# Tomography of flavoured leptogenesis with primordial blue gravitational
waves

Marco Chianese<EMAIL_ADDRESS>Dipartimento di Fisica ”Ettore
Pancini”, Università degli studi di Napoli “Federico II”, Complesso
Universitario Monte S. Angelo, I-80126 Napoli, Italy  INFN - Sezione di
Napoli, Complesso Universitario Monte S. Angelo, I-80126 Napoli, Italy
Satyabrata Datta<EMAIL_ADDRESS>Department of Physics and
Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 210023,
China  Rome Samanta<EMAIL_ADDRESS>Scuola Superiore Meridionale,
Università degli studi di Napoli “Federico II”, Largo San Marcellino 10, 80138
Napoli, Italy  INFN - Sezione di Napoli, Complesso Universitario Monte S.
Angelo, I-80126 Napoli, Italy  Ninetta Saviano<EMAIL_ADDRESS>INFN -
Sezione di Napoli, Complesso Universitario Monte S. Angelo, I-80126 Napoli,
Italy  Scuola Superiore Meridionale, Università degli studi di Napoli
“Federico II”, Largo San Marcellino 10, 80138 Napoli, Italy

###### Abstract

We explore a scenario where an early epoch of matter domination is driven by
the mass scale $M_{N}$ of the right-handed neutrinos, which also characterizes
the different flavour regimes of leptogenesis. Such a matter-domination epoch
gives rise to peculiar spectral imprints on primordial Gravitational Waves
(GWs) produced during inflation. We point out that the characteristic spectral
features are detectable in multiple frequency bands with current and future GW
experiments in case of Blue GWs (BGWs) described by a power-law with a
positive spectral index $(n_{T}>0)$ and an amplitude compatible with Cosmic
Microwave Background (CMB) measurements. We find that the three-flavour
leptogenesis regime with $M_{N}\lesssim 10^{9}~{}{\rm GeV}$ imprints BGWs more
prominently than the two-flavour and one-flavour regimes characterized by a
higher right-neutrino mass scale. In particular, a two-flavour (three-flavour)
leptogenesis regime is expected to leave distinct imprints in the mHz
($\mu$Hz) band. Moreover, we translate the current Big Bang Nucleosynthesis
(BBN) and LIGO limits on the GW energy density into constraints on the flavour
leptogenesis parameter space for different GW spectral index $n_{T}$.
Interestingly, a three-flavour leptogenesis regime can offer a unique signal
testable in the next LIGO run with a correlated signature in the PTA frequency
band with an amplitude comparable to the one expected from supermassive black
holes.

##  I introduction

Primordial Gravitational Waves (GWs) can be a unique probe of physics
operating at super high-energy scales that are otherwise unreachable, e.g.,
with terrestrial accelerators. Broadly, a high-energy theory/model can be
sensitive to GWs in two ways. First, the model can accommodate a source of GWs
whose amplitude and spectral features relate to the model parameters. Thus,
the model becomes testable with the GW searches. On the other hand, the model
can leave its imprints on GWs, regardless of how they are produced. This
article discusses the latter case. Let us assume a spectrum of primordial GWs,
which might be of inflationary origin [1, 2], and we aim at probing a post-
inflation high-energy model with those GWs . In a given scenario where the
underlying dynamics of GWs production are known, we expect specific spectral
features in the GWs today. Therefore, testing an independent model (which is
not related to the origin of GWs) would require finding detectable imprints in
those GWs caused exclusively by the model under consideration. This method
proposed as a tomographic search of BSM theories with GWs [3, 4] is similar to
X-ray tomography of an object, wherein the object is placed in front of an
X-ray source, and we let the X-ray pass through the object to know the
invisible internal properties of the object. Likewise, the GW tomography has
two main ingredients: the source, specifically the GW spectrum, and the
object, namely the high-energy model to be tested.

In this article, for GW spectrum, we consider primordial Blue Gravitational
Waves (BGWs) with $\Omega_{\rm GW}\sim f^{n_{T}}$ (here $\Omega_{\rm GW}$ is
the GWs energy density, $f$ is the frequency and $n_{T}>0$ is a spectral
index)111As we discuss later, in a realistic setting it should be a broken
power-law; otherwise the spectrum would not be cosmologically viable. which
might come, e.g., from inflation. For the high-energy model we want to probe,
we consider thermal leptogenesis, which is a process that generates the baryon
asymmetry of the universe [5]. BGWs are appealing by several considerations.
First, for $n_{T}>0$, the BGWs come with strong amplitude, and therefore, they
are easily detectable, given the projected sensitivities of the current and
planned GW detectors operating at different frequency bands. Second, the
recent finding of nHz stochastic GWs by the Pulsar Timing Arrays (PTAs) shows
a strong preference for a blue-tilted GW spectrum of primordial origin [6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16]. Specifically, a power-law fit to the 15 yrs
NANOGrav PTA data renders a large spectral index value; $n_{T}\sim 1.2-2.4$ at
$68\%$ CL [10]. Third, because of their potentially strong amplitude in the
sub-nHz and $\mu$Hz regions, primordial BGWs are among the very few candidates
that can be tested with Pulsar parameter drifts [17, 18] and possibly with
Lunar Laser Ranging (LLR) [19, 20]. Finally, as recently pointed out, BGWs
might affect large-scale structures by sourcing density perturbations at
second order [21]. Although we shall not provide any model that might produce
such BGWs, let us mention that the simplest single-field slow-roll inflation
models produce a nearly scale-invariant spectrum with very small amplitude.
Nonetheless, plenty of scenarios [22, 23, 24, 25, 26, 27, 28, 29, 30, 31] go
beyond the simplest one and predict BGWs that are detectable.

Leptogenesis [5] is a two-step process: a lepton asymmetry is created in the
first step, which is then processed to a baryon asymmetry via the sphaleron
transition [32]. Among several possibilities to generate lepton asymmetry [33,
34, 35], here we consider the one within the seesaw framework of light
neutrino masses. In this scenario, CP-violating and out-of-equilibrium decays
of right-handed (RH) neutrinos to lepton and Higgs doublet produce lepton
asymmetry. Typically, at a temperature $T\sim M_{N}$, where $M_{N}$ is the RH
neutrino mass scale, lepton asymmetry gets produced. Depending on the several
variants, a wide range of RH neutrino mass window, $M_{N}\in[\rm
MeV,10^{15}\rm GeV]$ [36, 37, 38, 39, 40, 41, 42, 43, 44, 45], might
facilitate a successful leptogenesis. As it is obvious,
$M_{N}\gtrsim\mathcal{O}(\rm TeV)$ scales are not reachable with collider
experiments, and so leptogenesis operating on those scales requires
alternative probes. A way to test such a high-scale scenario is provided by
the flavour effects [46, 47, 48] which connect it indirectly to low-energy
neutrino observables, specifically to the leptonic CP violation [49]. A
flavoured leptogenesis scenario can have three distinct regimes (as described
in the next section): one-flavour/vanilla regime (1FL) for $M_{N}\gtrsim
10^{12}$ GeV, two-flavour regime (2FL) for $10^{12}$ GeV $\gtrsim M_{N}\gtrsim
10^{9}$ GeV, and three-flavour regime (3FL) for $M_{N}\lesssim 10^{9}$ GeV.
Though connecting those regimes with low-energy neutrino observables is
possible, it is difficult to differentiate them observationally because of
plenty of free parameters in seesaw models. One way to distinguish them is to
impose further symmetries in the Lagrangian, e.g., discrete symmetries, to
reduce the number of parameters [50, 51]. Here, we shall show that,
alternatively, with the GW-tomography method, not only can such high scales be
probed, but different flavour regimes can potentially leave distinct imprints
on the BGW spectrum.

As outlined, given an independent source of GWs, testing a model requires
finding imprints on the GW spectrum caused by the model. Therefore, probing
different mass scales of leptogenesis (hence different flavour regimes)
requires finding characteristic GW spectral features dependent on RH masses.
We show that, though nontrivial, this can be done within some parameter space
in the seesaw models, which offers an RH neutrino mass-dependent matter epoch
affecting the standard propagation (in radiation domination) of primordial
BGWs. The idea is based on contemplating the origin of RH neutrino masses. Let
us suppose that, like any other Standard Model (SM) particles, the RH
neutrinos get their mass via a phase transition triggered by an SM singlet
scalar field. Once the field rolls down to its vacuum and attains its vacuum
expectation value, besides generating RH neutrino masses, it oscillates
coherently around the vacuum. Such oscillations persist long with a null
equation of state parameter (the field behaves like matter) when the scalar
field lives longer. A probable timeline in the scenario has been illustrated
in Fig. 1. Suppose the coupling of the scalar field to the RH neutrino plays a
pivotal role in determining its lifetime. In that case, the duration of the
matter domination can be controlled by the coupling and, therefore, by the RH
neutrino masses. This is the key aspect of our study. Irrespective of their
origin, as primordial gravitational waves pass through such a matter-dominated
phase, the beginning and the end of the matter-dominated phase get imprinted
on the final GW spectrum. Therefore, the final GW spectrum carries imprints of
the RH neutrino mass scale.

Within a concrete model of $U(1)_{B-L}$ phase transition222 In the seesaw
Lagrangian, $U(1)_{B-L}$ may naturally arise as a residual symmetry of many
Grand Unification Theories (GUT) [52, 53, 54, 55, 56]. triggered by a scalar
field $\Phi$ carrying $B-L$ charge, the key feature of our study is that a
peaked BGW spectrum gets distorted when it passes through such an RH neutrino
mass-dependent matter epoch and finally exhibits a double-peaked spectrum when
the scalar field lives longer. The locations of the low-frequency peak
($f_{\rm low}^{\rm peak}$) and the dip ($f^{\rm dip}$) in between are strongly
sensitive to RH neutrino masses. As the leptogenesis process enters from an
unflavoured to flavour regimes, these two characteristic frequencies shift
from a high to a low-frequency value, and the ratio $\Omega_{\rm GW}(f_{\rm
low}^{\rm peak})/\Omega_{\rm GW}(f_{\rm dip})$ increases. While the vanilla
leptogenesis scenario hardly imprints the GW spectrum, the two-flavour and the
three-flavour regimes exhibit characteristic spectral features in the GWs in
the mHz and $\mu$Hz bands. The scenario can accommodate a reasonably large
value of $n_{T}$ without contradicting any cosmological constraints and allows
the overall spectrum to span a wide range of frequencies accessible to the
current and planned GW detectors. Notably, in the PTA frequency band,
flavoured leptogenesis regimes allow GW amplitude comparable to supermassive
black hole binaries (SMBHB).

The rest of the paper is organized as follows. In Sec. II, we discuss the
scalar field dynamics and all the constraints the model must comply with. In
Sec. III, we present a detailed numerical study on how the BGW spectrum
contains information on the RH neutrino mass scale. We also discuss our model
in the context of the recent PTA results on nHz GWs. Finally, we draw our
conclusions in Sec. IV.

Figure 1: A schematic timeline showing the occurrence of the events considered
in this work. After inflation, the universe reheats at $T=T_{\rm R}$. Phase
transition happens, the scalar field oscillation begins, and RH neutrinos
become massive at $T=T_{c}$. At the temperature $T_{\rm lepto}\simeq M_{N}$,
the baryon asymmetry produced by the RH neutrinos freezes out. Starting from
$T_{\rm dom}$ until $T_{\rm dec}$, the scalar field dominates the universe’s
energy density and produces entropy around $T\simeq T_{\rm dec}$. The universe
becomes radiation-dominated after this. The big-bang nucleosynthesis and the
current temperature are denoted by $T_{\rm BBN}$ and $T_{0}$, respectively. A
peaked blue-tilted GW-spectrum generated during inflation when comes across
such a dynamical phase of $\Phi$, which also makes the RH neutrino massive and
whose decay width is determined by the RH neutrino mass scale, becomes a
double peak spectrum at present. The final spectrum carries imprints of the RH
neutrino mass scale and, therefore, different regimes of leptogenesis.

##  II scalar field dynamics and the scale of thermal Leptogenesis

The Lagrangian relevant to the discussion is given by

$\displaystyle-\Delta\mathcal{L}\subset
f_{Di\alpha}\overline{N_{Ri}}\tilde{H}^{\dagger}L_{\alpha}+\frac{1}{2}f_{Ni}\overline{N_{Ri}}\Phi
N_{Ri}^{C}+\frac{1}{4}\lambda_{H\Phi}|H|^{2}|\Phi|^{2}+V(\Phi,T)\,,$ (II.1)

where $f_{D}$ is the neutrino Dirac Yukawa coupling, $H(L)$ is the SM Higgs
(lepton doublet), $N_{R}$ is the RH neutrino field, $\Phi$ is a complex scalar
field with vacuum expectation value $v_{\Phi}$, and $V(\Phi,T)$ is the finite
temperature potential that determines the phase transition dynamics of $\Phi$.
Here, $N_{R}$ and $L$ have $B-L$ charge $-1$, whereas the same charges for
$\Phi$ and $H$ are $2$ and $0$, respectively. The Higgs portal coupling is in
general given by $\lambda_{H\Phi}\simeq\lambda_{H\Phi}^{\rm
tree}+\lambda_{H\Phi}^{\rm 1-loop}$ where $\lambda_{H\Phi}^{\rm 1-loop}$ is
generated by the first two terms in Eq. (II.1). We assume
$\lambda_{H\Phi}^{\rm tree}=0$ which is crucial to establish a connection
between the neutrino parameters and the lifetime of $\Phi$. The RH neutrinos
become massive ($M_{i}=f_{Ni}v_{\Phi}$) when $\Phi$ obtains its vacuum
expectation value $v_{\Phi}$. After the electroweak phase transition
($T\lesssim T_{\rm EW}$), when the Higgs settles to its minima $H=v_{h}$, the
first two terms generate light active neutrino mass $m_{\nu
i}\sim{f_{Di}}^{2}v_{h}^{2}/M_{i}$ via the standard type-I seesaw mechanism
[57, 58, 59, 60], whereas at $T\gg T_{\rm EW}$ the heavy neutrinos decay CP
asymmetrically to lepton doublets plus Higgs, and generate lepton asymmetry.
At a temperature $T\sim M_{i}$, lepton asymmetry produced by $i$th RH neutrino
freezes out; therefore, $M_{i}$ is typically the scale of thermal
leptogenesis. For the sake of simplicity, not focusing so much on the
hierarchy in $M_{i}$s, we consider a single scale for the rest of the
discussion, i.e. $M_{i}\equiv M_{N}=f_{N}v_{\Phi}$. In the high-scale
leptogenesis scenario, there are three distinct regimes of leptogenesis [46,
47, 48]. Generally, the RH neutrinos produce lepton doublets as a coherent
superposition of flavours: $\ket{L_{i}}=A_{i\alpha}\ket{L_{\alpha}}$, where
$\alpha=e,\mu,\tau$ and $A_{i\alpha}$ is the corresponding amplitude
determined by $f_{Di\alpha}$. For temperatures $T\gtrsim 10^{12}~{}{\rm GeV}$,
all the charged lepton Yukawa couplings are out of equilibrium (the
interaction strength is given by $\Gamma_{\alpha}\sim 5\times
10^{-3}~{}f_{L\alpha}T$, with $f_{L\alpha}$ being the charged lepton Yukawa
coupling) and do not participate in the leptogenesis process. This is the
unflavour or the vanilla regime of leptogenesis. In this regime, leptogenesis
is not sensitive to the neutrino mixing matrix, e.g. mixing angles and low-
energy CP phases. When the temperature drops down to $T\lesssim 10^{12}~{}{\rm
GeV}$, the $\tau$ flavoured charged lepton interaction comes into equilibrium
and breaks the coherent $\ket{L_{i}}$ state into $\tau$ and a composition of
$e+\mu$ flavour (two flavour regime). Likewise, the later composition is
broken once $T\lesssim 10^{9}~{}{\rm GeV}$, when the $\mu$ flavoured charged
lepton interaction comes into equilibrium (three flavour regime). In both
these flavoured regimes, leptogenesis may be sensitive to the low-energy
neutrino parameters. We will explore all these regimes and show that the
flavoured regime of leptogenesis leaves much more distinct imprints on BGWs
than the vanilla one, paving the path for a possible synergy between low-
energy neutrino physics and GWs.

As the temperature drops, the scalar field transits from $\Phi=0$ towards its
vacuum expectation value $\Phi=v_{\Phi}$. The finite temperature potential
that restores the symmetry at higher temperatures is given by [61, 62, 63, 64,
65]

$\displaystyle
V(\Phi,T)=D(T^{2}-T_{0}^{2})\Phi^{2}-ET\Phi^{3}+\frac{\lambda}{4}\Phi^{4}\,,$
(II.2)

with

$\displaystyle
D=\frac{3{g^{\prime}}^{2}+4\lambda}{24}\,,~{}~{}E=\frac{3{g^{\prime}}^{3}+{g^{\prime}}\lambda+3\lambda^{3/2}}{24\pi}\,,~{}~{}T_{0}=\frac{\sqrt{12\lambda}v_{\Phi}}{\sqrt{3{g^{\prime}}^{2}+4\lambda}}\,,$
(II.3)

where $g^{\prime}$ is the gauge coupling,333As mentioned in the introduction,
seesaw models with $U(1)_{B-L}$ gauge symmetry is well-motivated from GUT. In
that case, the scalar field carries a gauge charge. Therefore, the gauge
coupling appears naturally in the finite temperature potential. and the vacuum
expectation value $v_{\Phi}=\mu/\sqrt{\lambda}$ has been determined from the
zero temperature potential
$V(\Phi,0)=-\frac{\mu^{2}}{2}\Phi^{2}+\frac{\lambda}{4}\Phi^{4}$. The
structure of the finite temperature potential determines how the transition
proceeds. The last term in Eq. (II.2) generates a potential barrier causing a
secondary minimum at $\Phi\neq 0$, which at $T=T_{c}$ becomes degenerate with
the $\Phi=0$ one. At $T_{0}~{}(\lesssim T_{c})$, the potential barrier
vanishes, making the minimum at $\Phi=0$ a maximum [63]. The critical
temperature $T_{c}$ and the field value $\Phi_{c}=\Phi\left(T_{c}\right)$ are
given by [63, 66]

$\displaystyle T_{c}=T_{0}\frac{\sqrt{\lambda D}}{\sqrt{\lambda
D-E^{2}}}\,,~{}~{}\Phi_{c}=\sqrt{\frac{4D}{\lambda}(T_{c}^{2}-T_{0}^{2})}\,.$
(II.4)

A non-zero value of $E$ in Eq. (II.2) generally leads to a first-order
transition with a strength determined roughly by the order parameter
$\Phi_{c}/T_{c}$ [63]. Nonetheless, if $\Phi_{c}/T_{c}\ll 1$, the transition
is weakly first-order, which can be treated as a second-order transition
because the potential barrier disappears quickly. In this case, the transition
can be described by rolling of the field $\Phi$ from $\Phi=0$ to
$\Phi=v_{\Phi}$. In this article, we consider such a case working with the
values of $\lambda$ and ${g^{\prime}}$ so that $\Phi_{c}/T_{c}\ll 1$ is
fulfilled. In particular, we find that, for
${g^{\prime}}^{3}\lesssim\lambda\lesssim 1$ and ${g^{\prime}}\lesssim
10^{-2}$, the order parameter is $\Phi_{c}/T_{c}\lesssim 0.08$.444In Ref.[3],
it was checked numerically that the transition is a second-order for this
choice of parameters. Fixing the hierarchy between the couplings to be
$\lambda\simeq{g^{\prime}}^{3}$ (limiting condition), we also have
$T_{c}\simeq T_{0}=2\sqrt{g^{\prime}}v_{\Phi}$.

Once the field rolls down to the true vacuum, it oscillates around $v_{\Phi}$.
For a generic potential $V(\Phi)=\alpha\Phi^{\beta}$, the equation of state of
such a coherent oscillation can be computed as [3]

$\displaystyle\omega=(\beta-2)(\beta+2)^{-1}\,.$ (II.5)

Assuming the oscillation of the scalar field is driven by the dominant
quadratic term in the potential and expanding the zero temperature potential
around the true vacuum, we obtain $\alpha=\lambda v_{\Phi}^{2}$ and $\beta=2$.
Therefore, the scalar field behaves like matter ($\omega=0$). One can also
compute the angular frequency of oscillation, which is
$m_{\Phi}=\sqrt{2\lambda}v_{\Phi}$. If $\Phi$ is long-lived, these
oscillations persist and the universe goes through a significant period of
matter dominated epoch, leading to distinct imprints on the BGWs as we discuss
later.

Let us now constrain the RH neutrino mass scale (scale of leptogenesis)
$M_{N}=f_{N}v_{\Phi}$. First, the RH neutrino Yukawa coupling $f_{N}$ should
be large because we consider high-scale leptogenesis. However, in that case,
if $m_{\Phi}>2M_{N}$, $\Phi$ decay to RH neutrino pairs ($\Phi\rightarrow NN$)
would be too quick to provide matter domination. Therefore, we have to
restrict ourselves to the case $M_{N}>m_{\Phi}$. Second, RH neutrinos become
massive after the phase transition at $T=T_{c}$. Therefore, the scale of
leptogenesis $T_{\rm lepto}\sim M_{N}$ is bounded from above as $T_{\rm
lepto}\sim M_{N}\lesssim T_{c}$. For our choice
$\lambda\simeq{g^{\prime}}^{3}$, the constraint $m_{\Phi}<M_{N}\lesssim T_{c}$
from requiring long-lived $\Phi$ and successful leptogenesis thus becomes

$\displaystyle\sqrt{2g^{\prime}}g^{\prime}v_{\Phi}<M_{N}\lesssim
2\sqrt{g^{\prime}}v_{\Phi}\,.$ (II.6)

Notice that in this specific case the mass of $\Phi$ is suppressed by a factor
$\sqrt{g^{\prime}}$ compared to the
$M_{Z^{\prime}}=\sqrt{2}g^{\prime}v_{\Phi}$, thus kinematically forbidding the
$\Phi$ decays into $Z^{\prime}$ pairs. Two competitive decay channels of
$\Phi$ are $\Phi\rightarrow hh$ and $\Phi\rightarrow f\bar{f}V$, where $h$,
$f$, and $V$ are the SM Higgs, leptons and vector bosons, respectively. Both
the decays are radiative and therefore suppressed, making $\Phi$ long-lived.
The first decay channel, $\Phi\rightarrow hh$, is mediated by the neutrino
Dirac Yukawa $f_{D}$ and the coupling $f_{N}$ (see Fig. 2). Hence, it is
directly connected to light neutrino and leptogenesis parameter space. In the
limit $m_{\Phi}\gg m_{h}$, the decay rate is given by

$\displaystyle\Gamma_{\Phi}^{hh}\simeq\frac{({\lambda_{H\Phi}^{\rm
1-loop}})^{2}\,v_{\Phi}^{2}}{32\pi m_{\Phi}}=1.3\times
10^{-2}~{}\frac{f_{N}^{6}}{\lambda}\left(\frac{v_{\Phi}}{\rm
10^{13}~{}GeV}\right)^{2}\left(\frac{m_{\Phi}}{\rm 10^{8}~{}GeV}\right)~{}{\rm
GeV}\,,$ (II.7)

where the one-loop effective coupling $\lambda_{H\Phi}^{\rm 1-loop}\sim
f_{D}^{2}f_{N}^{2}/(2\pi^{2})$ (up to some logarithmic factor depending on the
renormalization scale [67, 68], which we ignore here) with
$f_{D}=\sqrt{m_{\nu}f_{N}v_{\Phi}/v_{h}^{2}}$ has been evaluated for
$m_{\nu}\simeq 0.01$ eV and $v_{h}=174$ GeV. On the other hand, the
$\Phi\rightarrow f\bar{f}V$ decay rate is given by [69]

$\displaystyle\Gamma_{\Phi}^{f\bar{f}V}\simeq\lambda{g^{\prime}}^{4}\left(\frac{m_{\Phi}}{\rm
10^{8}~{}GeV}\right)\,.$ (II.8)

Figure 2: Diagram generating leading-order radiative $\Phi HH$ coupling
through RH neutrino and Lepton mediation [69, 67, 68, 70].

Therefore, for $\Phi\rightarrow hh$ to dominate over $\Phi\rightarrow
f\bar{f}V$ (we need the matter dominated epoch driven by $\Phi$ to be
determined by the neutrino parameters), the following condition should be
satisfied

$\displaystyle f_{N}^{6}\left(\frac{v_{\Phi}}{\rm
10^{13}~{}GeV}\right)^{2}\gtrsim 7.8\times
10^{1}\lambda^{2}{g^{\prime}}^{4}\,.$ (II.9)

The general model has four independent parameters which are the $\Phi$ vacuum
expectation value $v_{\Phi}$ and the three couplings $f_{N}$, $\lambda$ and
$g^{\prime}$ (or equivalently the three masses $M_{N}$, $m_{\Phi}$ and
$M_{Z^{\prime}}$, respectively). However, for the sake of definiteness, we
hereafter assume $\lambda\simeq{g^{\prime}}^{3}$ (limiting condition for the
second-order phase transition) and $f_{N}\simeq{g^{\prime}}\ll 1$, which
implies the RH neutrino mass scale to be of the order of $Z^{\prime}$ boson
mass (i.e., $M_{N}\simeq M_{Z^{\prime}}\ll v_{\Phi}$). The last assumption
does not contradict Eq. (II.6) and confirms that any
$\mathcal{O}({g^{\prime}}^{4})$ radiative correction to the effective
potential is subdominant. With this consideration, Eq.s (II.7) and (II.9)
become

$\displaystyle\Gamma_{\Phi}^{hh}$ $\displaystyle\simeq$ $\displaystyle
1.3\times
10^{-11}~{}\left(\frac{g^{\prime}}{10^{-3}}\right)^{3}\left(\frac{v_{\Phi}}{\rm
10^{13}~{}GeV}\right)^{2}\left(\frac{m_{\Phi}}{\rm 10^{8}~{}GeV}\right)~{}{\rm
GeV\,,}$ (II.10) $\displaystyle{g^{\prime}}$ $\displaystyle\lesssim$
$\displaystyle 3.3\times 10^{-1}\left(\frac{v_{\Phi}}{\rm
10^{13}~{}GeV}\right)^{1/2}\,.$ (II.11)

The scalar field with initial vacuum energy
$\rho_{\Phi}\left(T_{c}\right)\simeq\lambda v_{\Phi}^{4}/4$ might dominate the
energy density for a period, and then inject entropy while decaying. We
account for this effect by solving the following Friedmann equations for the
energy densities of the scalar field ($\rho_{\Phi})$ and radiation
($\rho_{R}$):

$\displaystyle\frac{{\rm d}\rho_{R}}{{\rm
d}t}+4\mathcal{H}\rho_{R}=\Gamma_{\Phi}^{hh}\rho_{\Phi}\,,~{}~{}\frac{{\rm
d}\rho_{\Phi}}{{\rm
d}t}+3\mathcal{H}\rho_{\Phi}=-\Gamma_{\Phi}^{hh}\rho_{\Phi}\,,~{}~{}\frac{{\rm
d}s}{{\rm d}t}+3\mathcal{H}s=\Gamma_{\Phi}^{hh}\frac{\rho_{\Phi}}{T}\,,$
(II.12)

upon recasting them as

$\displaystyle\frac{{\rm d}\rho_{R}}{{\rm
d}z}+\frac{4}{z}\rho_{R}=0\,,~{}~{}\frac{{\rm d}\rho_{\Phi}}{{\rm
d}z}+\frac{3}{z}\frac{\mathcal{H}}{\tilde{\mathcal{H}}}\rho_{\Phi}+\Gamma_{\Phi}^{hh}\frac{1}{z\tilde{\mathcal{H}}}\rho_{\Phi}=0\,,$
(II.13)

where $z=T_{c}/T$, $\mathcal{H}$ is the Hubble parameter, $s$ is the entropy
density of the thermal bath, and from the third of Eq. (II.12), the
temperature-time relation has been derived as

$\displaystyle\frac{1}{T}\frac{{\rm d}T}{{\rm
d}t}=-\left(\mathcal{H}+\frac{1}{3g_{*s}(T)}\frac{{\rm d}g_{*s}(T)}{{\rm
d}t}-\Gamma_{\Phi}^{hh}\frac{\rho_{\Phi}}{4\rho_{R}}\right)=-\tilde{\mathcal{H}}\,.$
(II.14)

The amount of entropy production from the $\Phi$ decays can be computed by
solving

$\displaystyle\frac{{\rm d}a}{{\rm
d}z}=\left(1+\Gamma_{\Phi}^{hh}\frac{\rho_{\Phi}}{4\rho_{R}\tilde{\mathcal{H}}}\right)\frac{a}{z}\,,$
(II.15)

with $a$ being the scale factor, and then computing the ratio of
$\tilde{S}\sim a^{3}/z^{3}$, after and before the scalar field decays. The
amount of entropy production $\kappa=\tilde{S}_{\rm after}/\tilde{S}_{\rm
before}$ can be quantified as [3]

$\displaystyle\mathbb{\kappa}^{-1}\simeq\frac{\rho_{R}\left(T_{c}\right)}{\rho_{\Phi}\left(T_{c}\right)}\frac{T_{\rm
dec}}{T_{c}}\simeq\frac{3^{1/4}\left(\frac{30}{\pi^{2}g_{*}}\right)^{-3/4}T_{c}^{3}\sqrt{\Gamma_{\Phi}^{hh}\tilde{M}_{\rm
Pl}}}{V_{\rm eff}\left(0,T_{c}\right)}\,,$ (II.16)

where the decay temperature $T_{\rm dec}$ is computed by considering
$\Gamma_{\Phi}^{hh}\simeq\mathcal{H}$ and reads as

$\displaystyle T_{\rm
dec}\simeq\left(\frac{90}{\pi^{2}g_{*}}\right)^{1/4}\sqrt{\Gamma_{\Phi}^{hh}\tilde{M}_{\rm
Pl}}\,,$ (II.17)

with $\tilde{M}_{\rm Pl}=2.4\times 10^{18}~{}{\rm GeV}$ being the reduced
Planck constant, and $g_{*}\simeq 106.75$ is the effective degrees of freedom
that contribute to the radiation. Importantly, the scalar decays must occur
before BBN ($T_{\rm dec}\gtrsim T_{\rm BBN}\sim 10~{}{\rm MeV}$), implying
that the following condition must be satisfied

$\displaystyle g^{\prime}\gtrsim 5.6\times 10^{-5}\left(\frac{10^{11}{\rm
GeV}}{v_{\Phi}}\right)^{2/3}\,.$ (II.18)

Figure 3: Left: The entropy produced by the $\Phi$ decays on the $g^{\prime}-v_{\Phi}$ plane. In this plot, the right-handed neutrino mass scale reads as $M_{N}\sim g^{\prime}v_{\Phi}$. The white region in the top-left corner is excluded because $\Gamma_{\Phi}^{f\bar{f}V}>\Gamma_{\Phi}^{hh}$. The grey-shaded region in the bottom-left corner is excluded by BBN considerations (otherwise the BBN would occur in matter domination). Right: the evolution of the normalized energy density $\Omega_{i}=\rho_{i}/\rho_{\rm tot}$ for radiation (solid lines) and $\Phi$ scalar (dotted lines), and the total entropy (dashed lines) as a function of the inverse temperature. The colour codes for the benchmarks are the same as in the left panel plot. Because the ratio of the entropy after and before the decay is important, the initial value of $\tilde{S}$ has been chosen as $\tilde{S}(z=1)=1$. The parameter $\kappa$ in Eq. (II.16) is therefore $\kappa=\tilde{S}_{\rm after}=\tilde{S}\left(z\rightarrow\infty\right)$. Benchmark cases | $~{}~{}v_{\Phi}~{}[{\rm GeV}]~{}~{}$ | $g^{\prime}$ | $~{}~{}T_{\rm dom}~{}[{\rm GeV}]~{}~{}$ | $~{}~{}T_{\rm dec}~{}[{\rm GeV}]~{}~{}$
---|---|---|---|---
BP1 (1FL) | $10^{14.5}$ | $~{}~{}10^{-2}~{}~{}$ | $\sim 2.8\times 10^{8}$ | $\sim 2.1\times 10^{8}$
BP2 (2FL) | $10^{13.5}$ | $~{}~{}10^{-3}~{}~{}$ | $\sim 8.9\times 10^{5}$ | $\sim 3.7\times 10^{4}$
BP3 (3FL) | $10^{12.5}$ | $~{}~{}10^{-4}~{}~{}$ | $\sim 2.8\times 10^{3}$ | $\sim 6.6$
Table 1: The three benchmark points considered in the present paper. They
corresponds to different leptogenesis mass scale $M_{N}=10^{12.5}~{}{\rm
GeV}$, $M_{N}=10^{10.5}~{}{\rm GeV}$ and $M_{N}=10^{8.5}~{}{\rm GeV}$, and
therefore to different leptogenesis regimes, i.e., one-flavoured (or vanilla),
two-flavoured, and three-flavoured leptogenesis, respectively. In each flavour
regime, leptogenesis occurs in the standard radiation domination because
$T_{\rm lepto}\sim M_{N}>T_{\rm dom}$, where $T_{\rm dom}$ is the temperature
at which the scalar field starts to dominate.

In Fig. 3 we show the produced entropy on the allowed model parameter space
(left plot) and the evolution of the normalised energy densities
$\Omega_{i}=\rho_{i}/\rho_{\rm tot}$ and the entropy (right plot) for the
three sets of benchmark parameters (see Tab. 1): BP1
($g^{\prime}=10^{-2},~{}v_{\Phi}=10^{14.5}$ GeV), BP2
($g^{\prime}=10^{-3},~{}v_{\Phi}=10^{13.5}$ GeV), BP3
($g^{\prime}=10^{-4},~{}v_{\Phi}=10^{12.5}$ GeV). The benchmark points are
chosen such that each falls into a specific regime of leptogenesis: BP (1, 2,
3) are in one-flavoured (or vanilla), two-flavoured, and three-flavoured
regime, respectively. In the left plot, the white region in the top-left
corner is excluded because $g^{\prime}$ is bounded from above due to the
requirement of $\Gamma_{\Phi}^{hh}>\Gamma_{\Phi}^{f\bar{f}V}$ according to Eq.
(II.11), while BBN excludes the grey area according to Eq. (II.18). We find
that despite large initial energy, the entropy production is less for large
$v_{\Phi}$ because of the shorter lifetime of $\Phi$. We have also checked
that the analytical expression for the entropy production factor $\kappa$ in
Eq. (II.16) matches the numerical result (Fig. 3, right) with good accuracy.

Let us conclude this section with the following remarks: First, as it is
obvious from Fig. 3 (right), the scalar field energy density
$\rho_{\Phi}(T_{c})$ is always sub-dominant than the radiation energy density
at $T=T_{c}$. Therefore, the universe never encounters an exponential
expansion, i.e., a second period of inflation. Second, this article does not
present any detailed computation related to the baryon asymmetry (we present
an order of magnitude estimation in the next section). We, however, note that
the baryon asymmetry produced by the RH neutrinos will also be diluted due to
entropy production given in Eq. (II.16). Therefore, unlike the conventional
choice of a strong hierarchy in the RH neutrino masses, a more suitable option
would be to make RH neutrinos quasi-degenerate. This not only evades the issue
of entropy dilution by enhancing baryon asymmetry via the resonant-
leptogenesis mechanism [37], but also justifies our choice of a single RH
neutrino mass scale $M_{i}\equiv M_{N}$ throughout.

##  III Imprints of Flavour regimes of leptogenesis on BGWs

In what follows, we briefly discuss the GW production during inflation and
their propagation through multiple cosmological epochs until today. The
perturbed FLRW line element that describes GWs is given by

$\displaystyle{\rm d}s^{2}=a(\tau)\left[-{\rm
d}\tau^{2}+(\delta_{ij}+h_{ij}){\rm d}x^{\rm i}dx^{j})\right]\,,$ (III.1)

where $\tau$ is the conformal time, and $a(\tau)$ is the scale factor. The
transverse and traceless ($\partial_{i}h^{ij}=0$, $\delta^{ij}h_{ij}=0$) part
of $h_{ij}$ represents the GWs. Because the GWs are weak, $|h_{ij}|\ll 1$, the
following linearized evolution equation

$\displaystyle\partial_{\mu}(\sqrt{-g}\partial^{\mu}h_{ij})=16\pi
a^{2}(\tau)\mathcal{\pi}_{ij}$ (III.2)

is sufficient to study the propagation of the GWs. The quantity $\pi_{ij}$
represents an anisotropy stress tensor that couples to $h_{ij}$ as an external
source, which within a realistic cosmic setting only affects the GWs at large
scales (than those of PTAs), e.g., due to the free streaming of light
neutrinos [71, 72]. The Fourier space decomposition of $h_{ij}$ reads

$\displaystyle h_{ij}(\tau,\vec{x})=\sum_{\lambda}\int\frac{{\rm
d}^{3}\vec{k}}{(2\pi)^{3/2}}e^{i\vec{k}.\vec{x}}\epsilon_{ij}^{\lambda}(\vec{k})h_{\vec{k}}^{\lambda}(\tau)\,,$
(III.3)

where the index $\lambda=``+/-"$ represents two polarisation states. The
polarisation tensor $\epsilon_{ij}^{\lambda}(\vec{k})$ besides being
transverse and traceless, complies with the following conditions:
$\epsilon^{(\lambda)ij}(\vec{k})\epsilon_{ij}^{(\lambda^{\prime})}(\vec{k})=2\delta_{\lambda\lambda^{\prime}}$
and $\epsilon^{(\lambda)}_{ij}(-\vec{k})=\epsilon^{(\lambda)}_{ij}(\vec{k})$.
Assuming isotropy and similar evolution of each polarisation state, we replace
$h_{\vec{k}}^{\lambda}(\tau)$ with $h_{k}(\tau)$, where $k=|\vec{k}|=2\pi f$
with $f$ being the present frequency at $a_{0}=1$. Neglecting subdominant
contribution from $\pi_{ij}$, the GW propagation equation in the Fourier space
reads

$\displaystyle\ddot{h}_{k}+2\frac{\dot{a}}{a}\dot{h}_{k}+k^{2}h_{k}=0\,,$
(III.4)

where the dot represents a conformal time derivative. Using Eq. (III.3) and
Eq. (III.4), the energy density of the GWs can be computed as [73]

$\displaystyle\rho_{\rm GW}=\frac{1}{32\pi G}\int\frac{{\rm
d}k}{k}\left(\frac{k}{a}\right)^{2}T_{T}^{2}(\tau,k)P_{T}(k)\,,$ (III.5)

where $T_{T}^{2}(\tau,k)=|h_{k}(\tau)|^{2}/|h_{k}(\tau_{i})|^{2}$ is a
transfer function with $\tau_{i}$ as the initial conformal time. The quantity
$P_{T}(k)=k^{3}|h_{k}(\tau_{i})|^{2}/\pi^{2}$ characterizes the primordial
power spectrum and connects to the inflation models with specific forms. Here
we opt for a power-law parametrization of $P_{T}$:

$\displaystyle P_{T}(k)=rA_{s}(k_{*})\left(\frac{k}{k_{*}}\right)^{n_{T}}\,,$
(III.6)

where $r\lesssim 0.06$ [74] is the tensor-to-scalar-ratio, and $A_{s}\simeq
2\times 10^{-9}$ is the scalar perturbation amplitude at the pivot scale
$k_{*}=0.01~{}{\rm Mpc^{-1}}$. In the present analysis, we take $r=0.06$ and
consider the tensor spectral index $n_{T}$ constant plus blue-tilted
($n_{T}>0$). However, a scale dependence might arise owing to the higher order
corrections [75] depending on the inflation model, which we do not discuss
here.555We point out that the single field slow-roll inflation models
correspond to the consistency relation $n_{T}=-r/8$ [76], i.e., the spectral
index is slightly red-tilted ($n_{T}\lesssim 0$). The normalized GW energy
density pertinent to detection purposes is expressed as

$\displaystyle\Omega_{\rm GW}(k)=\frac{k}{\rho_{c}}\frac{{\rm d}\rho_{\rm
GW}}{{\rm d}k}\,,$ (III.7)

where $\rho_{c}=3\mathcal{H}_{0}^{2}/8\pi G$ with $\mathcal{H}_{0}\simeq
2.2\times 10^{-4}~{}\rm Mpc^{-1}$ being the Hubble constant. From Eq. (III.5),
the quantity $\Omega_{\rm GW}(k)$ is derived as

$\displaystyle\Omega_{\rm
GW}(k)=\frac{1}{12\mathcal{H}_{0}^{2}}\left(\frac{k}{a_{0}}\right)^{2}T_{T}^{2}(\tau_{0},k)P_{T}(k)\,,~{}~{}\tau_{0}=1.4\times
10^{4}{\rm~{}Mpc}\,.$ (III.8)

The transfer function can be computed analytically, matching numerical results
with a fair accuracy [77, 78, 79, 80, 81, 82]. When an intermediate matter
domination (recall that a RH neutrino mass-dependent intermediate matter
domination is the central theme of this article) is at play,
$T_{T}^{2}(\tau_{0},k)$ can be calculated as [81, 82]

$\displaystyle T_{T}^{2}(\tau_{0},k)=F(k)\,T_{1}^{2}(\zeta_{\rm
eq})\,T_{2}^{2}(\zeta_{\Phi})\,T_{3}^{2}(\zeta_{\Phi
R})\,T_{2}^{2}(\zeta_{R})\,,$ (III.9)

where the individual transfer functions read as

$\displaystyle T_{1}^{2}(\zeta)$ $\displaystyle=$ $\displaystyle
1+1.57\,\zeta+3.42\,\zeta^{2}\,,$ (III.10) $\displaystyle T_{2}^{2}(\zeta)$
$\displaystyle=$
$\displaystyle\left(1-0.22\,\zeta^{1.5}+0.65\,\zeta^{2}\right)^{-1}\,,$
(III.11) $\displaystyle T_{3}^{2}(\zeta)$ $\displaystyle=$ $\displaystyle
1+0.59\,\zeta+0.65\,\zeta^{2}\,,$ (III.12)

with $\zeta_{i}=k/k_{i}$. The modes $k_{i}$ given by

$\displaystyle k_{\rm eq}$ $\displaystyle=$ $\displaystyle 7.1\times
10^{-2}\Omega_{m}h^{2}{\rm Mpc^{-1}}\,\,\,{\rm
with}\,\,\,\Omega_{m}=0.31\,,\,\,\,h=0.7\,,$ (III.13) $\displaystyle k_{\Phi}$
$\displaystyle=$ $\displaystyle 1.7\times 10^{14}\left(\frac{g_{*s}(T_{\rm
dec})}{106.75}\right)^{1/6}\left(\frac{T_{\rm dec}}{10^{7}\rm GeV}\right){\rm
Mpc^{-1}}\,,$ (III.14) $\displaystyle k_{\Phi R}$ $\displaystyle=$
$\displaystyle 1.7\times 10^{14}\kappa^{2/3}\left(\frac{g_{*s}(T_{\rm
dec})}{106.75}\right)^{1/6}\left(\frac{T_{\rm dec}}{10^{7}\rm GeV}\right){\rm
Mpc^{-1}}\,,$ (III.15) $\displaystyle k_{R}$ $\displaystyle=$ $\displaystyle
1.7\times
10^{14}\kappa^{-1/3}\left(\frac{g_{*s}(T_{R})}{106.75}\right)^{1/6}\left(\frac{T_{R}}{10^{7}\rm
GeV}\right){\rm Mpc^{-1}}\,,$ (III.16)

re-enter the horizon at $T_{\rm eq}$ (the standard matter-radiation equality
temperature), at $T_{\rm dec}$ (the temperature at which the scalar field
decays), at $T_{\rm dom}$ (the temperature at which the scalar field starts to
dominate) and at $T_{R}$ (when the universe reheats after inflation),
respectively. In our computation, we use the analytical expression for
$\kappa$ presented in Eq. (II.16). Notice that we have to consider the
reheating after inflation precedes the phase transition of $\Phi$, i.e.,
$T_{R}>T_{c}$. For the following numerical analysis, we restrict ourselves to
the most conservative value $T_{R}\simeq T_{c}$, which does not contradict the
condition for thermal leptogenesis $T_{R}>M_{N}$ because $T_{c}>M_{N}$ (see
Eq. (II.6)). The quantity $F(k)$ in Eq. (III.9) is given by

$\displaystyle F(k)=\Omega_{m}^{2}\left(\frac{g_{*}(T_{k,\rm
in})}{g_{*0}}\right)\left(\frac{g_{*0s}}{g_{*s}(T_{k,\rm
in})}\right)^{4/3}\left(\frac{3j_{1}(k\tau_{0})}{k\tau_{0}}\right)^{2}\,,$
(III.17)

where $T_{k,\rm in}$ is the temperature corresponding to the horizon entry of
the $k$th mode, $j_{1}(k\tau_{0})$ is the spherical Bessel function,
$g_{*0}=3.36$, $g_{*0s}=3.91$, and an approximate form of the scale-dependent
$g_{*}$ is given by [83, 84, 82]

$\displaystyle g_{*(s)}(T_{k,\rm in})=g_{*0(s)}\left(\frac{A+{\tanh
k_{1}}}{A+1}\right)\left(\frac{B+{\tanh k_{2}}}{B+1}\right)\,,$ (III.18)

where

$\displaystyle
A=\frac{-1-10.75/g_{*0(s)}}{-1+10.75/g_{*0(s)}}\,,~{}~{}B=\frac{-1-g_{\rm
max}/10.75}{-1+g_{\rm max}/10.75}\,,$ (III.19)

and

$\displaystyle k_{1}=-2.5~{}\log_{10}\left(\frac{k/2\pi}{2.5\times
10^{-12}~{}{\rm
Hz}}\right)\,,~{}~{}k_{2}=-2.0~{}{\log}_{10}\left(\frac{k/2\pi}{6.0\times
10^{-9}~{}{\rm Hz}}\right)\,,$ (III.20)

with $g_{\rm max}$ being $\simeq 106.75$. With the above set of equations, we
now evaluate the normalized GW energy density $\Omega_{\rm GW}(k)$ in Eq.
(III.8) for the benchmark parameters considered in the previous section and
reported in Tab. 1. Before we present the numerical results, Let us point out
that the quantity $\Omega_{\rm GW}(k)h^{2}$ is constrained by two robust
bounds on the Stochastic Gravitational Wave Background (SGWB) placed by the
effective number of relativistic species during Big Bang Nucleosynthesis (BBN)
[85] and by the LIGO measurements [86]. In particular, the BBN constraint
reads

$\displaystyle\int_{f_{\rm low}}^{f_{\rm high}}{\rm d}f~{}f^{-1}\Omega_{\rm
GW}(f)h^{2}\lesssim 5.6\times 10^{-6}\Delta N_{\rm eff}\,,$ (III.21)

with $\Delta N_{\rm eff}\lesssim 0.2$. The lower limit of the integration is
the frequency that represents the mode entering the horizon at the BBN epoch,
which we take as $f_{\rm low}\simeq 10^{-10}$ Hz. The upper limit should be
the highest frequency of the GWs determined by the Hubble scale at the end of
inflation $f_{\rm high}=a_{\rm end}H_{\rm end}/2\pi$. For numerical purposes,
a choice $f_{\rm high}\simeq 10^{5}$ Hz would suffice as the spectrum falls
and the integration saturates at higher frequencies. For the LIGO constraint,
we rely on a crude estimation by considering a reference frequency $f_{\rm
LIGO}=35~{}{\rm Hz}$ and discarding the GWs having an amplitude more than
$6.8\times 10^{-9}$ at $f_{\rm LIGO}$ [87].

Figure 4: GW spectra for four different values of the tensor spectral index
$n_{T}$. Top-left: BP1 (one-flavour regime), top-right: BP2 (two-flavour
regime), bottom-left: BP3 (three-flavour regime). The shaded regions are
sensitivity curves of various GW experiments such as SKA [88], GAIA and THEIA
[89], $\mu$ARES [90], LISA [91], BBO [92], DECIGO [93], ET [94], CE [95], and
LIGO-O5 [96]. The black bands in the nHz region represent the NANOGrav 15yrs
data posteriors [6]. The dotted vertical lines (from left to right) correspond
to $f^{\rm peak}_{\rm low}$, $f_{\rm dip}$ and $f^{\rm peak}_{\rm high}$,
respectively. Bottom right: LIGO (solid) and BBN (dashed) constraints on the
$g^{\prime}$ and $v_{\Phi}$ plane for different values of $n_{T}$. Anything
above these lines is excluded. Note that LIGO constraint is much more stronger
than the BBN one, implying that a viable tomography of flavoured leptogenesis
with BGWs can be done only for $n_{T}\lesssim 0.8$.

In Fig. 4, we present the spectral features of the BGWs induced by the $\Phi$
matter domination set by the RH neutrino mass scale. In the top and bottom-
left panels, we show the normalized GW energy density as a function of the
frequency for different tensor spectral index $n_{T}$ for the three
leptogenesis benchmark points representing three different flavour regimes.
Notice that irrespective of the spectral index $n_{T}$, the overall spectrum
gets more suppressed for large entropy production. In addition, the red tilt
in the middle becomes more prominent because longer duration of $\Phi$
domination delays the horizon entry of scales
$k^{-1}\in\left[k_{\Phi}^{-1},k_{\Phi R}^{-1}\right]$, and the corresponding
amplitude within these $k$ values get reduced. Therefore, as the RH neutrino
mass scale decreases (e.g. going from BP1 to BP3), one gets a more distinct
double-peaked spectrum. As such, a three-flavour leptogenesis scenario (BP3:
bottom-left panel) would predict a more distinct double peak spectrum than a
two-flavour (BP2: top-right panel) or one-flavour (BP1: top-left panel)
scenario. In addition to that, starting from a one-flavour/vanilla regime, as
a leptogenesis scenario goes deep into the flavour regime, the low-frequency
peak shifts towards lower frequencies. As evident from Fig. 4, the low-
frequency peak for BP2 (two-flavour scenario) falls within the LISA (mHz)
sensitivity [91], whereas, for BP3 (three-flavour scenario), it falls in the
$\mu$Hz region, i.e., within the sensitivity range of THEIA [89] and $\mu$ARES
[90]. Using Eq.s (III.14), (III.16) and (II.16), below we present analytical
expressions for the three frequencies

$\displaystyle f_{\rm low}^{\rm peak}=5.47\times
10^{-6}\left(\frac{M_{N}}{10^{9}~{}\rm
GeV}\right)^{3/2}\left(\frac{g^{\prime}}{10^{-3}}\right)^{3/4}\left(\frac{g_{*s}(T_{\Phi})}{106.75}\right)^{1/6}~{}{\rm
Hz}\,.$ (III.22) $\displaystyle f_{\rm dip}\simeq 2.9\times
10^{-4}\left(\frac{M_{N}}{10^{9}~{}\rm
GeV}\right)^{7/6}\left(\frac{g^{\prime}}{10^{-3}}\right)^{7/12}\left(\frac{g_{*s}(T_{\Phi})}{106.75}\right)^{1/6}~{}{\rm
Hz}\,,$ (III.23) $\displaystyle f^{\rm peak}_{\rm high}\simeq 3\times
10^{2}\left(\frac{M_{N}}{10^{9}~{}\rm
GeV}\right)^{7/6}\left(\frac{g^{\prime}}{10^{-3}}\right)^{-5/12}\left(\frac{g_{*s}(T_{R})}{106.75}\right)^{1/6}~{}{\rm
Hz}\,,$ (III.24)

which determine the characteristic GW spectrum corresponding to the different
regimes (because they depend on $M_{N}$) of leptogenesis. In Fig. 5, we
concisely illustrate the above discussion.

Figure 5: Left: Three frequencies, $f_{\rm low}^{\rm peak}$, $f_{\rm dip}$,
and $f^{\rm peak}_{\rm high}$, describing the characteristic GW spectrum for
different regimes of leptogenesis. The LIGO constraint on SGWB [87] creates
the discontinuity in the curves at the end. Right: This plot quantitatively
describes that as the leptogenesis process enters deep into the flavour
regimes, the double peak feature in the GW spectrum becomes more prominent;
the ratio $\Omega_{\rm GW}(f_{\rm low}^{\rm peak})/\Omega_{\rm GW}(f_{\rm
dip})$ increases.

We now discuss the implications of the BBN and LIGO constraints on
$\Omega_{\rm GW}$. In the bottom-right plot in Fig. 4, we show such
constraints with solid (LIGO) and dashed (BBN) lines for different values of
$n_{T}$. The parameter space above these lines is excluded. Notice that these
constraints on $\Omega_{GW}$ correspond to bounds on the model parameter space
and, remarkably, on the RH neutrino mass scale. In particular, the LIGO
constraint is much stronger than BBN, ruling out the entire parameter space
under consideration for $n_{T}=0.8$. The allowed parameter space opens for
smaller values of $n_{T}$. We find that the three-flavour (two-flavour) regime
is allowed in case of $n_{T}=0.6$ ($n_{T}=0.5$), while for $n_{T}<0.5$ most of
the model parameter space evades all the GW bounds. An important aspect of the
obtained GW spectrum is that the overall amplitude increases with $n_{T}$, and
the spectrum spans a wide range of frequencies. Therefore, one way to predict
robustly testable GW signal is to fix a UV model that would predict a
particular value of $n_{T}$ ($\lesssim 0.8$ for the flavoured leptogenesis
model as discussed here), which is not our approach. Another way, which we
discuss here, is to take an IR approach by being agnostic about the source of
the BGWs and treating $n_{T}$ as a free parameter. In this case, if we are
able to fit a GW signal within a certain frequency band for a particular or a
range of $n_{T}$ values, we can robustly predict the GW spectrum with distinct
spectral features testable at a different frequency band. As such, as shown in
Fig. 4 (bottom-left), a three-flavour leptogenesis scenario can predict a
testable signal in the LIGO-O5 band for $n_{T}\gtrsim 0.6$. Therefore, if the
future LIGO run finds a stochastic GW signal aligned with our anticipation, a
three-flavour leptogenesis scenario would predict a signal with the similar
spectral slope in the LISA range, distinct spectral behaviour in the $\mu$Hz
range plus a testable blue tilted signal at the nHz frequencies. We can in
fact compare this signal in the nHz band with the recent PTA data. For this,
Let us first opt for the PTA parameterization for the $\Omega_{\rm GW}(f)$,
which reads:

$\Omega_{\rm GW}(f)=\Omega_{\rm yr}\left(\frac{f}{f_{\rm
yr}}\right)^{(5-\gamma)}\,,$ (III.25)

where $\Omega_{\rm yr}=2\pi^{2}A^{2}f_{\rm yr}^{2}/(3H_{0}^{2})$, $f_{\rm
yr}=1~{}{\rm yr^{-1}}\simeq 32~{}{\rm nHz}$ with $A$ and $\gamma$ being the
amplitude and spectral index. In the PTA frequency band, the $\Omega_{\rm
GW}(f)$ in Eq. (III.8) is basically $\Omega_{\rm GW}(f)\propto f^{n_{T}}$,
because in this range, $T_{1}(\xi)$ dominates, and we can approximate
$j_{1}(k\tau_{0})\simeq 1/k\tau_{0}$. As a result, the power spectrum
$P_{T}(k)$ in Eq. (III.6) determines the overall spectral shape. We can
therefore compare Eq. (III.25) and Eq. (III.8) to extract $A$ and $\gamma$. In
Fig. 6, we show the $A$ and $\gamma$ corresponding to BP3 with $n_{T}=0.6$
(Fig. 4, three-flavour leptogenesis scenario) with the blue triangle. The blue
contours represent the $68\%$ and $95\%$ CL posteriors of NANOGrav 15-yrs data
(here we chose NANOGrav posteriors for illustration),666Recently reported
global posteriors [97] that include results from all the PTAs are similar to
NANOGrav, because NANOGrav offers more statistically significant data among
all the PTAs. while the orange contours represent the $68\%$ and $95\%$ CL
posteriors predicted for GW-driven supermassive black hole binary populations
with circular orbits [10]. The black dashed lines represent the best-fit
spectral index value $\gamma=3.2$ of NANOGrav 15-yrs data and the analytically
predicted spectral index value $\gamma=13/3$ for GW-driven SMBHB models [98].
First, note that the spectral index is $\gamma=5-n_{T}=4.4$ (the blue
triangle) for our chosen GW spectrum. Although this value could be well inside
$99\%$ CL of NANOGrav data (we do not show the posterior here), the GW
spectrum predicts $A\sim A^{\rm SMBHB}$, which is in tension with the NANOGrav
data as shown in Fig. 6. Now, as discussed, with our BP3, a fit to an
observation of SGWB by LIGO-O5 would imply a robustly testable signal in the
nHz range but with $A\sim A^{\rm SMBHB}$. Therefore, if the recent PTA finding
persists, then future non-observation of nHz GWs with $A\sim A^{\rm SMBHB}$
would imply that BP3 (or, more generally, a part of our model parameter space)
would be ruled out. In fact, we think that the GW spectrum around BP3
($n_{T}=0.6$) is the most promising spectrum that the discussed flavoured
leptogenesis scenario offers. This is because, in this ballpark, GWs, apart
from having amplitudes $A\sim A^{\rm SMBHB}$ at nHz and $A\sim A^{\rm
LIGO-O5}$ at Hz, exhibit prominent and testable spectral features in the
$\mu$Hz and mHz bands. A higher value of $n_{T}$ saturates LIGO and BBN bounds
as shown in Fig. 4 (this is the reason that our scenario never produces a
signal with $A\sim A^{\rm PTAs}$), on the other hand, for smaller $n_{T}$, the
overall amplitude decreases, and the spectrum loses its testability.

Figure 6: Left: Comparison of GW amplitude and spectral indices: The blue
contours represent the $68\%$ and $95\%$ CL posteriors of NANOGrav 15-yrs
data. The orange contours represent the $68\%$ and $95\%$ CL posteriors
predicted for GW-driven supermassive black hole binary populations with
circular orbits [10]. The vertical dashed lines represent the best-fit values
$\gamma=3.2$ of NANOGrav 15-yrs data and analytically predicted value
$\gamma=13/3$ for GW-driven SMBHB models [98]. The blue triangle denotes $A$
and $\gamma$ corresponding to the BP3 with $n_{T}=0.6$ in our discussion,
representing a three-flavour regime of leptogenesis. Right: RH neutrino mass
splitting as a function of the RH neutrino mass scale $M_{N}$. The red curve
represents the mass splitting for the standard resonance leptogenesis scenario
without entropy production. The other three gauge coupling-dependent curves
represent the same but with an RH neutrino mass-dependent entropy production
scenario. All these curves are consistent with the observed value of the
baryon to photon ratio $\eta_{B}\simeq 6.3\times 10^{-10}$. For details, see
text.

We would like to conclude with the following remarks. First, a generic feature
of $\Omega_{\rm GW}\propto f^{n_{T}\gg 0}$ is that the spectrum does not span
a wide range of frequency due to the BBN bound on effective neutrino species.
Therefore, cosmologically viable inflationary BGWs require a low reheating
temperature after inflation. This has been discussed in detail in the
literature and recently in the context of PTA results [15]. However, when an
intermediate matter-dominated epoch is at play, the spectrum may span a wide
range of frequencies without violating any cosmological bounds because of the
entropy injection. It allows a high $T_{R}$, which is notably, also a natural
requirement for high-scale thermal leptogenesis, as we discuss here. Let us
also stress that our results depend on the parameterization of the primordial
power spectrum in Eq. (III.6). A different parameterization [82, 99, 100, 101]
would yield different results.

Second, since the last couple of years, there has been an effort to test
leptogenesis with primordial GWs, see, e.g., Refs. [69, 102, 103, 104, 105,
106, 107, 108, 109, 110, 111]. This article, nonetheless, for the first time,
attempts to probe flavour regimes of leptogenesis with GWs. We would also like
to point out a technical aspect of the leptogenesis computation, which differs
from the standard scenario. Because of the larger entropy production and
stronger baryon asymmetry dilution for smaller RH neutrino masses, our
scenario requires more quasi-degenerate RH neutrinos than the standard case as
the RH neutrino mass scale decreases. This can be understood in the following
way. The produced baryon-to-photon ratio in the resonance regime can be
written as [37]

$\displaystyle\eta_{B}=10^{-2}\varepsilon_{i}\zeta_{i}\kappa^{-1}\,\,\,{\rm
with}\,\,\,\varepsilon_{i}\simeq\frac{3M_{i}\overline{m}}{8\pi
v_{h}^{2}\delta}\,,$ (III.26)

where $\varepsilon_{i}$ is the CP asymmetry parameter, $\zeta_{i}\simeq
10^{-1}$ is the efficiency of lepton asymmetry production, $\overline{m}\simeq
0.1$ eV is the light neutrino mass scale, $\delta=(M_{j}-M_{i})/{M_{i}}$,
$v_{h}=174$ GeV is the SM Higgs vacuum expectation value, and $\kappa$ is the
injected entropy by $\Phi$ decay as defined in Eq. (II.16). We can obtain an
expression for $\delta$ in two cases. The standard case without entropy
production corresponds to $\kappa=1$. Given the observed value $\eta_{B}\simeq
6.3\times 10^{-10}$ [112], using Eq. (III.26), we obtain

$\displaystyle\delta_{S}$ $\displaystyle\simeq$ $\displaystyle 6.25\times
10^{-1}\left(\frac{M_{N}}{10^{9}\rm GeV}\right)\,,$ (III.27)
$\displaystyle\delta_{E}$ $\displaystyle\simeq$ $\displaystyle 4.5\times
10^{-3}\left(\frac{g^{\prime}}{10^{-3}}\right)^{1/4}\left(\frac{M_{N}}{10^{9}\rm
GeV}\right)^{3/2}\,,$ (III.28)

where the subscript $S(E)$ represents the mass splitting in the standard (with
entropy production) case. Here, we assume $M_{i}\equiv M_{N}$ and simplify
$\kappa$ in Eq. (II.16) in terms of our model parameters as

$\displaystyle\kappa\simeq
140\left(\frac{g^{\prime}}{10^{-3}}\right)^{-1/4}\left(\frac{M_{N}}{10^{9}\rm
GeV}\right)^{-1/2}.$ (III.29)

In Fig. 6 (right) we show $\delta_{E,\>S}$ as a function of $M_{N}$. Note that
while $\delta_{E}$ is always smaller than $\delta_{S}$, it becomes even
smaller as $M_{N}$ decreases. In this context, we can also make an interesting
observation that in this GW tomography method, more is the quasi-degeneracy in
the RH neutrino masses more distinct is the spectral feature in the GWs. This
is because, as discussed already, $\Omega_{\rm GW}(f_{\rm low}^{\rm
peak})/\Omega_{\rm GW}(f_{\rm dip})$ increases with the decrease of $M_{N}$.

Finally, because we consider a $U(1)$ gauge symmetry breaking, our scenario
must produce cosmic strings [113], which is another source of GWs [114]. We
nonetheless note that we work with unconventional parameter space, which in
the numerical simulations of the cosmic strings evolution are taken as
$g^{\prime}\sim~{}\mathcal{O}(1)$ and $\lambda\sim~{}\mathcal{O}(1)$ [115,
116]. If we, in any case, assume that for smaller values of these coupling,
the GW computation from cosmic strings holds, we might obtain interesting
spectral features in the cosmic strings radiated GWs because of the discussed
RH neutrino mass-dependent intermediate matter domination. This, however, is
beyond the scope of this paper, and we shall present a semi-analytical
description of the prospects of GWs from cosmic strings in our scenario in a
future publication.

##  IV Summary

In the present paper, for the first time, we show that flavour regimes of
thermal leptogenesis can be markedly tested with gravitational waves’ spectral
features. Flavour effects bridge high-scale leptogenesis and low-energy
neutrino physics and are important in leptogenesis computation. We opt for a
tomographic method, where gravitational waves originating from an independent
source, confront the leptogenesis model during their propagation in the early
universe and carry imprints of leptogenesis parameters on the final
gravitational waves spectrum. There are three distinct regimes of flavoured
leptogenesis characterized by the right-handed neutrino mass scale:
$M_{N}\gtrsim 10^{12}~{}{\rm GeV}$ (one-flavour/vanilla), $10^{9}~{}{\rm
GeV}\lesssim M_{N}\lesssim 10^{12}~{}{\rm GeV}$ (two-flavour), and
$M_{N}\lesssim 10^{9}~{}{\rm GeV}$ (three-flavour). Testing flavour regimes
with gravitational waves is thus equivalent to studying detectable spectral
features in the gravitational waves dependent on the mass scale $M_{N}$. This
is possible in the simplest and well-motivated realizations of seesaw
scenarios that facilitate a scalar field to give mass to the right-handed
neutrinos. Apart from generating right-handed neutrino masses, the scalar
field can be long-lived to provide a matter-dominated epoch with a lifetime
depending on $M_{N}$. When gravitational waves pass through such a matter-
dominated phase, the start, the end, and the duration of the matter-domination
get imprinted on the final gravitational waves spectrum. Thus in our case, the
spectral features in the GWs become dependent on $M_{N}$, hence on different
regimes of leptogenesis. While any gravitational waves that originate prior to
the scalar field domination can offer a viable tomography, here we consider
inflationary blue gravitational waves, which are now being widely discussed
after the recent finding of blue stochastic gravitational waves by the Pulsar
Timing Arrays, with an infra-red tail characterized by a simple power-law
$\Omega_{\rm GW}\sim f^{n_{T}}$. We show that owing to the scalar field
domination, the final gravitational waves exhibit a double peak spectrum, with
the characteristic frequencies depending on $M_{N}$. As the leptogenesis
process enters deep into the flavour regimes, the spectral features in the
gravitational waves become more prominent, and the characteristic frequencies
shift to lower values (see Figs. 4 and 5). As such, for a two-flavour (three-
flavour) regime, the low-frequency peak shows up in the mHz ($\mu$Hz) bands.
While the BBN constraint on the effective number of neutrino species and LIGO
bound on stochastic gravitational waves background restrict the spectral index
value $n_{T}\lesssim 0.8$ (see Fig. 4, bottom-right), a three-flavour
leptogenesis scenario offers the most promising signal for $n_{T}\sim
0.6-0.7$. Indeed, it shows an amplitude testable in the next LIGO run at
$f\sim$ Hz, characteristic spectral features in the mHz and $\mu$Hz bands (see
Fig. 4, bottom-left), plus a refutable gravitational wave strain at $f\sim$
nHz, comparable to the one expected from supermassive black holes (see Fig. 6,
left, the blue triangle).

## acknowledgements

The work of MC, RS and NS is supported by the research project TAsP
(Theoretical Astroparticle Physics) funded by the Istituto Nazionale di Fisica
Nucleare (INFN). The work of NS is further supported by the research grant
number 2022E2J4RK “PANTHEON: Perspectives in Astroparticle and Neutrino THEory
with Old and New messengers” under the program PRIN 2022 funded by the Italian
Ministero dell’Università e della Ricerca (MUR). The work of SD is supported
by the National Natural Science Foundation of China (NNSFC) under grant No.
12150610460.

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