# Superradiant Leptogenesis

Anish Ghoshal  Yuber F. Perez-Gonzalez  Jessica Turner

###### Abstract

We investigate how superradiance affects the generation of baryon asymmetry in
a universe with rotating primordial black holes, considering a scenario where
a scalar boson is coupled to the heavy right-handed neutrinos. We identify the
regions of the parameter space where the scalar production is enhanced due to
superradiance. This enhancement, coupled with the subsequent decay of the
scalar into right handed neutrinos, results in the non-thermal creation of
lepton asymmetry. We show that successful leptogenesis is achieved for masses
of primordial black holes in the range of order $O(0.1~{}{\rm g})-O(10~{}{\rm
g})$ and the lightest of the heavy neutrino masses, $M_{N}\sim
O(10^{12})~{}{\rm GeV}$. Consequently, regions of the parameter space, which
in the case of Schwarzchild PBHs were incompatible with viable leptogenesis,
can produce the observed matter-antimatter asymmetry.

##  1 Introduction

The origin of the matter-antimatter asymmetry remains one of the most
compelling and challenging open problems in cosmology and modern particle
physics. Detailed studies of antiparticles in cosmic rays [1], big-bang
nucleosynthesis (BBN) measurements [2] and very precise understanding of the
cosmic microwave background radiation (CMBR) from the Planck satellite [3]
provide us with stringent constraints on the number density of baryons per
entropy density, $Y_{B}=\frac{n_{b}-n_{\bar{{b}}}}{s}=8.8\times 10^{-11}$. To
explain such a matter-antimatter asymmetry dynamically, a baryogenesis
mechanism has to satisfy the three well-known (and necessary) Sakharov’s
conditions [4]: (i) baryon number B-violation, (ii) violation of C- and CP-
symmetries, and (iii) having particle interactions out of thermal equilibrium.
While there are many proposed mechanisms of baryogenesis, the most well-
studied are electroweak baryogenesis [5, 6, 7] and baryogenesis via
leptogenesis [8] from the decays of heavy right handed (RH) neutrinos which
are introduced as extension to the Standard Model (SM) to obtain light
neutrino masses via Type-I seesaw mechanism [9, 10, 11, 12].

It is well known that non-standard cosmologies affect these baryogenesis
mechanisms [13, 14, 15, 16, 17]. Primordial Black Holes (PBH) [18, 19] are a
rich non-standard cosmology that has received renewed interest after the
discovery of Gravitational Waves (GW) from black hole mergers by the LIGO-
Virgo-KAGRA collaboration [20, 21, 22, 23, 24, 25]. With masses as tiny as
$M_{\mathrm{BH}}\simeq 0.1$ g up to a few hundreds of solar masses, PBHs may
lead to interesting physical signatures111Long-lived PBHs with mass
($M_{\mathrm{BH}}\gtrsim 10^{15}$g) that may survive until today may
contribute to a significant energy budget of Dark Matter (DM)[26, 27, 28, 29,
30, 31, 32, 33, 34]. In particular, ultralight PBHs with
$M_{\mathrm{BH}}\lesssim 10^{9}$g, which completely evaporate before BBN
($T\sim 4$ MeV) can affect the production of dark matter [35, 36, 37, 38] or
dark radiation [39, 40], the generation of baryon asymmetry [41, 42, 43],
resonant GW spectrum induced by scalar perturbations [44], GW from cosmic
strings [45], GWs induced by ultralight primordial black hole domination and
evaporation [46, 47] and study of vacuum stability of the Standard Model (SM)
Higgs [48, 49]. While intermediate-mass black holes $10^{9}{\rm g}\lesssim
M_{\mathrm{BH}}\lesssim 10^{15}$g are subject to several constraints (e.g.,
from BBN [50, 51, 52]) ultralight PBHs ($0.1{\rm g}\lesssim
M_{\mathrm{BH}}\lesssim 10^{9}$g) face practically no cosmological or
astrophysical constraints. As such, we are free to consider their interplay
with new physics scenarios and how the presence of such light PBHs affects the
viable parameter space of the new physics model.

PBHs may affect leptogenesis in two primary ways: (a) the production of heavy
right handed neutrinos via the Hawking evaporation, which enhances the
efficiency of baryogenesis via leptogenesis in some areas of the parameter
space [41], and (b) entropy injection into the primordial plasma, which
somewhat reduces the efficiency of leptogenesis [42, 53, 54]. The black hole-
induced leptogenesis scenarios, based on a type-I seesaw mechanism [12, 10,
55, 9], have been well investigated in the literature [42, 56, 41, 53, 57] and
it was shown that how these two effects impact leptogenesis. Generally
speaking, heavy PBHs with initial masses exceeding
$\gtrsim\mathcal{O}(\rm{kg})$ are typically not efficient at producing heavy
right handed neutrinos with masses exceeding $10^{6}\,\rm{GeV}$, which are
suitable for thermal leptogenesis but instead lead to large entropy production
[42, 53]. Due to this, there is significant tension between such ultra-light
PBH-dominated pre-BBN Universe and thermal leptogenesis. However, this picture
changes when an initially spinning PBH and the presence of a scalar, which
couples to the RH neutrinos, is taken into account due to the effects of
superradiant instabilities. This is the subject of the present analysis.

More specifically, we consider a scenario where the RH neutrinos are coupled
to a scalar, for instance, in a Majoron model. The final lepton asymmetry
sourced from the PBHs is affected firstly due to the production of the scalar
and right handed neutrinos directly from PBH evaporation via Hawking
radiation, and their subsequent decays can source a lepton asymmetry.
Secondly, the PBHs are very efficient at producing photons, and such entropy
injections usually dilute the existing lepton asymmetry. Finally, and most
importantly, since we consider rotating PBHs, some of the energy and angular
momentum of the PBHs will be extracted to form bosonic clouds [58, 59, 60, 61,
62] of the scalar that couples to the right handed neutrinos. The subsequent
decays of this scalar can produce a lepton asymmetry leading to what we call
_superradiant leptogenesis_. We study the interplay of these three processes
and identify the regions of the parameter space that superradiance can impact
the final matter-antimatter asymmetry of the Universe. For other scenarios
involving superradiant production of SM and beyond-the-SM particles, see e.g.
[63, 64, 65, 66, 67]

The structure of this work is as follows: in Section 2, we discuss the Majoron
model we consider and follow in Section 3 with a short review of the relevant
primordial black hole physics and superradiance. In Section 4, we present the
Friedmann and Boltzmann equations of superradiant leptogenesis and discuss the
solutions of these equations in Section 5. Finally, we summarise and conclude
in Section 7. In this work, we consider natural units where $\hbar=c=k_{\rm
B}=1$, and define the Planck mass to be $M_{p}=1/\sqrt{G}=1.22\times
10^{19}\,{\rm GeV}$, with $G$ the gravitational constant.

##  2 Spontaneous $B-L$ symmetry breaking

Throughout, we consider a specific model that includes all the essential
components for superradiant leptogenesis. It extends the SM to include right
handed neutrinos and an additional scalar boson denoted as $\phi$. Notably,
$\phi$ will be produced via superradiance and is also responsible for
generating the RH neutrino masses. Specifically, the RH neutrino fields,
denoted as $N$, acquire heavy Majorana masses through the spontaneous breaking
of the global $B-L$ symmetry by $\phi$, often referred to as the Majoron. To
achieve this, we introduce an SM singlet complex scalar field called $\sigma$,
which carries a $B-L$ charge of $-2$. The Lagrangian is as follows:

$\displaystyle{\cal L}$ $\displaystyle={\cal
L}_{SM}+i\bar{N}\not{\partial}N+|\partial_{\mu}\sigma|^{2}-\left(Y\overline{L}\tilde{H}N+\frac{g}{2}\sigma\overline{N^{C}}N+\text{h.c}\right)-V(H,\sigma)\,,$
(2.1) $\displaystyle V(H,\sigma)$
$\displaystyle=\lambda_{H}(|H|^{2}-v_{ew}^{2})^{2}+\lambda_{\sigma}(|\sigma|^{2}-v_{B-L}^{2})^{2}+\lambda_{\sigma
H}(|\sigma|^{2}-v_{B-L}^{2})(|H|^{2}-v_{EW}^{2})\,,$

where $L$ is the leptonic doublet, $H$ denotes the Higgs doublet, and
$\tilde{H}=i\sigma_{2}H^{*}$. The symbols $Y$, $g$, $\lambda_{H}$,
$\lambda_{\sigma H}$, and $\lambda_{\sigma}$ represent dimensionless coupling
constants. Additionally, we have two dimensionful parameters, $v_{EW}$ and
$v_{B-L}$, which denote the electroweak and $B-L$ symmetry-breaking scales,
respectively. Notably, $g$, which parameterises the strength of the scalar
coupling to the RH neutrinos, will play a key role in the superradiant
leptogenesis process from PBH. Around the ${B-L}$ breaking vacuum, we
parameterise $\sigma$ by

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle(v_{B-L}+\rho/\sqrt{2})e^{i\phi/(\sqrt{2}v_{B-L})}\,,$ (2.2)

where $\rho$ is a real part of $\sigma$, and $\phi$ corresponds to the Nambu-
Goldstone boson, i.e. the Majoron. After $B-L$ symmetry breaking, $N$ acquires
the Majorana mass $M_{N}=gv_{B-L}$ and the Majoron acquiring a mass of order
$M_{\phi}\sim\lambda_{\sigma}v_{B-L}$ in the limit the Majoron-Higgs coupling
is vanishing, $\lambda_{\sigma H}=0$. Also, this symmetry breaking leads to
the formation of global cosmic strings which results in the generation of
gravitational waves as we will discuss in Section 6.2. After electroweak
symmetry breaking, the familiar type-I seesaw mechanism [9, 55, 10, 12]
provides mass to the light neutrinos, $m_{\nu}=v^{2}YM_{N}^{-1}Y^{T}$. In what
follows, we will assume that the Majoron mass always exceeds twice the right
handed neutrino masses, $M_{\phi}>2M_{N_{1}}$ where $M_{N_{1}}$ is the mass of
the lightest RH neutrino. While at least two RH neutrinos are required for the
successful generation of SM neutrino masses via type-I seesaw; we assume that
$M_{\phi}<M_{N_{2}},M_{N_{3}}$. Hence, the decays of $\phi$ to $N_{1}$ are
kinematically allowed while suppressed to $N_{2}$ and $N_{3}$. The decay width
of $\phi$ to RH neutrinos is

$\displaystyle\Gamma_{\phi\to NN}$ $\displaystyle=$
$\displaystyle\frac{g^{2}M_{\phi}}{16\pi}\sqrt{1-\frac{4M_{N}^{2}}{M_{\phi}^{2}}}\,,$
(2.3)

where, for simplicity, we denote $N_{1}\equiv N$ and $M_{N_{1}}\equiv M_{N}$
throughout the remainder of the paper. Let us note, however, that there could
be couplings between the Majoron and the other SM degrees of freedom. To
parametrise such a possibility in a model-independent way, we define the
branching ratio

$\displaystyle{\rm BR}\equiv\frac{\Gamma_{\phi\to{NN}}}{\Gamma_{\phi}},$ (2.4)

with $\Gamma_{\phi\to{\rm NN}}$ the decay width into RH neutrinos, and
$\Gamma_{\phi}$ the total $\phi$ decay width. Essentially, ${\rm BR}$ is
dependent on the parameter $\lambda_{\sigma H}$, so increasing or decreasing
$\lambda_{\sigma H}$ will increase or decrease ${\rm BR}$. From here onwards,
we treat the only independent variables for our analysis to be $M_{N}$,
$M_{\phi}$, $g$ and ${\rm BR}$.

##  3 Rotating Primordial Black Holes: Hawking Radiation and Superradiance

The formation PBHs during the Universe’s early stages of the Universe’s
evolution [68, 18] may occur when density fluctuations exceed a certain
threshold and reenter the Hubble horizon and collapse [69]. Alternatively, PBH
may form from the collapse of cosmic string loops, domain walls, or the
collision of bubbles [70]. Since we are interested in superradiance studies,
initially spinning PBH can be formed during aspherical collapse or in the
presence of large inhomogeneities during a matter-domination era, as shown in
Refs. [71, 72] or from collapse of domain walls involving gravitational
torques [73] or from strong first-order phase transitions [74]. We assume that
an initial density of PBHs was generated during a radiation-dominated period
after inflation. Considering the initial plasma temperature to be $T=T_{\rm
in}$, the initial monochromatic mass of the PBH, $M_{\rm in}$, is proportional
to the horizon mass [75, 76]:

$M_{\mathrm{in}}\equiv
M_{\mathrm{BH}}\left(T_{\mathrm{in}}\right)=\frac{4\pi}{3}\kappa\frac{\rho_{R}\left(T_{\mathrm{in}}\right)}{H^{3}\left(T_{\mathrm{in}}\right)}\,,$
(3.1)

where $\kappa=0.2$, $\rho_{R}\left(T_{\mathrm{in}}\right)$ is radiation energy
density at the time of PBH formation and $H$ is the Hubble rate at the time of
PBH formation. The initial energy density of PBHs can then be parametrised in
terms of

$\beta\equiv\frac{\rho_{\mathrm{BH}}\left(T_{\mathrm{in}}\right)}{\rho_{R}\left(T_{\mathrm{in}}\right)}=\frac{n_{\rm
BH}M_{\mathrm{in}}}{\rho_{R}\left(T_{\mathrm{in}}\right)}\,,$ (3.2)

where $\rho_{\mathrm{BH}}\left(T_{\mathrm{in}}\right)$ is the energy density
of the PBHs at the formation time, and $n_{\rm BH}$ is the initial number
density of PBHs. We assume that the population of PBHs are Kerr black holes
that carry non-zero total angular momentum, $J$ [77, 78, 73, 79, 80, 81, 82]
which further is characterised by the dimensionless quantity,
$a_{\star}=JM_{p}^{2}/M_{\rm BH}^{2}$ where $M_{\rm BH}$ the instantaneous PBH
mass. The PBH population is assumed to have an initial monochromatic spin
denoted by $a_{\star}^{\rm in}$.

The Hawking emission rate for any particle species is given by [83, 84]:

$\frac{\mathrm{d}^{2}\mathcal{N}_{i}}{\mathrm{~{}d}p\mathrm{~{}d}t}=\frac{g_{i}}{2\pi}\sum_{l=s_{i}}\sum_{m=-l}^{l}\frac{\mathrm{d}^{2}\mathcal{N}_{ilm}}{\mathrm{~{}d}p\mathrm{~{}d}t}\,,$
(3.3)

with

$\frac{\mathrm{d}^{2}\mathcal{N}_{ilm}}{\mathrm{~{}d}p\mathrm{~{}d}t}=\frac{\Gamma_{s_{i}}^{lm}\left(M,p,a_{\star}\right)}{e^{\left(E_{i}-m\Omega\right)/T_{\mathrm{BH}}}-(-1)^{2s_{i}}}\frac{p}{E_{i}}\,,$
(3.4)

where individual particle species $i$ is characterised by its three-momentum
$p$, total energy $E_{i}$, and internal degrees of freedom $g_{i}$, spin
$s_{i}$. $l$ and $m$ are the total and axial angular momentum quantum numbers,
respectively, and $\Omega$ represents the angular velocity of the black hole’s
horizon and is given by $\Omega=(a_{\star}/2GM_{\rm
BH})(1/(1+\sqrt{1-a_{\star}^{2}}))$. Finally,
$\Gamma_{s_{i}}^{lm}\left(M,p,a_{\star}\right)$ in the Hawking spectrum
represents the absorption probability of the $l,m$ mode. The evolution of the
spin and mass of a PBH without including superradiant effects is governed by
the following coupled equations [85, 86, 87, 36]

$\displaystyle\frac{dM_{\rm BH}}{dt}$ $\displaystyle=-\varepsilon\left(M_{\rm
BH},a_{\star}\right)\frac{M_{p}^{4}}{M_{\rm BH}^{2}}\,,$ (3.5)
$\displaystyle\frac{da_{\star}}{dt}$
$\displaystyle=-a_{\star}\left[\gamma\left(M_{\rm
BH},a_{\star}\right)-2\varepsilon\left(M_{\rm
BH},a_{\star}\right)\right]\frac{M_{p}^{4}}{M_{\rm BH}^{3}}\,,$ (3.6)

where $\varepsilon\left(M_{\rm BH},a_{\star}\right)$ and $\gamma\left(M_{\rm
BH},a_{\star}\right)$ are dimensionless evaporation functions related to the
power and torque emission, respectively, and are obtained numerically via, see
Refs. [36, 37]

$\displaystyle\varepsilon(M_{\rm BH},a_{\star})$
$\displaystyle=\sum_{i}\frac{g_{i}}{2\pi}\int_{0}^{\infty}\sum_{l=s_{i}}^{\infty}\sum_{m=-l}^{l}E\frac{\mathrm{d}^{2}\mathcal{N}_{ilm}}{\mathrm{d}p\mathrm{d}t}\,dE\,,$
(3.7) $\displaystyle\gamma(M_{\rm BH},a_{\star})$
$\displaystyle=\sum_{i}\frac{g_{i}}{2\pi}\int_{0}^{\infty}\sum_{l=s_{i}}^{\infty}\sum_{m=-l}^{l}m\frac{\mathrm{d}^{2}\mathcal{N}_{ilm}}{\mathrm{d}p\mathrm{d}t}\,dE\,.$
(3.8)

In addition to Hawking evaporation being an important source of particle
production, superradiance will play a key role in our scenario of interest.

Superradiance is an enhancement phenomenon that occurs for rotating black
holes, among other physical systems [88]. In a nutshell, a superradiant
instability occurs when the occupation number of massive bosonic fields
gravitationally bound to a black hole grows exponentially [58, 59, 60, 61,
88]. Some conditions are required for such a phenomenon to take place. First,
the Compton wavelength of a massive bosonic field should be comparable to the
BH gravitational radius in order for a bound state to be formed. This can be
parametrised via the gravitational fine-structure constant, which indicates
the strength of the gravitational binding of such a state [88, 89]

$\alpha=\frac{M_{\phi}M_{\mathrm{BH}}}{M_{p}^{2}}\approx
0.38\left(\frac{M_{\rm BH}}{10~{}{\rm
g}}\right)\left(\frac{M_{\phi}}{10^{13}~{}{\rm GeV}}\right),$ (3.9)

which naturally scales with the mass of the bosonic field and black hole.

The behaviour of the bounded state can be understood by solving the equation
of motion of the gravitationally coupled scalar field in the Kerr spacetime,
see e.g. Ref. [90]. The energies of these _quasinormal_ states will be
complex, $\omega=\omega_{R}+i\omega_{I}$, where the imaginary part indicates
whether the bound states decays, $\omega_{I}<0$, or if the occupation number
grows exponentially, $\omega_{I}>0$ [91, 92, 93, 94, 88, 90, 95, 96, 97, 62,
98, 99, 100]. Once the conditions for the formation of bound states are met, a
second requirement for superradiant instability to occur is that the real part
of the energy $\omega_{R}$ must be less than a critical value $\omega_{c}$,
$\omega_{R}<\omega_{c}$, with

$\displaystyle\omega_{c}=m\Omega\,,$ (3.10)

where $m$ is the azimuthal quantum number. Defining
$\Gamma_{\mathrm{sr}}=\omega_{I}$ as the decay or growth rate, we can
determine the time evolution of the number of particles $\mathcal{N}_{\phi}$
in the bound state through

$\displaystyle\frac{\mathrm{d}\mathcal{N}_{\phi}}{\mathrm{d}t}=\Gamma_{\mathrm{sr}}\,\mathcal{N}_{\phi}\,,$
(3.11)

where we have neglected, at this moment, the possibility that the scalar field
$\phi$ could decay into other particles. Moreover, note that if the particles
$\phi$ have significant self-interactions, the time evolution of the cloud
will be significantly modified, see Ref. [101]. For simplicity, we neglect the
effect of $\phi$ self-interactions. The generation of these particles depletes
the BH mass and spin. Using energy conservation arguments, we have the system
of equations for $M_{\rm BH}$ and $a_{\star}$,

$\displaystyle\frac{\mathrm{d}M_{\rm BH}}{\mathrm{d}t}$
$\displaystyle=-M_{\phi}\Gamma_{\mathrm{sr}}\,\mathcal{N}_{\phi}\,,$ (3.12a)
$\displaystyle\frac{\mathrm{d}a_{\star}}{\mathrm{d}t}$
$\displaystyle=-(\sqrt{2}-2\alpha
a_{\star})\Gamma_{\mathrm{sr}}\frac{\mathcal{N}_{\phi}}{G^{2}M_{\rm
BH}^{2}}\,.$ (3.12b)

Although there exist analytical approximations to determine the growth rate
$\Gamma_{\mathrm{sr}}$ for different limits, we will apply a numerical method
that determines such a rate for all values of $\alpha$ and the spin parameter
$a_{\star}$. For such a purpose, we follow the procedure established in Ref.
[90] to solve the Klein-Gordon equation in the Kerr background. To summarise,
the procedure is as follows. Using an ansatz in the form of a power series to
solve the equations obtained after a variable separation, one obtains a system
of algebraic equations for the bound state frequencies. Employing a root
finder for such an algebraic system, we thus obtain the frequencies $\omega$
that yield a bound state for a given set of quantum numbers $l,m$. Finally,
the growth rate is obtained by taking the imaginary part of the bound state
frequencies. As a cross-check, we have compared our results for the growth
rate of instabilities with those in Ref. [90] and found a good agreement.

##  4 Boltzmann Equations for Superradiant Leptogenesis

For the PBH mass and spin values that we investigate, the scalar $\phi$ will
be sourced by both Hawking evaporation and superradiance. In the case that
such a scalar is unstable and decays into right handed neutrinos, we will have
an additional source of non-thermal leptogenesis. In this section, we
investigate the interplay between the two mechanisms. The set of coupled
Boltzmann equations that governs the evolution of this system is

$\displaystyle aH\frac{dn_{\phi}^{\mathrm{sr}}}{da}$
$\displaystyle=\Gamma_{\mathrm{sr}}n_{\phi}^{\mathrm{sr}}-\Gamma_{\phi\to
NN}n^{\mathrm{sr}}_{\phi}$ (4.1) $\displaystyle aH\frac{dM_{\mathrm{BH}}}{da}$
$\displaystyle=-\varepsilon\frac{M_{p}^{4}}{M_{\mathrm{BH}}^{2}}-M_{\phi}\Gamma_{\mathrm{sr}}n_{\phi}^{\mathrm{sr}}$
$\displaystyle aH\frac{da_{\star}}{da}$
$\displaystyle=-a_{\star}[\gamma-2\varepsilon]\frac{M_{p}^{4}}{M_{\mathrm{BH}}^{3}}-\left[\sqrt{2}-2\alpha
a_{\star}\right]\Gamma_{\mathrm{sr}}n_{\phi}^{\mathrm{sr}}\frac{M_{p}^{2}}{M_{\mathrm{BH}}^{2}}$
$\displaystyle aH\frac{dn_{N}^{\mathrm{TH}}}{da}$
$\displaystyle=-\left(n_{N}^{\mathrm{TH}}-n_{N}^{\mathrm{eq}}\right)\Gamma_{N}^{T}$
$\displaystyle aH\frac{dn_{N}^{\mathrm{BH}}}{da}$
$\displaystyle=-n_{N}^{\mathrm{BH}}\Gamma_{N}^{\mathrm{BH}}+n_{\mathrm{BH}}\Gamma_{\mathrm{BH}\rightarrow
N}-n_{N}^{\mathrm{BH}}\Gamma_{\phi}^{\mathrm{BH}}+n_{\mathrm{BH}}\Gamma_{\mathrm{BH}\rightarrow\phi\rightarrow
N}$ $\displaystyle aH\frac{dn^{\phi}_{N}}{da}$ $\displaystyle=2\Gamma_{\phi\to
NN}n_{\phi}^{\mathrm{sr}}-\Gamma_{N}n^{\phi}_{N}$ $\displaystyle
aH\frac{dn_{B-L}}{da}$
$\displaystyle=\epsilon\left[\left(n_{N}^{\mathrm{TH}}-n_{N}^{\mathrm{eq}}\right)\Gamma_{N}^{T}+n_{N}^{\mathrm{BH}}\Gamma_{N}^{\mathrm{BH}}+n^{\phi}_{N}\Gamma_{N}\right]-\mathcal{W}n_{B-L}\,,$

where $a$ is the scale factor, $n_{\phi}^{\mathrm{sr}}$,
$n_{N}^{\mathrm{TH}}$, $n_{N}^{\mathrm{BH}}$, $n^{\phi}_{N}$ and $n_{B-L}$
denote the comoving number densities of $\phi$ produced from superradiance,
right handed neutrinos produced from the thermal bath, Hawking radiation,
scalar decays, and $B-L$ asymmetry respectively. The first line describes the
evolution of total $\phi$ numbers due to superradiant instability, including
the possible decay of $\phi$. The first (gain) term on the right-hand side
(RHS), $\Gamma_{\mathrm{sr}}n_{\phi}^{\mathrm{sr}}$, corresponds to the $\phi$
production rate per BH. The second (loss) term denotes the rate of decay of
these scalars into right handed neutrinos where we defined the scalar decay
width, $\Gamma_{\phi\to NN}$ in Eq. (2.3), and we approximate that the scalar
$\phi$ decays at rest [64].

The second and third lines describe the evolution of PBH mass and spin
discussed in Section 3, where we included terms coming from evaporation and
superradiance, see Eqs. (3.5) and (3.12). The fourth tracks the number density
of the right handed neutrinos produced from the thermal plasma,
$n_{N}^{\mathrm{TH}}$, where $\Gamma_{N}^{T}$ and $n_{N}^{\text{eq}}$ are the
thermally averaged decay rate and the equilibrium abundance of the right
handed neutrinos, respectively. Likewise, the fifth line tracks the number
density of the right handed neutrinos produced from the Hawking radiation. In
particular, the RHS has several gain and loss terms: the second term denotes
the right handed neutrinos sourced by Hawking evaporation, while the first
term provides the decay rate. We note that $\Gamma_{N}^{\mathrm{BH}}$ is the
decay width of the right handed, to leptons and Higgses, corrected by an
average inverse time dilatation factor,
$\Gamma_{N}^{\mathrm{BH}}\approx{\mathcal{K}_{1}\left(z_{\mathrm{BH}}\right)}/{\mathcal{K}_{2}\left(z_{\mathrm{BH}}\right)}\Gamma^{\text{rf}}_{N}$
with $\mathcal{K}_{1,2}(z)$ are modified Bessel functions of the first and
second kind, respectively, and we defined
$z_{\mathrm{BH}}=M_{N}/T_{\mathrm{BH}}$. We note, however, that for the
equations’ full numerical solution, we use the Hawking spectrum to compute the
thermal average; see for further details Refs. [42, 41]. There is an
additional loss and gain term from the secondary decays of scalars produced
via Hawking evaporation given by the third and fourth terms, respectively, and
we detail these in Appendix A.

The sixth line shows the dynamic evolution of the number density of right-hand
neutrinos from scalar decays, where a factor of two accounts for the scalar
decays into two right handed neutrinos. Finally, the last line involves the
evolution of ($B-L$) asymmetry generated due to $N$ decays and washout, where
the latter is denoted as $\mathcal{W}$. The CP-asymmetry parameter is denoted
by $\epsilon$ and is a function of the Yukawa matrix, $Y$ (see Eq. (2.1)). We
note that the Yukawa matrix is given by the usual Casas-Ibarra parametrisation
$Y=1/v\left(U\sqrt{{m}_{\nu}}R^{T}\sqrt{M}\right)$ [102]. Further, we assume
that the light neutrinos are normally ordered and approximated that the two
lightest masses to be equal, $m_{1}\approx m_{2}$, since $\Delta
m^{2}_{21}\ll\Delta m^{2}_{31}$. This approximation implies that the R-matrix
is a function by two real, $z=x+iy$, where we fix $x=\pi/4$ and $y=0.44$,
which ensures we are always in the strong washout regime and that $\epsilon$
is maximised allowing for the most conservative bound to be placed. Finally,
we fix the leptonic mixing angles, mass squared splitting, and CP-violating
phase at their best-fit values from global fit data [103] and fix the Majorana
phases to be CP-conserving. This set of coupled equations, together with the
Hubble expansion rate, are numerically solved using the facilities of ULYSSES
[104, 105] to explore the parameter space of superradiant leptogenesis.

##  5  Leptogenesis via superradiance

Figure 1:  Evolution of RHN number densities normalised to the initial photon
number density and the baryon-to-photon ratio $\eta_{B}$ for two benchmark
points, $M_{N}=4.7\times 10^{12}~{}{\rm GeV},M_{\phi}=2.05M_{N},M_{\rm
in}=10~{}{\rm g}$ (left) and $M_{N}=2.5\times 10^{12}~{}{\rm
GeV},M_{\phi}=10M_{N},M_{\rm in}=3.5~{}{\rm g}$ (right). We present the
different species contributing to the baryon asymmetry: thermal (purple),
primary from evaporation (blue), secondary from evaporation (green), and
superradiance (red). The baryon-to-photon ratio is presented for the whole
contribution (black) and from thermal leptogenesis only (black dashed). We
assume $a_{\star}^{\rm in}=0.999$, and $\beta=10^{-4}$. Figure 2:  Left:
baryon-to-photon ratio normalised to the observed value in the RHN mass vs.
initial PBH mass plane. The black line indicates the parameters consistent
with the observed baryon asymmetry, while the white dashed indicates the
contribution from evaporation and the dotted blue from superradiance. We
assume $a_{\star}^{\rm in}=0.999$, and $\beta=10^{-4}$. Right: $\eta_{B}$
normalised to the observed value in the initial PBH fraction $\beta$ vs. mass
plane. The black line indicates the parameters consistent with the observed
baryon asymmetry, while the white dashed indicates the contribution from
evaporation and the dotted blue from superradiance, assuming $M_{N}=7\times
10^{12}~{}{\rm GeV}$. The shaded region indicates the parameters excluded by
GW production [106].

Two example solutions for the coupled system of differential equations of Eq.
(4.1) are shown in Fig. 1. In both plots, $a_{\star}^{\rm in}=0.999$,
$\beta=10^{-4}$ while the left (right) plot the RH neutrino mass is fixed at
$M_{N}=4.7\times 10^{12}$ GeV ($2.5\times 10^{12}$ GeV) and
$M_{\phi}=2.05\times M_{N}$ ($10\times M_{N}$) with initial mass of PBH
$M_{\rm in}=10$ g (3.5 g). Further, we fix the coupling of $\phi$ to the RH
neutrinos to be $g=10^{-4}$ throughout. These two benchmark points lead to the
observed baryon asymmetry once superradiance is included. The purple line
shows the thermally produced RH neutrinos, while the blue and green lines show
the direct production of RH neutrinos from Hawking evaporation and those
produced from Hawking evaporation sourced $\phi$s that decay into RH
neutrinos. The red line shows the RH neutrino population sourced from the
superradiantly produced $\phi$ decays. Finally, the solid (dashed) lines
represent the total (solely thermal) baryon asymmetry. We observe from both
plots that the primary RH neutrino production via Hawking evaporation follows
the typical scaling, with a rapid increase in the production at the end of the
PBH evaporation. The scenario with the lighter PBH (right) shows a more
efficient production of RH neutrinos since the PBH temperature is higher. The
secondary production occurs at later times since this relies on the initial
production of the scalar $\phi$. The production of RH neutrinos from
superradiantly produced $\phi$ decays occurs at higher $z$ values for the
scenario with heavier $\phi$s (right plot) because a heavier $\phi$ leads to a
more significant superradiant instability growth rate and hence the bosonic
cloud forms, and decays, earlier. Moreover, since the right plot shows a
scenario where the scalar is an order of magnitude heavier than the RH
neutrino (as opposed to a factor of a few, as shown on the left plot), the
production time for RH neutrinos from this superradiance is shorter due to the
increase in the $\phi$ decay width. As the superradiant instability
exponentially amplifies $\phi$ production and its occupation, the baryon-to-
photon ratio (shown in solid black) increases by approximately two orders of
magnitude. However, this increase is diminished by the large entropy
injections towards the end of the PBHs lifetime. Therefore, superradiant
leptogenesis can increase the efficiency of leptogenesis in some areas of the
parameter space; however, the entropy dilution is always present, and we find
this increase in available parameter space is rather modest.

This increase in the parameter space is presented in Fig. 2 where the left
(right) plot shows the ratio of the predicted over the observed baryon-to-
photon ratio normalised to the observed is shown in the RH neutrino mass
($\beta$) vs initial PBH mass plane. The black line shows regions that are
consistent with the observed baryon asymmetry for PBH with $a_{\rm
ini}=0.999,\beta=10^{-4},$ GeV and $M_{\phi}=2.05M_{N}$. Each contribution
from Hawking evaporation and superradiance is also shown. We observe that the
effect of superradiant leptogenesis on increasing the viable parameter space
is localised in a horn-like region, indicated by the contour on the
$M_{N}-M_{\rm ini}$ plane that satisfies $\alpha
GM_{\mathrm{BH}}M_{\phi}\approx 0.2$ relation where superradiance is most
significant. From the right plot, this horn-like region is not sensitive to
$\beta$ (which parametrises the number density of the PBHs).

Figure 3:  Dependence of the regions consistent with the observed baryon
asymmetry on the branching ratio of the scalar into RHNs (left) and the mass
ratio between the scalar $\phi$ and the RHN mass $M_{N}$ (right). We assume
$a_{\star}^{\rm in}=0.999$, and $\beta=10^{-4}$.

The left plot of Fig. 3 shows the effect of varying the $\phi$ branching ratio
on the viable parameter space (red and green show the observed baryon
asymmetry produced for branching ratios of $1$ and $0.75$ respectively). As
expected, the parameter space shrinks for smaller branching ratios. Moreover,
the effect of increasing the mass difference between the Majoron and the RH
neutrino is shown on the right plot of Fig. 3. Naturally, increasing the mass
of the Majoron, for a fixed RHN mass, increases the decay width and this leads
to the more rapid decay of the superradiantly-produced Majoron cloud and
subsequently diminishes the viable parameter space.

So far, we have exclusively examined monochromatic distributions in both PBH
mass and angular momentum. The potential implications of more realistic
distributions within superradiant leptogenesis is worth considering.
Superradiance exhibits greater efficiency when $\alpha\sim 0.3$. Consequently,
introducing extended angular momentum distributions into our analysis is
expected to reduce the contribution to leptogenesis arising from superradiant
clouds. When we introduce extended spin distributions, we anticipate a notable
increase in the number of PBHs with spins below the nearly maximal value
$\alpha_{\star}^{\rm in}=0.999$ assumed previously. Consequently, fewer PBHs
will satisfy the necessary conditions to form $\phi$ superradiant clouds,
diminishing the overall effect we discussed earlier. However, if we consider
extended mass distributions, this could potentially mitigate such a reduction.
This is because we might have PBHs with masses greater than $M_{\rm in}$,
which could compensate for the smaller PBH spin values while maintaining
$\alpha$ constant. We nevertheless refrain from analysing this scenario
quantitatively, leaving it for future work.

##  6 Primordial Gravitational Waves probes of PBH leptogenesis

In this section, we explore the various ways PBHs contribute to generating
primordial gravitational waves, potentially leaving discernible imprints for
specific regions of the parameter space. Firstly, it is worth noting that the
formation of PBHs, characterised by significant curvature perturbations, can
induce gravitational waves, especially if these perturbations are of
inflationary origin [107] or from spectator fields present during inflation
[108, 109]. Secondly, PBHs are known to emit high-frequency gravitational
waves through Hawking evaporation, which serves as an additional source of
such waves [110]. Thirdly, the merger of PBHs results in the emission of
gravitational waves, thus presenting another avenue for GW production [111].
Fourthly, the evaporation of PBH may lead to unique GW spectral shapes emitted
via cosmic strings if present in the early universe [45, 112]. Finally, the
presence of inhomogeneities arising due to the distribution of PBHs also leads
to density fluctuations that can induce GWs in the form of isocurvature-
induced tensor perturbations [113, 114, 115] as well as resonant GW signals
arising due to first-order adiabatic scalar perturbation inducing second-order
gravitational waves due to PBH evaporation [46, 47]. Each of these
gravitational wave spectra have distinct characteristics, and we will focus
our discussion on the two primary sources most relevant to our specific
parameter space of interest.

###  6.1 Induced tensor perturbations

Once PBHs are formed, their distribution in space is random and act as sources
of inhomogeneities present during inflation in the form of isocurvature
perturbations [113]. If present in the early Universe, these inhomogeneities
lead to density fluctuations in the form of isocurvature perturbations. Later
on, when PBHs dominate the Universe’s energy density, the isocurvature
perturbations are converted to adiabatic perturbations that source tensor
perturbations at the second order. These tensor perturbations propagating as
GWs are probing the time of PBH formation. Next, the presence of adiabatic
scalar perturbation (either of inflationary origin or any other source) also
induces tensor perturbations and since these scalar modes are enhanced due to
the almost instantaneous evaporation of PBHs effectively changing the
background from matter-domination to radiation domination. This leads to
unique resonant GW enhance signals that may be detected. Combining these two
effects leads to a unique double-peaked GW spectrum with the present-day
dimensionless energy density given by [46, 47, 114, 116]

$\Omega_{\text{gw}}(t_{0},f)\simeq\Omega_{\text{gw}}^{\rm
peak}\left(\frac{f}{f^{\rm peak}}\right)^{11/3}\Theta\left(f^{\rm
peak}-f\right),$ (6.1)

where the peak of the spectrum is determined by the number density and initial
mass of the PBHs

$\Omega_{\text{gw}}^{\rm peak}\simeq 2\times
10^{-6}\left(\frac{\beta}{10^{-8}}\right)^{16/3}\left(\frac{M_{\mathrm{in}}}{10^{7}\rm
g}\right)^{34/9}.$ (6.2)

These analyses are valid only for distances larger than the mean separation
between PBHs, which in turn imposes an ultraviolet cutoff to the GW spectrum,
with $f_{\text{peak}}$ corresponding to the co-moving scale representing the
mean separation of PBHs at the time of PBH formation. Therefore appears the
$\Theta$-function in Eq. (6.1) with

$f^{\rm peak}\simeq 1.7\times 10^{3}\,{\rm
Hz}\,\left(\frac{M_{\mathrm{in}}}{10^{4}\rm g}\right)^{-5/6}.$ (6.3)

###  6.2 Gravitational Waves from global cosmic strings

The spontaneous breaking of the global $U(1)_{B-L}$ symmetry, as discussed in
Section 2, can lead to the generation of cosmic strings. After this symmetry
breaking, a network of horizon-size long strings can form whose length is
characterised by a correlation length $L=\sqrt{\mu/\rho_{\infty}}$ of loop
strings, where $\rho_{\infty}$ is the long-string energy density and $\mu$ is
the string tension. These strings can intersect and form loops that can
oscillate and generate gravitational waves whose energy density depends on the
string tension.

The cosmic string network can reach a scaling solution, where the string
energy density comprises a constant fraction of the Universe’s energy density.
As seen in several simulations, the size of a radiating loop at a cosmic time
$t$ is given by $l(t)=\alpha t_{i}-\Gamma G\mu(t-t_{i})$, where $l_{i}=\alpha
t_{i}$ is the initial size of the loop, $\Gamma\simeq 50$ [117], and
$\alpha\simeq 0.1$. The energy loss from a loop can written as a set of
normal-mode oscillations with frequencies $f_{k}=2k/l_{k}=a(t_{0})/a(t)f$,
where $k=1,2,3...\infty$. The $k$th mode GW density parameter is obtained as

$\displaystyle\Omega_{GW}^{(k)}(f)=\frac{2kG\mu^{2}\Gamma_{k}}{f\rho_{c}}\int_{t_{osc}}^{t_{0}}\left[\frac{a(t)}{a(t_{0})}\right]^{5}n\left(t,l_{k}\right)dt\,,$
(6.4)

where $t_{osc}$ is the oscillation time, $n\left(t,l_{k}\right)$ is the loop
number density which maybe estimated using the Velocity-dependent-One-Scale
(VOS) model. The quantity $\Gamma_{j}$ is given by $\Gamma_{j}=\frac{\Gamma
j^{-\delta}}{\zeta(\delta)}$, with $\delta=4/3$ for loops containing cusps
[118].

Detectable GWs from the global cosmic string loops arise from the most recent
epoch of cosmic evolution. The whole GW amplitude may grow with $\mu$, but in
the presence of a PBH-dominated era before the most recent radiation epoch at
$T=T_{\Delta}$, the GW spectrum at higher frequencies are given by
$\Omega_{\rm GW}^{(1)}(f\lesssim f_{\Delta})\sim f^{0}=\text{const}$ and
$\Omega_{\rm GW}^{(1)}(f\gtrsim f_{\Delta})\sim f^{-1}$, with $f_{\Delta}$ the
frequency of the spectral-break. This leads to unique GW spectra due to the
presence of PBH-domination era and its subsequent evaporation leading to the
onset of radiation era and can be tested via several upcoming GW detectors as
studied in detail in Ref. [45].

In addition, global cosmic strings efficiently contribute to $N_{\rm eff}$ via
Goldstone boson emission. However, the precise constraint is still debated
among the various groups performing lattice simulations (see e.g. Refs [119,
120, 121, 122] and Refs. [123, 124, 125]). To be safe from this we take the
upper bound to be $v_{B-L}\lesssim 10^{15}\,{\rm GeV}$ following the most
stringent among Ref. [126] and Ref. [125, 127] 222The non-observation of
B-modes in CMB provides another constraint on global strings from $H_{\rm
inf}\lesssim 3\times 10^{13}~{}\rm GeV$ (scale of inflation) [128] via the
maximum temperature of the Universe $T_{\rm max}\lesssim 4\times 10^{15}$ GeV
to $v_{B-L}\lesssim 10^{15}\,{\rm GeV}$.. This translates into a RH neutrino
mass of around $M_{N}=10^{11}$ GeV for $g=10^{-4}$ (see Fig. 3). For this
scenario, and PBH mass of $10^{2}-10^{3}$g, the corresponding GW spectrum will
be probed via the Einstein Telescope (ET) GW detector with four years of data-
taking and the signal-to-ratio needed to claim discovery is needed to be 10 or
larger as studied in details in Ref. [45]. However, with the same $v_{B-L}$
for larger $g$ values (that is, for larger $M_{N}$) we find that the
superradiance cloud decays too quickly to have any sizeable impact on the
lepton asymmetry generation. Nonetheless, it could happen that based on the
uncertainties related to the lattice simulation of global cosmic strings, and
the sensitivity reaches of the ET GW detector, we may also be able to probe
some regions of the parameter space shown in Fig. 3 via this GW spectral shape
and leave this for future dedicated studies. Further, if we had considered a
local $B-L$ symmetry333Also see Ref.[129] for a similar discussion, but
without PBH spin., then there are clear overlaps between the parameter space
shown in Fig. 6 of Ref. [45] and Fig. 3. However, such a theory also
introduces a heavy ZB-L vector boson, which would also have its superradiance
effects, and those signatures would require a separate dedicated study which
is beyond the scope of our present analysis.

##  7 Discussion and Conclusion

We have studied the impact of the superradiant primordial black holes on the
production of the baryon asymmetry of the Universe via high-scale
leptogenesis. We investigated how a Majoron model would be impacted in the
context of high-scale leptogenesis from spinning black holes. In particular,
the spinning PBHs’ evaporation is associated with particle production bursts
due to superradiance and an injection of scalar Majoron $\phi$, which decays
into right handed neutrinos. We found that such superradiance effects can
enhance the production of a lepton asymmetry beyond the region of the
parameter space in the minimal scenario, and this occurs where the
superradiance condition is satisfied, $\alpha GM_{\mathrm{BH}}M_{\phi}\approx
0.2$, see the horn-like region of Fig. 2. This effectively rescues some
regions of the parameter space otherwise, which did not yield viable high-
scale leptogenesis as the baryogenesis mechanism [42]. This additional viable
parameter space requires a PBH population with masses from $0.1$ to $10$ grams
with an initial abundance $\beta\gtrsim 10^{-7}$ (see Fig. 2). This impact on
the parameter space is also governed by the microphysics BSM parameters
involving RH neutrino masses and the coupling of the Marojon to SM, depicted
in terms of BR (see Fig. 3).

As described in Section 6, recent studies regarding presence of PBH-dominated
epoch [115, 113, 114, 106, 46, 47, 130, 131, 132] have shown that inevitable
scalar-induced gravitational waves resulting from adiabatic and isocurvature
perturbations (model-independent) can produce distinctive resonant peaks and
double-peaks in the GW spectrum, which can be used to probe the formation time
and decay time of PBHse. The unique spectral shapes of those GW are key to the
understanding if PBHs ever came to dominate the Universes’ energy density [46,
47]. If such spectra cover the regions where superradiance can increase the
number density of right handed neutrinos produced by PBHs (via intermittent
production of $\phi$) one may perhaps have a complementary test of
superradiant leptogenesis via primordial GW. However, such a study is beyond
the scope of the present analysis. Moreover, gauging the B-L symmetry will
have interesting consequences due to the presence of massive $Z_{B-L}$ boson,
which could have its own superradiance effects as well as the GW signatures
coming from local cosmic strings would show up as signatures in the GW
spectrum that LISA would be able to detect.

## Acknowledgments

This work has been funded by the UK Science and Technology Facilities Council
(STFC) under grant ST/T001011/1. 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. This work has
made use of the Hamilton HPC Service of Durham University. This work used the
DiRAC@Durham facility managed by the Institute for Computational Cosmology on
behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk), which is part of the
National eInfrastructure and funded by BEIS capital funding via STFC capital
grants ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC
operations grant ST/R000832/1.

##  Appendix A Secondary Decays

The PBHs produce RH neutrinos via Hawking radiation, which can decay and
produce a lepton asymmetry. The rate of this is given by:

$\Gamma_{\mathrm{BH}\rightarrow N}=\int\frac{d^{2}N_{N}}{dE_{N}dt}dE_{N}\,.$
(A.1)

However, the PBHs will also produce the scalar, $\phi$, via Hawking radiation
which can undergo a secondary decay to RHNs which will decay. These secondary
RHNs (denoted by superscript ’sec’) will have a slightly different energy
spectrum than the primaries. Their differential number density is given by

$\displaystyle\frac{d^{2}N^{\rm{sec}}_{N}}{dE_{N}dt}$ $\displaystyle=\int
dE_{\phi}\frac{d^{2}N_{\phi}}{dE_{\phi}dt}\frac{dN_{\phi\to NN}}{dE_{N}}\,,$
(A.2)

where the rate of the RHNs sourced from $\phi$ decays is found by integrating
the above differential number density over the energy of the RHNs:

$\Gamma_{\mathrm{BH}\rightarrow\phi\rightarrow
N}=\int\frac{d^{2}N^{\rm{sec}}_{N}}{dE_{N}dt}dE_{N}\,.$

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