# Spontaneous baryogenesis and generation of gravitational waves in a new
model of quintessential $\alpha$-attractor

Arunoday Sarkar<EMAIL_ADDRESS>Centre of Advanced Studies,
Department of Physics, The University of Burdwan,
Burdwan 713 104, India  Buddhadeb Ghosh<EMAIL_ADDRESS>Centre of
Advanced Studies, Department of Physics, The University of Burdwan,
Burdwan 713 104, India

(August 28, 2024)

###### Abstract

We apply a new version of quintessential $\alpha$-attractor inflaton potential
(arXiv: 2305.00230 [gr-qc]) to make a detailed analysis of the phenomenon of
_spontaneous baryogenesis_ in the post-inflationary kination period. In this
context, we compute various temperatures _viz.,_ the temperature at the end of
inflation (or start of kination) ($T_{\mathrm{end}}$), maximum thermalization
temperature ($T_{\mathrm{th}}$), freeze-out temperature ($T_{F}$), the
temperature of radiation domination ($T_{\mathrm{rad}}$), their respective
number of e-folds, the densities _viz.,_ total field energy density
($\rho_{\mathrm{kin}}$), radiation density ($\rho_{\mathrm{rad}}$) and the
freeze-out values of baryon-to-entropy ratio ($\eta_{F}$) at different mass
scales ($M$, $\tilde{M}$), as functions of $\alpha$. The numerical
calculations show that the baryon-to-entropy ratio is obtained as: for
$\alpha=0.29-0.30$, $\eta_{F}=8.7\times 10^{-11}-8.5\times 10^{-11}$ and for
$\alpha=4.2-4.3$, $\eta_{F}=8.5\times 10^{-11}-8.6\times 10^{-11}$. The
results are found to satisfy experimental bounds quite satisfactorily. We also
find a blue-tilted spectrum of relic gravitational waves (GW) of frequencies
lying in two narrow bands corresponding to the two regions of $\alpha$,
indicated above, _viz.,_ $f_{\mathrm{end}}=2.00\times 10^{10}-2.10\times
10^{10}$ Hz for lower region of $\alpha$ and $f_{\mathrm{end}}=2.12\times
10^{10}-2.11\times 10^{10}$ Hz for higher region of $\alpha$, during
transition from inflation to kination, which is supported by current
literature. The present-day peak values of the amplitudes of GWs emitted
during radiation domination,
($(\Omega_{\mathrm{GW},0}^{(\mathrm{RD})})_{\mathrm{peak}}$), are found to be
$\sim 10^{-7}$, which is consistent with the requirement for nucleosynthesis
and the associated root-mean-square values,
($\Omega_{\mathrm{GW},0}^{(\mathrm{RD})}\sim 10^{-18}$), conform to the
characteristic strain of the ongoing GW-detectors. In this way, a unified
picture of the roles of quintessential $\alpha$-attractor model, considered
here, in inflation, quintessence, spontaneous baryogenesis and the
gravitational waves production, emerges from the present study.

††preprint: APS/123-QED

##  I Introduction

The matter-antimatter asymmetry is an intriguing phenomenon in the frontiers
of particle physics and cosmology. There is no known exact mechanism to
generate the tiny baryon-to-entropy ratio $\eta\approx(8.6\pm 0.1)\times
10^{-11}$ [1, 2, 3, 4], which is a crucial requirement for big bang
nucleosynthesis (BBN) and is almost fixed from the early universe to the
present day. One can imagine that, due to some exotic process in the early
universe, baryon number conservation was violated, which led to the excess
number densities of particles over those of anti-particles. This process is
popularly known as Baryogenesis. In 1967, Andrei Sakharov proposed three
criteria [5], based on symmetry principles, in order to achieve the baryon
asymmetry: i) Baryon number ($B$) violation, ii) Charge conjugation ($C$) and
charge conjugation-parity ($CP$) violations and iii) thermal non-equilibrium
in presence of charge conjugation-parity-time reversal ($CPT$) invariance. The
first condition is trivial in order to achieve a baryon asymmetric universe
with $\Delta B\neq 0$. Since any stipulation of $C$ and $CP$ invariance would
be compatible with the $B$ invariance, $C$ and $CP$ violations are essentially
needed to ensure $B$ violation [6]. This is Sakharov’s second criterion. Now,
to maintain the excess of baryons the rate of $B$ violation should be very
small compared to the expansion rate of the universe, otherwise the $CPT$
invariance and hence the $T$ violation would bring back the universe in a
state of $\Delta B=0$, through thermal equilibrium. It is only possible by the
‘ _departure from thermal equilibrium_ ’, which is Sakharov’s third condition.
The last condition is important because in a phase of matter-antimatter
symmetry, no preferred direction of time can be made, but thermal out-of
equilibrium sets a unique direction of flow of time from the early universe to
the present day.

The Standard Model (SM) of particle physics can accommodate large $B$
violation [7] at high temperature by sphaleron transition [8] between the
degenerate $SU(2)$ vacua and very small CP violation [6]. However, it can not
meet Sakharov’s third criterion, as this requires a _Strong First Order
Electroweak Phase Transition_ (SFOEWPT) [9, 10, 11, 12], which is not possible
to achieve in SM [13, 14, 15] for experimentally favored Higgs mass [16, 17].

This was already evident in the large value ($\sim 10^{12}$) [7] of the ratio
($B$ violation rate/universe expansion rate), calculated in the SM. However, a
small value of the ratio consistent with the SFOEWPT can be obtained in a
number of _Beyond Standard Models_ (BSMs) (for a review, see Ref. [9]), at the
TeV scale.

Another angle of comparison between the SM and the BSMs is the fact that in
the former the Higgs mass is unstable against the bosonic and the fermionic
loop corrections, whereas in the latter it is not, due to the cancellation
effects. These theoretical findings make the situation somewhat confusing
because the experimental Higgs (mass $\sim 126$ GeV) in the LHC experiments
[18, 19] is SM-like. Only future collider experiments might resolve some of
these issues by either proving or disproving the BSMs.

An interesting alternative approach is to relax Sakharov’s third criterion by
invoking an effective formulation beyond conventional particle physics models
in order to generate the required baryon asymmetry in a scenario of thermal
equilibrium à la spontaneous baryogenesis (SB) [20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 4, 33, 34, 35, 36, 37, 38, 39, 40, 41]. This approach
is highly non-trivial because, generically, the average value of baryon number
density in thermal equilibrium is zero. However, a non-zero value can be
obtained even in thermal equilibrium, by the spontaneous breaking of the CPT
invariance [20] due to the kinetic energy of a rolling scalar field $\phi$.
This is particularly seen to manifest through the derivative coupling of the
scalar field with a non-conserving baryon current $J^{\mu}_{b}$ as
$\partial_{\mu}\phi J^{\mu}_{b}$ [4, 29]. The $\phi$ field acts as a pseudo-
Nambu-Goldstone boson [42, 43] in an approximate $U(1)$ symmetry [20, 44, 43].
(See for related studies in particle physics, [22, 45] and in inflationary
cosmology, [44, 43, 46].)

Nowadays, in the interface of particle physics and cosmology, this scenario is
popularly known as quintessential baryogenesis [47, 48, 49, 50, 51, 52] in the
paradigm of inflationary quintessence [53], since the idea of rolling scalar
field is inbuilt in this kind of inflationary framework. Quite interestingly,
here, the scalar field playing the roles of inflaton and quintessence, can
also trigger the process of baryogenesis. The quintessential inflaton field
when traverses through the kination regime [54] after inflation, particle
production starts to reheat the universe either gravitationally [55, 56, 57,
58, 59, 60] or through instant preheating (IP) [61, 62, 63]. These particles
interact with the scalar field leading to SB at a temperature between $\sim
10^{8}-10^{12}$ GeV, which is higher than that of radiation era ($\sim 10^{7}$
GeV). In this course, the $\phi$-field spends longer time in the kination
period than in inflation. Consequently, thermal equilibrium manifests before
radiation fills the universe. The rate of baryon non-conserving process
decreases, as the universe cools down due to Hubble expansion, which ceases to
act below a certain temperature $T_{F}$, called the freeze-out temperature.
The temperature at which baryogenesis takes place, should be above $T_{F}$
otherwise the CPT symmetry could be restored and baryon asymmetry would not
occur. The actual value of $T_{F}$ and the energy scale of baryon non-
conservation depend on the specific physical processes involved, provided that
the back reactions of the reheating particles on the scalar field dynamics are
negligibly small [64], ensuring that the field velocity $\partial_{0}\phi$
does not vanish.

The most clear evidence of quintessential SB scenario in the present and
future experiments would be the detection of the relic gravitational waves,
originated from the quantum fluctuations during inflation at sub-Hubble scale,
which re-enter the Hubble horizon at different epochs. During transition from
inflation to kination, due to a sharp change in the metric structure [47],
gravitational waves (GWs) of high frequencies $\sim 10^{10}$ Hz, called _blue-
tilted_ GWs are emitted, while during matter domination _red-tilted_ GWs of
frequencies $\sim 10^{-17}-10^{-19}$ Hz are emitted [65, 50]. The stochastic
signature of the former type of GWs are instrumental in the verification of
various BBN constraints [66, 67]. Overproduction of GWs would lead to spike in
their energy densities [68], which could be threatening to nucleosynthesis
process. These aspects specifically depend upon the particular modes of
particle production involved after inflation. Specifically, gravitational
reheating produces GW-energy too large to begin the nucleosynthesis, therefore
inefficient to satisfy the required BBN constraints [66, 67], whereas the IP
is more efficacious in this regard. Another important aspect is the amplitudes
of the GWs, called the _characteristic strain_ , which is extremely small
$\sim 10^{-18}$ (see figure 11), thereby making the detection processes too
difficult and complicated. In fact such a tiny value of the GW-amplitude
carries indirect signatures of small values of tensor-to-scalar ratio and the
amplitudes of CMB $B$ modes. That is why, the detection of primordial GWs are
the targets of many ongoing and forthcoming cosmological surveys. See section
IV.4 for details.

According to the current studies, the quintessential versions of
$\alpha$-attractors have been identified to be efficacious in explaining the
observational bounds of both inflation and dark energy. See Ref. [69] for
example. Recently in Ref. [70], the present authors have explored various
aspects of the quintessential inflationary regime for a specific model, by
constraining the parameter $\alpha$ and fixing the energy scales of inflation
and quintessence. They showed that for $0.1\leq\alpha\leq 4.3$, the model is
just appropriate for satisfying the Planck data with $68\%$ CL. Now, it will
be interesting to examine this new model in the context of SB. The present
paper, therefore, aims at revisiting the scenario of SB with the newly
obtained range of $\alpha$ and study the applicability of the new potential
[70] in estimating various constraints of quintessential inflationary
baryogenesis and the associated gravitational waves.

The outline of the present paper is as follows. It contains a preamble in
Section II, explaining the kination period through the definition of a number
of temperatures in the context of post-inflationary particle production.
Section III gives a brief review of SB. The new quintessential
$\alpha$-attractor inflaton potential is applied to analyze the baryogenesis
scenario, whose results are illustrated in Section IV. The spectra of relic
gravitational waves associated with this potential are also analyzed in the
same section (sub-section IV.4). Finally, Section V contains some concluding
remarks.

##  II The post-inflationary kination period

In the standard formulation of inflationary quintessence [53], the $\phi$
field enters into a kinetic energy dominated phase called _kination_ [54],
just after the end of inflation. This period of expansion can be characterized
by the scale factor ($a$)-dependent energy density [68]

$\rho_{\mathrm{kin}}(a)=\rho_{\mathrm{end}}\left(\frac{a_{\mathrm{end}}}{a}\right)^{6}$
(1)

and Hubble rate

$H_{\mathrm{kin}}(a)=\sqrt{\frac{\rho_{\mathrm{kin}}(a)}{3M_{p}^{2}}}=H_{\mathrm{end}}\left(\frac{a_{\mathrm{end}}}{a}\right)^{3}.$
(2)

Here, the subscript ‘end’ denotes the respective parameters at the point where
the inflation ends (and kination begins) and $M_{p}(=1/\sqrt{8\pi G})$ is the
reduced Planck mass. The constants $\rho_{\mathrm{end}}$ and
$H_{\mathrm{end}}$ are the inputs from a specific model of inflation,
depending on the nature of the potential $V(\phi)$. At the end of inflation,
$\phi$ reaches $\phi_{\mathrm{end}}$ and the corresponding value of the
potential, $V_{\mathrm{end}}=V(\phi_{\mathrm{end}})$, is obtained from
equating the first potential slow-roll parameter $\epsilon_{V}$ to unity.
Since in this period of expansion, the kinetic energy of the $\phi$ field
grows up to be comparable with the potential energy (_i.e._
$\dot{\phi}^{2}\sim V(\phi)$), we can write

$\rho_{\mathrm{kin}}=\frac{\dot{\phi}^{2}}{2}+V(\phi)\approx\frac{3}{2}\dot{\phi}^{2}.$
(3)

The end-value of the potential is therefore induced in $\rho_{\mathrm{end}}$
and $H_{\mathrm{end}}$ through the approximation,
$\dot{\phi}_{\mathrm{end}}^{2}\approx V_{\mathrm{end}}$ as

$H_{\mathrm{end}}=\sqrt{\frac{\rho_{\mathrm{end}}}{3M_{p}^{2}}}\approx\sqrt{\frac{V_{\mathrm{end}}}{2M_{p}^{2}}}.$
(4)

Now from Eqs. (1), (3) and (4) we get

$\frac{3}{2}\dot{\phi}^{2}\approx\frac{3}{2}V_{\mathrm{end}}\left(\frac{a_{\mathrm{end}}}{a}\right)^{6},$
(5)

which yields the field velocity of the form

$\dot{\phi}\approx\sqrt{V_{\mathrm{end}}}\left(\frac{a_{\mathrm{end}}}{a}\right)^{3}.$
(6)

Here the constants $\sqrt{V_{\mathrm{end}}}$ and $a_{\mathrm{end}}$ indicate
that the field dynamics depends on the particular inflationary model. However,
Ref. [68] shows that if the kinetic energy of the field is taken to be
extremely large ($\dot{\phi}^{2}\gg V(\phi)$) in comparison to the potential
energy (_i.e._ complete free-fall), then the dynamics of the field will be
model-independent. On the other hand, in our formalism, we have considered
kinetic energy to be comparable to the potential energy during kination, and
therefore we have always retained the effects of $V(\phi)$ during kination in
the constants like $V_{\mathrm{end}}$ and $a_{\mathrm{end}}$ following Ref.
[53], as it will be shown, the potential will have a significant role to play
in the process of SB.

Now, due to the expansion of the universe, $\dot{\phi}$ becomes significantly
small during kination with the increase in the scale factor $a$ and thereby
the total energy density diminishes rapidly until radiation takes over through
reheating associated with particle production. During the transition from
inflation to kination, due to a non-adiabatic change of the metric structure
[47], an easy-to-realize mechanism of reheating is gravitational particle
production (GPP) [55, 56, 57, 58, 59, 60]. But it is not so effective so far
as various constraints of BBN [63, 50] are concerned. For example, GPP
produces an insufficient amount of radiation density to start nucleosynthesis.
GPP has been therefore replaced by an efficient method, called _instant
preheating_ (IP) [61, 62, 63]. See Ref. [63] for details of this procedure,
specifically for $\alpha$-attractor. Generally, it is an instantaneous (within
one cycle of oscillation of the inflaton field), non-perturbative mechanism of
reheating due to the effects of ‘ _broad parametric resonance_ ’ (BPR) [71,
72, 73, 61], characterized by fast energy transfer from the inflaton field to
the superheavy massive particles produced, whose masses are typically $\sim
10^{17}-10^{18}$ GeV [61]. But in the case of quintessential inflation, as
there is no minimum where the inflaton field can oscillate, rather a
quintessential run-away, the BPR does not work. Just after the end of
inflation, almost instantaneously the non-oscillating inflaton field produces
superheavy particles, which again decay into fermions or bosons depending upon
the nature of interactions. The elementary particles thus created give rise to
a radiation density [74, 63, 50, 68]

$\rho_{\mathrm{rad}}(a)=\left[\frac{gM_{p}H_{\mathrm{end}}}{2\pi^{3/2}}\left(\frac{a_{\mathrm{end}}}{a}\right)^{2}\right]^{2},$
(7)

where $g(<1)$ is the positive coupling of IP. The associated temperature [75,
68] in unit of $M_{p}$ can be derived from Eq. (7) as,

$\begin{split}T(a)&=\left(\frac{30}{\pi^{2}g_{*}}\rho_{\mathrm{rad}}(a)\right)^{1/4}\\\
&=\left(\frac{15g^{2}M_{p}^{2}H_{\mathrm{end}}^{2}}{2\pi^{5}g_{*}}\right)^{1/4}\left(\frac{a_{\mathrm{end}}}{a}\right).\end{split}$
(8)

Here, $g_{*}$ denotes the total number of degrees of freedom of the
relativistic particles produced through IP. The factor in the first bracket
containing constant terms has the dimension of temperature. Therefore it can
be taken as a constant temperature term,

$T_{\mathrm{end}}=\left(\frac{15g^{2}M_{p}^{2}H_{\mathrm{end}}^{2}}{2\pi^{5}g_{*}}\right)^{1/4},$
(9)

such that,

$T=T_{\mathrm{end}}\left(\frac{a_{\mathrm{e}nd}}{a}\right).$ (10)

$T_{\mathrm{end}}$ actually denotes the temperature at the instant of ending
of inflation and beginning of kination at the scale factor
$a=a_{\mathrm{end}}$. It is model-dependent, since, according to Eqs. (4) and
(9) $T_{\mathrm{end}}\propto V_{\mathrm{end}}^{1/4}$.

Now, with the expansion of the universe, the decay rate with coupling $\beta$
[74, 76, 50] of the massive particles produced through IP decreases and let at
$a=a_{\mathrm{th}}$ it becomes equal to the Hubble rate of expansion (see Eq.
(2)). At this particular scale factor, thermal equilibrium commences and it is
found to be,

$a_{\mathrm{th}}=\left(\frac{2^{9/2}\pi^{6}T_{\mathrm{end}}^{4}}{g^{3}\beta^{4}M_{p}^{3}H_{\mathrm{end}}}\right)^{1/4}a_{\mathrm{end}},$
(11)

which is, when applied in Eq. (10), gives the temperature at the thermal
equilibrium,

$T_{\mathrm{th}}=\left(\frac{g^{3}\beta^{4}M_{p}^{3}H_{\mathrm{end}}}{2^{9/2}\pi^{6}}\right)^{1/4}.$
(12)

This is the maximum temperature of the universe after inflation that can be
attained in thermal equilibrium, called the _maximum thermalization
temperature_ [47]. After this event, the total energy density of the scalar
field becomes weaker and, as a consequence, radiation starts to emerge as a
major component of the universe.

Eq. (10) can be instrumental in evaluating the temperature during the era of
radiation domination. Let us call the temperature as $T_{\mathrm{rad}}$ when
the radiation domination starts and the corresponding scale factor as
$a_{\mathrm{rad}}$. At this scale factor, there will be a _cross-over_ between
two densities $\rho_{\mathrm{kin}}$ and $\rho_{\mathrm{rad}}$ (see Eqs. (1)
and (7)), which means that at this stage
$\rho_{\mathrm{kin}}(a_{\mathrm{rad}})=\rho_{\mathrm{rad}}(a_{\mathrm{rad}})$
(this happens at a particular value of e-folds) and thereafter
$\rho_{\mathrm{rad}}$ will exceed $\rho_{\mathrm{kin}}$. This aspect will be
illustrated in the specific case of $\alpha$-attractor potential in Section
IV. Now, from Eqs. (1) and (7) we get

$a_{\mathrm{rad}}=\left(\frac{12\pi^{3}}{g^{2}}\right)^{1/2}a_{\mathrm{end}}$
(13)

and from Eqs. (10) and (13) we have

$T_{\mathrm{rad}}=\left(\frac{g^{2}}{12\pi^{3}}\right)^{1/2}T_{\mathrm{end}}.$
(14)

Eq. (13) (Eq. (14)) shows that the ratio of the scale factors (temperatures)
in the radiation domination era and the corresponding end-value depends only
on the coupling constant of IP. In Section IV, this finding will play an
important role in analyzing the evolution of the universe during kination, for
the potential considered in the present paper.

The three temperatures $T_{\mathrm{end}}$, $T_{\mathrm{th}}$ and
$T_{\mathrm{rad}}$ are crucial in jointly framing the whole scenario of the
kination period. We can determine their values by doing a model-independent
estimate. We take the help of an empirical formula [77] relating the
inflationary energy scale (in the unit of reduced Planck mass) with the
tensor-to-scalar ratio $r$:

$V^{1/4}(r)\approx r^{1/4}\times\left(\frac{10^{16.5}}{2.43\times
10^{18}}\right)M_{p}.$ (15)

A good slow-roll inflaton potential always supports a small value of $r$
($r<0.064$ [78, 79]) and hence a small $\epsilon_{V}$, since
$r=16\epsilon_{V}$. Therefore, it will be reasonable to consider a rough
estimation $V\approx V_{\mathrm{end}}$, in a model-independent way. Thus, if
we take $r=0.05$, then from Eqs. (15) and (4), we obtain

$V_{\mathrm{end}}^{1/4}=6.15\times 10^{-3}M_{p}$ (16)

and

$H_{\mathrm{end}}=2.67\times 10^{-5}M_{p}.$ (17)

Choosing $\beta\sim 10^{-1}$, $g\sim 10^{-4}$ and $g_{*}\sim 10^{2}$ (based on
various literature of IP, for example, [61, 63]), we finally obtain

$T_{\mathrm{end}}=6.46\times 10^{-6}M_{p}=1.57\times
10^{13}\quad\mathrm{GeV},$ (18) $T_{\mathrm{th}}=5.92\times
10^{-7}M_{p}=1.44\times 10^{12}\quad\mathrm{GeV}$ (19)

and

$T_{\mathrm{rad}}=3.35\times 10^{-11}M_{p}=8.14\times
10^{7}\quad\mathrm{GeV}.$ (20)

Eqs. (18) - (20) show that $T_{\mathrm{end}}/T_{\mathrm{th}}\approx 10.9$ and
$T_{\mathrm{rad}}/T_{\mathrm{th}}\approx 5.6\times 10^{-5}$. Therefore it is
clear that thermal equilibrium is reached shortly after the end of inflation
and long before the advent of the radiation era, which is an essential
requirement for the SB.

##  III Basic framework of spontaneous baryogenesis

The relevant Lagrangian here is,

$\mathcal{L}_{\phi}=\left(-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)-\mathcal{L}_{\mathrm{eff}}\right),$
(21)

where the quintessential inflaton field $\phi$ is in the Friedmann metric
background $g_{\mu\nu}$ of signature $(-,+,+,+)$. In addition, $\phi$ is
coupled with a non-conserving baryon current $J^{\mu}$ by the following
$CP$-odd effective interaction [47, 4, 50, 52]

$\mathcal{L}_{\mathrm{eff}}=\frac{\lambda^{\prime}}{M}\partial_{\mu}\phi
J^{\mu},$ (22)

$\lambda^{\prime}$ being a dimensionless coupling constant and the mass scale
$M<M_{P}$ signifies a sub-Planckian cut-off of the effective theory. The
associated energy-momentum tensor can be obtained from the equation

$T^{\mu}_{\nu}=\delta^{\mu}_{\nu}\mathcal{L}_{\phi}-2g^{\mu\alpha}\frac{\partial\mathcal{L}_{\phi}}{\partial
g^{\alpha\nu}}.$ (23)

The scalar field $\phi$ is spatially homogenized by the cosmic expansion after
inflation. All primordial quantum fluctuations ($\delta\phi$) buried in $\phi$
within the Hubble sphere are transferred to density perturbations, whose
imprints are detected in the anisotropies of the cosmic microwave background.
The remaining $\phi$ field is now completely classical and time-dependent
only. Therefore, in Eq. (23) only $\mu=0$ component of $J^{\mu}$ _viz.,_
$J^{0}=n-\bar{n}$ will contribute, which measures the difference between the
number densities of baryons ($n$) and ant-baryons ($\bar{n}$). Following these
arguments from Eq. (23), the field density takes the form (see Appendix A)

$\begin{split}\rho&=-T_{0}^{0}=\left(\frac{\dot{\phi}^{2}}{2}+V(\phi)\right)+\left(-\frac{\lambda^{\prime}}{M}\dot{\phi}\left(n-\bar{n}\right)\right)\\\
&=\rho_{0}+\left(n\delta\mathcal{E}+\bar{n}\delta\bar{\mathcal{E}}\right),\end{split}$
(24)

where
$\delta\mathcal{E}=-\delta\bar{\mathcal{E}}=-\frac{\lambda^{\prime}\dot{\phi}}{M}$.
$\rho_{0}$ is the interaction-free part of the density $\rho$. The second term
indicates that, if the field velocity $\dot{\phi}$ is appreciably non-zero
(_i.e._ $\phi$ is a fast-rolling scalar field), then for each baryon (anti-
baryon), an extra amount of energy density
$\delta\mathcal{E}(\delta\bar{\mathcal{E}})$ would appear in the total energy
density budget of the universe. The extra positive and negative energy density
terms originate from the interaction term $\partial_{\mu}\phi J^{\mu}$
responsible for spontaneous $CPT$ symmetry breaking at a scale below the
Planck mass, as explained in Section I, which ultimately ensure the baryon-
excess ($n>\bar{n}$) in the universe. If, on the other hand, $CPT$ symmetry is
not broken in thermal equilibrium, there would be no baryon asymmetry _i.e._
$n=\bar{n}$. We shall calculate $n-\bar{n}$ in the next paragraph and show
that it is positive definite. At the thermal equilibrium, this additional
energy per baryon (anti-baryon) is recognized as an effective chemical
potential $\mu$ ($-\mu$) [4]. Alternative realizations of chemical potentials,
in this context, can be seen in Refs. [34, 35].

We explicitly compute the baryon excess by considering the statistical phase
space distribution function in thermal equilibrium,

$f(\epsilon,\mu)=\frac{1}{e^{\left(\frac{\epsilon-\mu}{T}\right)}+1}=\frac{1}{e^{\left(\frac{|\vec{p}|-\mu}{T}\right)}+1},$
(25)

of an ensemble of massless relativistic baryons of momentum $\vec{p}$ at a
particular energy level $\epsilon=|\vec{p}|$ with degeneracy $\tilde{g}$ as,

$\Delta
n=n-\bar{n}=\int\tilde{g}\frac{d^{3}\vec{p}}{(2\pi)^{3}}\left[f(\epsilon,\mu)-f(\epsilon,-\mu)\right],$
(26)

with the convention $c=\hbar=k_{B}=1$. Under the high temperature
approximation, $\mu/T\ll 1$, Eq. (26) reduces to (see Appendix B)

$\Delta n\approx\frac{\tilde{g}\mu
T^{2}}{6}=\frac{\tilde{g}\lambda^{\prime}\dot{\phi}T^{2}}{6M}>0.$ (27)

The temporal behavior of $\phi$, determined by $\dot{\phi}$, is encoded in the
solution of its equation of motion under the effective interaction of Eq. (22)
during kination in expanding background. We derive that equation of motion,
using Eq. (21) from the Euler-Lagrange equation

$\left[\partial_{\mu}\left(\frac{\partial}{\partial_{\mu}\phi}\right)-\frac{\partial}{\partial\phi}\right]\sqrt{-g}\mathcal{L}_{\phi}=0,$
(28)

as (see Appendix C)

$\left[1-\frac{\tilde{g}}{6}\left(\frac{\lambda^{\prime}M_{p}}{M}\right)^{2}\left(\frac{T}{M_{p}}\right)^{2}\right]\left(\ddot{\phi}+3H\dot{\phi}\right)+\frac{dV(\phi)}{d\phi}=0.$
(29)

The second term in the third bracket signifies the back-reaction on $\phi$ of
the coupling described in Eq (22). According to the _Carroll bound_ [64, 10]
$\frac{\lambda^{\prime}M_{p}}{M}<8$, and as described in Section I, the baryon
asymmetry operates at temperature $T$ lying in the range
$T_{F}<T<T_{\mathrm{th}}$, where $T_{\mathrm{th}}\sim 10^{-7}M_{p}$ (see Eq.
(19)), which implies $T/M_{p}\ll 1$. Therefore the effect of back-reaction
_vis-à-vis_ the second term is very small. Then from Eq. (29) the dynamical
equation of $\phi$ reduces to

$\ddot{\phi}+3H\dot{\phi}+\frac{dV(\phi)}{d\phi}\approx 0,$ (30)

which is the well-known Klein-Gordon equation of the scalar field $\phi$.

Now, in the kination regime, the field $\phi$ is predominantly controlled by
its kinetic energy density, which is approximately equal (_i.e._ comparable)
to the potential energy density $V(\phi)$ (see Eq. (3)). Therefore,
$\frac{dV(\phi)}{d\phi}\approx\frac{d\dot{\phi}^{2}}{d\phi}=0$ (since $\phi$
and $\dot{\phi}$ are mutually independent quantities) and consequently, the
potential gradient in Eq. (30) can be neglected. Thus we get

$\ddot{\phi}+3H\dot{\phi}\approx 0.$ (31)

Considering $H=\frac{\dot{a}}{a}$ in Eq. (31) we obtain,

$\frac{1}{a^{3}}\frac{d}{dt}\left(a^{3}\dot{\phi}\right)=0,$ (32)

whose solution is given by

$\dot{\phi}=\frac{C}{a^{3}},$ (33)

where $C$ is a constant. Using the approximation
$\dot{\phi}_{\mathrm{end}}=\dot{\phi}(a_{\mathrm{end}})\approx\sqrt{V_{\mathrm{end}}}$
indicated in Section II as initial condition, we obtain
$C=\sqrt{V_{\mathrm{end}}}\left(a_{\mathrm{end}}\right)^{3}$ and thus Eq. (33)
matches with Eq. (6). Therefore, under negligible amount of back-reaction, the
effective interaction gives the same equation of kination, as described in
Section II. This indicates that the approximations mentioned in Section II are
fairly consistent with the framework of SB.

According to Eqs. (6) and (10) the field velocity $\dot{\phi}$ can be
parameterized as a function of temperature as

$\dot{\phi}(T)=\sqrt{V_{\mathrm{end}}}\left(\frac{T}{T_{\mathrm{end}}}\right)^{3}.$
(34)

Eqs. (27) and (34) show that the baryon excess $\Delta n$ is only temperature
dependent. It is conveniently measured with respect to the entropy density $s$
(given in Refs. [4, 50, 52]) as

$\begin{split}\eta_{T}&=\left(\frac{\Delta
n}{s}\right)=\frac{\left(\frac{\tilde{g}\lambda^{\prime}\dot{\phi}T^{2}}{6M}\right)}{\left(\frac{2\pi^{2}g_{*}T^{3}}{45}\right)}\\\
&=\frac{15}{4\pi^{2}}\left(\frac{\lambda^{\prime}}{M}\right)\left(\frac{\tilde{g}}{g_{*}}\right)\left(\frac{\dot{\phi}(T)}{T}\right).\end{split}$
(35)

This is called the baryon-to-entropy ratio (BTER) at a finite temperature $T$.
Since $\dot{\phi}(T)\propto T^{3}$, $\eta_{T}\propto T^{2}$, _i.e._ $\eta_{T}$
decreases as the temperature of the universe decreases with expansion. But, as
explained in Section I, this will continue up to the freeze-out temperature
$T_{F}$. Below $T_{F}$ the baryon non-conserving processes fall out of thermal
equilibrium and no further baryons are generated. As a result, $\eta_{F}$
saturates at a freeze out value

$\eta_{F}=\frac{15}{4\pi^{2}}\left(\frac{\lambda^{\prime}}{M}\right)\left(\frac{\tilde{g}}{g_{*}}\right)\left(\frac{\dot{\phi}(T_{F})}{T_{F}}\right),$
(36)

which acts as a standard measure of the observed baryon asymmetry in the
present universe. Therefore it should conform with the experimental value of
$\eta$, mentioned in Section I.

Following Eqs. (4), (9), (10), (12) and (34) the freeze out value of the field
velocity takes the form

$\begin{split}\dot{\phi}(T_{F})&=\sqrt{2M_{p}^{2}}H_{\mathrm{end}}\left(\frac{T_{F}}{T_{\mathrm{end}}}\right)^{3}\\\
&=\sqrt{2M_{p}^{2}}\beta^{2}\left(\frac{\pi
g_{*}}{60}\right)^{3/4}\frac{T_{F}^{3}}{T_{\mathrm{th}}^{2}}.\end{split}$ (37)

We now finally obtain from Eqs. (36) and (37),

$\begin{split}\eta_{F}&=\frac{15\sqrt{2}}{4\pi^{2}}\left(\frac{\pi
g_{*}}{60}\right)^{3/4}\left(\frac{\tilde{g}}{g_{*}}\right)\left(\frac{\lambda^{\prime}M_{p}}{M}\right)\left(\frac{\beta
T_{F}}{T_{\mathrm{th}}}\right)^{2}\\\ &=1.8599\times
10^{-4}\times\left(\frac{\lambda^{\prime}M_{p}}{M}\right)\left(\frac{T_{F}}{T_{\mathrm{th}}}\right)^{2},\end{split}$
(38)

where $\tilde{g}\sim 1$ and we put the values of $g_{*}$ and $\beta$, used for
evaluating the temperatures of Eqs. (18) - (20). Here, $T_{\mathrm{th}}$ bears
the signature of the inflaton potential, while $M$ and $T_{F}$ are specific to
the particular baryon non-conserving process involved. Thus, in precise
evaluation of $\eta_{F}$ in light of the observed value, the data for both the
inflation and the baryon asymmetry phenomenology play the determining roles.
However, in order to keep the thermal equilibrium during baryogenesis, the
$T_{F}$ should lie within $T_{\mathrm{rad}}$ and $T_{\mathrm{th}}$. In fact,
by the Carroll bound, a rough estimate of $T_{F}$ can be computed as
$T_{F}=1.42\times 10^{-10}M_{p}=3.46\times 10^{8}$ GeV, if $T_{\mathrm{th}}$
and $\eta_{F}$ are allowed to take the model-independent value (see Eq. 19)
and the observational bound (mentioned in Section I), respectively.

##  IV Results and discussion

Let us begin with the following quintessential $\alpha$-attractor potential,
developed recently in Ref. [70]

$V(\phi)=V_{*}(n,\alpha)e^{-n\left(1-\tanh{\frac{\phi}{\sqrt{6\alpha}M_{p}}}\right)}.$
(39)

$V_{*}(n,\alpha)$ is the COBE/Planck-normalized inflationary energy scale,
which is a function of model parameters $n$ and $\alpha$ and has been derived
in Ref. [70]. The associated first potential slow-roll parameter is found to
be

$\epsilon_{V}(\phi)=\left[\left(\frac{n}{\sqrt{12\alpha}}\right)^{1/2}\operatorname{sech}{\frac{\phi}{\sqrt{6\alpha}M_{p}}}\right]^{4}.$
(40)

When $\phi=\phi_{\mathrm{end}}$, $\epsilon_{V}(\phi_{\mathrm{end}})=1$, from
which we obtain the end-value of $\phi$:

$\phi_{\mathrm{end}}=\sqrt{6\alpha}M_{p}\operatornamewithlimits{sech^{-1}}\left(\frac{\sqrt{12\alpha}}{n}\right)^{1/2}$
(41)

and the corresponding end-value expression of the potential of Eq.(39) is

$V_{\mathrm{end}}(n,\alpha)=V_{*}(n,\alpha)e^{-n\left[1-\sqrt{1-\left(\frac{\sqrt{12\alpha}}{n}\right)}\right]}.$
(42)

Figure 1: The beginning and the end of the energy scales of inflation in GeV
for $n=122$, plotted against $\alpha$ between $0.1$ and $4.3$. The small
variation of $V_{\mathrm{end}}^{1/4}$ near the value $4\times 10^{15}$ GeV is
shown in the inset.

Figure 1 shows the variations of $V_{*}^{1/4}$ and $V_{\mathrm{end}}^{1/4}$
with $\alpha$ from $\alpha=0.1$ to $4.3$ for $n=122$. The choices of $\alpha$
and $n$ are based upon the results in Ref. [70]. Depending upon the values of
$\alpha$, inflation occurs at a scale $V_{*}^{1/4}$, lying between $\sim
10^{15}-10^{16}$ GeV (see the green curve), but it ends at a scale
$V_{\mathrm{end}}^{1/4}$, which is strictly around $\sim 10^{15}$ GeV (see the
orange curve) for all allowed values of $\alpha$, which is also the point of
initialization of the kination period. As $\alpha$ increases, the difference
between the two energy scales _viz.,_ $V_{*}^{1/4}$ and
$V_{\mathrm{end}}^{1/4}$ also increases because of the _double pole behavior_
(discussed in Ref. [70]) of the inflaton potential of Eq. (39). When $\alpha$
is small, the potential is of slow-roll type, where the beginning and the end
of energy scales of inflation are almost of the same order. But as $\alpha$
increases, the potential becomes of the power-law type and the two energy
scales start to differ up to $\sim 10$ GeV. The order of magnitude of
$V_{\mathrm{end}}^{1/4}$ is instrumental in obtaining correct ranges of the
energy densities and temperatures of the universe during kination, as will be
described in next subsections.

###  IV.1 Evolution of the energy densities

The evolution of various energy densities in the kination period primarily
depends upon the ratio $a/a_{\mathrm{end}}$, which is measured through the
number of e-folds $N$, elapsed after the end of inflation,

$N\equiv\int_{0}^{N}dN^{\prime}=\int_{a_{\mathrm{end}}}^{a}d\ln{a^{\prime}}=\ln{\left(\frac{a}{a_{\mathrm{end}}}\right)}.$
(43)

Now, from Eqs. (1), (2), (4, (7) and (43) we get

$\left(\frac{\rho_{\mathrm{kin}}}{V_{\mathrm{end}}}\right)=\frac{3}{2}e^{-6N},$
(44)
$\left(\frac{H_{\mathrm{kin}}^{2}M_{p}^{2}}{V_{\mathrm{end}}}\right)=\frac{1}{2}e^{-6N}$
(45)

and

$\left(\frac{\rho_{\mathrm{rad}}}{V_{\mathrm{end}}}\right)=\frac{g^{2}}{8\pi^{3}}e^{-4N}.$
(46)

Figure 2: Evolution of the dimensionless post-inflationary energy densities
with the number of e-folds. At $N\approx 12.17$, the total energy density of
the quintessential inflaton field and that of radiation become equal and after
that radiation starts to dominate _i.e._ radiation era begins.

In figure 2 the behaviors of the energy density functions, stated above, are
described with the expansion of the universe, measured by the number of
e-folds $N$. It is clear that,
$\rho_{\mathrm{kin}}\sim\rho_{\mathrm{rad}}=10^{-32}V_{\mathrm{end}}\sim
10^{28}$ (Gev)4. This amount of energy density is sufficient for satisfying
the observational data of primordial nucleosynthesis [47, 52], which has
become possible for selecting the IP as a proper reheating mechanism instead
of the gravitational reheating or GPP. Such a choice therefore sets a perfect
stage for the successful SB process, which will be understood more clearly
later. Another interesting observation in figure 2 is that, there is a cross-
over between $\rho_{\mathrm{kin}}$ and $\rho_{\mathrm{rad}}$ at $N\approx
12.17$. The radiation density begins to be greater than the field density when
the expansion of the universe reaches $12.17$ numbers of e-folds after
inflation ends. Even more crucial is that, this value is model-independent
(which will be explained in the next sub-section) _i.e._ for all models, the
radiation domination occurs after $12.17$ e-folds from the end of inflation.
Only the magnitudes of the said energy densities at that specific number of
e-folds depend upon the potential concerned. This value of $N$ limits the
maximum amount of expansion, below which the SB takes place. Once the
radiation density becomes larger than the field density, the process of
baryogenesis stops. To get acquainted with this scenario with further clarity,
the evolution of temperatures of the universe at different phases should be
carefully examined.

###  IV.2 Evolution of the temperatures

In the left panel of figure 3, the evolution of $T/T_{\mathrm{end}}$ is shown
as a function of number of e-folds $N$, which is independent of the values of
$\alpha$. According to Eqs. (10) and (43) the temperature of the universe
falls exponentially after the completion of inflation, from
$T=T_{\mathrm{end}}$ (see Eq. (9)). In the right panel of figure 3, the
dependencies of $T_{\mathrm{end}}$ and $H_{\mathrm{end}}$ on $\alpha$ are
shown. $T_{\mathrm{end}}$ is $\sim 10^{12}$ GeV for all values of $\alpha$.
This is the temperature at the entry-level (since here $N=0$) of the kination
period.

Figure 3: _Left panel_ : Variation of temperature with number of e-folds $N$
starting from $T=T_{\mathrm{end}}$. _Right panel_ : The $\alpha$-dependencies
of $T_{\mathrm{end}}$ and $H_{\mathrm{end}}$ indicate that kination starts at
temperature $\sim 10^{12}$ GeV.

Now, as the universe enters into kination, thermal equilibrium is not attained
soon. It requires some time (or some extra e-folds), which is needed for the
initiation of the interaction of the field with the particles produced via
reheating. The maximum thermalization temperature $T_{\mathrm{th}}$ (see Eq.
(12)) determines the threshold of this equilibrium.

Figure 4: _Left panel_ : The number of e-folds required to attain the thermal
equilibrium at $T=T_{\mathrm{th}}$. _Right panel_ : The dependency of
$T_{\mathrm{th}}$ on $\alpha$ depicts the fact that thermal equilibrium starts
at a temperature around $10^{11}$ GeV.

If the $N(T_{\mathrm{th}})$ is the number of e-folds passed to reach the
thermal equilibrium, then from Eq. (10) and (43),

$N(T_{\mathrm{th}})=\ln{\left(\frac{T_{\mathrm{end}}}{T_{\mathrm{th}}}\right)},$
(47)

which is roughly $1.69-1.83$ depending on $\alpha$ chosen for the potential
and the associated maximum thermalization temperature is $\sim 10^{11}$ GeV
(see figure 4). Therefore, approximately after $2$ e-folds, when the
temperature drops from $10^{12}$ GeV to $10^{11}$ GeV, equilibrium is
achieved. After that, the quintessential baryogenesis takes place by the
spontaneous $CPT$ symmetry breaking, as explained in Sections I and III.
Therefore, $T_{\mathrm{th}}$ refers to the maximum possible temperature below
which baryogenesis takes place.

The baryon non-conserving process remains active until the radiation does not
overpower the total field density and thereafter, when the radiation density
crosses the field density at temperature $T_{\mathrm{rad}}$ (see Eq. (14)),
the process of baryogenesis stops. Thus, SB should occur above
$T_{\mathrm{rad}}$ and it therefore acts as the minimum possible temperature
above which baryogenesis must occur. Actually, the baryon number violating
process ceases to act, specifically at freeze-out temperature $T_{F}$, which
will be shown in next sub-section lying between $T_{\mathrm{th}}$ and
$T_{\mathrm{rad}}$.

Figure 5: _Left panel_ : The number of e-folds needed to begin the radiation
domination at $T=T_{\mathrm{rad}}$. _Right panel_ : The dependency of
$T_{\mathrm{rad}}$ on $\alpha$ depicts that the radiation domination is found
to be prevalent at temperature around $10^{7}$ GeV.

Let $T_{\mathrm{rad}}$ be attained at the cost of $N(T_{\mathrm{rad}})$ number
of e-folds, then from Eqs. (14) and (43)

$N(T_{\mathrm{rad}})=\ln{\left(\frac{T_{\mathrm{end}}}{T_{\mathrm{rad}}}\right)}=\frac{1}{2}\ln{\left(\frac{12\pi^{3}}{g^{2}}\right)}=12.1698.$
(48)

Therefore the radiation begins to dominate for the potential considered in Eq.
(39) exactly at $12.1698$ e-folds after inflation at the temperature around
$10^{7}$ GeV (see figure 5). Interestingly, this result matches with the one
at the ‘crossing point’ of energy densities in figure 2 corresponding to the
model considered here.

Up to now, it is easily realizable that, SB would occur at any temperature
between $\sim 10^{11}-10^{7}$ GeV with the number of e-folds lying in the
range $1.69-12.17$. However, the exact value of this temperature can be
estimated from the freeze-out temperature $T_{F}$, which is linked with the
BTER, calculated in next sub-section.

To get a rough estimate of $T_{F}$, if we assume that at the freeze-out point
$\eta_{F}$ is equal to the experimental value $\eta=8.6\times 10^{-11}$ in Eq.
(38), $\frac{\lambda^{\prime}M_{p}}{M}$ is restricted between $1$ and $8$
according to Carroll bound, say, $\frac{\lambda^{\prime}M_{p}}{M}=5$ and
$T_{\mathrm{th}}=7.5\times 10^{11}$ GeV, taken from figure 4, then within the
model concerned $T_{F}$ can be calculated to be $2.28\times 10^{8}$ GeV, which
is above $T_{\mathrm{rad}}$ ($\sim 10^{7}$ GeV, see figure 5). Now, in the
next sub-section it will be interesting to look at the conditions in which
$T_{F}$ acquires the value of this order.

###  IV.3 Calculations of $T_{F}$ and $\eta_{F}$

In order to compute the freeze-out temperature $T_{F}$, we consider a
$B$-violating and $(B-L)$-conserving process, of which the rate, derived from
an SM four-fermion non-renormalizable point interaction [47, 10, 50, 52, 75],
can be written as,

$\Gamma_{B-L}(T)=\frac{\bar{g}^{2}}{\tilde{M}^{4}}T^{5},$ (49)

where $\bar{g}<1$ is the coupling constant and $\tilde{M}$ indicates the
effective cut-off of the said interaction. At the specific temperature,
$T_{F}$, the rate $\Gamma_{B-L}$ becomes of the same order as that of the
Hubble expansion rate $H_{\mathrm{kin}}$ of the universe _i.e._
$\Gamma_{B-L}(T_{F})\sim H_{\mathrm{kin}}(T_{F})$ (called the _freeze-out
condition_) and beyond that no $B$-violation takes place. Consequently, the
baryon number saturates at the freeze-out value. Now, using Eqs. (2), (10) and
(49), we obtain

$T_{F}=\frac{H_{\mathrm{end}}^{1/2}\tilde{M}^{2}}{T_{\mathrm{end}}^{3/2}\bar{g}}.$
(50)

Figure 6: _Left column_ : Number of e-folds elapsed to attain the freezing of
baryon non-conserving process at different mass scales. _Right column_ : The
variations of freeze-out temperatures with $\alpha$ show that, freezing
happens at $\sim 10^{8}-10^{9}$ GeV for a given value of $\tilde{M}$ and
$\bar{g}=0.8$. For a particular value of $\alpha$, larger value of number of
e-folds corresponds to smaller value of $T_{F}$, which is consistent with the
evolution of temperature of the universe with its expansion, described in
previous sub-section. With the increase of the mass scale $\tilde{M}$, the
number of e-folds decreases and $T_{F}$ increases for a given value of
$\alpha$, which are also clear from Eqs. (50) and (51).

The required number of e-folds to reach the temperature $T_{F}$ can be
calculated from Eqs. (10) and (43) (as done before in the cases of
$N(T_{\mathrm{th}})$ and $N(T_{\mathrm{rad}})$) as,

$N(T_{F})=\ln{\left(\frac{T_{\mathrm{end}}}{T_{F}}\right)}.$ (51)

In figure 6 the freeze-out temperature $T_{F}$ (see Eq. (50)) and the
corresponding number of e-folds $N(T_{F})$ are plotted as functions of
$\alpha$ for various values of the mass-scale $\tilde{M}$. Here, $\tilde{M}$
is so chosen that $T_{F}$ could lie between $T_{\mathrm{th}}$ ($\sim 10^{11}$
GeV) and $T_{\mathrm{rad}}$ ($\sim 10^{7}$ GeV), as stated in previous sub-
section. Figure 6 highlights that, depending upon the values of $\tilde{M}$,
the freezing of $B$-violating process occurs at a temperature around
$10^{8}-10^{9}$ GeV when the universe has covered $6.3-10.2$ number of
e-folds, which lie in the midst of thermalization and radiation domination.

Figure 7: Variations of freeze-out values of BTER with $\alpha$ for three
values of $M$ and two sets of values of $\tilde{M}$ corresponding to $\sim
10^{10}$ GeV and $\sim 10^{11}$ GeV. For a particular values of $\alpha$ and
$\tilde{M}$, $\eta_{F}$ decreases with the increase in $M$. Similarly, for a
particular values of $\alpha$ and $M$, $\eta_{F}$ increases with the increase
in $\tilde{M}$.

Finally the task is to estimate the freeze-out value of BTER, $\eta_{F}$, from
Eq. (38). To compute the $\eta_{F}$ as function of $\alpha$, three values of
$M$ _viz.,_ $M=\lambda^{\prime}M_{p}/2=1.2\times 10^{18}$ GeV,
$\lambda^{\prime}M_{p}/4=6.07\times 10^{17}$ GeV,
$\lambda^{\prime}M_{p}/6=4.05\times 10^{17}$ GeV for $\lambda^{\prime}\approx
1$ and the estimated values of $T_{\mathrm{th}}$ and $T_{F}$ shown in figures
4 and 6, respectively, have been considered. The choices of $\tilde{M}$
remains the same as shown in figure 6. Two sets of $\tilde{M}$, each
containing three values, corresponding to $\tilde{M}\sim 10^{10}$ GeV and
$\tilde{M}\sim 10^{11}$ GeV are taken into account.

In figure 7, the obtained values of $\eta_{F}$ with the variation of $\alpha$
from $\alpha=0.1$ to $\alpha=4.3$ are shown. The left column corresponds to
the variations when $\tilde{M}\sim 10^{10}$ GeV and the right column comprises
that for $\tilde{M}\sim 10^{11}$ GeV. The results show that the cut-off scale
$M$ has no such significant effects, where as, $\tilde{M}$ affects $\eta_{F}$
significantly. When $\tilde{M}\sim 10^{10}$ GeV, $\eta_{F}$ is of the order of
$10^{-10}-10^{-11}$, which is required for obtaining the experimental bound.
Similarly, when $\tilde{M}\sim 10^{11}$ GeV, $\eta_{F}$ is of the order of
$10^{-7}-10^{-9}$, which is higher than that of the experimental bound.

For $\tilde{M}=2.7\times 10^{10}$ GeV and $M=4.05\times 10^{17}$ GeV (see the
blue curve of the first plot in left column of figure 7), two ranges of values
of $\eta_{F}$ are found to be closest to the experimental value,
$\eta\approx(8.6\pm 0.1)\times 10^{-11}$. They are $\eta_{F}=8.7\times
10^{-11}-8.5\times 10^{-11}$ for $\alpha=0.29-0.3$ respectively and
$\eta_{F}=8.5\times 10^{-11}-8.6\times 10^{-11}$ for $\alpha=4.2-4.3$
respectively. Therefore, there exists two ranges _viz.,_ $\alpha=0.29-0.3$ and
$\alpha=4.2-4.3$ corresponding to the lower and upper ends of $\alpha$,
respectively, in which the calculated BTER satisfies the experimental value
with specified error bar for the mass scales $\tilde{M}=2.7\times 10^{10}$ GeV
and $M=4.05\times 10^{17}$ GeV.

Table 1: Calculate values of $\eta_{F}$, $T_{\mathrm{th}}$ and $T_{F}$ for two ranges of values of $\alpha$, where $\eta_{F}$ satisfies the experimental requirement: $\eta\approx(8.6\pm 0.1)\times 10^{-11}$. $\alpha$ | $\eta_{F}/(10^{-11})$ | $T_{\mathrm{th}}/(10^{11})$ | $T_{F}/(10^{8})$ | $\frac{T_{\mathrm{th}}}{T_{F}}\times 10^{-3}$
---|---|---|---|---
$0.29-0.3$ | $8.7-8.5$ | $7.78-7.79$ | $2.175-2.17$ | $3.577-3.59$
$4.2-4.3$ | $8.5-8.6$ | $7.83-7.81$ | $2.15-2.17$ | $3.642-3.599$

Table 1 shows that, in the regime, where $\eta_{F}$ satisfies the experimental
value, the maximum thermalization temperature is found to be
$T_{\mathrm{th}}=7.78\times 10^{11}-7.79\times 10^{11}$ GeV for
$\alpha=0.29-0.3$ and $T_{\mathrm{th}}=7.83\times 10^{11}-7.81\times 10^{11}$
GeV for $\alpha=4.2-4.3$ (see figure 4). Similarly, the freeze-out temperature
is obtained as $T_{F}=2.175\times 10^{8}-2.17\times 10^{8}$ GeV for
$\alpha=0.29-0.3$ and $T_{F}=2.15\times 10^{8}-2.17\times 10^{8}$ GeV for
$\alpha=4.2-4.3$ (see the red curve in the first plot of second column of
figure 6). Here, $T_{\mathrm{th}}/T_{F}=3.577\times 10^{3}-3.59\times 10^{3}$
for $\alpha=0.29-0.3$ and $T_{\mathrm{th}}/T_{F}=3.642\times
10^{3}-3.599\times 10^{3}$ for $\alpha=4.2-4.3$, which are close to the value
given in Ref. [50]. In these two ranges of $\alpha$, the temperature at the
end of inflation (or the start of kination) $T_{\mathrm{end}}$ and the
temperature of radiation domination, $T_{\mathrm{rad}}$, are obtained as,
$T_{\mathrm{end}}=4.60\times 10^{12}-4.65\times 10^{12}$ GeV,
$T_{\mathrm{rad}}=2.38\times 10^{7}-2.39\times 10^{7}$ GeV for
$\alpha=0.29-0.3$ and $T_{\mathrm{end}}=4.66\times 10^{12}-4.63\times 10^{12}$
GeV, $T_{\mathrm{rad}}=2.42\times 10^{7}-2.4\times 10^{7}$ GeV for
$\alpha=4.2-4.3$ (see figures 3 and 5). Also, the symmetry breaking scale $M$
($\sim 10^{17}$ GeV) is found to be $10^{5}$ times larger than the Hubble rate
$H_{\mathrm{end}}$ ($\sim 10^{12}$ GeV) (see figure 3), which is an important
result according to Ref. [4].

Thus, from all the analyses done so far, it can be said that, for the
potential concerned in Eq. (39) the SB takes place at the temperature $T$
lying within the following inequality

$\begin{split}&T_{\mathrm{rad}}(\sim 10^{7}\mathrm{GeV})<T_{F}(\sim
10^{8}\mathrm{GeV})<T(\mathrm{GeV})\\\ &<T_{\mathrm{th}}(\sim
10^{11}\mathrm{GeV})<T_{\mathrm{end}}(\sim 10^{12}\mathrm{GeV})\\\ &<M(\sim
10^{17}\mathrm{GeV})<M_{p}(\sim 10^{18}\mathrm{GeV}).\end{split}$ (52)

###  IV.4 Spectra of the relic gravitational waves

Gravitational waves (GWs) are the results of tensor perturbations in
spatially-flat FLRW metric defined by (written in standard notations) [77]

$ds^{2}=-dt^{2}+a^{2}(t)\left(\delta_{mn}+h_{mn}\right)dx^{m}dx^{n}.$ (53)

$h_{mn}$ are the metric perturbations, which are transverse
($\partial_{m}h_{mn}=0$) having two crossed polarization states and trace-less
($h_{mm}=0$). The root-mean-square (RMS) value of the amplitudes of the GW
spectrum, $\Omega_{GW}(k,\tau)$, is measured as the ratio of logarithmic
derivative of GW energy density $\rho_{GW}$ with respect to the mode $k$ and
the conformal time ($\tau$)-dependent critical energy density
$\rho_{c}=3M_{p}^{2}H^{2}(\tau)$ [65, 52]. This can be transferred to its
present-day value $\Omega_{GW,0}$ [65, 52] by extracting the effects of
(dimensionless) inflationary tensor power spectrum $\Delta^{2}_{h}(k)$ (see
[80] for the calculations of mode-dependent tensor power spectrum) at present-
day, through a transfer function $T_{\mathrm{trans}}^{2}(k,\tau)$ [81, 82,
83]. This function encodes the post-inflationary evolution of a mode $k$ from
horizon re-entry to time ($\tau$) of observation. Now, a number of expressions
of $\Omega_{GW,0}$ exist depending upon the various epochs in which the modes
re-enter the horizon. Among them, only the kinetic-dominated (KD) and
radiation-dominated (RD) epochs are the subjects of interest in the present
work. They are given by [76, 65, 50, 52, 84],

$\Omega^{(\mathrm{RD})}_{GW,0}(k)=\left(\frac{\Omega_{\mathrm{rad},0}}{6\pi^{2}}\right)\left(\frac{H_{\mathrm{inf}}^{2}}{M_{p}^{2}}\right)\left(\frac{g_{*}}{g_{*,0}}\right)\left(\frac{g_{*s,0}}{g_{*s}}\right)^{4/3}$
(54)

for $k_{\mathrm{equal}}<k\leq k_{\mathrm{rad}}$ and

$\Omega^{(\mathrm{KD})}_{GW,0}(k)=\Omega^{(\mathrm{RD})}_{GW,0}(k)\left(\frac{k}{k_{\mathrm{rad}}}\right)$
(55)

for $k_{\mathrm{rad}}<k\leq k_{\mathrm{end}}$.
$\Omega_{\mathrm{rad},0}=9.13\times 10^{-5}$ [79, 78] is the present-day
density parameter of radiation. $g_{*s}$ and $g_{*}$ are relativistic degrees
of freedom corresponding to entropy density and radiation density
respectively, which are of the same order _i.e._ $10^{2}$ [52].
$g_{*s,0}=3.91$ and $g_{*,0}=3.36$ are their present values [79, 78]. The
subscripts ‘end’, ‘rad’ and ‘equal’ in $k$ refer to the modes taking re-entry
during the end of inflation (or start of kination), radiation era and matter-
radiation equality, respectively. Here, $H_{\mathrm{inf}}$ refers to the
inflationary Hubble parameter which is generically mode-dependent (see Ref.
[80]) and hence a little complicated. But it can be simplified for the
plateau-type potential considered here (see Eq. (39)), for which tensor-to-
scalar ratio is small, by approximating $H_{\mathrm{inf}}\approx
H_{\mathrm{end}}$, which is $k$-independent. Therefore the GW spectrum during
the radiation era is roughly scale-independent. It depends only on $\alpha$
through $H_{\mathrm{end}}$, while that for kinetic domination is slightly
blue-tilted, which will be examined in the next paragraph. In figure 8, we
plot the RMS values of the amplitudes of the present-day RD-GW spectrum
against $\alpha$, and the results show that they are $\sim 10^{-18}$, which is
indeed very small, lying within the projected ranges of various ongoing
experiments (see figure 11). The corresponding peak value [65, 52, 50] is
given by,

$\left(\Omega^{RD}_{GW,0}\right)_{\mathrm{peak}}=\Omega^{RD}_{GW,0}\left(\frac{T_{\mathrm{end}}}{T_{\mathrm{rad}}}\right)^{2}.$
(56)

In figure 9 the peak values of amplitudes of present-day RD-GW spectrum are
plotted against $\alpha$, in which the peak values are found to be $\sim
10^{-7}$, which satisfy the following constraint of BBN [66, 67]

$\left(\Omega^{RD}_{GW,0}\right)_{\mathrm{peak}}<2.46\times 10^{-6}.$ (57)

For $\alpha=0.29-0.3$ we get $\Omega^{(\mathrm{RD})}_{GW,0}=3.14\times
10^{-18}-3.2\times 10^{-18}$ and
$\left(\Omega^{RD}_{GW,0}\right)_{\mathrm{peak}}=1.16\times 10^{-7}-1.2\times
10^{-7}$. Similarly, for $\alpha=4.2-4.3$ we obtain
$\Omega^{(\mathrm{RD})}_{GW,0}=3.3\times 10^{-18}-3.23\times 10^{-18}$ and
$\left(\Omega^{RD}_{GW,0}\right)_{\mathrm{peak}}=1.22\times 10^{-7}-1.2\times
10^{-7}$.

Therefore, the potential considered in Eq. (39) is viable in making a reliable
connection between the GeV-scale - (spontaneous) baryogenesis and the BBN.

Figure 8: Plot of the RMS values of the amplitudes for the present-day RD-GW
spectrum against $\alpha$. Figure 9: Plot of the peak values of the amplitudes
for the present-day RD-GW spectrum against $\alpha$.

The GW frequencies [76, 65] having wave vectors $k_{\mathrm{rad}}$ and
$k_{\mathrm{end}}$ indicated in Eqs. (54) and (55) can be calculated using,

$f_{\mathrm{rad}}=1600\times\left(\frac{T_{\mathrm{rad}}}{10^{10}\mathrm{GeV}}\right)\mathrm{Hz}$
(58)

and

$f_{\mathrm{end}}=8900\times\left(\frac{H_{\mathrm{inf}}}{g\times
10^{4}\mathrm{GeV}}\right)^{1/2}\mathrm{Hz}.$ (59)

Figure 10: GW frequencies of the modes taking re-entry during radiation
domination and the end of inflation (_i.e._ start of kination).

Figure 10 shows that the obtained values of $f_{\mathrm{end}}$ and
$f_{\mathrm{rad}}$ are $\sim 10^{10}$ Hz and $\sim 10^{1}$ Hz respectively.
For $\alpha=0.29-0.3$ we get $f_{\mathrm{end}}=2\times 10^{10}-2.1\times
10^{10}$ Hz and $f_{\mathrm{rad}}=3.7-3.78$ Hz. Similarly, for
$\alpha=4.2-4.3$ we obtain $f_{\mathrm{end}}=2.12\times 10^{10}-2.11\times
10^{10}$ Hz and $f_{\mathrm{rad}}=3.8-3.78$ Hz. The $f_{\mathrm{end}}$
frequencies are large enough and hence they are often called the _blue tilted_
[85, 65, 50]. Ref. [50] shows that the frequencies of the modes entering the
horizon at present and during matter-radiation equality (corresponding to
$k_{\mathrm{equal}}$) are of the order of $10^{-19}$ Hz and $10^{-17}$ Hz
respectively, for which they are termed as the _red tilted_.

Therefore, during the transition from inflation to kination GWs of very high
frequencies (blue tilted) are emitted. This is because a sharp transition
takes place from periods of inflation to kination due to a change of metric
structure. Other transitions are smooth for which the corresponding emitted
GWs are of low frequencies (red tilted). However, all GWs have very low
amplitudes and therefore they are too feeble to detect.

In the present era of precision cosmology and multi-messenger astronomy [86],
several ground-based and space-based GW detectors are in operation to detect
the GW frequencies from very low end to very high end (see figure 11). For
example, Advanced-LIGO [87], Advanced-Virgo [88] and KAGRA [89] detect the GW-
strain $\sim 10^{-9}$ at roughly $100$ Hz. Again, the Laser Interferometer
Space Antenna (LISA) [90] and Deci-Hertz Interferometer Gravitational-wave
Observatory (DECIGO) [91] are made to sense the frequencies approximately from
$0.001$ Hz to $\sim 0.1$ Hz. Even weaker GWs of frequency like $\sim 10^{-8}$
Hz can be probed using Square Kilometer Array (SKA) [92], the largest radio
telescope, so far.

Figure 11: GW sensitivity curves of various detectors from several sources.
The plot has been generated on-line using http://gwplotter.com/ based on the
Ref. [93].

The stochastic signatures of blue-tilted frequencies are important in probing
deeply into the ‘pre-cosmic microwave background scenario’ and therefore
validating the paradigm of quintessential inflation, specifically the
kination. Unfortunately, at present, no GW detectors are able to detect $\sim
10^{10}$ Hz ($f_{\mathrm{end}}$) frequency. It needs more sensitivity and
sophistication. At least it is satisfactory that the frequencies $\sim 10$ Hz
($f_{\mathrm{rad}}$) are being detected in the ongoing experiments.

There is an interesting connection between the GWs and the theory of
inflation. Figure 11 shows that every experimental group has observed GWs of
strain $\sim 10^{-14}-10^{-24}$, which is undoubtedly too small. It is
because, according to Eqs. (4, (15) and (54) the strain is proportional to
tensor-to-scalar ratio $r$, which is very small ($<0.064$ [79]) for plateau-
type model of inflation. In a previous work [70] for the same potential
considered here, the $r$ is found to be $\sim 10^{-4}-10^{-2}$ for
$0.1\leq\alpha\leq 4.3$ at the pivot scale $k=0.002$ Mpc-1. Therefore the
observed tiny values of GW-strain indirectly support the framework of slow-
roll inflation, just like Planck observation [79]. However, this aspect will
be more prominent after the detection of $B$ modes with desired accuracy (see
[80, 70] for further discussions).

##  V Conclusions

In conclusion, we have

  1. 1.

applied a new model of quintessential $\alpha$-attractor inflaton potential to
study the scenario of spontaneous baryogenesis (SB) within $0.1\leq\alpha\leq
4.3$ and $n=122$,

  2. 2.

studied SB by deriving the expressions of energy density and equation of
motion of $\phi$ field, baryon excess and the freeze-out value of baryon-to-
entropy ratio (BTER) at finite temperature by incorporating the derivative
coupling of $\phi$ field with the non-conserving baryon current with a cut-off
scale $M$. This coupling, describing a CP-odd effective interaction, brings
extra terms in total field-energy density corresponding to the number
densities of particles and anti-particles. By a calculation of statistical
phase space distribution, the number density of particles is found to be
greater than that of anti-particles for a non-zero value of field velocity,
leading to the ‘baryon-excess’ of the universe. This baryon-excess is
translated into the computation of BTER. In this course, the effect of back-
reaction is shown to be negligible in the dynamical equation of motion of the
quintessential inflaton field,

  3. 3.

examined the evolution during kination of total field density
$\rho_{\mathrm{kin}}$ and radiation density $\rho_{\mathrm{rad}}$ of the
particles created through instant preheating (IP), with the expansion of the
universe, which is measured by the number of e-folds $N$, elapsed after the
end of inflation. The two energy densities are found to have same order of
magnitude, which is very necessary for a successful BBN process during matter
domination (which actually makes the IP a better reheating process rather than
GPP). At $N=12.1698$ there exists a cross-over between the above mentioned
energy densities, which acts as a transition point between starting and ending
of the SB process as well as between the domination of $\phi$-field and
radiation,

  4. 4.

derived the expressions of the temperature, $T_{\mathrm{end}}$ at the end of
inflation (or start of kination), the maximum thermalization temperature
$T_{\mathrm{th}}$, and the temperature of radiation domination
$T_{\mathrm{rad}}$, which have been used to examine the post-inflationary
kination period related to SB,

  5. 5.

analyzed the evolution of these three temperatures and their respective number
of e-folds with $\alpha$ within the range specified earlier. The order of
magnitude of the values of $T_{\mathrm{th}}$ are close to that of
$T_{\mathrm{end}}$ and too far from that of $T_{\mathrm{rad}}$ signifying the
fact that, thermal equilibrium attains just after the end of inflation and
long before the beginning of radiation era. This is needed to set a perfect
stage for the BBN in matter dominated era; because, even a slight deviation of
the order of the values of the said temperatures could be threatening for
obtaining the observed constraints of BBN,

  6. 6.

considered an effective non-renormalizable $4$-fermion point interaction with
a cut-off scale $\tilde{M}$ and the associated rate of $B$-violation
$\Gamma_{B-L}$ to compute the freeze-out temperature $T_{F}$ and the baryon-
to-entropy ratio $\eta_{F}$ with the variations of $\alpha$ for fixed values
of $M$ and $\tilde{M}$ using the ‘freeze-out condition’. The obtained values
of $T_{F}$ lie in between $T_{\mathrm{th}}$ and $T_{\mathrm{rad}}$ and much
below the cut-off mass-scale $\tilde{M}$, required for implementing the
effective $4$-fermi construct method. The ratio $T_{\mathrm{th}}/T_{F}$
becomes consistent with the value given in the current literature. On the
other hand, the $\eta_{F}$ values conform to the experimental bound for two
narrow ranges of $\alpha$ _viz.,_ $\alpha=0.29-0.3$ and $4.2-4.3$ and the
specific values of $M(\sim 10^{17}\leavevmode\nobreak\ \mathrm{GeV})$ and
$\tilde{M}(\sim 10^{10}\leavevmode\nobreak\ \mathrm{GeV})$. In between these
two ranges of $\alpha$, the values of $\eta_{F}$ slightly deviate from the
experimental value, keeping the order of magnitude same, and

  7. 7.

studied the variations of the amplitudes and the frequencies of the spectra of
relic gravitational waves with $\alpha$ for two types of transitions - one is
from inflation to kination ($KD$) and another is from $\phi$ domination to
radiation domination ($RD$). The calculated magnitudes of the present-day RMS
values of $RD-GW$ amplitudes, $\Omega_{GW,0}^{(\mathrm{RD})}$, are consistent
with the sensitivity profile of the ongoing $GW$-detectors. The corresponding
peak values, $(\Omega_{GW,0}^{(\mathrm{RD})})_{\mathrm{peak}}$, satisfy the
BBN constraint quite satisfactorily. A blue tilted $KD-GW$ spectrum is also
found to exist, which is an essential feature of the model of quintessential
inflation in the context of SB.

The new model of quintessential $\alpha$-attractor inflaton potential,
considered in this paper, therefore, appears to be efficacious in explaining
the baryon asymmetry at GeV-scale in the light of current cosmological surveys
and generation of ‘blue-spectrum’ of relic gravitational waves for
$\alpha=0.29-0.3$ and $\alpha=4.2-4.3$. According to Ref. [70], this model is
equally efficient in explaining early and late-time expansions of the universe
for $0.1\leq\alpha\leq 4.3$. Therefore, the combined results of the present
paper and Ref. [70] show that, the concerned potential is capable of unifying
inflation, baryogenesis and quintessence in a single framework.

###### Acknowledgements.

The authors acknowledge the University Grants Commission, The Government of
India for the CAS-II program in the Department of Physics, The University of
Burdwan. AS acknowledges The Government of West Bengal for granting him the
Swami Vivekananda fellowship.

##  Appendix A Density of $\phi$ field with the effective interaction

The Lagrangian of the $\phi$-field is given by

$\mathcal{L}_{\phi}=-\frac{1}{2}g^{\beta\gamma}\partial_{\beta}\phi\partial_{\gamma}\phi-V(\phi)-\frac{\lambda^{\prime}}{M}\partial_{\beta}\phi
J^{\beta}$ (60)

and the corresponding energy-momentum tensor can be calculated as

$\begin{split}T_{\nu}^{\mu}&=\delta_{\nu}^{\mu}\mathcal{L}_{\phi}-2g^{\mu\alpha}\frac{\partial\mathcal{L}_{\phi}}{\partial
g^{\alpha\nu}}\\\
&=\delta_{\nu}^{\mu}\left(-\frac{1}{2}g^{\beta\gamma}\partial_{\beta}\phi\partial_{\gamma}\phi-V(\phi)-\frac{\lambda^{\prime}}{M}\partial_{\beta}\phi
J^{\beta}\right)\\\
&+\left(g^{\mu\alpha}\partial_{\alpha}\phi\partial_{\nu}\phi+\frac{2\lambda^{\prime}}{M}g^{\mu\alpha}\partial_{\alpha}\phi
J_{\nu}\right)\\\
&=\delta_{\nu}^{\mu}\left(\frac{\dot{\phi}^{2}}{2}-V(\phi)-\frac{\lambda^{\prime}}{M}\dot{\phi}J^{0}\right)\\\
&+\left(g^{\mu
0}\dot{\phi}\partial_{\nu}\phi+\frac{2\lambda^{\prime}}{M}g^{\mu
0}\dot{\phi}J_{\nu}\right).\end{split}$ (61)

The time-time component of $T_{\nu}^{\mu}$ is,

$\begin{split}T_{0}^{0}&=\left(\frac{\dot{\phi}^{2}}{2}-V(\phi)-\frac{\lambda^{\prime}}{M}\dot{\phi}J^{0}\right)+\left(-\dot{\phi}^{2}+\frac{2\lambda^{\prime}}{M}\dot{\phi}J^{0}\right)\\\
&=-\frac{\dot{\phi}^{2}}{2}-V(\phi)+\frac{\lambda^{\prime}}{M}\dot{\phi}J^{0},\end{split}$
(62)

from which we obtain the density as

$\begin{split}\rho&=-T_{0}^{0}=\frac{\dot{\phi}^{2}}{2}+V(\phi)-\frac{\lambda^{\prime}}{M}\dot{\phi}J^{0}\\\
&=\frac{\dot{\phi}^{2}}{2}+V(\phi)-\frac{\lambda^{\prime}}{M}\dot{\phi}\left(n-\bar{n}\right).\end{split}$
(63)

##  Appendix B Derivation of baryon excess $\Delta n$

The Fermi-Dirac distribution function for massless relativistic baryons having
momentum $|\vec{p}|\equiv p$, energy $\epsilon=p$ and chemical potential $\mu$
is,

$f(p,\mu)=\frac{1}{e^{\left(\frac{p-\mu}{T}\right)}+1}=\frac{e^{-p/T}}{e^{-\mu/T}+e^{-p/T}}.$
(64)

Under high temperature approximation we have $\mu/T\ll 1$ and terms
$\mathcal{O}(\mu^{2}/T^{2})$ are neglected. Therefore,

$\begin{split}&f(p,\mu)\approx\frac{e^{-p/T}}{1-\left(\frac{\mu}{T}-e^{-p/T}\right)}\\\
&=e^{-p/T}\left[1-\left(\frac{\mu}{T}-e^{-p/T}\right)\right]^{-1}\\\
&=e^{-p/T}\left[1+\sum_{\mathbb{Z}\ni
n=1}^{\infty}\left(\frac{\mu}{T}-e^{-p/T}\right)^{n}\right]\\\
&=e^{-p/T}\Bigl{[}1+\sum_{\mathbb{Z}\ni
n=1}^{\infty}\sum_{m=0}^{n}{}^{n}C_{m}\left(\frac{\mu}{T}\right)^{n-m}\left(-e^{-p/T}\right)^{m}\Bigr{]}\\\
&\approx e^{-p/T}\left[1+\sum_{\mathbb{Z}\ni
n=1}^{\infty}{}^{n}C_{n-1}\left(\frac{\mu}{T}\right)\left(-e^{-p/T}\right)^{n-1}\right]\\\
&=e^{-p/T}+\frac{\mu}{T}\sum_{\mathbb{Z}\ni
n=1}^{\infty}n\left(-1\right)^{n-1}\left(e^{-np/T}\right).\end{split}$ (65)

Similarly, the same distribution function for anti-baryons is

$f(p,-\mu)=e^{-p/T}-\frac{\mu}{T}\sum_{\mathbb{Z}\ni
n=1}^{\infty}n\left(-1\right)^{n-1}\left(e^{-np/T}\right).$ (66)

Therefore,

$f(p,\mu)-f(p,-\mu)=\frac{2\mu}{T}\sum_{\mathbb{Z}\ni
n=1}^{\infty}n\left(-1\right)^{n-1}\left(e^{-np/T}\right).$ (67)

Thus, the baryon excess,

$\begin{split}&\Delta
n=n-\bar{n}=\int\tilde{g}\frac{d^{3}\vec{p}}{(2\pi)^{3}}\left[f(p,\mu)-f(p,-\mu)\right]\\\
&=\int_{0}^{\infty}\tilde{g}\frac{4\pi
p^{2}dp}{(2\pi)^{3}}\left[f(p,\mu)-f(p,-\mu)\right]\\\
&=\tilde{g}\frac{\mu}{\pi^{2}T}\sum_{\mathbb{Z}\ni
n=1}^{\infty}n(-1)^{n-1}\int_{0}^{\infty}p^{2}e^{-np/T}dp.\end{split}$ (68)

Putting $x=\frac{np}{T}$,

$\begin{split}&\Delta n=\tilde{g}\frac{\mu T^{2}}{\pi^{2}}\sum_{\mathbb{Z}\ni
n=1}^{\infty}\frac{(-1)^{n-1}}{n^{2}}\int_{0}^{\infty}x^{2}e^{-x}dx\\\
&=\tilde{g}\frac{\mu T^{2}}{\pi^{2}}\eta(2)\Gamma(3),\end{split}$ (69)

where $\eta(2)$ is the _Dirichlet eta function_ , which is related to _Riemann
zeta function_ as $\eta(2)=\left(1-2^{1-2}\right)\zeta(2)$. Putting
$\zeta(2)=\frac{\pi^{2}}{6}$ and $\Gamma(3)=2!$ we finally obtain

$\Delta n=\frac{\tilde{g}\mu
T^{2}}{6}=\frac{\tilde{g}\lambda^{\prime}\dot{\phi}T^{2}}{6M}.$ (70)

##  Appendix C Equation of motion of $\phi$ with the effective interaction

We have the Lagrangian for the $\phi$ field

$\mathcal{L}_{\phi}=-\frac{1}{2}g^{\beta\gamma}\partial_{\beta}\phi\partial_{\gamma}\phi-V(\phi)-\frac{\lambda^{\prime}}{M}\partial_{\beta}\phi
J^{\beta},$ (71)

where the metric is composed of usual Friedmann metric components $g_{00}=-1$
and $g_{ii}=a^{2}$ such that $\sqrt{-g}=a^{3}$. Now,

$\begin{split}&\frac{\partial\mathcal{L}_{\phi}}{\partial(\partial_{\mu}\phi)}=-\frac{1}{2}g^{\beta\gamma}\left(\delta_{\beta}^{\mu}\partial_{\gamma}\phi+\partial_{\beta}\phi\delta_{\gamma}^{\mu}\right)-\frac{\lambda^{\prime}}{M}\delta_{\beta}^{\mu}J^{\beta}\\\
&=-g^{\mu\beta}\partial_{\beta}\phi-\frac{\lambda^{\prime}}{M}\delta_{\beta}^{\mu}J^{\beta}=-g^{\mu
0}\dot{\phi}-\frac{\lambda^{\prime}}{M}\delta_{0}^{\mu}J^{0},\end{split}$ (72)
$\begin{split}&\partial_{\mu}\left(\sqrt{-g}\frac{\partial\mathcal{L}_{\phi}}{\partial(\partial_{\mu}\phi)}\right)\\\
&=-\frac{\partial_{\mu}g}{2\sqrt{-g}}\frac{\partial\mathcal{L}_{\phi}}{\partial(\partial_{\mu}\phi)}+\sqrt{-g}\partial_{\mu}\left(\frac{\partial\mathcal{L}_{\phi}}{\partial\left(\partial_{\mu}\phi\right)}\right)\\\
&=\frac{6a^{5}\dot{a}}{2a^{3}}\left(\dot{\phi}-\frac{\lambda^{\prime}}{M}J^{0}\right)+a^{3}\left(\ddot{\phi}-\frac{\lambda^{\prime}}{M}\partial_{0}J^{0}\right)\\\
&=a^{3}\left[\left(\ddot{\phi}-\frac{\lambda^{\prime}}{M}\partial_{0}J^{0}\right)+3H\left(\dot{\phi}-\frac{\lambda^{\prime}}{M}J^{0}\right)\right]\\\
&=a^{3}\left[\left(\ddot{\phi}-\frac{\tilde{g}\lambda^{\prime
2}T^{2}}{6M^{2}}\ddot{\phi}\right)+3H\left(\dot{\phi}-\frac{\tilde{g}\lambda^{\prime
2}T^{2}}{6M^{2}}\dot{\phi}\right)\right]\\\
&=a^{3}\left[1-\frac{\tilde{g}}{6}\left(\frac{\lambda^{\prime}M_{p}}{M}\right)^{2}\left(\frac{T}{M_{p}}\right)^{2}\right]\left(\ddot{\phi}+3H\dot{\phi}\right)\end{split}$
(73)

and

$\frac{\partial\left(\sqrt{-g}\mathcal{L}_{\phi}\right)}{\partial\phi}=-a^{3}\frac{\partial
V}{\partial\phi}.$ (74)

Therefore, the equation of motion becomes

$\begin{split}&\left[\partial_{\mu}\left(\frac{\partial}{\partial(\partial_{\mu}\phi)}\right)-\frac{\partial}{\partial\phi}\right]\sqrt{-g}\mathcal{L}_{\phi}=0\\\
\mathrm{or,}&\left[1-\frac{\tilde{g}}{6}\left(\frac{\lambda^{\prime}M_{p}}{M}\right)^{2}\left(\frac{T}{M_{p}}\right)^{2}\right]\left(\ddot{\phi}+3H\dot{\phi}\right)+\frac{\partial
V}{\partial\phi}=0.\end{split}$ (75)

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11affiliationtext: School of Computer Science, University of Auckland,
Auckland, New Zealand22affiliationtext: School of Biological Sciences,
University of Auckland, Auckland, New Zealand

# Bayesian inference of the climbing grade scale

Alexei Drummond  Alex Popinga