# Cogenesis of Baryon Asymmetry and Gravitational Dark Matter from PBH

Basabendu Barman  Debasish Borah  Suruj Jyoti Das  and Rishav Roshan

###### Abstract

We propose a scenario where dark matter (DM) with a wide mass range from a few
keV to PeV can be produced solely from evaporating primordial black holes
(PBH), while being consistent with the required free streaming length for
structure formation. If DM does not have any other interactions apart from
gravity and the universe has a PBH dominated phase at early epoch, then PBH
evaporation typically leads to overproduction of DM in this mass range. By
incorporating this gravitational DM within a Type-I seesaw scenario with three
right handed neutrinos (RHN), we bring the abundance of PBH generated DM
within observed limits by late entropy injection due to decay of one of the
RHNs, acting as the diluter. The diluter, due to its feeble coupling with the
bath particles, gets produced primarily from the PBH evaporation thereby
leading to the second stage of early matter domination after the end of PBH
dominated era. The other two RHNs contribute to the origin of light neutrino
mass and also lead to the observed baryon asymmetry via leptogenesis with
contributions from both thermally and PBH generated RHNs. The criteria of DM
relic and baryon asymmetry can be satisfied simultaneously if DM mass gets
restricted to a ballpark in the MeV-GeV regime with the requirement of
resonant leptogenesis for heavier DM mass in order to survive the large
entropy dilution at late epochs.

PI/UAN-2022-714FT

##  1 Introduction

Observational evidences from astrophysics and cosmology based experiments
suggest that we live in a baryon asymmetric universe whose matter content is
dominated by a non-luminous, non-baryonic form of matter known as dark matter
(DM) [1, 2]. The observed baryon asymmetry of the universe (BAU) is
quantitatively quoted in terms of the ratio of excess baryons over antibaryons
and photons [2]

$\eta_{B}=\frac{n_{B}-n_{\overline{B}}}{n_{\gamma}}\simeq 6.2\times 10^{-10}.$
(1.1)

Such an asymmetry needs to be generated dynamically as any pre-existing
asymmetry is likely to be wiped out by the exponential expanding phase during
cosmic inflation. However, the Standard Model (SM) fails to satisfy the
criteria, known as Sakharov’s conditions [3], for dynamical generation of BAU
requiring the presence of beyond standard model (BSM) physics. On the other
hand, the particle nature of DM is not yet known with none of the SM particles
being fit to be DM candidate. Among different BSM proposals for generating
BAU, baryogenesis via leptogenesis [4] is one of the most popular scenarios.
In such a scenario, a non-zero asymmetry is first created in the lepton sector
first and then gets converted into baryon asymmetry via $(B+L)$-violating
electroweak sphaleron transitions [5]. Interestingly, most leptogenesis
framework can also accommodate non-zero neutrino mass and mixings [1], another
observed phenomena which the SM fails to address. Similarly, among different
BSM proposals for DM, the weakly interacting massive particle (WIMP) paradigm
is one of the most well studied scenarios. In WIMP framework, a particle
having mass and interactions around the electroweak ballpark can lead to the
observed DM abundance after undergoing thermal freeze-out in the early
universe [6].

While leptogenesis remains a high scale phenomena with limited observational
scopes111Probing high scale leptogenesis via stochastic gravitational wave
observations is possible in certain scenarios, as pointed out in [7]., WIMP DM
has promising direct detection prospects due to sizeable interactions among DM
and SM particles. However, no such signatures have been found in direct
detection experiments so far, motivating the particle physics community to
explore other viable alternatives. It is also important to note that all the
observational evidences supporting the presence of DM in the universe are
based on purely gravitational interactions only. Thus, it is appealing to
consider the possibility of DM production in the early universe via a
mechanism relying on gravitational interactions only [8]. DM production via
minimal coupling to gravity has been discussed in several works [9, 10, 11,
12, 13, 14, 15, 16, 17]. Similarly, non-minimal coupling to gravity can also
lead to DM production, as discussed already in several works including [18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. In this work, we consider another
alternative: production of gravitational DM from evaporation of primordial
black holes (PBH) assuming the latter to dominate the energy density of the
universe at some stage. Contrary to earlier works where either light or
superheavy DM production from evaporating PBH was shown, here we show the
possibility of a much wider range allowing the intermediate mass DM: from a
few keV to PeV. This is achieved by considering the presence of three heavy
right handed neutrinos (RHN) which also takes part in Type-I seesaw mechanism
[29, 30, 31, 32] of neutrino mass generation. DM with mass in the intermediate
regime mentioned above usually gets overproduced from PBH evaporation and can
be brought within Planck limits by late entropy injection from one of the
right handed neutrinos which is long-lived and couples feebly to SM neutrinos
resulting in vanishingly small lightest active neutrino mass. The other two
RHNs can not only reproduce neutrino oscillation data via Type-I seesaw but
also can produce the baryon asymmetry of the universe via leptogenesis.

This paper is organised as follows. In section 2, we briefly discuss our basic
setup consisting of three right handed neutrinos, a particle DM candidate with
only gravitational interactions and a PBH dominated phase. In section 3, we
provide some analytical estimates for PBH generated DM and leptogenesis from
PBH generated RHN decay. In section 4, we discuss the details of our numerical
analysis and finally conclude in section 5.

##  2 The Basic Framework

As mentioned before, we consider a simple BSM framework where the SM is
extended by three generations of right handed neutrinos $N_{i}\,(i=1,2,3)$ and
a dark matter particle, all of which are singlets under the SM gauge symmetry.
Although three copies of RHNs are considered in typical Type-I seesaw model,
two are sufficient to fit light neutrino data. However, we require three
copies to realise the cogenesis of DM and baryon asymmetry as we discuss in
upcoming sections. The interaction Lagrangian reads

$\displaystyle-\mathcal{L}\supset\frac{1}{2}\,M_{N}\,\overline{N^{c}}N+y_{N}\,\overline{N}\,\tilde{H}^{\dagger}\,\ell+{\rm
h.c.}\,,$ (2.1)

where we have suppressed the generation index and the RHNs are supposed to be
mass diagonal. The SM leptons are denoted by $\ell$, while $\tilde{H}\equiv
i\,\sigma_{2}\,H^{\star}$, $H$ being the SM Higgs and $\sigma_{i}$ are the
Pauli spin matrices. The Dirac Yukawa coupling of neutrinos get fixed via
Casas-Ibarra parametrisation (see Appendix. C), after using the best-fit
values of light neutrino parameters [1]. Since DM is assumed to have only
gravitational interactions, it only has a mass term, the exact form of which
depends on the particle nature of DM particle. In our numerical analysis, for
simplicity, we will consider a real singlet scalar $S$ of mass
$m_{\text{DM}}$, to be a potential DM candidate222We assume the strength of
the portal interaction $S^{2}\,\left|H\right|^{2}$ is absent by construction
to ensure only gravitational production of DM. Considering DM with a different
spin does not alter the outcome of our analysis. which is produced purely from
PBH evaporation. One of the RHNs, $N_{3}$ in the present scenario, is
considered to be long lived such that its decay rate is suppressed compared to
the other two RHNs. As a consequence, the universe undergoes a
$N_{3}$-dominated epoch even after PBH is completely evaporated. The other two
RHNs, namely $N_{1,2}$ can undergo CP-violating out of equilibrium decay,
producing lepton asymmetry which can then be converted into the baryon
asymmetry of the universe, thanks to the electroweak sphaleron transition.
Sphaleron interactions are in equilibrium in the temperature range between
$\sim 100$ GeV and $10^{12}$ GeV, and they convert a fraction of a non-zero
$B-L$ asymmetry into a baryon asymmetry via

$\displaystyle Y_{B}\simeq
a_{\text{sph}}\,Y_{B-L}=\frac{8\,N_{F}+4\,N_{H}}{22\,N_{F}+13\,N_{H}}\,Y_{B-L}\,,$
(2.2)

where $N_{F}$ is the number of fermion generations and $N_{H}$ is the number
of Higgs doublets, which in our case $N_{F}=3\,,N_{H}=1$ and $a_{\text{sph}}$
turns out to be 28/79. In leptogenesis, where purely a lepton asymmetry is
generated, $B-L=-L$. This is converted into the baryon asymmetry via sphaleron
transition [33, 34]. Finally, the observed baryon asymmetry of the universe is
given by [35]

$\displaystyle\eta_{B}=\frac{n_{B}-n_{\overline{B}}}{n_{\gamma}}\simeq
6.2\times 10^{-10}\,,Y_{B}\simeq 8.7\times 10^{-11}\,.$ (2.3)

Figure 1: Dominant component of the energy density of the universe at
different epoch (time runs from left to right).

During the decay of $N_{3}$ entropy is injected into the radiation bath [36]
leading to dilution of the DM and baryon number density produced from PBH.
Role of such late entropy injection in tackling DM over-abundance has been
studied by several authors in different contexts [37, 38, 39, 40, 41, 42, 43,
44, 45]. In Fig. 1 we schematically show the components that dominate the
energy density of the universe at different epoch. As the time flows from the
left to the right, hence, as expected, the Universe is radiation dominated
(RD) at the end of reheating. Once the PBH formation happens, the PBH energy
density can dominate, which ends as the PBH evanescence is complete. The
universe is then dominated by RHN produced from the PBH. Finally, as the RHN
decays away, the usual radiation domination era again commences. Note that, in
addition to DM, the diluter $N_{3}$ is also dominantly produced from PBH
evaporation as its coupling with SM leptons is tiny due to the requirement of
its long-lifetime.

##  3 PBH and Cogenesis

Primordial black holes, originally proposed by Hawking [46, 47], can have very
interesting cosmological signatures [48, 49] (a recent review of PBH may be
found in [50]). While PBH with suitable mass range can itself be a DM
candidate, we are interested in its ultra light mass regime where it is not
long-lived enough to be DM but can play non-trivial role in production of DM
as well as baryon asymmetry. Role of PBH evaporation on DM genesis has been
studied in several works [51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63,
64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75]. Similarly, the role of PBH
evaporation on baryogenesis was first pointed out in [46, 49] and has been
studied subsequently by several authors in different contexts [76, 77, 78, 79,
51, 80, 81, 82, 83, 84, 85, 86, 87]. In a few related works, the possibility
of both DM and RHN production from PBH evaporation was considered [78, 51, 59,
88, 61, 80, 82, 83, 89, 71, 70, 90]. In the present work, we consider DM
production solely from PBH evaporation and prevent it from being overproduced
by late entropy dilution from one of the RHNs while the other two RHNs produce
the baryon asymmetry via leptogenesis. Before proceeding to the detailed
numerical analysis, we first show the key features of DM and RHN production
from PBH evaporation to set the motivation clear.

We assume PBHs to have formed after inflation during the era of radiation
domination. Assuming radiation domination, the mass of the black hole from
gravitational collapse is typically close to the value enclosed by the post-
inflation particle horizon and is given by [78, 61]

$m_{\text{BH}}=\frac{4}{3}\,\pi\,\gamma\,\Bigl{(}\frac{1}{\mathcal{H}\left(T_{\text{in}}\right)}\Bigr{)}^{3}\,\rho_{\text{rad}}\left(T_{\text{in}}\right)\,$
(3.1)

with

$\rho_{\text{rad}}\left(T_{\text{in}}\right)=\frac{3}{8\,\pi}\,\mathcal{H}\left(T_{\text{in}}\right)^{2}\,M_{\text{pl}}^{2}\,$
(3.2)

where $\mathcal{H}$ is the Hubble parameter, and $\gamma\simeq 0.2$ is a
numerical factor which contains the uncertainty of the PBH formation. As
mentioned earlier, PBHs are produced during the radiation dominated epoch,
when the SM plasma has a temperature $T=T_{\text{in}}$ which is given by

$T_{\text{in}}=\Biggl{(}\frac{45\,\gamma^{2}}{16\,\pi^{3}\,g_{\star}\left(T_{\text{in}}\right)}\Biggr{)}^{1/4}\,\sqrt{\frac{M_{\text{pl}}}{m_{\text{BH}}(T_{\text{in}})}}\,M_{\text{pl}}\,.$
(3.3)

Once formed, PBH can evaporate by emitting Hawking radiation [46, 47]. A PBH
can evaporate efficiently into particles lighter than its instantaneous
temperature $T_{\text{BH}}$ defined as [47]

$T_{\rm BH}=\frac{1}{8\pi\,G\,m_{\rm BH}}\approx
1.06~{}\left(\frac{10^{13}\;{\rm g}}{m_{\rm BH}}\right)~{}{\rm GeV}\,,$ (3.4)

where $G$ is the universal gravitational constant. The mass loss rate can be
parametrised as [91]

$\frac{dm_{\text{BH}}(t)}{dt}=-\frac{\mathcal{G}\,g_{\star}\left(T_{\text{BH}}\right)}{30720\,\pi}\,\frac{M_{\text{pl}}^{4}}{m_{\text{in}}(t)^{2}}\,,$
(3.5)

where $\mathcal{G}\sim 4$ is the grey-body factor. Here we ignore the
temperature dependence of $g_{\star}$ during PBH evolution, valid in the pre-
sphaleron era. On integrating Eq. (3.5) we end up with the PBH mass evolution
equation as

$m_{\text{BH}}(t)=m_{\text{in}}(T_{\text{in}})\Bigl{(}1-\frac{t-t_{\text{in}}}{\tau}\Bigr{)}^{1/3}\,,$
(3.6)

with

$\tau=\frac{10240\,\pi\,m_{\text{in}}^{3}}{\mathcal{G}\,g_{\star}(T_{\text{BH}})\,M_{\text{pl}}^{4}}\,,$
(3.7)

as the PBH lifetime. Here onward we will use $m_{\text{in}}(T_{\text{in}})$
simply as $m_{\text{in}}$. The evaporation temperature can then be computed
taking into account
$H(T_{\text{evap}})\sim\frac{1}{\tau^{2}}\sim\rho_{\text{rad}}(T_{\text{evap}})$
as

$T_{\text{ev}}\equiv\Bigl{(}\frac{45\,M_{\text{pl}}^{2}}{16\,\pi^{3}\,g_{\star}\left(T_{\text{evap}}\right)\,\tau^{2}}\Bigr{)}^{1/4}\,.$
(3.8)

However, if the PBH component dominates at some point the total energy density
of the universe, the SM temperature just after the complete evaporation of
PBHs is: $\overline{T}_{\text{evap}}=2/\sqrt{3}\,T_{\text{evap}}$ [53].

The initial PBH abundance is characterized by the dimensionless parameter
$\beta$ that is defined as

$\beta\equiv\frac{\rho_{\text{BH}}\left(T_{\text{in}}\right)}{\rho_{\text{rad}}\left(T_{\text{in}}\right)}\,,$
(3.9)

that corresponds to the ratio of the initial PBH energy density to the SM
energy density at the time of formation. Note that, $\beta$ steadily grows
until PBH evaporation since the PBH energy density scales like non-
relativistic matter $\sim a^{-3}$, while the radiation energy density scales
as $\sim a^{-4}$. Therefore, an initially radiation-dominated universe will
eventually become matter-dominated if the PBHs are still around. The condition
of PBH evanescence during radiation domination can be expressed as [61]

$\beta<\beta_{\text{crit}}\equiv\gamma^{-1/2}\,\sqrt{\frac{\mathcal{G}\,g_{\star}(T_{\text{BH}})}{10640\,\pi}}\,\frac{M_{\text{pl}}}{m_{\text{in}}}\,,$
(3.10)

where $\beta_{c}$ is the critical PBH abundance that leads to early matter-
dominated era. Note that for simplicity, we consider a monochromatic mass
function of PBHs implying all PBHs to have identical masses. Additionally, the
PBHs are assumed to be of Schwarzschild type without any spin and charge. The
gravitational waves (GW) induced by large-scale density perturbations laid by
PBHs could lead to a backreaction problem [92, 53], that can be avoided if the
energy contained in GWs never overtakes the one of the background universe or
in other words if

$\beta<10^{-4}\,\Bigl{(}\frac{10^{9}\text{g}}{m_{\text{in}}}\Bigr{)}^{1/4}\,.$
(3.11)

Since PBH evaporation produces all particles, including radiation that can
disturb the successful predictions of BBN, hence we require
$T_{\text{evap}}>T_{\text{BBN}}\simeq 4$ MeV. This can be translated into an
upper bound on the PBH mass. On the other hand, a lower bound on PBH mass can
be obtained from the CMB bound on the scale of inflation [35] :
$\mathcal{H}_{I}\equiv\mathcal{H}(T_{\text{in}})\leq 2.5\times
10^{-5}\,M_{\text{pl}}$, where
$\mathcal{H}(T_{\text{in}})=\frac{1}{2\,t_{\text{in}}}$ with
$t(T_{\text{in}})=\frac{m_{\text{in}}}{M_{\text{pl}}^{2}\,\gamma}$ (as
obtained from Eq. (3.1)). Using these BBN and CMB bounds together, we have a
window for allowed initial mass for PBH that reads $0.1\,\text{g}\lesssim
m_{\text{in}}\lesssim 3.4\times 10^{8}\,\text{g}$. The range of PBH masses
between these bounds is at present generically unconstrained [50]. While PBH
can evaporate by Hawking radiation, it can be stable on cosmological scales if
sufficiently heavy, potentially giving rise to some or all of DM [93]. In
order not to spoil the structure formation, a DM candidate which is part of
the thermal bath or produced from the thermal bath should have mass above a
few keV in order to give required free-streaming of DM as constrained from
Lyman-$\alpha$ flux-power spectra [94, 95, 96]. Such light DM of keV scale
leads to a warm dark matter (WDM) scenario having free-streaming length within
that of cold and hot DM. If such light DM is also produced from PBH
evaporation, it leads to a potential hot component in total DM abundance,
tightly constrained by observations related to the CMB and baryon acoustic
oscillation (BAO) leading to an upper bound on the fraction of this hot
component with respect to the total DM, depending on the value of DM mass
[97]333A conservative $10~{}\%$ upper bound on such hot dark matter (HDM)
component [53] can lead to similar constraints on DM mass along with PBH
initial fraction. . The lower bound on DM mass from Lyman-$\alpha$ can be
found in Appendix B.

Figure 2: The DM is over-abundant in the blue shaded region, for $\zeta=\xi=1$
(left) and in the blue (orange) shaded $\zeta=\xi=10(100)$ (right) considering
the DM to be scalar. The grey shaded regions are discarded due to limits on
PBH mass from CMB (lower limit), BBN (upper limit) and Lyman-$\alpha$ (see
text). Observed DM relic is achievable only in the white region in the top
right corner (left), whereas non-zero values of $\xi$ opens up a window in the
bottom (right).

We consider DM of spin $s$ with only gravitational interaction such that it
can be produced solely from PBH evaporation. Along with the DM, the PBH also
emits right handed neutrinos $N_{i}$ with $i\in 1,2,3$. The number of any
particle $X$ radiated during the evaporation of a single PBH

$\mathcal{N}_{X}=\frac{g_{X,H}}{g_{\star,H}(T_{\text{BH}})}\begin{cases}\frac{4\,\pi}{3}\,\Bigl{(}\frac{m_{\text{in}}}{M_{\text{pl}}}\Bigr{)}^{2}&\text{for
}m_{X}<T_{\text{BH}}^{\text{in}}\,,\\\\[8.0pt]
\frac{1}{48\,\pi}\,\Bigl{(}\frac{M_{\text{pl}}}{m}\Bigr{)}^{2}&\text{for
}m_{X}>T_{\text{BH}}^{\text{in}}\,,\end{cases}\,,$ (3.12)

where

$g_{\star,H}(T_{\text{BH}})\equiv\sum_{i}\omega_{i}\,g_{i,H}\,;g_{i,H}=\begin{cases}1.82&\text{for
}s=0\,,\\\ 1.0&\text{for }s=1/2\,,\\\ 0.41&\text{for }s=1\,,\\\ 0.05&\text{for
}s=2\,,\\\ \end{cases}$ (3.13)

with $\omega_{i}=2\,s_{i}+1$ for massive particles of spin $s_{i}$,
$\omega_{i}=2$ for massless species with $s_{i}>0$ and $\omega_{i}=1$ for
$s_{i}=0$. At temperatures $T_{\text{BH}}\gg T_{\text{EW}}\simeq 160$ GeV, PBH
evaporation emits the full SM particle spectrum according to their
$g_{\star,H}$ weights, while at temperatures below the MeV scale, only photons
and neutrinos are emitted. For $T_{\text{BH}}\gg 100$ GeV (corresponding to
$m_{\text{BH}}\ll 10^{11}$ g), the particle content of the SM corresponds to
$g_{\star,H}\simeq 108$. The DM yield produced by evaporation is directly
related to the BH abundance at evaporation as

$Y_{\text{DM}}(T_{0})=\frac{n_{\text{DM}}}{s}\Big{|}_{T_{0}}=\,\mathcal{N}_{X}\,\frac{n_{\text{BH}}}{s}\Big{|}_{T_{\text{ev}}}\,.$
(3.14)

Now, PBH abundance at evaporation for PBH domination can be obtained using the
first Friedmann equation as

$n_{\text{BH}}(T_{\text{ev}})=\frac{1}{6\,\pi}\,\frac{M_{P}^{2}}{m_{\text{BH}}^{2}\,\tau^{2}}\equiv\frac{1}{6\,\pi}\,\left(\frac{\mathcal{G}\,g_{\star,H}}{10640\,\pi}\right)^{2}\,\frac{M_{P}^{10}}{m_{\text{BH}}^{7}}\,.$
(3.15)

Then, the DM relic abundance
$\Omega_{\text{DM}}\,h^{2}=\frac{m_{\text{DM}}\,s_{0}}{\rho_{c}}\,Y_{\text{DM}}\left(T_{0}\right)\,,$
in the present epoch reads

$\Omega_{\text{DM}}\,h^{2}=\mathbb{C}(T_{\text{ev}})\begin{cases}\frac{1}{\pi^{2}}\,\sqrt{\frac{M_{P}}{m_{\text{BH}}}}\,m_{\text{DM}}&\text{for
}m_{\text{DM}}<T_{\text{BH}}^{\text{in}}\,,\\\\[8.0pt]
\frac{1}{64\,\pi^{4}}\left(\frac{M_{P}}{m_{\text{BH}}}\right)^{5/2}\,\frac{M_{P}^{2}}{m_{\text{DM}}}&\text{for
}m_{\text{DM}}>T_{\text{BH}}^{\text{in}}\,,\end{cases}\,$ (3.16)

with
$\mathbb{C}(T_{\text{ev}})=\frac{s_{0}}{\rho_{c}}\,\frac{1}{\zeta}\,\frac{g_{X,H}}{g_{\star,H}}\,\frac{5}{g_{\star
s}(T_{\text{ev}})}\,\,\left(\frac{\pi^{3}\,g_{\star}(T_{\text{ev}})}{5}\right)^{3/4}\,\sqrt{\frac{\mathcal{G}\,g_{\star,H}}{10640\,\pi}}$.
Here $\zeta$ parametrizes a possible entropy production after PBH evaporation
until now, i.e.,
$\zeta\,\left(sa^{3}\right)_{\text{evap}}=\left(sa^{3}\right)_{0}$. Note that,
for the heavy DM case relic abundance has an inverse dependence on the DM
mass, implying heavy DM leads to under abundance. The over abundant region for
DM with spin $s=0$ produced from PBH evaporation is shown in the left panel of
Fig. 2 for $\zeta=\xi=1$, i.e., considering no entropy injection at any given
epoch. Here we note that in the majority of the parameter space the DM is over
abundant irrespective of their spins and only $m_{\text{DM}}\gtrsim 10^{10}$
GeV can lead to right abundance for $m_{\text{in}}\gtrsim 10^{6}$ g [78, 72].
Here we would like to mention that for DM with different spins this over
abundant region only slightly changes. Particularly, the low DM mass region is
in conflict with the WDM limit. An obvious way to overcome this tension is to
consider that the DM produced from PBH evaporation does not constitute the
whole DM abundance. But since we are considering PBH is the only source of all
of the DM, therefore this tension can be alleviated by considering entropy
non-conservation [78, 61]. This is shown in the right panel of Fig. 2, where
$\zeta=\xi=10$ (in blue) and $\zeta=\xi=100$ (in yellow) is assumed444Another
way is to significantly increase $g_{\star,H}$ as mentioned in [61].. Here we
see the low DM mass window can be resurrected simply because
$\Omega_{\text{DM}}\propto\,\zeta^{-1}\,m_{\text{DM}}$555For simplicity we
consider $\zeta=\xi$, which is, of course not a mandate..

Figure 3: Constraints on RHN mass from the requirement of obtaining
$Y_{B}^{\text{obs}}$ considering $\zeta=1$ (left) and $\zeta=10$ (right). All
the coloured regions are discarded from the bounds derived in Eq. (3.18). The
vertical black dashed line corresponds to (from left to right) the bound from
the scale of inflation (CMB), sphaleron transition and BBN. The cyan dashed
diagonal straight line corresponds to
$T_{\text{BH}}^{\text{in}}=m_{\text{in}}$. The white triangular region in the
middle is the region that is allowed (see text).

The RHNs emitted during PBH evaporation can undergo CP-violating decays,
generating lepton asymmetry. This lepton asymmetry is then further converted
into the observed baryon asymmetry of the Universe via sphaleron transition
[78]. It is possible to analytically derive the mass range of RHNs (and PBH)
emitted from PBH evaporation that can provide the observed baryon asymmetry.
In the Type-I seesaw mechanism, the quantity $\epsilon$ has an upper bound
[98, 99]

$\epsilon\lesssim\frac{3}{16\,\pi}\,\frac{M_{1}\,m_{\nu,\text{max}}}{v^{2}}\,,$
(3.17)

where $v=246$ GeV is the SM Higgs VEV and $m_{\nu,\text{max}}$ is the mass of
the heaviest light neutrino. On the other hand, the final asymmetry produced
from PBH evaporation is given by
$Y_{B}^{\text{obs}}=n_{B}/s\Big{|}_{T_{0}}=\frac{1}{\zeta}\,\mathcal{N}_{X}\,\epsilon\,a_{\text{sph}}\,Y_{B}\Big{|}_{T_{\text{evap}}}\simeq
8.7\times 10^{-11}$ [35], where $a_{\text{sph}}\simeq 1/3$ and $T_{0}$ is the
present temperature of the Universe. These together constrain the mass of the
RHN produced from PBH evaporation both from above and from below [78]

$M_{1}\begin{cases}>\frac{4\,g_{\star,H}(T_{\text{BH}}^{\text{in}})}{g_{X}\,a_{\text{sph}}}\,\zeta\,\frac{Y_{B}^{0}}{Y_{B}^{\text{evap}}}\,\frac{v^{2}\,M_{P}^{2}}{m_{\nu}\,m_{\text{in}}^{2}}&\text{for
}M_{1}<T_{\text{BH}}^{\text{in}}\,;\\\\[8.0pt]
<\frac{g_{X}\,a_{\text{sph}}}{256\,\pi^{2}\,g_{\star,H}}\,\frac{1}{\zeta}\,\frac{Y_{B}^{\text{evap}}}{Y_{B}^{0}}\,\frac{M_{P}^{2}\,m_{\nu}}{v^{2}}&\text{for
}M_{1}>T_{\text{BH}}^{\text{in}}\,,\end{cases}$ (3.18)

To ensure non-thermal production of baryon asymmetry it is also necessary to
consider $M_{1}>T_{\text{evap}}$ [78] that leads to

$M_{1}\gtrsim 3\times
10^{-3}\,\left[\mathcal{G}^{2}\,g_{\star}(T_{\text{evap}})\left(\frac{M_{P}^{5}}{m_{\text{in}}^{3}}\right)^{2}\right]^{1/4}\,,$
(3.19)

otherwise for $M_{1}<T_{\text{evap}}$, the RHNs produced from PBH evaporation
are in thermal bath and then washout processes are in effect. Finally, in
order for lepton asymmetry to be sufficiently generated from RHNs produced
from PBH evaporation, one requires evaporation to be over before sphaleron
transition $T_{\text{evap}}\gtrsim T_{\text{EW}}$, which translates into
$m_{\text{in}}\lesssim 3\times 10^{5}\,\text{g}$. In Fig. 3 we summarize the
bounds on RHN mass, required to produce the observed baryon asymmetry. The
tiny triangular white part is the only window where
$Y_{B}=Y_{B}^{\text{obs}}$. This region typically corresponds to $0.1\lesssim
m_{\text{in}}\lesssim 20$ g and $10^{12}\lesssim M_{1}\lesssim 10^{17}$ GeV
when entropy is assumed to be conserved, i.e., $\zeta=1$. Note that, this
region shrinks for larger $\xi=10$, as shown in the right panel. This is
however expected since a larger $\zeta$ allows a larger entropy injection
(from $T_{\text{evap}}$ to $T_{0}$), diluting the asymmetry produced. Thus,
while a larger $\zeta$ can provide breathing space for lighter DM (cf., Fig.
2), but in turn tightens the allowed parameter space for observed baryon
asymmetry. It is therefore clear, satisfying both of them simultaneously needs
a careful choice of $\zeta$, i.e., the entropy injection, such that not only
right DM abundance for lighter DM is obtained, but baryon asymmetry should
also not get too much diluted. It is interesting to note that the heavy DM
mass $(\gtrsim 10^{10}~{}\rm GeV)$ region although remains viable even in the
absence of entropy injection, it is not possible to satisfy the observed
asymmetry in those regions as they typically correspond to massive PBH
$(\gtrsim 10^{4}~{}\rm g$), leading to under production of asymmetry (cf. Fig.
3).

##  4 Numerical Analysis

To this end we have analytically established that in order to open up the low
mass DM window (which is otherwise over abundant) it is necessary to have a
substantial entropy injection $(\gtrsim\mathcal{O}(10))$. However, the same
entropy injection reduces the viable parameter space for observed baryon
asymmetry. Thus, it is rather difficult to satisfy both DM abundance and
correct asymmetry together, and one has to stick to the region of
comparatively lighter PBH mass to achieve both. In this section we will
investigate the viability of the analytical results by solving a set of
coupled Boltzmann equations (BEQ) numerically.

In order to compute the final lepton (baryon) and DM yield, we numerically
solve the set of coupled BEQs in Appendix A. Since we are interested in PBHs
with mass $\lesssim\mathcal{O}(1)$ g (where leptogenesis from PBH dominates),
with typical evaporation temperature
$T_{\text{evap}}\lesssim\mathcal{O}(10^{11})$ GeV, we do not include the
$\Delta L=2$ washout processes in the BEQ as such processes go out of
equilibrium at temperatures $T\lesssim 6\times 10^{12}$ GeV as shown in [86]
and thus have no influence on final asymmetry666 In the standard radiation
dominated early Universe, if the lightest RHN mass exceeds $\sim 10^{15}$ GeV
and heaviest active neutrino mass is greater than $\sim 0.1$ eV, $\Delta L=2$
washout processes erase the lepton asymmetry.. We consider $\beta=10^{-3}$,
such that PBH dominate the energy density at some epoch and particle
production takes place during PBH domination. The temperature of the thermal
bath as a function of the scale factor is shown in the left panel of Fig. 4,
where we can clearly see the effect of entropy dilution in two different
epochs. The first one takes place at $a\sim 10^{6}$ when the PBH evaporation
is completed, while the second one at a later epoch corresponding to $a\sim
10^{15}$ when the decay of $N_{3}$ is completed. Note that, prior to $N_{3}$
domination, the universe is again dominated by radiation energy density for a
brief period of time. This is clearly visible from the right panel plot where
we have shown the energy density of radiation (red), PBH (blue) and $N_{3}$
(black) as functions of the scale factor $a$. The cyan-shaded regions indicate
the two different epochs of early matter domination. Here one can see that the
PBH energy density falls sharply at $a\sim 10^{6}$ denoting the end of PBH
domination. Afterward the Universe goes through radiation domination, that is
being overtaken by the $N_{3}$ energy density (second matter dominated era) at
$a\sim 10^{9}$. Finally that ends at $a\sim 10^{15}$ with the complete decay
of $N_{3}$ into radiation. It is important to clarify that in these plots we
have fixed the $N_{3}$ mass and adjusted its decay width accordingly such that
it never achieves equilibrium with the SM bath (hence long-lived) by tuning
the lightest active neutrino mass that we consider to be a free parameter.

Figure 4:  Evolution of temperature of the thermal bath $T$ as a function of
the scale factor $a$ (left) and energy densities of radiation, PBH and $N_{3}$
as a function of the scale factor (right). We take $m_{in}=1$ g,
$M_{1}=10^{13}$ GeV, $M_{3}=10^{12}$ GeV, $m_{\text{DM}}=1$ GeV, with $N_{3}$
decay width adjusted to be $\Gamma_{3}=1.3\times 10^{-11}$ GeV (see text).
Figure 5: Evolution of DM (left) and baryon yield (right panel) for two
different values of PBH masses shown in two different colours. We consider
$m_{\text{DM}}=1$ GeV, $M_{1}=10^{13}$ GeV, $M_{3}=10^{12}$ GeV, while keeping
$N_{3}$ decay $\Gamma_{3}=1.3\times 10^{-11}$ GeV to be fixed. Figure 6:
Evolution of DM (left) and baryon yield (right panel) for two different
$M_{3}$ values, shown in two different colours. We consider $m_{\text{DM}}=1$
GeV, $m_{\text{in}}=1$ g, and the decay width $\Gamma_{3}=1.3\times 10^{-11}$
GeV is kept fixed. Figure 7: Evolution of DM yield (left panel) and baryon
asymmetry yield (right panel) for choice of parameters which together satisfy
observed DM relic and baryon asymmetry. We choose $m_{\text{DM}}=1$ MeV,
$M_{1}=10^{13}$ GeV, $M_{3}=10^{8}$ GeV, $m_{in}=1$g, and $\Gamma_{3}=4\times
10^{-15}$ GeV.

As advocated in the last section, DM overproduced from PBH evaporation gets
diluted to right abundance by introducing entropy injection at late epochs. In
this case $N_{3}$ decay is responsible for adequate entropy dilution. The
effect of entropy injection on the DM and baryon yield is demonstrated in Fig.
5, considering two different PBH masses. In the left panel we see DM is first
overproduced at the end of PBH evaporation around $a\sim 10^{6}$. The DM yield
then starts diminishing once $N_{3}$ starts decaying at later epoch near
$a\sim 10^{12}$. Once $N_{3}$ decay is complete, the DM yield saturates close
to the Planck 2018 limit for $m_{\text{DM}}=1$ GeV. As we already noticed in
Fig. 3, entropy injection has a destructive effect on the baryon asymmetry
since it dilutes the asymmetry generated from RHN decay as well. This is again
established in the right panel, where we see the generated yield for asymmetry
becomes under-abundant after the completion of $N_{3}$ decay for a fixed
$M_{3}$. For a fixed PBH mass the effect of varying $M_{3}$ is shown in Fig.
6, where in the left panel we see that a heavier $M_{3}$ is capable of
producing the right DM abundance for a fixed DM mass since a heavier $N_{3}$
results in larger entropy dilution. This affects $Y_{B}$ as well, as one can
see from the right panel of Fig. 6. Therefore, one has to make a careful
choice of PBH and diluter mass in order to satisfy both DM abundance and right
baryon asymmetry. In Fig. 7 we show a specific benchmark point which gives
rise to right asymptotic yield for the DM and that of baryon asymmetry
simultaneously, in agreement with observations. Here we choose a DM of mass
$m_{\text{DM}}=1$ MeV, while $M_{3}=10^{8}$ GeV with a PBH of mass 1 g.

Figure 8: Left: Parameter space allowed by relic density in $\Delta
S-m_{\text{DM}}$ plane, where PBH mass is scanned over the range: {0.5-5} g.
The colour code is with respect to $Y_{B}$, considering vanilla high scale
leptogenesis. The black vertical dashed line corresponds to the conservative
bound from WDM. The green arrows denote the net allowed parameter space.
Right: Same as left, but considering resonant leptogenesis.

The entropy released due to the decay of long-lived heavy $N_{3}$ can be
expressed analytically as [36, 38]

$\Delta S\simeq\left[1+2.95\,\left(\frac{2\,\pi^{2}\,g_{\star
s}(T)}{45}\right)^{\frac{1}{3}}\,\left(\frac{Y_{3}^{2}\,M_{3}^{2}}{M_{P}\,\Gamma_{3}}\right)^{\frac{2}{3}}\right]^{\frac{3}{4}}\,,$
(4.1)

where $Y_{3}=n_{N_{3}}/s$ is the initial yield of $N_{3}$ before the onset of
the second matter dominated era due to $N_{3}$ and $\Gamma_{3}$ is its decay
width. Note that the amount of entropy injection is inversely proportional to
$\Gamma_{3}$ as expected. Now, $\Gamma_{3}$ cannot be arbitrarily small as
$N_{3}$ has to decay before BBN. This gives an upper bound on $\Delta\,S$,
around $10^{11}$, which translates into an upper bound on the dark matter
mass, $m_{\rm DM}\lesssim 10^{6}$ GeV. This is nevertheless a significant
improvement as DM masses all the way till $m_{\rm DM}\sim 10^{10}$ GeV, if
solely produced from PBH evaporation, remain disallowed in the usual scenario
without late entropy injection, either due to overproduction or WDM limits, as
shown in the left panel plot of Fig. 2. It is important to specify here that
$\Delta S$ can be related to the lightest neutrino mass $m_{\nu}^{1}$, since
the value of the $N_{3}$ Yukawas are determined by $m_{\nu}^{1}$. For
instance, we find that for $m_{\rm in}=1$ g, and $M_{3}<T_{\rm BH}^{\rm in}$,
$1\lesssim\Delta S\lesssim 10^{6}$ typically corresponds to lightest neutrino
mass in the range $10^{-10}\,{\rm eV}\lesssim m_{\nu}^{1}\lesssim
10^{-24}\,{\rm eV}$. Such tiny values of the lightest neutrino mass and hence
tiny Dirac Yukawa couplings of $N_{3}$ are expected since $N_{3}$ has to be
sufficiently long-lived. Now, in order to explore the viable parameter space
we perform a numerical scan over the DM mass $m_{\text{DM}}$ in keV-GeV range
and on the PBH mass $m_{\text{in}}:\\{0.5-5\\}$ g, fixing the leptogenesis
scale $M_{1}$ at $10^{13}$ GeV. In the left panel of Fig. 8 we show the relic
density allowed parameter space in $\Delta S-m_{\text{DM}}$ plane. As already
established earlier, light DM with mass below a certain upper limit is
disallowed from the Lyman-$\alpha$ bound on WDM, which are shown by the red
points on the left side of black vertical dashed line. However, a substantial
amount of entropy injection can improve this situation as already shown in the
right panel plot of Fig. 2. Thus, DM mass
$m_{\text{DM}}\gtrsim\mathcal{O}$(MeV) is allowed for $\Delta
S\gtrsim\mathcal{O}(10)$. The parameter space is also capable of explaining
the observed baryon asymmetry in vanilla leptogenesis framework for
$10\lesssim\Delta S\lesssim 100$, which agrees with earlier observations in
[78]. We project the most conservative WDM bound that disallows DM mass
$m_{\text{DM}}\lesssim 0.3$ MeV. Note that, as foretold, with the increase in
$\Delta S$, the baryon asymmetry decreases (cf. Fig. 3). Thus, cogenesis of
right DM abundance and baryon asymmetry is possible with an entropy production
of the order of $\Delta S\lesssim\mathcal{O}(100)$.

In order to overcome the large entropy dilution at late epochs, we also check
if the lepton asymmetry can be significantly overproduced. It is well
established that the requirement on the heavy neutrino mass scale for
successful leptogenesis can be significantly relaxed if (at least) two of the
mass eigenvalues $M_{i}$, say for $i=1,2$ are quasi-degenerate, i.e., $\Delta
M=M_{2}-M_{1}\ll\overline{M}=(M_{1}+M_{2})/2$ [100, 101]. In such a case of
resonant leptogenesis, the CP-violating decay asymmetry originating from the
interference of tree and self-energy contributions-the so-called
$\epsilon$-type or indirect CP violation is resonantly enhanced, and dominates
over the contribution from vertex-corrections, or the so-called
$\epsilon^{\prime}$-type or direct CP violation. Since a larger CP asymmetry
results in larger baryon asymmetry, hence in this case one can allow larger
entropy injection compared to the standard vanilla leptogenesis scenario. This
in turn helps in relaxing the bound on DM mass, allowing heavier masses
compared to the vanilla leptogenesis scenario. This is what we can see from
the right panel of Fig. 8, where we find the allowed parameter space
corresponds to $\Delta S\gtrsim\mathcal{O}(100)$. Note that for $\Delta
S\gtrsim 100$, the WDM limit becomes relatively relaxed and thus the whole
parameter space opens up for correct DM relic.

##  5 Conclusions

We have proposed a scenario where gravitational dark matter is produced from
evaporating primordial black holes. Except for the superheavy mass window
$m_{\text{DM}}\gtrsim 10^{10}$ GeV, DM production solely from PBH evaporation
leads to overabundance if PBH dominates the energy density of the universe at
early epochs. While lighter mass window $m_{\text{DM}}\lesssim\mathcal{O}(1)$
keV gives correct relic, it faces severe constraints from the requirement of
structure formation. We particularly focus on this keV to $10^{10}$ GeV mass
window of gravitational DM and incorporate it within a Type-I seesaw framework
with three right handed neutrinos responsible for generating light neutrino
masses. While DM in this mass window gets overproduced from PBH evaporation,
late entropy injection from decay of one of the RHNs (acting as a diluter) can
bring the DM abundance within observed limits. As the late entropy dilution
must occur before the epoch of BBN in order not to disturb the successful
prediction for light nuclei abundance, we impose the upper bound on diluter
lifetime which gets translated into an upper bound on entropy injection,
allowing DM mass upto $\sim 1$ PeV. Along with DM, the diluter also gets
dominantly produced from PBH evaporation as its couplings to the SM particles
remain suppressed from the requirement of long lifetime needed for sufficient
entropy release due to its decay. This effectively leads to two different
stages of early matter domination: first from PBH and then from the diluter.
The other two RHNs can have sizeable couplings with SM leptons thereby
generating the required neutrino mass and mixing. We consider the production
of these RHNs both from the bath as well as PBH and show that their subsequent
CP violating out-of-equilibrium decays can lead to successful leptogenesis as
well. Since the lepton asymmetry is required to be overproduced initially in
order to survive the subsequent entropy dilution, the DM parameter space gets
squeezed from a few keV-PeV window to a smaller range around MeV-GeV ballpark
from the requirement of producing both DM relic as well as the baryon
asymmetry of the universe, with PBH mass $m_{\text{BH}}\lesssim 5$ g. The
long-lived nature of $N_{3}$ necessarily pushes the lightest active neutrino
mass $m_{\nu}^{1}$ to vanishingly small values ($\lesssim 10^{-10}$ eV). Thus,
the effective neutrino mass will be very much out of reach from ongoing
tritium beta decay experiments like KATRIN [102], whereas any positive
detection of $m_{\nu}^{1}$ in future experiments might be able to falsify our
scenario. While gravitational DM has no scope of direct detection, the
required PBH mass as well as multiple stages of early matter domination can
have interesting observable consequences, specially in the context of
gravitational wave observations [103, 104, 105, 92, 106, 107, 108, 45], which
we leave for future studies.

##  Appendix A Coupled Boltzmann Equations

The relevant coupled Boltzmann equations (BEQ) in the present framework reads
[81, 83, 85]

$\displaystyle\frac{dm_{\text{BH}}}{da}=-\frac{\kappa}{a\,\mathcal{H}}\,\epsilon(m_{\text{BH}})\,\left(\frac{1\text{g}}{m_{\rm
BH}}\right)^{2}\,,$
$\displaystyle\frac{d\widetilde{\rho}_{R}}{da}=-\frac{\epsilon_{\text{SM}}(m_{\text{BH}})}{\epsilon(m_{\text{BH}})}\,\frac{a}{m_{\text{BH}}}\,\frac{dm_{\text{BH}}}{da}\,\widetilde{\rho}_{\text{BH}}+\frac{a}{\mathcal{H}}\Gamma_{3}M_{3}\widetilde{n}_{N_{3}}^{\text{BH}}\,,$
$\displaystyle\frac{d\widetilde{\rho}_{\text{BH}}}{da}=\frac{1}{m_{\text{BH}}}\,\frac{dm_{\text{BH}}}{da}\,\widetilde{\rho}_{\text{BH}}\,,$
$\displaystyle
a\mathcal{H}\frac{d\widetilde{n}_{N_{1}}^{T}}{da}=-\left(\widetilde{n}_{N_{1}}^{T}-\widetilde{n}_{N_{1}}^{\text{eq}}\right)\,\Gamma^{T}_{N_{1}}\,,$
$\displaystyle
a\mathcal{H}\frac{d\widetilde{n}_{N_{1}}^{\text{BH}}}{da}=-\widetilde{n}_{N_{1}}^{\text{BH}}\,\Gamma_{N_{1}}^{\text{BH}}+\Gamma_{\text{BH}\to
N_{1}}\,\frac{\widetilde{\rho}_{\text{BH}}}{m_{\text{BH}}}\,,$ (A.1)

along with

$\displaystyle a\mathcal{H}\frac{d\widetilde{N}_{B-L}}{da}=\epsilon_{\Delta
L}\,\Biggl{[}\left(\widetilde{n}_{N_{1}}^{T}-\widetilde{n}_{N_{1}}^{\text{eq}}\right)\,\Gamma^{T}_{N_{1}}+\widetilde{n}_{N_{1}}^{\text{BH}}\,\Gamma_{N_{1}}^{\text{BH}}\Biggr{]}-\mathcal{W}\,\widetilde{N}_{B-L}\,,$
$\displaystyle
a\mathcal{H}\frac{d\widetilde{n}_{N_{3}}^{\text{BH}}}{da}=\Gamma_{\text{BH}\to
N_{3}}\,\frac{\widetilde{\rho}_{BH}}{m_{BH}}-\Gamma_{3}\widetilde{n}_{N_{3}}^{\text{BH}}$
$\displaystyle
a\mathcal{H}\frac{d\widetilde{n}_{DM}^{\text{BH}}}{da}=\Gamma_{\text{BH}\to
DM}\,\frac{\widetilde{\rho}_{BH}}{m_{BH}}$
$\displaystyle\frac{dT}{da}=-\frac{T}{\Delta}\Biggl{[}\frac{1}{a}+\frac{\epsilon_{\text{SM}}(m_{\text{BH}})}{\epsilon(m_{\text{BH}})}\,\frac{1}{m_{\text{BH}}}\,\frac{dm_{\text{BH}}}{da}\,\frac{g_{\star}(T)}{g_{\star
s}(T)}\,a\,\frac{\widetilde{\rho}_{BH}}{4\,\widetilde{\rho}_{R}}+\frac{\Gamma_{3}M_{3}}{3\mathcal{H}~{}s~{}a^{4}}\widetilde{n}_{N_{3}}^{\text{BH}}\Biggr{]}\,.$
(A.2)

Here, $\widetilde{\rho}_{R}$ and $\widetilde{\rho}_{\text{BH}}$ denotes the
comoving energy density of radiation and PBH, $\mathcal{H}$ is the Hubble
parameter, whereas $\widetilde{n}_{N1}$, $\widetilde{n}_{N3}$ and
$\widetilde{n}_{DM}$ represent the comoving number densities of $N_{1}$,
$N_{3}$ and DM respectively with $\mathcal{W}$ being the term responsible for
washout. Suffix ‘T’ and ‘BH’ denotes the thermal and non-thermal contributions
respectively. Note that we are considering a scenario of DM with only
gravitational interactions, hence it doesn’t get produced from the thermal
bath. Also, as we will see, $N_{3}$ plays the role of diluting the DM relic
and hence have to be long-lived. This results in suppressed Yukawas and hence
no thermal production. Finally, $\widetilde{N}_{B-L}$ indicates the comoving
B-L asymmetry.

##  Appendix B Lyman-$\alpha$ Constraint

Due to their large initial momentum, DM particles could have a large free-
streaming length leading to a suppression on the structure formation at small
scales. In the present scenario where DM has no interactions with the SM or
with itself, the DM momentum simply redshifts, and its value $p_{0}$ at
present is [78]

$p_{0}=\frac{a_{\text{evap}}}{a_{0}}\,p_{\text{evap}}\simeq\frac{a_{\text{evap}}}{a_{\text{eq}}}\frac{\Omega_{R}}{\Omega_{m}}\,\langle
E_{\text{evap}}\rangle=\langle
E_{\text{eq}}\rangle\,\frac{\Omega_{R}}{\Omega_{m}}\,,$ (B.1)

where

$\langle E_{\text{eq}}\rangle=\langle
E_{\text{evap}}\rangle\,\frac{a_{\text{evap}}}{a_{\text{eq}}}\approx\frac{T_{\text{BH}}}{\xi}\,\frac{T_{\text{eq}}}{T_{\text{evap}}}\,\left[\frac{g_{\star,s}(T_{\text{eq}})}{g_{\star,s}(T_{\text{evap}}}\right]^{\frac{1}{3}}\,,$
(B.2)

where in the last line we have assumed entropy injection from PBH evaporation
to matter-radiation equality:
$\xi\,\left(sa^{3}\right)_{\text{evap}}=\left(sa^{3}\right)_{\text{eq}}$. A
lower bound on the DM mass can be obtained from the upper bound on a typical
velocity of warm DM at the present time. Taking $v_{\text{DM}}\lesssim
1.8\times 10^{-8}$ [61] for $m_{\text{DM}}\simeq 3.5$ keV [94], we obtain

$\displaystyle m_{\text{DM}}\gtrsim 10^{4}\,\langle E_{\text{eq}}\rangle\equiv
10^{4}\,\frac{T_{\text{BH}}}{\xi}\,\frac{T_{\text{eq}}}{T_{\text{evap}}}\,\left[\frac{g_{\star,s}(T_{\text{eq}})}{g_{\star,s}(T_{\text{evap}}}\right]^{\frac{1}{3}}\,,$
(B.3)

where $a_{\text{eq}}\equiv\frac{\Omega_{R}}{\Omega_{m}}\simeq 1.8\times
10^{-4}$.

##  Appendix C Casas-Ibarra Parametrization

As the neutral component of the SM Higgs doublet acquires a VEV leading to the
spontaneous breaking of the SM gauge symmetry, neutrinos in the SM obtain a
Dirac mass that can be written as

$\displaystyle m_{D}=\frac{y_{N}}{\sqrt{2}}v.$ (C.1)

The Dirac mass $m_{D}$ together with the RHN bare mass $M_{N}$, can explain
the nonzero light neutrino masses with the help of Type-I seesaw [31, 29]
where the light-neutrino masses can be expressed as,

$\displaystyle m_{\nu}\simeq m_{D}^{T}~{}M^{-1}~{}m_{D}.$ (C.2)

The mass eigenvalues and mixing are then obtained by diagonalising the light-
neutrino mass matrix as

$\displaystyle m_{\nu}=\mathcal{U}^{*}m_{\nu}^{d}\mathcal{U}^{\dagger}\,,$
(C.3)

with $m_{\nu}^{d}=dia(m_{1},m_{2},m_{3})$ consisting of the mass eigenvalues
and $\mathcal{U}$ representing the Pontecorvo-Maki-Nakagawa-Sakata matrix
[1]777The charged lepton mass matrix is considered to be diagonal.. One can
parameterise the complex structure of the neutrino Yukawa matrix that is
required for generating the CP violating decays of the RHNS using Casas-Ibarra
(CI) parametrisation [109] as,

$\displaystyle
y_{N}=\frac{\sqrt{2}}{v}\sqrt{M}~{}\mathbb{R}~{}\sqrt{m_{\nu}^{d}}~{}\mathcal{U}^{\dagger}\,,$
(C.4)

where $\mathbb{R}$ is a complex orthogonal matrix
$\mathbb{R}^{T}\mathbb{R}=I$, which we choose as

$\displaystyle\mathbb{R}=\begin{pmatrix}0&\cos{z}&\sin{z}\\\
0&-\sin{z}&\cos{z}\\\ 1&0&0\end{pmatrix}\,,$ (C.5)

where $z=a+ib$ is a complex angle. The diagonal light neutrino mass matrix
$m_{\nu}^{d}$ is calculable using the best fit values of solar and atmospheric
mass obtained from the latest neutrino oscillation data [1]. Now, the elements
of Yukawa coupling matrix $y_{N}$ for a specific value of $z$, can be obtained
for different choices of the heavy neutrino masses. For example, with
$M_{1}=10^{13}$ GeV, $M_{2}=50\,M_{1}$, $M_{3}=10^{9}$ GeV,
$m_{\nu}^{1}=10^{-15}$ eV and $\\{a,b\\}=\\{2.77,0.36\\}$ we obtain the
following structure

$\displaystyle
y_{N}=\left(\begin{array}[]{ccc}-0.037\,-0.007\,i&-0.002\,-0.031\,i&0.064\,-0.026\,i\\\
-0.033\,+0.197\,i&-0.682\,+0.0126\,i&-0.592\,-0.152\,i\\\ 1.508\times
10^{-10}&-6.764\times 10^{-11}-1.313\times 10^{-11}i&7.30\times
10^{-11}-1.490\times 10^{-11}\,i\end{array}\right)\,,$ (C.9)

which satisfies the light neutrino mass, as well as produces desired CP
asymmetry.

## Acknowledgements

BB received funding from the Patrimonio Autónomo - Fondo Nacional de
Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José
de Caldas (MinCiencias - Colombia) grant 80740-465-2020. This project has
received funding /support from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No
860881-HIDDeN. RR was supported by the National Research Foundation of Korea
(NRF) grant funded by the Korean government (NRF-2020R1C1C1012452).

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