4}$. So, the energy density of the produced particles
evolves as

$\displaystyle\rho_{\chi}(\tau)=\bar{\rho}_{\chi}\left(\frac{\bar{a}}{a(\tau)}\right)^{3},$
(262)

where $\bar{\rho}_{\chi}\sim 10^{44}\mbox{ GeV}^{4}\sim 10^{-30}M_{pl}^{4}$
and $\bar{a}$ are, respectively, the energy density of the produced particles
and the value of the scale factor at the end of the non-adiabatic phase.
Finally, the energy density of the background at this moment is given by
$\bar{\rho}_{\varphi}=3\bar{H}^{2}M_{pl}^{2}\sim 10^{57}\mbox{ GeV}^{4}\sim
10^{-17}M_{pl}^{4}$, showing that the energy density of the produced particles
is sub-leading at the beginning of kination, but will eventually be dominant
because, during the kination regime, the energy density of the inflaton field
decreases as $a^{-6}$.

Once again, two different situations arise:

  1. 1.

Decay of the superheavy particles into lighter ones before the end of
kination.

In this case, the energy density of the the inflaton field and the one of the
relativistic plasma, when the decay is finished, i.e., when ${\Gamma}\sim
H_{dec}=\bar{H}\left(\frac{\bar{a}}{a_{dec}}\right)^{3}\sim
10^{10}\left(\frac{\bar{a}}{a_{dec}}\right)^{3}$ GeV $\sim
10^{-8}\left(\frac{\bar{a}}{a_{dec}}\right)^{3}M_{pl}$ , is given by

$\displaystyle\rho_{\varphi,dec}=3{\Gamma}^{2}M_{pl}^{2}\quad\mbox{and}\quad\rho_{\chi,dec}\sim
10^{-29}\left(\frac{\bar{a}}{a_{dec}}\right)^{3}M_{pl}^{4}\sim 10^{-21}\Gamma
M_{pl}^{3}.$ (263)

Imposing that the end of the decay precedes the end of kination, that means,
$\rho_{\chi,dec}\leq\rho_{\varphi,dec}$, one gets $\Gamma\geq 10^{-21}M_{pl}$,
and, since the decay is after the beginning of the kination and for our LQI
model $H_{kin}\sim 10^{-8}M_{pl}$ one gets $\Gamma\leq H_{kin}\cong 4\times
10^{-8}M_{pl}$. So, the following bound for the decay rate is obtained,

$\displaystyle 10^{-21}M_{pl}\leq\Gamma\leq 10^{-8}M_{pl}.$ (264)

Finally, the reheating temperature, i.e., the temperature of the universe when
the relativistic plasma in thermal equilibrium starts to dominate, which
happens when $\rho_{\varphi,reh}\sim\rho_{\chi,reh}$, can be calculated as
follows: Since after the decay the evolution of the respective energy
densities is given by

$\displaystyle\rho_{\varphi,reh}=\rho_{\varphi,dec}\left(\frac{a_{dec}}{a_{reh}}\right)^{6},\qquad\rho_{\chi,reh}=\rho_{\chi,dec}\left(\frac{a_{dec}}{a_{reh}}\right)^{4},$
(265)

one has
$\frac{\rho_{\chi,dec}}{\rho_{\varphi,dec}}=\left(\frac{a_{dec}}{a_{reh}}\right)^{2}$
and, thus, the reheating temperature will be

$\displaystyle
T_{reh}=\left(\frac{30\rho_{\chi,reh}}{\pi^{2}g_{reh}}\right)^{1/4}=\left(\frac{30\rho_{\chi,dec}}{\pi^{2}g_{reh}}\right)^{1/4}\sqrt{\frac{\rho_{\chi,dec}}{\rho_{\varphi,dec}}}\cong
10^{-16}\left(\frac{M_{pl}}{\Gamma}\right)^{1/4}M_{pl},$ (266)

where $g_{reh}=106.75$ is the effective number of degrees of freedom for the
Standard Model. So, taking into account the bound (294) the reheating
temperature ranges between $10^{3}$ GeV and $10^{7}$ GeV.

  2. 2.

Decay of the superheavy particles into lighter ones after the end of kination.

In the case that the decay of the $\chi$-field is after the end of kination,
one has to impose ${\Gamma}\leq H_{end}$. Now, taking into account that at the
end of kination

$\displaystyle
H^{2}_{end}=\frac{2\rho_{\varphi,end}}{3M_{pl}^{2}}\quad\mbox{and}\quad\rho_{\varphi,end}=\bar{\rho}_{\varphi}\left(\frac{\bar{a}}{a_{end}}\right)^{6}=\frac{\bar{\rho}_{\chi}^{2}}{\bar{\rho}_{\varphi}},$
(267)

where it is used that the kination ends when
${\rho}_{\chi,end}\sim{{\rho}_{\varphi,end}}$, meaning
$\left(\frac{\bar{a}}{a_{end}}\right)^{3}=\frac{\bar{\rho}_{\chi}}{\bar{\rho}_{\varphi}}$.
So, the condition ${\Gamma}\leq H_{end}$ leads to the bound

$\displaystyle\Gamma\leq 10^{-22}M_{pl}.$ (268)

On the other hand, assuming once again instantaneous thermalization, the
reheating temperature (i.e., the temperature of the universe when the
thermalized plasma starts to dominate) will be obtained when all the
superheavy particles decay, i.e., when $H\sim\Gamma$, obtaining

$\displaystyle
T_{reh}=\left(\frac{30}{\pi^{2}g_{reh}}\right)^{1/4}\rho_{\chi,dec}^{1/4}=\left(\frac{90}{\pi^{2}g_{reh}}\right)^{1/4}\sqrt{{\Gamma}M_{pl}}~{},$
(269)

where one has to take into account that, after the end of the kination regime,
the energy density of the produced particles dominates the one of the inflaton
field.

Consequently, since the BBN epoch occurs at the $1$ MeV regime and taking once
again $g_{reh}=106.75$, one can find that, in that case, the reheating
temperature is bounded by

$\displaystyle 1\mbox{ GeV}\leq T_{reh}\leq 10^{7}\mbox{ GeV }.$ (270)

###  VI.2 Instant preheating

In this subsection we review the work of Felder, Kofman and Linde fkl  (see
also fkl0 ), where Instant Preheating for QI scenarios was introduced and
discussed in detail.

Essentially, instant reheating is based on the interacting part of the
Lagrangian density, namely $-\frac{1}{2}g^{2}\varphi^{2}\chi^{2}$, where g is
a coupling constant, which due to the phase transition between the end of
inflation and the beginning of kination ceases to be in the vacuum state to
produce superheavy massive particles. These created particles will interact
decaying into light ones, becoming a relativistic plasma whose energy density
eventually dominates those of the background and as a consequence the universe
will thermalize and will reheat.

To deal with particle production, note that, in Fourier space, the $k$-mode of
the quantum field $\chi$ satisfies the equation of a time dependent harmonic
oscillator

$\displaystyle{\chi}_{k}^{\prime\prime}+\omega_{k}^{2}(\tau){\chi}_{k}=0,$
(271)

where the derivative is with respect to the conformal time $\tau$ and the
square of the frequency is given by

$\displaystyle\omega_{k}^{2}(\tau)=k^{2}+a^{2}(\tau)\left[m_{\chi}^{2}+g^{2}\varphi^{2}(\tau)+\left(\xi-\frac{1}{6}\right)R(\tau)\right],$
(272)

being once again $m_{\chi}$ the bare mass of the quantum field, $R(\tau)$ the
scalar curvature and $\xi$ the coupling constant with gravity.

To clarify ideas about preheating (i.e., particle production) we consider the
simplest toy model

$\displaystyle
V(\varphi)=\left\\{\begin{array}[]{ccc}\frac{1}{2}m^{2}\varphi^{2}&\mbox{for}&\varphi<0\\\
0&\mbox{for}&\varphi\geq 0,\end{array}\right.$ (275)

although the reasoning will serve for a general class of Quintessential
Inflation potentials.

Near the phase transition, which actually corresponds to the beginning of
kination ($\varphi_{kin}=0$), one has
$\varphi(\tau)\cong\varphi^{\prime}_{kin}\tau$. Then, if we simplify choosing
a conformal coupling $\xi=\frac{1}{6}$, we can approximate the frequency
$\omega_{k}(\tau)$ by
$\sqrt{k^{2}+a^{2}_{kin}(m_{\chi}^{2}+g^{2}(\varphi^{\prime}_{kin})^{2}\tau^{2})}$,
where we have disregarded the expansion of the universe taking
$a(\tau)=a_{kin}$ in order to obtain the well-known overbarrier problem in
scattering theory nikishov , which can be analytically solved. Effectively,
the $\beta_{k}$-Bogoliubov coefficient is related with the reflection
coefficient via the formula (see popov ; marinov ; haro  and meyer ; fedoryuk
for a mathematical explanation)

$\displaystyle|\beta_{k}|^{2}=e^{-Im\int_{\gamma}\omega_{k}(\tau)d\tau},$
(276)

where $\gamma$ is a closed path in the complex plan that contains the two
turning points $\tau_{\pm}=\pm
i\frac{\sqrt{k^{2}+a^{2}_{kin}m_{\chi}^{2}}}{ga_{kin}\varphi^{\prime}_{kin}}.$

A simple calculation yields that the number of particles in the $k$-mode is
given by haro

$\displaystyle
n_{k}\equiv|\beta_{k}|^{2}=e^{-\frac{\pi(k^{2}+a^{2}_{kin}m_{\chi}^{2})}{ga_{kin}\varphi^{\prime}_{kin}}}.$
(277)

Then, the number density of produced particles is given by in terms of the
cosmic time $t$ Birrell , namely

$\displaystyle
n_{\chi}(t)\equiv\frac{1}{2\pi^{2}a^{3}(t)}\int_{0}^{\infty}k^{2}|\beta_{k}|^{2}dk=\frac{(g\dot{\varphi}_{kin})^{\frac{3}{2}}}{8\pi^{3}}\left(\frac{a_{kin}}{a(t)}\right)^{3}e^{-\frac{\pi
m_{\chi}^{2}}{g\dot{\varphi}_{kin}}}.$ (278)

Analogously, the energy density of the produced particles is given by Birrell

$\displaystyle\rho_{\chi}(t)\equiv\frac{1}{2\pi^{2}a^{4}(t)}\int_{0}^{\infty}\omega_{k}(t)k^{2}|\beta_{k}|^{2}dk.$
(279)

Then, at the phase transition one has

$\displaystyle\rho_{\chi,kin}=\frac{g^{2}\dot{\varphi}^{2}_{kin}}{4\pi^{4}}e^{-\frac{\pi
m_{\chi}^{2}}{g\dot{\varphi}_{kin}}},$ (280)

and at late times
$\rho_{\chi}(t)\cong\sqrt{m_{\chi}^{2}+g^{2}\varphi^{2}(t)}n_{\chi}(t)$, which
means that at late times the $\chi$-particles acquire an effective mass equal
to

$\displaystyle m_{eff}(t)\equiv\sqrt{m_{\chi}^{2}+g^{2}\varphi^{2}(t)}.$ (281)

Now, we will see that three constraints must be imposed fkl  in order to have
a viable model:

  1. 1.

If we do not want an exponential suppression of the energy density, one has to
choose a bare mass satisfying $m_{\chi}\leq\sqrt{g\dot{\varphi}_{kin}}$. In
fact, for the sake of simplicity, we will take $m_{\chi}=0$, and thus, the
effective mass is given by $m_{eff}=g|\varphi|$.

  2. 2.

For masses -the effective mass of the field $\chi$ is $g|\varphi|$\- greater
than the Hubble parameter the vacuum polarization energy density due to the
field $\chi$, which for $H\ll g|\varphi|$ can be calculated using the WKB
approximation, is of the order $\frac{H^{6}}{g^{2}\varphi^{2}}$ kaya . This
quantity is smaller than the energy density of the background ($\sim
H^{2}M_{pl}^{2}$) when the pivot scale leaves the Hubble radius, meaning that
the polarization effects will not affect the last stages of inflation.
Therefore, since $H\leq\sqrt{\frac{V}{3M_{pl}^{2}}}$, for a quadratic
potential the condition $H\ll g|\varphi|$ is accomplished imposing
$g\gg\frac{m}{M_{pl}}$.

  3. 3.

The energy density of the produced $\chi$-particles cannot dominate before its
decay into light particles, which will form the relativistic plasma, because,
if so, the force driving the inflation back to $\varphi=0$ will not disappear
and the inflaton field would not continue its movement forward up to $\infty$.
Effectively, the interaction term $\frac{1}{2}g^{2}\varphi^{2}\chi^{2}$
entails that after the phase transition the inflaton field satisfies the
equation

$\displaystyle\ddot{\varphi}+3H\dot{\varphi}=-g^{2}\chi^{2}\varphi.$ (282)

When the energy density of the $\chi$-particles is sub-dominant the right-hand
side of (282) is negligible and the field rolls towards $\infty$, but when it
is dominant the right-hand side ceases to be negligible, meaning that the
inflaton is under the action of the quadratic potential
$V(\varphi)=\frac{1}{2}g^{2}\varphi^{2}\chi^{2}$, so the inflaton field will
roll down to zero, which may produce a new inflationary phase.

Therefore, to avoid this situation, we have to calculate when the energy
density of the background and the field $\chi$ are of the same order, that is,

$\displaystyle\rho_{\varphi}(t)\sim\rho_{\chi}(t)\Longleftrightarrow
3H^{2}(t)M_{pl}^{2}\sim g\varphi(t)n_{\chi}(t).$ (283)

To obtain these quantities we use that for the model presented here, after the
phase transition the universe enters in a kination regime and one has

$\displaystyle\dot{H}=-3H^{2}\Longleftrightarrow
H(t)=\frac{1}{3t}\Longleftrightarrow
a(t)=a_{kin}\left(\frac{t}{t_{kin}}\right)^{\frac{1}{3}}=a_{kin}\left({3H_{kin}t}\right)^{\frac{1}{3}},$
(284)

and taking into account the Raychaudhuri equation
$\dot{H}=-\frac{\dot{\varphi}^{2}}{2M_{pl}^{2}}$, we get

$\displaystyle\varphi(t)=M_{pl}\int_{t_{kin}}^{t}\sqrt{-2\dot{H}(s)}ds=M_{pl}\int_{H(t)}^{H_{kin}}\sqrt{\frac{-2}{\dot{H}(H)}}dH=\sqrt{\frac{2}{3}}M_{pl}\ln(3H_{kin}t).$
(285)

Now, using that $\rho_{\chi}(t)\cong g\varphi(t)n_{\chi}(t)$, one gets

$\displaystyle\rho_{\chi}(t)\cong 3\times
10^{-3}g^{5/2}M_{pl}^{5/2}H^{1/2}_{kin}\frac{1}{t}\ln(3H_{kin}t),\quad\mbox{and}\quad\rho_{\varphi}(t)\cong
3\times 10^{-1}M_{pl}^{2}\frac{1}{t^{2}}.$ (286)

Then, both quantities are of the same order when

$\displaystyle t\sim\bar{t}\equiv\frac{10^{2}}{g^{5/2}\sqrt{H_{kin}M_{pl}}}.$
(287)

And we obtain an important constraint for this theory: the decaying time,
i.e., when the particles have decayed into a relativistic plasma, must be
smaller than $\bar{t}$, in order for the back-reaction to be subdominant so
that the inflaton field rolls monotonically towards $\infty$.

On the other hand, assuming as usual that there is no substantial drop of
energy between the end of inflation and the phase transition time, and using
that the value of the power spectrum of the curvature perturbation when the
pivot scale leaves the Hubble radius is given by btw
${\mathcal{P}}_{\zeta}\cong\frac{H_{*}^{2}}{8\pi^{2}M_{pl}^{2}\epsilon_{*}}\sim
2\times 10^{-9}$, one obtains

$\displaystyle m^{2}\sim 3\times 10^{-9}\pi^{2}(1-n_{s})^{2}M_{pl}^{2},$ (288)

where we have used that for our model one has
$\epsilon_{*}=\frac{2M_{pl}^{2}}{\varphi^{2}_{*}}\cong\frac{1-n_{s}}{4}$,
being $n_{s}$ the spectral index. Then, since the recent observational data
constrain the value of the spectral index to be $n_{s}=0.968\pm 0.006$ Planck
, taking its central value one gets $m\sim 5\times 10^{-6}M_{pl}$ and, as a
consequence, $H_{kin}\sim H_{end}\sim\frac{m\varphi_{end}}{\sqrt{6}M_{pl}}\sim
3\times 10^{-6}M_{pl}$. Thus, for values of $g\leq 10^{-2}$, one gets
$\bar{t}\geq 10^{10}M_{pl}^{-1}$ and $\varphi(\bar{t})\sim M_{pl}$, meaning
that for times $t\in[10^{6}M_{pl}^{-1},\bar{t}]$ the value of the inflaton
field remains close to $M_{pl}$ and the effective mass of the $\chi$-field
will approximately be $gM_{pl}$.

Since the decay must be before than $\bar{t}$, the condition $t_{dec}<\bar{t}$
, where $t_{dec}\cong\frac{1}{3\Gamma}$ is the time when the field $\chi$
decayed, i.e., $H_{dec}\sim{\Gamma}$, leads to the relation
$g<10^{2}\left(\frac{\Gamma}{M_{pl}}\right)^{2/5}$, which together with the
condition $g\gg\frac{m}{M_{pl}}\sim 10^{-6}$ constrains the value of the decay
rate to satisfy $\Gamma\gg 10^{-20}M_{pl}\sim 10^{-2}$ GeV.

Let’s now calculate the temperature at the reheating time, which as we have
already seen is given by

$\displaystyle T_{reh}\sim g_{reh}^{-1/4}\rho_{\chi,reh}^{1/4}\sim
g_{reh}^{-1/4}\rho_{\chi,dec}^{1/4}\sqrt{\frac{\rho_{\chi,dec}}{\rho_{\varphi,dec}}}.$
(289)

Taking into account that, if
$t_{dec}>10^{6}M_{pl}^{-1}\Longrightarrow\Gamma<10^{-6}M_{pl}$, the effective
mass is approximately $gM_{pl}$, one has

$\displaystyle\rho_{\varphi,dec}=3{\Gamma}^{2}M_{pl}^{2}\quad\mbox{and}\quad\rho_{\chi,dec}\sim
gM_{pl}n_{\chi,dec}\sim 10^{-2}g^{5/2}\sqrt{H_{kin}M_{pl}}M_{pl}^{2}{\Gamma},$
(290)

getting that

$\displaystyle T_{reh}\sim
10^{-2}g^{15/8}H^{3/8}_{kin}M_{pl}^{7/8}{\Gamma}^{-1/4}\sim
10^{-4}g^{15/8}\left(\frac{M_{pl}}{\Gamma}\right)^{1/4}M_{pl}.$ (291)

The values of the parameters $g$ and $\Gamma$ must satisfy the constraint
$10^{-6}\ll g<10^{2}\left(\frac{\Gamma}{M_{pl}}\right)^{2/5}$, and choosing
for instance $g\sim 10^{-4}$ and $\Gamma\sim 10^{-12}M_{pl}$ one obtains a
reheating temperature around $10^{9}$ GeV.

Finally, one can conclude that the value of the parameters involved in the
theory must accomplish all the following requirements:

  1. 1.

$10^{-6}\ll g<10^{2}\left(\frac{\Gamma}{M_{pl}}\right)^{2/5}$. (The back-
reaction is not important at the last stages of inflation and the $\chi$-field
decays before the end of kination)

  2. 2.

$10^{-20}M_{pl}\ll\Gamma<10^{-6}M_{pl}$. (The decay ends when the value of the
inflaton field is of the order of the reduced Planck mass).

  3. 3.

$10^{-19}<g^{15/8}\left(\frac{M_{pl}}{\Gamma}\right)^{1/4}<10^{-7}$.
(Reheating temperatures guaranteeing the BBN success).

###  VI.3  Curvaton reheating in Quintessential Inflation

In this section we review the so-called curvature reheating mechanism in
Quintessential Inflation. To do that we follow FL , but taking into account
that for the authors of that paper, contrary to our convention, the reheating
temperature is the temperature of the universe when the curvaton field has
totally decayed in relativistic particles. Recall that the universe is really
reheated when the energy density of inflaton is of the same order than the one
of the decay products, provided that the curvaton field had previously
decayed.

We assume that the potential of the curvaton field, namely $\sigma$, is
quadratic, $V(\sigma)=\frac{1}{2}m_{\sigma}^{2}\sigma^{2}$, where the mass of
the curvaton is chosen to be smaller than the value of the Hubble parameter at
the end of inflation $m_{\sigma}\ll H_{end}$. At this moment the curvaton
field is in a slow-roll regime because the condition $m_{\sigma}\ll
H_{end}\Longleftrightarrow V_{\sigma\sigma}\ll H_{end}^{2}$ means that the
curvaton potential is flat enough at the end of inflation LW . Then, in order
to avoid a second inflationary stage now driven by the curvaton, one has to
impose that its energy density is subdominant when the curvaton starts to
oscillate, which happens when $m_{\sigma}\cong H$ LW  (see also the section
$5.4.1$ of mukhanovbook  for a detailed discussion of the quadratic
potential). Then,

$\displaystyle\rho_{\sigma}(t_{osc})\equiv\rho_{\sigma,osc}<\rho_{\varphi,osc}=3H^{2}_{osc}M_{pl}^{2},$
(292)

where $t_{osc}$ is the time when the curvaton starts to oscillate. Taking into
account that $H_{osc}\cong m_{\sigma}$ and since at the beginning of the
oscillations, i.e., at the end of the slow-roll period for the curvaton,
$H\dot{\sigma}\sim V_{\sigma}\Longrightarrow\dot{\sigma}_{osc}\sim
m_{\sigma}\sigma_{osc}$, one has $\rho_{\sigma,osc}\cong
m_{\sigma}^{2}\sigma^{2}_{osc}$ obtaining the bound
$\sigma^{2}_{osc}<3M_{pl}^{2}$.

Here we consider a Quintessential Inflation potential, as the Peebles-Vilenkin
one or similar, according to which the universe enters in a kination regime
immediately after the end of inflation. Then, after the phase transition, the
energy density of the background evolves as $a^{-6}$, and those of the
curvaton as $a^{-3}$ because during the oscillation regime the effective
Equation of State parameter for a power law potential
$V(\sigma)=V_{0}\left(\frac{\sigma}{M_{pl}}\right)^{2n}$ is given by
$w_{eff}\cong\frac{n-1}{n+1}$ turner .

Now, let ${\Gamma}$ be the decay rate of the curvaton. Then, there are two
different situations:

  1. 1.

The curvaton decays when it was subdominant.

In this first case, the curvaton decays into radiation (recall that the
thermalization is nearly instantaneous) at a time ${t_{dec}}$ satisfying
$H_{dec}\sim{\Gamma}$, and thus, one has

$\displaystyle\rho_{\sigma,dec}<\rho_{\varphi,dec}\Longrightarrow\rho_{\sigma,osc}\frac{{\Gamma}}{m_{\sigma}}<3{\Gamma}^{2}M_{pl}^{2},$
(293)

where we have used that the energy density of the curvaton decays as $a^{-3}$,
that the universe is in the kination phase (the Hubble parameter also decays
as $a^{-3}$) and $H_{osc}\cong m_{\sigma}$. Then, since
$\rho_{\sigma,osc}\cong m_{\sigma}^{2}\sigma^{2}_{osc}$ and ${\Gamma}\cong
H_{dec}\leq H_{osc}\cong m_{\sigma}$, one gets the constraint

$\displaystyle\frac{\sigma^{2}_{osc}}{3M_{pl}^{2}}\leq\frac{{\Gamma}}{m_{\sigma}}\leq
1.$ (294)

Now, to obtain the reheating temperature, that is, when
$\rho_{\varphi,reh}\sim\rho_{\sigma,reh}$, one gets

$\displaystyle T_{reh}\sim g_{reh}^{-1/4}\rho_{\sigma,reh}^{1/4}\sim
g_{reh}^{-1/4}\rho_{\sigma,dec}^{1/4}\sqrt{\frac{\rho_{\sigma,dec}}{\rho_{\varphi,dec}}}\sim\frac{\rho_{\sigma,dec}^{3/4}}{M_{pl}{\Gamma}}\sim\frac{m_{\sigma}^{3/4}|\sigma(t_{osc})|^{3/2}}{M_{pl}{\Gamma}^{1/4}},$
(295)

where we have used that
$\rho_{\sigma,dec}=\rho_{\sigma,osc}\left(\frac{a_{osc}}{a_{dec}}\right)^{3}=\rho_{\sigma,osc}\frac{H_{dec}}{H_{osc}}\cong
m\sigma^{2}_{osc}{\Gamma}$.

Then, using the bound (294) we can see that the reheating temperature is
constrained to be in the range

$\displaystyle\frac{m_{\sigma}^{1/2}|\sigma_{osc}|^{3/2}}{\sqrt{3}M_{pl}}\leq
T_{reh}\leq\frac{m_{\sigma}^{1/2}|\sigma_{osc}|}{{3}^{1/4}M^{1/2}_{pl}}.$
(296)

On the other hand, when the decay of the curvaton takes place when it is
subdominant, the power spectrum of the curvature perturbation is given by LU ;
ABM
${\mathcal{P}}_{\zeta}=\frac{1}{1296\pi^{2}}\frac{m_{\sigma}^{2}}{{\Gamma}^{2}}\frac{H_{*}^{2}\sigma_{*}^{2}}{M^{4}_{pl}}$,
which from the bound (294) leads to

$\displaystyle\frac{1}{1296\pi^{2}}\frac{H_{*}^{2}\sigma_{*}^{2}}{M^{4}_{pl}}\leq{\mathcal{P}_{\zeta}}\leq\frac{1}{144\pi^{2}}\frac{H_{*}^{2}}{\sigma_{*}^{2}},$
(297)

where we have used that before the oscillations the curvaton rolls slowly and,
thus, $\sigma_{osc}\sim\sigma_{*}$. Now, taking into account that
${\mathcal{P}_{\zeta}}\sim 2\times 10^{-9}$, one gets the bounds

$\displaystyle\frac{H_{*}}{|\sigma_{*}|}\geq 2\times 10^{-3}\quad\mbox{ and
}\quad{H_{*}}{|\sigma_{*}|}\leq 5\times 10^{-3}M_{pl}^{2}.$ (298)

Then, choosing $H_{*}\sim 4\times 10^{-5}M_{pl}$, which for $n_{s}\cong 0.96$
is the value of the Hubble parameter when the pivot scale leaves the Hubble
radius if the curvaton field is not present ha , and taking into account that
the condition $H_{*}\ll|\sigma_{*}|$ must be satisfied in order to guarantee
the Gaussianity of the curvature perturbation LW , one can safely take
$|\sigma_{*}|\sim 10^{-2}M_{pl}$, which agrees with the bound
$\sigma_{*}^{2}\sim\sigma^{2}_{osc}<3M_{pl}^{2}$.

For these values, the equation (296) becomes

$\displaystyle 5\times 10^{-4}\sqrt{m_{\sigma}M_{pl}}\leq T_{R}\leq 7\times
10^{-3}\sqrt{m_{\sigma}M_{pl}},$ (299)

which means that in order to get the nucleosynthesis bounds one has to choose
light masses of the curvaton, namely $m_{\sigma}\leq 10^{-14}M_{pl}\sim
2\times 10^{4}$ GeV.

  2. 2.

The curvaton decays when the curvaton field dominated the universe.

Assuming once again instantaneous thermalization, since the curvaton decays
when it dominates, that is, when $\rho_{\varphi,dec}\leq\rho_{\sigma,dec}$,
the reheating time will occur at the decay time ($H_{reh}=H_{dec}\sim\Gamma$).

In this case the condition

$\displaystyle
3M_{pl}^{2}\Gamma^{2}=\rho_{\sigma,reh}<\rho_{\sigma,osc}=m_{\sigma}^{2}\sigma_{osc}^{2},$
(300)

together with $H_{reh}<H_{osc}$, leads to the constraints

$\displaystyle\frac{{\Gamma}^{2}}{m_{\sigma}^{2}}<\frac{\sigma^{2}_{osc}}{3M_{pl}^{2}}\qquad\mbox{and}\qquad\frac{{\Gamma}}{m_{\sigma}}<1.$
(301)

Next, in this case, the reheating temperature is $T_{reh}\sim
g_{reh}^{-1/4}\rho_{\sigma,reh}^{1/4}\sim\sqrt{M_{pl}{\Gamma}}$ and the
constraint leads to the bound

$\displaystyle T_{reh}\leq\sqrt{\frac{m_{\sigma}}{M_{pl}}}|\sigma_{osc}|.$
(302)

On the other hand, when the curvaton decays after its domination, the power
spectrum of the curvature perturbation is given by LW

$\displaystyle{\mathcal{P}}_{\zeta}\cong\frac{1}{9\pi^{2}}\frac{H_{*}^{2}}{\sigma_{*}^{2}}\sim
2\times 10^{-9}\Longrightarrow\frac{H_{*}}{|\sigma_{*}|}\sim 4\times
10^{-4}M_{pl}.$ (303)

Then, choosing as in the previous case $H_{*}\sim 4\times 10^{-5}M_{pl}$, one
has $|\sigma_{*}|\sim 10^{-1}M_{pl}$, and, since the curvaton rolls slowly
before the oscillations, one can safely take
$|\sigma_{osc}|\sim|\sigma_{*}|\sim 10^{-1}M_{pl}$, which satisfies the bound
(301).

Finally, from these values and the equation (302), one can conclude that only
for curvaton light masses satisfying $m_{\sigma}\leq 5\times
10^{-16}M_{pl}\sim 10^{3}$ GeV a reheating temperature compatible with the BBN
success is obtained.

##  VII Impact of the production of Gravitational Waves in the BBN success

###  VII.1 Overproduction of GWs

It is well-known that a reheating due to the gravitational production of light
particles is incompatible with the overproduction of Gravitational Waves
because both satisfy the same equation, and thus, they scale in the same way.
For this reason it is impossible that at the reheating time the ratio of the
energy density of the GWs to the energy density of the relativistic plasma was
smaller that $10^{-2}$, which is essential to prevent any complication in the
BBN success pv . In fact, the minimum value of this ratio, which is obtained
when the light particles are minimally coupled with gravity ($\xi=0$), is of
the order $1/N_{s}$, where $N_{s}$ is the number of light fields. As has been
explained in pv  in a minimal GUT theory $N_{s}=4$, which corresponds to the
electroweak Higgs doublet, and only in supersymmetric theories $N_{s}$ is of
the order $10^{2}$ (see the explanation given by Peebles-Vilenkin in pv ).

However, as we will see in this subsection, for our LQI model, when the
reheating is due to the gravitational creation of heavy particles, the
overproduction of GW’s do not modify the BBN.

So, to show this, first of all we recall that the production of Gravitational
Waves during the phase transition from the end of inflation to the beginning
of kination is Giovannini99

$\displaystyle\rho_{GW}(\tau)\cong\frac{H_{kin}^{4}}{2\pi^{3}}\left(\frac{a_{kin}}{a(\tau)}\right)^{4}\cong
10^{-2}H_{kin}^{4}\left(\frac{a_{kin}}{a(\tau)}\right)^{4}.$ (304)

The success of the BBN demands that the ratio of the energy density of GWs to
the one of the produced particles at the reheating time satisfies attractor1

$\displaystyle\frac{\rho_{GW,reh}}{\rho_{\chi,reh}}\leq 10^{-2}.$ (305)

First of all we see that the constraint (305) is overpassed when the decay of
the superheavy massive particles is previous to the end of kination.
Effectively, if the decay occurs before the end of kination one has

$\displaystyle\frac{\rho_{GW,reh}}{\rho_{\chi,reh}}=\frac{\rho_{GW,dec}}{\rho_{\chi,dec}}$
(306)

because the light relativistic particles evolve as the GW’s. Now, taking into
account that during kination the energy density of the background scales as
$a^{-6}$, it yields that

$\displaystyle\left(\frac{a_{kin}}{a_{dec}}\right)^{4}=\left(\frac{\rho_{\varphi,dec}}{\rho_{\varphi,kin}}\right)^{2/3},$
(307)

and then, since in LQI one has $H_{kin}\sim 4\times 10^{-8}M_{pl}$ and using
the results obtained previously (see the formula (263), we get

$\displaystyle\frac{\rho_{GW,reh}}{\rho_{\chi,reh}}\cong
10^{-1}\left(\frac{\Gamma}{M_{pl}}\right)^{1/3}.$ (308)

Therefore, the bound (305) is overpassed when

$\displaystyle\Gamma\leq 7\times 10^{-4}M_{pl},$ (309)

which is completely compatible within the bound (294).

On the contrary, when the decay of superheavy particles is produced after the
end of kination, and assuming once again instantaneous thermalization, the
reheating time will coincide with the decay one. Then, since
$\rho_{\chi,dec}=3{\Gamma}^{2}M_{pl}^{2}$ and

$\displaystyle
H_{dec}=H_{end}\left(\frac{a_{end}}{a_{dec}}\right)^{3/2}\Longrightarrow\left(\frac{a_{end}}{a_{dec}}\right)^{3/2}=\sqrt{\frac{3}{2}}\frac{\Gamma
M_{pl}\sqrt{\bar{\rho}_{\varphi}}}{\bar{\rho}_{\chi}},$ (310)

where we have used that
$H_{end}=\sqrt{\frac{2}{3}}\frac{\bar{\chi,\rho}}{M_{pl}\sqrt{\bar{\rho}_{\varphi}}}$,
we will have

$\displaystyle\rho_{GW,dec}=\rho_{GW,end}\left(\frac{a_{end}}{a_{dec}}\right)^{4}=\rho_{GW,end}\left(\sqrt{\frac{3}{2}}\frac{\Gamma
M_{pl}\sqrt{\bar{\rho}_{\varphi}}}{\bar{\rho}_{\chi}}\right)^{8/3}$ (311)

and, using that

$\displaystyle\left(\frac{a_{kin}}{a_{end}}\right)^{4}=\frac{\rho_{\varphi,end}}{\rho_{\varphi,kin}}\quad\mbox{and}\quad\rho_{\varphi,end}=\frac{\bar{\rho}_{\chi}^{2}}{\bar{\rho}_{\varphi}},$
(312)

we get

$\displaystyle\rho_{GW,dec}=\left(\frac{2}{16}\right)^{1/3}\left(\frac{H_{kin}}{M_{pl}}\right)^{2}\left(\frac{\bar{\rho}_{\varphi}}{\bar{\rho}_{\chi}^{2}}\right)^{1/3}(\Gamma
M_{pl})^{8/3}\cong 10^{-2}\Gamma^{8/3}M_{pl}^{4/3},$ (313)

and thus,

$\displaystyle\frac{\rho_{GW,reh}}{\rho_{\chi,reh}}=\frac{\rho_{GW,dec}}{\rho_{\chi,dec}}\cong
3\times 10^{-3}\left(\frac{\Gamma}{M_{pl}}\right)^{2/3}\leq 10^{-17},$ (314)

where we have used the bound (301), so the constraint (305) is clearly
overpassed.

###  VII.2 BBN constraints from the logarithmic spectrum of GWs

It is well-known that during inflation GWs are produced (known as primordial
GWs, in short PGWs) and in the post-inflationary period, i.e., during
kination, the logarithmic spectrum of GWs, namely $\Omega_{GW}$ defined as
$\Omega_{GW}\equiv\frac{1}{\rho_{c}}\frac{d\rho_{GW}(k)}{d\ln k}$ (where
$\rho_{GW}(k)$ is the energy density spectrum of the produced GWs;
$\rho_{c}=3H_{0}^{2}M_{pl}^{2}$, where $H_{0}$ is the present value of the
Hubble parameter, is the so-called critical density) scales as $k^{2}$ rubio ,
producing a spike in the spectrum of GWs at high frequencies. Then, so that
GWs do not destabilize the BBN, the following bound must be imposed (see
Section 7.1 of maggiore ),

$\displaystyle I\equiv h_{0}^{2}\int_{k_{BBN}}^{k_{end}}\Omega_{GW}(k)d\ln
k\leq 10^{-5},$ (315)

where $h_{0}\cong 0.678$ parametrizes the experimental uncertainty to
determine the current value of the Hubble constant and $k_{BBN}$, $k_{end}$
are the momenta associated to the horizon scale at the BBN and at the end of
inflation respectively. As has been shown in Giovannini1 , the main
contribution of the integral (315) comes from the modes that leave the Hubble
radius before the end of the inflationary epoch and finally re-enter during
kination, that means, for $k_{end}\leq k\leq k_{kin}$, where
$k_{end}=a_{end}H_{end}$ and $k_{kin}=a_{kin}H_{kin}$. For these modes one can
calculate the logarithmic spectrum of GWs as in Giovannini  (see also rubio ;
Giovannini2 ; Giovannini3 ; Giovannini:2016vkr  where the graviton spectra in
quintessential models have been reassessed, in a model-independent way, using
numerical techniques),

$\displaystyle\Omega_{GW}(k)=\tilde{\epsilon}\Omega_{r}h^{2}_{GW}\left(\frac{k}{k_{end}}\right)\ln^{2}\left(\frac{k}{k_{kin}}\right),$
(316)

where $h^{2}_{GW}=\frac{1}{8\pi}\left(\frac{H_{kin}}{M_{pl}}\right)^{2}$ is
the amplitude of the GWs; $\Omega_{r}\cong 2.6\times 10^{-5}h_{0}^{-2}$ is the
present density fraction of radiation, and the quantity $\tilde{\epsilon}$,
which is approximately equal to $0.05$ for the Standard Model of particle
physics, takes into account the variation of massless degrees of freedom
between decoupling and thermalization (see rubio ; Giovannini1  for more
details). As has been derived in Giovannini1 , the specific form of the
expression above comes from the behavior of the Hankel functions for small
arguments. Now, plugging expression (316) into (315) and disregarding the sub-
leading logarithmic terms, one finds

$\displaystyle
2\tilde{\epsilon}h_{0}^{2}\Omega_{r}h^{2}_{GW}\left(\frac{k_{kin}}{k_{end}}\right)\leq
10^{-5}\Longrightarrow
10^{-2}\left(\frac{H_{kin}}{M_{pl}}\right)^{2}\left(\frac{k_{kin}}{k_{end}}\right)\leq
1\Longrightarrow 10^{-17}\left(\frac{k_{kin}}{k_{end}}\right)\leq 1$ (317)

because in our LQI model we have $H_{kin}\sim 4\times 10^{-8}M_{pl}$.

To calculate the ratio $k_{kin}/k_{end}$, we will have to study the following
two different situations:

  1. 1.

When the decay of superheavy particles occurs before the end of kination.

In this case the reheating time coincides with the end of kination, so a
simple calculation leads to

$\displaystyle\frac{k_{kin}}{k_{end}}=\frac{k_{kin}}{k_{reh}}.$ (318)

Using the formulas

$\displaystyle\rho_{\varphi,reh}=\rho_{\varphi,kin}\left(\frac{a_{kin}}{a_{reh}}\right)^{6}\qquad\mbox{and}\qquad\rho_{\chi,reh}=\rho_{\chi,dec}\left(\frac{a_{dec}}{a_{reh}}\right)^{4},$
(319)

we get

$\displaystyle\frac{a_{kin}}{a_{reh}}=\left(\frac{\rho_{\chi,dec}}{\rho_{\varphi,kin}}\right)^{1/6}\left(\frac{a_{dec}}{a_{reh}}\right)^{2/3}=\left(\frac{\rho_{\chi,dec}}{\rho_{\varphi,kin}}\right)^{1/6}\left(\frac{\rho_{\chi,dec}}{\rho_{\varphi,dec}}\right)^{1/3},$
(320)

where we have used the relation
$\left(\frac{a_{dec}}{a_{reh}}\right)^{2}=\frac{\rho_{\chi,dec}}{\rho_{\varphi,dec}}$.
Then, taking into account that
$H_{reh}=\Gamma\left(\frac{a_{dec}}{a_{reh}}\right)^{3}$, we obtain

$\displaystyle\frac{k_{kin}}{k_{end}}=\frac{k_{kin}}{k_{reh}}=\frac{H_{kin}a_{kin}}{H_{reh}a_{reh}}=\frac{H_{kin}}{\Gamma}\frac{\rho_{\varphi,dec}}{\rho_{\chi,dec}}\left(\frac{\rho_{\varphi,dec}}{\rho_{\varphi,kin}}\right)^{1/6}.$
(321)

Finally, using our previous results

$\displaystyle\rho_{\varphi,dec}=3\Gamma^{2}M_{pl}^{2},\quad\rho_{\chi,dec}\sim
10^{-21}\Gamma
M_{pl}^{3}\quad\mbox{and}\quad\rho_{\varphi,kin}=3H_{kin}^{2}M_{pl}^{2},$
(322)

we arrive at

$\displaystyle\frac{k_{kin}}{k_{end}}\sim
10^{6}\left(\frac{\Gamma}{M_{pl}}\right)^{1/3},$ (323)

and consequently the constraint becomes

$\displaystyle\left(\frac{\Gamma}{M_{pl}}\right)^{1/3}\leq 10,$ (324)

which is obviously overpassed for all the viable values of $\Gamma$, i.e, for
all values between $10^{-21}M_{pl}$ and $10^{-8}M_{pl}$.

  2. 2.

When the decay of superheavy particles occurs after the end of kination.

Since during the kination period we have
$H_{end}=H_{kin}\left(\frac{a_{kin}}{a_{end}}\right)^{3}$, one gets

$\displaystyle\frac{k_{kin}}{k_{end}}=\left(\frac{H_{kin}}{H_{end}}\right)^{2/3}$
(325)

and, taking into account that

$\displaystyle
H_{end}=\sqrt{\frac{2\bar{\rho}_{\chi}^{2}}{3M_{pl}^{2}\bar{\rho}_{\varphi}}}\sim
3\times 10^{-22}M_{pl}\qquad\mbox{and}\qquad H_{kin}\sim 4\times
10^{-8}M_{pl},$ (326)

the bound is completely overpassed.

##  VIII Conclusions

In the present review we have dealt with some important Quintessential
Inflation scenarios, namely the original Peebles-Vilenkin one -the first
Quintessential Inflation model introduced at the end of the 90’s- and some of
its improved versions, a kind of exponential scenarios, the Lorentzian
Quintessential Inflation and also Quintessential Inflation in the context of
$\alpha$-attractors. Studying these models, first of all we have checked that
at early times they provide results -spectral index of scalar perturbations,
ratio of tensor to scalar perturbations, number of e-folds from the horizon
crossing to the end of inflation- agreeing with those of the standard
inflation and matching with the current observational data. Next, the models
should contain a phase transition from the end of inflation to the beginning
of kination where the adiabatic evolution of the universe must be broken in
order to have enough particle production to reheat the universe after its
cooling produced during inflation. Once we have checked that the models
contain this phase transition we have chosen the reheating mechanism, in our
case the gravitational particle production of superheavy particles, although
in this review we have also studied other mechanisms such as the Instant
Preheating or the Curvaton reheating, and we have calculated numerically the
energy density of the produced particles and the corresponding reheating
temperature, whose maximum value is for the Lorentzian Quintessential
Inflation model around $10^{7}$ GeV.

We have also dealt with the overproduction of Gravitational Waves, produced as
well during the phase transition from the end of inflation to the beginning of
kination, showing that in the case of Lorentzian Quintessential Inflation -and
also for the $\alpha$-attractors in the context of QI- all the bounds imposed
are overpassed, so they do not disturb the Big Bang Nucleosynthesis success.

Finally, we have found the numerical value of the other parameter (these
models normally only depend on two parameters, the first one is determined
using the power spectrum of scalar perturbations when the pivot scale leaves
the Hubble radius), imposing that the model matches with the current
observational data at the present time. To do it, we have solved numerically
the corresponding dynamical system imposing initial conditions, which are
determined from the values of the inflaton field and its first derivative
during the slow-roll period, at the beginning of the radiation era. In this
way, we have checked numerically that, at late times, the models enter in a
dark energy regime (via quintessence) able to explain the current
observational data.

## Acknowledgments

The investigation of J. de H. has been supported by MINECO (Spain) grant
MTM2017-84214-C2-1-P, and in part by the Catalan Government 2017-SGR-247.
L.A.S. thanks the School of Mathematical Sciences (Queen Mary University of
London) for the support provided.

## References

  * (1) A. Guth, The inflationary universe: a possible solution to the horizon and flatness problems, Phys. Rev. D 23, 347 (1981).
  * (2) A. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. B 108, 389 (1982).
  * (3) A. Albrecht and P.J. Steinhardt, Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking, Phys. Rev. Lett. 48, 1220 (1982).
  * (4) G. Chibisov and V. Mukhanov, Galaxy formation and phonons, Mon. Not. Roy. Astron. Soc. 200, 535 (1982).
  * (5) A. A. Starobinsky, Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations, Phys. Lett. B 117, 175 (1982).
  * (6) A. H. Guth and S-Y. Pi, Fluctuations in the New Inflationary Universe, Phys. Rev. Lett. 49, 1110 (1982).
  * (7) J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Spontaneous creation of almost scale-free density perturbations in an inflationary universe, Phys. Rev. D 28, 679 (1983).
  * (8) A. D. Linde, Scalar Field Fluctuations in Expanding Universe and the New Inflationary Universe Scenario, Phys. Lett. B 116, 335 (1982).
  * (9) P. A. R. Ade et al [Planck Collaboration], Planck 2015 results. XX. Constraints on inflation, Astron. Astrophys. 594, A20 (2016) [arXiv:1502.02114 [astro-ph.CO]].
  * (10) E. J. Copeland, M. Sami and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D 15, 1753 (2006) [arXiv:hep-th/0603057].
  * (11) A. G. Riess et al, Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astron. J.116, 1009 (1998) [arXiv:astro-ph/9805201].
  * (12) S. Perlmutter et al, Measurements of Omega and Lambda from $42$ High-Redshift Supernovae, Astrophys. J. 517, 565 (1999) [arXiv:astro-ph/9812133].
  * (13) B. Spokoiny, Deflationary Universe Scenario, Phys. Lett. B 315, 40 (1993) [arXiv:gr-qc/9306008].
  * (14) P.J.E. Peebles and A. Vilenkin, Quintessential inflation, Phys.Rev. D 59, 063505 (1999) [arXiv:astro-ph/9810509].
  * (15) P.J.E. Peebles and B. Ratra, Cosmology with a time-variable cosmological ”constant”, Astrophys. J. Lett. 352, L17 (1988).
  * (16) S. Tsujikawa, Quintessence: A Review, Class. Quant. Grav. 30, 214003 (2013) [arXiv:1304.1961 [gr-qc]].
  * (17) E.I. Guendelman, R. Herrera, R., P. Labrana, E. Nissimov and S. Pacheva, Emergent Cosmology, Inflation and Dark Energy, Gen. Rel. Grav. 47, 10 (2015) [arXiv:1408.5344 [gr-qc]].
  * (18) J. de Haro, J. Amorós and S. Pan, A simple nonsingular inflationary quintessential model, Phys. Rev. D 93, 084018 (2016) [arXiv:1601.08175 [gr-qc]].
  * (19) J. de Haro and E. Elizalde, Inflation and late-time acceleration from a double-well potential with cosmological constant, Gen. Rel. Grav. 48, 77 (2016) [arXiv:1602.03433 [gr-qc]].
  * (20) J. de Haro, On the viability of quintessential inflationary models from observational data, Gen. Rel. Grav. 49, 6 (2017) [arXiv:1602.07138 [gr-qc]].
  * (21) L. Aresté Saló and J. de Haro, Quintessential inflation at low reheating temperatures, Eur. Phys. J. C 77, 798 (2017) [arXiv:1707.02810 [gr-qc]].
  * (22) J. Haro and S. Pan, Bulk viscous quintessential inflation, Int. J. Mod. Phys. D 27, 1850052 (2018) [arXiv:1512.03033 [gr-qc]].
  * (23) Md. Wali Hossain, R. Myrzakulov, M. Sami and E. N. Saridakis, A class of quintessential inflation models with parameter space consistent with BICEP2, Phys. Rev. D 89, 123513 (2014) [arXiv:1404.1445 [gr-qc]]
  * (24) C.Q. Geng, Md. Wali Hossain, R.Myrzakulov, M. Sami and E.N. Saridakis, Quintessential inflation with canonical and noncanonical scalar fields and Planck 2015 results, Phys. Rev. D 92, 023522 (2015) [arXiv:1502.03597 [gr-qc]].
  * (25) Md. Wali Hossain, R. Myrzakulov, M. Sami and E. N. Saridakis, Unification of inflation and dark energy à la quintessential inflation Int. J. Mod. Phys. D 24, no. 05, 1530014 (2015) [arXiv:1410.6100 [gr-qc]].
  * (26) Md. Wali Hossain, R. Myrzakulov, M. Sami and E. N. Saridakis, Evading Lyth bound in models of quintessential inflation, Phys. Lett. B 737, 191-195 (2014) [arXiv:1405.7491 [gr-qc]].
  * (27) C. Q. Geng, C. C. Lee, M. Sami, E. N. Saridakis and A. A. Starobinsky, Observational constraints on successful model of quintessential Inflation, JCAP 06, 011 (2017) [arXiv:1705.01329 [gr-qc]].
  * (28) J. de Haro, J. Amorós and S. Pan, Simple inflationary quintessential model II: Power law potentials, Phys. Rev. D 94, 064060 (2016) [arXiv:1607.06726 [gr-qc]].
  * (29) K. Dimopoulos, Towards a model of Quintessential Inflation, Nucl. Phys. Proc. Suppl 95, 70 (2001) [arXiv:astro-ph/0012298]
  * (30) K. Dimopoulos and J. W. F. Valle, Modeling Quintessential Inflation, Astropart. Phys. 18, 287 (2002) [arXiv:astro-ph/0111417].
  * (31) M. Giovannini, Low-scale Quintessential Inflation, Phys. Rev. D 67, 123512 (2003) [arXiv:hep-ph/0301264].
  * (32) J. Haro, W. Yang and S. Pan, Reheating in quintessential inflation via gravitational production of heavy massive particles: A detailed analysis, JCAP 01, 023 (2019) [arXiv:1811.07371 [gr-qc]].
  * (33) L. Kofman, A. Linde and A. Starobinsky, Reheating after Inflation, Phys. Rev. Lett. 73, 3195-3198 (1994) [arXiv:hep-th/9405187].
  * (34) L. Kofman, A. Linde and A. Starobinsky, Towards the Theory of Reheating After Inflation, Phys. Rev. D 56, 3258-3295 (1997) [arXiv:hep-ph/9704452].
  * (35) P.B. Greene, L. Kofman, A. Linde and A. Starobinsky, Structure of Resonance in Preheating after Inflation, Phys. Rev. D 56, 6175 (1997).
  * (36) Y. Shtanov, J. Traschen and R. Brandenberger, Universe Reheating after Inflation, Phys. Rev. D 51, 5438 (1995) [arXiv:hep-ph/9407247].
  * (37) B.A. Bassett and S. Liberati, Geometric Reheating after Inflation, Phys. Rev. D 58, 021302 (1998) [arXiv:hep-ph/9709417].
  * (38) M. Joyce, Electroweak Baryogenesis and the Expansion Rate of the Universe, Phys. Rev. D 55, 1875 (1997) [arXiv:hep-ph/9606223].
  * (39) L. Parker, Particle Creation in Expanding Universes, Phys. Rev. Lett. 21 , 562 (1968); Quantized Fields and Particle Creation in Expanding Universes. I, Phys. Rev. 183, 1057 (1969); Quantized Fields and Particle Creation in Expanding Universes. II, Phys. Rev. D 3, 346 (1970).
  * (40) V. M. Folov, S. G. Mamayev and V. M. Mostepanenko, On the difference in creation of particles with spin $0$ and $1/2$ in isotropic cosmologies, Phys. Lett. A 55, 7 (1976).
  * (41) A.A. Grib, B.A. Levitskii and V.M. Mostepanenko, Particle creation from vacuum by a nonstationary gravitational field in the canonical formalism, Theoreticheskaya i Matematicheskaya Fizika 19, 59 (1974).
  * (42) A.A. Grib, S.G. Mamayev and V.M. Mostepanenko, Particle creation from vacuum in homogeneous isotropic models of the Universe, Gen. Rel. Grav. 7, 535 (1976); The creation of particles from a vacuum in a nonsteady isotropic universe, Soviet Physics Journal 17, 1700-1704 (1974).
  * (43) L.H. Ford, Phys. Gravitational particle creation and inflation, Rev. D 35, 2955 (1987).
  * (44) Y.B. Zeldovich and A.A. Starobinsky, Rate of particle production in gravitational fields, JETP Lett. 26, 252 (1977).
  * (45) T. Damour and A. Vilenkin, String Theory and Inflation, Phys. Rev. D 53, 2981 (1995) [arXiv:hep-th/9503149].
  * (46) M. Giovannini, Gravitational waves constraints on post-inflationary phases stiffer than radiation, Phys. Rev. D 58, 083504 (1998) [arXiv:hep-ph/9806329].
  * (47) K. Dimopoulos and C. Owen, Quintessential Inflation with $\alpha$-attractors, JCAP 06, 027 (2017) [arXiv:1703.00305 [gr-qc]]
  * (48) Y. Akrami, R. Kallosh, A. Linde and V. Vardanyan, Dark energy, $\alpha$-attractors, and large-scale structure surveys, JCAP 06, 041 (2018) [arXiv:1712.09693 [hep-th]].
  * (49) G. Felder, L. Kofman and A. Linde, Instant Preheating, Phys. Rev. D 59, 123523 (1999) [arXiv:hep-ph/9812289]
  * (50) G. Felder, L. Kofman and A. Linde, Inflation and Preheating in NO models, Phys. Rev. D 60, 103505 (1999) [arXiv:hep-ph/9903350].
  * (51) K. Dimopoulos, L.D. Wood and C. Owen, Instant Preheating in Quintessential Inflation with $\alpha$-Attractors. Phys. Rev. D 97, 063525 (2018) [arXiv:1712.01760 [astro-ph.CO]].
  * (52) B. Feng and M. Li, Curvaton Reheating in Non-oscillatory Inflationary Models, Phys. Lett. B 564, 169-174 (2003) [arXiv:hep-ph/0212213].
  * (53) A. Agarwal, S. Bekov and K. Myrzakulov, Quintessential Inflation and curvaton reheating, [arXiv:1807.03629 [gr-qc]].
  * (54) K. Dimopoulos and T. Markkanen, Non-minimal gravitational reheating during kination, JCAP 06, 021 (2018) [arXiv:1803.07399 [gr-qc]].
  * (55) D. J. H. Chung, E.W. Kolb and A. Riotto, Superheavy dark matter, Phys. Rev. D 59, 023501 (1998) [arXiv:hep-ph/9802238].
  * (56) D. J. H. Chung, P. Crotty, E. W. Kolb and A. Riotto, On the gravitational production of superheavy dark matter, Phys. Rev. D 64, 043503 (2001) [arXiv:hep-ph/0104100].
  * (57) N.D. Birrell and P.C.W. Davies, Massive particle production in anisotropic space-times, J. Phys. A: Math. Gen. 13, 2109 (1980)
  * (58) S. Hashiba and J. Yokoyama, Gravitational reheating through conformally coupled superheavy scalar particles, JCAP 01, 028 (2018) [arXiv:1809.05410 [gr-qc]].
  * (59) A. A. Grib, S.G. Mamayev and V. M. Mostepanenko, Vacuum Quantum Effects in Strong Fields, Friedmann Laboratory Publishing for Theoretical Physics, St. Petersburg (1994).
  * (60) S. Panda, M. Sami and I. Thongkool, Reheating the D-brane universe via instant preheating, Phys. Rev. D 81, 103506 (2010) [arXiv:0905.2284 [hep-th]].
  * (61) D.G. Figueroa and E.H. Tanin, Inconsistency of an inflationary sector coupled only to Einstein gravity, JCAP 10, 050 (2019) [arXiv:1811.04093 [astro-ph.CO]].
  * (62) G. F. Giudice, E. W. Kolb and A. Riotto, Largest temperature of the radiation era and its cosmological implications, Phys. Rev. D 64, 023508 (2001) [arXiv:hep-ph/0005123].
  * (63) T. Hasegawa, N. Hiroshima, K. Kohri, R.S.L.Hansen, T. Tram and S. Hannestad, MeV-scale reheating temperature and thermalization of oscillating neutrinos by radiative and hadronic decays of massive particles, JCAP 12, 012 (2019) [arXiv:1908.10189 [hep-ph]].
  * (64) J. Ellis, D.V. Nanopoulos and S. Sarkar, The cosmology of decaying gravitinos, Nuc. Phys. B 259, 175-188 (1985).
  * (65) J. Ellis, A. Linde and D. Nanopoulos, Inflation can save the gravitino, Phys. Lett. B 118, 59 (1982).
  * (66) M. Maggiore, Gravitational Wave Experiments and Early Universe Cosmology, Phys. Rept. 331, 283 (2000) [arXiv:gr-qc/9909001].
  * (67) B.A. Bassett, S. Tsujikawa and D. Wands, Inflation Dynamics and Reheating, Rev.Mod.Phys. 78, 537 (2006) [arXiv:astro-ph/0507632].
  * (68) Y. Akrami et al., Planck 2018 results. X. Constraints on inflation, [arXiv:1807.06211 [astro-ph.CO]].
  * (69) Y. Akrami et al., Planck 2018 results. VI. Cosmological parameters, [arXiv:1807.06209 [astro-ph.CO]].
  * (70) K. Dimopoulos, L. Donaldson Wood and C. Owen, Instant Preheating in Quintessential Inflation with $\alpha$-Attractors, Phys. Rev. D 97, 063525 (2018) [arXiv:1712.01760 [astro-ph.CO]].
  * (71) A.R. Liddle and S.M. Leach, How long before the end of inflation were observable perturbations produced?, Phys. Rev. D 68, 103503 (2003) [arXiv:astro-ph/0305263].
  * (72) E. J. Copeland, M. Sami and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D 15, 1753 (2006) [arXiv:hep-th/0603057].
  * (73) P. J. Steinhardt, L. M. Wang and I. Zlatev, Cosmological tracking solutions, Phys. Rev. D 59,123504 (1999) [arXiv:astro-ph/9812313]
  * (74) L. A. Ureña-López and T. Matos, A New cosmological tracker solution for quintessence, Phys. Rev. D 62, 081302 (2000) [arXiv:astro-ph/0003364].
  * (75) J. Haro, J. Amorós and S. Pan, The Peebles - Vilenkin quintessential inflation model revisited, Eur.Phys.J. C79, no.6, 505 (2019) [arXiv:1901.00167 [gr-qc]].
  * (76) D. Benisty and E.I. Guendelman, Lorentzian Quintessential Inflation, Int. J. Mod. Phys. D 29, 2042002 (2020) [arXiv:2004.00339 [astro-ph.CO]].
  * (77) D. Benisty and E.I. Guendelman, Quintessential Inflation from Lorentzian slow-roll”, Eur. Phys. J. C80, 577 (2020) [arXiv:2006.04129 [astro-ph.CO]]
  * (78) J. Martin, C. Ringeval and V. Vennin, Shortcomings of New Parametrizations of Inflation, Phys. Rev. D 94, 123521 (2016) [arXiv:1609.04739 [astro-ph.CO]].
  * (79) A. D. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. B 108, 389 (1982)
  * (80) R. Kallosh, A. Linde and D. Roest, Superconformal Inflationary $\alpha$-Attractors, JHEP 11, 198 (2013) [arXiv:1311.0472 [hep-th]].
  * (81) A. Kehagias, A. M. Dizgah and A. Riotto, Comments on the Starobinsky Model of Inflation and its Descendants, Phys. Rev. D89, 043527 (2014) [arXiv:1312.1155 [hep-th]].
  * (82) M. Braglia, W. T. Emond, F. Finelli, A. E. Gumrukcuoglu, K. Koyama, Unified framework for Early Dark Energy from $\alpha$-attractors, Phys. Rev. D102, 083513 (2020) [arXiv:2005.14053 [astro-ph]].
  * (83) L. Aresté Saló, D. Benisty, E. I. Guendelman and J. de Haro, Quintessential Inflation and Cosmological See-Saw Mechanism: Reheating and Observational Constraints, JCAP 07, 007 (2021) [arXiv:2102.09514 [astro-ph.CO]]
  * (84) K. Dimopoulos and J. W. F. Valle, Modeling Quintessential Inflation, Astropart.Phys. 18, 287-306 (2002) [arXiv:astro-ph/0111417].
  * (85) K. Dimopoulos, The Curvaton Hypothesis and the eta-problem of Quintessential Inflation, with and without Branes, Phys.Rev. D 68, 123506 (2003) [arXiv:astro-ph/0212264].
  * (86) K. Dimopoulos and L. Donaldson-Wood, Warm Quintessential Inflation, Phys. Lett. B 796, 26-31 (2019) [arXiv:1906.09648 [gr-qc]].
  * (87) K. Dimopoulos and S. Sánchez López, Quintessential inflation in Palatini $f(R)$ gravity, Phys. Rev. D 103, 043533 (2021) [arXiv:2012.06831 [gr-qc]].
  * (88) D. Bettoni, A. Lopez-Eiguren and J. Rubio, Hubble-induced phase transitions on the lattice with applications to Ricci reheating, (2021) [arXiv:2107.09671 [hep-ph]].
  * (89) D. Bettoni and J. Rubio, Hubble-induced phase transitions: Walls are not forever JCAP 01 002 (2020) 002 [arXiv:1911.03484 [astro-ph.CO]].
  * (90) D. Bettoni, G. Domènech and J. Rubio, Gravitational waves from global cosmic strings in quintessential inflation JCAP 1902, 034 (2019) [arXiv:1810.11117 [astro-ph.CO]].
  * (91) D. Bettoni and J. Rubio, Quintessential Affleck-Dine baryogenesis with non minimal couplings, Phys.Lett. B784, 122-129 (2018) [arXiv:1805.02669 [astro-ph.CO]].
  * (92) K. Dimopoulos, M. Karciauskas and C. Owen, Quintessential inflation with a trap and axionic dark matter, Phys. Rev. D100, 083530 (2019) [arXiv:1907.04676 [hep-ph]].
  * (93) J. C. Bueno Sanchez and K. Dimopoulos, Curvaton reheating allows TeV Hubble scale in NO inflation, JCAP 0711, 007 (2007) [arXiv:0707.3967 [hep-ph]].
  * (94) Z. Kepuladze and M. Maziashvili, A new take on the inflationary quintessence, Phys. Rev. D103, 063540 (2021) [arXiv:2102.09203 [astro-ph]].
  * (95) C. van de Bruck, K. Dimopoulos, C. Longden and C. Owen, Gauss-Bonnet-coupled Quintessential Inflation, [arXiv:1707.06839 [astro-ph]].
  * (96) K. Kleidis and V.K. Oikonomou, A Study of an Einstein Gauss-Bonnet Quintessential Inflationary Model, Nucl. Phys B948, 114765 (2019) [arXiv:1909.05318 [gr-qc]].
  * (97) R. Rosenfeld and J. A. Frieman, A Simple Model for Quintessential Inflation, JCAP 0509, 003 (2005) [arXiv:astro-ph/0504191].
  * (98) M.C. Bento, R. Gonzalez Felipe and N.M.C. Santos, A Simple Quintessence Model, presented at the ”seventh Alexander Friedmann international seminar on gravitation and cosmology” (2008) [arXiv:0809.1679 [astro-ph]].
  * (99) M.C. Bento, R. Gonzalez Felipe and N.M.C. Santos, Brane assisted quintessential inflation with transient acceleration, Phys. Rev.D77, 123512 (2008) [arXiv:0801.3450 [astro-ph]].
  * (100) A.A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B91, Pages 99-102 (1980).
  * (101) A. Einstein, Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, Sitzungsberichte der Königlich Preussischen Akadamie der Wissenschaften (1917).
  * (102) N.D. Birrell and C.P.W. Davies, Quantum Fields in Curved Space (Cambridge: Cambridge University Press) (1982).
  * (103) J. de Haro and L. Aresté Saló, Reheating constraints in quintessential inflation, Phys. Rev. D 95, 123501 (2017) [arXiv:1702.04212 [gr-qc]].
  * (104) R. Allahverdi, R. Brandenberger, F.-Y. Cyr-Racine and A. Mazumdar, Reheating in Inflationary Cosmology: Theory and Applications, Ann. Rev. Nucl. Part. Sci. 60, 27 (2010) [arXiv:1001.2600 [hep-th]].
  * (105) J. de Haro, J. Amorós and S. Pan, A simple nonsingular inflationary quintessential model, Phys. Rev. D 93, 084018 (2016) [arXiv:1601.08175 [gr-qc]].
  * (106) A. Linde, Particle Physics and Inflationary Cosmology, Contemp. Concepts Phys. 5, 1-362 (2005) [arXiv:hep-th/0503203].
  * (107) J. Haro, Gravitational particle production: a mathematical treatment, J. Phys. A: Mat. Theor. 44 205401 (2011).
  * (108) S. Winitzki, Cosmological particle production and the precision of the WKB approximation, Phys. Rev. D 72, 104011 (2005) [arXiv:gr-qc/0510001].
  * (109) T. S. Bunch, Adiabatic regularisation for scalar fields with arbitrary coupling to the scalar curvature, J. Phys. A 13, 1297 (1980).
  * (110) Ya B. Zeldodovich and A. A. Starobinsky, Particle production and vacuum polarization in an anisotropic gravitational field, Sov. Phys JETP 34, 1159 (1972).
  * (111) J. de Haro, S. Pan, L. Aresté Saló, Understanding gravitational particle production in quintessential inflation, JCAP 06, 056 (2019) [arXiv:1903.01181 [gr-qc]].
  * (112) L. Aresté Saló and J. de Haro, Gravitational particle production of heavy massive particles in Quintessential Inflation: A numerical analysis (2021) [arXiv:2108.10795 [gr-qc]].
  * (113) A.I. Nikishov, Barrier scattering in field theory removal of Klein paradox, Nuc. Phys. B 21, 346-358 (1970).
  * (114) V.S. Popov, Pair production in a variable external field (Quasiclassical approximation), Soviet-Physics-JETP 34, 709-718 (1972).
  * (115) S.M. Marinov and V.S. Popov, Electron-positron pair creation from vacuum induced by variable electric field, Fortschritte der Physik 25, 373-400 (1977).
  * (116) J. Haro, Pair production in a uniform electric field, Int. Jour. Theor. Phys. 42, 531-547 (2003).
  * (117) M.V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, Berlin (1993).
  * (118) R.E. Meyer, Exponential asymptotic, SIAM Review 22, 213-224 (1980).
  * (119) A. Kaya and M. Tarman, Stress-Energy Tensor of Adiabatic Vacuum in Friedmann-Robertson-Walker Spacetimes, JCAP 04, 040 (2011) [arXiv:1104.5562 [gr-qc]]
  * (120) D. H. Lyth and D. Wands, Generating the curvature perturbation without an inflaton, Phys. Lett. B524, 5-14 (2002) [arXiv:hep-ph/0110002].
  * (121) V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press (2005).
  * (122) M. S. Turner, Coherent scalar-field oscillations in an expanding universe, Phys. Rev. D 28, 1243 (1983).
  * (123) A. R. Liddle and L. A. Ureña-Lopez, Curvaton reheating: an application to braneworld inflation, Phys. Rev. D 68, 043517 (2003) [arXiv:astro-ph/0302054].
  * (124) M. Giovannini, Production and detection of relic gravitons in quintessential inflationary models, Phys. Rev. D 60, 123511 (1999) [arXiv:astro-ph/9903004].
  * (125) J. Rubio and C. Wetterich, Emergent scale symmetry: Connecting inflation and dark energy, Phys. Rev. D 96, 063509 (2017) [arXiv:1705.00552 [gr-qc]].
  * (126) M. Giovannini, Production and detection of relic gravitons in quintessential inflationary models, Phys. Rev. D 60, 123511 (1999) [arXiv:astro-ph/9903004].
  * (127) M. Giovannini, Thermal history of the plasma and high-frequency gravitons, Class. Quant. Grav. 26, 045004 (2009) [arXiv:0807.4317 [astro-ph]].
  * (128) M. Giovannini, Stochastic backgrounds of relic gravitons, T$\Lambda$CDM paradigm and the stiff ages, Phys. Lett. B 668, 44 (2008) [arXiv:0807.1914 [astro-ph]].
  * (129) M. Giovannini, The first observations of wide-band interferometers and the spectra of relic gravitons, Phys. Lett. B 759, 528 (2016) [arXiv:1603.09217 [astro-ph.CO]].