# Reheating process in the $R^{2}$ inflationary model
with the baryogenesis scenario

Hyun Jeong  Research Center for the Early Universe (RESCEU), Graduate School
of Science, The University of Tokyo, Tokyo 113-0033, Japan Department of
Physics, Graduate School of Science, The University of Tokyo, Tokyo 113-0033,
Japan Kohei Kamada  Research Center for the Early Universe (RESCEU), Graduate
School of Science, The University of Tokyo, Tokyo 113-0033, Japan Department
of Physics, Graduate School of Science, The University of Tokyo, Tokyo
113-0033, Japan Alexei A. Starobinsky  L. D. Landau Institute for Theoretical
Physics RAS, Chernogolovka, Moscow region 142432, Russian Federation Kazan
Federal University, Kazan 420008, Republic of Tatarstan, Russian Federation
Jun’ichi Yokoyama  Research Center for the Early Universe (RESCEU), Graduate
School of Science, The University of Tokyo, Tokyo 113-0033, Japan Department
of Physics, Graduate School of Science, The University of Tokyo, Tokyo
113-0033, Japan Kavli Institute for the Physics and Mathematics of the
Universe (Kavli IPMU), WPI, UTIAS, The University of Tokyo, Kashiwa, Chiba
277-8568, Japan Trans-Scale Quantum Science Institute, The University of
Tokyo, Tokyo 113-0033, Japan

###### Abstract

Post-inflationary evolution and (re)heating of the viable inflationary model,
the $R^{2}$ one, is made more realistic by including the leptogenesis scenario
into it. For this purpose, right-handed Majorana neutrinos with a large mass
are added to the matter sector of the Standard Model to explain the neutrino
oscillation experiments and the baryon asymmetry of the Universe. We have
found parameters that characterize this model: non-minimal coupling of the
Higgs field $\xi$ and the mass of the right-handed Majorana neutrino
$M_{N_{\alpha}}$. We have analyzed the effect of these parameters on the
reheating process and the resultant physical quantities: spectral indices and
baryon asymmetry.

##  1 Introduction

The inflationary scenario [1, 2, 3, 4, 5] is now a successfully established
paradigm for the early Universe, which not only solved the horizon and
flatness problems but also made definite predictions for primordial
cosmological perturbations that can be directly tested by observations. A lot
of inflationary models were proposed historically but observational data have
already excluded most of them of them including once popular ones. Still a
number of successful inflationary models including the original $R^{2}$, or
Starobinsky, one [1] remain which satisfy all presently available data. The
most important and discriminatory among these predictions are the spectral
index $n_{\mathrm{s}}(k)$ of the Fourier power spectrum of scalar
perturbations and ratio of the power spectra of tensor and scalar ones $r(k)$.
While the generic and observationally confirmed prediction of all viable slow-
roll inflationary models, that do not require consideration of post-
inflationary evolution, is that both $|n_{\mathrm{s}}(k)-1|$ and $r(k)$ are
small and weakly scale-dependent, while $r(k)$ is of the order of
$|n_{\mathrm{s}}(k)-1|$ or less, calculation of these indices with the
accuracy better than 10% required by observations requires analyzing
(re)heating process, which turns the inflationary stage into the radiation-
dominated one by decay of an inflaton (dubbed the scalaron in the $R^{2}$
model) into ultra-relativistic particles and anti-particles and radiation.
Thus, the detailed study of the reheating process is very important for the
determination of the observationally consistent theory of inflation which
describes our Universe.

In the successful $R^{2}$ inflationary model [1], see also [6] for more
details including reheating and [7] for the final primordial spectra of scalar
and tensor perturbations, as well as further developments in [8, 9, 10], the
gravity sector is modified such that the $R^{2}$ term is added to the
Einstein-Hilbert action where $R$ is the Ricci scalar. This was suggested long
time ago by the consideration of the renormalization of an average value of
the energy-momentum tensor of quantum matter fields in an external classical
curved space-time, see e.g. [11, 12, 13] and also [14]. However, the smallness
of the measured Fourier power spectrum of primordial scalar perturbations
requires the dimensionless coefficient ($\hbar=c=1$) of this term to be
unexpectedly very large: $\approx 5.1\cdot 10^{8}$ [7, 15, 16]111This value is
given for the number of e-folds $N=55$ of the observable part of inflation
between the pivot scale
$k_{0}=$0.05\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}^{-1}$$ and the end of
inflation, otherwise it has to be multiplied by $(N/55)^{2}$.. On the other
hand, no observable effect suggests that the coefficient in front of the Weyl
tensor squared term in the effective action, also required for the one-loop
renormalizability, can be so large and even may be significantly different
from unity. This provides us with a possibility to use the inflationary
solution of the model in the regime when the $R^{2}$ term dominates the $R$
(Einstein) one while the undesired (due to the ghost appearance would be taken
in full) Weyl squared term still may be considered perturbatively.

If the main channel of scalaron decay after $R^{2}$ inflation is to light
scalar particles minimally or weakly (but not conformally) coupled to gravity,
this model has a very mild reheating process in contrast to other inflationary
models such as Higgs inflation [17, 18, 19, 20, 21, 22, 23, 24], and the
calculated spectral indices $(n_{\mathrm{s}},r)$ are in agreement with the
Cosmic Microwave Background (CMB) observations [25]. In fact, the dependence
of these indices on the number of e-folds $N$ during Higgs inflation is
essentially the same as for $R^{2}$ inflation, see [26] for the explanation of
this fact. It is only the actual value of $N$ that is larger for Higgs
inflation by several percents due to faster reheating after it caused by large
non-minimal coupling of the Higgs boson to gravity ($\xi\approx 4.5\cdot
10^{4}\sqrt{\lambda}$ for $N=55$). Furthermore, it has been argued that in the
$R^{2}$ inflationary model, dark matter and the right-handed Majorana neutrino
can be generated by scalaron decay, so it can be made compatible with present
matter content of the Universe [27, 28].

However, a more comprehensive study of reheating process after inflation is
needed to derive the values of $N$ and $n_{s}-1$ with accuracy of about $1\%$
or better expected in future CMB and large-scale structure observations. For
this purpose, more possible channels of scalaron decay after inflation into
particles and antiparticles of different quantum matter fields suggested by
modern extensions of the Standard Model (SM) of elementary particles have to
be investigated. So far, mostly the simplest channels like decay into light
scalar particles minimally or non-minimally coupled to gravity have been
considered. The enhancement of production of a certain matter field via
inflaton decay means the suppression of production of other matter fields.
Thus, branching ratios of different decay channels become very important for
the determination of observables such as the spectral indices, as well as the
amounts of baryon asymmetry and dark nonbaryonivc matter in the Universe.

An adjustment of the branching ratios can be done by controlling additional
parameters of the inflationary model characterizing the effective coupling of
inflaton (irrespective of its nature) to other particles in some extension of
SM. In the case of the $R^{2}$ inflationary model with the SM matter sector,
as shown in [29], the non-minimal coupling parameter of the Higgs field $\xi$
plays the role of regulating the branching ratio between the scalaron decay
into the Higgs boson and the decay into the gauge boson. When the Higgs field
is conformally coupled to the Ricci scalar, the reheating is mainly realized
by the scalaron decay into the gauge bosons rather than that into the Higgs
boson, that leads to deviation of the reheating temperature and the spectral
indices from the well-known values.

In this paper, we analyze how the reheating process of the successful $R^{2}$
inflationary model is modified when we add the successful leptogenesis
scenario to it by introducing supermassive right-handed Majorana neutrinos
[30]. In this case, the $R^{2}$ model is equipped with the SM matter fields
and the three generations of the right-handed Majorana neutrinos. Thus, the
masses of the Majorana neutrinos $M_{N_{\alpha}}$ are the new model parameters
in addition to the non-minimal coupling $\xi$. We analyzed how the reheating
process and the resultant physical quantities change by varying these
parameters.

This paper is organized as follows. In section 2, we introduce the $R^{2}$
inflationary model and all the matter fields in the extended SM of elementary
particles. The decay rates of scalaron into matter fields are calculated
there, too. The parameters that characterize this model are the non-minimal
coupling $\xi$ and the masses of the right-handed Majorana neutrinos
$M_{N_{\alpha}}$. In this paper, we restrict ourselves to the two typical
values of $\xi$: minimal coupling ($\xi=0$) and conformal coupling
($\xi=-1/6$). In section 3, we analyze the effect of the non-minimal coupling
$\xi$ on the reheating process by numerically solving the set of Boltzmann
equations. In section 4, we analyze the effect of the Majorana mass
$M_{N_{\alpha}}$ on the reheating process by solving the Boltzmann equations.
We see that the mass dependence appears only when the Higgs field is
conformally coupled. In section 5, we discuss the parameter dependence of the
physical quantities such as the spectral indices and the baryon asymmetry. As
a result, we find that the mass dependence strongly appears in the conformally
coupled case but not so much in the minimally coupled one. In section 6, we
summarize the results obtained and discuss directions of future research in
this area.

##  2 Decays during the reheating

We extend the $R^{2}$ inflationary model [1] to incorporate leptogenesis [30]
with heavy right-handed Majorana neutrinos, which induce the type-I see-saw
mechanism. During the reheating epoch, there are three main candidates into
which scalaron can decay: Higgs bosons, gauge bosons, and Majorana
neutrinos.222The decay rates into other massless SM fermions do not have to be
cared because their decay rates through the free field Lagrangian is zero (see
eq. (12)) and the decay channel through the Yukawa interaction is phase space
suppressed.  In this section, we will see that the model parameters that
govern the reheating process are the non-minimal coupling $\xi$ and the mass
of the right-handed Majorana fermion $M_{N_{\alpha}}$.

###  2.1 Decays of the scalaron

Lagrangians are defined in the Jordan frame as follows.

$S_{\text{Gravity}}^{\text{JF}}=\frac{M_{\text{G}}^{2}}{2}\int
d^{4}x\sqrt{-g}\left(R+\frac{R^{2}}{6M^{2}}\right),$ (1)

$\begin{split}S_{\text{SM}}^{\text{JF}}&=\int
d^{4}x\sqrt{-g}\left[-g^{\mu\nu}D_{\mu}\mathcal{H}^{\dagger}D_{\nu}\mathcal{H}-m_{h}^{2}\mathcal{H}^{\dagger}\mathcal{H}\right.\\\
&\left.-\lambda\left(\mathcal{H}^{\dagger}\mathcal{H}\right)^{2}+\xi
R\mathcal{H}^{\dagger}\mathcal{H}-\frac{1}{4}\sum_{F}F^{a}_{\mu\nu}F^{a\mu\nu}+\cdots\right],\end{split}$
(2)

$\begin{split}S_{\text{Majorana}}^{\text{JF}}&=\sum_{\alpha}\int
d^{4}x\sqrt{-g}\\\
&\left[iN^{\dagger}_{\alpha}\bar{\sigma}^{\mu}\nabla_{\mu}N_{\alpha}-\frac{1}{2}M_{N_{\alpha}}\left(N_{\alpha}N_{\alpha}+N^{\dagger}_{\alpha}N^{\dagger}_{\alpha}\right)\right].\end{split}$
(3)

Here, $\mathcal{H}$ and $N_{\alpha}$ are the SU(2) doublet Higgs field and the
right-handed Majorana neutrino field (two-component spinor), respectively, and
$D_{\mu}$ is the general coordinate covariant and gauge covariant derivative.
In (1),
$M_{\text{G}}=$2.4\text{\times}{10}^{18}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
is the reduced Planck mass and $M=1.3\times
10^{-5}M_{\text{G}}=$3.1\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
is the mass of the scalaron, which is unambiguously related to the magnitude
of the primordial Fourier power spectrum of scalar perturbations determined
from the measured CMB temperature and polarization fluctuations [15]. In (2),
$\sum_{F}$ means the sum over U(1)Y, SU(2)W and SU(3)c gauge fields in SM. The
non-minimal coupling of the Higgs field to Ricci scalar $\xi
R\mathcal{H}^{\dagger}\mathcal{H}$, which is necessary for the renormalization
of the ultraviolet divergence of the quantum Higgs field on the classical
curved space-time, is introduced. In (3), $\alpha$ runs over the three
generations of the Majorana neutrino. $\nabla_{\mu}$ is the covariant
derivative defined in the curved space, which includes the spin connection
[13]. $\bar{\sigma}^{\mu}=\left(\bm{1},-\bm{\sigma}\right)$ is the four-vector
Pauli matrices.

It can be easily seen that this model has a mechanism of inflation by doing
the following redefinition of the metric tensor field333We can see this fact
even in the Jordan frame when we examine the classical equation of motion [1,
6]. :

$g_{\mu\nu}\to\tilde{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu}=\exp\left(\sqrt{\frac{2}{3}}\frac{\phi}{M_{\text{G}}}\right)g_{\mu\nu}.$
(4)

Then, those actions are transformed into the Einstein frame [10, 31], and the
new degree of freedom $\phi$ (scalaron) can be extracted from the $R^{2}$
term:

$S_{\text{Gravity}}^{\text{EF}}=\int d^{4}x\sqrt{-\tilde{g}}\\\
\left[\frac{M_{\text{G}}^{2}}{2}\tilde{R}-\frac{1}{2}\tilde{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right].$
(5)

The potential $V(\phi)$ is calculated as

$V(\phi)=\frac{3M^{2}M_{\text{G}}^{2}}{4}\left[1-\exp\left(-\sqrt{\frac{2}{3}}\frac{\phi}{M_{\text{G}}}\right)\right]^{2}.$
(6)

So, slow-roll inflation can take place for large field values of $\phi$. After
the end of inflation, the scalaron starts to oscillate around the origin of
the potential and begins to decay into matter fields. The decay channels can
be found from the action expanded up to the linear order in $\phi$ in the
Einstein frame444Strictly speaking, this is not the Einstein frame because we
have the term $\xi\tilde{R}\tilde{\mathcal{H}}^{\dagger}\tilde{\mathcal{H}}$
other than the Einstein-Hilbert term. Nevertheless, we call this frame
Einstein frame just for convenience.  :

$\begin{split}S_{\text{SM}}^{\text{EF}}&\simeq\int d^{4}x\sqrt{-\tilde{g}}\\\
&\left[-\tilde{g}^{\mu\nu}\partial_{\mu}\tilde{\mathcal{H}}^{\dagger}\partial_{\nu}\tilde{\mathcal{H}}-m_{h}^{2}\tilde{\mathcal{H}}^{\dagger}\tilde{\mathcal{H}}+\frac{2}{\sqrt{6}}m_{h}^{2}\tilde{\mathcal{H}}^{\dagger}\tilde{\mathcal{H}}\frac{\phi}{M_{\text{G}}}\right.\\\
&+\left.\xi\tilde{R}\tilde{\mathcal{H}}^{\dagger}\tilde{\mathcal{H}}-\lambda\left(\tilde{\mathcal{H}}^{\dagger}\tilde{\mathcal{H}}\right)^{2}-2\sqrt{6}\left(\xi+\frac{1}{6}\right)\tilde{g}^{\mu\nu}\frac{\partial_{\mu}\phi}{M_{\text{G}}}\tilde{\mathcal{H}}\partial_{\nu}\tilde{\mathcal{H}}\right.\\\
&-\left.\left(\xi+\frac{1}{6}\right)\tilde{g}^{\mu\nu}\frac{\partial_{\mu}\phi\partial_{\nu}\phi}{M_{\text{G}}^{2}}\tilde{\mathcal{H}}^{\dagger}\tilde{\mathcal{H}}+\cdots\right],\end{split}$
(7)

$\begin{split}S_{\text{Majorana}}^{\text{EF}}&\simeq\sum_{\alpha}\int
d^{4}x\sqrt{-\tilde{g}}\\\
&\left[i\tilde{N}^{\dagger}_{\alpha}\tilde{\bar{\sigma}}^{\mu}\tilde{\nabla}_{\mu}\tilde{N}_{\alpha}-\frac{1}{2}M_{N_{\alpha}}\left(\tilde{N}_{\alpha}\tilde{N}_{\alpha}+\tilde{N}^{\dagger}_{\alpha}\tilde{N}^{\dagger}_{\alpha}\right)\right.\\\
&+\left.\frac{1}{2\sqrt{6}}\frac{M_{N_{\alpha}}}{M_{\text{G}}}\phi\left(\tilde{N}_{\alpha}\tilde{N}_{\alpha}+\tilde{N}^{\dagger}_{\alpha}\tilde{N}^{\dagger}_{\alpha}\right)\right].\end{split}$
(8)

Fields with a tilde are the Weyl-transformed fields555In this paper, the Weyl
transformation refers to the scaling of the metric tensor field, on the other
hand, the conformal transformation is the coordinate transformation which
induces the Weyl scaling of the metric. defined by
$\tilde{\mathcal{H}}=\Omega^{-1}\mathcal{H}$ and
$\tilde{N_{\alpha}}=\Omega^{-3/2}N_{\alpha}$. In (7) and (2), there are terms
that were absent in the Jordan frame because in general, the matter action is
not Weyl invariant. These are rewritten as follows to the lowest order in
$\phi$:

$S_{\text{matter}}^{\text{EF}}=\frac{1}{\sqrt{6}}\int
d^{4}x\sqrt{-\tilde{g}}\frac{\phi}{M_{\text{G}}}T^{\mu}_{\mu}+O(\phi^{2}),$
(9)

There are also quantum contributions to the trace part of the energy-momentum
tensor (9), which come from the fact that there are no regularization schemes
that hold conformal invariance simultaneously. Using the dimensional
regularization, the quantum contribution, or anomaly is calculated as [32, 29,
33, 34]:

$\begin{split}&\left(T^{\mu}_{\mu}\right)_{\text{quantum}}=\epsilon\left(-\frac{1}{4}\sum_{\tilde{F}}\tilde{F}^{a}_{\mu\nu}\tilde{F}^{a\mu\nu}-\lambda\left(\tilde{\mathcal{H}}^{\dagger}\tilde{\mathcal{H}}\right)^{2}\right)\\\
&\xrightarrow{\epsilon\to
0}-\sum_{\tilde{F}}\frac{\beta(\alpha_{\tilde{F}})}{32\pi^{2}}\tilde{F}^{a}_{\mu\nu}\tilde{F}^{a\mu\nu}-\beta(\lambda)\left(\tilde{\mathcal{H}}^{\dagger}\tilde{\mathcal{H}}\right)^{2},\end{split}$
(10)

Note that
$\tilde{F}^{a}_{\mu\nu}\tilde{F}^{a\mu\nu}=F^{a}_{\mu\nu}F^{a\mu\nu}$. Based
on those actions in the Einstein frame, decay rates of scalaron are calculated
as follows.666There are also contributions from the running of the non-minimal
coupling $\xi$. This contributes to the anomaly of the Higgs field. However,
this contribution is negligible. In the minimally coupled case, the anomaly
contribution is far subdominant compared to the classical contribution (11).
In the case of the conformal coupling, the classical decay rate is so small
that there is a possibility that the quantum contribution can come into play,
but the beta function at the conformal coupling is $0$ at the one-loop level.
There are inhomogeneous terms at the higher-loop level, but this contribution
is expected to be small enough to be neglected [35, 36]. We thank A. Kamada
for clarifying this point.

$\Gamma_{\phi\to h}\simeq\frac{M^{3}\sqrt{1-\frac{4m_{h}^{2}}{M^{2}}}}{48\pi
M_{\text{G}}^{2}}\left[-\left(1+6\xi\right)+\frac{2m_{h}^{2}}{M^{2}}\right]^{2},$
(11)

$\Gamma_{\phi\to
N_{\alpha}}=\frac{M}{96\pi}\left(\frac{M_{N_{\alpha}}}{M_{\text{G}}}\right)^{2}\left(1-\frac{4M_{N_{\alpha}}^{2}}{M^{2}}\right)^{\frac{3}{2}},$
(12)

$\Gamma_{\phi\to
g}\simeq\sum_{F}\frac{b_{\alpha_{F}}^{2}\alpha_{F}^{2}\mathcal{N}_{\alpha_{F}}}{768\pi^{3}}\frac{M^{3}}{M_{\text{G}}^{2}},$
(13)

where the subscripts $h,N_{\alpha}$ and $g$ stand for the complex Higgs
doublet, Majorana neutrinos, and gauge bosons, respectively, and
$\mathcal{N}_{\alpha_{F}}$ is the number of gauge bosons for corresponding
interactions: $\mathcal{N}_{\alpha_{Y}}=1,\ \mathcal{N}_{\alpha_{W}}=3,\
\mathcal{N}_{\alpha_{c}}=8$. $b_{\alpha_{F}}$ is the first coefficient of the
beta functions for the corresponding gauge fields in SM:
$b_{\alpha_{Y}}=41/6,\ b_{\alpha_{W}}=-19/6,\ b_{\alpha_{c}}=-7$. The decay
rate (12) can be calculated by using two-component formalism [37, 38, 39], and
this is half of the decay rate into four component Dirac spinor, which is due
to the fact that the degree of freedom into which the scalaron can decay is
decreased by half.777The formulas (11) and (12) in the limit $m_{h}\to
0,~{}M_{N_{\alpha}}\to 0$ for scalaron decay into light conformally and
minimally coupled scalars and fermions were present already in the early
papers [1, 6]. Note that there is no decay of scalaron into two gravitons in
the absence of SM interaction ($\alpha_{F}=0)$ [40, 41]. This decay still
becomes possible due to the term proportional to the square of the Weyl tensor
in the conformal trace anomaly (10) [40], but it can be neglected since its
rate is typically suppressed by the ratio $M^{4}/M_{\text{G}}^{4}$ compared to
(11). There are also loop contributions to the $\Gamma_{\phi\to g}$, but they
can be neglected, too, because the mass of the Higgs boson is very small [34].

The dots in (2) represent other interactions in SM such as Yukawa interactions
and these will lead to four-leg interactions in the Einstein frame (dots in
(7)). The scalaron decay through these couplings are phase space suppressed,
so these contributions can be neglected. This suppression is the same for the
decay via the second anomaly term in (10).

The decay rates of scalaron (11) (12) (13) are calculated and shown in Fig. 1
together with the mass dependence of the decay rate into the Majorana neutrino
(12). This decay rate takes maximum when
$M_{N_{\alpha}}\simeq$1.0\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.
In calculating the decay rate into the gauge bosons, the values of the gauge
couplings in (13) are evaluated at the energy scale $M/2$ using the results of
[42].

Figure 1: The decay rates of the scalaron into various fields together with
the mass dependence are shown. The red dashed line and the blue dotted line
represent the decay rate into the Higgs field with the minimal coupling
($\xi=0$) and the conformal coupling ($\xi=-1/6$), respectively. The orange
solid line and the green dash-dot line represent the decay rate into the
right-handed Majorana neutrino and the gauge fields.

When $\xi=0$, the decay rate into Higgs bosons is by far the largest. This is
qualitatively because decay rates are determined by the degree of breaking of
Weyl invariance, and this is violently broken by the kinetic term of the Higgs
scalar field.888The decay to Higgs field dominates the reheating process
unless $|\xi+1/6|<0.007$ according to [29]. In this subsection, the production
of the matter fields was described by the decay of the scalaron in the
Einstein frame, but this can also be understood as the gravitational particle
production in the Jordan frame [6, 8].

###  2.2 Decay of the Majorana neutrino

The reheating process is realized not only by the decay of scalaron into gauge
bosons and Higgs bosons but also that of Majorana neutrino into leptons and
Higgs bosons. The decay rate of Majorana neutrino into radiation is given by:

$\Gamma_{N_{1}\to R(\text{radiation})}=\Gamma_{N_{1}\to
lh}+\Gamma_{N_{1}\to\bar{l}h}=\frac{M_{N_{1}}}{8\pi}\sum_{\alpha}|y_{1\alpha}|^{2}.$
(14)

We used the summation convention over the repeated indices here. Here, three
generations of the Majorana neutrino are considered. $M_{N_{\alpha}}$ and
$y_{\alpha\beta}$ are the mass and components of the Yukawa coupling matrix,,
respectively. These Majorana neutrinos are also assumed to explain the masses
of three active neutrinos through the see-saw mechanism (Type I). The mass
matrix for the active neutrinos are calculated as follows:

$m_{\alpha\beta}=-y_{\alpha\gamma}\frac{v^{2}}{2M_{\gamma}}y_{\gamma\beta}$
(15)

where $v=$246\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ is the vacuum
expectation value of the Higgs field. We define $m_{1}<m_{2}<m_{3}$ as the
eigenvalues of this mass matrix (15). The following assumptions are made for
simplicity.

  * •

It is assumed that three active neutrinos have the normal mass hierarchy:
$m_{1}\ll m_{\text{sol}}$, $m_{2}\simeq
m_{\text{sol}}=$0.009\text{\,}\mathrm{eV}$$, $m_{3}\simeq
m_{\text{atm}}=$0.05\text{\,}\mathrm{eV}$$. Here, $m_{\text{sol}}$ and
$m_{\text{atm}}$ are the values of the mass difference determined by the solar
neutrino experiments and atmospheric neutrino experiments,, respectively [43].

  * •

This hierarchy is assumed to be explained by the mass hierarchy of three
right-handed neutrinos: $M_{N_{1}}\ll M_{N_{2}}\ll M_{N_{3}}$. That is, all
the components of the Yukawa coupling matrix are of the same order.

  * •

The mass of the right-handed Majorana neutrino is so heavy that the
leptogenesis occurs non-thermally. In this scenario, throughout the history of
the Universe, the right-handed Majorana neutrino is never thermalized.

  * •

It is postulated that only the lightest $N_{1}$ are produced by the decay of
the scalaron during reheating. This can be achieved by the following mass
configuration (see (12) and Fig. 1): $M_{N_{1}}<M/2<M_{N_{2}}<M_{N_{3}}$.

These assumptions lead

$m_{3}\sim\sum_{\alpha}\frac{|y_{1\alpha}|^{2}}{2M_{N_{1}}}v^{2}\sim
m_{\text{atm}}\simeq$0.05\text{\,}\mathrm{e}\mathrm{V}$.$ (16)

Using this relation, (14) is re-expressed as

$\Gamma_{N_{1}\to R}=\frac{M_{1}^{2}m_{\text{atm}}}{4\pi v^{2}}.$ (17)

##  3 The effect of the curvature coupling

In the previous section, we have seen that the model parameters that govern
the inflaton decay in this model are $\xi$ and $M_{N_{1}}$. Here, we analyze
the effects of non-minimal coupling $\xi$ in detail. There are two typical
values of the curvature coupling $\xi$: the minimal coulpling ($\xi=0$) and
the conformal coupling ($\xi=-1/6$). In the minimally coupled case, the decay
rate into the Higgs particle (11) is so strong that Higgs particles are
produced mainly. On the other hand, in the conformally coupled case, the decay
rate into the Higgs particles is strongly suppressed, and the scalaron decays
into Majorana neutrino and the gauge bosons.

In this section, we numerically confirm this by solving the Boltzmann
equations below while the mass of the Majorana fermion $M_{N_{1}}$ is fixed to
be $1.0\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ so that
the decay rate of scalaron into the Majorana fermion takes the maximum (see
(12) and Fig. 1).

$\frac{d\rho_{\phi}}{dt}=-3H\rho_{\phi}-\Gamma_{\phi\to
N_{1}}\rho_{\phi}-\Gamma_{\phi\to g}\rho_{\phi}-\Gamma_{\phi\to
h}\rho_{\phi},$ (18)

$\frac{d\rho_{N_{1}}}{dt}=-3H\rho_{N_{1}}+\Gamma_{\phi\to
N_{1}}\rho_{\phi}-\Gamma_{N_{1}\to R}\rho_{N_{1}},$ (19)

$\frac{d\rho_{R}}{dt}=-4H\rho_{R}+\Gamma_{\phi\to
g}\rho_{\phi}+\Gamma_{\phi\to h}\rho_{\phi}+\Gamma_{N_{1}\to R}\rho_{N_{1}},$
(20)

$H^{2}=\frac{\rho_{\phi}+\rho_{r}+\rho_{N_{1}}}{3M_{\text{G}}^{2}}.$ (21)

where $\rho_{\phi}$, $\rho_{N_{1}}$, $\rho_{R}$ and $H$ are the energy density
of the inflaton, the lightest right-handed Majorana neutrino, radiation and
the Hubble parameter, respectively.

###  3.1 The minimally coupled case

For numerical calculation, the above equations are rewritten with the
dimensionless variables: $\bar{a}=a/a_{I},\ f=\rho_{\phi}M^{-4}\bar{a}^{3},\
n=\rho_{N_{1}}M_{N_{1}}^{-1}M^{-3}\bar{a}^{3},\ r=\rho_{R}M^{-4}\bar{a}^{4}$.
Here, $a_{I}$ is the scale factor at the end of inflation. With these
variables, we can rewrite the set of Boltzmann equations as follows [44].

$\frac{df}{d\bar{a}}=-\frac{\sqrt{3}M_{\text{G}}\left(\Gamma_{\phi\to
N_{1}}+\Gamma_{\phi\to g}+\Gamma_{\phi\to
h}\right)f\bar{a}}{M^{2}\sqrt{r+f\bar{a}+n(M_{N_{1}}/M)\bar{a}}},$ (22)

$\frac{dn}{d\bar{a}}=\frac{\sqrt{3}M_{\text{G}}\left(\Gamma_{\phi\to
N_{1}}(M/M_{N_{1}})f-\Gamma_{N_{1}\to
R}n\right)\bar{a}}{M^{2}\sqrt{r+f\bar{a}+n(M_{N_{1}}/M)\bar{a}}},$ (23)

$\frac{dr}{d\bar{a}}=\frac{\sqrt{3}M_{\text{G}}\left((\Gamma_{\phi\to
g}+\Gamma_{\phi\to h})f+\Gamma_{N_{1}\to
R}(M_{N_{1}}/M)n\right)\bar{a}^{2}}{M^{2}\sqrt{r+f\bar{a}+n(M_{N_{1}}/M)\bar{a}}}.$
(24)

Once the mass of the Majorana neutrino $M_{N_{1}}$ is determined, we are ready
to solve the Boltzmann equations with the initial conditions,

$f(\bar{a}=1)=3H_{\text{inf}}^{2}M_{\text{G}}^{2}M^{-4}\ \text{and}\ \
n(1)=r(1)=0.$ (25)

where $H_{\text{inf}}=M/2$ is the Hubble parameter at the end of the
inflation. The result of numerical calculation of Boltzmann equation when
$M_{N_{1}}$ is
$1.0\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ and the
Higgs field is minimally coupled to scalar curvature $R$ is shown in Fig. 2.

Figure 2: Evolution of the energy densities of several fields as functions of
the rescaled scale factor $\bar{a}$. Here, we assume that the Higgs field is
minimally coupled to gravity. The blue dashed line, the orange solid line and
the green dotted line represent the energy densities of the inflaton,
radiation and the right-handed Majorana neutrino, respectively.

During the inflaton-dominated epoch, $f$ remains approximately constant
because the decay products’ energy density is negligible compared to that of
the inflaton. Thus, $\rho_{\phi}\propto\bar{a}^{-3}$. In this epoch, the
energy density of radiation is [45]

$\rho_{R}\simeq\frac{4}{5}\Gamma_{\phi\to h}M^{3}\bar{a}^{-\frac{3}{2}}.$ (26)

As for the Majorana neutrino $N_{1}$, (23) can be solved as

$\begin{split}&n=\frac{3\Gamma_{\phi\to
N_{1}}M_{\text{G}}^{2}}{4\Gamma_{N_{1}\to R}M_{N_{1}}M}\\\
&-\frac{M}{2\Gamma_{N_{1}\to R}}\exp\left[-\frac{4\Gamma_{N_{1}\to
R}}{3M}\left(\bar{a}^{\frac{3}{2}}-1\right)+\ln\left(\frac{3\Gamma_{\phi\to
N_{1}}M_{\text{G}}^{2}}{2M_{N_{1}}M^{2}}\right)\right].\end{split}$ (27)

When $n$ is small and the decay term in RHS of (23) is negligible, the energy
density of the Majorana neutrino can be approximated as

$\rho_{N_{1}}\simeq\Gamma_{\phi\to
N_{1}}\left(\frac{M}{M_{N_{1}}}\right)M_{\text{G}}^{2}M_{N_{1}}\bar{a}^{-\frac{3}{2}}.$
(28)

Gradually, the effect of the decay of Majorana neutrino into radiation cannot
be ignored after some growth of its energy density $n$. After
$\bar{a}_{t}\simeq$8.8\text{\times}{10}^{2}\text{\,}$$, the source and decay
terms in (23) are balanced, so the energy density $n$ becomes constant and
$\rho_{N}\propto\bar{a}^{-3}$. The comparison of the first and second terms in
RHS of (23) are shown in Fig. 3. It can be seen that the matching period
$\bar{a}_{t}\simeq$8.8\text{\times}{10}^{2}\text{\,}$$ is consistent with the
moment at which the power-law of the energy density of the Majorana neutrino
changes in Fig. 2. This transition period $\bar{a}_{t}$ is actually a function
of the mass of the Majorana fermion $M_{N_{1}}$. It depends on the mass of the
Majorana fermion as

$\bar{a}_{t}=\left[-\frac{3}{2}\left(\frac{2}{3}+\frac{M}{2\Gamma_{N_{1}\to
R}}\ln\left[\frac{3\Gamma_{\phi\to
N_{1}}M_{\text{G}}^{2}}{2M_{N_{1}}M^{2}}\right]\right)\right]^{\frac{2}{3}}$
(29)

which is obtained by substituting (27) into $dn/d\bar{a}=0$ and solving for
$\bar{a}_{t}$. This power-law transition period (29) is a decreasing function
of the mass of the Majorana fermion. This is reasonable because the greater
the mass $M_{N_{1}}$, the larger the density $\rho_{N_{1}}$ produced by the
inflaton decay (see (12)) and the longer it takes to decrease by the decay of
the Majorana neutrino.

Figure 3: The decay or source contribution to the energy density of Majorana
neutrino is shown with respect to $\bar{a}$. The blue solid line and the
orange dashed line represent the first term and the second term in the RHS of
(23).

As the energy density of radiation $\rho_{R}$ decreases slower than that of
inflaton $\rho_{\phi}$, there is a moment when the former reaches the latter:
$\rho_{R}=\rho_{\phi}$. From this moment, the Universe becomes dominated by
the radiation, which leads to the different behavior of the scale factor. We
define the reheating temperature $T_{R}$ as the temperature at this transition
moment. Using Stefan-Boltzman’s law, $\rho_{R}=(\pi^{2}g_{r}/30)T^{4}$, the
reheating temperature is calculated as
$T_{R}=$3.26\text{\times}{10}^{9}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ by
taking into account the degree of freedom of radiation at the reheating:
$g_{r}=106.75$. Similarly, the reheating temperature in the original $R^{2}$
inflation model without heavy Majorana neutrinos is given by
$T_{R}=$3.22\text{\times}{10}^{9}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.
This enhancement is reasonable because the scalaron gets two additional
channels to radiation and the reheating becomes slightly more efficient. This
enhancement of order of one percent in the reheating temperature can also be
checked by $\sqrt{\left(\Gamma_{\phi\to h}+\Gamma_{\phi\to
N_{1}}+\Gamma_{\phi\to g}\right)/\Gamma_{\phi\to h}}\simeq 1.01$, in which we
used the fact that the reheating temperature is proportional to the root of
the total decay rate of the scalaron. Because of the low reheating temperature
$T_{R}\ll M_{N_{1}}$, the right-handed neutrions are not thermalized, and
hence leptogenesis occurs non-thermally.

It is important to confirm that thermalization has been achieved by the end of
reheating so that the reheating temperature calculated above makes sense.
Thermalization proceeds mainly through the inelastic $2\to 3$ scatterings in
the t-channel of gauge bosons, which is more efficient than the $2\to 2$
scattering [46, 47]. This leads to thermal equilibrium. Thermalization rate
$\Gamma_{\text{th}}$ is estimated as follows:

$\Gamma_{\text{th}}\sim\alpha^{3}\left(\frac{M_{\text{G}}}{M}\right)\Gamma_{\phi\to
g}.$ (30)

This exceeds the Hubble rate around the transition to the radiation-dominated
epoch, so the reheating temperature we have calculated makes sense.

###  3.2 The conformally coupled case

The second natural choice for the curvature coupling parameter $\xi$ is
conformal coupling $\xi=-1/6$, which preserves the conformal invariance of the
Higgs kinetic term in this case.999It is well-motivated in the context of
supergravity, too. Thus, we can suppress the decay of the scalaron to the
Higgs particles101010The decay rate through the violation of conformal
invariance by the Higgs mass term is negligible. (see Eq. (11)) and the
branching ratio of the decay into the Majorana neutrino is enhanced.
Therefore, in this case, the decay rate $\Gamma_{\phi\to h}$ is negligible.

Following the same procedure discussed in 3.1, the result is shown in Fig. 4,
again fixing the mass
$M_{N_{1}}=$1.0\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
and assuming $M_{N_{2}},M_{N_{3}}>M/2$.

Figure 4: Results of numerical calculation of Boltzmann equations when the
Higgs field is conformally coupled to $R$. The blue dashed line, the orange
solid line and the green dotted line represent the energy density of the
inflaton, the radiation and the Majorana neutrino, respectively.

From Fig. 4, it can be seen that $\rho_{R}$ is more suppressed in the
conformally coupled case than in the minimally coupled case (Fig. 2). As a
result, the transition period into the radiation-dominated era becomes later
compared to the minimally coupled case. The corresponding reheating
temperature is calculated as
$T_{R}=$5.11\text{\times}{10}^{8}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$111111If
we take the mass of the Majorna field
$M_{1}=$3.0\text{\times}{10}^{12}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$,$1.5\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
so that decay rates (12) and (13) coincide with each other, the corresponding
reheating temperature is
$T_{R}=$2.98\text{\times}{10}^{8}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.
This decrease is reasonable because, in the limit $\Gamma_{\phi\to N_{1}}\to
0$, the reheating temperature must decrease to
$1.4\text{\times}{10}^{8}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ according to
[29], in which the right-handed Majorana neutrino is absent.. In this model,
the Universe is reheated mainly by the subsequent decay of the produced
Majorana neutrino into radiation, unlike the minimally coupled case.

##  4 The effect of the Majorana fermion mass

So far, we have discussed the reheating process by fixing the mass of the
Majorana neutrino. In this section, we discuss the dependence on the mass of
the Majorana neutrino on the reheating process. As we have seen, in the
minimally coupled case, the reheating is governed mainly by the Higgs
particles, thus the variation in the $M_{N_{1}}$ changes the reheating
temperature very little.

On the other hand, in the conformally coupled case, the reheating is realized
mainly by the production and the decay of the Majorana neutrino, thus the
Majorana mass dependence appears. We compare the RHS of (24) below, and we can
see that the contribution from the decay of the Majorana neutrino is the
largest if
$M_{N_{1}}=$1.0\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
as seen in Fig. 5. However, this result depends on the mass of the Majorana
fermion $M_{N_{1}}$. If we decrease the mass of the Majorana femrion, the
contribution from the gauge bosons exceeds that from the decay of the Majorana
neutrino when the Majorana mass $M_{N_{1}}$ is below
$M_{N_{1}}=$1.0\text{\times}{10}^{12}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
which corresponds to the fact that the decay rate into the Majorana neutrino
becomes far sub-dominant: $\Gamma_{\phi\to N_{1}}\ll\Gamma_{\phi\to g}$.

Figure 5: The contribution to radiation from each decay mode (see (24)) is
calculated. The blue dashed line, the orange dotted line and the green dash-
dot line represent the contribution from the decay of the Majorana neutrino
$N_{1}$ with mass
$M_{N_{1}}=$1.0\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
$M_{N_{1}}=$1.0\text{\times}{10}^{12}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
and
$M_{N_{1}}=$1.0\text{\times}{10}^{11}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.
The red solid line represent the contribution from the inflaton decay into the
gauge bosons when the Majorana mass
$M_{N_{1}}=$1.0\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.
As for the contribution from the gauge bosons, the change in the Majorana mass
only results in the change in the moment of the transition to the exponential
decay.

The consequence of this can be seen in the behavior of $\rho_{R}$ on
$M_{N_{1}}$ near the transition of power law. We show this behavior in Fig. 6.
This mass dependence cannot be found in the minimally coupled case.

As we lower the mass of the Majorana neutrino, the energy density at the
transition to radiation domination becomes smaller, resulting in the lower
reheating temperature. The difference leads to the difference in reheating
temperature. This leads to the difference in the spectral indices, which will
be discussed in 5.1.

Figure 6: The blue dashed line, the orange dotted line, the green solid line
and the red dash-dot line represent the evolution of the energy density of
radiation $\rho_{R}$ with different masses:
$M_{N_{1}}=$1.0\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
$M_{N_{1}}=$1.0\text{\times}{10}^{12}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
$M_{N_{1}}=$1.0\text{\times}{10}^{11}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
and
$M_{N_{1}}=$1.0\text{\times}{10}^{10}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
respectively.

However, it can be seen that the mass dependence disappears below around
$1.0\text{\times}{10}^{12}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ because the
decay rate of Majorana fermion into radiation becomes so small (see Fig. 5)
that the reheating is switched to be governed by the scalaron decay into gauge
bosons rather than by the decay of the Majorana fermion. In this case, the
only relevant decay channel is the decay into gauge bosons as we have
suppressed the decay channel into Higgs bosons and Majorana fermions by taking
conformal coupling and taking the small mass, respectively.

This mass dependence of the transition period results in the mass dependence
of the reheating temperature $T_{R}$. This is approximately described as
121212This can be obtained by naively taking the equality $\Gamma_{\phi}=H$.

$T_{R}\simeq 1.1\times
10^{9}\times\left(\frac{M_{N_{1}}}{${10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$}\right)\left(1-\frac{4M_{N_{1}}^{2}}{M^{2}}\right)^{\frac{3}{4}}$\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
(31)

when
$$2.0\text{\times}{10}^{12}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$<M_{N_{1}}<$1.5\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
and the decay rate $\Gamma_{\phi\to N_{1}}$ exceeds the decay rate
$\Gamma_{\phi\to g}$ (see Fig. 1). If this condition fails $\Gamma_{\phi\to
g}>\Gamma_{\phi\to N_{1}}$, the mass dependence of the reheating temperature
disappears:
$T_{R}\simeq$2.2\text{\times}{10}^{8}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.
This approximated expression almost coincides with the numerical results, but
we will use the numerical results for the accurate calculation of the spectral
indices.

##  5 Parameter dependence of the obsevables

We have seen that the reheating process heavily depends on the non-minimal
coupling. The dependence on $M_{N_{1}}$ is negligiblly small in the minimally
coupled case, because the scalaron mainly decays into the Higgs particles, and
the branching ratio into the Majorana neutrino becomes very small. On the
contrary, the mass dependence does appear in the conformally coupled case for
$M_{N_{1}}>$1.0\text{\times}{10}^{12}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
because the decay into the Higgs particles is suppressed and the branching
ratio into the Majorana fermion becomes large. In the previous section, we
confirmed this by numerically solving the Boltzmann equations.

In this section, we discuss the above argument by calculating the parameter
dependence of the observables, namely the spectral indices and the baryon
asymmetry.

###  5.1 Spectral indices

The period of the transition from the matter-dominated epoch to the radiation-
dominated epoch determines the duration for the perturbation corresponding to
the CMB pivot scale
$k_{\text{CMB}}=$0.002\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}^{-1}$$ to
experience the inflationary period, which is measured by the e-folding number
$N$ [25, 48, 49]

$N\simeq
57-\frac{1}{3}\log\frac{${10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$}{T_{r}}.$
(32)

where $l_{p}$ represents the Planck length and the subscripts $0$, $r$, $e$
and $*$ denote the value at present time, the end of reheating, the end of
inflation and the moment when the perturbation crossed the horizon. Here, we
defined the end of the inflation as the moment when the slow-roll parameter
$\epsilon=M_{\text{G}}^{2}(V^{\prime}/V)^{2}/2=1$ becomes one. When we derive
(32), we used the fact that the change of the equation of state is
instantaneous, which can be confirmed numerically.

This e-folding number $N$ can also be expressed as the integral of $\phi$:

$N=\frac{1}{M_{\text{G}}^{2}}\int_{\phi_{e}}^{\phi_{*}}d\phi\frac{V[\phi]}{V^{\prime}[\phi]}.$
(33)

The slow-roll parameters for the evaluation of the properties of the
cosmological perturbation measured by the CMB are calculated by this field
value $\phi_{*}$ because the curvature perturbation is conserved after the
horizon crossing:

$\left\\{\begin{array}[]{ll}\epsilon&=\left.\frac{1}{2}M_{\text{G}}^{2}\left(\frac{V^{\prime}(\phi)}{V(\phi)}\right)\right|_{\phi=\phi_{*}},\\\
\eta&=\left.M_{\text{G}}^{2}\left(\frac{V^{\prime\prime}(\phi)}{V(\phi)}\right)\right|_{\phi=\phi_{*}},\\\
\zeta^{2}&=\left.M_{\text{G}}^{4}\left(\frac{V^{\prime}(\phi)V^{\prime\prime\prime}(\phi)}{V(\phi)^{2}}\right)\right|_{\phi=\phi_{*}}.\end{array}\right.$
(34)

The spectral indices are calculated by using these slow roll parameters as

$\left\\{\begin{array}[]{ll}n_{s}&=1+2\eta-6\epsilon-\frac{2}{3}\eta^{2}+0.374\zeta^{2},\\\
n_{T}&=-2\epsilon,\\\ r&=16\epsilon.\end{array}\right.$ (35)

We show the mass dependence of the spectral indices in the minimally coupled
case and the conformally coupled case in Fig. 7. In addition to these, we show
the values of the Higgs inflation model for comparison131313Here, we assume
that the energy of the inflaton is immediately converted to the energy of the
radiation after the end of inflation due to the violent preheating [23, 24]..

Figure 7: The blue solid line and the orange solid line represent the mass
dependence of the spectral indices $(n_{s},r)$ in the minimally coupled case
and the conformally coupled case, respectively. The red point represent the
value in the case of the Higgs inflation. In the upper graph, we
simultaneously plotted the $68\%$ and $95\%$ CL region for the spectral
indices from TT, TE, EE$+$lowE$+$lensing$+$BK15$+$BAO data [25]. In the lower
graph, for the conformally coupled case, we re-plotted the value of
$(n_{s},r)$ when we take different values for the Majorana mass:
$M_{N_{1}}=$1.0\text{\times}{10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
$M_{N_{1}}=$1.0\text{\times}{10}^{12}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
$M_{N_{1}}=$1.0\text{\times}{10}^{11}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
and
$M_{N_{1}}=$1.0\text{\times}{10}^{10}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
correspond to the circle, the triangle, the vertical line and the horizontal
line.

We can see that the mass dependence disappears in the minimally coupled case.
On the other hand, the mass dependence is manifest in the conformally coupled
case and as we diminish the mass of the Majorana fermion, gradually the main
role of the reheating switches to the decay into gauge bosons, so the mass
dependence disappears below around
$M_{N_{1}}\simeq${10}^{11}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ as we have
seen in section 4. In order to distinguish the right-handed neutrino mass with
the future CMB observation, we require the accuracy of $\mathcal{O}(10^{-4})$
for $n_{s}$ and of $\mathcal{O}(10^{-5})$ for $r$.

###  5.2 Baryon asymmetry

Leptogenesis is one of the most promising baryogenesis scenarios [30]. In this
scenario, lepton asymmetry, which is produced when the heavy Majorana neutrino
decays, is converted to the baryon asymmetry by the subsequent sphaleron
process [50, 51, 52, 53]. The net lepton asymmetry produced by one $N_{1}$
particle decay is expressed as:

$\delta\equiv\frac{\Gamma(N_{1}\to
lh)-\Gamma(N_{1}\to\bar{l}h)}{\Gamma(N_{1}\to lh)+\Gamma(N_{1}\to\bar{l}h)}.$
(36)

This asymmetry comes from the interference terms between the tree-level
diagram and the one-loop diagram. Including both one-loop vertex and self-
energy correction, we obtain [30, 54, 55]

$\begin{split}\delta&\simeq-\frac{3M_{1}}{16\pi y_{1\rho}y_{\rho
1}^{*}}\text{Im}\left[y_{1\alpha}y_{1\beta}\left(y_{\gamma\alpha}^{*}\frac{1}{M_{N_{\gamma}}}y_{\gamma\beta}^{*}\right)\right]\\\
&\equiv-\frac{3\delta_{\text{eff}}}{16\pi y_{1\rho}y_{\rho
1}^{*}}\left|\left(y_{1\alpha}y_{\alpha
2}^{\dagger}\right)^{2}\frac{M_{N_{1}}}{M_{N_{2}}}+\left(y_{1\alpha}y_{\alpha
3}^{\dagger}\right)^{2}\frac{M_{N_{1}}}{M_{N_{3}}}\right|\\\
&\simeq-\frac{3\delta_{\text{eff}}}{16\pi}y_{1\alpha}y_{\alpha
1}^{*}\frac{M_{N_{1}}}{M_{N_{2}}}\\\
&\simeq\frac{3}{8\pi}\frac{M_{N_{1}}^{2}m_{3}}{M_{N_{2}}v^{2}}\delta_{\text{eff}},\end{split}$
(37)

In the first line in (37), we used $M_{N_{\gamma\neq 1}}\gg M_{N_{1}}$ for the
approximation. In the second line in, we defined the effective CP phase in the
Yukawa matrix:

$\delta_{\text{eff}}\equiv\frac{\text{Im}\left[\left(y_{1\alpha}y_{\alpha
2}^{\dagger}\right)^{2}\frac{M_{N_{1}}}{M_{N_{2}}}+\left(y_{1\alpha}y_{\alpha
3}^{\dagger}\right)^{2}\frac{M_{N_{1}}}{M_{N_{3}}}\right]}{\left|\left(y_{1\alpha}y_{\alpha
2}^{\dagger}\right)^{2}\frac{M_{N_{1}}}{M_{N_{2}}}+\left(y_{1\alpha}y_{\alpha
3}^{\dagger}\right)^{2}\frac{M_{N_{1}}}{M_{N_{3}}}\right|}.$ (38)

So, this quantity is less than one by definition: $\delta_{\text{eff}}\leq 1$.
In the third line in (37), we used the asuumption that the all the components
in the Yukawa matrix are the same order and $M_{N_{2}}\ll M_{N_{3}}$, which
are mentioned in 2.2.

The matching condition for the baryon asymmetry $n_{B}/s\simeq 0.87\times
10^{-10}$ imposes restriction on the mass prameters. The lepton asymmetry is
generated through the non-thermal leptogenesis [55, 56] because of the low
reheating temperature and the large Majorana fermion mass, which realizes
sizable gravitational particle production. Hence, there is no wash-out effect:

$\frac{dn_{L}a^{3}}{dt}=\delta\cdot\Gamma_{N\to R}n_{N}a^{3}.$ (39)

In terms of a rescaled variable: $\bar{n}_{L}\equiv n_{L}\bar{a}^{3}M^{-3}$,
we find

$\frac{d\bar{n}_{L}}{d\bar{a}}=\delta\cdot\frac{\sqrt{3}M_{\text{G}}\Gamma_{N_{1}\to
R}n\bar{a}}{M^{2}\sqrt{r+f\bar{a}+n(M_{N_{1}}/M)\bar{a}}}.$ (40)

The generated lepton asymmetry $n_{L}$ can be calculated by integrating (40).
The lepton asymmetry is most generated around the end of the reheating, which
is far long before the period where the shaleron process to occur. After
thermalization, $n_{L}/s$ is given by

$\frac{n_{L}}{s}=\left(\frac{3^{5}\cdot
5}{2^{7}\pi^{2}g_{r}\rho_{R}^{3}}\right)^{\frac{1}{4}}\cdot\frac{\bar{n}_{L}M^{3}}{\bar{a}^{3}}.$
(41)

This lepton number density per entropy density is conserved quantity after
reheating ($\because n_{L}\propto a^{-3}$) and this is converted to the baryon
asymmetry when the sphaleron process occurs:

$\frac{n_{B}}{s}\simeq-\frac{28}{79}\frac{n_{L}}{s}.$ (42)

We calculated the mass dependence of the resultant ratio of the baryon
asymmetry to the entropy $n_{B}/\left(s\delta\right)$ in the conformally
coupled case and in the minimally coupled case in Fig. 8. We can see that the
produced baryon asymmetry is larger in the case of the conformally coupled
case, which is due to the enhancement of the branching ratio of
$\Gamma_{\phi\to N_{1}}$.

Figure 8: The baryon asymmetry to the entropy $n_{B}/\left(s\delta\right)$ is
plotted with respect to the Majorana mass $M_{N_{1}}$. The blue solid line and
the orange dashed line represent the mass dependence of
$n_{B}/\left(s\delta\right)$ in the conformally coupled case and in the
minimally coupled case.

Furthermore, the mass dependence is larger in the conformally coupled case
than in the minimally coupled case because the decay into the Higgs field is
suppressed and the branching ratio into the Majorana fermion becomes larger.

Next, we discuss the value of the mass $M_{N_{1}}$ at which
$n_{B}/\left(s\delta\right)$ takes the maximum. In the minimally coupled case,
the entropy is governed by the Higgs particles, not by the decay products by
the Majorana neutrino, so is less affected by the change in mass $M_{N_{1}}$.
Hence, $n_{B}/\left(s\delta\right)$ gets maximum near the mass at which the
$\Gamma_{\phi\to N_{1}}$ (see (12)) gets maximum
($M_{N_{1}}\simeq${10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$). On the
other hand, in the conformally coupled case, the entropy is governed by the
decay products of the Majorana neutrinos, so is affected by the change in the
mass $M_{N_{1}}$. Thus, the mass $M_{N_{1}}$ at which
$n_{B}/\left(s\delta\right)$ gets maximum deviates from
$M_{N_{1}}\simeq${10}^{13}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.

The different behavior in Fig. 8 means that there is a difference in the
restriction in the parameter we have. We show the limitation on the mass
prameters: $M_{N_{1}}$ and $M_{N_{2}}$ in Fig. 9.

Figure 9: The allowed region for the mass prameters are shown. The blue ray-
shaded region and the orange dot-shaded region represent the conformally
coupled case and the minimally coupled case. In the green shaded region, the
assumption we made is broken and the resonant leptogenesis can occur, so the
result is not reliable there. The mass configuration must be on the red solid
line in order to be consistent with the solar neutrino oscillation experiment.

From Fig. 8, one can read off the required $\delta$ to explain the baryon
asymmetry of the Universe for each $M_{N_{1}}$. The upper-bound of $M_{N_{2}}$
can be obtained by the condition $\delta_{\text{eff}}\leq 1$, which is shown
in Fig. 9. We can see that the allowed range for the mass parameters
$\delta_{\text{eff}}$ is larger in the conformally coupled case due to the
enhanced efficiency in the leptogenesis. We simultaneously showed the region
where our analysis is not reliable
$\left|M_{N_{2}}-M_{N_{1}}\right|<M_{N_{1}}$ because we used the assumption
$M_{N_{\gamma\neq 1}}\gg M_{N_{1}}$ in the first line of (37). In general, the
resonant leptogenesis [57] can occur in this region, so further analysis is
required. We also showed the mass configuration in which $m_{2}\simeq
m_{\text{sol}}=$0.009\text{\,}\mathrm{e}\mathrm{V}$$ is realized. This
condition constraints the mass of the second lightest right-handed Majorana
neutrino $M_{N_{2}}$. This constraint is strongly dependent on the democratic
assumption on the neutrino Yukawa matrix. It is highly probable that these
constraints on the Majorana masses $M_{N_{\alpha}}$ can change in other
assumptions for the Yukawa matrix structure because this results in the change
in the expression for the decay rate $\Gamma_{N\to R}$ (see (14)) and the net
lepton asymmetry $\delta$ (see (37)), which strongly influences our whole
analysis above.

##  6 Conclusion and Discussion

We have analyzed the $R^{2}$ inflation model with the extended matter sector,
which possesses the leptogenesis scenario, which is a more realistic
cosmological model. We conducted a comprehensive study of the reheating
process by considering all of the matter fields, by identifying the relevant
channels and by solving the evolution equation of the system. There are two
parameters that characterize this model: the non-minimal coupling $\xi$ and
the mass of the Majorana fermion $M_{N_{1}}$. The effect of the parameter
$\xi$ is more relevant in the reheating process because it regulates the decay
rate of the scalaron into the Higgs field. In the minimally coupled case
($\xi=0$), the decay rate $\Gamma_{\phi\to h}$ is so large that the reheating
is mainly realized by this decay, so the reheating temperature is higher and
is determined only by this decay rate, and the Majorana mass dependence does
not come into play. On the other hand, in the conformally coupled case, the
decay rate $\Gamma_{\phi\to h}$ is much suppressed, thus the decay into the
right-handed Majorana neutrino and into gauge bosons becomes relevant. Which
decay is more relevant is determined by the Majorana mass $M_{N_{1}}$, so the
mass dependence appears in this case.

We have confirmed the above statement by seeing the mass dependence of the
reheating process and the observables: the spectral indices and the baryon
asymmetry. As we have seen in Fig. 7 and Fig. 8, the mass dependence appears
in the conformally coupled case and less appears in the minimally coupled
case. Furthermore, we showed the limitation on the mass parameters by
considering the consistency with the value of the baryon asymmetry to entropy
ratio estimated from the observation.

There is still a remaining task that have to be tackled. As discussed in the
section 5.2, the assumption for the leptogenesis in 2.2 can be modified. This
leads to different decay rate $\Gamma_{N_{1}\to R}$ and different net lepton
asymmetry $\delta$, so the above analysis will change. We need to do the
exhaustive analysis to examine all the possiblities in the leptogenesis
scenario and to get the thorough constraints on the Majorana mass and the
Yukawa matrix.

For future work, the analysis for the continuous $\xi$ can be interesting,
especially in the large coupling limit $\xi\to\infty$. As it has been studied
in the Higgs inflation model [17, 18, 19, 20, 21, 22, 23, 24], it can be
expected that some violent production of the right-handed Majorana neutrino
will occur via the see-saw Lagrangian. Because of the high reheating
temperature, leptogenesis can occur thermally in contrast to the case of our
research. In any case, the leptogenesis in the Mixed $R^{2}$-Higgs
inflationary model [58, 26] will be the subject for the future work.

## Acknowledgements

We thank A. Kamada, Y. Watanabe for useful discussion. The work of H. J. is
supported by the Forefront Physics and Mathematics Program to Drive
Transformation (FoPM). A. A. S. was partially supported by the RSF grant
21-12-00130. J. Y. is supported by JSPS KAKENHI Grant No. 20H05639 and Grant-
in-Aid for Scientific Research on Innovative Areas 20H05248. K. K. is
supported by JSPS KAKENHI, Grant-in-Aid for Sci- entific Research No. (C)
JP19K03842.

## References

  * [1] Alexei A Starobinsky.  A new type of isotropic cosmological models without singularity.  Physics Letters B, 91(1):99–102, 1980.
  * [2] Katsuhiko Sato.  First-order phase transition of a vacuum and the expansion of the universe.  Monthly Notices of the Royal Astronomical Society, 195(3):467–479, 1981.
  * [3] Alan H Guth.  Inflationary universe: A possible solution to the horizon and flatness problems.  Physical Review D, 23(2):347, 1981.
  * [4] Andrei D Linde.  A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems.  Physics Letters B, 108(6):389–393, 1982.
  * [5] Andreas Albrecht and Paul J Steinhardt.  Cosmology for grand unified theories with radiatively induced symmetry breaking.  Physical Review Letters, 48(17):1220, 1982.
  * [6] Alexei A Starobinsky.  Nonsingular Model of the Universe with the Quantum Gravitational De Sitter Stage and its Observational Consequences.  In Proceedings of the Second Seminar “Quantum Theory of Gravity”, Moscow Oct. 1981, INR Press, Moscow (1982), p. 58-72, 1981.
  * [7] Alexei A Starobinsky.  The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy.  Sov. Astron. Lett., 9:302–304, 1983.
  * [8] Alexander Vilenkin.  Classical and Quantum Cosmology of the Starobinsky Inflationary Model.  Phys. Rev. D, 32:2511, 1985.
  * [9] Milan B Mijic, Michael S Morris, and Wai-Mo Suen.  The R**2 Cosmology: Inflation Without a Phase Transition.  Phys. Rev. D, 34:2934, 1986.
  * [10] Kei-ichi Maeda.  Inflation as a Transient Attractor in R**2 Cosmology.  Phys. Rev. D, 37:858, 1988.
  * [11] Ya B Zeldovich and A A Starobinsky.  Particle production and vacuum polarization in an anisotropic gravitational field.  Sov. Phys. JETP, 34:1159–1166, 1972.
  * [12] Leonard Parker and S A Fulling.  Adiabatic regularization of the energy momentum tensor of a quantized field in homogeneous spaces.  Phys. Rev. D, 9:341–354, 1974.
  * [13] N D Birrell and P C W Davies.  Quantum fields in curved space.  Cambridge University press, 1984.
  * [14] H. Nariai and K. Tomita.  On the removal of initial singularity in a big-bang universe in terms of a renormalized theory of gravitation. 2. criteria for obtaining a physically reasonable model.  Prog. Theor. Phys., 46:776–786, 1971.
  * [15] Thomas Faulkner, Max Tegmark, Emory F Bunn, and Yi Mao.  Constraining f (r) gravity as a scalar-tensor theory.  Physical Review D, 76(6):063505, 2007.
  * [16] Tibério de Paula Netto, Ana M Pelinson, Ilya L Shapiro, and Alexei A Starobinsky.  From stable to unstable anomaly-induced inflation.  The European Physical Journal C, 76(10):1–14, 2016.
  * [17] Jorge L Cervantes-Cota and H Dehnen.  Induced gravity inflation in the standard model of particle physics.  Nuclear Physics B, 442(1-2):391–409, 1995.
  * [18] Fedor Bezrukov and Mikhail Shaposhnikov.  The standard model higgs boson as the inflaton.  Physics Letters B, 659(3):703–706, 2008.
  * [19] A O Barvinsky, A Yu Kamenshchik, and A A Starobinsky.  Inflation scenario via the Standard Model Higgs boson and LHC.  JCAP, 11:021, 2008.
  * [20] Fedor Bezrukov, Dmitry Gorbunov, and Mikhail Shaposhnikov.  On initial conditions for the hot big bang.  Journal of Cosmology and Astroparticle Physics, 2009(06):029, 2009\.
  * [21] A O Barvinsky, A Yu Kamenshchik, C Kiefer, A A Starobinsky, and C Steinwachs.  Asymptotic freedom in inflationary cosmology with a non-minimally coupled Higgs field.  JCAP, 12:003, 2009.
  * [22] Juan Garcia-Bellido, Daniel G Figueroa, and Javier Rubio.  Preheating in the standard model with the higgs inflaton coupled to gravity.  Physical Review D, 79(6):063531, 2009.
  * [23] Yohei Ema, Ryusuke Jinno, Kyohei Mukaida, and Kazunori Nakayama.  Violent preheating in inflation with nonminimal coupling.  Journal of Cosmology and Astroparticle Physics, 2017(02):045, 2017\.
  * [24] Matthew P DeCross, David I Kaiser, Anirudh Prabhu, Chanda Prescod-Weinstein, and Evangelos I Sfakianakis.  Preheating after multifield inflation with nonminimal couplings. i. covariant formalism and attractor behavior.  Physical Review D, 97(2):023526, 2018.
  * [25] Y. Akrami et al.  Planck 2018 results. X. Constraints on inflation.  Astron. Astrophys., 641:A10, 2020.
  * [26] Minxi He, Alexei A Starobinsky, and Jun’ichi Yokoyama.  Inflation in the mixed Higgs-$R^{2}$ model.  JCAP, 05:064, 2018.
  * [27] D S Gorbunov and A G Panin.  Scalaron the mighty: producing dark matter and baryon asymmetry at reheating.  Physics Letters B, 700(3-4):157–162, 2011.
  * [28] D S Gorbunov and A G Panin.  Free scalar dark matter candidates in r2-inflation: the light, the heavy and the superheavy.  Physics Letters B, 718(1):15–20, 2012.
  * [29] Dmitry Gorbunov and Anna Tokareva.  R2-inflation with conformal sm higgs field.  Journal of Cosmology and Astroparticle Physics, 2013(12):021, 2013\.
  * [30] Masataka Fukugita and Tsutomu Yanagida.  Barygenesis without grand unification.  Physics Letters B, 174(1):45–47, 1986.
  * [31] Kei-ichi Maeda.  Towards the einstein-hilbert action via conformal transformation.  Physical Review D, 39(10):3159, 1989.
  * [32] Yuki Watanabe.  Rate of gravitational inflaton decay via gauge trace anomaly.  Physical Review D, 83(4):043511, 2011.
  * [33] A Morozov.  Anomalies in gauge theories.  Soviet Physics Uspekhi, 29(11):993, 1986.
  * [34] Ayuki Kamada.  On Scalaron Decay via the Trace of Energy-Momentum Tensor.  JHEP, 07:172, 2019.
  * [35] Ayuki Kamada and Takumi Kuwahara.  Lessons from $T^{\mu}_{~{}\mu}$ on inflation models: Two-loop renormalization of $\eta$ in the scalar QED.  Phys. Rev. D, 103(11):116001, 2021.
  * [36] Ayuki Kamada and Takumi Kuwahara.  Lessons from $T^{\mu}_{~{}\mu}$ on inflation models: Two-scalar theory and Yukawa theory.  Phys. Rev. D, 101(9):096012, 2020.
  * [37] Herbi K Dreiner, Howard E Haber, and Stephen P Martin.  Two-component spinor techniques and feynman rules for quantum field theory and supersymmetry.  Physics Reports, 494(1-2):1–196, 2010.
  * [38] Kenneth M Case.  Reformulation of the majorana theory of the neutrino.  Physical Review, 107(1):307, 1957.
  * [39] Palash B Pal.  Dirac, majorana, and weyl fermions.  American Journal of Physics, 79(5):485–498, 2011.
  * [40] A A Starobinsky.  Evolution of small perturbations of isotropic cosmological models with one-loop quantum gravitational corrections.  JETP Lett., 34:438–441, 1981.
  * [41] Y Ema, R Jinno, K Mukaida, and K Nakayama.  Gravitational particle production in oscillating backgrounds and its cosmological implications.  Phys. Rev. D, 94:063517, 2016.
  * [42] Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, Gian F. Giudice, Filippo Sala, Alberto Salvio, and Alessandro Strumia.  Investigating the near-criticality of the Higgs boson.  JHEP, 12:089, 2013.
  * [43] MC Gonzalez-Garcia, Michele Maltoni, Jordi Salvado, and Thomas Schwetz.  Global fit to three neutrino mixing: critical look at present precision.  Journal of High Energy Physics, 2012(12):1–24, 2012.
  * [44] Daniel J H Chung, Edward W Kolb, and Antonio Riotto.  Production of massive particles during reheating.  Physical Review D, 60(6):063504, 1999.
  * [45] Edward W. Kolb and Michael S. Turner.  The Early Universe, volume 69.  Nature Publishing Group UK London, 1990.
  * [46] Sacha Davidson and Subir Sarkar.  Thermalisation after inflation.  Journal of High Energy Physics, 2000(11):012, 2000.
  * [47] Rouzbeh Allahverdi, Robert Brandenberger, Francis-Yan Cyr-Racine, and Anupam Mazumdar.  Reheating in inflationary cosmology: theory and applications.  Annual Review of Nuclear and Particle Science, 60:27–51, 2010.
  * [48] Daniel Baumann.  Cosmology.  Cambridge University Press, 2022.
  * [49] Katsuhiko Sato and Jun’ichi Yokoyama.  Inflationary cosmology: First 30+ years.  International Journal of Modern Physics D, 24(11):1530025, 2015\.
  * [50] N. S. Manton.  Topology in the Weinberg-Salam Theory.  Phys. Rev. D, 28:2019, 1983.
  * [51] Frans R Klinkhamer and Nicholas S Manton.  A saddle-point solution in the weinberg-salam theory.  Physical Review D, 30(10):2212, 1984.
  * [52] Peter Arnold and Larry McLerran.  Sphalerons, small fluctuations, and baryon-number violation in electroweak theory.  Physical Review D, 36(2):581, 1987.
  * [53] V A Kuzmin, V A Rubakov, and M E Shaposhnikov.  On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe.  Phys. Lett. B, 155:36, 1985.
  * [54] Laura Covi, Esteban Roulet, and Francesco Vissani.  Cp violating decays in leptogenesis scenarios.  Physics Letters B, 384(1-4):169–174, 1996.
  * [55] T Asaka, Koichi Hamaguchi, M Kawasaki, and T Yanagida.  Leptogenesis in inflaton decay.  Physics Letters B, 464(1-2):12–18, 1999.
  * [56] Takehiko Asaka, Holger Bech Nielsen, and Yasutaka Takanishi.  Non-thermal leptogenesis from the heavier majorana neutrinos.  Nuclear Physics B, 647(1-2):252–274, 2002.
  * [57] Apostolos Pilaftsis and Thomas E J Underwood.  Resonant leptogenesis.  Nucl. Phys. B, 692:303–345, 2004.
  * [58] Yohei Ema.  Higgs Scalaron Mixed Inflation.  Phys. Lett. B, 770:403–411, 2017.