# Cosmological $\Lambda$ converts to reheating energy and cold dark matter

She-Sheng Xue<EMAIL_ADDRESS><EMAIL_ADDRESS>ICRANet, Piazzale della
Repubblica, 10-65122, Pescara,
Physics Department, Sapienza University of Rome, P.le A. Moro 5, 00185, Rome,
Italy

###### Abstract

Suppose that the early Universe starts with a quantum spacetime originated
cosmological $\Lambda$-term at the Planck scale $M_{\rm pl}$. The cosmological
energy density $\rho_{{}_{{}_{\Lambda}}}$ drives inflation and simultaneously
reduces its value to create the pair-energy density $\rho_{{}_{{}_{M}}}$ via
the continuous pair productions of massive fermions and antifermions. The
decreasing $\rho_{{}_{{}_{\Lambda}}}$ and increasing $\rho_{{}_{{}_{M}}}$, in
turn, slows down the inflation to its end when the pair production rate
$\Gamma_{M}$ is larger than the Hubble rate $H$ of inflation. A large number
of massive pairs is produced and reheating epoch starts. In addition to the
Einstein equation and energy-conservation law, we introduce the Boltzmann-type
rate equation describing the number of pairs produced from (annihilating to)
the spacetime, and reheating equation describing massive unstable pairs decay
to relativistic particles and thermodynamic laws. This forms a close set of
four independent differential equations uniquely determining $H$,
$\rho_{{}_{\Lambda}}$, $\rho_{{}_{M}}$ and radiation-energy density
$\rho_{{}_{R}}$, given the initial conditions at inflation end. Numerical
solutions demonstrate three episodes of preheating, massive pairs dominate and
genuine reheating. Results show that $\rho_{{}_{\Lambda}}$ can efficiently
convert to $\rho_{{}_{M}}$ by producing massive pairs, whose decay accounts
for reheating $\rho_{{}_{R}}$, temperature and entropy of the Big-Bang
Universe. The stable massive pairs instead account for cold dark matter. Using
CMB and baryon number-to-entropy ratio measurements, we constrain the
effective mass of pairs, Yukawa coupling and degeneracies of relativistic
particles. As a result, the obtained inflation $e$-folding number, reheating
scale, temperature and entropy are in terms of the tensor-to-scalar ratio in
the theoretically predicated range $0.042\lesssim r\lesssim 0.048$,
consistently with current observations.

###### Contents

  1. I Introduction
  2. II Einstein equation and pair production
    1. II.1 Generalized equation for the Friedmann Universe
    2. II.2 Particle-antiparticle pair production from spacetime
    3. II.3 Naturally resultant inflation and CMB observations
      1. II.3.1 Inflation start scale determined by CMB measurements
      2. II.3.2 Inflation end constraint on $n_{s}\\!-\\!r$ relation and reheating initial condition
  3. III Cosmic rate equation for pairs and spacetime
    1. III.1 Particle-antiparticle pair annihilation and decay
    2. III.2 Boltzmann rate equation for particle-antiparticle pairs and spacetime
    3. III.3 A close set of fundamental equations in the reheating epoch
  4. IV Different episodes in the reheating epoch
    1. IV.1 preheating episode: ${\mathcal{P}}$-episode
      1. IV.1.1 Numerical solutions of cosmological $\rho_{{}_{\Lambda}}$ converted to matter $\rho_{{}_{M}}$
      2. IV.1.2 Threshold of produced pair mass and degeneracy for $\rho_{{}_{M}}>\rho_{{}_{\Lambda}}$
      3. IV.1.3 Minimal comoving radius $(Ha)^{-1}$ location
    2. IV.2 Massive pairs domination: ${\mathcal{M}}$-episode
      1. IV.2.1 Spacetime and pairs are coupled in horizon $H$ evolution
      2. IV.2.2 Preliminary discussions on CPT symmetry and increasing entropy
    3. IV.3 Relativistic particles domination: ${\mathcal{R}}$-episode of the genuine reheating
      1. IV.3.1 Massive pairs decay to relativistic particles
      2. IV.3.2 Energy densities of massive pairs and relativistic particles
      3. IV.3.3 Temperature and entropy of reheating epoch
      4. IV.3.4 Reheating scale factor and initial condition for standard cosmology
  5. V Observations to fix reheating temperature and mass scale
    1. V.1 Reheating temperature and entropy vs tensor-to-scalar ratio $r\lesssim 0.048$
    2. V.2 Reheating $\rho_{{}_{R}}\gg\rho_{{}_{\Lambda}}$ and lower limit of tensor-to-scalar ratio $r\gtrsim 0.042$
  6. VI Massive cold dark matter $\Omega^{0}_{c}$
    1. VI.1 Massive cold dark matter produced in reheating
    2. VI.2 Massive cold dark matter couples to the spacetime in its evolution
  7. VII Summary and remark reheating epoch
  8. VIII Acknowledgment

##  I Introduction

In the standard model of modern cosmology ($\Lambda$CDM), the cosmological
constant, inflation, reheating, dark matter and coincidence problem have been
long-standing basic issues for decades. The inflation inflation0  is a
fundamental epoch and the reheating RevReheating  is a critical mechanism,
which transitions the Universe from the cold massive state left by inflation
to the hot Big Bang BigBang . The cosmic microwave background (CMB)
observations have been attempting to determine a unique model of inflation and
reheating. On the other hand, what is the crucial role that the cosmological
$\Lambda$ term play in inflation and reheating, and what is the essential
reason for the coincidence of dark-matter dominate matter density and the
cosmological $\Lambda$ energy density. There are various models and many
efforts, that have been made to approach these issues, and readers are
referred to review articles and professional books, for example, see Refs.
Peebles ; kolb ; book ; Inflation_R ; Inflation_higgs ; reviewL ; Bamba:2012cp
; Nojiri:2017ncd ; Coley2019 ; prigogine1989 ; prigogine1989+ ; Khlopov ;
wangbin ; martin ; wuyueliang2012 ; wuyueliang2016 ; axioninf ; xue2000 ;
xuecos2009 ; xueNPB2015 .

Suppose that the quantum gravity originates the cosmological term $\Lambda\sim
M^{2}_{\rm pl}$ at the Planck scale. The initial state of the Universe is an
approximate de Sitter spacetime of the horizon
$H_{\circ}\approx(\Lambda/3)^{1/2}$ without any matter. The cosmological
$\Lambda$ energy density drives the spacetime inflation with the scale factor
$a(t)\approx e^{H_{\circ}t}$. On the other hand, de Sitter spacetime is
unstable against spontaneous particle creations pairproduction ;
pairproductionCold ; CKR1998 ; CKR2019 ; parametric ; Ema2018 . By reducing
its value, the cosmological $\Lambda$ term creates very massive pairs of
fermions and antifermions $m\sim M_{\rm pl}$ for matter content.

We adopt two independent equations of Friedmann and energy conservation law to
completely determine the cosmological energy density
$\rho_{{}_{{}_{\Lambda}}}$ and inflation rate $H$ xue2020MPLA ,

$\displaystyle H^{2}=\frac{8\pi G}{3}(\rho_{{}_{\Lambda}}+\rho_{{}_{M}}),$ (1)
$\displaystyle\dot{H}=-\frac{8\pi G}{2}(1+\omega_{{}_{M}})\rho_{{}_{M}},$ (2)

where the matter-energy density $\rho_{{}_{{}_{M}}}$ and its equation of state
$\omega_{{}_{M}}$ are calculated by using pair productions. The cosmological
energy density $\rho_{{}_{{}_{\Lambda}}}$ governs the spacetime inflation rate
$H$ and in the meantime produces the matter-energy density
$\rho_{{}_{{}_{M}}}$, whose back reaction, in turn, slows down inflation to
its end. The obtained results are in agreement with CMB observations. Suppose
that after reheating the matter-energy density is much larger than the
cosmological energy density, we show due to such back reaction that the
cosmological term $\rho_{{}_{\Lambda}}$ tracks down the matter term
$\rho_{{}_{M}}$ from the reheating end to the radiation-matter equilibrium,
then it varies very slowly, $\rho_{{}_{\Lambda}}\propto$ constant,
consistently leading to the cosmic coincidence in the present time. The
detailed discussions and results of such scenario $\tilde{\Lambda}$CDM have
been presented there.

We focus our attention on the detailed structure of reheating epoch to
understand how the cosmological energy density $\rho_{{}_{{}_{\Lambda}}}$
converts to the matter- energy density $\rho_{{}_{{}_{M}}}$ of massive pairs.
Some of these massive pairs are unstable and decay to relativistic particles
of radiation-energy density $\rho_{{}_{{}_{R}}}$. Others are stable, possibly
accounting for the dark matter particles. For purpose of showing these, in
addition to the aforementioned Einstein equation and conservation law, we
study and solve other two independent equations of the Boltzmann-type: (i) the
rate equation for massive particle-antiparticle pairs productions and
annihilations; (ii) the reheating equation for massive pairs decay to
relativistic particles, including the use of thermodynamic laws. These four
independent equations allow us to completely determine the main properties in
the reheating epoch. The obtained results are expressed in terms of the
tensor-to-scalar ratio $0.042\lesssim r\lesssim 0.048$, they are consistent
with current observations.

We organise this article as follow. In Sec. II, we briefly summarise the
scenario $\tilde{\Lambda}$CDM and previous results of the pre-inflation and
inflation epochs. In Sec. III, we discuss the cosmic rate equation determining
the relation between the matter density $\rho_{{}_{M}}$ in the Einstein
equations and the pair density $\rho^{H}_{{}_{M}}(H)$ produced by the pair-
production process. In Secs. IV and V, we present the detailed discussions of
three episodes of the reheating epoch, and calculations of relevant physical
quantities, compared with observations. The studies of cold dark matter
particles are arranged in Sec. VI. The article ends with a summary of the
results and remarks.

In this article, $G=M^{-2}_{\rm pl}$ is the the Newton constant, $M_{\rm pl}$
is the Planck scale and reduced Planck scale $m_{\rm
pl}\equiv(8\pi)^{-1/2}M_{\rm pl}=2.43\times 10^{18}$GeV.

##  II Einstein equation and pair production

To proceed with more detailed discussions and calculations on the reheating
epoch, for readers’ convenience, we present the brief and necessary summaries
of the scenario $\tilde{\Lambda}$CDM and its applications to the pre-inflation
and inflation epochs.

Figure 1: This figure is duplicated from Fig. 1 in Ref. Mielczarek2011 .
Schematic evolution of the Hubble radius $H^{-1}$ and the physical length
scale $\lambda(a)$, where physically interested scale
$\lambda_{0}=\lambda(a_{0})$ at the present time $a_{0}=1$ crossed the Hubble
horizon at the early time $a_{4}$. Setting the scale factor $a_{4}=a_{*}$ at
the inflation scale $H^{-1}_{*}$ fixed by the CMB pivot scale
$\lambda_{0}=\lambda_{*}=k_{*}^{-1}$. The pre-inflation $a>a_{*}$, the
inflation $a_{4}<a<a_{3}=a_{\rm end}$ and the inflation end $a_{3}=a_{\rm
end}$, the reheating $a_{3}<a<a_{2}=a_{{}_{R}}$, the genuine reheating at
$a_{{}_{R}}$ and the recombination at $a_{1}=a_{\rm eq}$.

###  II.1 Generalized equation for the Friedmann Universe

The Universe evolution from the inflation to the reheating is rather complex.
In order to study the different episodes of the reheating epoch after the
inflation, we rewrite the generalised equations (1) and (2) for the Friedmann
Universe as,

$\displaystyle H^{2}=(3m^{2}_{\rm
pl})^{-1}(\rho_{{}_{\Lambda}}+\rho_{{}_{M}}+\rho_{{}_{R}}),$ (3)
$\displaystyle\dot{H}=-(3/2)(3m^{2}_{\rm
pl})^{-1}\left[(1+\omega_{{}_{M}})\rho_{{}_{M}}+(1+\omega_{{}_{R}})\rho_{{}_{R}}\right].$
(4)

where the Hubble rate $H\equiv\dot{a}/a$ of the scale factor $a(t)$ variation
and $\dot{H}\equiv dH/dt$. The energy density $\rho_{{}_{M}}$ is a specific
notation for massive pairs of $\omega_{{}_{M}}\approx 0$. The energy density
$\rho_{{}_{R}}$ a specific notation for relativistic particles of
$\omega_{{}_{R}}\approx 1/3$. Their Equations of States are defined as
$\omega_{{}_{M,R}}=p_{{}_{M,R}}/\rho_{{}_{M,R}}$ and $p_{{}_{M,R}}$ is the
pressure of massive pairs or relativistic particles respectively. The energy
density $\rho_{{}_{\Lambda}}\equiv\Lambda^{2}/(8\pi G)$ and
$\omega_{{}_{\Lambda}}\equiv-1$ are attributed to the cosmological
$\Lambda$-term.

Moreover, we introduce the expanding time scale $\tau_{{}_{H}}$ and the
$\epsilon$-rate representing the rate of the $H$ variation in time:

$\displaystyle\tau_{{}_{H}}\equiv
H^{-1};\quad\epsilon\equiv-\frac{\dot{H}}{H^{2}}=\frac{3}{2}\frac{\rho_{{}_{M}}+(4/3)\rho_{{}_{R}}}{\rho_{{}_{\Lambda}}+\rho_{{}_{M}}+\rho_{{}_{R}}},$
(5)

whose values characterise different epochs in Universe evolution. Note that in
the inflation epoch the $\epsilon$-rate is analogous to the so-called slow-
roll parameter. As a convenient unit for calculations and expressions, we
adopt the reduced Planck scale $m_{\rm pl}\equiv(8\pi G)^{-1/2}=1$, unless
otherwise stated. In order to illustrate the pre-inflation, inflation and
different episodes of the reheating epoch studied in this article, a schematic
description of the Universe evolution for the standard cosmological scenario
is presented in Fig. 1.

###  II.2 Particle-antiparticle pair production from spacetime

In this section, we briefly recall how to calculate the energy density
$\rho^{H}_{{}_{M}}$ of the matter produced from the spacetime horizon $H$ by
the pair production of particles $F$ and antiparticles $\bar{F}$:

$\displaystyle{\mathcal{S}}\Rightarrow F+\bar{F},$ (6)

where ${\mathcal{S}}$ indicates the spacetime. Such pair production is
considered to be a semi-classical process of producing particles and
antiparticles in the slowly time-varying horizon $H$. In the physical regime,
where produced $F$ and $\bar{F}$ particle masses $m$ are much larger than the
horizon $H$ ($m/H\gg 1$), i.e., they are well inside the horizon, we
approximately obtain the averaged number, mass densities and pressure of
massive pairs produced from $t=0$ to $t\gtrsim 2\pi H^{-1}$ xue2020MPLA ,

$\displaystyle\rho^{H}_{{}_{M}}$ $\displaystyle\approx$ $\displaystyle 2\,\chi
m^{2}H^{2}(1+s),$ (7) $\displaystyle p^{H}_{{}_{M}}$ $\displaystyle\approx$
$\displaystyle(s/3)\rho^{H}_{{}_{M}},\quad s\approx 1/2(H/m)^{2}\ll 1$ (8)

$\omega_{{}_{M}}=p_{{}_{M}}/\rho_{{}_{M}}\approx s/3$, the pair number density
$n^{H}_{{}_{M}}\approx\rho^{H}_{{}_{M}}/(2m)$ and pair-production rate

$\displaystyle\Gamma_{M}$ $\displaystyle\approx$ $\displaystyle-(\chi
m/4\pi)(H^{-1}dH/dx)=(\chi m/4\pi)\epsilon,$ (9)

where the theoretical coefficient $\chi\approx 1.85\times 10^{-3}$. These
massive “non-reletivistic” pairs are produced in the thin shell with the
thickness $(\chi m)^{-1}$ on the horizon. The pair-production rate
$\Gamma_{M}$ (9) is proportional to the $\epsilon$-rate (5) of the Horizon
variation in time.

In each epoch of the Universe evolution, we simply introduce the unique mass
parameter $m$ to effectively characterize and describe the total contribution
from all kinds of particle-antiparticle pairs (dark matter and/or usual
matter) and their degeneracies to the produced pairs’ energy density (7) and
pressure (8),

$\displaystyle\rho^{H}_{{}_{M}}$ $\displaystyle\approx$ $\displaystyle 2\,\chi
H^{2}m^{2};\quad m^{2}\equiv\sum_{f}g_{d}^{f}m^{2}_{f},$ (10)

where $g_{d}^{f}$ and $m_{f}$ are the degeneracy and mass of the particle of
the flavor $f$, and the $\sum_{f}$ sums up all flavors produced. The pair-
production energy density (7) and rate (9) show that the pair production
process is in favor of massive pairs whose wavelengths are inside the Horizon
$H^{-1}$. The inequality $m_{f}^{2}>H^{2}$ implies that the degeneracy
$g^{f}_{d}$ should be small in the epoch of large $H^{2}$ value, whereas the
such degeneracy $g^{f}_{d}$ should be large in the epoch of small $H^{2}$
value. Therefore the effective mass parameter $m$ in general depends on the
epoch. The values of the mass parameter $m$ in different epochs are determined
by observations.

At the end of this section, we present some discussions on the non-
exponentially suppressed number and energy densities (7) of very massive
particle-antiparticle pairs production and annihilation in cosmological
evolution. They are approximately obtained in the static case “$H=$ const” and
leading orders of adiabatic approximation to smoothly varying background
field. In the pioneering work CKR1998 , the authors show that the
nonadiabaticity and discontinuous transition in the cosmic scale factor $a(t)$
evolution result in the number and energy densities of superheavy particles
produced fall off with a finite power of $H/m$ for $m\gg H$, see Eqs.
(12,14,15) in their article. This situation is very different from the
exponentially suppressed density ($\sim e^{-m/H}$) of superheavy particles
produced in static or adiabatically evolutionary Universe. Besides, the
authors show the superheavy particle production in cosmologically interesting
quantities, such as dark matter relic abundance, which is plotted in terms of
$m/H$, see in their Fig. 2 the solid line for the inflationary epoch
discontinuously into the matter dominate epoch. Moreover, their recent article
CKR2019  discusses the gravitational production of super-Hubble-mass particles
from rapid background field oscillations and the exponential suppression can
be avoided. Such situation can be realized in the parametric resonance
parametric  and the $2\rightarrow 2$ scattering case Ema2018 . In our scenario
under consideration, the nonadiabaticity is mainly due to the back and forth
oscillating processes ${\mathcal{S}}\Leftrightarrow\bar{F}+F$ of productions
(6) and annihilation (20) of massive particle and antiparticle pairs. The
oscillating frequency is proportional to pairs’ mass $m\gg H$. The
cosmological evolution and pair-production vacuum evolution are certainly non-
adiabatic and discontinuous, since the cosmic scale factor $a(t)$, the Hubble
rate $H(t)$ and their time derivatives are back reacted by producing massive
pairs and these pairs annihilation and decay. It happens particularly in the
transitions from one cosmic epoch to another, see Fig. 1. The detailed studies
will be presented in a separate paper.

###  II.3 Naturally resultant inflation and CMB observations

In the pair-production process, the cosmological term $\rho_{{}_{\Lambda}}$
and the horizon $H$ must decreases because the gravitational energy of the
spacetime has to pay for the energy gain due to massive pair production and
pairs’ kinetic energy. This back reaction of pair productions on the spacetime
has been taken into account by the Einstein equation (3) and generalised
conservation law (4). In the pre-inflation ($a<a_{4}$) and inflation
($a_{4}\leq a\leq a_{3}$) epochs, as shown in Fig. 1, we approximately adopt
$\rho_{{}_{M}}\approx\rho^{H}_{{}_{M}}$, namely all produced pairs’ energy
density $\rho^{H}_{{}_{M}}$ contribute to the matter-energy density
$\rho_{{}_{M}}$ in the Einstein equations, neglecting the pair annihilation
$F+\bar{F}\Rightarrow{\mathcal{S}}$. The reasons are: (i) the pair energy
density $\rho^{H}_{{}_{M}}$ is small, particles and antiparticles are spaced
rapidly far apart in the inflating spacetime $H\gtrsim\Gamma_{M}$; (ii) the
pair annihilation is a self-quench process, as its rate is proportional to the
pair density.

####  II.3.1 Inflation start scale determined by CMB measurements

We recall the results xue2020MPLA  of the inflation epoch ($a_{4}\leq a\leq
a_{3}$ in Fig. 1), when $H>\Gamma_{M}$, $\rho_{{}_{\Lambda}}\gg\rho_{{}_{M}}$,
$\rho_{{}_{R}}\approx 0$, and
$\epsilon\approx(3/2)\rho_{{}_{M}}/\rho_{{}_{\Lambda}}\ll 1$. Using
$\rho_{{}_{M}}\approx\rho^{H}_{{}_{M}}$ (7) and Eq. (4), we obtain the slowly
decreasing

$H=H_{*}e^{-\chi m_{*}^{2}x}=H_{*}e^{-\chi m_{*}^{2}N}$ (11)

in terms of the $e$-folding variable $x=\ln(a/a_{*})=N$. The $m_{*}$ is the
mass parameter in the inflation epoch. The initial scale $H_{*}$ corresponds
to the interested mode of the pivot scale $k_{*}$ crossed the horizon
$(k_{*}=H_{*}a_{*})$ for CMB power spectra observations. At this pivot scale,
the scalar, tensor power spectra and their ratio are

$\displaystyle\Delta^{2}_{{}_{\mathcal{R}}}=\frac{1}{8\pi^{2}}\frac{H^{2}_{*}}{m^{2}_{\rm
pl}\,\epsilon
c_{s}};\quad\Delta^{2}_{h}=\frac{2}{\pi^{2}}\frac{H^{2}_{*}}{m^{2}_{\rm
pl}};\quad
r\equiv\frac{\Delta^{2}_{h}}{\Delta^{2}_{{}_{\mathcal{R}}}}=16\,\epsilon
c_{s},$ (12)

where $c_{s}<1$ due to the Lorentz symmetry broken by the time dependence of
the background book . Equation (12) relates to two CMB observational values:
the spectral index $n_{s}\approx 0.965$ and the scalar amplitude
$A_{s}=\Delta^{2}_{{}_{\mathcal{R}}}\approx 2.1\times 10^{-9}$ at
$k_{*}=0.\,05\,({\rm Mpc})^{-1}$ Planck2018 . As a result, we determine the
mass parameter $m_{*}$ and $\epsilon$-rate value $\epsilon^{*}$
($n_{s}=1-2\epsilon^{*}$) at the inflation scale $H_{*}$

$\displaystyle m_{*}$ $\displaystyle\approx$ $\displaystyle 3.08\,m_{\rm
pl},\quad\epsilon^{*}=\chi m^{2}_{*}\approx 1.75\times 10^{-2},$ (13)
$\displaystyle H_{*}$ $\displaystyle=$ $\displaystyle 3.15\times
10^{-5}\,(r/0.1)^{1/2}m_{\rm pl},$ (14)

where $r$ is the scalar-tensor-ratio and the dimensionless notation $\chi
m^{2}_{*}\equiv\chi(m_{*}/m_{\rm pl})^{2}$ for $m_{\rm pl}=1$ is used. The
energy-density ratio of pairs and cosmological term is given by

$\displaystyle\frac{\rho^{*}_{{}_{M}}}{\rho^{*}_{{}_{\Lambda}}}\approx\frac{2\chi(m_{*}H_{*})^{2}}{3(m_{\rm
pl}H_{*})^{2}}=\frac{2}{3}\chi m^{2}_{*}\approx 1.17\times 10^{-2},$ (15)

and $H^{2}_{*}\approx\rho^{*}_{{}_{\Lambda}}/(3m^{2}_{\rm pl})$. This shows
that the pairs’ contribution $\rho_{{}_{M}}$ to the Hubble horizon (3) is
indeed negligible, compared with the cosmological term contribution
$\rho_{{}_{\Lambda}}$ in the inflation epoch. Despite its smallness, the
pairs’ energy density makes the Hubble rate $H$ slowly decrease.

####  II.3.2 Inflation end constraint on $n_{s}\\!-\\!r$ relation and
reheating initial condition

The transition from the inflation epoch $H>\Gamma_{M}$ to the reheating epoch
$\Gamma_{M}>H$ is physically continuous. We can estimate the inflation ending
scale $H_{\rm end}$ by using $H_{\rm end}\lesssim\Gamma_{M}$. Approximately
using $2\epsilon\approx 2\chi m^{2}\approx(1-n_{s})$ without any parameter,
Equations (9) and (11) yield the inflation ending scale factor $a_{\rm end}$
and $e$-folding number $N_{\rm end}$,

$\displaystyle N_{\rm end}$ $\displaystyle\gtrsim$
$\displaystyle\ln\left(\frac{a_{\rm
end}}{a_{*}}\right)=\frac{2}{1-n_{s}}\ln\left[\frac{7.91\times
10^{-4}\,(r/0.1)^{1/2}}{(1-n_{s})^{3/2}\,(\chi/2)^{1/2}}\right].$ (16)

From. Eq. (11), the inflation ending scale $H_{\rm end}$ is

$\displaystyle H_{\rm end}$ $\displaystyle=$ $\displaystyle H_{*}e^{-\chi
m_{*}^{2}N_{\rm end}}$ (17) $\displaystyle\approx$ $\displaystyle
H_{*}e^{-(1-n_{s})N_{\rm end}/2}\approx(0.42,0.35)H_{*},$

for the $e$-folding number $N_{\rm end}=(50,60)$ and the tensor-to-scalar
ratio $r=(0.037,0.052)$. The numerical result $N_{\rm end}$ (16) depend on the
CMB measurements of $r$ and $n_{s}$. The $r\\!-\\!n_{s}$ relation (16) in
terms of $N_{\rm end}=50,60$ is shown in Fig. 2 to compare with the constrains
from CMB measurements.

In the inflation epoch, the scale factor increases from the beginning $a_{*}$
to the end $a_{\rm end}$

$\displaystyle\Delta_{3}\equiv\frac{a_{3}}{a_{4}}=\frac{a_{\rm
end}}{a_{*}}=e^{N_{\rm end}}.$ (18)

where $a_{4}=a_{*}$, $a_{3}=a_{\rm end}$, the pre-inflation $a<a_{4}$, the
inflation $a_{*}<a<a_{\rm end}$, and the pivot length scale is the
$\lambda_{*}=k^{-1}_{*}=\lambda_{0}=k^{-1}_{0}$, as shown in Fig. 1 for the
schematic evolution of the Hubble radius $H^{-1}$ and physically interested
wavelength $\lambda(a)$ for the standard cosmological scenario.

Figure 2: On the Figure 28 of the Planck 2018 results Planck2018  for
constraints on the tensor-to-scalar ratio $r$, the parameter-free ($n_{s}-r$)
relation xue2020MPLA  is plotted as two QFC curves respectively representing
$N_{\rm end}=60$ and $N_{\rm end}=50$ are consistently inside the blue zone
constrained by several observational data sets. The real values of $r$ ratio
should be below the curves due to the nature of inequality (16).

Equations (14,15,17), $\chi\ll 1$ and $\epsilon^{*}\ll 1$ (13) show that the
$H$-variation is very small in the inflation epoch, implying

$\displaystyle H^{2}_{\rm end}=\frac{\rho^{\rm end}_{{}_{\Lambda}}+\rho^{\rm
end}_{{}_{M}}}{3m^{2}_{\rm pl}}\gtrsim\frac{\rho^{\rm
end}_{{}_{\Lambda}}}{3m^{2}_{\rm pl}};\quad\frac{\rho^{\rm
end}_{{}_{M}}}{\rho^{\rm end}_{{}_{\Lambda}}}\ll 1,$ (19)

namely, the cosmological term $\rho^{\rm end}_{{}_{\Lambda}}\approx
3m^{2}_{\rm pl}H^{2}_{\rm end}$ is still dominant over the pair energy density
$\rho^{\rm end}_{{}_{M}}\approx 2\chi m^{2}_{*}H^{2}_{\rm end}$ at the
inflation end. We consider the ratio $\rho^{\rm end}_{{}_{M}}/\rho^{\rm
end}_{{}_{\Lambda}}$ (15) and the scale $H_{\rm end}$ (17) as initial
conditions of the reheating epoch ($a_{3}\leq a\leq a_{2}$ in Fig. 1) to be
studied in Sec. IV.

##  III Cosmic rate equation for pairs and spacetime

The inflation epoch ends and reheating epoch starts. The transitioning process
from one to another cannot be instantaneous and must be very complex, due to
the large density of particle-antiparticle pairs and the back reactions of
microscopic and macroscopic processes. One of them is that pairs annihilate
back to the spacetime, besides they are produced from the spacetime. This
means that the produced pairs’ energy density $\rho^{H}_{{}_{M}}$ is not the
same as the matter energy density $\rho_{{}_{M}}$ in the Einstein equation
$\rho^{H}_{{}_{M}}\not=\rho_{{}_{M}}$. They are related by the the Boltzmann
type rate equation of pairs and spacetime, that is studied in this section and
shown to be important for understanding the transition between the inflation
end and reheating start in next Sec. IV.

###  III.1 Particle-antiparticle pair annihilation and decay

We focus on the dynamics and kinematics of particle-antiparticle pairs after
they are produced. From the microscopical points of view, the particle-
antiparticle pairs can in turn annihilate to the spacetime, i.e., the inverse
process of the production process (6)

$\displaystyle F+\bar{F}\Rightarrow{\mathcal{S}}.$ (20)

Such back and forth processes

$\displaystyle{\mathcal{S}}\Leftrightarrow\bar{F}F,$ (21)

can be regarded as particle-antiparticle emissions and absorptions of the
spacetime. As shown in Eqs. (5) and (9), the macroscopic time scale
$\tau_{{}_{H}}=H^{-1}$ of the spacetime expansion is much longer than the time
scale $\Gamma^{-1}_{M}$ of microscopic pair productions and annihilations.
Therefore, the CPT symmetry of local field theories should be held for
microscopic processes, we consider that the pair-annihilation rate is the same
as the pair-production rate

$\displaystyle\Gamma^{{}^{\rm Anni}}_{M}=\Gamma^{{}^{\rm
Prod}}_{M}=\Gamma_{M},$ (22)

although the Universe expansion violates the $T$-symmetry of time translation
and reflection.

Henceforth, we introduce the mass parameter $\hat{m}$ to represent not only
the effective masses, but also the effective degeneracies of pairs produced in
the reheating epoch. That is the substitution $m\rightarrow\hat{m}$ in Eqs.
(7,8) and the rate (9) becomes

$\displaystyle\Gamma_{M}$ $\displaystyle\approx$
$\displaystyle(\chi\hat{m}/4\pi)\epsilon.$ (23)

These back and forth processes of massive pairs production and annihilation
are due to purely gravitational interactions. Some of the massive pairs,
however, can also carry the quantum numbers of gauge interactions. Therefore,
in addition to their annihilation to the spacetime (20), these “unstable”
massive pairs via gauge interactions decay to relativistic particles
$\bar{\ell}~{}\ell$, which are much lighter than massive pairs $\bar{F}F$
themselves,

$\displaystyle\bar{F}F\Rightarrow\bar{\ell}~{}\ell.$ (24)

These relativistic particles $\bar{\ell}~{}\ell$ represent elementary
particles in the standard model (SM) of particle physics, and possible
massless or massive sterile particles (neutrinos) for the dark matter, as well
as composite particles xue2016_2  or particles of other theories beyond SM. In
general, the decay rate of massive pairs is proportional to the pair mass
$\hat{m}$ and can be parametrised as

$\displaystyle\Gamma^{{}^{\rm
de}}_{M}=g^{2}_{{}_{Y}}\hat{m},\quad(\bar{F}F\Rightarrow\bar{\ell}~{}\ell)$
(25)

where $g_{{}_{Y}}$ is the Yukawa coupling between the massive pairs and
relativistic particles. It is important to note that the decay rate
$\Gamma^{{}^{\rm de}}_{M}$ (25) depends not only on the Yukawa coupling
$g_{{}_{Y}}$, but also on the phase space of final particles, to which massive
pairs decay. The some of spacetime produced massive pairs cannot decay to
relativistic particles $\bar{\ell}~{}\ell$, namely $\Gamma^{{}^{\rm
de}}_{M}\approx 0$, and they would account for the cold massive dark matter.

Taking into account both pair-production and pair-annihilation processes, we
study the semi-classical rate equation for the pair energy density
$\rho_{{}_{M}}$ based on the conservation of the total pair numbers in the
Universe evolution obeying Einstein equations (3) and (4).

###  III.2 Boltzmann rate equation for particle-antiparticle pairs and
spacetime

We adopted the phase space density (the distribution function in phase space)
is spatially homogenous and isotropic, and integrating over phase space, we
have the pair number density depending only on the time $n_{{}_{M}}(t)$, so
that the Liouville operator in the phase space for the kinematic part is just
$d(a^{3}n_{{}_{M}})/dt=a^{3}\dot{n}_{{}_{M}}(t)+3Ha^{3}n_{{}_{M}}(t)$.
Adopting the usual approach kolb ; rate  at the semi-classical level, we use
the cosmic rate equation of the Boltzmann type for the pair number density
$n_{{}_{M}}$,

$\displaystyle\frac{dn_{{}_{M}}}{dt}+3Hn_{{}_{M}}$ $\displaystyle=$
$\displaystyle\Gamma_{M}\left(n_{{}_{M}}^{H}-n_{{}_{M}}\right)-\Gamma_{M}^{{}^{\rm
de}}n_{{}_{M}},$ $\displaystyle\frac{d\rho_{{}_{M}}}{dt}+3H\rho_{{}_{M}}$
$\displaystyle=$
$\displaystyle\Gamma_{M}\left(\rho_{{}_{M}}^{H}-\rho_{{}_{M}}\right)-\Gamma_{M}^{{}^{\rm
de}}\rho_{{}_{M}},$ (26)

where the second line is due to the massive pairs $\rho_{{}_{M}}\approx
2\hat{m}n_{{}_{M}}$ and $\rho^{H}_{{}_{M}}\approx 2\hat{m}n^{H}_{{}_{M}}$.
These equations effectively describe the pair dynamics of the back and forth
processes (21) and decay processes (25) in the Universe evolution. The term
$3Hn_{{}_{M}}$ of the time scale $(3H)^{-1}$ represents the spacetime
expanding effect on the pair density $n_{{}_{M}}$. It should be emphasized
that $\Gamma_{M}n_{{}_{M}}^{H}$ is the source term of pair productions from
the space time, and $\Gamma_{M}n_{{}_{M}}$ is the depletion term of pair
annihilations into the space time. The spacetime horizon and particle-
antiparticle pairs are coupled via the back and forth processes (21). The pair
production and annihilation rates are assumed to be equal to $\Gamma_{M}$ (22)
in the detailed balance term

$\displaystyle\Gamma^{\rm Prod}_{M}n_{{}_{M}}^{H}-\Gamma^{\rm
Anni}_{M}n_{{}_{M}}=\Gamma_{M}\left(n_{{}_{M}}^{H}-n_{{}_{M}}\right),$ (27)

in the RHS of the cosmic rate equation (26).

###  III.3 A close set of fundamental equations in the reheating epoch

In addition to the cosmic rate equation (26), there is another Boltzmann
equation from the conservation of the radiation energy of relativistic
particles from massive particle decays. Massive pairs decay (25) to
relativistic particles $\bar{\ell}~{}\ell$, whose energy density is
$\rho_{{}_{R}}$. The evolution of such radiation energy density
$\rho_{{}_{R}}$ can be obtained by the energy conservation law, see for
example Ref. kolb ,

$\displaystyle d(a^{3}\rho_{{}_{R}})$ $\displaystyle=$ $\displaystyle-
p_{{}_{R}}d(a^{3})-d(a^{3}\rho_{{}_{M}})$ (28) $\displaystyle=$
$\displaystyle-\frac{\rho_{{}_{R}}}{3}d(a^{3})+(a^{3}\rho_{{}_{M}})\Gamma^{{}^{\rm
de}}_{M}dt,$

where $d(a^{3}\rho_{{}_{M}})=-(a^{3}\rho_{{}_{M}})\Gamma^{{}^{\rm de}}_{M}dt$
is the massive pair energy that converts to the radiation energy of
relativistic particles. This leads to the crucial reheating equation

$\displaystyle\dot{\rho}_{{}_{R}}+4H\rho_{{}_{R}}=\Gamma^{{}^{\rm
de}}_{M}\rho_{{}_{M}},$ (29)

in the reheating epoch.

As a consequence, we have a close set of four ordinary differential equations
to uniquely determine the time evolutions of the Hubble rate $H$, the
cosmological term $\rho_{{}_{\Lambda}}$, massive pairs’ energy density
$\rho_{{}_{M}}$ and relativistic particles’ energy density $\rho_{{}_{R}}$.
They are the cosmic rate equation (26) for $\rho_{{}_{M}}$, the reheating
equation (29) for $\rho_{{}_{R}}$, Einstein equations (3) and (4) for $H$ and
$\rho_{{}_{\Lambda}}$, In addition, there are three algebraic relations: the
pair-production rate $\Gamma_{M}$ (23), the pair-decay rate $\Gamma^{{}^{\rm
de}}_{M}$ (25) and the spacetime evolution $\epsilon$-rate (5). For further
analysis, we recast these equations as the Einstein equations

$\displaystyle h^{2}$ $\displaystyle=$
$\displaystyle\Omega_{{}_{\Lambda}}+\Omega_{{}_{M}}+\Omega_{{}_{R}},$ (30)
$\displaystyle\frac{dh^{2}}{dx}$ $\displaystyle=$
$\displaystyle-3\Omega_{{}_{M}}-4\Omega_{{}_{R}},$ (31)

and the cosmic rate equations

$\displaystyle\frac{d\Omega_{{}_{M}}}{dx}+3\Omega_{{}_{M}}$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{M}}{H}\left(\Omega_{{}_{M}}^{H}-\Omega_{{}_{M}}\right)-\frac{\Gamma^{{}^{\rm
de}}_{M}}{H}\Omega_{{}_{M}},$ (32)
$\displaystyle\frac{d\Omega_{{}_{R}}}{dx}+4\Omega_{{}_{R}}$ $\displaystyle=$
$\displaystyle\frac{\Gamma^{{}^{\rm de}}_{M}}{H}\Omega_{{}_{M}},$ (33)

where, instead of the cosmic time $t$, we adopt the cosmic $e$-folding
variable $x=\ln(a/a_{\rm end})$ and
$d(\cdot\cdot\cdot)/dx=d(\cdot\cdot\cdot)/(Hdt)$ for the sake of simplicity
and significance in physics.

In order to study the reheating epoch, we adopt in Eqs. (30-33) the scale
factor $a_{\rm end}$ at the end of the inflation, corresponding the
normalisations $h\equiv H/H_{\rm end}$,
$\Omega_{{}_{\Lambda,M,R}}\equiv\rho_{{}_{\Lambda,M,R}}/\rho_{c}^{\rm end}$
and

$\displaystyle\Omega^{H}_{{}_{M}}\equiv\frac{\rho^{H}_{{}_{M}}}{\rho_{c}^{\rm
end}}=\frac{2}{3}\chi\hat{m}^{2}h^{2},\quad\rho_{c}^{\rm
end}\equiv\frac{3H^{2}_{\rm end}}{(8\pi G)},$ (34)

in unit of the inflation ending scale $H_{\rm end}$ and the corresponding
characteristic density $\rho_{c}^{\rm end}$. Using the rates $\Gamma_{M}$ (23)
and $\Gamma^{{}^{\rm de}}_{M}$ (25), we write the ratios in Eqs. (32) and
(33),

$\displaystyle\frac{\Gamma_{M}}{H}=\left(\frac{\chi}{4\pi}\right)\left(\frac{\hat{m}}{H_{\rm
end}}\right)\frac{\epsilon}{h};\quad\frac{\Gamma^{{}^{\rm
de}}_{M}}{H}=g^{2}_{{}_{Y}}\left(\frac{\hat{m}}{H_{\rm
end}}\right)\frac{1}{h},$ (35)

which represent the rates $\Gamma_{M}$ and $\Gamma^{{}^{\rm de}}_{M}$ of the
microscopic processes (21) and (25) compared with the Hubble rate $H$ of the
macroscopic expansion of the spacetime. Moreover, we rewrite the
$\epsilon$-rate (36) of time-varying horizon $H$ as

$\displaystyle\epsilon$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{H}\frac{dH}{dx}=\frac{3}{2}\frac{\Omega_{{}_{M}}+(4/3)\Omega_{{}_{R}}}{\Omega_{{}_{\Lambda}}+\Omega_{{}_{M}}+\Omega_{{}_{R}}},$
(36)

to characterise the different episodes of the reheating epoch.

Equations (30-36) can be numerically integrated, provided that the initial
conditions (19), see Sec. II.3, are given at the beginning of the reheating
epoch. In order to understand the reheating physics encoded, it is worthwhile
to find the asymptotic solutions to physically characterise particular
episodes in the entire reheating epoch. This epoch is represented by the scale
factor changing from the inflation end $a_{3}=a_{\rm end}$ to the genuine
reheating $a_{2}=a_{{}_{R}}$ in Fig. 1, the schematic diagram of the Universe
evolution.

##  IV Different episodes in the reheating epoch

In the reheating epoch, general speaking, the horizon $h$ and the cosmological
term $\Omega_{{}_{\Lambda}}$ decreases, as the matter content
$\Omega_{{}_{M}}$ or $\Omega_{{}_{R}}$ increases, meanwhile the ratio
$\Gamma_{M}/H$ (35) and the $\epsilon$-rate (36) increase. To gain the insight
into the physics first, we use the $\epsilon$-rate values (36) to characterize
the different episodes in the reheating epoch. In each episode, the
$\epsilon$-rate slowly varies in time, we approximately have the time scale of
the spacetime expansion

$\displaystyle H^{-1}\approx\epsilon t.$ (37)

In the transition from one episode to another, the $\epsilon$-rate
significantly changes its value. Using the characteristic $\epsilon$ values
$\epsilon\ll 1,\epsilon\approx 3/2,\epsilon\approx 2$, we identify the
following three different episodes ${\mathcal{P}}$-episode,
${\mathcal{M}}$-episode and ${\mathcal{R}}$-episode in the reheating epoch.
The ${\mathcal{P}}$-episode and the ${\mathcal{M}}$-episode have some
similarities to the preheating phase in usual inflation models RevReheating ;
BigBang .

###  IV.1 preheating episode: ${\mathcal{P}}$-episode

The preheating ${\mathcal{P}}$-episode is a transition from the inflation end
to the reheating start. In this episode, the pair production rate $\Gamma_{M}$
(23) is larger than the Hubble rate $H$, that is still much larger than the
pair decay rate $\Gamma^{{}^{\rm de}}_{M}$ (25),

$\displaystyle\Gamma_{M}>H\gg\Gamma^{{}^{\rm
de}}_{M},\quad\rho_{{}_{\Lambda}}>\rho_{{}_{M}}\gg\rho_{{}_{R}}.$ (38)

The radiation energy density of relativistic particles is completely
negligible $\rho_{{}_{R}}\approx 0$, compared with the massive pairs’ energy
density $\rho_{{}_{M}}$ and cosmological one $\rho_{{}_{\Lambda}}$. The
studies presented in this and next sections are also relevant to the “stable”
massive pairs which do not decay to relativistic particles $\bar{\ell}\ell$
(25), i.e. $\Gamma^{{}^{\rm de}}_{M}=0$, via gauge interactions.

####  IV.1.1 Numerical solutions of cosmological $\rho_{{}_{\Lambda}}$
converted to matter $\rho_{{}_{M}}$

After the inflation end $\Gamma_{M}\gtrsim H$, the pair-production rate
$\Gamma_{M}$ (23) increases as the $\epsilon$-rate

$\displaystyle\epsilon$ $\displaystyle\approx$
$\displaystyle\frac{3}{2}\frac{\rho_{{}_{M}}}{\rho_{{}_{\Lambda}}+\rho_{{}_{M}}}=\frac{3}{2}\frac{\Omega_{{}_{M}}}{\Omega_{{}_{\Lambda}}+\Omega_{{}_{M}}}.$
(39)

As a result, the massive pairs are significantly produced and pairs have the
large density $n_{{}_{M}}$ and rate $\Gamma_{M}n_{{}_{M}}$ to annihilate into
the spacetime (20). The spacetime horizon and pairs are thus coupled, and the
back and forth processes (21) have to be considered by the rate equation (26),

$\displaystyle\dot{\rho}_{{}_{M}}+3H\rho_{{}_{M}}$ $\displaystyle=$
$\displaystyle\Gamma_{M}\left(\rho_{{}_{M}}^{H}-\rho_{{}_{M}}\right)$ (40)

where the decay of massive pairs to relativistic particles is neglected. The
reheating qquation (33) is then not relevant, and the basic equations (30),
(31) and (32) reduce to

$\displaystyle h^{2}$ $\displaystyle=$
$\displaystyle\Omega_{{}_{\Lambda}}+\Omega_{{}_{M}},$ (41) $\displaystyle
dh^{2}/dx$ $\displaystyle=$ $\displaystyle-3\Omega_{{}_{M}},$ (42)
$\displaystyle d\Omega_{{}_{M}}/dx+3\Omega_{{}_{M}}$ $\displaystyle=$
$\displaystyle(\Gamma_{M}/H)\left(\Omega_{{}_{M}}^{H}-\Omega_{{}_{M}}\right).$
(43)

These equations uniquely determine the evolutions of the Hubble rate $H$,
pairs’ energy densities $\rho_{{}_{M}}$ and cosmological term
$\rho_{{}_{\Lambda}}$. The ratio $\Gamma_{M}/H$ (35) of the pair-production
rate and the Hubble rate has to be larger than one, $\Gamma_{M}/H>1$ in Eq.
(43).

Figure 3:  (Color Online). In a few $e$-folding number $x=\ln(a/a_{\rm end})$,
(a) the Hubble rate drops rapidly; (b) the pair energy density exceeds the
energy density of the cosmological term; (c) the ratio $\Gamma_{M}/H>1$
increases rapidly; (d) the $\epsilon$-rate of $H$ variation increases in the
transition from $\epsilon\ll 1$ (inflation epoch) to the asymptotic value
$\epsilon\sim{\mathcal{O}}(1)$ (${\mathcal{M}}$-episode, see Sec. IV.2). These
illustrations are plotted with the initial condition (44) and parameter
$(\hat{m}/m_{\rm pl})=27.7$.

We are in the position of solving these equations for the
${\mathcal{P}}$-episode. Using the follow values at the inflation end $H_{\rm
end}$ (17) and energy density ratio $\rho^{\rm end}_{{}_{M}}/\rho^{\rm
end}_{{}_{\Lambda}}\ll 1$ (19), see Sec. II.3,

$\displaystyle\Omega^{\rm end}_{{}_{M}}=4.7\times 10^{-3},~{}{\rm
so~{}that}~{}~{}(\Gamma_{M}/H)_{\rm end}\approx 1$ (44)

as the initial conditions for starting the ${\mathcal{P}}$-episode, we
numerically integrate Eqs. (30,31) and (32), by selecting values of the mass
parameter $\hat{m}/m_{\rm pl}$.

The numerical solutions are plotted in Figures 3 and 4 in terms of the
$e$-folding variable $x=\ln(a/a_{\rm end})$, where the scale factor $a_{\rm
end}$ correspondingly to the scale $H_{\rm end}$ (17) at the inflation end.
These solutions show an important result that the cosmological energy density
$\rho_{{}_{\Lambda}}$ is significantely converted to the matter energy density
$\rho_{{}_{M}}$, as the pair-production rate $\Gamma_{M}$ increases and
becomes larger than the Hubble rate $H$. In more details, we list that in the
${\mathcal{P}}$-episode the physical quantities vary in time as follow,

  1. (i)

the Hubble rate $h$ decreases rapidly in a few $e$-folding number, as the
energy density $\rho_{{}_{M}}$ of produced pairs becomes dominate over the
energy density $\rho_{{}_{\Lambda}}$ of the cosmological term, see Fig. 3 (a);

  2. (ii)

the pair energy density $\rho_{{}_{M}}$ increases at the expense of the energy
density $\rho_{{}_{\Lambda}}$ of the cosmological term, eventually
$\rho_{{}_{M}}$ exceeds and dominates $\rho_{{}_{\Lambda}}$, see Fig. 3 (b);

  3. (iii)

the ratio $\Gamma_{M}/H>1$ (38) of the pair-production rate $\Gamma_{M}$ and
the Hubble rate $H$ increases and becomes much larger than unity
($\Gamma_{M}/H\gg 1$), see Fig. 3 (c);

  4. (iv)

the $H$ varying $\epsilon$-rate (39) increases from $\epsilon\ll 1$ to
$\epsilon\sim{\mathcal{O}}(1)$, indicating the transition from the inflation
end to the preheating ${\mathcal{P}}$-episode, and it then approaches to an
asymptotic value, see Fig. 3 (d).

In Figure 4 (left), we plot the energy densities $\Omega_{{}_{\Lambda}}$ and
$\Omega_{{}_{M}}$ are plotted as functions of the horizon $h^{2}$,
corresponding to Figures (a) and (b) in Fig. 3. It shows that two branches of
asymptotic solutions respectively for $h^{2}>0.9$, $h^{2}<0.9$

$\displaystyle\Omega_{{}_{\Lambda}}$ $\displaystyle\sim$
$\displaystyle\alpha_{g}h^{2},\quad\Omega_{{}_{\Lambda}}\sim\alpha_{s}h^{2};\quad\alpha_{g}\gg\alpha_{s},$
(45)

$\Omega_{{}_{M}}+\Omega_{{}_{\Lambda}}=h^{2}$ and the turning point is about
$h^{2}\approx 0.98$ at which $\Omega_{{}_{M}}$ exceed $\Omega_{{}_{\Lambda}}$
and the rapid $\rho_{{}_{\Lambda}}\Rightarrow\rho_{{}_{M}}$ converting process
takes place. The characteristic behaviour (45) in the reheating epoch is the
same as $\Omega_{{}_{\Lambda}}\propto h^{2}$ in the pre-inflation and
inflation epoch, as well as the radiation and matter dominate epochs in the
standard cosmology discussed in Ref. xue2020MPLA .

Figure 4 (right) shows that in the preheating ${\mathcal{P}}$-episode, the
ratio $\rho_{{}_{M}}/\rho_{{}_{\Lambda}}$ rapidly increases in a few
$e$-folding numbers from $\rho_{{}_{M}}/\rho_{{}_{\Lambda}}\ll 1$ at the
inflation end to a value
$\rho_{{}_{M}}/\rho_{{}_{\Lambda}}\gtrsim{\mathcal{O}}(1)$. As physically
expected, these results show in the ${\mathcal{P}}$-episode, the fast decrease
of the Hubble rate $H$ that becomes much smaller than the pair-production rate
$\Gamma_{M}$; the convention of the energy $\rho_{{}_{\Lambda}}$ to the
matter-energy $\rho_{{}_{M}}$ that finally becomes dominant in the evolution.
These properties of ${\mathcal{P}}$-episode are very different from the
properties of the inflation epoch. The ${\mathcal{P}}$-episode end can be
defined at

$\displaystyle\rho_{{}_{\Lambda}}\approx\rho_{{}_{M}},\quad a\gtrsim a_{\rm
end}$ (46)

where $a\approx 1.11a_{\rm end}$, from Figs. 3 and 4 (right). This shows that
the preheating ${\mathcal{P}}$-episode is a very brief transition episode. The
latter approached asymptotic values of $\epsilon$-rate and
$\rho_{{}_{M}}/\rho_{{}_{\Lambda}}$, shown in Figs. 3 (d) and 4 (right),
indicate another ${\mathcal{M}}$-episode that will be discussed soon.

Figure 4:  (Color Online). Left: The energy densities $\Omega_{{}_{\Lambda}}$
(blue) and $\Omega_{{}_{M}}$ (orange) are plotted as functions of the horizon
$h^{2}$, corresponding to Figures (a) and (b) in Fig. 3. Right: The ratio
$\Omega_{{}_{M}}/\Omega_{{}_{\Lambda}}=\rho_{{}_{M}}/\rho_{{}_{\Lambda}}$
varies in the transition from $\rho_{{}_{M}}/\rho_{{}_{\Lambda}}\ll 1$
(inflation epoch) to $\rho_{{}_{M}}/\rho_{{}_{\Lambda}}\gg 1$, approaching to
a constant (massive pair dominated ${\mathcal{M}}$-episode). This ratio
$\rho_{{}_{M}}/\rho_{{}_{\Lambda}}$ is plotted for selected values
$\hat{m}/m_{\rm pl}=27.7,24.6,18.5$, corresponding to the solid black line,
green dashed line, and red dotted line. These illustrations are plotted with
the initial conditions $H_{\rm end}$ (17) and (44).

####  IV.1.2 Threshold of produced pair mass and degeneracy for
$\rho_{{}_{M}}>\rho_{{}_{\Lambda}}$

Moreover, another important result is that the numerical calculations shows
how these solutions depend on the pair mass parameter $\hat{m}$ (10)
introduced for the reheating epoch:

  1. (a)

For large mass parameters $\hat{m}/m_{\rm pl}>20$, the pair energy density
$\rho_{{}_{M}}$ exceeds the energy density $\rho_{{}_{\Lambda}}$ of the
cosmological term and the asymptotic value of the ratio
$\rho_{{}_{M}}/\rho_{{}_{\Lambda}}>1$,

$\displaystyle\hat{m}/m_{\rm pl}>20,\quad\rho_{{}_{M}}/\rho_{{}_{\Lambda}}>1.$
(47)

The physical explanation is that the effective degeneracy $g_{d}$ (10) of
massive pairs produced in the reheating epoch has to be large enough so that
the production rate $\Gamma_{M}$ (9) is much larger than the Hubble rate
$\Gamma_{M}\gg H$ and the conversion from $\rho_{{}_{\Lambda}}$ to
$\rho_{{}_{M}}$ is efficient. This case (47) corresponds to the real physical
situation that the most relevant matter of the Universe is generated in the
reheating epoch, and afterwards the Universe expansion starts the radiation
dominated epoch $\rho_{{}_{R}}\gg\rho_{{}_{M}}\gg\rho_{{}_{\Lambda}}$ of the
standard cosmology;

  2. (b)

For small mass parameters $\hat{m}/m_{\rm pl}\lesssim 20$, the pair energy
density $\rho_{{}_{M}}$ never exceeds the energy density $\rho_{{}_{\Lambda}}$
of the cosmological term, namely the ratio
$\rho_{{}_{M}}/\rho_{{}_{\Lambda}}<1$ is always smaller than one. This case
should correspond to the unphysical situation that the Universe inflation
would have never completely ended, i.e., $H^{2}$ is always dominated by the
cosmological term $\rho_{{}_{\Lambda}}$.

We will discuss the mass parameter $\hat{m}$ for the real physical situation
(a) to see how its values relate to the CMB observations through the ratio
$\rho_{{}_{M}}/\rho_{{}_{\Lambda}}>1$ in Sec. V.

Observe that the mass parameter $\hat{m}$ of the reheating epoch is larger
than the mass parameter $m_{*}$ (13) of the inflation epoch, i.e.,
$\hat{m}>m_{*}$. From the viewpoint of the pair production, the pair mass
scale in the reheating epoch should be smaller than that in the inflation
epoch, since the horizon $H$ of the reheating epoch is smaller than that of
the inflation epoch. Therefore, this implies that the effective degeneracies
$g^{f}_{d}$ (10) of pairs produced in the reheating epoch $\Gamma_{M}/H>1$ is
much larger than the effective degeneracies of pairs produced in the inflation
epoch $\Gamma_{M}/H<1$.

Equation (39) shows that the asymptotic value of the Horizon variation
$\epsilon$-rate (5) relates to the ratio $\rho_{{}_{M}}/\rho_{{}_{\Lambda}}$
asymptotic value, see Figs. 3 (d) and 4 (right). For large mass parameter
$\hat{m}/m_{\rm pl}\gtrsim 27.7$, the asymptotic value of the ratio
$\rho_{{}_{M}}/\rho_{{}_{\Lambda}}\gg 1$, the $\epsilon$-rate (39) approaches
to the constant $\epsilon\approx\epsilon_{{}_{M}}=3/2$. This shows the episode
of massive pairs domination: ${\mathcal{M}}$-episode.

####  IV.1.3 Minimal comoving radius $(Ha)^{-1}$ location

Before discussing the ${\mathcal{M}}$-episode, we would like to mention the
turning point at which the Universe acceleration vanishes $\ddot{a}=0$,

$\displaystyle
2\rho_{{}_{\Lambda}}=(1+3\,\omega_{{}_{M}})\,\rho_{{}_{M}}-\\!(1\\!+\\!3\omega_{{}_{R}})\rho_{{}_{R}},$
(48)

which is obtained from the $1\\!-\\!1$ component of the Einstein equation

$\displaystyle
2\frac{dH}{dt}\\!+\\!2H^{2}\\!=\frac{2\ddot{a}}{a}=\\!\Big{[}2\rho_{{}_{\Lambda}}\\!-\\!(1\\!+\\!3\omega_{{}_{M}})\rho_{{}_{M}}-\\!(1\\!+\\!3\omega_{{}_{R}})\rho_{{}_{R}}\Big{]}.$
(49)

At this turning point, the Universe stops acceleration $\ddot{a}>0$ and starts
deceleration $\ddot{a}<0$. The turning point occurs at
$\rho_{{}_{\Lambda}}=\rho_{{}_{M}}/2$ for $\omega_{{}_{M}}\approx 0$ and
$\rho_{{}_{R}}\approx 0$. This tells us the balance point of the competition
between the cosmological term $\rho_{{}_{\Lambda}}$ and matter term
$\rho_{{}_{M}}$ in the ${\mathcal{P}}$-episode.

On the other hand, the minimal value of the comoving radius $(aH)^{-1}$
locates at

$\displaystyle d(aH)^{-1}/dt=0\quad\Rightarrow\quad\dot{H}+H^{2}=0.$ (50)

From Eqs. (3) and (4), we obtain

$\displaystyle\rho_{{}_{\Lambda}}=\rho_{{}_{M}}/2+\rho_{{}_{R}}\approx\rho_{{}_{M}}/2,\quad\epsilon=\epsilon^{\rm
min}_{{}_{\Lambda}}=1,$ (51)

coinciding with the turning point (48). Namely at the minimal comoving radius
$(aH)^{-1}$, the Universe stops acceleration $\ddot{a}>0$ and starts
deceleration $\ddot{a}<0$, beginning the reheating epoch and standard
cosmology. This is indeed the case for large mass parameter $(\hat{m}/m_{\rm
pl})>20$ and the ratio $\rho_{{}_{M}}/\rho_{{}_{\Lambda}}$ becomes larger than
2. The numerical results (Fig. 3) show that this turning/minimal point
$\epsilon_{\rm min}=1$ locates at $x_{\rm min}\approx 1.7\times 10^{-2}$ and
$a_{\rm min}\approx a_{\rm end}\times\exp~{}(1.7\times 10^{-2})=1.02~{}a_{\rm
end}$.

While, the turning/minimal point $\epsilon_{\rm min}=1$ is never reached, for
the cases of the small mass parameter $(\hat{m}/m_{\rm pl})<20$ and the ratio
$\rho_{{}_{M}}/\rho_{{}_{\Lambda}}$ is always smaller than 2, see Fig. 4
(right). The reason is that there is no enough matter of massive pairs
produced to balance the cosmological term and slow down the Universe
acceleration. As a consequence, the Universe keeps acceleration $\ddot{a}>0$
and does not run into the epoch of the standard cosmology. Therefore, the mass
parameter range $(\hat{m}/m_{\rm pl})<20$ should be excluded.

###  IV.2 Massive pairs domination: ${\mathcal{M}}$-episode

After the ${\mathcal{P}}$-episode transition, the reheating epoch is in the
$\mathcal{M}$-episode of massive pair domination. The $\mathcal{M}$-episode is
characterised by

$\displaystyle\rho_{{}_{M}}$ $\displaystyle\gg$
$\displaystyle\rho_{{}_{\Lambda}}\gg\rho_{{}_{R}},\quad\Gamma_{M}>H>\Gamma^{{}^{\rm
de}}_{M},$ (52)

so that the radiation energy density $\rho_{{}_{R}}$ is negligible in the
Einstein equations (3-5) and the cosmic rate equation (26). The $H$ variation
$\epsilon$-rate $\epsilon_{{}_{M}}$ is a constant, shown as an asymptotic
value $\epsilon_{{}_{M}}\approx 3/2$ in Fig. 3 (d) for
$\rho_{{}_{M}}/\rho_{{}_{\Lambda}}\gg 1$, see Fig. 4 (right). In this episode,
the Hubble rate $H$ and scale factor $a(t)$ vary as

$\displaystyle H^{-1}\approx\epsilon_{{}_{M}}t,\quad a(t)\sim
t^{1/\epsilon_{{}_{M}}},$ (53)

and $h^{2}\approx\Omega_{M}$, the pair energy density
$\Omega_{M}\propto(a/a_{\rm end})^{-2\epsilon_{{}_{M}}}$ drops as in the
matter dominated universe, analogously to the scenario Starobinsky1980 .

####  IV.2.1 Spacetime and pairs are coupled in horizon $H$ evolution

In addition to the large pair energy density $\rho_{{}_{M}}$, the pair
production/annihilation rate $\Gamma_{M}$ is much larger than the Hubble rate
$H$, i.e., $\Gamma_{M}/H\gg 1$, see Fig. 3 (c). The back and forth processes
of pair production and annihilation $\bar{F}F\Leftrightarrow\mathcal{S}$ are
important, as described by the cosmic rate equation (40) with the detailed
balance term $D_{M}$,

$\displaystyle
D_{M}\equiv\Gamma_{M}\left(\Omega_{{}_{M}}^{H}-\Omega_{{}_{M}}\right)$ (54)

and the characteristic time scale $\tau_{{}_{M}}$,

$\displaystyle\tau^{-1}_{{}_{M}}\equiv\frac{\Gamma_{M}}{\Omega_{{}_{M}}}\left(\Omega_{{}_{M}}^{H}-\Omega_{{}_{M}}\right),$
(55)

which is actually the time period of back and forth
$\bar{F}F\Leftrightarrow\mathcal{S}$ oscillating processes. This macroscopic
time scale is much smaller than the macroscopic expansion time scale
$\tau_{{}_{H}}=H^{-1}$, $\tau_{{}_{M}}\ll\tau_{{}_{H}}$. In this situation,
the microscopic back and forth process
(${\mathcal{S}}\Leftrightarrow\bar{F}F$) is much faster than the horizon
expanding process, thus the space time and massive pairs are completely
coupled each other via these back and forth processes.

Therefore, the back and forth oscillating process
$\bar{F}F\Leftrightarrow\mathcal{S}$ can build a local chemical equilibrium of
the quantum-number and energy equipartition

$\displaystyle\rho_{{}_{M}}\Leftrightarrow\rho^{H}_{{}_{M}};\quad\mu_{F}+\mu_{\bar{F}}=\mu_{\rm
spacetime}$ (56)

between massive pairs and the space time. The chemical potential of particles
is opposite to the chemical potential of antiparticles, i.e.,
$\mu_{F}=-\mu_{\bar{F}}$, so that particle and antiparticle pairs have zero
chemical potential $\mu_{\rm pair}=\mu_{F}+\mu_{\bar{F}}=0$. The chemical
equilibrium (56) leads to the space time “chemical potential” is zero, i.e.,
$\mu_{\rm spacetime}=0$. In this case, the detailed balance term (54) for the
oscillations $\rho_{{}_{M}}\Leftrightarrow\rho^{H}_{{}_{M}}$ in the
microscopic time scale $\tau_{{}_{M}}$ should vanishes, in the sense of its
time-averaged

$\displaystyle\langle\rho_{{}_{M}}-\rho^{H}_{{}_{M}}\rangle=0,$ (57)

over the macroscopic time $\tau_{{}_{H}}\gg\tau_{{}_{M}}$. This means that
$\rho_{{}_{M}}\approx\rho^{H}_{{}_{M}}$ in the macroscopic time scale
$\tau_{{}_{H}}$. The cosmic rate equation approximately becomes

$\displaystyle\frac{d\rho_{{}_{M}}}{dt}+3H\rho_{{}_{M}}$
$\displaystyle\approx$ $\displaystyle 0,$ (58)

whose solution is $\rho_{{}_{M}}\propto a^{-3}$. This is consistent with the
solution to Eq. (3) for the massive pair domination
$\rho_{{}_{M}}\gg\rho_{{}_{\Lambda}}$ and $\rho_{{}_{M}}\gg\rho_{{}_{R}}$,
yielding $H^{2}\sim\rho_{{}_{M}}\propto a^{-3}$. This is also self-consistent
with the pair-production formula (7)
$\rho_{{}_{M}}\approx\rho^{H}_{{}_{M}}=\chi\hat{m}^{2}H^{2}\propto a^{-3}$.

In order to verify these discussions, we check the solution (57) or (58)
analytically and numerically. The approximately analytical solution
$\rho_{{}_{M}}\approx\rho^{H}_{{}_{M}}=\chi\hat{m}^{2}H^{2}$ averaged over the
time $\tau_{{}_{H}}$ consistently obeys the cosmic rate equation (58),

$\displaystyle\langle\dot{\rho}^{H}_{{}_{M}}\rangle=\langle
2\chi\hat{m}^{2}H\dot{H}\rangle=-\langle
2H\rho^{H}_{{}_{M}}\epsilon\rangle\approx-3H\rho^{H}_{{}_{M}},$ (59)

where $\langle\epsilon\rangle\approx\epsilon_{{}_{M}}=3/2$, and the detailed
balance term (54) vanishes. Numerical results quantitatively show the same
conclusion: $\rho_{{}_{M}}$ approaches to $\rho^{H}_{{}_{M}}$, and
$n_{{}_{M}}\approx\rho_{{}_{M}}/2\hat{m}$ approaches to
$n^{H}_{{}_{M}}\approx\rho^{H}_{{}_{M}}/2\hat{m}$

$\displaystyle\rho_{{}_{M}}\approx\rho^{H}_{{}_{M}}=2\chi\hat{m}^{2}H^{2},\quad
n_{{}_{M}}\approx n^{H}_{{}_{M}}=\chi\hat{m}H^{2},$ (60)

see Fig. 5 (a); correspondingly the detailed balance term (54) vanishes, see
Fig. 5 (b). The detailed balance solution (60) is valid in the matter dominate
evolution, and it is peculiar for stable massive pairs, which have no gauge
interactions except gravitation one.

Figure 5:  (Color Online). In a few $e$-folding number $x=\ln(a/a_{\rm end})$,
(a) the energy density $\rho^{H}_{{}_{M}}$ (34) (orange) of the spacetime
produced pairs and the pair energy density of $\rho_{{}_{M}}$ (blue) as the
solution to the cosmic rate equation (40), showing
$\rho_{{}_{M}}\approx\rho^{H}_{{}_{M}}$; (b) the detailed balance term $D_{M}$
(54) vanishes. These illustrations are plotted with the initial condition (44)
and parameter $(\hat{m}/m_{\rm pl})=27.7$.

####  IV.2.2 Preliminary discussions on CPT symmetry and increasing entropy

We observe that in the ${\mathcal{M}}$-episode, due to $\Gamma_{M}\gg H$ and
$\tau_{{}_{M}}\ll\tau_{{}_{H}}$, the spacetime and pairs are tightly coupled
each other and the local chemical equilibrium (56) can be established. This
indicates that the CPT symmetry (22) is approximately preserved in the cosmic
rate equation (26) involving the microscopical process
$\mathcal{S}\Leftrightarrow\bar{F}F$. On the other hand, the $T$-symmetry or
time-reversal symmetry $t\rightarrow-t$ is macroscopically violated in the
Universe expansion with increasing time and entropy.

This is not contradictory for the following reasons. The number and energy
density of pair productions is slightly larger than that of pair
annihilations, $\rho^{H}_{{}_{M}}\gtrsim\rho_{{}_{M}}$. The pair production
$\mathcal{S}\rightarrow\bar{F}F$ produce the entropy of pairs. Instead, the
pair annihilation $\bar{F}F\rightarrow\mathcal{S}$ eliminate the entropy of
pairs. The net entropy of pairs produced is very small as pairs are very
massive and the difference $\rho^{H}_{{}_{M}}-\rho_{{}_{M}}$ is very small.
For this tiny production of pairs’ entropy, the Universe consistently
undergoes an entropically favourable expansion.

In the case that the number and energy densities of pair productions and
annihilations are exactly the same $\rho_{{}_{M}}\equiv\rho^{H}_{{}_{M}}$, a
local thermodynamic equilibrium (LTE) can be built between the spacetime and
pairs, associating with the Hawking temperature $T_{H}=H/2\pi$ for a De Sitter
spacetime of an exactly constant $H$. In this case, no pairs’ entropy is
produced and the total entropy of pairs is conserved. Pairs’ temperature and
entropy render the physical senses of the spacetime entropy and temperature,
which is not in the scope of this article.

Figure 6:  (Color Online). In a few $e$-folding number $x=\ln(a/a_{\rm end})$,
(a) the blue line $h^{2}$ and orange line $\Omega_{{}_{R}}$, the Hubble rate
drops more rapidly than the case neglecting $\Omega_{{}_{R}}$ Fig. 3 (a); (b)
$\Omega_{{}_{R}}$, $\Omega_{{}_{M}}$ and $\Omega_{{}_{\Lambda}}$ are lines
green, blue and orange; (c) the $\epsilon$-rate of $H$ variation increases in
the transition from $\epsilon\ll 1$ (${\mathcal{P}}$-episode) to the
asymptotic value $\epsilon\sim{\mathcal{O}}(1)$ (${\mathcal{M}}$-episode) and
approaches to $\epsilon=2$ of the radiation domination
(${\mathcal{R}}$-episode); (d) the ratios of $\Omega_{{}_{M}}/\Omega_{{}_{R}}$
(orange) and $\Omega_{{}_{\Lambda}}/\Omega_{{}_{R}}$ (blue), recalling
$h^{2}=\Omega_{{}_{\Lambda}}+\Omega_{{}_{M}}+\Omega_{{}_{R}}$. These
illustrations are plotted with the initial conditions (44) and $\Omega^{\rm
end}_{{}_{R}}=0.0$; the parameter $(\hat{m}/m_{\rm pl})=27.7$ and
$g^{2}_{{}_{Y}}=10^{-9}$.

###  IV.3 Relativistic particles domination: ${\mathcal{R}}$-episode of the
genuine reheating

After the ${\mathcal{M}}$-episode of massive pairs domination, the massive
pairs’ decay term $\Gamma^{{}^{\rm de}}_{M}\rho_{{}_{M}}$ in the cosmic rate
equation (40) starts to dominate, when the time $t\gtrsim\tau_{{}_{R}}$, where
$\tau_{{}_{R}}=(\Gamma^{{}^{\rm de}}_{M})^{-1}$ is the characteristic time
scale of massive pairs decay to relativistic particles, producing tremendous
amounts of entropy. The reheating epoch starts its genuine reheating episode,
i.e., ${\mathcal{R}}$-episode. We discuss this episode following the line of
Ref. kolb .

####  IV.3.1 Massive pairs decay to relativistic particles

To study this ${\mathcal{R}}$-episode, we numerically solve the closed set of
the basic equations (30-33) and the reheating equation (33) by taking into
account relativistic particles of the radiation energy density
$\Omega_{{}_{R}}$ and the decay term,

$\displaystyle R_{M}=\Gamma_{M}^{{}^{\rm de}}\Omega_{{}_{M}},$ (61)

so as to obtain the energy densities $\Omega_{{}_{R}}$, $\Omega_{{}_{M}}$,
$\Omega_{{}_{\Lambda}}$ and the horizon $H$ as functions of the $e$-folding
number $x$ (time $t$). The initial condition of of the radiation energy
density $\Omega_{{}_{R}}=\Omega^{\rm end}_{{}_{R}}=0$ is chosen at the
inflation end $a_{\rm end}$, in addition to the initial conditions (44). The
numerical results are reported in Figures 6, which show that in the $H^{2}$
(30) the radiation energy density $\Omega_{{}_{R}}$ of relativistic particles
increases from the negligible contribution to the dominated contribution,
compared with $\Omega_{{}_{M}}$ and $\Omega_{{}_{\Lambda}}$.

This phenomenon can be understood by comparing the decay term $R_{M}$ (61)
with the detailed balance term $D_{M}$ (54) in the cosmic rate equation (32).
The $\rho_{{}_{R}}$ is negligible when $D_{M}>R_{M}$, while the
$\rho_{{}_{R}}$ is dominant when $R_{M}>D_{M}$, and the transition from the
one to another occurs approximately at $R_{M}\gtrsim D_{M}$ , where
$\rho_{{}_{R}}\lesssim h^{2}$, as shown in Figs. 7 (a) and (b).

More precisely, it is the comparison between the characteristic time scale
$\tau_{{}_{M}}$ (55) of the pair back and forth process
$F\bar{F}\Leftrightarrow{\mathcal{S}}$ (21) and the characteristic time scale
$\tau_{{}_{R}}$ of the pair decay process $F\bar{F}\Rightarrow\ell\bar{\ell}$
(25),

$\displaystyle\tau^{-1}_{{}_{R}}\equiv\frac{R_{M}}{\Omega_{{}_{M}}}=\Gamma^{{}^{\rm
de}}_{M}=g^{2}_{{}_{Y}}\hat{m}.$ (62)

When $\tau_{{}_{M}}<\tau_{{}_{R}}$, the process
$F\bar{F}\Leftrightarrow{\mathcal{S}}$ is faster thus dominates, whereas
$\tau_{{}_{R}}<\tau_{{}_{M}}$, the process $F\bar{F}\Rightarrow\ell\bar{\ell}$
is faster thus dominates. In Figs. 7 (c) and (d), two time scales
$\tau_{{}_{M}}$ and $\tau_{{}_{R}}$ are plotted as dimensionless quantities
$\tau_{{}_{R}}/\tau_{{}_{H}}=(\Gamma_{M}^{{}^{\rm de}}/H)^{-1}$ and
$\tau_{{}_{M}}/\tau_{{}_{H}}=(\Gamma_{M}/H)^{-1}$ to show two episodes:

  1. (i)

$\tau_{{}_{R}}/\tau_{{}_{H}}>1$ and $\tau_{{}_{M}}/\tau_{{}_{H}}<1$
($D_{M}>R_{M}$), indicating the ${\mathcal{M}}$-episode and the decay process
$F\bar{F}\Rightarrow\ell\bar{\ell}$ (25) being irrelevant;

  2. (ii)

$\tau_{{}_{R}}/\tau_{{}_{H}}<1$ and $\tau_{{}_{M}}/\tau_{{}_{H}}>1$
($D_{M}<R_{M}$); indicating the ${\mathcal{R}}$-episode and the process
$F\bar{F}\Leftrightarrow{\mathcal{S}}$ (21) being irrelevant.

The separatrix of two episodes, i.e., the crossing point of two blue and
orange lines in Figs. 7, roughly gives the scale factor $a_{{}_{R}}$ at which
the genuine reheating occurs, i.e., ${\mathcal{R}}$-episode.

The reheating scale factor $a_{{}_{R}}$ value depends on the Yukawa coupling
$g_{{}_{Y}}$, the larger $g_{{}_{Y}}$ and the smaller $a_{{}_{R}}$, as it
should be. This is shown by the left column (a,c) and the right column (b,d)
of Figs. 7. Around this point $a_{{}_{R}}$, Figures 6 (b) and (d) show
$\Omega_{{}_{R}}\gg\Omega_{{}_{M}}\gg\Omega_{{}_{\Lambda}}$, and Fig. 6 (c)
shows $\epsilon$-rate (36) approaches to two ($\epsilon\rightarrow 2$),
indicating the genuine reheating occurrence. Note that at this point
$a_{{}_{R}}$ the numerical calculations of the basic equations (30-33) run
into the stiffness system of step size being effective zero. However, in the
case of genuine reheating starts, the analytical solution to these basic
equations can be found and studied in the next section.

Figure 7:  (Color Online). Plotted as a function of the $e$-folding variable
$x=\ln(a/a_{\rm end})$ for the same initial conditions and parameters as
adopted in Figures 6, the detailed balance term $D_{M}/H$ (54) (blue) and the
decay term $R_{M}/H$ (61) (orange); two time scales
$\tau_{{}_{M}}/\tau_{{}_{H}}$ (55) (blue) and $\tau_{{}_{R}}/\tau_{{}_{H}}$
(62) (orange). Left column (a) and (c) for $g^{2}_{{}_{Y}}=10^{-9}$ and
$a_{{}_{R}}\gtrsim 20.1a_{\rm end}$; Right column (b) and (d) for
$g^{2}_{{}_{Y}}=10^{-6}$ and $a_{{}_{R}}\gtrsim 1.8a_{\rm end}$.

####  IV.3.2 Energy densities of massive pairs and relativistic particles

After the $\mathcal{M}$-episode, the decay term $R_{M}$ (61) prevails over the
detailed balance term $D_{M}$ (54) in the cosmic rate equation (32), massive
pairs undergo the process of decay to relativistic particles, rather than the
process of annihilating to the spacetime. The spacetime and pairs are
decoupled in time evolution. It starts the $\mathcal{R}$-episode of the
genuine reheating and radiation energy domination, which is characterised by

$\displaystyle\rho_{{}_{R}}$ $\displaystyle\gg$
$\displaystyle\rho_{{}_{M}}\gg\rho_{{}_{\Lambda}},\quad\epsilon\rightarrow\epsilon_{{}_{R}}\approx
2,$ (63)

and $\Gamma^{{}^{\rm de}}_{M}/H>1$, as shown in Figs. 6. As a result,
Equations (30) and (31) or Eq. (36) gives

$\displaystyle H^{-1}\approx\epsilon_{{}_{R}}t,\quad
a(t)/a_{{}_{R}}\approx(t/\tau_{{}_{R}})^{1/\epsilon_{{}_{R}}},$ (64)

where the period of massive particles decay $\tau_{{}_{R}}=(\Gamma^{{}^{\rm
de}}_{M})^{-1}$ (62) is the reheating time scale and $a_{{}_{R}}$ is the scale
factor at the genuine reheating. Following the line presented in Ref. kolb ,
we discuss how the annihilation/decay of $F\bar{F}$ pairs transfers their mass
energy to relativistic particles, and calculate the radiation energy density
$\rho_{R}$, entropy $S$ and temperature $T$ of relativistic particles.

Since massive pairs predominately decay to relativistic particles, the
detailed balance term $D_{M}$ (54) is negligible, and the cosmic rate equation
(32) reduces to,

$\displaystyle\frac{d(a^{3}\rho_{{}_{M}})}{dt}$ $\displaystyle=$
$\displaystyle
a^{3}\frac{d\rho_{{}_{M}}}{dt}+3Ha^{3}\rho_{{}_{M}}\approx-\tau^{-1}_{{}_{R}}a^{3}\rho_{{}_{M}},$
(65) $\displaystyle\Rightarrow$
$\displaystyle\rho_{{}_{M}}\approx\rho_{{}_{M}}(a_{{}_{R}})\left(\frac{a}{a_{{}_{R}}}\right)^{-3}\exp-t/\tau_{{}_{R}}.$
(66)

The reheating equation (33) becomes

$\displaystyle\frac{d(a^{4}\rho_{{}_{R}})}{dt}$ $\displaystyle=$
$\displaystyle
a^{3}\frac{\rho_{{}_{M}}(a_{{}_{R}})}{\tau_{{}_{R}}}\left(\frac{a}{a_{{}_{R}}}\right)^{-3}\exp-t/\tau_{{}_{R}}.$
(67)

In theory it requires the the time integration from the initial time
$t_{i}(a_{i})\ll\tau_{{}_{R}}$ when $\rho_{{}_{R}}(a_{i})=0$ to the final time
$t_{f}\gg\tau_{{}_{R}}$ to obtain the radiation energy density $\rho_{{}_{R}}$
of relativistic particles,

$\displaystyle\rho_{{}_{R}}$ $\displaystyle=$
$\displaystyle\left(\frac{a_{{}_{R}}}{a}\right)^{4}\frac{\rho_{{}_{M}}(a_{{}_{R}})}{\tau_{{}_{R}}}\int^{t_{f}}_{t_{i}}\left(\frac{t}{\tau_{{}_{R}}}\right)^{1/\epsilon_{{}_{R}}}e^{-t/\tau_{{}_{R}}}dt\approx
0.89\left(\frac{a_{{}_{R}}}{a}\right)^{4}\rho_{{}_{M}}(a_{{}_{R}}).$ (68)

Through their gauge and/or other induced interactions, these relativistic
particles $\bar{\ell}\ell$ including sterile particles and other particles
beyond the SM, are quickly thermalised at a very high temperature $T_{\rm
RH}$, due to their high number and energy densities. The local thermalization
time scale is very short than the expansion time scale $\tau_{{}_{H}}=H^{-1}$,
and thus the local thermal equilibrium (LTE) is built.

####  IV.3.3 Temperature and entropy of reheating epoch

On the other hand, the second law of thermodynamics applied to a comoving
volume element yields kolb

$\displaystyle
dS=\frac{dQ}{T}=-\frac{d(a^{3}\rho_{{}_{M}})}{T}\approx\frac{a^{3}\rho_{{}_{M}}}{T}\tau_{{}_{R}}^{-1}dt,$
(69)

where $dQ$ is the pair mass energy and $dS$ is the entropy of relativistic
particles produced from massive pairs decay. Therefore, in a comoving volume,
the entropy and energy densities of relativistic particles at the thermal
state of temperature $T$ are given by,

$\displaystyle\rho_{{}_{R}}=\frac{\pi^{2}}{30}g_{*}T^{4},\quad
S=\frac{2\pi^{2}}{45}g_{*}a^{3}T^{3},\quad\rho_{{}_{R}}=\frac{3}{4}\left(\frac{45}{2\pi^{2}g_{*}}\right)^{1/3}S^{4/3}a^{-4},$
(70)

where the appropriately time-averaged degeneracy (effective) $g_{*}$ over the
decay period $\tau_{{}_{R}}$ counts for the total number of effectively
massless degrees of freedom, those species share common temperature $T$. Using
the formula for the entropy (70), Eq. (69) can be written as,

$\displaystyle
S^{1/3}\dot{S}=\left(\frac{2\pi^{2}g_{*}}{45}\right)^{1/3}a^{4}\rho_{{}_{M}}\tau^{-1}_{{}_{R}}.$
(71)

Integrating this equation over the $F\bar{F}$ pair decay interval from the
initial scale factor $a_{i}$ to the reheating scaling factor
$a_{{}_{R}}>a_{i}$ leads to an approximate solution kolb

$\displaystyle S^{4/3}_{R}$ $\displaystyle=$ $\displaystyle
1.09\left(\frac{4}{3}\rho_{{}_{M}}(a_{i})a^{4}_{{}_{R}}\right)\left(\frac{16\pi^{3}g_{*}\rho_{{}_{M}}(a_{i})}{135M^{2}_{\rm
pl}}\right)^{1/3}\tau^{2/3}_{{}_{R}},$ $\displaystyle\Rightarrow S_{R}$
$\displaystyle\approx$ $\displaystyle
1.32(16\pi^{3}g_{*}/135)^{1/4}a^{3}_{{}_{R}}\rho_{{}_{M}}(a_{i})(\tau_{{}_{R}}/M_{\rm
pl})^{1/2},$ (72)

where Eq. (66) is adopted and the initial entropy $S_{i}(a_{i})\approx 0$, as
massive pairs’ entropy is approximately zero. The integration is performed
over the period of massive particles decay characterized by the time scale
$\tau_{{}_{R}}$. In theory it requires the integrating from the initial time
$t_{i}(a_{i})\ll\tau_{{}_{R}}$ to the final time
$t(a_{{}_{R}})\gg\tau_{{}_{R}}$, when the entropy of relativistic particles is
significantly increased. In practice, $a_{i}\lesssim a_{{}_{R}}$ and
$t_{f}\gtrsim\tau_{{}_{R}}$ are approximately adopted in Eq. (72) for the
reason that the entropy $S_{R}$ is mainly produced in the reheating time
$\tau_{{}_{R}}$ and around the scale factor $a_{{}_{R}}(\tau_{{}_{R}})$.

At the reheating time scale $\tau_{{}_{R}}$ and scale factor $a_{{}_{R}}$, the
reheating scale $H_{\rm RH}$ can be obtained by either $\tau_{{}_{R}}$ from
the Friedmann equation (30) or the reheating temperature $T_{\rm RH}\equiv
T(t=\tau_{{}_{R}})$ from the thermalization (70) kolb :

$\displaystyle H^{2}_{\rm RH}$ $\displaystyle\equiv$ $\displaystyle
H^{2}(t=\tau_{{}_{R}})\approx\frac{1}{4}\tau^{-2}_{{}_{R}},$ (73)
$\displaystyle H^{2}_{\rm RH}$ $\displaystyle\approx$
$\displaystyle\frac{8\pi}{3M^{2}_{\rm
pl}}\rho_{{}_{R}}\approx\frac{8\pi}{3M^{2}_{\rm
pl}}\left(\frac{\pi^{2}g_{*}}{30}T^{4}_{\rm RH}\right),$ (74)

leading to the reheating temperature

$\displaystyle T_{\rm RH}\approx 0.55g_{*}^{-1/4}(M_{\rm
pl}/\tau_{{}_{R}})^{1/2}=0.55(g_{{}_{Y}}^{2}/g^{1/2}_{*})^{1/2}(\hat{m}/M_{\rm
pl})^{1/2}M_{\rm pl},$ (75)

and the all-important entropy per comoving volume,

$\displaystyle S_{R}$ $\displaystyle\approx$ $\displaystyle
1.32\left(\frac{16\pi^{3}}{135}\right)^{1/4}(g_{*}^{1/2}/g^{2}_{{}_{Y}})^{1/2}(\hat{m}/M_{\rm
pl})^{1/2}a^{3}_{{}_{R}}\rho_{{}_{M}}(a_{i})/\hat{m},$ (76)

and $s_{R}\equiv S_{R}/a^{3}_{{}_{R}}$. Analytical Eqs. (73) and (74)
physically mean that at the the genuine reheating (i) $H_{\rm
RH}\approx\Gamma^{{}^{\rm de}}_{M}/2=g_{{}_{Y}}^{2}\hat{m}/2$ the Hubble rate
is the same order of the pair decay rate, (ii) $H^{2}_{\rm
RH}\approx\rho_{{}_{R}}/3m^{2}_{\rm pl}$ the energy of relativistic particles
is predominate. These results depend on the effective degeneracy $g_{*}$ (70)
and the decay rate $\tau^{-1}_{{}_{R}}=\Gamma^{{}^{\rm
de}}_{M}=g_{{}_{Y}}^{2}\hat{m}$ (25) of massive pairs $F\bar{F}$ to
relativistic particles $\ell\bar{\ell}$.

Indeed, our numerical calculations show the consistency of the approximation
$a_{i}\lesssim a_{{}_{R}}$ used in Eq. (72) and the agreement with the
approximate analytical solutions (73) and (75). From Figs. 6 and 7, we find
that the reheating predominately takes place around the reheating scale factor
$a_{{}_{R}}$, at which $\tau_{{}_{R}}\sim\tau_{{}_{M}}$. At this reheating
scale factor $a_{{}_{R}}$, we indeed find the ratios $\tau_{{}_{R}}/H_{\rm
RH}\approx\tau_{{}_{M}}/H_{\rm RH}\sim{\mathcal{O}}(1)$ are about the order of
unity. Moreover, from Fig. 6 ($a$) we obtain the reheating scale

$\displaystyle H_{\rm RH}\approx 3.16\times 10^{-4}H_{\rm end}\approx
6.1\times 10^{9}{\rm GeV},\quad a_{{}_{R}}\approx 20.1a_{\rm end}$ (77)

for the case $g^{2}_{{}_{Y}}=10^{-9}$ and $\hat{m}=27.7m_{\rm pl}$, where
$H_{\rm end}$ (17) is used. While $H_{\rm RH}\sim 10^{-1}H_{\rm end}=1.9\times
10^{12}{\rm GeV}$ (the plot is not present), and $a_{{}_{R}}\approx
1.8~{}a_{\rm end}$ for the case $g^{2}_{{}_{Y}}=10^{-6}$. Note that the value
$H_{\rm end}\approx 7.95\times 10^{-6}m_{\rm pl}$ corresponds to $N_{\rm
end}=60$ and $r=0.052$ in Eq. (17).

####  IV.3.4 Reheating scale factor and initial condition for standard
cosmology

Our study shows that the reheating epoch composes the preheating
${\mathcal{P}}$-episode, the ${\mathcal{M}}$-episode of massive pairs
domination, and the genuine reheating ${\mathcal{R}}$-episode, and it lasts
for the period from $a_{\rm end}=a_{3}$ to $a_{{}_{R}}=a_{2}$, see Fig. 1.
From the numerical results $\epsilon$-rate of Figs. 6 (c) and
$\rho_{{}_{M}}\gg\rho_{{}_{\Lambda}}$ of Fig. 4 (right) for the mass parameter
$\hat{m}>20m_{\rm pl}$ (47), we observe that the ${\mathcal{M}}$-episode
$\epsilon\approx 3/2$ lasts last much longer time than ${\mathcal{P}}$-episode
$\epsilon\ll 1$ and the ${\mathcal{R}}$-episode $\epsilon\approx 2$. This
implies that in the reheating epoch the $\rho_{{}_{\Lambda}}\approx
3m^{2}_{\rm pl}H^{2}_{\rm end}$ energy density of the spacetime converts to
the $\rho_{{}_{M}}$ energy density of massive pairs, which then converts to
the $\rho_{{}_{R}}\approx 3m^{2}_{\rm pl}H^{2}_{\rm RH}$ energy density of
relativistic particles at the the genuine reheating (69). This is the case
that the reheating epoch ends at the scale factor $a_{{}_{R}}$ with the
condition $\rho_{{}_{\Lambda}}\ll\rho_{{}_{M}}\ll\rho_{{}_{R}}$ to initiate
the standard cosmological scenario.

In order to estimate the the scale factor change $a_{{}_{R}}/a_{\rm end}$ in
the reheating epoch, we approximately use the conservation law (58) for the
massive pair domination

$\displaystyle\Delta_{2}\equiv\frac{a_{2}}{a_{3}}=\frac{a_{{}_{R}}}{a_{\rm
end}}\approx\left(\frac{\rho^{i}_{{}_{M}}}{\rho^{f}_{{}_{M}}}\right)^{1/3}\approx\frac{1}{\pi}\left(\frac{45}{4}\right)^{1/3}\left(\frac{H^{2}_{\rm
end}M_{\rm pl}^{2}}{g_{*}T^{4}_{\rm RH}}\right)^{1/3},$ (78)

assuming the initial pair energy density $\rho^{i}_{{}_{M}}\approx\rho^{\rm
end}_{{}_{\Lambda}}\approx 3m^{2}_{\rm pl}H^{2}_{\rm end}$ (19) at the
beginning of the reheating epoch, and the final one
$\rho^{f}_{{}_{M}}\approx\rho_{{}_{R}}\approx 3m^{2}_{\rm pl}H^{2}_{\rm RH}$
at the end of the reheating epoch, in virtue of Eqs. (70) and (74).

##  V Observations to fix reheating temperature and mass scale

Following the method proposed by Ref. Mielczarek2011 ; Mielczarek2010 , we fix
the reheating temperature by the CMB observations. Considering the
cosmological evolution of the physical wavelength $\lambda(a)$ and wavenumber
$k(a)$

$\displaystyle\lambda(a)=\lambda_{0}\frac{a}{a_{0}},\quad
k(a)=k_{0}\frac{a_{0}}{a}\quad\lambda(a)=1/k(a),$ (79)

where the comoving wavenumber $k_{0}=k(a_{0})$ and wavelength
$\lambda_{0}=1/k_{0}$ are constants in the cosmological evolution. The total
increase of the scale factor from the horizon crossing $a_{4}$ to the present
measurement $a_{0}=1$,

$\displaystyle\Delta_{\rm
tot}=\frac{a_{0}}{a_{4}}=\frac{\lambda_{0}}{\lambda(a_{4})}=\frac{(H)_{\rm
cross}}{k_{0}},$ (80)

where $\lambda(a_{4})=(H^{-1})_{\rm cross}$, or $k_{0}=a_{4}(H)_{\rm cross}$
and $(H)_{\rm cross}$ is the value of the Hubble parameter when the mode
$\lambda(a)=1/k(a)$ crossed the horizon at the scale factor $a_{4}$ during the
inflation or pre-inflation epoch, see Fig. 1. As an example, the pivot scale
$k_{0}=k_{*}=0.\,05\,({\rm Mpc})^{-1}$ for the CMB observations Planck2018 ,
correspondingly to the scale factor $a_{4}=a_{*}$ and $(H)_{\rm cross}=H_{*}$
(14) at the inflation scale.

Figure 8:  (Color Online). Fixed the observed spectral index $n_{s}=0.965$, in
terms of the tensor-to-scalar ratio $r$, we plot the inflation end $e$-folding
number $N_{\rm end}$ (16) and the reheating temperature $T_{\rm RH}$ (84).
These plots refer to their lower limits, due to the nature of inequality (16).
The real values of $N_{\rm end}$ and $T_{\rm RH}$ should be slightly above the
curves for a given $r$ value. The left plot $N_{\rm end}$ is the supplement to
Fig. 2, since the same equation (16) is used.

###  V.1 Reheating temperature and entropy vs tensor-to-scalar ratio
$r\lesssim 0.048$

Using the scalar spectrum $\Delta^{2}_{\mathcal{R}}$ (12) and the total scale
factor $\Delta_{\rm tot}$ (80) at the CMB pivot scale $k_{*}$, one arrives at

$\displaystyle\Delta_{\rm tot}=\frac{M_{\rm
pl}}{k_{*}}\sqrt{\pi\epsilon_{*}A_{s}}=\frac{M_{\rm
pl}}{\sqrt{2}~{}k_{*}}\sqrt{\pi(1-n_{s})A_{s}}.$ (81)

On the other hand, as illustrated in Fig. 1, $\Delta_{\rm
tot}=\Delta_{3}\Delta_{2}\Delta_{1}\Delta_{0}$, where
$\Delta_{3}=(a_{3}/a_{4})$ (18) and $\Delta_{2}=(a_{2}/a_{3})$ (78) are
computed, whereas $\Delta_{1}=(a_{1}/a_{2})=(g_{*}/2)^{1/3}(T_{\rm RH}/T_{\rm
rec})$ and $\Delta_{0}=(a_{0}/a_{1})=1+z_{\rm rec}$ are given in terms of the
temperature $T_{\rm rec}=T_{\rm CMB}(1+z_{\rm rec})$ and redshift $z_{\rm
rec}$ at the recombination Mielczarek2011 ,

$\displaystyle\Delta_{1}\Delta_{0}\equiv\frac{a_{0}}{a_{2}}=\frac{a_{0}}{a_{{}_{R}}}\approx(g_{*}/2)^{1/3}(T_{\rm
RH}/T_{\rm CMB}),$ (82)

as a result,

$\displaystyle\Delta_{\rm tot}=e^{N_{\rm
end}}\frac{1}{\pi}\left(\frac{45}{4}\right)^{1/3}\left(\frac{H^{2}_{\rm
end}M_{\rm pl}^{2}}{g_{*}T^{4}_{\rm RH}}\right)^{1/3}\frac{T_{\rm RH}}{T_{\rm
CMB}}\left(\frac{g_{*}}{2}\right)^{1/3}.$ (83)

Equations (81) and (83) are independent of the effective reheating degeneracy
$g_{*}$ and yield the reheating temperature

$\displaystyle\frac{T_{\rm RH}}{M_{\rm pl}}$ $\displaystyle=$
$\displaystyle\left(\frac{45}{2^{3/2}}\right)\frac{e^{3N_{\rm
end}}}{\pi^{9/2}}[(1-n_{s})A_{s}]^{-3/2}\left(\frac{k_{*}}{T_{\rm
CMB}}\right)^{3}\left(\frac{H_{\rm end}}{M_{\rm pl}}\right)^{2},$ (84)

in terms of the CMB observations $T_{\rm CMB}=2.725~{}{\rm K}=2.348\times
10^{-4}$ eV and $k_{*}=0.05{\rm Mpc}^{-1}$ ( ${\rm Mpc}^{-1}=6.39\times
10^{-30}$eV), as well as and $N_{\rm end}$ (16) and $H_{\rm end}$ (17), whose
values depend on the CMB measurements $A_{s}$, $n_{s}$ and $r$, see Sec. II.3.

Given the observed the scalar amplitude $A_{s}=2.1\times 10^{-9}$ and spectral
index $n_{s}=0.965$, the inflation ending $e$-folding number $N_{\rm end}$
(16) and the reheating temperature $T_{\rm RH}$ (84) are plotted in Fig. 8 as
functions of the tensor-to-scalar ratio $r$ without any free parameter. Figure
8 shows that their values are $N_{\rm end}\approx(50,60)$ and $T_{\rm
RH}/M_{\rm pl}\approx(5.5\times 10^{-13},1.1\times 10^{0})$ in the range
$r\approx(0.037,0.052)$. This $r$ range is consistent with the observational
constrain on the upper limit of the the tensor-to-scalar ratio $r<0.1$ or
$r<0.065$ Planck2018 .

Figure 9:  (Color Online). Fixed the observed spectral index $n_{s}=0.965$, as
functions of the tensor-to-scalar ratio $r$, we plot (Left) the inflation
scale $H_{*}$ (14) (blue), inflation end scale $H_{\rm end}$ (17) (green), and
reheating scale $H_{\rm RH}$ (74) (orange) in unit of the Planck scale $M_{\rm
pl}$; (Right) the entropy $S_{\rm patch}$ (87) within the physical patch
$H^{-3}_{\rm RH}$ at the reheating end $a_{{}_{R}}$. Note that $H_{\rm
RH}\propto g_{*}^{1/2}$ and $S_{\rm patch}\propto g_{*}^{-5/2}$ depend on the
effective degeneracy $g_{*}$ of relativistic particles in reheating. Here we
adopt $g_{*}\simeq 10^{2}$ for the standard model of particle physics,
including sterile neutrinos.

After obtaining the reheating temperature at the end $a_{{}_{R}}$ of
reheating, we calculate the entropy $S_{\rm patch}$ produced within the
physical patch of the volume $H^{-3}_{\rm RH}$, which evolves from the initial
patch of the volume $H^{-3}_{*}$ at the start $a_{*}=1$ of inflation. The
patch grows by a scale factor of Eqs. (18) and (78),

$\displaystyle
a^{3}_{{}_{R}}=(\Delta_{3}\Delta_{2})^{3}\approx\frac{45e^{3N_{\rm
end}}}{4\pi^{3}}\left(\frac{H^{2}_{\rm end}M_{\rm pl}^{2}}{g_{*}T^{4}_{\rm
RH}}\right)=e^{3N_{\rm end}}\left(\frac{H^{2}_{\rm end}}{H^{2}_{\rm
RH}}\right).$ (85)

The entropy per comoving volume $S_{R}$ (76) at the end of reheating can be
expressed as,

$\displaystyle S_{R}$ $\displaystyle\approx$ $\displaystyle 2.2\times
10^{6}\left(\frac{16\pi^{3}}{135}\right)^{1/4}\frac{3a^{3}_{{}_{R}}}{8\pi}H^{2}_{\rm
end}M_{\rm pl}.$ (86)

where we use $\rho^{i}_{{}_{M}}(a_{i})\approx\rho^{\rm
end}_{{}_{\Lambda}}\approx 3m^{2}_{\rm pl}H^{2}_{\rm end}$ (78) and the
constrain (89) below. The entropy $S_{\rm patch}$ within the physical patch
$H^{-3}_{\rm RH}$ is given by,

$\displaystyle S_{\rm patch}=H^{-3}_{\rm RH}S_{R}$ $\displaystyle\approx$
$\displaystyle 3.64\times 10^{5}~{}e^{3N_{\rm end}}\left(\frac{H^{4}_{\rm
end}M_{\rm pl}}{H^{5}_{\rm RH}}\right),$ (87)

which is a function of $r$ and $g_{*}$. In Fig. 9 (left), we plot $S_{\rm
patch}$ by using $N_{\rm end}$ (16), $H_{\rm end}$ (17) and $H_{\rm RH}$ (74).
It shows that the calculated entropy accords with the observational vale
$S_{\rm patch}\sim 10^{88}$ around $r\sim 0.045$. To have a better
understanding how the physical patch horizon $H_{*}>H_{\rm end}>H_{\rm RH}$
evolves, we plot in the same Fig. 9 (right) all characteristic Hubble scales
from the inflation to the reheating: the inflation scale $H_{*}$ (14),
inflation end scale $H_{\rm end}$ (17), and reheating scale $H_{\rm RH}$ (74)
in unit of the Planck scale $M_{\rm pl}$. It shows that the unphysical
situation $H_{\rm RH}>H_{\rm end}$ occurs when $r>0.047$. Therefore the
$r>0.047$ range should be excluded and this theoretical upper limit is
consistent with the observational one $r<0.065$ Planck2018 . Due to the
dependence of $H_{\rm RH}$ on $g_{*}$ and the approximations adopted in these
preliminary calculations, we conservatively suggest a theoretical upper limit
of the tensor-to-scalar ratio $r\lesssim 0.048$, which demands elaborated
numerical calculations to precisely fix it.

These results show that the model $\tilde{\Lambda}$CDM we have been discussing
so far is in accordance with observations. This implies that the precisely
measuring $r$-value is uniquely not only to determine the $e$-folding number
of the inflation, the reheating temperature, all characteristic Hubble scales
and produced entropy, but also the tensor-to-scalar ratio upper and lower
limits. The theoretical lower limit will be obtained in the next section.

Figure 10:  (Color Online). Using the observed spectral index $n_{s}=0.965$,
we plot the mass parameter $\hat{m}/M_{\rm pl}$ (left) and the effective
Yukawa coupling $(g_{{}_{Y}}^{2}/g^{1/2}_{*})$ (right), as functions of the
tensor-to-scalar ratio $r$ in the same range $(0.042,0.048)$ where the
physically sensible values $N_{\rm end}$ and $T_{\rm RH}$ are also plotted in
Fig. 8.

###  V.2 Reheating $\rho_{{}_{R}}\gg\rho_{{}_{\Lambda}}$ and lower limit of
tensor-to-scalar ratio $r\gtrsim 0.042$

In the scenario $\tilde{\Lambda}$CDM, there are two parameters to describe the
properties of the reheating epoch: (i) the effective mass parameter
$\hat{m}/M_{\rm pl}$ physically represents spacetime produced pairs’ masses
and their degeneracies; (ii) the effective Yukawa coupling
$(g_{{}_{Y}}^{2}/g^{1/2}_{*})$ represents pairs’ decay strength to
relativistic particles of degeneracies $g_{*}$. We need at least two
independent observations to see their possible values and to check the self-
consistency and self-sufficiency of the scenario.

To determine these two parameters of the scenario $\tilde{\Lambda}$CDM, from
Eqs. (75) and (84), we obtain one constrain,

$\displaystyle\left(\frac{T_{\rm RH}}{M_{\rm
pl}}\right)^{2}=0.3~{}(g_{{}_{Y}}^{2}/g^{1/2}_{*})~{}(\hat{m}/M_{\rm pl}).$
(88)

Another constrain on these two parameters

$\displaystyle(g_{{}_{Y}}^{2}/g^{1/2}_{*})~{}(M_{\rm pl}/\hat{m})\approx
3.6\times 10^{-13},$ (89)

comes from the combination of the reheating temperature $T_{\rm RH}$ (75) and
the ratio $T_{\rm RH}/\hat{m}\approx 3.3\times 10^{-7}$ obtained from the
observed baryon number-to-entropy ratio $n_{B}/s$, which will be duly
discussed in another article xuebaryon .

In Fig. 10, we numerically plot the constrains (88) and (89) on two parameters
$\hat{m}/M_{\rm pl}$ and $(g_{{}_{Y}}^{2}/g^{1/2}_{*})$ as a function of the
tensor-to-scalar ratio $r$ in the range $(0.042,0.048)$. However, on the other
hand, we can constrain values of these two parameters from the theoretical
point view, so as to provide an insight into the lower limit of values
$r\not=0$, a priori  to observations. Recall that in Sec. IV.1.2 we point out
the theoretical threshold $\hat{m}>20m_{\rm pl}=4M_{\rm pl}$ (47) for
$\rho_{{}_{M}}\gg\rho_{{}_{\Lambda}}$ that is necessary for the Universe
evolution proceeding the standard cosmological scenario. Applying this
theoretical threshold to the constrains of Eqs. (88) and (89) or numerical
results Fig. 10, we definitely conclude that the tensor-to-scalar ratio
$r\not=0$ and conservatively suggest its lower limit should be around $0.042$
in the present preliminary calculations. More elaborated numerical
calculations are required to precisely determine the $r$ lower limit. The
preliminarily theoretical estimate $r$-range $[0.042,0.048]$ requires more
elaborated numerical analyses. Nevertheless this $r$-range is relevant to the
measurements by the next generation CMB observations, such as CMB-S4 which
measures $r\gtrsim 10^{-3}$ CMB-S4 . In Ref. newbound_r , the authors obtain
the most recent constrain the tensor-to-scalar ratio $r<0.044$ by using Planck
data.

In the theoretically estimated $r$-range $[0.042,0.048]$, it is shown that the
inflation $e$-folding number $58\gtrsim N_{\rm end}\gtrsim 54$ and the
reheating temperature $10^{-3}\gtrsim T_{\rm RH}/M_{\rm pl}\gtrsim 10^{-8}$
from the numerical results presented in Fig. 8; the inflation scale
$H_{*}/M_{\rm pl}\approx 4.0\times 10^{-6}$, inflation end scale $H_{\rm
end}/M_{\rm pl}\approx 1.5\times 10^{-6}$, whereas $10^{-14}\gtrsim H_{\rm
RH}/M_{\rm pl}\gtrsim 10^{-4}$ and the entropy $10^{120}\gtrsim S_{\rm
patch}\gtrsim 10^{76}$ from the numerical results presented in Fig. 9.

In the theoretically estimated $r$-range $[0.042,0.048]$, the effective pair
mass and degeneracy parameter $10^{-1}\lesssim\hat{m}/M_{\rm pl}\lesssim
10^{4}$ and the effective Yukawa coupling
$10^{-14}\lesssim(g_{{}_{Y}}^{2}/g^{1/2}_{*})\lesssim 10^{-9}$. If
$g_{*}\gtrsim 10^{2}$ for the standard model of particle physics, including
sterile neutrinos, $10^{-13}\lesssim g_{{}_{Y}}^{2}\lesssim 10^{-8}$. We check
back the parameters used in Figs. 6 and 7 are $(\hat{m}/m_{\rm pl})=27.7$ and
$g^{2}_{{}_{Y}}=10^{-9}$. This indicates that the model $\tilde{\Lambda}$CDM
we have been discussing so far is self consistent and self-contained as a
theoretical framework.

##  VI Massive cold dark matter $\Omega^{0}_{c}$

Among massive fermion-antifermion pairs purely gravitationally generated from
the spacetime, some of them possess no other or extremely weak gauge
interactions, except gravitational ones. Henceforth, they can be stable
against the decay to relativistic particles $\bar{\ell}\ell$ (24), and
possibly have lifetimes longer than the Universe life. Thus these stable
massive pairs can be candidates of massive cold dark matter in the
$\Lambda$CDM model and contribute to the value $\Omega^{0}_{c}$ observed
today. We consider that after produced at the reheating, stable massive pairs
evolve in two different possibilities: (i) they decouple from the spacetime
and evolve as a non-relativistic fluid; (ii) they couple with the spacetime
via ${\mathcal{S}}\Leftrightarrow\bar{F}F$ from the reheating to the present
time.

###  VI.1 Massive cold dark matter produced in reheating

The gravitational productions of massive bosonic particles in the reheating
can play an important role in explaining the massive cold dark matter
pairproductionCold ; CKR1998 ; CKR2019 ; parametric ; Ema2018 . Following this
line we consider the massive fermion pair energy density (10) produced in the
reheating, that can be in general decomposed to the unstable and stable pair
contributions:

$\displaystyle\rho^{H}_{{}_{M}}$ $\displaystyle\approx$ $\displaystyle 2\,\chi
H^{2}\hat{m}^{2};\quad\hat{m}^{2}\equiv\sum_{f}g_{d}^{f}m^{2}_{f}+\sum_{f^{\prime}}g_{d}^{f^{\prime}}m^{2}_{f^{\prime}}.$
(90)

The contributions from the stable pairs of masses $m_{f^{\prime}}$ and
degeneracies $g_{d}^{f^{\prime}}$ are

$\displaystyle\rho^{H{\rm cold}}_{{}_{M}}\equiv 2\,\chi\hat{m}^{2}_{\rm
cold}H^{2}=(\hat{m}^{2}_{\rm
cold}/\hat{m}^{2})\rho^{H}_{{}_{M}},\quad\hat{m}^{2}_{\rm
cold}\equiv\sum_{f^{\prime}}g_{d}^{f^{\prime}}m^{2}_{f^{\prime}}$ (91)

where the effective mass and degeneracy parameter $\hat{m}_{\rm cold}$ is
introduced for stable massive pairs and $\hat{m}_{\rm cold}<\hat{m}$. The
stable pair number density $n^{H{\rm cold}}_{{}_{M}}\approx\rho^{H{\rm
cold}}_{{}_{M}}/2\hat{m}_{\rm cold}$. Recall that these stable massive pairs
follow the detailed balance solution (60), namely $\rho^{\rm
cold}_{{}_{M}}\approx\rho^{{\rm cold}H}_{{}_{M}}\approx\chi\hat{m}^{2}_{\rm
cold}H^{2}_{\rm RH}$ at the reheating.

After these stable pairs produced, they couple with the spacetime in the
$\mathcal{M}$-episode, see Sec. IV.2.1, and then decouple in the
$\mathcal{R}$-episode of the reheating epoch. From the reheating epoch to the
present time, they evolve as a non-relativistic fluid and conserve their
numbers

$\displaystyle n^{\rm cold}_{{}_{M}}(a_{0})a^{3}_{0}=n^{\rm
cold}_{{}_{M}}(a_{{}_{R}})a^{3}_{{}_{R}},$ (92)

per comoving volume. In connection with observations, the relic abundance of
Eq. (92) can be defined as

$\displaystyle\Omega^{0}_{c}\equiv\frac{\rho^{\rm
cold}_{{}_{M}}(a_{0})a^{3}_{0}}{\rho_{c}^{0}a^{3}_{0}}=\Omega^{0}_{{}_{R}}\frac{\rho^{\rm
cold}_{{}_{M}}(a_{0})a^{3}_{0}}{\rho^{0}_{{}_{R}}a^{3}_{0}},$ (93)

where $\Omega^{0}_{{}_{R}}=\rho^{0}_{{}_{R}}/\rho_{c}^{0}$, the radiation
energy density $\rho^{0}_{{}_{R}}$ and the critical density
$\rho_{c}^{0}=3m^{2}_{\rm pl}H^{2}_{0}$ corresponding to the present horizon
$H_{0}$ and scale factor $a_{0}$.

On the other hand, the entropy $S(a)=s(a)a^{3}$ per comoving volume from the
reheating epoch to the present time is conserved,

$\displaystyle S_{R}=S_{0};\quad
S_{0}=\frac{2\pi^{2}}{45}g^{*}_{0}a^{3}_{0}T^{3}_{0},\quad
S_{R}=\frac{2\pi^{2}}{45}g^{*}_{{}_{R}}a^{3}_{{}_{R}}T^{3}_{\rm RH}$ (94)

and the present temperature $T_{0}=T_{\rm CMB}=2.348\times 10^{-4}$ eV,
yielding

$\displaystyle(g^{*}_{0})^{1/3}a_{0}T_{0}=(g^{*}_{{}_{R}})^{1/3}a_{{}_{R}}T_{\rm
RH},$ (95)

where $g^{*}_{0}$ and $g^{*}_{{}_{R}}\equiv g_{*}$ are the effective
degeneracies of relativistic particles respectively at the present time and
reheating. Using the relationship (70) of radiation entropy and energy
density, the conservations of particle number (92) and entropy (95), we obtain
the relic abundance of the cold dark matter,

$\displaystyle\Omega^{0}_{c}=\Omega^{0}_{{}_{R}}\left(\frac{4}{3}\frac{\hat{m}_{\rm
cold}}{T_{0}}\right)\frac{n^{\rm
cold}_{{}_{M}}(a_{{}_{R}})a^{3}_{{}_{R}}}{S_{R}(a_{{}_{R}})}.$ (96)

Moreover, using the entropy $S_{R}$ (76) per comoving volume and the constrain
(89) from the observed baryon number-to-entropy ratio, we have,

$\displaystyle\frac{n_{{}_{M}}(a_{{}_{R}})a^{3}_{{}_{R}}}{S_{R}(a_{{}_{R}})}\approx\frac{n^{H}_{{}_{M}}(a_{{}_{R}})a^{3}_{{}_{R}}}{S_{R}}\approx
4.55\times 10^{-7}\left(\frac{135}{16\pi^{3}}\right)^{1/4}.$ (97)

As a result, the cold dark matter relic abundance is given by

$\displaystyle\Omega^{0}_{c}=4.4\times
10^{-7}~{}\Omega^{0}_{{}_{R}}\left(\frac{\hat{m}_{\rm cold}}{T_{0}}\right).$
(98)

The observed values $\Omega^{0}_{c}h^{2}_{0}\approx 0.12$,
$\Omega^{0}_{{}_{R}}h^{2}_{0}\approx 4.31\times 10^{-5}$ and baryon component
$\Omega^{0}_{b}h^{2}_{0}\approx 2.24\times 10^{-2}$ Planck2018 , the factor
$h^{2}$ here comes from the convention $H_{0}=100h_{0}({\rm km/sec/Mpc})$. The
result (98) can possibly be of right order of magnitude, if the effective mass
and degeneracy parameter of the stable massive pairs $\hat{m}_{\rm
cold}/T_{0}\sim{\mathcal{O}}(10^{10})$. It is not physically expected that the
composition of stable pairs in total pairs produced is so small, $\hat{m}_{\rm
cold}\ll\hat{m}$, where $\hat{m}/M_{\rm pl}\sim 10$, see Sec. V.2 and Fig. 10
(left).

###  VI.2 Massive cold dark matter couples to the spacetime in its evolution

We turn to another possibility that after produced at the reheating, the
stable massive pairs always couple with the spacetime via the process
${\mathcal{S}}\Leftrightarrow\bar{F}F$ (21), and follow the cosmic rate
equation (43), rather than the free expansion of a non-relativistic fluid.

For the matter dominate case $H^{2}\approx(8\pi/3M_{\rm pl})\rho_{{}_{M}}$, we
show in Sec. IV.2.1 the detailed balance solution $\rho^{\rm
cold}_{{}_{M}}\approx 2\chi\hat{m}^{2}_{\rm cold}H^{2}$ (60) to the cosmic
rate equation (43). However, in the radiation dominate epoch after the
reheating, one needs to find the numerical solution to the close set of Eqs.
(30)-(33). We do not attempt to find the entire evolution history of the cold
dark matter $\rho^{\rm cold}_{{}_{M}}$ in this article. Instead, we focus on
its evolution in the more recent epoch when the radiation component is
negligible and the cosmological term starts to dominate over matter. In this
case, Equations (30)-(33) can be approximately recasted as

$\displaystyle h^{2}$ $\displaystyle=$
$\displaystyle\Omega_{{}_{\Lambda}}+\Omega^{\rm cold}_{{}_{M}},$ (99)
$\displaystyle\frac{dh^{2}}{dx}$ $\displaystyle=$ $\displaystyle-3\Omega^{\rm
cold}_{{}_{M}},$ (100) $\displaystyle\frac{d\Omega^{\rm
cold}_{{}_{M}}}{dx}+3\Omega^{\rm cold}_{{}_{M}}$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{M}}{H}\left(\Omega_{{}_{M}}^{H{\rm
cold}}-\Omega^{\rm cold}_{{}_{M}}\right),$ (101)

in unit of the critical density $\rho_{c}^{0}=3m^{2}_{\rm pl}H^{2}_{0}$ at the
present horizon $H_{0}$ and scale factor $a_{0}$. Based on the fact that the
cosmological term $\Omega_{{}_{\Lambda}}$ very slowly varies and
$\Omega_{{}_{\Lambda}}\gtrsim\Omega_{{}_{M}}$, the approximate solution to
Eqs. (99-101) can be found at this preliminarily stage,

$\displaystyle\Omega^{\rm cold}_{{}_{M}}\approx\Omega_{{}_{M}}^{H{\rm
cold}}=\frac{16\pi}{3}\chi\left(\frac{\hat{m}}{M_{\rm
pl}}\right)^{2}h^{2};\quad h^{2}=\left(\frac{H}{H_{0}}\right)^{2},$ (102)

similarly to the detailed balance solution (60). As a result, the cold dark
matter relic abundance at the present time is

$\displaystyle\Omega^{0}_{c}\approx\frac{16\pi}{3}\chi\left(\frac{\hat{m}_{\rm
cold}}{M_{\rm pl}}\right)^{2},\quad\chi\approx 1.85\times 10^{-3}.$ (103)

Recall the discussions on the ratio $\hat{m}/M_{\rm pl}\approx 5\sim 10$
around the range of the tensor-to-scalar ratio $r\approx 0.044$, see Sec. V.2
and Fig. 10 (left). Thus we expect that the compositions of stable pairs and
unstable pairs are comparable, i.e., $\hat{m}_{\rm cold}\lesssim\hat{m}$, and
the cold dark matter relic abundance (102) can be of right order of magnitude,
compared with observed value $\Omega^{0}_{c}\approx 0.3$.

We further numerically find the time-varying solution to Eqs. 99,100 and 101
by considering slow $\Omega_{{}_{\Lambda}}$ variation in time from the
radiation/matter dominate epoch $\Omega_{{}_{R,M}}\gg\Omega_{{}_{\Lambda}}$ to
the present epoch $\Omega_{{}_{M}}\sim\Omega_{{}_{\Lambda}}$. The solution is
consistent with the generalized Friedmann equation xueNPB2015

$\displaystyle h^{2}$ $\displaystyle\approx$
$\displaystyle\Omega^{0}_{{}_{M}}(1+z)^{3-\delta_{{}_{M}}}+\Omega^{0}_{{}_{\Lambda}}(1+z)^{\delta_{{}_{\Lambda}}},$
(104)

where $\Omega^{0}_{{}_{M}}$ and $\Omega^{0}_{{}_{\Lambda}}$ are respectively
present values of matter and cosmological terms. The effective parameters
$|\delta_{{}_{M}}|\ll 1$ and $|\delta_{{}_{\Lambda}}|\ll 1$
($\delta_{{}_{M}}\delta_{{}_{\Lambda}}>0$) describe the tiny coupling between
$\Omega_{{}_{\Lambda}}$ and $\Omega_{{}_{M}}$ via the $\rho_{{}_{M}}^{H{\rm
cold}}$ (91) and cosmic rate equation (101). Among the radiation and matter
content $\Omega_{{}_{R,M}}$, the massive cold dark matter
$\Omega_{{}_{M}}^{\rm cold}$ plays the main role in interacting with
$\Omega_{{}_{\Lambda}}$. The reason is that in the cosmic rate equation (101),
the rate $\Gamma_{M}\propto m$ (9) and density $\rho_{{}_{M}}^{H{\rm
cold}}\propto m^{2}_{\rm cold}$ (91) are proportional to the masses of
superheavy cold dark matter particles. In the evolution epoch when
$\Omega_{{}_{M}}>\Omega_{{}_{\Lambda}}$, the cosmic rate equation (101) shows
that $\Omega_{{}_{M}}$ decreases faster than $(1+z)^{3}$ ($\delta_{{}_{M}}<0$)
and $\Omega_{{}_{\Lambda}}$ increases ($\delta_{{}_{\Lambda}}<0$) from large
redshifts to low redshifts. This implies the cold dark matter annihilation to
the spacetime ($\Omega_{{}_{\Lambda}}$). The more details will be presented in
a separated paper. On the other hand, the data analysis show the generalized
Friedmann equation (104) has the better fit to observational data than the
$\Lambda$CDM ($\delta_{{}_{M}}=\delta_{{}_{\Lambda}}=0$) clement , and greatly
relieved the $\Lambda$CDM $H_{0}$ tensions zhangxin .

In this article, we leave these two possibilities of cold dark matter
evolution as open questions for further discussions, since more theoretical
studies are required and the contributions from bosonic candidates for cold
dark matter particles have not been considered. We end this section by adding
the following comments. Among relativistic particle $\bar{\ell}\ell$ from
unstable massive pairs decay, there are massive and massless sterile particles
of weak interacting with other SM particles. The smaller their interacting
strengths are, the earlier they decouple from the thermal equilibrium bath of
relativistic particles, and freely evolve as relativistic or non-relativistic
fluids, accounting for the catalogue of warm dark matter in $\Lambda$CDM. We
cannot determine their properties in this article.

##  VII Summary and remark reheating epoch

We would like to make a summary to close this section. After the
$\rho_{{}_{\Lambda}}$-dominated inflation $H>\Gamma_{M}$, where the pair
production density $\rho^{H}_{{}_{M}}$ is negligible, the reheating epoch
starts $\Gamma_{M}\gtrsim H$, $\rho^{H}_{{}_{M}}$ becomes large and pair
annihilation to spacetime and decay to relativistic particles are important.
Therefore the cosmic rate equation (26) of the Boltzmann type governing the
processes $\bar{F}F\Leftrightarrow{\mathcal{S}}$ and
$\bar{F}F\rightarrow\bar{\ell}\ell$ are relevant. This is anothor dynamical
equation in addition to two Einstein equations (3,4) and the reheating
equation (29) from the energy conservation.

Described by the horizon $H$ and three cosmological constitutions
$\rho_{{}_{\Lambda,M,R}}$, the reheating epoch is determined by a close system
of four dynamical equations (30-33) and pair production density
$\rho^{H}_{{}_{M}}$ (34), production rate $\Gamma_{M}$ and decay rate
$\Gamma^{{}^{\rm de}}_{M}$ (35). Numerically solving this system, we find
three characteristic episodes:

  1. (i)

the ${\mathcal{P}}$-episode of the transition from the inflation end to the
reheating start, the pair-production rate is much larger than the Hubble rate
($\Gamma_{M}\gg H$), the cosmological energy density $\rho_{{}_{\Lambda}}$
quickly decreases and converts to the matter-energy density $\rho_{{}_{M}}$ of
massive pairs produced, as a consequence
$\rho_{{}_{\Lambda}}\ll\rho_{{}_{M}}$;

  2. (ii)

the ${\mathcal{M}}$-episode of massive pairs domination and their back and
forth interaction to the spacetime ($F\bar{F}\Leftrightarrow{\mathcal{S}}$),
when the cosmic rate equation plays an essential role in build a local
chemical equilibrium of the quantum-number and energy equipartition between
massive pairs and spacetime;

  3. (iii)

the ${\mathcal{R}}$-episode of the genuine reheating
$\rho_{{}_{R}}\gg\rho_{{}_{M}}$ through massive pairs predominately decaying
to relativistic particles quickly thermalised, the reheating temperature and
entropy are computed in terms of the CMB measured scalar amplitude $A_{s}$ and
spectral index $n_{s}$, as well as the tensor-to-scalar ratio $r$.

As a result, using the CMB measurements at pivot scale $k_{*}=0.05({\rm
Mpc})^{-1}$ and the Hubble scale $H_{*}$ at the beginning of inflation, we
calculate the Hubble scale $H_{\rm end}$ and energy densities $\rho^{\rm
end}_{{}_{\Lambda,M}}$ at the end of inflation after $e$-folding number
$N_{\rm end}$. These are treated as initial conditions for studying the
reheating epoch, and we calculate the Hubble scale $H_{\rm RH}$, temperature
$T_{\rm RH}$ and entropy $S_{\rm patch}$ at the genuine reheating episode.
Based on the reheating temperature (84) determined by CMB measurements and the
observed baryon number-to-entropy ratio, we consistently constrain the
effective pair mass parameter $(\hat{m}/M_{\rm pl})$ and the effective Yukawa
coupling $(g_{{}_{Y}}^{2}/g^{1/2}_{*})$ in the theoretical framework of
$\tilde{\Lambda}$CDM. Thus no free parameter is adjustable.

We present these results as functions of the tensor-to-scalar ratio $r$, and
show they are in accordance with observations. As a result, the final results
are expressed as functions of the tensor-to-scalar ratio $r$ and their
numerical values are in accordance with the CMB observations so far. Moreover,
from purely theoretical points of views, we preliminarily limit the $r$ values
in the range $0.042\lesssim r\lesssim 0.048$. The upper limit is consistent
with observations, while the lower limit shows $r\not=0$ definitely, to be
fixed by next-generation experiments, such as CMB-S4 CMB-S4 .

There are two kinds of massive fermion-antifermion pairs gravitationally
produced: (i) unstable pairs that couple and decay to relativistic particles;
(ii) stable pairs that do not couple to other particles except gravity and
possibly have a lifetime longer than the Universe. Considering stable massive
pairs produced in the reheating and following the cosmic rate equation (43)
for the back and forth processes ${\mathcal{S}}\Leftrightarrow\bar{F}F$, we
discuss the sable massive pairs as cold dark matter candidates and calculate
their relic abundance in connection with observations. We preliminarily find
that in the matter dominate epoch the cold dark matter-energy density
$\rho^{\rm cold}_{{}_{M}}\propto H^{2}$ in its evolution. More detailed
analyses are required.

We would like to mention that at the reheating start
$\rho_{{}_{\Lambda}}\gg\rho_{{}_{M}}\gg\rho_{{}_{R}}$, the rapid converting
process $\rho_{{}_{\Lambda}}\Rightarrow\rho_{{}_{M}}\Rightarrow\rho_{{}_{R}}$
leads to $\rho_{{}_{\Lambda}}\ll\rho_{{}_{M}}\ll\rho_{{}_{R}}$, and most
relevant mass-energy and entropy of Universe had been produced by the end of
reheating. This violent dynamical process could lead to the emission of
primordial gravitational wave, see review RevReheating ; Maggiore1999 ;
Guzzetti2016  and references therein.

After the reheating epoch, the radiation dominated and entropy conserved the
epoch of the standard cosmology starts. In standard cosmology epoch, the pairs
production (7) and (8) become small and their contributions to the Hubble
scale $H$ and its variation are strongly suppressed and negligible, compared
with the matter produced in the reheating. However, the pair’s production
provides an indirect interaction between cosmological energy density
$\rho_{{}_{\Lambda}}$ and matter density $\rho_{{}_{M}}$ in evolution, which
can possibly account for the cosmological coincidence xue2020MPLA .

In this article, we have mainly addressed basic issues in the reheating epoch
and cold dark matter candidates and presented some preliminary analysis and
results. Further studies are still required and more elaborately numerical
computations are very inviting, based on ongoing observations. Nevertheless,
we expect that this theoretical scenario and present results provide some
further understandings of the phenomena of the reheating and cold dark matter
in the standard model $\Lambda$CDM for the modern cosmology.

##  VIII Acknowledgment

Author thanks Drs. Yu Wang and Rahim Moradi for indispensable numerical
assistances of using Python and mathematics.

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