# Inflationary Gravitational Wave Spectral Shapes
as test for Low-Scale Leptogenesis

Zafri A Borboruah<EMAIL_ADDRESS>Indian Institute of Technology Bombay,
Mumbai 400076, Maharashtra, India  Anish Ghoshal<EMAIL_ADDRESS>Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland  Lekhika Malhotra
<EMAIL_ADDRESS>Indian Institute of Technology Bombay, Mumbai
400076, Maharashtra, India  Urjit Yajnik<EMAIL_ADDRESS>Indian Institute
of Technology Gandhinagar, Gandhinagar 382055, India  Indian Institute of
Technology Bombay, Mumbai 400076, Maharashtra, India

###### Abstract

We study thermal and non-thermal resonant leptogenesis in a general setting
where a heavy scalar $\phi$ decays to right-handed neutrinos (RHNs) whose
further out-of-equilibrium decay generates the required lepton asymmetry.
Domination of the energy budget of the Universe by the $\phi$ or the RHNs
alters the evolution history of the primordial gravitational waves (PGW), of
inflationary origin, which re-enter the horizon after inflation, modifying the
spectral shape. The decays of $\phi$ and RHNs release entropy into the early
Universe while nearly degenerate RHNs facilitate low and intermediate scale
leptogenesis. We show that depending on the coupling $y_{R}$ of $\phi$ to
radiation species, RHNs can achieve thermal abundance before decaying, which
gives rise to thermal leptogenesis. A characteristic damping of the GW
spectrum resulting in two knee-like features or one knee-like feature would
provide evidence for low-scale thermal and non-thermal leptogenesis
respectively. We explore the parameter space for the lightest right-handed
neutrino mass $M_{1}\in[10^{2},10^{14}]$ GeV and washout parameter $K$ that
depends on the light-heavy neutrino Yukawa couplings $\lambda$, in the weak
($K<1$) and strong ($K>1$) washout regimes. The resulting novel features
compatible with observed baryon asymmetry are detectable by future experiments
like LISA and ET. By estimating signal-to-noise ratio (SNR) for upcoming GW
experiments, we investigate the effect of the scalar mass $M_{\phi}$ and
reheating temperature $T_{\phi}$, which depends on the $\phi-N$ Yukawa
couplings $y_{N}$.

###### Contents

  1. I Introduction
  2. II Thermal Leptogenesis
    1. II.1 Conditions for Departure from Thermal Equilibrium
    2. II.2 CP Asymmetry
    3. II.3 Lower Bounds on Heavy Neutrino Masses
    4. II.4 Resonant Leptogenesis
  3. III Non-thermal Leptogenesis via scalar decay
    1. III.1 Set Up
    2. III.2 The Boltzmann Equations
  4. IV Classification of Scenarios
    1. IV.1 Analytical Estimates of Equilibrium Temperature
    2. IV.2 Dilution Factor from Entropy Injection
  5. V Gravitational Waves of Inflationary Origin
    1. V.1 Primordial spectrum in standard and non-standard cosmology
    2. V.2 Two-step entropy injection
    3. V.3 Signal-to-noise ratio (SNR)
    4. V.4 Bounds from BBN and CMB on dark radiation
  6. VI Results
    1. VI.1 Example spectra for different cases
    2. VI.2 Estimates for Signal to Noise Ratio
    3. VI.3 Compatibility with NANOGrav
  7. VII Discussion & Conclusion
  8. A Change of Variable
  9. B Thermalization of RHNs

##  I Introduction

The Standard Model (SM) of particle physics is unable to explain tiny neutrino
masses and the observed matter-antimatter asymmetry of the Universe. Neutrino
oscillation experiments involving solar Super-Kamiokande:2001bfk ; Super-
Kamiokande:2002ujc ; SNO:2002tuh ; Super-Kamiokande:2005mbp ; Super-
Kamiokande:2016yck ; Borexino:2015axw , atmospheric IceCube:2017lak ;
ANTARES:2018rtf  and reactor KamLAND:2008dgz ; T2K:2011ypd ;
DoubleChooz:2011ymz ; T2K:2013ppw  neutrinos provide evidence that neutrinos
have mass and the flavor states mix due to the propagation of multiple mass
eigenstates. This however only quantifies the squared mass differences of the
neutrinos and not the absolute mass scales involved. On the other hand, the
$\beta$-decay experiment KATRIN KATRIN:2021uub  gives a stringent direct limit
on the absolute value of neutrino mass scale, $m_{\nu}<0.8$ eV.

On the cosmological frontiers, measurements of the Cosmic Microwave Background
radiation (CMBR) by Planck 2018 Planck:2018vyg  and large-scale structure
(LSS) constrain the sum of all neutrino masses to $\sum_{i}m_{\nu_{i}}<0.12$
eV Aghanim:2018eyx ; eBOSS:2020yzd . On the other hand, the observed baryon
asymmetry of the Universe (BAU) Zyla:2020zbs ; Aghanim:2018eyx , often
expressed in terms of the baryon to photon ratio is, according to the Planck
data Aghanim:2018eyx ,

$\eta_{B}^{\rm CMB}=\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}}=(6.21\pm 0.16)\times
10^{-10}$ (1)

which is in agreement with the values of abundances of light elements
extracted from BBN (Big Bang Nucleosynthesis) data Fields:2019pfx .

Just augmenting the Standard Model (SM) with two or more right-handed Majorana
neutrinos (RHNs), one can explain the tiny neutrino masses, generated via the
well-known Type-I seesaw mechanism Minkowski:1977sc ; Yanagida:1979as ;
Glashow:1979nm ; Mohapatra:1979ia , while baryon asymmetry can be explained
using the baryogenesis via leptogenesis mechanism Fukugita:1986hr , everything
tied together under one umbrella. However for this mechanism of baryogenesis
to be successful, the lightest RHN mass is required to exceed the Davidson-
Ibarra bound, $M_{1}\gtrsim 10^{9}$ GeV, assuming the RHN mass spectrum is
hierarchical, and also assuming no lepton flavor effects Nardi:2006fx ;
Abada:2006fw ; Abada:2006ea ; Davidson:2002qv ; Giudice:2003jh . This destroys
any hope of direct detection of the high-scale seesaw and leptogenesis as they
lie well beyond the energy reach of the current or foreseeable future
laboratory experiments or astrophysical observations making it impossible to
test such new physics.

Nonetheless, certain indirect signatures of new physics like lepton number
violation processes through neutrinoless double beta decay Cirigliano:2022oqy
or CP violation in neutrino oscillation Endoh:2002wm  are searched for in
laboratories. On the other hand, theoretical constraints on the low energy
values of couplings, consistent with UV-completion such as $SO(10)$ Grand
Unified Theories (GUT) DiBari:2008mp ; Bertuzzo:2010et ; Buccella:2012kc ;
Altarelli:2013aqa ; Fong:2014gea ; Mummidi:2021anm ; Patel:2022xxu  or with
the requirement of electroweak (EW) vacuum (meta)stability in the early
Universe Ipek:2018sai ; Croon:2019dfw  allow us to pose some bounds on the
large parameter space involved in seesaw and leptogenesis.

The discovery of Gravitational Waves (GWs) from black hole mergers by LIGO and
Virgo collaboration LIGOScientific:2016aoc ; LIGOScientific:2016sjg  and
evidence for measurements of stochastic GW background by pulsar timing array
(PTA) observations Carilli:2004nx ; Janssen:2014dka ; Weltman:2018zrl ;
EPTA:2015qep ; EPTA:2015gke ; NANOGrav:2023gor ; NANOGrav:2023hvm  have lead
to several new physics cases to detect GWs of primordial origins as well.
Particularly in the context of baryogenesis via leptogenesis, cosmological
pathways to probe such high-scale physics involve predictions of the CMB
spectral indices Ghoshal:2022fud  or gravitational waves from local cosmic
strings Dror:2019syi ; Saad:2022mzu ; DiBari:2023mwu ; Blasi:2020wpy , global
cosmic strings Fu:2023nrn , domain walls Barman:2022yos ; King:2023cgv ,
nucleating and colliding vacuum bubbles Dasgupta:2022isg ; Borah:2022cdx ,
other topological defects Dunsky:2021tih , graviton bremmstrahlung
Ghoshal:2022kqp , inflationary tensor perturbations propagating as GW
Berbig:2023yyy ; Borboruah:2024eal , primordial blackholes Perez-
Gonzalez:2020vnz ; Datta:2020bht ; JyotiDas:2021shi ; Barman:2021ost ;
Bernal:2022pue ; Bhaumik:2022pil ; Bhaumik:2022zdd ; Ghoshal:2023fno , and
exploring the oscillatory features of the primordial non-gaussianity and
curvature bi-spectrum and tri-spectrum Cui:2021iie ; Fong:2023egk .

Under this circumstance, we need more such probes of BSM physics and form a
synergy between cosmological observations and laboratory experiments involving
seesaw and low-scale leptogenesis. In this paper, we take a first step in this
direction and investigate a general scenario where a heavy scalar denoted by
$\phi$ may decay to create an initial non-thermal abundance of right-handed
neutrinos (RHN) whose subsequent decays lead to leptogenesis satisfying Eq.
(1). We will be concerned with Gravitational Waves that are generated due to
quantum tensor fluctuations of the metric during inflation Grishchuk:1974ny ;
Starobinsky:1979ty ; Rubakov:1982df ; Guzzetti:2016mkm . These GWs can act as
a book-keeping of the expansion history of our Universe Seto:2003kc ;
Boyle:2005se ; Boyle:2007zx ; Kuroyanagi:2008ye ; Nakayama:2009ce ;
Kuroyanagi:2013ns ; Jinno:2013xqa ; Saikawa:2018rcs ; Chen:2024roo . Since the
detailed time evolution of the Hubble rate during the expansion determines
various PGW frequencies that are red-shifted to the present day, thus allowing
us to investigate standard and non-standard thermal history of the universe in
the pre-BBN era Bernal:2020ywq ; Nakayama:2008ip ; Nakayama:2008wy ;
Kuroyanagi:2011fy ; Buchmuller:2013lra ; Buchmuller:2013dja ; Jinno:2014qka ;
Kuroyanagi:2014qza ; Ghoshal:2023sfa ; Ghoshal:2022ruy ; Chen:2024roo . In
this context, since we are interested in low scale seesaw and leptogenesis, we
will study a scenario where $\phi$ dominates the energy budget of the Universe
which leads to interesting GW spectral shapes. On one hand, $\phi$ decay will
facilitate low-scale leptogenesis and on the other hand, it is also
responsible for the generation of GW spectral shapes that can be measured by
various GW detectors. In particular, we consider an epoch of intermediate
matter domination by $\phi$ particles affecting the post-inflationary
evolution of the Universe Scherrer:1984fd ; Kolb:1990vq ; Bezrukov:2009th .

We will study GW probes of both thermal and non-thermal low scale leptogenesis
scenarios. The leptogenesis scale is brought down mainly by the resonance
effect that enhances the CP parameter $\epsilon$. Non-thermal leptogenesis is
achieved via the scalar decay to RHNs. Non-thermal production of RHN creates
an excess initial abundance of RHNs after the scalar has decayed, compared to
the thermal case. Depending on reheating temperature $T_{\phi}$ and washout
parameter $K$, the final efficiency factor $\kappa_{f}$ may get enhanced
compared to the thermal case due to this excess initial abundance Hahn-
Woernle:2008tsk ; Zhang:2023oyo . Unlike Berbig:2023yyy , here the RHNs need
not dominate the Universe since the scalar is assumed to dominate the energy
density of the Universe which distorts inflationary GWs. Therefore we do not
need to stick to only the $K\ll 1$ condition of Berbig:2023yyy  for RHN
domination. Note that RHNs can still dominate after $\phi$ decays, depending
on parameters.

The decays occur very far away from thermal equilibrium, therefore it is bound
to release a large amount of entropy dump, which in turn dilutes the energy
density of primordial GWs that enter the horizon before the decay. We show
that if the scalar couples directly to the radiation bath, under appropriate
conditions RHNs produced non-thermally from $\phi$ thermalize quickly with the
radiation bath, in which case entropy is injected in two stages, one from
scalar decay and the other from RHN decay if they dominate the energy budget
of the Universe before decaying. In this case, two knee-like features occur in
the GW spectrum due to the scalar decay which we show for the first time. We
will show the capability of several upcoming GW detectors like LISA, DECIGO,
BBO, etc. in probing the parameter space for such leptogenesis, particularly
in regions where it is impossible to test via laboratory or astrophysical
experiments.

The paper is organized as follows: In Sec. II, we give a brief overview of the
thermal leptogenesis mechanism and the Davidson-Ibarra bound on heavy neutrino
masses. We consider resonant leptogenesis to bring the leptogenesis scale
down. In Sec. III, we discuss the non-thermal leptogenesis scenario from
scalar decay and the relevant Boltzmann equations. In Sec. IV, we identify
three classes of scenarios relevant to GW observations, two of which
correspond to non-thermal and one to thermal leptogenesis. We provide
analytical approximations of matter-radiation equilibrium temperature and
dilution factor from entropy injection for these classifications. Then in Sec.
V we discuss primordial gravitational waves from inflation with and without an
intermediate matter domination epoch. We also introduce a transfer function to
encapsulate a 2-step entropy injection. This section is followed by our
results in Sec. VI. Finally, we conclude this paper in Sec. VII.

##  II Thermal Leptogenesis

Before discussing the effect of scalar field reheating on leptogenesis let us
first we give a brief review of standard thermal leptogenesis in this section.
Initially proposed by Fukugita and Yanagida in 1986 Fukugita:1986hr ;
Yanagida:2005hz , thermal leptogenesis hinges on the out-of-equilibrium decay
of RHNs, which are thermally produced through scattering processes in the
early Universe from the thermal bath of the primordial SM plasma. The
Lagrangian is given by,

$\mathcal{L}_{\rm thermal}={\cal L}_{\rm
SM}+i\overline{N}\not{\partial}N-\left(\lambda\overline{L}\tilde{H}N+\frac{M_{N}}{2}\overline{N^{C}}N+\text{h.c}\right)$
(2)

where $\mathcal{L}_{\rm SM}$ is the SM Lagrangian, $N$ represents RHNs,
$\tilde{H}=i\sigma_{2}H^{*}$ where $H$ is the SM Higgs doublet and $L$ is the
SM leptonic doublet. Here $M_{N}=\text{diag}[M_{1},M_{2},M_{3}]$ is the RHN
Majorana mass matrix and $\lambda$ encompasses the light-heavy Yukawa
couplings. Due to the Majorana mass term of RHNs, after electroweak symmetry
breaking (EWSB), the familiar Type-I seesaw mechanism Yanagida:1979as
provides mass to the light neutrinos, $m_{\nu}=v^{2}\lambda
M_{N}^{-1}\lambda^{T}$, where $v\sim 174$ GeV is the vacuum expectation value
(VEV) of the SM Higgs. On the other hand, RHNs of Majorana nature, can decay
into both leptons and anti-leptons through the Yukawa coupling matrix
$\lambda$, thereby violating lepton number (LNV),

$N_{1}\rightarrow H+L\quad\text{and}\quad N_{1}\rightarrow
H^{\dagger}+L^{\dagger}$ (3)

Typically, this scenario assumes a hierarchical heavy neutrino mass spectrum,
where the masses of the heavier RH neutrinos, $M_{2}$ and $M_{3}$, are much
greater than the lightest RHN mass, $M_{1}$. This condition allows the
lightest neutrino, $N_{1}$ to generate the necessary lepton asymmetry when CP
is violated in their decay, after washing out any asymmetry created by $N_{2}$
and $N_{3}$. The CP asymmetry, denoted as $\epsilon_{1}$, is given by:

$\epsilon_{1}=\frac{\Gamma_{N_{1}\rightarrow HL}-\Gamma_{N_{1}\rightarrow
H^{\dagger}L^{\dagger}}}{\Gamma_{N_{1}}^{rf}}$ (4)

where $\Gamma_{N_{1}\rightarrow Hl}$ is the decay width of the
$N_{1}\rightarrow H+l$ decay and

$\Gamma_{N_{1}}^{rf}=\Gamma_{N_{1}\rightarrow HL}+\Gamma_{N_{1}\rightarrow
H^{\dagger}L^{\dagger}}=\frac{|\lambda\lambda^{\dagger}|_{11}}{8\pi}M_{1}$ (5)

is the total decay width of $N_{1}$ in the rest frame of the neutrino. The
lepton asymmetry can be re-written as $B-L$ asymmetry denoted by the number
density $n_{B-L}$, which is subsequently converted into the baryon asymmetry
through $B+L$ conserving electroweak sphaleron processes Harvey:1990qw ;
Khlebnikov:1988sr ; Luty:1992un . When there is no initial $B-L$ asymmetry,
the final baryon asymmetry, denoted as baryon-to-photon number density ratio
$\eta_{B}$, is expressed as Buchmuller:2004nz ,

$\eta_{B}\equiv\frac{n_{B}-n_{\overline{B}}}{n_{\gamma}}=-\frac{3}{4}\frac{a_{\text{sph}}}{f}\,\epsilon_{1}\kappa_{f}\equiv-d\,\epsilon_{1}\kappa_{f}\simeq
0.96\times 10^{-2}\,\epsilon_{1}\kappa_{f}$ (6)

where $a_{\text{sph}}=28/79$ Arnold:1987mh ; Khlebnikov:1988sr ; Harvey:1990qw
is the efficiency of the sphaleron processes of converting $B-L$ asymmetry
into baryon asymmetry and $f=2387/86$ is the dilution factor due to production
of photons since the onset of leptogenesis till recombination assuming an
isentropic expansion of the Universe. And $\kappa_{f}$ is the efficiency
parameter Buchmuller:2004nz , which is independent of $\epsilon_{1}$ and which
parameterizes the efficiency of lepton asymmetry production depending on the
initial $N_{1}$ abundance and washout. It can be calculated by solving the
relevant Boltzmann equations governing the evolution of the $N_{1}$ abundance
and $B-L$ asymmetry Nardi:2007jp ; Basboll:2006yx . Its final value is
normalized to unity if thermal initial abundance of $N_{1}$ is assumed and
there is no washout.

###  II.1 Conditions for Departure from Thermal Equilibrium

One of the key conditions for successful baryogenesis, as outlined by
Sakharov, is the departure from thermal equilibrium. In the context of
leptogenesis, this condition is satisfied if the total decay rate of $N_{1}$
is smaller than the Hubble expansion rate of the Universe when the temperature
is approximately equal to the mass of $N_{1}$:

$\Gamma_{N_{1}}^{rf}<\mathcal{H}(T\sim M_{1})$ (7)

where $\mathcal{H}(T\sim
M_{i})\equiv\mathcal{H}(M_{i})=\sqrt{\frac{8\pi^{3}g_{\ast}}{90}}\frac{M_{i}^{2}}{M_{\text{pl}}}$
is the Hubble parameter with $M_{\text{pl}}=1.22\times 10^{19}$ GeV and
$g_{*}$ is the relativistic degrees of freedom which is approximately equal to
$106.75$ at the high temperatures under discussion. It can be shown that the
above condition is satisfied when the effective neutrino mass, defined by,

$\tilde{m}_{1}=\frac{(m_{D}^{\dagger}m_{D})_{11}}{M_{1}}$ (8)

where $m_{D}=\lambda\,v$ is the Dirac mass matrix with $v=\braket{H}\simeq
174$ GeV being the vacuum expectation value (VEV) of the Standard Model Higgs,
is smaller than the equilibrium neutrino mass, defined as,

$m_{\ast}=\frac{16\pi^{5/2}}{3\sqrt{5}}g_{*}^{1/2}\frac{v^{2}}{M_{\rm
pl}}\simeq 1.1\times 10^{-3}\text{ eV}$ (9)

The ratio is called the decay parameter,

$K\equiv\frac{\Gamma_{N_{1}}^{rf}}{\mathcal{H}(M_{1})}=\frac{\tilde{m}_{1}}{m_{\ast}}$
(10)

which determines whether $N_{1}$ decays are in thermal equilibrium or not. The
case $K<1$ is called the weak washout regime, where Yukawa interactions of
$N_{1}$ with light neutrinos via the SM Higgs are weak, hence washout of any
initial asymmetry via reverse processes is small. Therefore leptogenesis is
unavoidable in this regime. However in the so called strong washout regime
($K>1$), although Yukawa interactions are strong, leptogenesis can still occur
and the final asymmetry depends on the interplay between decay, inverse decay
and scattering of RHNs Buchmuller:2004nz .

It turns out the final efficiency parameter $\kappa_{f}$ can be estimated
analytically for both weak and strong washout regime Buchmuller:2004nz . In
the strong washout regime, $\kappa_{f}$ is estimated by,

$\kappa_{f}(K)\simeq(2\pm 1)\times 10^{-2}\left(\frac{0.01\text{
eV}}{m_{*}\,K}\right)^{1.1\pm 0.1}$ (11)

while in the weak washout regime, $\kappa_{f}$ depends on the initial
abundance of $N_{1}$. For thermal initial abundance, it is approximately 1 in
the weak regime while for vanishing initial abundance, it is estimated as,

$\kappa_{f}(K)\simeq\frac{9\pi^{2}K^{2}}{64},\quad(K<1)$ (12)

For thermal initial $N_{1}$ abundance, $\kappa_{f}$ can be generically written
for all $K$ as,

$\displaystyle\kappa_{f}(K)$
$\displaystyle\simeq\frac{2}{z_{B}(K)K}\left(1-e^{\frac{1}{2}z_{B}(K)K}\right)$
(13) $\displaystyle\text{where }z_{B}(K)$ $\displaystyle\simeq
1+\frac{1}{2}\text{ ln}\left(1+\frac{\pi K^{2}}{1024}\left[\text{
ln}\left(\frac{3125\pi K^{2}}{1024}\right)\right]^{5}\right)$ (14)

In Fig. 2 we show $\kappa_{f}$ as a function of washout parameter $K$ for
thermal (red solid line) and vanishing (blue dashed line) initial abundance of
$N_{1}$. Depending on the CP parameter $\epsilon_{1}$ and $K$, the final
baryon asymmetry can be calculated from Eq. (6).

###  II.2 CP Asymmetry

The CP asymmetry $\epsilon_{1}$ is determined by the interference between the
tree level and 1-loop diagrams Flanz:1994yx ; Covi:1996wh  and can be
calculated from $lH\rightarrow lH$ scattering processes Buchmuller:1997yu .
The relevant Feynmann diagrams giving tree level, vertex and self-energy
contributions are shown in Fig. 1.

Figure 1: Feynmann diagrams of $N\rightarrow lH$ decay process at the (a)
tree-level and (b)-(c) 1-loop level. (b) represents the vertext contribution
and (c) represents self-energy contribution.

The maximal asymmetry is obtained for $M_{2,3}\gg M_{1}$ and light neutrino
masses $m_{1}\sim\tilde{m}_{1}$, $m_{3}=\sqrt{m_{\text{atm}}^{2}+m_{1}^{2}}$
Buchmuller:2003gz ; Hahn-Woernle:2008tsk ; Buchmuller:2004tu ; Davidson:2002qv
; Hambye:2003rt ,

$\epsilon_{1}^{\text{max}}=\frac{3}{16\pi}\frac{M_{1}}{v^{2}}(m_{3}-m_{1})\simeq
10^{-6}\left(\frac{M_{1}}{10^{10}\text{
GeV}}\right)\left(\frac{m_{3}-m_{1}}{0.05\text{ eV}}\right)$ (15)

###  II.3 Lower Bounds on Heavy Neutrino Masses

Putting the maximal CP asymmetry parameter $\epsilon_{1}^{\text{max}}$ in Eq.
(6) gives the maximal baryon asymmetry $\eta_{B}^{\text{max}}$, which needs to
be larger than the observed asymmetry $\eta_{B}^{\text{CMB}}$, i.e.
$\eta_{B}^{\text{max}}\geq\eta_{B}^{\text{CMB}}$. Then one finds the Davidson-
Ibarra bound on $M_{1}$ using Eq. (6) Davidson:2002qv ; Buchmuller:2002rq ;
Buchmuller:2004nz ,

$M_{1}\geq 6.4\times 10^{8}\text{
GeV}\left(\frac{\eta_{B}^{\text{CMB}}}{6\times
10^{-10}}\right)\left(\frac{0.05\text{
eV}}{m_{3}-m_{1}}\right)\kappa_{f}^{-1}$ (16)

Assuming the final efficiency factor $\kappa_{f}=1$ gives the $3\sigma$ bound
on the minimum reheating temperature $T_{\rm RH}$ at the end of inflation
Buchmuller:2004tu ,

$T_{\text{RH}}\gtrsim M_{1}\gtrsim 4\times 10^{8}\text{ GeV}$ (17)

This stems from the condition that $T_{\rm RH}$ needs to be larger than the
initial temperature of leptogenesis $T_{i}\sim M_{1}$ when the RHN decay
process becomes out-of-equilibrium (see Eq. (7)) Buchmuller:2005eh .

###  II.4 Resonant Leptogenesis

Thermal leptogenesis operates at high temperatures, which in turn pushes the
inflationary reheating temperature $T_{\rm RH}$ to a high scale, as shown in
Eq. (16) and (17). Such high scales involving RHN masses $M_{i}\gtrsim 10^{9}$
GeV are difficult to test in the lab. Also a high neutrino mass is in conflict
with Electroweak naturalness condition which requires RHN masses
$M_{i}\lesssim 10^{7}$ GeV Vissani:1997ys ; Clarke:2015gwa ; Datta:2022tab . A
lower reheating temperature from inflation $T_{\rm RH}$ is motivated in
different inflationary scenarios like Drees:2021wgd ; Ghoshal:2022zwu . Even
in some SUGRA models Kawasaki:2004qu , reheating temperature is found to be as
low as $10^{6}$ GeV which contradicts a high $T_{\text{RH}}$, favouring low-
scale-leptogenesis. Motivated by these reasons, and for lab testability, the
scale of leptogenesis can be brought down by several orders in the so called
resonant regime which focuses on nearly degenerate heavy neutrinos leading to
pronounced CP violation in their decays. It can be shown that if the mass
difference between the two lightest right-handed neutrinos is of the order of
the decay widths of them, i.e. $M_{2}-M_{1}\sim\Gamma_{N_{2}}^{rf}$ the CP
parameter $\epsilon_{1}$ is enhanced via the self-energy contribution
Pilaftsis:1997dr ; Hambye:2003rt  in the loop level diagrams shown in Fig 1.

The maximum CP asymmetry with resonant effect due to quasi-degenerate right-
handed neutrinos $N_{1}$ and $N_{2}$ can be written as Berbig:2023yyy ;
Hambye:2003rt ,

$\epsilon=\text{Min}\left[\epsilon_{1}^{\text{max}}\frac{S_{2}\cdot
m_{3}-m_{1}}{m_{3}-m_{1}},1\right],\quad\text{where
}S_{2}=\frac{M_{2}}{2\Gamma_{N_{2}}^{rf}}\quad\text{and
}M_{2}-M_{1}=\frac{\Gamma_{N_{2}}^{rf}}{2}$ (18)

where, $\epsilon_{1}^{\rm max}$ is the maximal asymmetry in non-resonant case
given by Eq. (15). It was shown that within the resonant regime the lowest RHN
mass $M_{1}$ can be as low as 1 TeV Pilaftsis:1997jf . In Pilaftsis:2005rv
this scale is shown to reach upto the electroweak symmetry breaking scale
$\sim 100$ GeV, below which sphaleron processes are suppressed and the lepton
asymmetry can not be effectively converted to baryon asymmetry. From here
onwards we will consider resonant leptogenesis with $M_{1}\gtrsim 100$ GeV for
our analysis. Note that in some alternate leptogenesis scenarios, e.g. via
neutrino oscillations, BAU can be generated for even for RHN mass values $\sim
100$ MeV Klaric:2020phc .

##  III Non-thermal Leptogenesis via scalar decay

In vanilla leptogenesis, RHNs are in thermal equilibrium at production.
However, it is possible to produce RHNs non-thermally from inflaton
Asaka:1999yd ; Giudice:2003jh ; Hahn-Woernle:2008tsk ; Zhang:2023oyo  or a
generic scalar decay Datta:2022tab ; Nemevsek:2023yjl . Such a scenario can
lead to excess initial abundance of RHNs compared to their thermal abundance.
As a result the final efficiency factor, $\kappa_{f}$ can be enhanced upto
$\mathcal{O}(2)$ which relaxes the Davidson-Ibarra bound on RHN mass given in
Eq. 16. Note that the RHNs can also be produced thermally from the scalar
which is similar to thermal leptogenesis. We consider the Lagrangian
Ghoshal:2022fud ,

${\cal L_{\rm nonthermal}}=\mathcal{L}_{\rm
thermal}-\left(\frac{y_{N}}{2}\phi\overline{N^{C}}N+\frac{y_{R}}{2}\phi\overline{f}f+\text{h.c}\right)$
(19)

where $f$ represents a Dirac fermion which is in thermal equilibrium with the
radiation bath. This kind of Lagrangian can be obtained by considering for
example a gauged $U(1)$ extension Ghoshal:2022fud ; Ghoshal:2022hyc  of the SM
where the vector-like fermions $f$ can couple to the SM sector via the $U(1)$
gauge boson. We will assume the scalar mass to always exceed twice the right-
handed neutrino masses, $M_{\phi}>2M_{1}$ where $M_{1}$ is the mass of the
lightest RH neutrino. The decay width of $\phi$ to RH neutrinos is
Croon:2019dfw ; Ghoshal:2022fud ,

$\Gamma_{\phi\to
N_{i}N_{i}}\simeq\frac{y_{N_{i}}^{2}M_{\phi}}{16\pi}\sqrt{1-\frac{4M_{i}^{2}}{M_{\phi}^{2}}}$
(20)

where $i=1,2,3$ represents the 3 RH neutrinos. On the other hand, decay width
of the scalar to radiation is given by Ghoshal:2023phi ; Ghoshal:2022fud ,

$\displaystyle\Gamma_{\phi\to R}$ $\displaystyle\simeq$
$\displaystyle\frac{y_{R}^{2}\,M_{\phi}}{8\pi}$ (21)

where subscript $\phi\rightarrow R$ denotes $\phi$ decaying to radiation via
the Dirac fermions $f$ in Eq. (19) which are in thermal equilibrium with the
SM bath. We define the branching ratios,

$\displaystyle\text{BR}(\phi\to{N_{i}N_{i}})\equiv\frac{\Gamma_{\phi\to{N_{i}N_{i}}}}{\Gamma_{\phi}},\quad\text{BR}(\phi\to{R})\equiv\frac{\Gamma_{\phi\to{R}}}{\Gamma_{\phi}}$
(22)

where $\Gamma_{\phi}=\sum_{i}\Gamma_{\phi\rightarrow
N_{i}N_{i}}+\Gamma_{\phi\rightarrow R}$ is the total decay width of $\phi$.
Essentially, ${\rm BR}(\phi\to{N_{i}N_{i}})$ is dependent on the coupling
$y_{R}$, so decreasing $y_{R}$ will increase ${\rm BR}(\phi\to{N_{i}N_{i}})$.
If $y_{R}\sim 0$ then $BR(\phi\to{N_{i}N_{i}})\sim 1$ which leads to maximum
non-thermal RHN production from $\phi$ hence maximum lepton asymmetry.

We assume a resonant leptogenesis scenario with $M_{1}\sim M_{2}\ll M_{3}$
such that the scalar dominantly decays to the two lightest RHNs, $N_{1}$ and
$N_{2}$. The resonance effect makes sure that enough asymmetry is generated
for all $M_{1}$ values starting from 100 GeV, to explain the BAU. RHNs can be
in thermal equilibrium with the radiation bath either via their Yukawa
couplings $\lambda$ or via the 2-to-2 scattering process
$N_{i}N_{i}\leftrightarrow\overline{f}f$ via $\phi$ for non-vanishing values
of $y_{R}$. See appendix B for more details. During this paper, we take
$y_{R}=0$ when discussing non-thermal leptogenesis and only consider non-zero
$y_{R}$ when demonstrating thermal leptogenesis. We treat the independent
variables for our analysis to be $M_{i}$, $M_{\phi}$, $\lambda$ and
$y_{N_{i}}$. Later in Sec. III we will parameterize $\lambda$ and $y_{N_{i}}$
in terms of the decay temperatures of the scalar $T_{\phi}$ and right-handed
neutrinos $T_{N_{i}}$.

We consider the parameter space of the scalar where it dominates the energy
budget of the Universe before decaying, leading to a phase of matter dominated
pre-BBN era. The RHNs produced from scalar decay, as we shall see, may or may
not further dominate the energy density depending on the parameters used.
Their out-of-equilibrium decay to relativistic SM particles will inject
considerable amount of entropy into the Universe. This dampens the primordial
gravitational waves from inflation that re-enters the horizon during the early
matter domination epoch. We show such signature of low-scale leptogenesis in
primordial gravitational waves can be detected in various GW experiments such
as BBO, DECIGO, $\mu$-ARES, ET, LISA, CE etc.

###  III.1 Set Up

We consider the scalar $\phi$ was in thermal equilibrium with the primordial
plasma after the end of inflation either via its direct coupling with SM
particles, or via indirect interactions, e.g. coupling with inflaton field in
a specific inflationary model or vector-like fermions in a gauged $U(1)$
extension of the SM mentioned above. Then the scalar froze out at some
temperature $T_{f}$ and eventually it dominates the energy budget of the
Universe at temperature $T_{\rm dom}$. If the freeze out happens while the
scalar particles were still relativistic we obtain the maximal $\phi$
abundance Bezrukov:2009th ; Kolb:1990vq ,

$\frac{n_{\phi}}{s}\Big{|}_{f}=\frac{45\,\zeta(3)}{2\pi^{4}\,g_{*S}}$ (23)

where subscript $f$ denotes the time of freeze-out, $n_{\phi}$ is the number
density of $\phi$ and the entropy density is given by,

$s=\frac{2\pi^{2}}{45}g_{*S}T^{3}$ (24)

The effective degrees of freedom related to entropy is denoted by $g_{*S}$
which we take to be equal to $g_{*}=106.75$ at high temperatures for
simplicity in our analysis. After freeze-out, $\phi$ eventually becomes non-
relativistic after which its energy density red-shifts as matter and dominates
over the radiation density starting from temperature Giudice:1999fb ;
Kolb:1990vq ,

$T_{\rm dom}\sim\frac{M_{\phi}}{g_{*}}\approx 0.01M_{\phi}$ (25)

The $\phi$-dominated phase starts at temperature $T_{\rm dom}$ and ends at
some temperature decay temperature $T_{\phi}$; we always assume
$T_{\phi}<T_{\rm dom}$ such that we get a period of time in early Universe of
scalar era. The dynamics of $\phi$ and its decay products into RHNs and
radiation can be obtained by solving the relevant Boltzmann equations which we
will describe in Sec. III.2.

We start with Eq. (20) and (21), the partial decay widths of $\phi$ to
lightest RHNs and radiation and consider only the two lightest RHNs
$N_{1},N_{2}$. The branching ratios are given by (22). As long as mass of the
scalar $M_{\phi}>2M_{1,2}$, the scalar can decay to $N_{1}$ and $N_{2}$.

The RHNs $N_{1},N_{2}$ further decay to lepton-Higgs pairs with the thermally
averaged decay widths,

$\Gamma_{N_{i}}=\Gamma_{N_{i}}^{rf}\,\frac{K_{1}(z)}{K_{2}(z)}\sim\mathcal{H}(M_{i})\,K\,\frac{K_{1}(z)}{K_{2}(z)},$
(26)

where
$\mathcal{H}(M_{i})=\sqrt{\frac{8\pi^{3}g_{\ast}}{90}}\frac{M_{i}^{2}}{M_{\text{pl}}}$
is the Hubble parameter with $M_{\text{pl}}=1.22\times 10^{19}$ GeV.
$K_{1}(z)$,$K_{2}(z)$ are the modified bessel functions of first and second
kind and $z=M_{1}/T$.

The decay temperatures of the scalar or the RHNs are calculated assuming that
the energy density of scalar/RHN is converted instantaneously into radiation
Hahn-Woernle:2008tsk ,

$T_{\phi}=\left(\frac{90}{8\pi^{3}g_{\ast}(T_{\phi})}\right)^{\frac{1}{4}}\sqrt{\Gamma_{\phi}M_{\text{pl}}}$
(27)
$T_{N_{i}}=\left(\frac{90}{8\pi^{3}g_{\ast}(T_{N_{i}})}\right)^{\frac{1}{4}}\sqrt{\Gamma_{N_{i}}^{rf}M_{\text{pl}}}=M_{i}\sqrt{K}$
(28)

where $\Gamma_{\phi}\sim\Gamma_{\phi\rightarrow
N_{1}N_{1}}+\Gamma_{\phi\rightarrow N_{2}N_{2}}+\Gamma_{\phi\rightarrow R}$ is
the total decay width of $\phi$ and $g_{\ast}(T)$ is the relativistic degree
of freedom at the corresponding temperature $T$ which is taken to be $\sim
106.75$. Using Eq. (20) (21) in Eq. (27) and considering Eq. (25), we can
write the condition $T_{\phi}<T_{\rm dom}$ in terms of couplings as,

$y_{N_{i}}^{2}+y_{R}^{2}\lesssim\frac{M_{\phi}}{M_{\rm pl}}$ (29)

where $i=1,2$ and we have considered $M_{1},M_{2}\ll M_{\phi}$. We can expect
that the latest radiation dominated era starts at either $T_{\phi}$ or
$T_{N_{1}}$ depending on which one is lower. Notice that reheating temperature
$T_{\text{RH}}$ after inflation now need not be larger than $M_{1}$ as in the
thermal case. Rather $T_{\rm RH}$ only needs to be roughly larger than $T_{\rm
dom}$ now, since RHNs are produced dominantly at $T_{\phi}<T_{\rm dom}$ from
$\phi$ decay, regardless of their masses.

In Table 1, we summarize all the necessary conversion formulas for the
parameters used in our analysis in terms of the couplings and masses of the
scalar and RHNs.

Variable | Symbol | Definition
---|---|---
RHN decay width in the rest frame | $\Gamma_{N_{i}}^{rf}$ | $\frac{|\lambda|^{2}M_{i}}{8\pi}$
Scalar decay width to RHNs | $\Gamma_{\phi\rightarrow N_{i}N_{i}}$ | $\frac{y_{N_{i}}^{2}M_{\phi}}{16\pi}\sqrt{1-\frac{4M_{i}^{2}}{M_{\phi}^{2}}}$
Scalar decay width to radiation | $\Gamma_{\phi\rightarrow R}$ | $\frac{y_{R}^{2}\,M_{\phi}}{8\pi}$
Total decay width of scalar | $\Gamma_{\phi}$ | $\Gamma_{\phi\rightarrow N_{1}N_{1}}+\Gamma_{\phi\rightarrow N_{2}N_{2}}+\Gamma_{\phi\rightarrow R}$
Scalar decay temperature | $T_{\phi}$ | $\left(\frac{90}{8\pi^{3}g_{\ast}(T_{\phi})}\right)^{\frac{1}{4}}\sqrt{\Gamma_{\phi}M_{\text{pl}}}$
|  | $\sim 0.05\sqrt{y_{N_{1}}^{2}+y_{R}^{2}}\left(\frac{100}{g_{*}(T_{\phi})}\right)^{\frac{1}{4}}\sqrt{M_{\phi}M_{\rm pl}}$
Washout parameter | $K$ |  $\frac{\tilde{m}_{1}}{m_{*}}$=$\frac{\tilde{m}_{1}}{1.1\times 10^{-3}\,\text{eV}}=\frac{|\lambda|^{2}_{11}v^{2}/M_{1}}{1.1\times 10^{-3}\,\text{eV}}$
RHN decay temperature | $T_{N_{1}}$ | $\left(\frac{90}{8\pi^{3}g_{\ast}(T_{\phi})}\right)^{\frac{1}{4}}\sqrt{\Gamma_{N_{1}}^{rf}M_{\text{pl}}}\sim M_{1}\sqrt{K}$
Table 1: Definition of the relevant parameters used for our analysis in terms
of couplings and masses of the scalar and RHNs. Here $v=174$ GeV is the
electrweak VEV, $\lambda$ is the Yukawa coupling of RHNs with active neutrinos
via SM Higgs. Here we also considered resonant scenario, i.e. $y_{N_{1}}\sim
y_{N_{2}}\ll y_{N_{3}}$.

###  III.2 The Boltzmann Equations

The evolution of the energy densities of matter and radiation and the number
density of $B-L$ asymmetry are determined by the Boltzmann equations,

$\displaystyle\dot{\rho}_{\phi}$
$\displaystyle=-3\mathcal{H}\rho_{\phi}-\Gamma_{\phi}\,\rho_{\phi}$ (30)
$\displaystyle\dot{\rho}_{N_{1}}$
$\displaystyle=-3\mathcal{H}\rho_{N_{1}}+\Gamma_{\phi\rightarrow
N_{1}N_{1}}\,\rho_{\phi}-\Gamma_{N_{1}}(\rho_{N_{1}}-\rho_{N_{1}}^{eq})$
$\displaystyle\dot{\rho}_{N_{2}}$
$\displaystyle=-3\mathcal{H}\rho_{N_{2}}+\Gamma_{\phi\rightarrow
N_{2}N_{2}}\,\rho_{\phi}-\Gamma_{N_{2}}(\rho_{N_{2}}-\rho_{N_{2}}^{eq})$
$\displaystyle\dot{n}_{B-L}$
$\displaystyle=-3\mathcal{H}n_{B-L}-\epsilon\sum_{i=1}^{2}\Gamma_{N_{i}}(n_{N_{i}}-n_{N_{i}}^{eq})-\Gamma_{ID}\,n_{B-L}$
$\displaystyle\dot{\rho}_{R}$
$\displaystyle=-4\mathcal{H}\rho_{R}+\Gamma_{\phi\rightarrow
R}\,\rho_{\phi}+\sum_{i=1}^{2}\Gamma_{N_{i}}(\rho_{N_{i}}-\rho_{N_{i}}^{eq})$

where $\mathcal{H}$ is the Hubble parameter in cosmic time and dot denotes
derivative with respect to cosmic time and $\rho^{eq}_{X}$ and $n_{X}^{eq}$
represents the equilibrium energy and number density of the species $X$
respectively. $\Gamma_{ID}$ is the inverse decay rate. See Appendix A for
details. Here we have considered only decays and inverse decays and neglected
scattering processes of the RHNs and the scalar. Also, for simplicity here we
considered only non-relativistic RHN production in the equations for
$N_{1,2}$, which may not be guaranteed when $M_{\phi}\gg M_{1,2}$.

While solving the Boltzmann equations we take the coupling $y_{R}=0$ defined
in Eq. (19) such that $\Gamma_{\phi\rightarrow R}\sim 0$ in order to maximize
the RHN production from $\phi$ and hence the lepton asymmetry. Later we will
consider the $y_{R}>0$ scenario which is similar to thermal leptogenesis. We
start solving the Boltzmann equations from the time when the scalar energy
density overtakes the radiation energy density. Thus the initial energy
density of the scalar is equal to the radiation density Kolb:1990vq ,

$\rho_{\phi}(T_{\rm dom})=\rho_{R}(T_{\rm
dom})=\frac{\pi^{2}}{30}g_{\ast}(T_{\rm dom})\,T_{\rm dom}^{4}$ (31)

We take $T_{\rm dom}$ from Eq. (25). The initial densities of RHNs are taken
to be free parameters smaller than $\rho_{\phi}(T_{\rm dom})$ while initial
$B-L$ asymmetry is taken to be negligible. Further details on the numerical
solution of the Boltzmann equations and calculation of final $\kappa_{f}$
value as-well-as maximum available temperature are given in Appendix A. In
Fig. 2 along with the thermal case we also show the enhancement in
$\kappa_{f}$ in non-thermal leptogenesis from $\phi$ decay by solving the
Boltzmann equations (30). The solid magenta line, dashed green line and the
dash-dotted purple line shows the final efficiency factor as a function of the
washout parameter for a special case, $M_{1}\sim T_{\phi}$ Hahn-
Woernle:2008tsk . Here we took $M_{1}=10^{5}$ GeV, $M_{\phi}=10^{8}$ GeV and
$T_{\phi}=10^{5},5\times 10^{4},10^{4}$ GeV. Efficiency factor is maximum at
$K\sim 0.3$ for $M_{1}\sim T_{\phi}$. We note that for $T_{\phi}\ll M_{1}$,
the final efficiency factor becomes independent of $K$.

At this point we are able to numerically calculate the final baryon asymmetry
in our model. However we are not interested in the exact amout of asymmetry
generated, since our goal is studying GW probe of leptogenesis. In the next
section we discuss classification of scenarios based on numerical results and
analytical approximations, relevant for GW sensitivity. We make sure that for
all cases, observed BAU is successfully generated.

Figure 2: Analytical estimate of final efficiency factor $\kappa_{f}$ as a
function of washout parameter $K$ in thermal and numerical solutions to the
Boltzmann equations (30) in non-thermal leptogenesis. Red solid curve
represents thermal initial abundance of $N_{1}$ and blue dashed curve
represents vanishing initial $N_{1}$ abundance in thermal leptogenesis. Here
we used Eq. (13) and (14) for the blue curve while for the red curve we use
Eq. (12) for $K\ll 1$ and interpolate it to the thermal case in the $K\gg 1$
regime. The solid magenta, dashed green and dash-dotted purple lines represent
non-thermal leptogenesis with $M_{1}=10^{5}$ GeV, $M_{\phi}=10^{8}$ GeV for
different $T_{\phi}$ values.

##  IV Classification of Scenarios

In this section, we identify two general non-thermal scenarios relevant for GW
observations along the lines of Zhang:2023oyo . Additionally we introduce a
thermal leptogenesis scenario with two-step entropy injection. We present
analytical estimates of the equilibrium temperature when matter domination
ends, and the dilution factor from entropy injection for each of the three
scenarios. In all the cases, the resonantly enhanced final asymmetry is higher
than the observed asymmetry of the Universe for $M_{1}\gtrsim 100$ GeV as
discussed in the previous sections. In brief, the three scenarios are,

  1. 1.

Case (a): Instantaneous RHN decay - Portrayed in Fig. 3, we classify this
scenario by requiring $\Gamma_{N}\gg\Gamma_{\phi}$ which implies $T_{\rm
N_{1}}\gg T_{\phi}$. This means that the RHNs are mostly produced at
temperatures around $T_{\phi}$, when the scalar completely decays away and
radiation starts dominating again. Those RHNs instantaneously decay to the SM
bath. Even in the strong washout case $K\gg 1$, RHNs do not have time to
thermalize with the radiation bath, thus the RHN decay process is strongly
out-of-equilibrium. The evolution of the energy densities of different matter
and radiation are shown in Fig. 3 for both weak ($K\ll 1$) and strong ($K\gg
1$) washout regime. We can see that for strong washout, the RHN density is
lower than the weak washout case due to higher decay rate. From GW
perspective, this case is only dependent of the total decay width of $\phi$
and independent of partial decay widths to RHNs and radiation, i.e.
independent of the coupling $y_{R}$ given in Eq. (19). However the amount of
leptogenesis depends on $y_{R}$.

  2. 2.

Case (b): RHN Domination - For this scenario we need $y_{R}\sim 0$, and
$\Gamma_{N}\ll\Gamma_{\phi}$ implying $T_{N_{1}}\ll T_{\phi}$ which means the
RHN decay is delayed after production. As shown in Fig. 4, RHNs start to
dominate the energy density as soon as the scalar decays away, with no
radiation domination in between. When the RHNs decay out-of-equilibrium,
radiation domination starts. If $K\ll 1$ the RHNs never thermalize with the
radiation bath as seen in Fig. 4 (Left). If $K\gtrsim 1$, the RHNs eventually
thermalize with the radiation bath due to their strong Yukawa interactions
with the SM particles as shown in Fig. 4 (Right).

  3. 3.

Case (c): Thermal leptogenesis - For large enough $y_{R}$ and for
$T_{N_{1}}\ll T_{\phi}$, the RHNs produced from $\phi$ can quickly thermalize
with the radiation bath and attain thermal abundance via 2-to-2 scattering
process $N_{i}N_{i}\leftrightarrow\phi\leftrightarrow\overline{f}f$ (see
Appendix B for details). This scenario is similar to thermal leptogenesis. The
scalar will decay to both RHNs and radiation that will usher in an era of
radiation domination, with the RHNs contributing to the radiation. In general,
RHNs will also be produced thermally from the radiation bath via the
scattering process. After attaining thermal abundance, eventually the RHNs
freeze-out when the scattering process decouples, and then decay at
$T_{N_{1}}$. Depending on their decay widths, these RHNs may or may not
dominate the energy density prior to its decay. We are interested in the
scenario when they do dominate. The necessary condition for that is
Giudice:1999fb ; Berbig:2023yyy ,

$T_{N_{1}}<T_{\rm dom}^{N_{1}}\lesssim 2\%M_{1}$ (32)

which is the fermionic version of Eq. (25). The scalar dynamics adds nothing
new to the final lepton asymmetry compared to the traditional thermal
leptogenesis scenario. On the other hand, entropy injection now takes place in
two stages: first at around $T_{\phi}$ when $\phi$ decays and secondly at RHN
decay temperature $T_{N_{1}}$. Hence this case corresponds to 2-step entropy
injection contrary to the non-thermal case, where entropy is injected in 1
step. A requirement for two-step entropy injection is,

$T_{\phi}\gg T_{\rm dom}^{N_{1}}$ (33)

such that after their non-thermal production from $\phi$, RHNs get enough time
to thermalize then freeze-out before they start to dominate.

Figure 3: The evolution of the energy densities of the scalar $\rho_{\phi}$,
RHN $\rho_{N}=\rho_{N_{1}}+\rho_{N_{1}}$ and radiation, $\rho_{R}$ with
respect to the radiation temperature. Here we have taken
$M_{1}=10^{5},M_{\phi}=3M_{1},T_{\phi}=10^{3}$ GeV for weak washout $K=2\times
10^{-4}$ (Left) and strong washout $K=50$ (Right). Both correspond to Case (a)
mentioned at the beginning of Sec. IV. We can see RHNs instantaneously decay
after production. The bottom panels in each of the images show the amount of
entropy injected, $S=sa^{3}$ where $s\propto T^{3}$ is the entropy density
defined in Eq. (24) and $a$ is the scale factor.

Figure 4: Same as Fig. 3 but for RHN domination scenario, as described in Case
(b) of Sec. IV. For weak washout, $K\ll 1$ (Left), RHNs are always out-of-
equilibrium. For strong washout, $K\gg 1$ (Right), RHNs get thermalized later
due to their strong Yukawa interactions. The final asymmetry can be calculated
numerically by solving the Boltzmann equations. The bottom panels show the
entropy injection.

###  IV.1 Analytical Estimates of Equilibrium Temperature

Equilibrium temperature is the temperature at which matter domination ends,
kickstarting the latest radiation dominated era. From the discussion above and
from Fig. 3 and Fig. 4, we see that it is around the decay temperatures
$T_{\phi}$ or $T_{N_{1}}$ depending on which one is smaller,

$T_{\text{dec.}}\sim\text{Min}[T_{\phi},T_{N_{1}}]$ (34)

We use this temperature in Eq. (48) for calculating the damped GW spectrum
from inflation. Note that the exact equilibrium temperature is a little lower
than the decay temperature since the decay is not instantaneous. However for
our considerations, $T_{\rm dec.}$ is a reasonable estimate of it.

If we have thermal RHN scenario discussed in Case (c) above, provided the RHNs
dominate before decaying, there are two stages of entropy injection, one from
scalar decay at $T_{\phi}$ and another from RHN decay at $T_{N_{1}}$. In that
case, we have an additional radiation dominated era between the two matter
dominated epochs. Hence we take the two equilibrium temperatures to be
$T_{\phi}$ and $T_{N_{1}}$ in this scenario.

###  IV.2 Dilution Factor from Entropy Injection

Both scalar and RHN decay injects entropy into the Universe if the decay
products thermalize quickly with the radiation bath. The general formula for
the dimensionless dilution factor from the entropy injection by a dominating
species $\chi$ decaying to relativistic particles is given by Scherrer:1984fd
; Berbig:2023yyy

$\Delta=\frac{s(T_{\text{after}})a^{3}(T_{\text{after}})}{s(T_{\text{before}})a^{3}(T_{\text{before}})}=\left(1+2.95\left(\frac{2\pi^{2}\left<g_{*}(T)\right>}{45}\right)^{\frac{1}{3}}\frac{(\frac{n_{\chi}}{s}M_{\chi})^{\frac{4}{3}}}{(M_{\text{pl}}\Gamma_{\chi})^{\frac{2}{3}}}\right)^{\frac{3}{4}}$
(35)

where $s$ is the entropy density, $T_{\text{before/after}}$ represents
temperature before/after the decay process and $\frac{n_{\chi}}{s}$ is the
initial abundance of the particle $\chi$.

For our analysis, we estimate the dilution factor using the formula in (35)
for the 3 classifications we made:

Case (a): In this case, the RHNs decay out-of-equilibrium instantaneously to
relativistic particles at $T_{\phi}$. Therefore we can assume that it is the
scalar which is directly decaying to the radiation bath releasing entropy. The
dilution factor $\Delta$ can be given by Eq. (35) with $\chi$ replaced by
$\phi$. The initial abundance is given by Eq. (23). In the large entropy
injection approximation $\Delta\gg 1$, using Eq. (27) we can write the
dilution factor to be,

$\Delta\simeq 3.7\times 10^{9}\left(\frac{M_{\phi}}{10^{15}\text{
GeV}}\right)\left(\frac{\text{TeV}}{T_{\phi}}\right)$ (36)

Figure 5: Suppressed gravitational spectra of a benchmark Case (a) scenario
when RHNs decay instantaneously. Here we have taken $r=0.035$, $n_{T}=0.5$,
$T_{\rm RH}=10^{10}$ GeV, $K=10^{3}$, and varied $T_{\phi}$ and $M_{\phi}$.
Gray line shows the standard GW spectrum without any intermediate matter
domination. Black solid and dashed lines correspond to $M_{\phi}=10^{9}$ and
$3\times 10^{9}$ GeV respectively. The extreme dampening at frequencies above
$\sim 10$ Hz is due to inflaton decay.

Case (b): RHNs dominate the energy budget of the Universe as soon as they are
produced. The energy density stored in $\phi$ species is first transferred to
the RHNs and then dumped into the radiation bath when RHNs decay at
$T_{N_{1}}$. Hence, this scenario can be imagined as if $\phi$ itself is
decaying to SM particles at $T_{N_{1}}$.

The dilution factor in that case will be same as Case (a), except
$\Gamma_{\phi}$ will be replaced by $\Gamma_{N_{1}}^{rf}$ in Eq. (35) to
specify the elongated period of matter domination compared to Case (a). Note
that considering only the lightest RHN $N_{1}$ is sufficient here. Again in
the large entropy injection limit $\Delta\gg 1$, one can write the dilution
factor as,

$\Delta\simeq 3.7\times 10^{9}\left(\frac{M_{\phi}}{10^{15}\text{
GeV}}\right)\left(\frac{10^{5}\text{
GeV}}{M_{1}}\right)\sqrt{\frac{10^{-4}}{K}}$ (37)

Figure 6: Same as Fig. 5 but for $n_{T}=0$. Solid and dashed black lines
correspond to $M_{\phi}=10^{9}$ and $3\times 10^{9}$ GeV respectively.

Case (c): As discussed earlier, within suitable parameter choices, one can
obtain thermal leptogenesis with two-step entropy injection (Appendix B). The
dilution factor from scalar decay $\Delta_{\phi}$ is again estimated by Eq.
(36) as in Case (a), while the dilution factor for RHNs, $\Delta_{N}$ is
calculated assuming an initial RHN abundance just before their decay to be
Bezrukov:2009th ,

$\frac{n_{N}}{s}=\frac{135\,\zeta(3)}{4\pi^{4}\,g_{\ast S}}$ (38)

which is based on the assumption that RHNs were relativistic when they froze
out, analogous to Eq. (23). In the large entropy injection limit $\Delta_{N}$
is given as,

$\Delta_{N}\simeq 1109\,\sqrt{\frac{10^{-10}}{K}}$ (39)

Lastly, if the RHNs do not dominate (see Eq. (32)), we only have 1-step
entropy injection from scalar decay and corresponding dilution factor
$\Delta_{\phi}$. In that case, GW experiments can not probe the leptogenesis
scale, since no suppression of the inflationary GW spectrum occurs at
$T_{N_{1}}$.

##  V Gravitational Waves of Inflationary Origin

###  V.1 Primordial spectrum in standard and non-standard cosmology

Primordial inflation, the early exponential expansion of the Universe, is
believed to have transitioned into an epoch known as reheating. This
transition resulted in the creation of a high-energy plasma of radiation,
following the dynamics of the Standard Model of particle physics. During
inflation, gravitational waves (GWs) from quantum tensor fluctuations of the
metric were generated. Initially, while outside the cosmological horizon,
these GWs exhibited constant amplitudes. However, as they re-entered the
horizon in the radiation-dominated era, their amplitudes underwent damping.
The propagation of GW modes that re-enter the horizon is controlled by a
transfer function.

Figure 7: Same as Fig. 5 but for a benchmark Case (b), “RHN domination”
scenario. Here we take $T_{\phi}=10^{7}$ GeV, $r=0.035$, $n_{T}=0.5$ and vary
the washout parameter $K$ along with $M_{1}$ and $M_{\phi}$. The spectral
shape depends on the leptogenesis scale $T_{N_{1}}=M_{1}\sqrt{K}$ as shown.
Again, solid and dashed black lines represent $M_{\phi}=10^{9}$ and $3\times
10^{9}$ GeV respectively.

The current power spectrum of GWs, denoted as $\Omega_{GW}(k)$ can be
expressed as a function of the wave number, $k=2\pi f$ where $f$ is the
frequency Berbig:2023yyy ,

$\Omega_{GW}(k)=\frac{1}{12}\left(\frac{k}{a_{0}H_{0}}\right)^{2}T_{T}^{2}(k)P_{T}^{\text{prim.}}(k)$
(40)

with $a_{0}=1$ representing the present-day scale factor and $H_{0}\approx
2.2\times 10^{-4}$ Mpc-1 being the current Hubble expansion rate Datta:2022tab
. $P_{T}^{\text{prim.}}(k)$ is the primordial tensor power spectrum from
inflation, and $T_{T}^{2}(k)$ is the transfer function that characterizes the
propagation of GWs in the background of a Friedmann-Lemaitre-Robertson-Walker
(FLRW) Universe after the temperature at the time of horizon re-entry,
$T_{\text{in}}$ given by Nakayama:2008wy :

$T_{\text{in}}=5.8\times
10^{6}\,\text{GeV}\left(\frac{106.75}{g_{*}(T_{\text{in}})}\right)^{1/6}\left(\frac{k}{10^{14}\,\text{Mpc}^{-1}}\right)$
(41)

The conventional parameterization for the inflationary tensor power spectrum,
$P_{T}^{\text{prim.}}(k)$ typically involves expressing its amplitude
($A_{T}$) and spectral index ($n_{T}$) with respect to the pivot scale
$k_{*}=0.05\,\text{Mpc}^{-1}$, as outlined in the Planck 2018 analysis
Planck:2018jri .

$P_{T}^{\text{prim.}}(k)=A_{T}(k_{*})\left(\frac{k}{k_{*}}\right)^{n_{T}}$
(42)

where the amplitude $A_{T}(k_{*})$ is related to the scalar power spectrum via
the tensor-to-scalar-ratio $r\leq 0.035$ BICEP:2021xfz :

$A_{T}(k_{*})=A_{S}(k_{*})\,r,\quad A_{S}(k_{*})=2.0989\times 10^{-9}$ (43)

where $A_{S}(k_{*})$ is scalar power spectrum Planck:2018jri . For our
analysis we fix the value of $r=0.035$. The spectral index $n_{T}$ plays a
critical role in characterizing the inflationary tensor power spectrum. While
standard single-field slow-roll inflation predicts a red-tilted spectrum,
where $n_{T}$ satisfies the consistency relation $n_{T}=-r/8$ Liddle:1993fq ,
other scenarios, including blue-tilted spectra ($n_{T}>0$), are well-motivated
in various cosmological models Brandenberger:2006xi ; Baldi:2005gk ;
Kobayashi:2010cm ; Calcagni:2004as ; Calcagni:2013lya ; Cook:2011hg ;
Mukohyama:2014gba ; Kuroyanagi:2020sfw .

Figure 8: Same as Fig. 5 but for Case (c), thermal leptogenesis, portraying
2-step entropy injection and a knee-like feature around $f\sim 10^{-2}$ Hz due
to $\phi$ decay, in addition to the one due to RHN decay at lower frequencies.
Here, we take $r=0.035$, $n_{T}=0.5$, $M_{\phi}=10^{13}$ GeV,
$T_{\phi}=10^{5}$ GeV and $R_{\rm RH}=10^{15}$ GeV. The effect of $K$ is
shown. The solid and dashed line correspond to $M_{1}=10^{5}$ and $3\times
10^{5}$ GeV.

Numerical computation and analytical fitting gives the transfer function
$T_{T}^{2}(k)$ as:

$T_{T}^{2}(k)=\Omega_{m}^{2}\left(\frac{g_{*}(T_{\text{in}})}{g_{*}^{0}}\right)\left(\frac{g_{*S}^{0}}{g_{*S}(T_{\text{in}})}\right)^{4/3}\left(\frac{3j_{1}(z_{k})}{z_{k}}\right)^{2}F(k)$
(44)

Here, $g_{*}^{0}=3.36$ and $g_{*S}^{0}=3.91$ are the present-day values of the
effective degrees of freedom. $\Omega_{m}=0.31$ Datta:2022tab  is the total
matter density. The Bessel function $j_{1}(z_{k})$ describes the damping of GW
amplitudes after horizon re-entry with $z_{k}\equiv k\tau_{0}$ and
$\tau_{0}=2/H_{0}$. In the limit $z_{k}\gg 1$, relevant for the frequencies of
interest, we can replace $j_{1}(z_{k})$ for $1/(\sqrt{2}z_{k})$. Finally, the
fitting function $F(k)$ in standard cosmology is given by Kuroyanagi:2014nba ,

$F(k)_{\text{standard}}=T_{1}^{2}\left(\frac{k}{k_{\text{eq.}}}\right)T_{2}^{2}\left(\frac{k}{k_{\text{RH}}}\right)$
(45)

If there was an intermediate matter domination epoch, the fitting function is
modified Kuroyanagi:2014nba ,

$F(k)_{\text{IMD}}=T_{1}^{2}\left(\frac{k}{k_{\text{eq.}}}\right)T_{2}^{2}\left(\frac{k}{k_{\text{dec.}}}\right)T_{3}^{2}\left(\frac{k}{k_{\text{dec.
S}}}\right)T_{2}^{2}\left(\frac{k}{k_{\text{RH S}}}\right)$ (46)

Here we introduce:

$\displaystyle k_{\text{eq.}}$ $\displaystyle=7.1\times
10^{-2}\,\text{Mpc}^{-1}\cdot\Omega_{m}h^{2}$ (47) $\displaystyle
k_{\text{dec.}}$ $\displaystyle=1.7\times
10^{14}\,\text{Mpc}^{-1}\left(\frac{g_{*S}(T_{\text{dec.}})}{g_{*S}^{0}}\right)^{1/6}\left(\frac{T_{\text{dec.}}}{10^{7}\,\text{GeV}}\right)$
(48) $\displaystyle k_{\text{RH}}$ $\displaystyle=1.7\times
10^{14}\,\text{Mpc}^{-1}\left(\frac{g_{*S}(T_{\text{RH}})}{g_{*S}^{0}}\right)^{1/6}\left(\frac{T_{\text{RH}}}{10^{7}\,\text{GeV}}\right)$
(49) $\displaystyle k_{\text{dec. S}}$
$\displaystyle=k_{\text{dec.}}\Delta^{2/3}$ (50) $\displaystyle k_{\text{RH
S}}$ $\displaystyle=k_{\text{RH}}\Delta^{-1/3}$ (51)

where $T_{\text{dec.}}$ is defined by (34) and we set $h=0.7$. The entropy
dilution factor $\Delta$ is the dilution factor and the fit functions read,

$\displaystyle T_{1}^{2}(x)$ $\displaystyle=1+1.57x+3.42x^{2}$ (52)
$\displaystyle T_{2}^{2}(x)$
$\displaystyle=\left(1-0.22x^{3/2}+0.65x^{2}\right)^{-1}$ (53) $\displaystyle
T_{3}^{2}(x)$ $\displaystyle=1+0.59x+0.65x^{2}$ (54)

At this point we can plot the spectrum $\Omega_{\rm GW}h^{2}$ as a function of
current frequency, using Eq. (40). We discuss the features in these plots
accounting for $\phi$ and $N$ domination in Sec. VI.1. The suppression of the
GW spectrum happens above a frequency $f_{\rm sup}$, given by,

$f_{\text{sup}}\simeq 2.7\times
10^{-8}\;\text{Hz}\;\left(\frac{T_{\text{dec.}}}{{\rm GeV}}\right)$ (55)

The suppression factor is given by Seto:2003kc ,

$R_{\text{sup}}=\frac{\Omega_{\text{GW}}^{\text{IMD}}}{\Omega_{\text{GW}}^{\text{standard}}}\simeq\frac{1}{\Delta^{4/3}}$
(56)

Figure 9: Same as Fig. 8 but here we take $M_{1}=10^{6}$ GeV, $K=10^{-8}$ and
show the dependence of the spectral shape on $T_{\phi}$ and $M_{\phi}$. The
solid and dashed lines represent $M_{\phi}=10^{13}$ and $3\times 10^{13}$ GeV
respectively.

###  V.2 Two-step entropy injection

Our Case (c) discussed in Sec. IV has two stages of entropy injection. We
propose a modified transfer function to incorporate the two stages along the
lines of Kuroyanagi:2014nba , by assuming the form of the fitting function to
be,

$F(k)_{\text{IMD}}^{\text{2-step}}=T_{1}^{2}\left(\frac{k}{k_{\text{eq.}}}\right)T_{2}^{2}\left(\frac{k}{k_{\text{dec.}}^{\phi}}\right)T_{3}^{2}\left(\frac{k}{k_{\text{dec.S}}^{\phi}}\right)T_{2}^{2}\left(\frac{k}{k_{\text{dec.}}^{N}}\right)T_{3}^{2}\left(\frac{k}{k_{\text{dec.S}}^{N}}\right)T_{2}^{2}\left(\frac{k}{k_{\text{RH
S}}^{\text{2-step}}}\right)$ (57)

where $k_{\text{dec.}}^{\phi},k_{\text{dec.}}^{N}$ are related to the decay
temperatures $T_{\phi}$ and $T_{N_{1}}$ respectively and
$k_{\text{dec.S}}^{\phi},k_{\text{dec.S}}^{N}$ are related to the domination
temperatures of $\phi$ and $N_{1}$ respectively. We define them as,

$\displaystyle k_{\text{dec.}}^{\phi}$ $\displaystyle=1.7\times
10^{14}\,\text{Mpc}^{-1}\left(\frac{g_{*S}(T_{\phi})}{g_{*S}^{0}}\right)^{1/6}\left(\frac{T_{\phi}}{10^{7}\,\text{GeV}}\right)$
(58) $\displaystyle k_{\text{dec.}}^{N}$ $\displaystyle=1.7\times
10^{14}\,\text{Mpc}^{-1}\left(\frac{g_{*S}(T_{N_{1}})}{g_{*S}^{0}}\right)^{1/6}\left(\frac{T_{N_{1}}}{10^{7}\,\text{GeV}}\right)$
(59) $\displaystyle k_{\text{dec.S}}^{\phi}$
$\displaystyle=k_{\text{dec.}}^{\phi}\Delta_{\phi}^{2/3}$ (60) $\displaystyle
k_{\text{dec.S}}^{N}$ $\displaystyle=k_{\text{dec.}}^{N}\Delta_{N}^{2/3}$ (61)
$\displaystyle k_{\text{RH S}}^{\text{2-step}}$
$\displaystyle=k_{\text{RH}}(\Delta_{\phi}\Delta_{N})^{-1/3}$ (62)

where $\Delta_{\phi},\Delta_{N}$ are the dilution factors from $\phi$ and RHN
decays respectively. The necessary conditions to use this transfer function
are $M_{\phi}\gg M_{1},T_{\phi}>T_{N_{1}}$ and Eq. (32) as discussed earlier.
A proper way to obtain the 2-step transfer function requires numerical fitting
which we left as a future direction. In this case we have two suppression
frequencies corresponding to scalar and RHN decays, analogous to Eq. (55),

$f_{\rm sup}^{\phi}\simeq 2.7\times
10^{-8}\;\text{Hz}\;\left(\frac{T_{\phi}}{{\rm GeV}}\right),\quad f_{\rm
sup}^{N}\simeq 2.7\times 10^{-8}\;\text{Hz}\;\left(\frac{T_{N_{1}}}{{\rm
GeV}}\right),\quad$ (63)

And similar to Eq. (56), the suppression factors corresponding to the first
($\phi$ decay) and the second ($N$ decay) step of entropy injection are given
by,

$R_{\text{sup}}^{\rm 1st}\simeq\frac{1}{(\Delta_{\phi}\Delta_{N})^{4/3}},\quad
R_{\text{sup}}^{\rm 2nd}\simeq\frac{1}{\Delta_{N}^{4/3}}$ (64)

Figure 10: Projected sensitivities of different experiments in the RHN
parameter space for $n_{T}=0$ (Left) and $n_{T}=0.5$ (Right). Here
$\tilde{m}_{1}$ is the effective neutrino mass defined by Eq. (8) and
$m_{*}=1.1\times 10^{-3}$ eV is the equilibrium neutrino mass defined in Eq.
(9). For both plots we took $M_{\phi}$= $10^{15}$ GeV , $T_{\phi}$ = $10^{10}$
GeV, $r=0.035$. The colored regions associated with each experiment implies
sensitivity with SNR $>10$. As expected, a higher $n_{T}$ increases overall
SNR. See main text for detailed description and analysis.

Figure 11: Same as Fig. 10 but for $n_{T}=0$, $T_{\phi}$ = $10^{6}$ GeV and
$M_{\phi}$= $10^{15}$ GeV (Left) and $10^{10}$ GeV (Right). For a fixed
$T_{\phi}$, the left plot is expected to have smaller overall SNR values due
to higher $M_{\phi}$. Here $m_{*}=1.1\times 10^{-3}$ eV.

###  V.3 Signal-to-noise ratio (SNR)

Interferometers are instruments that measure displacements in terms of
dimensionless strain-noise denoted as $h_{\text{GW}}(f)$. This strain-noise is
related to the amplitude of gravitational waves (GWs) and can be converted
into an energy density, expressed by the formula:

$\Omega_{\text{exp}}(f)h^{2}=\frac{2\pi^{2}f^{2}}{3H_{0}^{2}}h_{\text{GW}}(f)^{2}h^{2}$
(65)

Here, $H_{0}$ represents the present-day Hubble rate ($H_{0}=h\times
100\frac{\text{km/s}}{\text{Mpc}}$). To assess the likelihood of detecting the
primordial GW background, we calculate the signal-to-noise ratio (SNR) using
the experimental sensitivity $\Omega_{\text{exp}}(f)h^{2}$, either given or
projected. The SNR is determined by the following formula:

$\text{SNR}\equiv\sqrt{\tau}\int_{f_{\text{min}}}^{f_{\text{max}}}df\left(\frac{\Omega_{\text{GW}}(f)h^{2}}{\Omega_{\text{exp}}(f)h^{2}}\right)^{2}$
(66)

In this analysis, we use $h=0.7$ and an observation time of $\tau=4$ years. A
detection threshold of SNR $\geq 10$ is applied.

In the spectrum plots we display the sensitivity curves for different ongoing
and future GW experiments. They can be grouped as:

  * •

Ground-based interferometers: These detectors, such as LIGO/VIRGO
LIGOScientific:2016aoc ; LIGOScientific:2016sjg ; LIGOScientific:2017bnn ;
LIGOScientific:2017vox ; LIGOScientific:2017ycc ; LIGOScientific:2017vwq ,
aLIGO/aVIRGO LIGOScientific:2014pky ; VIRGO:2014yos ; LIGOScientific:2019lzm ,
AION Badurina:2021rgt ; Graham:2016plp ; Graham:2017pmn ; Badurina:2019hst ,
Einstein Telescope (ET) Punturo:2010zz ; Hild:2010id , and Cosmic Explorer
(CE) LIGOScientific:2016wof ; Reitze:2019iox , use interferometric techniques
on the Earth’s surface to detect gravitational waves.

  * •

Space-based interferometers: Space-based detectors like LISA Baker:2019nia ,
BBO Crowder:2005nr ; Corbin:2005ny ; Cutler:2009qv , DECIGO, U-DECIGO
Seto:2001qf ; Yagi:2011wg , AEDGE AEDGE:2019nxb ; Badurina:2021rgt , and
$\mu$-ARES Sesana:2019vho  are designed to detect gravitational waves from
space, offering different advantages over ground-based counterparts.

  * •

Recasts of star surveys: Surveys such as GAIA/THEIA Garcia-Bellido:2021zgu
utilize astrometric data from stars to indirectly infer the presence of
gravitational waves.

  * •

Pulsar timing arrays (PTA): PTA experiments like SKA Carilli:2004nx ;
Janssen:2014dka ; Weltman:2018zrl , EPTA EPTA:2015qep ; EPTA:2015gke , and
NANOGRAV NANOGRAV:2018hou ; Aggarwal:2018mgp ; NANOGrav:2020bcs  use precise
timing measurements of pulsars to detect gravitational wave signatures.

  * •

CMB polarization: Experiments like Planck 2018 Planck:2018jri ; Planck:2018vyg
, BICEP 2/Keck BICEP:2021xfz  as computed by Clarke:2020bil , and LiteBIRD
Hazumi:2019lys  focus on detecting gravitational wave imprints in the
polarization of the cosmic microwave background radiation.

  * •

CMB spectral distortions: Projects like PIXIE, Super-PIXIE Kogut:2019vqh , and
VOYAGER2050 Chluba:2019kpb  aim to detect gravitational wave signatures
through spectral distortions in the cosmic microwave background.

The blue violine lines in the spectrum plots are the most recent data from
NANOGrav.

###  V.4 Bounds from BBN and CMB on dark radiation

The gravitational wave energy density should be smaller than the bound on dark
radiation parameterized as the number of additional neutrinos $\Delta N_{\rm
eff}$ Luo:2020fdt ; Maggiore:1999vm ,

$\int_{f_{\rm min}}^{\infty}\frac{df}{f}\Omega_{\rm GW}(f)h^{2}\leq 5.6\times
10^{-6}\Delta N_{\rm eff}$ (67)

We neglect the frequency dependence of this bound and set $\Omega_{\rm GW}\leq
5.6\times 10^{-6}\Delta N_{\rm eff}$ for the GW spectra that we calculate. BBN
puts a bound on $\Delta N_{\rm eff}^{\rm BBN}\simeq 0.4$ Cyburt:2015mya .
Planck plus BAO observations set the bound $\Delta N_{\rm eff}^{\rm
Plack+BAO}\simeq 0.28$ Planck:2018vyg . Projected bounds from future
experiments are $\Delta N_{\rm eff}^{\rm Proj.}=0.014$ for CMB-HD CMB-
HD:2022bsz , $\Delta N_{\rm eff}^{\rm Proj.}=0.05$ for CMB-Bharat CMB-bharat ,
$\Delta N_{\rm eff}^{\rm Proj.}=0.06$ for CMB Stage IV doi:10.1146/annurev-
nucl-102014-021908  and NASA’s PICO mission Alvarez:2019rhd , $\Delta N_{\rm
eff}^{\rm Proj.}\lesssim 0.12$ for CORE CORE:2017oje , the South Pole
Telescope SPT-3G:2014dbx  and Simons observatory SimonsObservatory:2018koc .

##  VI Results

###  VI.1 Example spectra for different cases

Fig. 5 illustrates the damped gravitational wave (GW) spectra for Case (a)
discussed in Sec. IV, assuming spectral index $n_{T}=0.5$ and $r=0.035$,
demonstrating the influence of $T_{\phi}$ and $M_{\phi}$, alongside
sensitivity curves of current and future GW experiments. The blue violin lines
are taken from recent NANOGrav 15yr dataset NANOGrav:2023hvm ;
NANOGrav:2023gor . Different horizontal lines on the plot are coming from the
current or projected $\Delta N_{\rm eff}$ bounds from BBN, Planck 2018 and CMB
surveys Berbig:2023yyy . In Case (a) scenario, characterized by the condition
$T_{\phi}<T_{N_{1}}$, we set the minimum $T_{\phi}$ to be approximately 100
GeV to accommodate sphaleron effects within the electroweak scale. The gray
solid line marked “No IMD” is the standard spectrum with no intermediate
matter dominated phase while the black solid and dashed lines are the spectral
shapes in presence of the matter dominated epoch. Higher frequencies in the
figure correspond to earlier times, showcasing suppression beyond the critical
frequency $f_{\rm sup}$ due to entropy injection given by Eq. (55).
Suppression at frequencies higher than $f_{\rm sup}\sim 10^{-5}-0.1$ Hz are
attributed to scalar and right-handed neutrino (RHN) decay, while suppression
beyond $\sim 10$ Hz is a consequence of inflaton decay terminating at the
reheating temperature $T_{\rm RH}=10^{10}$ GeV. Lower values of $T_{\phi}$
shift the suppression frequency to the left and increase dilution factor. The
solid and dashed spectral lines represent $M_{\phi}=10^{9}$ and $3\times
10^{9}$ GeV respectively. A higher $M_{\phi}$ means more dilution.
Importantly, GW signatures remain invariant across weak and strong washout
regimes, as they solely depend on the scalar dynamics in this case. Fig. 6 is
same as Fig. 5 but shown for $n_{T}=0$.

Figure 12: Same as Fig. 10 but for $T_{\phi}=100$ GeV and $n_{T}=0.5$,
$M_{\phi}=10^{15}$ GeV. Here $m_{*}=1.1\times 10^{-3}$ eV.

Fig. 7 presents the GW signature of long-lived RHNs in Case (b) for different
values of the washout parameter $K$ and lightest RHN mass $M_{1}$. Once again
the solid and dashed spectral lines correspond to $M_{\phi}=10^{9}$ and
$3\times 10^{9}$ GeV respectively. We choose the combination
$T_{N_{1}}=M_{1}\sqrt{K}$ in a way such that $100\;\text{GeV}\leq
T_{N_{1}}<T_{\phi}$, where we have taken $T_{\phi}=10^{7}$ GeV. Other
parameters are set as: $r=0.035,n_{T}=0.5,T_{\rm RH}=10^{10}$ GeV. We see that
the role of $T_{\phi}$ seen in Fig. 5 is taken up by $T_{N_{1}}$ in this case,
as expected.

Case (c) mirrors thermal leptogenesis along with 2-step entropy injection as
discussed earlier. Figures 8 and 9 depict the GW signatures of this scenario.
For these plots, we make sure that the thermalization condition given in Eq.
(84) is satisfied assuming $y_{N_{i}}=y_{R}$, such the two-step entropy
injection scenario is applicable. We use the 2-step transfer function
utilizing Eq. (57). Notably, the scalar’s effect manifests as an additional
dip in the spectrum, a knee-like feature, positioned between the dips induced
by inflaton decay and RHN decay. Thus we have two knee-like features, one from
$\phi$ decay and another from RHN decay. In Fig. 8 we show the dependency of
the spectral shape on $M_{1}$ and $K$, fixing $T_{\phi}=10^{5}$ GeV,
$M_{\phi}=10^{13}$ GeV, $T_{\rm RH}=10^{15}$ GeV for $n_{T}=0.5$. The solid
and dashed lines correspond to $M_{1}=10^{5}$ and $3\times 10^{5}$ GeV
respectively. Note that for $T_{N_{1}}<100$ GeV, efficient leptogenesis is not
possible as sphalerons are turned off.

Finally in Fig. 9, we show the dependency of the spectrum on $T_{\phi}$ and
$M_{\phi}$ fixing $M_{1}=10^{6}$ GeV, $K=10^{-8}$ such that leptogenesis scale
is $T_{N_{1}}=100$ GeV. Other parameters are same as in Fig. 8. The solid and
dashed lines here represent $M_{\phi}=10^{13}$ and $3\times 10^{13}$ GeV
respectively. The entropy injection from scalar is independent of leptogenesis
scale but it reduces the projected SNR for different experiments via the
additional suppression as seen in the plots.

Figure 13: Same as Fig. 10 but for $T_{\phi}=100$ GeV and $n_{T}=0$,
$M_{\phi}=10^{5}$ GeV. Here $m_{*}=1.1\times 10^{-3}$ eV.

Figure 14: Same as Fig. 10 but demonstrating thermal leptogenesis (Case (c) in
Sec. IV) with two-step entropy injection for $n_{T}=0$ (Left) and $n_{T}=0.5$
(Right). Here $m_{*}=1.1\times 10^{-3}$ eV. For both plots we took $M_{\phi}$=
$10^{15}$ GeV , $T_{\phi}$ = $10^{10}$ GeV, $r=0.035$. The green shaded region
represents instantaneous RHN decay (Case (a)). Two-step entropy injection is
not possible in the brown shaded region as $T_{\phi}$ is less than RHN
domination temperature $T_{\rm dom}^{N_{1}}\sim 2\%M_{1}$. See main text for
detailed description and analysis.

Figure 15: Same as Fig. 14 but for $T_{\phi}=10^{6}$ GeV.

###  VI.2 Estimates for Signal to Noise Ratio

The next set of Figures 10, 11, 12, 13, 14, 15 demonstrates the projected
future sensitivities of U-DECIGO, DECIGO, BBO, $\mu$-ARES, LISA, CE and ET
experiments in terms of signal to noise ratio (SNR) given in Eq. (66), in the
parameter space of the RHN for different values of
$n_{T},M_{\phi},T_{\phi},T_{\rm RH}$. Figures 10, 11, 12, 13 correspond to
non-thermal leptogenesis with black dashed line separating the Case (a) and
Case (b) regions. Figures 14 and 15 demostrate the thermal leptogenesis with
two-step entropy injection scenario, Case (c), with green region specifying
Case (a) and brown region corresponding to $T_{\phi}<T_{\rm dom}^{N_{1}}$,
where two-step entropy injection is not possible as described in Sec IV.

The gray shaded region is excluded from the requirement that $T_{\rm eq.}$
should be larger than the BBN temperature of a few MeV. In Fig. 14 and 15 we
are not interested in $K\gtrsim 10^{-4}$ where RHN domination is not possible,
i.e. Eq. (32) is not satisfied, thus GW is not connected to leptogenesis in
this region. In all the SNR plots, the colored regions associated with
different experiments can be probed by that experiment with SNR $\geq 10$. In
all these SNR plots we choose $T_{\rm RH}=10^{16}$ GeV in order to eliminate
its effect on the SNR. In vanilla leptogenesis scenario, vacum stability of SM
Higgs gives an upper bound on the lightest RHN mass $M_{1}\lesssim 10^{14}$
GeV Casas:1999cd ; Elias-Miro:2011sqh  and we assume it would be approximately
the same in our model Fu:2023nrn . Hence we take upper limit on $M_{1}$ to be
around $10^{14}$ GeV. On the other hand, the perturbativity limit of the model
requires the Yukawa couplings $\lambda_{ij}<4\pi$, which along with the
necessity to produce at least one mass eigenstate $m_{\nu}=0.05$ eV gives
another bound, $M_{2}\lesssim 10^{15}$ GeV. Hence, in our quasi-degenerate RHN
model, we find the perturbativity bound $M_{1}\lesssim 10^{15}$ GeV. The
perturbativity limit of the light-heavy neutrino mixing $y_{N_{i}}<4\pi$ is
always satisfied in our model.

In Fig. 10, we pick a high scalar mass and decay temperature
$M_{\phi}=10^{15}$ GeV, $T_{\phi}=10^{10}$ GeV and show the projected
sensitivities of different experiments for $n_{T}=0$ (Left) and $n_{T}=0.5$
(Right). We see that increasing the spectral index $n_{T}$ also increases the
GW sensitive region by increasing the SNR as expected. For $n_{T}=0$, LISA and
ET will not be able to probe leptogenesis but for $n_{T}=0.5$ both the
experiments will be sensitive to a large portion of the parameter space. In
the Case (a) region of instantaneous RHN decay, the SNR will not depend on
$M_{1}$ and $K$ but only on $M_{\phi}$ and $T_{\phi}$. In the Case (b) region,
RHNs decay at $T_{N_{1}}=M_{1}\sqrt{K}$. Hence decreasing $M_{1}$ or $K$
results in more suppression and lower SNR.

In Fig. 11, we show the sensitivities for $n_{T}=0$, $T_{\phi}$ = $10^{6}$ GeV
and $M_{\phi}$= $10^{15}$ GeV (Left) and $10^{10}$ GeV (Right). Both the left
and right plots show almost the same sensitive area in the parameter space but
suppression in the left plot is expected to be higher due to higher mass
scalar decay. In both the plots we see the Case (a) region covering more space
due to lower $T_{\phi}$ value compared to Fig. 10. We also notice that CE is
no longer sensitive for $n_{T}=0,T_{\phi}=10^{6}$ GeV in Fig. 11 (Left) while
it is sensitive for $T_{\phi}=10^{10}$ GeV in Fig. 10 (Left).

In Fig. 12, we show the effect of a low scalar decay temperature
$T_{\phi}=100$ GeV assuming $M_{\phi}=10^{15}$ GeV. We note that lowering
$T_{\phi}$ increases the area of the region of instantaneous RHN decay (Case
(a)) in the parameter space, and for $n_{T}=0.5$ most of the parameter space
is accessible to DECIGO, BBO, $\mu-$ARES and U-DECIGO. We find no experiment
to be sensitive to leptogenesis for $n_{T}=0$ in this case. Since
$T_{\phi}=100$ GeV, below this scale efficient leptogenesis is not possible in
the Case (b) region in this figure.

Fig. 13 shows the shape of the sensitive regions for $M_{\phi}=10^{5}$ GeV and
$T_{\phi}=100$ GeV assuming $n_{T}=0$. We see that even for $n_{T}=0$,
UDECIGO, DECIGO, BBO, $\mu-$ARES can probe leptogenesis with $T_{\phi}=100$
GeV, given small mass $M_{\phi}$ of the scalar. A small scalar mass injects
less amount of entropy when they decay as a result of which the suppression is
small, hence SNR is larger.

In Fig. 14 and 15 we show the SNR contours in the Case (c) region of two-step
entropy injection assuming a non-vanishing coupling $y_{R}$. We make sure that
the thermalization condition (84) is satisfied for given parameter choices as
well as the condition for two-stages of entropy injection, $T_{\phi}>T_{\rm
dom}^{N_{1}}$ described in Sec. IV. In Fig. 14 we chose $M_{\phi}=10^{15}$
GeV, $T_{\phi}=10^{10}$ GeV and $T_{\rm RH}=10^{16}$ GeV and demosntrate the
effect of the spectral index $n_{T}=0$ (Left) and $n_{T}=0.5$ (Right). Note
that instantaneous RHN decay scenario (Case (a)) is still possible in thermal
scenario, denoted by the green shaded region. Fig. 15 is the same as Fig. 14
but for $T_{\phi}=10^{6}$ GeV.

We calculate the SNR contours for all these GW experiments taking various
values of $M_{\phi}$ and $T_{\phi}$ for $n_{T}=0$ and $0.5$, which we do not
show in this paper. However, based on these results and Fig. 10-15, in Tables
2 and 3, we summarize which key GW experiments can probe the parameter space
$M_{1}$ vs $K$, associated to thermal and non-thermal leptogenesis in the
future with threshold SNR $\geq 10$, for these parameter choices. We see that
thermal leptogenesis is possible only for high scalar mass $M_{\phi}\sim
10^{15}$ GeV which U-DECIGO, BBO and $\mu-$ARES will be able to probe even for
$n_{T}=0$ while LISA and ET requires $n_{T}\gtrsim 0.5$.

$M_{\phi}\,$ (GeV) |  $T_{\phi}\,$ (GeV) | U-DECIGO | BBO |  $\mu-$ARES | LISA | ET | CE
---|---|---|---|---|---|---|---
$10^{15}$ | $10^{10}$ | Non-Th, Th | Non-Th, Th | Non-Th, Th | - | - | Non-Th, Th
$10^{6}$ | Non-Th, Th | Non-Th, Th | Non-Th, Th | - | - | -
$10^{2}$ | - | - | - | - | - | -
$10^{10}$ | $10^{6}$ | Non-Th | Non-Th | Non-Th | - | - | -
$10^{2}$ | - | - | - | - | - | -
$10^{5}$ | $10^{2}$ | Non-Th | Non-Th | Non-Th | - | - | -
Table 2: Future sensitivity of different key experiments to probe the RHN parameter space ($M_{1}$ vs. $K$) with threshold SNR $>10$, given benchmark values of $M_{\phi}$ and $T_{\phi}$ for $n_{T}=0$. Non-Th and Th implies sensitivity of the specific experiment to non-thermal and thermal leptogenesis. Note that thermal scenario requires Eq. (84) to be satisfied. Here we considered $T_{\rm RH}=10^{16}$ GeV and maximal tensor-to-scalar ratio $r=0.035$. $M_{\phi}\,$ (GeV) |  $T_{\phi}\,$ (GeV) | U-DECIGO | BBO |  $\mu-$ARES | LISA | ET | CE
---|---|---|---|---|---|---|---
$10^{15}$ | $10^{10}$ | Non-Th, Th | Non-Th, Th | Non-Th, Th | Non-Th, Th | Non-Th, Th | Non-Th, Th
$10^{6}$ | Non-Th, Th | Non-Th, Th | Non-Th, Th | Non-Th, Th | Non-Th, Th | Non-Th, Th
$10^{2}$ | Non-Th | Non-Th | Non-Th | - | - | -
$10^{10}$ | $10^{6}$ | Non-Th | Non-Th | Non-Th | Non-Th | Non-Th | Non-Th
$10^{2}$ | Non-Th | Non-Th | Non-Th | Non-Th | Non-Th | Non-Th
$10^{5}$ | $10^{2}$ | Non-Th | Non-Th | Non-Th | Non-Th | Non-Th | Non-Th
Table 3: Same as Table. 2 but for $n_{T}=0.5$. We see that most of the
parameter space $M_{\phi}$ vs $T_{\phi}$ will be probed by future GW
experiments if $n_{T}=0.5$.

###  VI.3 Compatibility with NANOGrav

Several papers NANOGrav:2023hvm ; Vagnozzi:2023lwo ; Datta:2023xpr ;
Ellis:2023oxs  discuss the possibility of inflationary gravitational waves to
be the source of the stochastic background detected in the recent NANOGrav
data NANOGrav:2023gor . The NANOGrav collaboration NANOGrav:2023hvm  states
the best fit is found to be around $n_{T}\sim 2.61\pm 0.85$, $r\sim
10^{-14.06\pm 5.82}$ and $T_{\rm RH}\sim 1$ GeV. However at such low reheating
temperatures, efficient leptogenesis as we discussed in the paper is not
possible due to sphaleron constraints.

##  VII Discussion & Conclusion

Leptogenesis at low and intermediate energy scales provides an attractive
scenario for baryogenesis, linking the observed the matter-antimatter
asymmetry to the neutrino sector and seesaw scale. In this analysis, we offer
a probe of this scenario via observations of the stochastic GW background that
originates during inflation (which is responsible for the creation of
primordial density perturbation of the Universe) and propagates in the post-
inflationary era to reach us at the present time. Particularly we showed that
detecting a characteristic GW spectral shape will represent a smoking gun for
a RHN or $\phi$ matter-dominated era in the early Universe. We present a
minimal model realization of non-thermal leptogenesis, for which we found that
a large fraction of the viable parameter space will be probed in future GW
experiments. For the lightest RHN mass $M_{1}$ and washout parameter $K$
involving light-heavy neutrino Yukawa couplings $\lambda$, we find that
$M_{1}$ in the range $100$ GeV $\lesssim M_{1}\lesssim 10^{14}$ GeV and $K$
for extremely weak washout ($K\sim 10^{-20}$) and for strong washout ($K\sim
10^{4}$) regime consistent with successful baryogenesis, novel GW shapes will
be probes by U-DECIGO, DECIGO, BBO, $\mu-$ARES, LISA, CE, ET etc. By
estimating signal-to-noise ratio (SNR) for GW experiments, we investigated the
effect of the scalar mass $M_{\phi}$ and decay temperature $T_{\phi}$ which
depends on the $\phi-N$ Yukawa couplings $y_{N}$.

The key interpretations of our results are summarized below:

  * •

The overall SNR is larger for a larger spectral index $n_{T}$ (compare for
example Fig. 10 Left and Right plots).

  * •

For vanishing Yukawa coupling $y_{R}$ defined in the Lagrangian (19), we
obtain non-thermal leptogenesis which can be probed in future GW experiments
such as U-DECIGO, BBO etc. (see spectra in Fig. 5, 7 and SNR in Fig. 10, 11,
12, 13).

  * •

Thermal leptogenesis with two-step entropy injection is possible with a non-
zero $y_{R}$ (Eq. (83) and Fig. 14, 15). We propose the two-step entropy
injection transfer function via Eq. (57) (see spectra in Fig. 8 and 9). Such
two step will be detected in U-DECIGO, BBO, $\mu-$ARES etc. for $n_{T}=0$ and
LISA, ET and CE as well for $n_{T}=0.5$.

  * •

If $T_{N_{1}}<T_{\phi}$, then lower values of $M_{1}$ and $K$ reduce SNR and
therefore challenging to test for all experiments (Fig. 10, 11, 12, 13, 14,
15).

  * •

If $T_{\phi}<T_{N_{1}}$ in the non-thermal scenario, a lower $T_{\phi}$ value
decreases the SNR and frequency of suppression $f_{\rm sup}$ given in Eq.
(55), making it more difficult to observe (see spectra in Fig. 5,6). However,
suitable low values of $T_{\phi}$ increases the overall GW sensitive region in
the parameter space of the lightest RHN (compare Case (a) regions in Fig. 10,
11 and 12). The Case (a) scenario is discussed in Sec. IV.

  * •

A higher $M_{\phi}$ means larger entropy injection which decreases the overall
SNR values for all experiments (see spectra in Fig. 5, 6, 7, 9). $M_{\phi}$
also sets the upper bound on $M_{1}$ in our model, i.e. $M_{1}\leq
M_{\phi}/2$. We are interested in the parameter space where $\phi$ is long
lived in order to dominate the energy Budget if the Universe (see discussion
following Eq. (25) and Eq. (29)).

  * •

In Case (a) described in Sec. IV, where RHNs decay instantaneously after
production from $\phi$ decay, the leptogenesis scale is $\sim T_{\phi}$ which
means GW experiments can probe leptogenesis even for strong washout $K>1$
where RHNs do not dominate the energy budget of the Universe (Fig. 10, 11, 12,
13). This is unlike thermal leptogenesis Berbig:2023yyy  where the RHN
domination criterion $K\lesssim 10^{-4}$ given in Eq. (33) must be satisfied
for GW sensitivity.

  * •

Table 2 and 3 summarize which key experiments can probe thermal and non-
thermal leptogenesis for $n_{T}=0$ and $0.5$ respectively, given benchmark
values of the scalar mass $M_{\phi}$ and scalar decay temperature $T_{\phi}$.

If the characteristic features of the GW spectral shapes proposed in this
study are observed one may look to target additional observations in order to
distinguish between a RHN or scalar $\phi$ dominated pre-BBN era and other
forms of early matter domination. Particularly in low-scale leptogenesis RHN
masses of GeV-TeV could be searched in typical Heavy Neutral Lepton Searches
(HNL) search experiments (see Beacham:2019nyx ; Abdullahi:2022jlv ;
Middleton:2022dio  for current experimental limits) at the particle physics
laboratories. In this manner we can complement GW searches with laboratory
searches in the same BSM parameter space. however such a study is beyond the
reach of the present paper and we plan to explore this in future. RH
neutrinos, if they exist should also show up in experiments such as neutrino-
less double beta decay Dolinski:2019nrj ; Gomez-Cadenas2023  or lepton number
violating processes Li:2021fvw ; Cai:2017mow  and therefore provide us with a
myriad of pathways to independently verify the existence of an early RHN-
domination or scalar domination era. This leads to a unique and exciting
opportunity to form synergies between GW searches and laboratory and CMB
searches.

###### Acknowledgements.

ZAB is supported by a DST-INSPIRE fellowship. LM is supported by a UGC
fellowship. Authors thank Prof. Kai Schmidt for his valuable comments on the
NANOGrav best fit.

##  Appendix A Change of Variable

It is convenient to use comoving variables while solving the Boltzmann
equations. We adopt the same prescription as in Chung:1998rq ; Hahn-
Woernle:2008tsk ,

$E_{\phi,N_{i}}=\rho_{\phi,N_{i}}a^{3},\quad\tilde{N}_{i}=n_{i}a^{3}=\frac{\rho_{N_{i}}}{M_{i}}a^{3},\quad\tilde{N}_{\text{B-L}}=n_{\text{B-L}}a^{3},\quad
R=\rho_{R}a^{4},\quad S=s\,a^{3}$ (68)

where $a$ is the scale factor of the Universe. Instead of time $t$, we write
the Boltzmann equations in terms of the ratio of scale factor $a$ to its
initial value $a_{I}$, given by,

$y=\frac{a}{a_{I}}$ (69)

We take $a_{I}=1$ without loss of generality. The expansion rate can be
written as,

$H=\sqrt{\frac{8\pi(a_{I}E_{\phi}\,y+a_{I}E_{N_{1}}\,y+a_{I}E_{N_{2}}\,y+R)}{3M_{\text{pl}}^{2}\,a_{I}^{4}\,y^{4}}}$
(70)

The temperature can be written as,

$T=\left[\frac{30\,R}{\pi^{2}g_{\ast}a_{I}^{4}y^{4}}\right]^{\frac{1}{4}}$
(71)

However this is not the actual temperature of the Universe during the early
matter dominated epoch. The maximum possible temperature can be approximated
as,

$T_{\text{tot}}=\left[\frac{30\,(R+E_{\phi}+E_{N_{1}}+E_{N_{2}})}{\pi^{2}g_{\ast}a_{I}^{4}y^{4}}\right]^{\frac{1}{4}}$
(72)

The redshift parameter $z$ is defined in terms of $M_{1}$,

$z=\frac{M_{1}}{T}=M_{1}a_{I}\left[\frac{\pi^{2}g_{\ast}}{30\,R}\right]^{\frac{1}{4}}y$
(73)

The equilibrium energy densities are given by Hahn-Woernle:2008tsk ; Hahn-
Woernle:2009jyb ,

$\displaystyle E_{N_{i}}^{eq}$
$\displaystyle=\rho_{N_{i}}^{eq}a^{3}=\frac{a_{I}^{3}M_{i}^{4}y^{3}}{\pi^{2}}\left[\frac{3}{z^{2}}K_{2}(z)+\frac{1}{z}K_{1}(z)\right],\quad\tilde{N}_{i}^{eq}=\frac{E_{N_{i}}^{eq}}{M_{i}},$
$\displaystyle n_{N_{i}}^{eq}$
$\displaystyle=\frac{E_{N_{i}}^{eq}}{M_{i}y^{3}},\quad
n_{R}^{eq}=\frac{3\zeta(3)}{4\pi^{2}}\times
2\left(\frac{M_{1}}{z(y)}\right)^{3}$ (74)

The inverse decay rate can be written as Buchmuller:2004nz ,

$\Gamma_{ID}=\frac{1}{2}\sum_{i=1}^{2}\Gamma_{N_{i}}\frac{n_{N_{i}}^{eq}}{n_{R}^{eq}}$
(75)

Now we can write the Boltzmann equations given in Eq. (30) in terms of the
comoving energy densities as,

$\displaystyle\frac{dE_{\phi}}{dy}$
$\displaystyle=-\frac{\Gamma_{\phi}}{\mathcal{H}}\frac{E_{\phi}}{y},$
$\displaystyle\frac{dE_{N_{1}}}{dy}$
$\displaystyle=\frac{\Gamma_{\phi\rightarrow
N_{1}N_{1}}}{\mathcal{H}}\frac{E_{\phi}}{y}-\frac{\Gamma_{N_{1}}}{\mathcal{H}y}\left(E_{N_{1}}-E_{N_{1}}^{eq}\right),$
$\displaystyle\frac{dE_{N_{2}}}{dy}$
$\displaystyle=\frac{\Gamma_{\phi\rightarrow
N_{2}N_{2}}}{\mathcal{H}}\frac{E_{\phi}}{y}-\frac{\Gamma_{N_{2}}}{\mathcal{H}y}\left(E_{N_{2}}-E_{N_{2}}^{eq}\right),$
(76) $\displaystyle\frac{d\tilde{N}_{B-L}}{dy}$
$\displaystyle=-\sum_{i=1}^{2}\frac{\Gamma_{N_{i}}}{\mathcal{H}y}\left[\epsilon\left(\tilde{N}_{i}-\tilde{N}^{eq}_{i}\right)+\frac{1}{2}\frac{n_{N_{i}}^{eq}}{n_{R}^{eq}}\tilde{N}_{B-L}\right],$
$\displaystyle\frac{dR}{dy}$ $\displaystyle=\frac{\Gamma_{\phi\rightarrow
R}}{\mathcal{H}}E_{\phi}+\sum_{i=1}^{2}\frac{\Gamma_{N_{i}}}{\mathcal{H}}\left(E_{N_{i}}-E_{N_{i}}^{eq}\right)$

By solving the above equations, one can calculate the efficiency parameter to
be,

$\kappa_{f}=-\frac{4}{3}\epsilon^{-1}R^{-3/4}\tilde{N}_{\text{B-L}}\left[\frac{\pi^{4}g_{\ast}^{3/4}}{30^{3}\zeta(3)^{4}}\right]$
(77)

##  Appendix B Thermalization of RHNs

For extremely weak washout $K\ll 1$, i.e. small light-heavy Yukawa couplings
$\lambda$, RHNs can never attain thermal equilibrium with the radiation bath
via decay and inverse decay. Here we show that in presence of a non-vanishing
coupling $y_{R}$ with radiation, they can be thermalized through the 2-to-2
scattering process $N_{i}N_{i}\leftrightarrow\overline{f}f$ via $\phi$. We are
interested in thermalization of $N_{1}$ during the $\phi-$dominated epoch. The
necessary condition is,

$\Gamma_{\rm SC}\gtrsim\mathcal{H}(T)$ (78)

where $\Gamma_{\rm SC}$ is the scattering width of the 2-to-2 scattering
process and $\mathcal{H}(T)\sim\frac{T^{4}}{M_{\rm pl}T_{\phi}^{2}}$ is the
Hubble parameter at temperature $T$ during the $\phi-$dominated epoch
Mambrini:2021zpp ; Garcia:2020eof . The scattering rate can be approximated
from the scattering cross section $\sigma_{c}$ Ghoshal:2023phi ,

$\Gamma_{\rm SC}=\sigma_{\rm c}\,|v_{\rm rel}|\,n^{\rm eq}$ (79)

where $v_{\rm rel}$ is the relative velocity between the projectile and target
particles and $n^{\rm eq}$ is the equilibrium number density of RHNs. Here we
assumed that the cross section is velocity independent. During $\phi$
dominated era, $T\gg m_{f}$ where $m_{f}$ is the mass of the fermion. Hence
fermions are relativistic with energy $\sim T$. We also assume the RHNs
produced from $\phi$ are relativistic as well. Since $\phi$ is non-thermal,
its tempertature need not be same as the radiation temperature, nevertheless
we approximate it to be $\mathcal{O}(T)$, such that $|v_{\rm rel}|=c=1$ and
$n^{\rm eq}\sim T^{3}$ Dolgov:2009zj . The total cross-section in the center-
of-mass frame of the scalar is given by,

$\sigma_{c}\sim\int\frac{\overline{|\mathcal{M}|^{2}}}{(E_{1}+E_{2})^{2}}d\Omega$
(80)

where $E_{1}\sim E_{2}\sim T$ are the energies of the incident particles and
$\mathcal{M}$ is the Feynmann aplitude with over-line implying average over
spin or polarization states. The integral is performed over solid angle
$\Omega$. We have considered the magnitude of the 3-momenta of the incoming
and outgoing particles to be equal. At tree level we get,

$\overline{|\mathcal{M}|^{2}}\sim\left(y_{R}\,y_{N_{1}}\right)^{2}$ (81)

Then the scattering rate is given by,

$\Gamma_{\rm SC}\sim\left(y_{R}\,y_{N_{1}}\right)^{2}T$ (82)

Therefore Eq. 78 becomes,

$y_{R}\,y_{N_{1}}\gtrsim\sqrt{\frac{T^{3}}{M_{\rm pl}\,T_{\phi}^{2}}}$ (83)

Taking $T=T_{\phi}$ in the above equation, one obtains the approximate
condition for the RHNs to be in thermal equilibrium till $T_{\phi}$. Then at a
lower temperature, the RHNs decouples and freezes out until their decay. For
example, for RHNs to freeze out at around $T_{\phi}\sim 10^{5}$ GeV we need
$y_{R}\,y_{N_{1}}\sim 10^{-7}$.

The two-step entropy injection scenario is possible when Eq. (83) along with
the condition for $\phi$ domination given in (29) are satisfied
simultaneously. Solving both of them simultaneously, we obtain the condition
on scalar mass for the two-step entropy injection scenario,

$M_{\phi}\gtrsim 2\sqrt{T_{\phi}\,M_{\rm pl}}+M_{\rm
pl}\left(y_{N_{1}}-y_{R}\right)^{2}\gtrsim 2\sqrt{T_{\phi}\,M_{\rm pl}}$ (84)

where the last inequality is the minimum requirement for $M_{\phi}$ for which
thermal scenario is possible when $y_{N_{1}}\sim y_{R}$.

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