Luis C. Bermúdez-Bustamante et al. Dust formation during the interaction of
binary stars by common envelope.

Planetary Nebulae: a Universal Toolbox in the Era of Precision Astrophysics
2023 10.1017/xxxxx 384

Proceedings IAU Symposium

# Dust formation during the interaction of binary stars by common envelope.

Luis C. Bermúdez-Bustamante1,2 0000-0002-3629-6259 Orsola De Marco1,2
0000-0002-1126-869X Lionel Siess3 0000-0001-6008-1103 Daniel J. Price4
0000-0002-4716-4235 Miguel González-Bolívar1,2 0000-0002-5939-9269  Mike Y. M.
Lau4,5,6 0000-0002-6592-2036 Chunliang Mu1,2 0000-0003-1848-6507 Ryosuke
Hirai4,5 0000-0002-8032-8174 Taïssa Danilovich4,7 0000-0002-1283-6038 Mansi
Kasliwal8 0000-0002-5619-4938 1School of Mathematical and Physical Sciences,
Macquarie University, Balaclava Road, North Ryde, Sydney, NSW 2109, Australia
2Astrophysics and Space Technologies Research Centre, Macquarie University,
Balaclava Road, North Ryde, Sydney, NSW 2109, Australia
3Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles
(ULB), CP 226, 1050 Brussels, Belgium
4School of Physics and Astronomy, Monash University, Clayton, Victoria 3800,
Australia
5OzGrav: The ARC Centre of Excellence for Gravitational Wave Discovery,
Australia
6Heidelberger Institut für Theoretische Studien, Schloss-Wolfsbrunnenweg 35,
69118 Heidelberg, Germany
7ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D),
Clayton 3800, Australia
8Division of Physics, Mathematics, and Astronomy, California Institute of
Technology, Pasadena, CA 91125, USA

###### Abstract

We performed numerical simulations of the common envelope (CE) interaction
between two intermediate-mass asymptotic giant branch (AGB) stars and their
low-mass companions. For the first time, formation and growth of dust in the
envelope is calculated explicitly. We find that the first dust grains appear
as early as $\sim$1–3 yrs after the onset of the CE, and are smaller than
grains formed later. As the simulations progress, a high-opacity dusty shell
forms, resulting in the CE photosphere being up to an order of magnitude
larger than it would be without the inclusion of dust. At the end of the
simulations, the total dust yield is $\sim 8.2\times 10^{-3}$ ($\sim 2.2\times
10^{-2}$ ) for a CE with a 1.7 (3.7 ) AGB star. Dust formation does not
substantially lead to more mass unbinding or substantially alter the orbital
evolution.

###### keywords:

stars: AGB and post-AGB, stars: winds, outflows, (stars:) binaries (including
multiple): close

††editors: O. De Marco, A. Zijlstra, R. Szczerba, eds.

##  1 Introduction

Unstable mass transfer from an expanding, more massive star to a compact star
leads to a CE interaction (Ivanova et al., 2013). The envelope of the giant
eventually surrounds both the donor star’s core and the companion. At the end
of this phase, the envelope may be ejected and a binary system with a much
smaller orbital separation or a merger is left behind.

The exact source(s) of energy driving the expansion and ejection of the
envelope is not fully identified, and there are problems when considering only
orbital energy and angular momentum transfer from the stars to the envelope
(but see Valsan et al., 2023), although the work done by recombination energy
released in inner layers of the donor likely plays a major role (e.g., Ivanova
et al., 2015; Reichardt et al., 2020; Lau et al., 2022).

The CE interaction leads to the formation of systems such as cataclysmic
variables, X-ray binaries or progenitors of type Ia supernovae (Paczynski,
1971; Ivanova et al., 2013; De Marco & Izzard, 2017). It is also likely
responsible for at least a fraction of intermediate luminosity transients
(Kasliwal et al., 2011), such as luminous red novae (e.g., Blagorodnova et
al., 2017), some of which produce dust (Tylenda et al., 2011; Nicholls et al.,
2013). From the above, dust might act as an additional driving mechanism,
altering the appearance of transient objects by increasing their opacity.

Previous studies on dust in interacting binary stars were conducted in 1D (Lü
et al., 2013), analyzed envelope thermodynamic properties without explicitly
calculating dust formation (Glanz & Perets, 2018; Reichardt et al., 2019),
calculated dust formation but in post-processing (Iaconi & De Marco, 2019;
Iaconi et al., 2020), or focused on the phase before the common envelope
(Bermudez et al., 2020).

In González-Bolívar et al. (2023, hereafter paper I) we performed CE
simulations with intermediate-mass AGB stars, using a simplified formulation
by Bowen (1988) to calculate the dust opacity. We find that dust does not
significantly increase the mass loss, but it does increase the envelope
opacity. Unfortunately, the Bowen formulation does not provide us with
information about the properties of the dust, such as its mass or grains size.
For this reason, we perform CE simulations with an explicit calculation of the
dust formation process.

##  2 Numerical Setup

We use the same stellar models as in Paper I, calculated with the 1D stellar
evolution code of (e.g., Paxton et al., 2011). The first model is an AGB star,
evolved from the main sequence, with a total mass of 1.7 (0.56 in the core), a
radius of 260 , undergoing its seventh thermal pulse. The second model is an
AGB star with a total mass of 3.7 (0.72 in the core), a radius of 343 ,
experiencing its fourth thermal pulse. Both models feature a 0.6 companion
star initially positioned at the critical distance for triggering Roche lobe
overflow, i.e., at a distance of 550 (637 ) from the 1.7 (3.7 ) AGB star.

We use the smoothed particle hydrodynamics (SPH) code (Price et al., 2018)
with $1.37\times 10^{6}$ SPH particles as was done by González-Bolívar et al.
(2022). We also use ’s implementation of ’s OPAL/SCVH equation of state tables
(as done by Reichardt et al., 2020) to account for recombination energy. We
assume the formation of carbon-rich dust with a C/O ratio of 2.5, despite both
AGB stars having a C/O ratio of 0.32 when our models ceased. The C/O ratio is
not anticipated to impact the AGB convective structure.

We use the treatment described in Siess et al. (2022) for carbon dust
nucleation and growth, which takes place in two steps: i) the formation of
seed particles from the gas phase in the so-called nucleation stage, followed
by ii) a growth phase where monomers (dust building blocks) accumulate on the
seed particles to reach macroscopic dimensions. The treatment is based on the
theory of moments presented by Gail et al. (1984, and subsequent papers).

##  3 Dust properties and its impact on orbital and thermodynamics properties

###  3.1 Opacity of dust grains

Figure 1: Dust opacity versus distance from the centre of mass for the 1.7
(left) and 3.7 (right) models after 20 years of simulation. The color bar
indicates $t_{\rm nuc}$, the time in years after the start of the simulations,
when dust nucleation occurred for each SPH particle.

Figure 1 shows a snapshot111Movies corresponding to the figures are available
at: https://tinyurl.com/y455avdj. of the dust opacity as a function of
distance from the center of mass (the color bar depicts nucleation time
$t_{\rm nuc}$ as explained in the caption) at 20 years from the start of the
simulation. In the inner regions where there is no dust, the opacity is
constant, at a user-defined value of 2$\times$10-4 cm2 g-1. This value is
appropriate for recombined gas, but too low for the ionised regions where
electron scattering opacities are of the order of 0.5 cm2 g-1. However, this
is not important for now as these inner hot regions do not experience dust
formation, which happens $\sim$1–3 yrs from the start of the simulation, in a
shell with radii of $\sim$10–20 au. When dust forms, the opacity rapidly
increases to values of $\sim$17–20 cm2 g-1 and decreases to almost zero again
at larger radii. The dust forming shell radius expands over time and at 20
years it is $\sim$130 au (160 au) from the 1.7 (3.7 ) model. The dust forming
shell thickness is also variable.

###  3.2 Dust size

Figure 2: Average dust grains size versus distance for the 1.7 (top) and 3.7
(bottom) model, after 44 years of simulations. Same color bar as in Figure 1.

Figure 2 shows a snapshot of dust grains size as a function of distance (same
color bar as in Figure 1). Early dust grains (yellow colour in Figure 2) have
a narrow range in size around $\sim$0.02–0.04 $\mu$m, and late dust grains
(darker colours) have a lager size between $\sim$0.03 and 1 $\mu$m. In the 1.7
(3.7 ) model, the largest dust grains are formed at $\sim$12 yrs ($\sim$15
yrs). Interestingly, at the very end of the 3.7 simulation ($\sim$44 years),
dust formation resumes in a thin shell near the centre, but at a low rate.

Depending on $t_{\rm nuc}$, dust grains grew to different maximum sizes. Our
formalism does not include dust destruction, so once a grain stops growing,
its size remains constant. However, what is plotted in Figure 2 is the average
grain size in a given SPH particle, which can contain many dust grains because
it has a set total mass, e.g., $1.7\times 10^{27}$ g or $8.3\times 10^{-7}$ in
the 1.7 simulation. As such, the average dust grain size in an SPH particle
can decrease if new small grains are formed, even after previously-formed
grains have become large.

###  3.3 Dust mass

Figure 3: Dashed lines show the total dust mass over time in CE simulations.
Horizontal lines represent the maximum carbon available in the envelope for
1.7 (black solid) and 3.7 (red solid) AGB stars.

Figure 3 shows that for the 1.7 (3.7 ) model, the total dust mass increases
steadily until it plateaus at $\sim$32 yrs ($\sim$38 yrs) with a value of
$\sim$8.2$\times$10-3 ($\sim$2.2$\times$10-2 ). Although it is difficult to
establish a pattern with only two stars, we see that the more massive the
donor star, the longer it takes for the total amount of dust to plateau.

After $\sim$40 yrs, the 1.7 (3.7 ) model has produced roughly four (ten) times
more dust than during the first $\sim$20 yrs. The drastic increase and
subsequent plateauing of dust mass is a consequence of a large fraction of the
expanding envelope reaching temperatures below the condensation threshold.
Because of that, nucleation is efficient (i.e., there is a large number of
dust seeds) and the addition of monomers contribute to dust growth, producing
the increase in the dust mass. However, with further gas expansion the
nucleation rate decreases, fewer monomers are available and dust production is
consequently reduced.

Compared to simulations by Iaconi et al. (2020) for carbon grains, the amount
of dust formed in our 1.7 (3.7 ) model is four (ten) times larger for a star
twice (four times) as massive. In Iaconi et al. (2020), the total dust mass
plateaus at $\sim 2.2\times 10^{-3}$ after $\sim$14 yrs for a 0.88 RGB star,
which started off with a radius of 83 . It is difficult to comment on whether
these differences are due to the different type of stars, to the fact that
Iaconi et al. (2020) post-processed the dust formation in their star, or to
the nucleation formalism used.

###  3.4 Orbital evolution, unbound mass and comparison with non-dusty
simulations

Figure 4: Left panel: orbital separation over time in CE simulations with a
1.7 AGB star. Dust formation is explicitly calculated (blue lines), or assumed
with a maximum dust opacity of $5~{}\mathrm{cm^{2}\,g^{-1}}$ (red lines) or
$15~{}\mathrm{cm^{2}\,g^{-1}}$ (green lines) as in Paper I. Also shown is a CE
simulation without dust (yellow lines). The inset displays the orbital
separation at the end of the in-spiral phase. Right panel: Mechanical bound
mass (solid lines) and thermal bound mass (dashed lines) over time.

For all simulations, the dynamic in-spiral phase ends after $\sim$7.5 yr
(Figure 4, left panel). The orbital separation decreases faster in models
where dust opacity is calculated either explicitly (blue line), or using the
Bowen opacity approximation (green and red lines), than in the model with no
dust (yellow line). Radiative forces have a significant impact on accelerating
the envelope, which interact through gravitational friction with the stars. On
the other hand, the final (plateau) separation is similar, being $\sim$42
(blue line) and $\sim$33 (yellow line), although the latter is still
decreasing somewhat.

In the right panel of Figure 4, “mechanical” bound mass (solid lines) is the
mass of SPH particles whose sum of kinetic and gravitational potential energy
is negative, while “thermal” bound mass (dashed lines) is the mass of SPH
particles whose sum of kinetic, gravitational and thermal energy (not
including recombination energy) is less than zero. The bound mass decreases
faster in models with explicit dust formation (blue lines) than in models (i)
without dust (yellow lines) or (ii) with a maximum Bowen opacity of 5
$\mathrm{cm}^{2}\mathrm{g}^{-1}$ (red lines); however, when the latter
increases to $15~{}\mathrm{cm}^{2}\mathrm{g}^{-1}$ (green lines), the
evolution of the bound mass is similar. High-opacity dust is located too far
in the wind222See movies of dust opacity versus distance at
https://tinyurl.com/y455avdj. to generate effective driving on a dynamical
timescale required to aid with envelope ejection, because at those distances,
the radiative flux is greatly diluted and the material is mostly unbound
already. We conclude that dust formation does not lead to substantially more
mass unbinding, although $\sim 10^{-2}$ of dust is produced.

##  4 The photospheric size

Figure 5: Photosphere size over time in the orbital (solid lines) and the
“polar” plane (dashed lines) for the 3.7 model with and without dust (blue and
orange, respectively). Shaded areas depict uncertainty.

Figure 5 shows the average radius, in the polar and orbital directions, of the
stellar surface where the optical depth, calculated using the Planck-mean
opacity, becomes 1 (i.e. the photosphere). We note that after $\sim$5 yrs the
size of the photosphere is an order of magnitude larger when dust is included.
This is expected, since dust opacity will be high enough for the expanding
envelope to become optically thick. We also note that in the dusty model, the
equatorial photosphere grows approximately one year before the polar
photosphere, effectively indicating that there is a “hole” at the poles where
an observer would see deeper into the object. However, this hole is filled
rapidly and the size of the photosphere in the two orthogonal directions
becomes very comparable, indicating an approximately spherical object.

##  5 Summary

We have carried out simulations of CE interactions of two intermediate-mass
AGB stars (1.7 and 3.7 ) with a 0.6 companion. To test the dust grains
properties, simulations include a self-consistent treatment of dust formation
with a C/O ratio of 2.5 (by number). Our main conclusions are:

  1. 1.

Dust formation starts $\sim$1–3 yrs after the onset of the CE and leads to the
formation of a dusty shell that moves outward. At $\sim$20 yrs the shell has a
thickness of $\sim 40$ au and it is located at $\sim 130$ au ($\sim 160$ au)
in the 1.7 (3.7 ) model.

  2. 2.

Dust begins as tiny seeds and grows via monomer addition. The growth halts
once the dust moves away along with the expanding envelope. Dust grains formed
at $\sim$1–3 yrs are smaller ($\sim$0.02–0.04 $\mu$m) than dust grains formed
after $\sim$10 yrs ($\sim$0.03–1 $\mu$m).

  3. 3.

The amount of dust rises and plateaus just below the theoretical maximum,
yielding $\sim 8.2\times 10^{-3}$ ($\sim 2.2\times 10^{-2}$ ) for the 1.7 (3.7
) model.

  4. 4.

High-opacity dust is located too far in the wind to generate effective driving
on a dynamical timescale required to aid with envelope ejection. Because of
that, although $\sim 10^{-2}$ of dust is produced, it does not lead to
substantially more mass unbinding.

  5. 5.

Dust greatly impacts the optical appearance of the CE. At $\sim$12 yrs, the
photosphere is an order of magnitude larger if dust is included.

Movies corresponding to the figures are available at:
https://tinyurl.com/y455avdj.

OD, LS, MK and LB acknowledge support through the Australian Research Council
Discovery Program grant DP210101094. LS is senior research associates from
F.R.S.- FNRS (Belgium). TD is supported in part by the Australian Research
Council through a Discovery Early Career Researcher Award (DE230100183). This
work was supported in part by Oracle Cloud credits and related resources
provided by Oracle for Research. This work was performed in part on the OzSTAR
national facility at Swinburne University of Technology. The OzSTAR program
receives funding in part from the Astronomy National Collaborative Research
Infrastructure Strategy (NCRIS) allocation provided by the Australian
Government, and from the Victorian Higher Education State Investment Fund
(VHESIF) provided by the Victorian Government. Some of the simulations were
undertaken with the assistance of resources and services from the National
Computational Infrastructure (NCI), which is supported by the Australian
Government. This research was supported in part by the Australian Research
Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO
3D), through project number CE170100013.

## References

  * Bermudez et al. (2020) Bermúdez-Bustamante L. C., García-Segura G., Steffen W., Sabin L., 2020, MNRAS, 493, 2606.
  * Blagorodnova et al. (2017) Blagorodnova N., Kotak R., Polshaw J., Kasliwal M. M., Cao Y., Cody A. M., Doran G. B., et al., 2017, yCat, J/ApJ/834/107.
  * Bowen (1988) Bowen G. H., 1988, ApJ, 329, 299.
  * De Marco & Izzard (2017) De Marco O., Izzard R. G., 2017, PASA, 34, e001.
  * Gail et al. (1984) Gail H.-P., Keller R., Sedlmayr E., 1984, A&A, 133, 320
  * Glanz & Perets (2018) Glanz H., Perets H. B., 2018, MNRAS, 478, L12.
  * González-Bolívar et al. (2023) González-Bolívar M., Bermúdez-Bustamante L. C., De Marco O., Siess L., Price D. J., Kasliwal M., 2023, arXiv, arXiv:2306.16609.
  * González-Bolívar et al. (2022) González-Bolívar M., De Marco O., Lau M. Y. M., Hirai R., Price D. J., 2022, MNRAS, 517, 3181.
  * Iaconi & De Marco (2019) Iaconi R., De Marco O., 2019, MNRAS, 490, 2550.
  * Iaconi et al. (2020) Iaconi R., Maeda K., Nozawa T., De Marco O., Reichardt T., 2020, MNRAS, 497, 3166.
  * Ivanova et al. (2013) Ivanova N., Justham S., Chen X., De Marco O., Fryer C. L., Gaburov E., Ge H., et al., 2013, A&ARv, 21, 59.
  * Ivanova et al. (2015) Ivanova N., Justham S., Podsiadlowski P., 2015, MNRAS, 447, 2181.
  * Kasliwal et al. (2011) Kasliwal M. M., Cenko S. B., Kulkarni S. R., Ofek E. O., Quimby R., Rau A., 2011, ApJ, 735, 94.
  * Lau et al. (2022) Lau M. Y. M., Hirai R., González-Bolívar M., Price D. J., De Marco O., Mandel I., 2022, MNRAS, 512, 5462.
  * Lü et al. (2013) Lü G., Zhu C., Podsiadlowski P., 2013, ApJ, 768, 193.
  * Nicholls et al. (2013) Nicholls C. P., Melis C., Soszynski I., Udalski A., Szymanski M. K., Kubiak M., Pietrzynski G., et al., 2013, MNRAS, 431, L33.
  * Paczynski (1971) Paczyński B., 1971, ARA&A, 9, 183.
  * Paxton et al. (2011) Paxton B., Bildsten L., Dotter A., Herwig F., Lesaffre P., Timmes F., 2011, ApJS, 192, 3.
  * Price et al. (2018) Price D. J., Wurster J., Tricco T. S., Nixon C., Toupin S., Pettitt A., Chan C., et al., 2018, PASA, 35, e031.
  * Reichardt et al. (2019) Reichardt T. A., De Marco O., Iaconi R., Tout C. A., Price D. J., 2019, MNRAS, 484, 631.
  * Reichardt et al. (2020) Reichardt T. A., De Marco O., Iaconi R., Chamandy L., Price D. J., 2020, MNRAS, 494, 5333.
  * Siess et al. (2022) Siess L., Homan W., Toupin S., Price D. J., 2022, A&A, 667, A75.
  * Tylenda et al. (2011) Tylenda R., Hajduk M., Kamiński T., Udalski A., Soszyński I., Szymański M. K., Kubiak M., et al., 2011, A&A, 528, A114.
  * Valsan et al. (2023) Valsan V., Borges S. V., Prust L., Chang P., 2023, MNRAS, 526, 5365.


MS-TP-23-46

# Wash-in leptogenesis after the evaporation of primordial black holes

Kai Schmitz<EMAIL_ADDRESS>Institute for Theoretical Physics,
University of Münster, 48149 Münster, Germany  Xun-Jie Xu<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China

(August 28, 2024)

###### Abstract

Wash-in leptogenesis is a powerful mechanism to generate the baryon asymmetry
of the Universe that treats right-handed-neutrino interactions on the same
footing as electroweak sphaleron processes: as mere spectator processes acting
on the background of chemical potentials in the Standard Model plasma.
Successful wash-in leptogenesis requires this chemical background to be
$CP$-violating, which can be achieved by violating any of the more than ten
global charges that are conserved in the Standard Model at very high
temperatures. In this paper, we demonstrate that the primordial charge
asymmetries required for wash-in leptogenesis can be readily produced by
evaporating primordial black holes (PBHs). Our argument is based on the fact
that the Hawking radiation emitted by PBHs contains more or less any state in
the particle spectrum. Therefore, if heavy states with $CP$-violating decays
are present in the ultraviolet, PBH evaporation will unavoidably lead to the
production of these states. We illustrate this scenario by means of a simple
toy model where PBH evaporation leads to the production of heavy particles
that we call asymmetrons and whose decay results in a primordial charge
asymmetry for right-handed electrons, which in turn sets the initial
conditions for wash-in leptogenesis. We focus on the parameter region where
the decay of the initial thermal asymmetron abundance occurs long before PBH
evaporation and only results in a negligible primordial charge asymmetry. PBH
evaporation at later times then serves as a mechanism to resurrect the
asymmetron abundance and ensure the successful generation of the baryon
asymmetry after all. We conclude that PBHs can act as asymmetry-producing
machines that grant access to whatever $CP$-violating physics may be present
in the ultraviolet, rekindling it at lower energies where it can be
reprocessed into a baryon asymmetry by right-handed neutrinos.

##  I Introduction

The Standard Model (SM) fails to provide an explanation for the observed
baryon asymmetry of the Universe, usually expressed in terms of the baryon-to-
photon ratio, $\eta_{B}^{0}=n_{b}^{0}/n_{\gamma}^{0}\simeq 6.1\times 10^{-10}$
Planck:2018vyg ; ParticleDataGroup:2022pth , which hence provides clear
evidence for new physics. A popular scenario for the dynamical generation of
the baryon asymmetry in the early Universe consists of thermal leptogenesis
Fukugita:1986hr , which, in its simplest form, is based on the charge–parity
($CP$) and baryon-minus-lepton number ($B\\!-\\!L$) violating out-of-
equilibrium decays of heavy right-handed neutrinos (RHNs). These decays first
generate a primordial $B\\!-\\!L$ asymmetry, which is subsequently partially
converted to a baryon asymmetry by the chemical transport in the SM plasma,
including electroweak sphalerons Kuzmin:1985mm . Baryogenesis via leptogenesis
is an attractive scenario that links the generation of the matter–antimatter
asymmetry in the early Universe to new physics in the neutrino sector that may
be within the reach of laboratory experiments Buchmuller:2005eh ; Chun:2017spz
; Bodeker:2020ghk . Thermal leptogenesis in its standard formulation, however,
comes with several restrictions that render its experimental exploration very
challenging. For instance, in order to achieve a sufficient amount of $CP$
violation in RHN decays, the mass of the lightest RHN mass eigenstate is
bounded from below, $M_{1}\gtrsim 10^{9}\,\textrm{GeV}$ Davidson:2002qv ;
Buchmuller:2002rq . Similarly, the Yukawa interactions that couple RHNs to SM
lepton–Higgs pairs must not be too strong, since otherwise wash-out effects
will erase any previously generated asymmetry and hence spoil the success of
leptogenesis.

There are numerous proposals for alternative leptogenesis scenarios at lower
energies, notably resonant leptogenesis Pilaftsis:2003gt ; Pilaftsis:2005rv ;
Dev:2017wwc  and leptogenesis via RHN oscillations Akhmedov:1998qx ;
Drewes:2017zyw  (see Refs. Klaric:2020phc ; Klaric:2021cpi  for recent work on
the relation between these two models). These scenarios partially rely on
special choices of parameter values (e.g., a highly degenerate RHN mass
spectrum) and continue to exploit the $CP$ violation in the RHN sector in
order to generate a $B\\!-\\!L$ asymmetry. By contrast, the recently proposed
mechanism of wash-in leptogenesis Domcke:2020quw ; Domcke:2022kfs  follows a
different approach; it dispenses with $CP$ violation in the RHN sector
altogether and separates the energy scales of $CP$ violation and $B\\!-\\!L$
violation.

Wash-in leptogenesis generalizes thermal leptogenesis to nontrivial chemical
background configurations in the early Universe and treats RHN interactions on
the same footing as electroweak sphalerons: as mere spectator processes that
reprocess the chemical potentials of the SM particle species in the thermal
bath. The action of the electroweak sphalerons on the chemical composition of
the SM plasma then results in the usual violation of baryon-plus-lepton number
$B\\!+\\!L$, while the action of the RHN interactions results in the violation
of $B\\!-\\!L$. Together, these two effects are sufficient for the generation
of a primordial baryon asymmetry.

$CP$ violation in the RHN sector is irrelevant for wash-in leptogenesis.
Instead, wash-in leptogenesis is based on the idea that new $CP$-violating
dynamics at high energies are responsible for the generation of primordial
charge asymmetries, which are then reshuffled in a $B\\!-\\!L$-violating
fashion by RHN interactions at low energies. In this sense, the idea of wash-
in leptogenesis denotes a general mechanism that can be used as a building
block in a complete model of baryogenesis; but it does not represent a
complete baryogenesis scenario on its own. In other words, wash-in
leptogenesis requires an ultraviolet (UV) completion that provides an
explanation for the origin of the nontrivial chemical background that the RHN
interactions are supposed to act on — a mechanism for chargegenesis in the
language of Refs. Domcke:2022kfs . An interesting and viable scenario for such
a UV completion of wash-in leptogenesis is axion inflation Anber:2015yca ;
Jimenez:2017cdr ; Domcke:2019mnd ; Domcke:2022kfs , i.e., models of inflation
where inflation is driven by a pseudoscalar axion field that spontaneously
breaks $CP$ invariance by means of its nonzero and time-dependent background
field value. If the axion field couples to the SM hypercharge gauge field, the
$CP$ violation induced by the dynamics of the rolling axion field is reflected
in the dual production of (A) maximally helical hypermagnetic fields and (B)
fermionic charge asymmetries in accordance with the chiral anomalies of the SM
fermion currents. These charge asymmetries then provide the necessary initial
conditions for wash-in leptogenesis at lower temperatures.

Wash-in leptogenesis after axion inflation is, however, only one example
scenario among countless other possibilities. In fact, wash-in leptogenesis
only requires that at least one of various global charges that are conserved
in the SM at high temperatures be violated. Depending on the temperature scale
of wash-in leptogenesis, there are up to eleven suitable charges whose
violation can set the right initial conditions for wash-in leptogenesis,

$\displaystyle q_{e}\,,\quad q_{2B_{1}-B_{2}-B_{3}}\,,\quad q_{u-d}\,,\quad
q_{d-s}\,,\quad q_{B_{1}-B_{2}}\,,$ (1) $\displaystyle q_{\mu}\,,\quad
q_{u-c}\,,\quad q_{\tau}\,,\quad q_{d-b}\,,\quad q_{B}\,,\quad q_{u}\,,$

see Refs. Domcke:2020quw ; Domcke:2022kfs  for more details. This
characteristic feature of wash-in leptogenesis renders it an extremely
flexible and general mechanism that can lead to the successful generation of
the baryon asymmetry in a plethora of models. The purpose of this paper is to
highlight one such class of models, which has not yet been discussed in the
literature: wash-in leptogenesis after the evaporation of primordial black
holes (PBHs) Escriva:2022duf .

The key observation behind this scenario is that the Hawking radiation emitted
by evaporating PBHs contains particles across the whole particle spectrum,
including heavy states that are potentially never produced thermally or that
already freeze out at very high temperatures. In order to generate the baryon
asymmertry of the Universe, it is then enough to assume that there is at least
one state in the particle spectrum whose interactions result in a sufficient
amount of $CP$ violation, such that it can generate the right initial
conditions for wash-in leptogenesis. We shall refer to this particle as the
asymmetron. Given a population of very light PBHs in the early Universe, the
evaporation of these PBHs will unavoidably result in the production of
asymmetrons, whose interactions then yield the charge asymmetries required for
wash-in leptogenesis. In our scenario, PBH evaporation thus plays the role of
the chargegenesis mechanism that serves as the UV completion of wash-in
leptogenesis.

Our scenario is related to baryogenesis Hawking:1974rv ; Carr:1976zz ;
Barrow:1990he ; Baumann:2007yr ; Morrison:2018xla ; Hooper:2020otu  and
leptogenesis Perez-Gonzalez:2020vnz ; Datta:2020bht ; JyotiDas:2021shi ;
Bernal:2022pue ; Calabrese:2023key ; Khan:2023myt  models that rely on PBH
evaporation in order to produce an abundance of particles whose decays either
violate baryon number $B$ or lepton number $L$ (i.e., RHNs in the latter
case). Meanwhile, our scenario is more general than these earlier proposals.
In our case, we merely need to assume that the $CP$-violating asymmetron
interactions produce any of the eleven global charges in Eq. (1), which are
conserved in the SM at high temperatures; we do not need to identify the
asymmetron with, say, a RHN and we do not need to assume that the asymmetry
produced in asymmetron interactions corresponds to $B$ or $L$ right away.
Instead, PBHs play the role of generic asymmetry-producing machines in our
scenario that provide a portal to whatever $CP$-violating physics may be
present at high energies. PBH evaporation grants access to these
$CP$-violating dynamics and leads to a nontrivial chemical equilibrium in one
form or another, which is all it takes to realize wash-in leptogenesis.

In this paper, we will illustrate the basic idea of wash-in leptogenesis after
PBH evaporation by means of a simple toy model, in which PBH evaporation gives
rise to an abundance of asymmetrons whose decays result in a primordial
asymmetry between right-handed electrons and left-handed positrons. We expect
that this toy model can be easily generalized to more complex scenarios,
involving other primordial charge asymmetries or even combinations thereof.
The rest of the paper is organized as follows: We begin in Sec. II by
introducing our toy model. Then, in Sec. III, we provide a few analytical
estimates, before we study the full system of Boltzmann equations in Sec. IV.
In Sec. V, we perform a systematic scan of parameter space, before we finally
conclude in Sec. VI.

##  II  A simple scenario

The idea of wash-in leptogenesis can be illustrated by considering a
particularly simple scenario in the early universe when the temperature is
above ${\cal O}(100)$ TeV, at which the electron Yukawa interaction is not
equilibrated.

Therefore, any primordial asymmetry of the right-handed electron ($e_{R}$)
produced at higher temperatures can be retained until ${\cal O}(100)$ TeV. If
the wash-out effect due to for example RHN interactions is strong, the $B-L$
asymmetry is then mainly determined by wash-in leptogenesis, which generates
Domcke:2020quw

$q_{B-L}=-\frac{3}{10}q_{e}\thinspace,$ (2)

where $q_{e}=n_{e_{R}}-n_{\overline{e_{R}}}$ with $n_{e_{R}}$ and
$n_{\overline{e_{R}}}$ the number densities of $e_{R}$ and $\overline{e_{R}}$,
and $q_{B-L}=\sum_{f}Q_{B-L}^{(f)}(n_{f}-n_{\overline{f}})$ with $n_{f}$ and
$Q_{B-L}^{f}$ the number density and the $B-L$ charge of particle $f$.

Equation (2) implies that if there is a mechanism at high temperatures giving
rise to nonzero $q_{e}$, there should be $B-L$ asymmetry at low temperatures.
To generate nonzero $q_{e}$, we consider PBHs which can emit particles via
Hawking radiation quite generically, independent of the particle interactions.
In particular, they can emit extremely heavy particles that could otherwise
not be produced in the early universe due to the finite reheating temperature.
For a given PBH of mass $m_{{\rm BH}}$, the rate of emission is

$\displaystyle\frac{d^{2}N_{i}}{dtdE}$
$\displaystyle\approx\frac{g_{i}}{2\pi}\cdot\frac{v_{i}}{\exp(E/T_{{\rm
BH}})\pm 1}\thinspace,$ (3) $\displaystyle T_{{\rm BH}}$
$\displaystyle=\frac{m_{{\rm pl}}^{2}}{8\pi m_{{\rm BH}}}\thinspace,$ (4)

where $d^{2}N_{i}/dtdE$ denotes the number of particle $i$ emitted within the
differential time $dt$ and the differential energy $dE$; $g_{i}$ and $v_{i}$
are the multiplicity and graybody factor of particle $i$; and $m_{{\rm
pl}}=1.22\times 10^{19}$ GeV is the Planck mass. The “$\pm$” sign takes $+$ or
$-$ for fermions and bosons respectively.

The Hawking radiation of a PBH in general emits particles and antiparticles
equally. To generate the asymmetry of $e_{R}$, we introduce a heavy particle
$X$ that can decay asymmetrically to $e_{R}$ and $\overline{e_{R}}$. For this
reason, we dub it “the asymmetron”. The asymmetron can be either a Majorana
fermion or a real scalar, as it needs to be its own antiparticle. We assume
that it is a Majorana fermion with the following interaction:

${\cal L}\supset yXe_{R}S\thinspace,$ (5)

where $S$ is a scalar with the opposite SM charges of $e_{R}$, i.e., $e_{R}$
and $S$ possess hypercharges $Y_{e_{R}}=-1$ and $Y_{S}=+1$. As an asymmetron,
$X$ needs to decay asymmetrically. Hence we introduce the parameter of
asymmetric decay:

$\epsilon\equiv\frac{\Gamma_{X\to
e_{R}S}-\Gamma_{X\to\overline{e_{R}}S^{*}}}{\Gamma_{X\to
e_{R}S}+\Gamma_{X\to\overline{e_{R}}S^{*}}}\thinspace,$ (6)

where $\Gamma$ denotes the decay width of the process indicated in the
subscript. The asymmetric decay can be attained by for example assigning a
flavor structure to Eq. (5) with $CP$ phases, similar to the asymmetric decay
of right-handed neutrinos. In general, we expect $\epsilon\lesssim
y^{2}/(8\pi)$ if all Yukawa couplings in the flavorful generalization are
${\cal O}(y)$. Much smaller $\epsilon$ is possible by tuning down the $CP$
phases in the flavor structure.

##  III Low temperature, High mass
— an analytical estimate

One of the most prominent features of PBHs is that they can emit heavy
particles with masses much higher than the environmental temperature. For
heavy asymmetrons produced at a relatively late epoch of the universe when
$T\ll m_{X}$, their decay is highly out-of-equilibrium, which is important for
the produced asymmetry to evade thermal washout. Due to this feature, within a
certain range of the parameter space, the asymmetry $q_{e}$ would
predominantly depend on the yield of $X$ from PBH evaporation.

This is the scenario we will concentrate on in what follows. To make the
discussion more concrete, let us set a benchmark:

$\displaystyle m_{{\rm BH0}}=10\ {\rm g}\thinspace,\ m_{X}=10^{12}\ {\rm
GeV}\thinspace,$ (7) $\displaystyle y=1\thinspace,\ \beta=10^{-6}\thinspace,\
\epsilon=-0.017\times\frac{y^{2}}{8\pi}\thinspace,$ (8)

where $m_{{\rm BH0}}$ is the initial mass of PBHs, and $\beta\equiv\rho_{{\rm
BH0}}/\rho_{{\rm tot}}$ with $\rho_{{\rm BH0}}$ the initial energy density of
PBHs and $\rho_{{\rm tot}}$ the total energy density of the thermal bath. For
simplicity, we assume that all PBHs have the same mass. The numeric factor for
$\epsilon$ is set at 0.017 such that the produced asymmetry meets the observed
value, as we will see later.

Figure 1 shows the evolution of the number density of $X$ produced from PBHs
for this benchmark. The result is obtained from our numerical calculation in
Sec. IV. Analytically, we can also understand the behavior of the curve well
and provide a quite accurate estimate of the result.

Figure 1: Evolution of the asymmetron number density, $n_{X}$, and the
electron asymmetry, $q_{e}$, for the benchmark in Eqs. (7)-(8). Note that here
$n_{X}$ and $q_{e}$ denote only the contributions of asymmetrons produced from
PBHs, while the thermal asymmetron abundance is also present (the dashed
line), though it leads to negligible asymmetry due to strong washout. The
dotted curve represents the value of $10^{-14}m_{\text{BH}}/m_{\text{BH}0}$.

According to Eq. (4), the initial temperature of the PBHs in Eq. (7) is
$T_{\text{BH}0}\approx 1.06\times 10^{12}$ GeV, which is comparable to
$m_{X}$. Hence they can efficiently emit $X$ particles. As the PBHs keep
evaporating, the PBH temperature and hence the emission rate keeps increasing.
Eventually the PBHs will fully evaporates with the lifetime given by111See
Appendix A for a review of the basic aspects of PBH.

$\tau_{{\rm BH}}=\frac{m_{{\rm BH}0}^{3}}{3g_{{\rm
BH}}m_{\text{pl}}^{4}}\thinspace,$ (9)

where $g_{{\rm BH}}\approx 7.5\times 10^{-3}$ if all SM degrees of freedom are
included. By comparing $\tau_{{\rm BH}}$ with the time of the universe,
$t\approx 1/(2H)$ where $H$ is the Hubble parameter, we obtain that the PBHs
should fully evaporate when the universe cools down to

$T_{{\rm eva}}\approx 10^{9}\ \text{GeV}\times\left(\frac{10\ {\rm g}}{m_{{\rm
BH0}}}\right)^{3/2}\thinspace.$ (10)

The formation of PBHs is related to cosmological perturbations at the horizon
scale ($H^{-1}$), leading to $m_{{\rm BH0}}\propto 4\pi H^{-3}\rho_{{\rm
tot}}/3\propto T^{-6}\cdot T^{4}\propto T^{-2}$. Hence smaller PBHs are formed
at higher temperatures. More specifically, the temperature of the universe at
which the PBHs with initial mass $m_{{\rm BH0}}$ are formed is given by
Bernal:2022pue

$T_{0}\approx 1.35\times 10^{15}\ \text{GeV}\times\left(\frac{10\ {\rm
g}}{m_{{\rm BH0}}}\right)^{1/2}\thinspace.$ (11)

So the entire PBH life for the given benchmark spans from $T_{0}\sim 10^{15}$
GeV to $T_{{\rm eva}}\sim 10^{9}$ GeV.

Despite that these PBHs keep evaporating during the period from $10^{15}$ to
$10^{9}$ GeV, their masses stay almost static until the very last moment
shortly before $T_{\text{eva}}$—see the dotted curve in Fig. 1. Consequently,
the emission rate of $X$ is almost a constant at $T\gg T_{{\rm eva}}$. This
leads to the flatness of the blue curve in Fig. 1 for $m_{X}/T\in[4\times
10^{-2},\ 7\times 10^{2}]$.

The flat part of the curve can be estimated by assuming the balance between
the production of $X$ from PBHs and the depletion due to $X$ decay:

$n_{\text{BH}}\Gamma_{\text{BH}\to
X}=n_{X}\Gamma_{X}\frac{m_{X}}{2T_{\text{BH}}}\thinspace,$ (12)

where $n_{\text{BH}}$ denotes the number density of PBHs,
$\Gamma_{\text{BH}\to X}$ denotes the emission rate of $X$ per PBH, and
$\Gamma_{X}=y^{2}m_{X}/(16\pi)$ is the decay rate of $X$ at rest. The PBH
number density $n_{\rm BH}$ is determined by rescaling its initial value,
$n_{{\rm BH}}=n_{{\rm BH0}}\,a_{0}^{3}/a^{3}$ with $n_{{\rm
BH0}}=\beta{\rho_{\rm tot}}/{m_{\text{BH0}}}$. For $\Gamma_{\text{BH}\to X}$,
we take $\Gamma_{\text{BH}\to X}\approx 0.012\,T_{{\rm BH}}$—see Appendix A
for details. The $m_{X}/(2T_{{\rm BH}})$ factor accounts for the time dilution
effect of relativistic $X$ decay, since the average value of $m_{X}/E$ is
around $0.46\,m_{X}/T_{{\rm BH}}$.

Solving Eq. (12) with respect to $n_{X}$, we obtain

$\frac{n_{X}}{n_{\gamma}}\approx 0.28\frac{\beta
m_{\text{pl}}^{4}T_{0}}{m_{{\rm BH0}}^{3}m_{X}^{2}y^{2}}\thinspace,$ (13)

where $n_{\gamma}=2\zeta(3)T^{3}/\pi^{2}$ is the photon number density. For
the benchmark values in Eqs. (7)-(8), Eq. (13) gives $n_{X}/n_{\gamma}\approx
4.7\times 10^{-14}$, agreeing well with the flat part of the blue curve in
Fig. 1.

At high temperatures ($m_{X}/T\lesssim 30$), the number density $n_{X}$ in Eq.
(13) is much lower than its thermal equilibrium value, $n_{X}^{{\rm eq}}$,
which is plotted in Fig. 1 as the black dashed curve. In this regime, any
asymmetry produced via $X$ decay can be very efficiently washed out by thermal
processes ($e_{R}S\leftrightarrow X\leftrightarrow\overline{e_{R}}S^{*}$).
Therefore, we regard the gray region encompassed by the dashed curve as the
strong washout region. Within this region, the asymmetry $q_{e}$ generated
from $X$ decay is strongly suppressed. Only when $n_{X}$ stretches out of the
gray region, the generated asymmetry can be effectively retained, as can be
seen from the orange curve in Fig. 1.

Therefore, the final contribution of PBH evaporation to the asymmetry can be
estimated by

$q_{e}\approx n_{{\rm BH}}\epsilon\int_{t_{\text{eq}}}^{t_{{\rm
eva}}}\Gamma_{\text{BH}\to X}dt\thinspace,$ (14)

where $t_{{\rm eq}}$ and $t_{{\rm eva}}$ denote the moments when the blue
curve crosses the dashed curve and when the PBHs fully evaporate,
respectively. The result is

$\frac{q_{e}}{n_{\gamma}}\approx
0.0073\epsilon\beta\frac{m_{\text{pl}}^{2}T_{0}}{m_{{\rm BH0}}\left(T_{{\rm
BH}}^{{\rm eq}}\right)^{2}}\thinspace,$ (15)

where $T_{{\rm BH}}^{{\rm eq}}$ denotes the PBH temperature at $t=t_{{\rm
eq}}$. Taking the benchmark values in Eqs. (7)-(8), we obtain
$q_{e}/n_{\gamma}\approx-1.6\times 10^{-7}$, agreeing well with the orange
curve in Fig. 1.

To account for the observed baryon asymmetry,
$\eta_{B}\equiv(n_{B}-n_{\overline{B}})/s\approx 8.6\times 10^{-11}$ where
$s\approx 7.04n_{\gamma}$ for the present universe or $192.2n_{\gamma}$ when
all SM species are relativistic, wash-in leptogenesis requires
$q_{e}/n_{\gamma}=192.2\eta_{B}\times(-10/3)\times(79/28)=-1.6\times 10^{-7}$.
Therefore, the above benchmark produces exactly the observed baryon asymmetry.

If the asymmetron mass is well above the initial temperature of the PBHs,
$m_{X}\gg T_{{\rm BH}0}$, then we also need to take into account that the
emission of $X$ becomes significant only when the PBH temperature increases to
$T_{{\rm BH}}\gtrsim{\cal O}(m_{X})$. In practice, we find that it is a good
approximation to replace $T_{{\rm BH}}^{{\rm eq}}\to\max(T_{{\rm BH}}^{{\rm
eq}},\ m_{X}/3.6)$ in Eq. (15) to account for the heavy mass suppression.

Here we would like to compare our calculation to a similar calculation in Ref.
Bernal:2022pue , which also analytically estimated the production of heavy
generic particles ($X$) from PBHs and the associated asymmetry. In Ref.
Bernal:2022pue , the authors assume that all $X$ particles decay only after
the PBHs fully evaporate, while here we assume that the decay is rapid,
causing a balance between the production and depletion of $X$. The two
different scenarios correspond to small and large $y$. While we do find that
there is a limited space (see Sec. V) for the former scenario to generate the
required asymmetry predominantly from PBHs, we mainly focus on the latter as
the viable parameter space is significantly larger. In fact, our Eq. (15) can
be readily applied to the scenario with small $y$ and long-lived $X$, because
Eq. (14) does not involve any assumptions on the evolution of $n_{X}$. As we
have checked, Eq. (15) approximately reproduces the analytical result of Ref.
Bernal:2022pue  in the weak washout regime.

##  IV The Boltzmann approach

Figure 2: Numerical solutions of the Boltzmann equations. From left to right
panels, we select three three representative examples to demonstrate i) strong
washout with late PBH evaporation; ii) strong washout with early PBH
evaporation; iii) weak washout. The labels “TH” and “PBH” denote thermal and
PBH contributions, and “TH+PBH” denotes their sum. The black dotted and dashed
lines in the lower panels are analytical estimates given by Eqs. (15) and
(20).

Figure 3: The viable parameter space in the $y$-$m_{X}$ plane. The gray and
colored regions lead to insufficient and sufficient production of the $q_{e}$
asymmetry, respectively. The color bars indicate the value of
$\log_{10}(\Delta_{2}/\Delta_{1})$ where $\Delta_{1,2}$ denote the TH and PBH
contributions to the asymmetry.

Figure 4: The viable parameter space in the $m_{{\rm BH0}}$-$m_{X}$ plane. The
color coding is the same as Fig. 4.

Our analytical approach in Sec. III allows us to understand the evolution of
the PBH-produced asymmetron $X$ qualitatively, and to estimate the resulting
asymmetry accurately within certain ranges. Nevertheless, a numerical approach
based on solving Boltzmann equations is desirable for more accurate and
comprehensive calculations.

The Boltzmann equations governing the evolution of $X$ and the electron
asymmetry read

$\displaystyle\frac{dY_{1}}{da}$
$\displaystyle=\frac{\Gamma_{X}r_{1}}{Ha}\left[Y^{{\rm eq}}-Y_{1}\right],$
(16) $\displaystyle\frac{dY_{2}}{da}$
$\displaystyle=\frac{1}{Ha}\left[a^{3}n_{\text{BH}}\Gamma_{\text{BH}\to
X}-Y_{2}\Gamma_{X}r_{2}\right],$ (17) $\displaystyle\frac{d\Delta_{1}}{da}$
$\displaystyle=\frac{\Gamma_{X}}{Ha}\left[\epsilon r_{1}\left(Y_{1}-Y^{{\rm
eq}}\right)-\frac{r_{1}}{2}\frac{n_{X}^{\text{eq}}}{n_{e_{R}}^{\text{eq}}}\Delta_{1}\right],$
(18) $\displaystyle\frac{d\Delta_{2}}{da}$
$\displaystyle=\frac{\Gamma_{X}}{Ha}\left[\epsilon
r_{2}Y_{2}-\frac{r_{1}}{2}\frac{n_{X}^{\text{eq}}}{n_{e_{R}}^{\text{eq}}}\Delta_{2}\right],$
(19)

where $Y\equiv n_{X}a^{3}$ and $\Delta\equiv q_{e}a^{3}$ with $a$ the scale
factor. The subscripts 1 and 2 denoting the thermal and PBH contributions,
respectively. The superscript “eq” indicates equilibrium values. The
relativistic factors $r_{1,2}\equiv\langle m_{X}/E\rangle_{1,2}$ account for
the time dilution of relativistic $X$ decay.

The last terms in Eqs. (18) and (19) are the so-called washout terms. They are
proportional to $\Delta_{1,2}$ and impose exponential suppression on the
produced asymmetry until the coefficient
$n_{X}^{\text{eq}}/n_{e_{R}}^{\text{eq}}$ becomes negligibly small.

Equations (16) and (18) can be solved analytically by making a few
approximations in the strong and weak washout regimes and combing the results
via analytical interpolation Buchmuller:2004nz . The result is

$\frac{\Delta_{1}}{n_{\gamma}a^{3}}=\frac{3}{8}\epsilon\kappa_{f},$ (20)

where $\kappa_{f}$ is a complicated function of $K\equiv\Gamma_{X}/H|_{T\to
m_{X}}$. Its full expression is given in Appendix B. In the weak washout
regime ($K\ll 1$), $\kappa_{f}$ reduces to $9\pi^{2}K^{2}/64$. Equations (20)
with the full expression of $\kappa_{f}$ can provide a rather accurate
estimate of the thermal contribution to $q_{e}$—see the black dashed lines in
the lower panels of Fig. 2.

Numerically solving the Boltzmann equations is straightforward. Figure 2 shows
the numerical solutions for three examples representing three different
regimes, as discussed below.

$\blacksquare$ (i): strong washout with late PBH evaporation.

In the left panels, we set $m_{{\rm BH0}}=1$ gram, $\beta=10^{-4}$,
$m_{X}=10^{13}$ GeV, $y=1$ and $\epsilon=y^{2}/(8\pi)$. The $K$ value of this
example is large, $K\approx 1.4\times 10^{3}\gg 1$. As is shown in the plot,
the large coupling $y$ leads to strong washout within ${\cal O}(0.1)\lesssim
m_{X}/T\lesssim{\cal O}(10)$.222The sharp drop at $m_{X}/T\approx 0.4$ is
caused by $q_{e}$ crossing zero. The thermal production of the asymmetry is
washed out rapidly after $X$ enters equilibrium and before $n_{X}^{{\rm eq}}$
is exponentially suppressed, while the PBH production of the asymmetry can
avoid this strong washout if the evaporation lasts sufficiently long.

$\blacksquare$ (ii): strong washout with early PBH evaporation.

For comparison, in the middle panels we show another example with $m_{{\rm
BH0}}=1$ gram, $\beta=10^{-3}$, $m_{X}=10^{11}$ GeV, $y=1$ and
$\epsilon=y^{2}/(8\pi)$. Due to the smaller $m_{X}$, the PBHs fully evaporate
within the strong washout regime. Hence the PBH production of the asymmetry is
almost entirely washed out.

$\blacksquare$ (iii): weak washout.

If the coupling is sufficiently weak, $X$ cannot reach thermal equilibrium at
temperatures relevant for the production of asymmetry. This is in the weak
washout regime. In the right panels, we plot an example with $m_{{\rm BH0}}=1$
gram, $\beta=10^{-4}$, $m_{X}=10^{13}$ GeV, $y=10^{-3}$ and
$\epsilon=y^{2}/(8\pi)$. The $K$ value of this example is small, $K\approx
3.5\times 10^{-4}\ll 1$, leading to only weak washout of the thermal and PBH
contributions to the asymmetry. As is shown in the lower right panel, the
asymmetry $|q_{e}|$ produced from PBHs is many orders of magnitude higher than
the thermal one. This is a noteworthy difference, given that the thermal and
PBH productions of $n_{X}$ shown in the upper panel are roughly comparable.
The reason for this substantial difference is that the PBH curve for $n_{X}$
deviates more from the equilibrium curve than the thermal one.

From the three examples, we can see that the PBH lifetime is crucial for the
asymmetry produced from PBHs to evade strong washout and to dominate over the
thermal production. Note, however, that their lifetime cannot be too long if
PBHs are employed in wash-in leptogenesis, which requires that some SM yukawa
interactions are not equilibrated. For wash-in leptogenesis via $|q_{e}|$, it
requires that the PBHs should evaporate before the electron yukawa interaction
reaches equilibrium, i.e. $T_{{\rm eva}}\gtrsim 100$ TeV, corresponding to
$m_{\text{BH0}}\lesssim 4.6$ kg.

##  V Full scan of the parameter space

To identify the viable parameter space, we numerically solve the Boltzmann
equations for $m_{\text{BH0}}\in[0.1,\ 10^{3}]$ gram, $m_{X}\in[10^{7},\
10^{16}]$ GeV, and $y\in[10^{-3},\ 1]$. This allows us to perform a scan of
the parameter space. The results are presented in Figs. 4 and 4, where the
colored/gray regions produce sufficient/insufficient baryon asymmetry.

The upper bound of $m_{{\rm BH}0}$ is set at $10^{3}$ gram according to the
discussion at the end of Sec. IV. The lower bound is set at $0.1$ gram because
the formation of lighter PBHs would require $T_{0}\gtrsim 10^{16}$ GeV, which
would be above the energy scale of inflation constrained by the Planck 2018
data Planck:2018jri .

The asymmetry produced by PBHs is proportional to $\beta$ and $\epsilon$. For
$\beta$, we set $\beta=2\times 10^{-5}\times(1\ {\rm g}/m_{{\rm BH0}})$ so
that the universe is always radiation dominated during PBH evaporation. Larger
$\beta$ is possible but the evolution involves PBH domination—see e.g.
Bernal:2022pue . If one further increases $\beta$ to values above $1.1\times
10^{-6}\times(10^{4}\ \text{g}/m_{{\rm BH0}})^{17/24}$, it would lead to an
overly large contribution to the cosmological $N_{{\rm eff}}$ parameter via
gravitational waves Domenech:2020ssp . As for $\epsilon$, we only require that
it is below $y^{2}/(8\pi)$. In practice, we set $\epsilon$ at this upper limit
and compute the resulting asymmetry. If it is above the required value, then
we consider it as a viable sample point, since the produced asymmetry can be
readjusted to the desired value by reducing $\epsilon$.

In Fig. 4 we present the parameter space in the the $y$-$m_{X}$ plane with
$m_{{\rm BH0}}$ fixed at $0.1$, $1$, $10$ and $1000$ gram. In these plots,
there are some red or yellow regions in which the asymmetry is dominantly
produced by PBH. Typically these regions are in the regime of strong washout
with late PBH evaporation, corresponding to regime (i) discussed in Sec. IV.
However, for $m_{{\rm BH0}}=0.1$ gram, there is a small yellow region around
$y\sim 2\times 10^{-2}$ and $m_{X}\sim 3\times 10^{14}$ GeV. This regime
actually corresponds to the weak washout regime. Note that as $m_{{\rm BH0}}$
increases, the viable parameter space for weak washout quickly vanishes, while
the viable parameter space in the strong washout regime remains robust.

In Fig. 4, we fix $y$ at given values and scan over $m_{X}$ and $m_{{\rm
BH0}}$. The left panel in Fig. 4 implies that for $y=1$, late PBH evaporation
generally leads to sufficient production of $q_{e}$. For example, $X$ emitted
from PBHs with $m_{{\rm BH0}}=10^{3}$ g can have a quite broad mass range,
$m_{X}\in[10^{7},\ 10^{11}]$ GeV, to support sufficient and dominant PBH
production of $q_{e}$. This mass range shrinks as $m_{{\rm BH0}}$ decreases,
which is expected since smaller $m_{{\rm BH0}}$ implies earlier PBH
evaporation. When $y$ is below certain values, the region favored by the
regime of strong washout with late PBH evaporation vanishes quickly, as is
shown in the right panel in Fig. 4. This is caused by the suppression of the
asymmetric decay parameter $\epsilon$ which is proportional to $y^{2}$.

##  VI Summary and Conclusions

Wash-in leptogenesis offers a novel mechanism for generating the baryon
asymmetry of the universe by utilizing possible primordial charge asymmetries
not necessarily limited to the commonly considered cases of $B$, $L$ or $B-L$
charges. The primordial charge asymmetries required for wash-in leptogenesis
can be readily produced by evaporating primordial black holes (PBHs).

To demonstrate the viability of this idea, we consider a rather simple
scenario, in which PBHs emit heavy particles (dubbed as the asymmetrons) that
can asymmetrically decay to electrons and thus generate the primordial
electron asymmetry, $q_{e}$. If the asymmetrons are emitted by PBHs at a
relatively late epoch of the universe, the generated asymmetry can effectively
evade strong wash-out, as illustrated by Fig. 1 for a specific benchmark.

By solving the Boltzmann equations with both PBH and thermal contributions
taken into account, we find that there are three very different regimes: (i)
strong washout with late PBH evaporation, (ii) strong washout with early PBH
evaporation, and (iii) weak washout. For each regime, we demonstrate an
example in Fig. 2. Our results reveal that the PBH lifetime is crucial for the
asymmetry produced from PBHs to evade strong washout and to dominate over the
thermal production.

We further conduct a full scan of the parameter space (see Figs. 4 and 4) and
find that strong washout with late PBH evaporation is the most favored regime.
The former only requires that the coupling of the asymmetron with the electron
is not suppressed, while the latter requires relatively large PBH masses,
corresponding to longer lifetimes. For instance, an ${\cal O}(1)$ asymmetron
coupling combined with an initial mass of $10^{3}$ g for the PBHs can easily
lead to sufficient primordial asymmetry before wash-in leptogenesis, if the
asymmetron mass is within $[10^{7},\ 10^{11}]$ GeV.

In conclusion, our research demonstrated that PBHs in joint with wash-in
leptogenesis can act as a very generic and efficient “asymmetry-producing
machine”, since any asymmetry injected by PBHs into the SM thermal bath at a
sufficiently early epoch would eventually be reprocessed by wash-in
leptogenesis and converted to the baryon asymmetry. Our novel approach
features invulnerability to strong washout and the capability to employ more
generic $CP$ violating sources. This opens new avenues for exploring the
origin of the baryon asymmetry.

###### Acknowledgements.

The authors would like to thank the organizers of the CERN Neutrino Platform
Pheno Week in March 2023, where this project was initiated. The work of K. S.
is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research
Training Group, GRK 2149: Strong and Weak Interactions — from Hadrons to Dark
Matter. X.-J. X. is supported in part by the National Natural Science
Foundation of China under grant No. 12141501 and also supported by CAS Project
for Young Scientists in Basic Research (YSBR-099).

##  Appendix A Basic formulae for PBH evaporation

Schwarzschild PBHs radiate particles at the rate given by Eq. (3). The
graybody factor $v_{i}$ in Eq. (3) is suppressed for $T_{{\rm BH}}\ll m_{i}$
where $m_{i}$ is the mass of the particle emitted. For $T_{{\rm BH}}\gg
m_{i}$, it behaves like blackbody radiation. The actual dependence of the
graybody factor on the energy and the mass can be quite complicated (see e.g.
Cheek:2021odj ; Masina:2021zpu ; Auffinger:2020afu  for recent discussions).
In this work, we approximate the graybody factor as follows Bernal:2022pue :

$v_{i}=\frac{27E^{2}}{64\pi^{2}T_{{\rm BH}}^{2}}\cdot\Theta\thinspace,$ (21)

where

$\Theta\equiv\begin{cases}1&m_{i}/T_{{\rm BH}}<x_{{\rm cri.}}\\\
0&m_{i}/T_{{\rm BH}}>x_{{\rm cri.}}\end{cases}\thinspace,\ x_{{\rm
cri.}}\approx 3.6\thinspace.$ (22)

With the graybody factor in Eq. (21), we can compute the energy loss rate of a
PBH:

$\frac{dm_{{\rm BH}}}{dt}=-\sum_{i}\int\frac{d^{2}N_{i}}{dtdE}EdE=g_{{\rm
BH}}\frac{m_{{\rm pl}}^{4}}{m_{{\rm BH}}^{2}}\thinspace,$ (23)

where $g_{{\rm BH}}\approx 7\times 10^{-5}g_{\star}$ and $g_{\star}$ counts
all bosonic ($\times 1$) and fermionic ($\times 7/8$) degrees of freedom that
are lighter than $T_{{\rm BH}}$. The PBHs considered in this work typically
have $T_{\text{BH}}\gg$TeV so all SM particles should be included as massless
degrees of freedom. Assuming the dominance of the SM contribution to $g_{{\rm
BH}}$, we can treat $g_{{\rm BH}}$ as a constant: $g_{{\rm BH}}\approx
7.5\times 10^{-3}$.

For constant $g_{{\rm BH}}$, Eq. (23) has the following analytical solution:

$m_{{\rm BH}}=m_{{\rm BH}0}\left(1-3g_{{\rm
BH}}m_{\text{pl}}^{4}\frac{t-t_{0}}{m_{{\rm BH}0}^{3}}\right)^{1/3}.$ (24)

where $t_{0}$ and $m_{{\rm BH}0}=m_{{\rm BH}}(t_{0})$ denote the initial time
and mass of the PBH. Eq. (24) implies that the PBH mass vanishes at
$t-t_{0}=\tau_{{\rm BH}}$ with

$\tau_{{\rm BH}}=\frac{m_{{\rm BH}0}^{3}}{3g_{{\rm
BH}}m_{\text{pl}}^{4}}\thinspace,$ (25)

which is the lifetime of the PBH.

The emission rate of $X$ from a PBH, according to Eqs (3) and (21), is given
by

$\Gamma_{\text{BH}\to
X}\equiv\int\frac{d^{2}N_{X}}{dtdE}dE\approx\frac{81\zeta(3)}{256\pi^{3}}T_{{\rm
BH}}\cdot\Theta\approx 0.012T_{{\rm BH}}\cdot\Theta\thinspace.$ (26)

If $X$ decays rapidly after the emission, then the energy spectrum is
proportional to Eqs (3).

If $X$ is sufficiently long-lived, then the energy spectrum is computed by

$\frac{dN_{X}}{dE}=\int\frac{d^{2}N_{X}}{dtdE}dt=\int_{T_{{\rm
BH0}}}^{\infty}\frac{d^{2}N_{X}}{dtdE}\frac{8\pi m_{{\rm
BH}}^{4}}{g_{\text{BH}}m_{\text{pl}}^{6}}dT_{{\rm BH}}\thinspace.$ (27)

Assuming the Boltzmann statistics and $T_{{\rm BH}}\gtrsim m_{X}/x_{{\rm
cri.}}$, we obtain

$\frac{dN_{X}}{dE}=\frac{81m_{\text{pl}}^{2}\left[1-e^{-x}\left(1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\right)\right]}{8192\pi^{6}g_{\text{BH}}T_{{\rm
BH0}}^{3}x^{3}}\thinspace,x\equiv E/T_{{\rm BH0}}\thinspace.$ (28)

Note that $1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}=e^{x}+{\cal
O}(x^{5})$ so at small $x$, Eq. (28) is proportional to
$x^{5}e^{-x}/x^{3}\approx x^{2}e^{-x}$. At $x\gg 1$, Eq. (28) implies

$\frac{dN_{X}}{dE}\approx\frac{81m_{\text{pl}}^{2}}{8192\pi^{6}g_{\text{BH}}T_{{\rm
BH0}}^{3}x^{3}}\thinspace,(\text{for}\ x\gg 1)\thinspace.$ (29)

Therefore, unlike usual thermal distributions which is suppressed by $e^{-x}$
at large $x$, Eq. (28) features a very “hard” spectrum at high energies.

In Sec. IV, we introduced the $r_{2}$ factor, which is evaluated as follows:

$r_{2}\equiv\left\langle\frac{m_{X}}{E}\right\rangle_{2}=\frac{\int\frac{m_{X}}{E}fd^{3}p}{\int
fd^{3}p}=\begin{cases}0.46\frac{m_{X}}{T_{{\rm BH}}}&\text{for short-lived
}X\\\ \frac{1}{3}\frac{m_{X}}{T_{{\rm BH}0}}&\text{for long-lived
}X\end{cases}.$ (30)

Here $f$ takes $\frac{d^{2}N_{X}}{dtdE}$ in Eq. (3) for short-lived $X$, or
$\frac{dN_{X}}{dE}$ in Eq. (28) for long-lived $X$.

##  Appendix B Analytical solutions of the Boltzmann equations

The Boltzmann equations considered in this work can be written into the
following form

$\frac{dY(a)}{da}=X(a)-F(a)Y(a)\thinspace,$ (31)

where $X(a)$ and $F(a)$ denote some generic functions of $a$. It is known that
differential equations in the form of Eq. (31) has the following solution:

$Y(a^{\prime})=G(a^{\prime})\int_{0}^{a^{\prime}}\frac{X}{G}da\thinspace,\ \
G(a^{\prime})\equiv e^{-\int_{0}^{a^{\prime}}Fda}\thinspace.$ (32)

In principle, Eq. (32) can be used to analytically solve the Boltzmann
equations. In practice, one has to take various approximations when applying
it to specific processes. For the thermal production of asymmetry considered
in this work, we can adopt results from a similar calculation in Ref.
Buchmuller:2004nz —see Eqs. (62), (63), (68), and (69) therein. The result is

$\frac{q_{e}}{n_{\gamma}}=\frac{3}{8}\epsilon\kappa_{f}\thinspace,$ (33)

where the full expression of the $\kappa_{f}$ function is given by

$\kappa_{f}(K)\approx
2\frac{1-e^{-\frac{2}{3}z_{B}K\overline{N}}}{z_{B}K}-2e^{-\frac{2}{3}N}\left(e^{\frac{2}{3}\overline{N}}-1\right),$
(34)

with

$\displaystyle z_{B}$ $\displaystyle\approx 1+\frac{1}{2}\log\left[1+\frac{\pi
K^{2}}{1024}\left(\log\frac{3125\pi K^{2}}{1024}\right)^{5}\right],$ (35)
$\displaystyle N$ $\displaystyle=\frac{9\pi K}{16}\thinspace,$ (36)
$\displaystyle\overline{N}$
$\displaystyle=\frac{N}{\left(1+\sqrt{4N/3}\right)^{2}}\thinspace.$

## References

  * (1) Planck Collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6, [1807.06209]. [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * (2) Particle Data Group Collaboration, R. L. Workman et al., Review of Particle Physics, PTEP 2022 (2022) 083C01.
  * (3) M. Fukugita and T. Yanagida, Baryogenesis Without Grand Unification, Phys. Lett. B 174 (1986) 45–47.
  * (4) V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov, On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe, Phys. Lett. B 155 (1985) 36.
  * (5) W. Buchmuller, R. D. Peccei, and T. Yanagida, Leptogenesis as the origin of matter, Ann. Rev. Nucl. Part. Sci. 55 (2005) 311–355, [hep-ph/0502169].
  * (6) E. J. Chun et al., Probing Leptogenesis, Int. J. Mod. Phys. A 33 (2018), no. 05n06 1842005, [1711.02865].
  * (7) D. Bodeker and W. Buchmuller, Baryogenesis from the weak scale to the grand unification scale, Rev. Mod. Phys. 93 (2021), no. 3 035004, [2009.07294].
  * (8) S. Davidson and A. Ibarra, A Lower bound on the right-handed neutrino mass from leptogenesis, Phys. Lett. B 535 (2002) 25–32, [hep-ph/0202239].
  * (9) W. Buchmuller, P. Di Bari, and M. Plumacher, Cosmic microwave background, matter - antimatter asymmetry and neutrino masses, Nucl. Phys. B 643 (2002) 367–390, [hep-ph/0205349]. [Erratum: Nucl.Phys.B 793, 362 (2008)].
  * (10) A. Pilaftsis and T. E. J. Underwood, Resonant leptogenesis, Nucl. Phys. B 692 (2004) 303–345, [hep-ph/0309342].
  * (11) A. Pilaftsis and T. E. J. Underwood, Electroweak-scale resonant leptogenesis, Phys. Rev. D 72 (2005) 113001, [hep-ph/0506107].
  * (12) B. Dev, M. Garny, J. Klaric, P. Millington, and D. Teresi, Resonant enhancement in leptogenesis, Int. J. Mod. Phys. A 33 (2018) 1842003, [1711.02863].
  * (13) E. K. Akhmedov, V. A. Rubakov, and A. Y. Smirnov, Baryogenesis via neutrino oscillations, Phys. Rev. Lett. 81 (1998) 1359–1362, [hep-ph/9803255].
  * (14) M. Drewes, B. Garbrecht, P. Hernandez, M. Kekic, J. Lopez-Pavon, J. Racker, N. Rius, J. Salvado, and D. Teresi, ARS Leptogenesis, Int. J. Mod. Phys. A 33 (2018), no. 05n06 1842002, [1711.02862].
  * (15) J. Klarić, M. Shaposhnikov, and I. Timiryasov, Uniting Low-Scale Leptogenesis Mechanisms, Phys. Rev. Lett. 127 (2021), no. 11 111802, [2008.13771].
  * (16) J. Klarić, M. Shaposhnikov, and I. Timiryasov, Reconciling resonant leptogenesis and baryogenesis via neutrino oscillations, Phys. Rev. D 104 (2021), no. 5 055010, [2103.16545].
  * (17) V. Domcke, K. Kamada, K. Mukaida, K. Schmitz, and M. Yamada, Wash-In Leptogenesis, Phys. Rev. Lett. 126 (2021), no. 20 201802, [2011.09347].
  * (18) V. Domcke, K. Kamada, K. Mukaida, K. Schmitz, and M. Yamada, Wash-in leptogenesis after axion inflation, JHEP 01 (2023) 053, [2210.06412].
  * (19) M. M. Anber and E. Sabancilar, Hypermagnetic Fields and Baryon Asymmetry from Pseudoscalar Inflation, Phys. Rev. D 92 (2015), no. 10 101501, [1507.00744].
  * (20) D. Jimenez, K. Kamada, K. Schmitz, and X.-J. Xu, Baryon asymmetry and gravitational waves from pseudoscalar inflation, JCAP 1712 (2017), no. 12 011, [1707.07943].
  * (21) V. Domcke, B. von Harling, E. Morgante, and K. Mukaida, Baryogenesis from axion inflation, JCAP 10 (2019) 032, [1905.13318].
  * (22) A. Escrivà, F. Kuhnel, and Y. Tada, Primordial Black Holes, 2211.05767.
  * (23) S. W. Hawking, Black hole explosions, Nature 248 (1974) 30–31.
  * (24) B. J. Carr, Some cosmological consequences of primordial black-hole evaporations, Astrophys. J. 206 (1976) 8–25.
  * (25) J. D. Barrow, E. J. Copeland, E. W. Kolb, and A. R. Liddle, Baryogenesis in extended inflation. 2. Baryogenesis via primordial black holes, Phys. Rev. D 43 (1991) 984–994.
  * (26) D. Baumann, P. J. Steinhardt, and N. Turok, Primordial Black Hole Baryogenesis, hep-th/0703250.
  * (27) L. Morrison, S. Profumo, and Y. Yu, Melanopogenesis: Dark Matter of (almost) any Mass and Baryonic Matter from the Evaporation of Primordial Black Holes weighing a Ton (or less), JCAP 05 (2019) 005, [1812.10606].
  * (28) D. Hooper and G. Krnjaic, GUT Baryogenesis With Primordial Black Holes, Phys. Rev. D 103 (2021), no. 4 043504, [2010.01134].
  * (29) Y. F. Perez-Gonzalez and J. Turner, Assessing the tension between a black hole dominated early universe and leptogenesis, Phys. Rev. D 104 (2021), no. 10 103021, [2010.03565].
  * (30) S. Datta, A. Ghosal, and R. Samanta, Baryogenesis from ultralight primordial black holes and strong gravitational waves from cosmic strings, JCAP 08 (2021) 021, [2012.14981].
  * (31) S. Jyoti Das, D. Mahanta, and D. Borah, Low scale leptogenesis and dark matter in the presence of primordial black holes, JCAP 11 (2021) 019, [2104.14496].
  * (32) N. Bernal, C. S. Fong, Y. F. Perez-Gonzalez, and J. Turner, Rescuing high-scale leptogenesis using primordial black holes, Phys. Rev. D 106 (2022), no. 3 035019, [2203.08823].
  * (33) R. Calabrese, M. Chianese, J. Gunn, G. Miele, S. Morisi, and N. Saviano, Limits on light primordial black holes from high-scale leptogenesis, Phys. Rev. D 107 (2023), no. 12 123537, [2305.13369].
  * (34) S. Khan and R. Adhikari, Primordial Black Hole Leptogenesis in Supersymmetry, 2304.02604.
  * (35) W. Buchmuller, P. Di Bari, and M. Plumacher, Leptogenesis for pedestrians, Annals Phys. 315 (2005) 305–351, [hep-ph/0401240].
  * (36) Planck Collaboration, Y. Akrami et al., Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641 (2020) A10, [1807.06211].
  * (37) G. Domènech, C. Lin, and M. Sasaki, Gravitational wave constraints on the primordial black hole dominated early universe, JCAP 04 (2021) 062, [2012.08151]. [Erratum: JCAP 11, E01 (2021)].
  * (38) A. Cheek, L. Heurtier, Y. F. Perez-Gonzalez, and J. Turner, Primordial black hole evaporation and dark matter production. I. Solely Hawking radiation, Phys. Rev. D 105 (2022), no. 1 015022, [2107.00013].
  * (39) I. Masina, Dark Matter and Dark Radiation from Evaporating Kerr Primordial Black Holes, Grav. Cosmol. 27 (2021), no. 4 315–330, [2103.13825].
  * (40) J. Auffinger, I. Masina, and G. Orlando, Bounds on warm dark matter from Schwarzschild primordial black holes, Eur. Phys. J. Plus 136 (2021), no. 2 261, [2012.09867].