#  Tension between HST/JWST and $\Lambda$CDM Cosmology, PBH, and Antimatter in
the Galaxy

A.D. Dolgov<EMAIL_ADDRESS>Novosibirsk State University, Novosibirsk, 630090,
Russia  Bogolyubov Laboratory of Theoretical Physics, JINR, Dubna, 141980,
Russia

###### Abstract

Recent data released by James Webb Space Telescope (JWST) and, somewhat
earlier, the data presented by Hubble Space Telescope (HST) are commonly
understood as a strong indication for breaking of the canonical $\Lambda$CDM
cosmology. It is argued in the presented work that massive primordial black
holes (PBH) could seed galaxy and quasar formation in the very young universe
as it has been conjectured in our paper of 1993 and resolve the tension
induced by the JWST and the HST data with the standard cosmology. This point
of view is presently supported by several recent works. The proposed mechanism
of PBH formation leads to the log-normal mass spectrum of PBHs and predicts
abundant antimatter population of our Galaxy, Milky Way. Both these
predictions are in excellent agreement with astronomical observations.

##  I Introduction

During last decade observations made by Hubble Space Telescope (HST), see e.g.
hst-1 ; hst-2 ; hst-3  and very recently by James Webb Space Telescope (JWST)
jwst-1 ; jwst-2 ; jwst-3 ; jwst-4  have led to the surprising conclusion that
the early universe, younger than one billion years, is densely populated by
well developed galaxies, quasars (supermassive black holes), gamma-bursters,
and heavy elements (heavier than helium). These striking results were taken by
the community as absolutely incompatible with the canonical $\Lambda$CDM
cosmology, especially after release of the JWST data. In fact already
observations of HST could be a sufficient cause for anxiety, not only with
respect to the early universe but also to the contemporary very old universe
almost 15 billion years old. The troubling situation in the present day
universe as well as in the universe with redshifts $z=6-10$ are summarised in
review AD-UFN . The state of art is emphatically characterised as crisis in
cosmology that is believed to hit strong blow to the conventional $\Lambda$CDM
picture.

However, the resolution of the above mentioned problems was suggested in our
papers DS  (DS) and DKK  (DKK). long before these problems arose. In these
works a new mechanism of massive primordial black hole (PBH) formation was
worked out that could lead to their efficient creation with the masses in the
range from a fraction of the solar mass up to billion solar masses.

An essential input of DS and DKK papers is the suggestion of an inverted
formation mechanism of galaxies and their central black holes. Usually it is
assumed that supermassive BHs (SMBHs), that are observed in centres of all
large galaxies, are created by matter accretion to the density excess in the
galactic centre, but the estimated necessary time is much longer than the
universe age, even for the contemporary universe, with the age about 15
billion years, to say nothing about the 20 times younger universe at $z\sim
10$.

On the opposite, as it was conjectured in refs. DS ; DKK , supermassive black
holes were created first in the early universe at prestellar epoch, that’s why
they are caller primordial, and later they have SEEDED galaxy formation.

Model DS/DKK is verified by a very good agreement of the calculated log-normal
mass spectrum of PBH with observations and by discovery of abundant antimatter
population in the Galaxy envisaged according to DS and DKK. The model also
predicts an early formation of galaxies, seeded by PBH, quasars (alias SMBH),
rich chemistry (heavy elements), and dust in the early universe.

##  II A few words about HST and JWST observations

The orbit of HST is at the distance of 570 km from the Earth. The orbit of
JWST is much larger, it is about $1.5\times 10^{6}$ km. The mirror of HST has
diameter equal to 2.4 m, while JWST has 2.7 time larger one and
correspondingly the area of JWST mirror is approximately 7.4 times larger. In
fig. 1 the images of HST and JWST are presented.

Figure 1:

HST operates in optical wave length range, for example 450 nm, corresponding
to blue light. It has also a possibility to catch the signal in the infrared
range with the wave length 0.8-2.5 microns. JWST has high sensitivity to
infrared radiation with the wave length 0.6 - 28,5 micron. It allows to
penetrate deep into the early universe, up to redshifts $z\sim 15$.

Accidentally HST and JWST observed the same galaxy at $z=12$, see fig. 2. This
coincidence is a strong argument in favour of the reliable operation of these
two very different instruments.

Figure 2:

Comparison of the JWST data and theoretical expectation of the $\Lambda$CDM
cosmology is depicted in fig. 3. Theoretical expectations (colorored dots at
$z=15$) are noticeably below observations.

Figure 3:

##  III Spectral measurements and puzzles of the early universe

Only continuum in micron range was measured by JWST till February. That raised
justified doubts on accuracy of the redshifts determination of the observed
galaxies. Now numerous observations of spectra of different elements
excellently confirm the early data. For example according to ref. spectrum-z11
the JWST NIRCam 9-band near-infrared imaging of the luminous $z=10.6$ galaxy
GN-z11 from the JWST Advanced Deep Extragalactic Survey (JADES) proved that
the spectral energy distribution (SED) is entirely consistent with the
expected form of the high-redshift galaxy.

In a simultaneous work spectrum-z11-2  the spectroscopy of GN-z11, the most
luminous candidate $z>10$ Lyman break galaxy is presented. The nitrogen lines
are clearly observed. Quoting the authors: ”The spectroscopy confirms that
GN-z11 is a remarkable galaxy with extreme properties seen 430 Myr after the
Big Bang.”

Another example of spectral measurements by a different instrument: age of
most distant galaxy is confirmed with Oxygen observation. The radio telescope
array ALMA (Atacama Large Millimeter Array) has pin-pointed the exact cosmic
age of a distant JWST-identified galaxy, GHZ2/GLASS-z12, at 367 million years
after the Big Bang alma-z12 . The observations of the a spectral emission line
emitted by ionized Oxygen near the galaxy, red-shifted according to its age in
the early universe, confirms the JWST data. This data show that the JWST is
able to look out to record distances and, quoting the authors, heralds a leap
in our ability to understand the formation of the earliest galaxies in the
Universe.

A population of red candidate massive galaxies (stellar mass $>10^{10}$ solar
masses) at $7.4\lesssim z\lesssim 9.1$, 500–700 Myr after the Big Bang,
including one galaxy with a possible stellar mass of $\sim 10^{11}M_{\odot}$,
too massive to be created in so early universe, is observed in imp-gal .
Authors conclude that according to the ’science’ it is impossible to create so
well developed galaxies. ”May be they are supermassive black holes of the kind
never seen before. That might mean a revision of usual understanding of black
holes.”

Figure 4:

Clearly these ”black holes of the kind never seen before” nicely fit the
assertion that they are primordial, as suggested in refs. DS ; DKK .

Recent observation by ALMA alma-6.8  of an extremely massive reionization-era
galaxy with $M_{*}=1.7\times 10^{11}M_{\odot}$ at z = 6.853 with active
galactic nuclei (AGN) that have huge luminosity suggests that this object is
powered by ${\sim 1.6\times 10^{9}M_{\odot}}$ black hole if accreting closely
to the Eddington limit. It is nearly impossible to create so massive BH in the
early universe. But supermassive primordial black hole could easily feed such
monster

##  IV Rich chemistry in the early universe

According to the standard lore, light elements, deuterium, helium, and tiny
traces of lithium are created from the primordial protons and neutrons in the
early universe roughly during the first 100 seconds. This process is called
big bang nucleosynthesis (BBN). Heavier elements, the so called metals
(everything heavier than helium-4 are called metals in astrophysics) are
created through stellar nucleosynthesis. Next, supernova explosions populate
interstellar medium with metals.

Unexpectedly high abundances of heavy elements (high metallicity) are observed
in the very early universe by HST and JWST. For example, as reported in ref.
peng-metals , a study of a strongly lensed galaxy SPT0418-47 revealed its
mature metallicity, amounts of elements heavier than helium and hydrogen, such
as carbon, oxygen and nitrogen. According to the estimate of the team the
amount is comparable to that of the sun, which is more than 4 billion years
old and inherited most of its metals from previous generations of stars that
had 8 billion years to build them up. Analysis using optical strong line
diagnostics suggests that galaxy SPT0418-SE has near-solar elemental
abundance, while the ring appears to have supersolar metallicity O/H and N/O.

One more example of well developed chemistry, that demands too long evolution
if produced by the conventional mechanism is presented in ref. cameron .
Observations of GN-z11 with JWST/NIRSpec disclosed numerous oxygen, carbon,
nitrogen, and helium emission lines at $z=10.6$. The data prefers (N/O),
greater than 4 times solar and the derived $C/O\approx 30$ solar. Nitrogen
enhancement in GN-z11 cannot be explained by enrichment from metal-free
Population III stars. The suggested explanation is that yields from runaway
stellar collisions in a dense stellar cluster or a tidal disruption event
provide promising solutions to give rise to these unusual emission lines at
$z=10.6$, and explain the resemblance between GN-z11 and a nitrogen-loud
quasar.

High abundances of heavy elements may be a result of BBN with baryon-to-gamma
ratio close to unity, as it takes place in the DS DS  and DKK DKK  model, see
below.

##  V Seeding of galaxy formation by PBH

###  V.1 Seeding of early galaxies

The hypothesis pioneered by DS DS  and DKK DKK , that galaxy formation is
seeded by SMBH allows to understand the presence of SMBH in all large and
several small galaxies accessible to observation. This mechanism explains how
the galaxies observed by JWST in the very young universe might be created.
Presently it is rediscovered in several recent works.

As is stated in ref. liu , the recent observations with JWST have identified
several bright galaxy candidates at $z\gtrsim 10$, some of which appear
unusually massive (up to $\sim 10^{11}\ \rm M_{\odot}$). Such early formation
of massive galaxies is difficult to reconcile with standard $\Lambda\rm CDM$
predictions demanding very high star formation efficiency (SFE), possibly even
in excess of the cosmic baryon mass budget in collapsed structures. With an
idealized analysis based on linear perturbation theory and the Press-Schechter
formalism, the observed massive galaxy candidates can be explained, with lower
SFE than required in $\Lambda\rm CDM$, if structure formation is accelerated
by massive ($\gtrsim{10^{9}\ \rm M_{\odot}}$) PBHs that enhance primordial
density fluctuations.

Observations made by JWDST (and HST) of high-redshift quasars reveal that many
supermassive black holes were in place less than 700 Million years after the
Big Bang. In particular, in ref. bogdan-1  the detection of an X-ray-luminous
quasar powered by SMBH with the mass $\sim 4\times 10^{7}M_{\odot}$ in a
gravitationally-lensed galaxy, identified by JWST at $z\approx 10.3$, is
reported. As is stated by the authors, this mass is comparable to the inferred
stellar mass of its host galaxy, in contrast to the usual examples from the
local universe where mostly the BH mass is $\sim 0.1$% of the host galaxy’s
stellar mass. The combination of such a high BH mass and large BH-to-galaxy
stellar mass ratio $\sim 500$ Myrs after the Big Bang is consistent with a
picture wherein such BHs originated from heavy seeds. Let stress again, that
this detection suggests that early supermassive black holes originate from
heavy seeds.

However, the origin of the first BHs, that started the seeding, remains a
mystery. According to the authors. the seeds of the first BHs are postulated
to be either light i.e., $(10-100)M_{\odot}$ remnants of the first stars or
heavy i.e., $(10^{4}-10^{5})M_{\odot}$, originating from direct collapse of
gas clouds. The latter hypothesis is questionable, but a supermassive
primordial black hole would perfectly work.

In a subsequent paper seed-2  a support to the heavy seeding channel for the
formation of supermassive BHs within the first billion years of cosmic
evolution is also proposed. As is mentioned in this work, ”the James Webb
Space Telescope is now detecting early black holes (BHs) as they originate
from seeds to supermassive BHs. Recently Bogdan et al bogdan-1  reported the
detection of an X-ray luminous supermassive black hole, UHZ-1, with a
photometric redshift at $z>10$. Such an extreme source at this very high
redshift provides new insights on seeding and growth models for BHs given the
short time available for formation and growth. The resulting ratio of
$M_{BH}/M^{*}$ remains two to three orders of magnitude higher than local
values, thus lending support to the heavy seeding channel for the formation of
supermassive BHs within the first billion years of cosmic evolution.

###  V.2  Seeding of globular clusters and dwarf galaxies

The idea of seeding of globular clusters and dwarf galaxies by primordial
black holes was worked out in ref. AD-KP-dwarfs . Primordial IMBHs with masses
of a few thousand solar mass can explain their formation, poorly understood
otherwise. In the last several years such IMBHs inside globular clusters are
observed. Similar features are true for dwarfs. In particular the seeding of
dwarfs by intermediate mass BHs is confirmed by the recent data. For instance
in the dwarf galaxy SDSS J1521+1404 the BH is discovered with the mass $M\sim
10^{5}M_{\odot}$ seed-dwarf-1 . For the first time, astronomers have spotted
evidence of a pair of dwarf galaxies featuring GIANT black holes on a
collision course with each other. In fact, they haven’t just found just one
pair – they’ve found two.

Another recent example dwarf-2  of intermediate-mass black holes is the
finding of episodic, large-scale and powerful jet activity in a dwarf galaxy
SDSS J090613.77+561015.2. This can be explained by an intermediate-mass black
hole (IMBH) with a mass of $M_{BH}=3.6^{+5.9}_{-2.3}\times 10^{5}M_{\odot}$,
Such huge black hole surely could not be created by accretion but vice versa
might seed the formation of the dwarf.

##  VI Peculiar stars in the Galaxy

###  VI.1 Ancient stars

A discovery of primordial stars in globular cluster M92 prim-stars  was
recently announced. The absolute age of the globular cluster M92 was evaluated
and found to be practically equal to the universe age, ${t_{M92}=13.8\pm
0.75}$ Gyr. As it is stated in the paper, possibly these stars came to us from
JWST epoch or even from the earlier one.

Similar declaration of pristine stars in the Galaxy was made almost at the
same time pristine . An international team of researchers, Pristine Inner
Galaxy Survey (PIGS) team, has obtained the largest set of detailed
observations yet of the oldest stars in the center of our Galaxy, the Milky
Way. Some of the stars that were born in the first billion years after the Big
Bang are still around today.

In fact, extremely old stars in the Galaxy were discovered considerably
earlier. As it is asserted in ref. old-star-1 , new more accurate methods of
determination of stellar ages led to discovery of surprisingly old stars.
Employing thorium and uranium abundances in comparison with each other and
with several stable elements the age of metal-poor, halo star BD+17o 3248 was
estimated as $13.8\pm 4$ Gyr. For comparison the age of inner halo of the
Galaxy is 11.4 ± 0.7 Gyr halo-age

The age of a star in the galactic halo, HE 1523-0901, was estimated to be
about 13.2 Gyr. First time many different chronometers, such as the U/Th,
U/Ir, Th/Eu and Th/Os ratios to measure the star age have been employed old-
star-2 .

And now, the most surprising star which is older than the Universe. Metal
deficient high velocity subgiant in the solar neighborhood HD 140283 has the
age 14.46 ± 0.31 Gyr [28]. The determined central value of the age exceeds the
universe age by two standard deviations if the Hubble parameter is low, H =
67.3 (according to the CMB analysis) and $t_{U}=13.8$ Gyr, if H = 74
(according to the traditional methods), and $t_{U}=12.5$ Gyr. The age of this
star exceeds the universe age more than by 10 $\sigma$.

In our model DS ; DKK  not only primordial black holes could be formed but, if
the bubbles with high baryon-to-photon ratio are not sufficiently massive,
compact stellar kind objects could be created. Such ”stars” might look older
than they are because they would be enriched with heavy elements mimicking
larger age.

###  VI.2 Fast moving stars

Several stars are discovered in the Galaxy with unusually high velocity much
larger than the galactic virial velocity, that is about 200 km/sec. There are
several very fast pulsars in the Galaxy, but their origin is evident. Pulsars
are the results of supenova explosions and a small angular asymmetry in the
emitted radiation could create a strong kick, which would accelerate a pulsar
up to $10^{3}$ km/sec. The observed fast stars look normal, except of very
high velocity, about 500 km/sec.

In ref. vennes  a a discovery of a low mass white dwarf, LP 40-365, was
reported, that travels at a velocity greater than the Galactic escape velocity
and whose peculiar atmosphere is dominated by intermediate mass elements.
According to the authors these properties suggest that it could be the
predicted leftover remains from a type Iax supernova. On the other hand, it
can naturally be a primordial star with high initial abundances of heavy
elements.

Let us mention several more discoveries of other high velocity stars in the
Galaxy hattori ; marchetti . The authors argue that these stars could be
accelerated by a population of Iintermediate mass black holes (IMBHs) in
Globular clusters, if there is sufficient number of IMBHs. So many IMBHs were
not expected but the recent data reveal more and more of them in contrast to
conventional expectations and in agreement with refs. DS ; DKK .

As it is noted in ref. ruffini  observations of stellar remnants linked to
Type Ia and Type Iax supernovae are necessary to fully understand their
progenitors and explain the origin of their high speed. Multiple progenitor
scenarios predict a population of kicked donor remnants and partially-burnt
primary remnants, both moving with relatively high velocity. But only a
handful of examples consistent with these two predicted populations have been
observed. It is reported in ref. ruffini  that the likely first known example
of an unbound white dwarf that is consistent with being the fully-cooled
primary remnant to a Type Iax supernova is LP 93-21. The candidate, LP 93-21,
is travelling with a galactocentric velocity of $v_{gal}\approx 605$ km/sec,
and is gravitationally unbound to the Milky Way. The authors claim ruling out
its extragalactic origin. The Type Iax supernova ejection scenario is
consistent with its peculiar unbound trajectory, given the observed anomalous
elemental abundances. This discovery reflects recent models that suggest
stellar ejections likely occur often.

Let us repeat here that extragalactic primordial star presumably populating
the galactic halo, according to assertion of papers DS ; DKK , very well fits
the observations made in ref. ruffini

###  VI.3 Stars with unusual chemistry

An unusually red star was observed in planetary system through microlensing
event MOA-2011-BLG-291 bennet . The host star and planet masses are estimated
as $M_{host}=0.15^{+027}_{-0.10}M_{\odot}$ and
$m_{planet}=18^{+34}_{-12}M_{\oplus}$. The source star that is redder (or
brighter) than the bulge main sequence. The favoured interpretation by the
authors is that the source star is a lower main sequence star at a distance of
$4.9\pm 1.3$ kpc in the Galactic disk. According to the authors, the life-time
of main sequence star with the solar chemical content is larger than the
universe age already for $M<0.8M_{\odot}$. It implies the primordial origin of
the registered star with already evolved chemistry.

May it be a primordial helium star? There could be stars dominated by helium,
even purely helium stars, in our scenario DS ; DKK .

##  VII Pulsar humming

If a pulsar moves in any way, orbiting around a star, the relative motion of
the pulsar causes the pulses to shift slightly. These shifts can be measured
with extreme accuracy. The observations are so precise, pulsars were used to
measure the orbital decay of binary systems as indirect evidence of
gravitational waves long before they are observed directly.

Unexpectedly high number of SMBH binaries are presumably observed through
distortion of the pulsar timing by emission of gravitational waves puls-hum .
The NANOGrav 15 yr data set shows evidence for the presence of a low-frequency
gravitational-wave background. While many physical processes can source such
low-frequency gravitational waves, but most natural possibility seems to be
that the signal as coming from a population of supermassive black hole (SMBH)
binaries distributed throughout the Universe puls-hum .

It is difficult to explain such huge number of SMBH binaries. However, this
can be naturally expected if these SMBHs are primordial.

##  VIII Possible types of black holes

###  VIII.1 BH classification by mass

There is the following conventional division of black holes by their masses:
1\. Supermassive black holes (SMBH): $M=(10^{6}-10^{10})M_{\odot}$ (the record
mass is about $10^{11}M_{\odot}$).
2\. Intermediate mass black holes (IMBH): $M=(10^{2}-10^{5})M_{\odot}$.
3\. Solar mass black holes: masses from a fraction of $M_{\odot}$ up to
$100M_{\odot}$.
The origin of most of these BHs is unclear in the traditional approach, except
maybe of the BHs with masses of a few solar masses, that might be
astrophysical. Highly unexpected was a great abundance of IMBH which are
copiously appearing in observations during last few years.

The assumption that (almost) all black holes in the universe are primordial
strongly reduce or even eliminate the tension between the data and expected
numbers of black holes.

###  VIII.2 BH classification by formation mechanism

1\. Astrophysical black holes, created by the collapse of a star that
exhausted its nuclear fuel. The expected masses should start immediately above
the neutron star mass, i.e. about ${3M_{\odot}}$, but noticeably below
$100M_{\odot}$. Instead we observe that the BH mass spectrum in the galaxy has
maximum at ${M\approx 8M_{\odot}}$ with the width $\bm{\sim(1-2)M_{\odot}}$.
The result is somewhat unexpected but an explanations in the conventional
astrophysical frameworks might be possible. Recently LIGO/Virgo discovered
black holes with masses close to $100M_{\odot}$. Their astrophysical origin
was considered unfeasible due to huge mass loss in the process of collapse.
Now some, quite exotic, formation mechanisms are suggested.
2\. BH formed by accretion on the mass excess in the galactic center. In any
large galaxy there exists a supermassive BH (SMBH) at the center, with masses
varying from several millions of $M_{\odot}$ (e,g, Milky Way) up to almost
hundred billions $M_{\odot}$. However, the conventional accretion mechanisms
are not efficient enough to create such monsters during the universe life-
time, ${t_{U}\approx 14.6}$ Gyr. At least 10-fold longer time is necessary, to
say nothing about SMBH in 10 times younger universe.
3\. Primordial black holes (PBHs) created during pre-stellar epoch. The idea
of the primordial black hole (PBH) i.e. of black hole which have been formed
in the early universe prior to star formation, was first put forward by Ya.B.
Zeldovich and I.D. Novikov ZN-PBH . According to their arguments, if the
density contrast in the early universe inside the bubble with radius equal to
the cosmological horizon might accidentally happen to be large,
$\bm{\delta\varrho/\varrho\approx 1}$, then that piece of volume would be
inside its gravitational radius i.e. it became a PBH, that decoupled from the
cosmological expansion. Subsequently this mechanism was elaborated by S.
Hawking SH-BH  and B.Carr and S.Hawking Carr-SH

##  IX PBH and inflation

In earlier works the predicted masses of PBH were quite low, more or less
equal to the mass of the matter inside cosmological horizon at the moment ob
PBH formation. Inflation allowed for formation of PBH with very large masses.
It was first applied to PBH creation by Dolgov and Silk DS , and a year later
by Carr, Hilbert, and Lidsey carr-infl , and soon after that by Ivanov,
Naselsky, and Novikov INN .

Presently inflationary mechanism of PBH production is commonly used. It allows
to create PBH with very high masses, but the predicted spectrum is multi-
parameter one and quite complicated. The only exception is the log-normal
spectrum of refs. DS ; DKK  which is verified by observations in excellent
agreement.

##  X Black Dark Matter

The first suggestion PBH might be dark matter ”particles” was made by S.
Hawking in 1971 SH-DM  and later by G. Chapline in 1975 chaplin , who noticed
that low mass PBHs might be abundant in the present-day universe and their
energy density could be comparable to the energy density of dark matter. In
the latter paper the scale independent spectrum of cosmological perturbations
was assumed, thus leading to the flat PBH mass spectrum in log interval:

$\displaystyle{dN=N_{0}(dM/M)}$ (1)

with maximum mass ${M_{max}\lesssim 10^{22}}$ g, which hits the allowed mass
range. The next proposal of BH-dominated dark matter was made in ref. DS  that
even was contained in the title ”Baryon isocurvature fluctuations at small
scales and baryonic dark matter,” with much larger and more realistic black
hole masses, close to $10M_{\odot}$.

###  X.1 Bounds on BH energy density

The constraints on the density of black holes were reviewed by Carr and Kuhnel
BH-limits  and the results are presented in Fig. 5 for monochromatic mass
spectrum of PBHs.

Figure 5:

Caption to Fig 5: Constraints on $f(M)$ for a monochromatic mass function,
from evaporations (red), lensing (blue), gravitational waves (GW) (gray),
dynamical effects (green), accretion (light blue), CMB distortions (orange)
and large-scale structure (purple). Evaporation limits from the extragalactic
gamma-ray background (EGB), the Voyager positron flux (V) and annihilation-
line radiation from the Galactic centre (GC). Lensing limits from microlensing
of supernovae (SN) and of stars in M31 by Subaru (HSC), the Magellanic Clouds
by EROS and MACHO (EM) and the Galactic bulge by OGLE (O). Dynamical limits
from wide binaries (WB), star clusters in Eridanus II (E), halo dynamical
friction (DF), galaxy tidal distortions (G), heating of stars in the Galactic
disk (DH) and the CMB dipole (CMB). Large scale structure constraints(LSS).
Accretion limits from X-ray binaries (XB) and Planck measurements of CMB
distortions (PA). The incredulity limits (IL) correspond to one PBH per
relevant environment (galaxy, cluster, Universe). There are four mass windows
(A, B, C, D) in which PBHs could have an appreciable density.

###  X.2 Lifting the bounds on the black hole fraction in dark matter

At the beginning of this secrtion it would be proper to quote Bernard Carr’s
words at 2019 : ”all limits are model dependent and have caveats.”

There are several papers, where authors looking and finding ways to eliminate
or weaken the limits on the number density of black holes.

In ref. boehm  it is argued that primordial black holes in the mass range
$(30-100)M_{\odot}$ could be the dark matter carriers since they might escape
microlensing and cosmic microwave background constraints. They are however
subject to the constraints from the binary merger rate observed by the LIGO
and Virgo experiments. The authors argue that in realistic situation the
masses of black holes in expanding universe depend upon time and this leads to
a suppression of binary formation. Hence they conclude that this effect
reopens the possibility for dark matter in the form of LIGO-mass PBHs.

In ref. frampton  the authors have opened the window for PBH with masses in
the range $(10^{2}-10^{5})M_{\odot}$ to make full or significant contribution
to cosmological dark matter. They claim that the derivation of the accepted
bound that excluded considerable contribution of PBH in this mass range is
based on oversimplified accretion model.

As is argued in ref. rubin , PBHs can form clusters. Dynamical interactions in
PBH clusters offers additional channel for the orbital energy dissipation thus
increasing the merging rate of PBH binaries, and the constraints on the
fraction of PBH in dark matter can be weaker than that obtained by assuming a
homogeneous PBH space distribution.

A recent analysis performed in paper eroshenko  permits to conclude that a
possible clustering of PBH could significantly reduce efficiency of the merger
process and the final rate of gravitational wave bursts in some parameter
range. As a result the fraction of PBH in dark matter could be as large as
unity without distortion of LIGO/Virgo observational data.

##  XI Observations of black holes

A possibility of black hope existence was ingeniously discovered in 1783 by
John Michell, an English country parson, famous for many other discoveries in
physics. He noticed that there could be stellar bodies having the second
cosmic velocity larger than the speed of light. Since such objects neither
shine nor reflect light, it would be impossible to observe them directly.
Michell called such, not emitting light stars as ”dark stars”. According to
his understanding a single dark star would be invisible, but if a double
system of a dark and a usual star is formed, one may identify dark star
observing the other one rotating around ”nothing”. This is one of possible
ways to observe black holes at the present time.

However, all that happened to be not absolutely true, or possibly even
entirely wrong. BHs evaporate and shine (Hawking radiation), though nobody yet
saw it. The most powerful sources of radiation (quasars) are supermassive
black holes, point-like objects radiate as thousands galaxies through
ultrarelativistic particle collision in the process of matter accretion. Near-
solar mass BHs are observed through X-rays from accreting surrounding matter.
Black holes may act as gravitational lenses, that’s how MACHOs and some other
BHs are discovered. Observation of the stellar motion around supposed black
hole permits to identify the latter as e.g. supermassive black hole in our
Galaxy was discovered. All these methods only allow to determine the mass
inside central volume. According to theory of General Relativity, a huge mass
in a small volume must form a black hole. However, strictly speaking BH
existence is not proven by all these methods.

The first direct proof of black hole existence was the registration of
gravitational waves from a pair of coalescing massive bodies by
LIGO/Virgo/Kagra. The data explicitly shows that the the coalescence is indeed
between two black holes, because the best fit to the form of the signal is
achieved under assumption of the Schwarzschild metric that according to GR
describes non-charged and (almost) non-rotating black hole. The observations
permit to determine the masses of two coalescing BHs, their spins, and the
mass of the final black hole.

##  XII  Gravitational waves from BH binaries

###  XII.1 Are the GW sources primordial BHs?

As it is argued e.g. in paper BDPP , discovery of gravitational waves (GW) by
LIGO interferometer strongly indicates that the sources of GW are primordial
black holes. In fact here is general agreement between several groups of
theorists, that the gravitational waves discovered by LIGO/Virgo
interferometers originated from PBH binaries. We discuss this issue here
following our paper BDPP . There are three features that indicate that the
sources of GWs should most naturally be primordial black holes:
1\. Origin of heavy BHs (with masses ${\sim 30M_{\odot}}$). To form so heavy
BHs, the progenitors should have ${M>100M_{\odot}}$ and a low metal abundance
to avoid too much mass loss during the evolution. Such heavy stars might be
present in young star-forming galaxies but they are not observed in the
necessary amount. Recently there emerged much more striking problem because of
the observation of BH with ${M\sim 100M_{\odot}}$. Formation of such black
holes in the process of stellar collapse was considered to be strictly
forbidden. On the other hand, primordial black holes with the observed by LIGO
masses may be easily created with sufficient density.
2\. Formation of BH binaries from the original stellar binaries.  Stellar
binaries are formed from common interstellar gas clouds and are quite frequent
in galaxies. If BH is created through stellar collapse, a small non-sphericity
would result in a huge velocity of the BH and the binary would be destroyed.
BH formation from PopIII stars and subsequent formation of BH binaries with
tens of ${M_{\odot}}$ is estimated to be small. The problem of the binary
formation is simply solved if the observed sources of GWs are the binaries of
primordial black holes. They were at rest in the comoving volume, and when
inside the cosmological horizon they were gravitationally attracted and might
loose energy due to dynamical friction or interaction with third body in the
early universe. The probability for them to become gravitationally binded is
probably high enough. The conventional astrophysical scenario is not excluded
but less natural.
3\. Low spins of the coalescing BHs. The low values of the BH spins sae
observed in GW150914 and in almost all (except for three) other events. It
strongly constrains astrophysical BH formation from close binary systems.
Astrophysical BHs are expected to have considerable angular momentum,
nevertheless the dynamical formation of double massive low-spin BHs in dense
stellar clusters is not excluded, though difficult. On the other hand, PBH
practically do not rotate, because vorticity perturbations in the early
universe are vanishingly small. Still, individual PBH forming a binary
initially rotating on elliptic orbit could gain collinear spins about 0.1 -
0.3, rising with the PBH masses and eccentricity Post-Mit ; PKM . This result
is in agreement with the GW170729 LIGO event produced by the binary with
masses ${50M_{\odot}}$ and ${30M_{\odot}}$ and GW190521.

To summarise: each of the mentioned problems might be solved in the
conventional frameworks but it looks much simpler to assume that the
LIGO/Virgo/Kagra sources are primordial black holes.

###  XII.2 Chirp mass distribution

It is well known that two rotating gravitationally bound massive bodies emit
gravitational waves. In quasi-stationary inspiral regime, the radius of the
orbit and the rotation frequency are approximately constant, to be more exact,
slowly decreasing, and the GW frequency is twice the rotation frequency. The
luminosity of the GW radiation in inspiral regime is:

$\displaystyle\bm{{L=\frac{32}{5}\,m_{Pl}^{2}\left(\frac{M_{c}\,\omega_{orb}}{m_{Pl}^{2}}\right)^{10/3}\,,}}$
(2)

where $M_{1}$, $M_{2}$ are the masses of two bodies in the binary system and
$\bm{M_{c}}$ is the so called chirp mass:

$\displaystyle\bm{{M_{c}=\frac{(M_{1}\,M_{2})^{3/5}}{(M_{1}+M_{2})^{1/5}}\,,}}$
(3)

and

$\displaystyle\bm{{\omega^{2}_{orb}=\frac{M_{1}+M_{2}}{m_{Pl}^{2}R^{3}}\,.}}$
(4)

In ref. dkmppss  the available data on the chirp mass distribution of the
black holes in the coalescing binaries in O1-O3 LIGO/Virgo runs are analyzed
and compared with theoretical expectations based on the hypothesis that these
black holes are primordial with log-normal mass spectrum. The results are
presented in Fig. 6. The inferred best-fit mass spectrum parameters,
$M_{0}=17M_{\odot}$ and $\bm{\gamma=0.9}$, fall within the theoretically
expected range and shows excellent agreement with observations. On the
opposite, binary black hole formation based on massive binary star evolution
require additional adjustments to reproduce the observed chirp mass
distribution, see Fig. 7. Similar value of the parameters are obtained in
refs. Raidal ; Liu , see also KAP-Mit .

Figure 6: Model distribution $F_{PBH}(<M)$ with parameters $M_{0}\approx
17M_{\odot}$ and $\bm{\gamma\sim 1}$ for two best Kolmogorov-Smirnov tests.
EDF= empirical distribution function. Figure 7: Cumulative distributions
$F(<M)$ for several astrophysical models of binary BH coalescences.

So we may conclude that PBHs with log-normal mass spectrum perfectly fit the
data. Astrophysical black holes seem to be disfavoured.

A new analysis of the Ligo-Virgo-Kagra data was performed recently in ref.
KAP-Mit . The authors concluded that the chirp-mass distribution of LVK GWTC-3
BH+BH binaries with distinct two bumps can be explained by two different
populations of BH+BH binaries:
1) the low-mass bump at $M_{0}\sim 10M_{\odot}$ due to the astrophysical BH+BH
formed in the local Universe from the evolution of massive binaries
2) the PBH binaries with log-normal mass spectrum with $M_{0}\simeq
10M_{\odot}$ and $\gamma\simeq 10$. The central mass of the PBH distribution
is larger than the expected PBH mass at the QCD phase transition ($\sim
8M_{\odot}$) but still can be accommodated with the mass of the cosmological
horizon provided that the temperature $T_{QCD}\sim 70$ MeV, possible for non-
zero chemical potential at QCD p.t.

Figure 8:

The observed (blue step-like curve) and model (red solid curve) distribution
function of the chirp-masses of coalescing binary BHs from the LVK GWTC-3
catalogue. The model includes almost equal contributions from coalescences of
astrophysical binary BHs (green dashed curve) and primordial BHs with the
initial log-normal mass spectrum with parameters $M_{0}=33M_{\odot}$,
$\gamma=10$ \- with such $\gamma$ heavier PBH practically are not created.

##  XIII Cosmic antimatter

###  XIII.1 Anti-history

The father of antimatter is justly admitted to be Paul Dirac. In his Nobel
Lecture at December 12, 1933 “Theory of electrons and positrons”, dedicated to
his prediction of positrons, he foresaw that there could be antistars and
possibly antiworlds: ”… It is quite possible that… these stars being built up
mainly of positrons and negative protons. In fact, there may be half the stars
of each kind. The two kinds of stars would both show exactly the same spectra,
and there would be no way of distinguishing them by present astronomical
methods.”

Now we expect that there may be some presumably small fraction of antistars in
the Galaxy. Since they are immersed into interstellar gas consisting of
matter, they can be detected due to excess of gamma radiation with energy of a
several hundred MeV, originating from annihilation of the interstellar gas on
the surface of antistar.

Situation is different if an antistar ”lives” in a distant antigalaxy. Still
it is possible in principle to distinguish a star from an antistar through
rather subtle effects considered in ref. DVN . First of all the spectra of the
emitted radiation are not exactly the same, even if CPT is unbroken and the
polarisation of radiation from weak decays could be a good indicator and
lastly the types of emitted neutrinos versus antineutrinos from supernovae or
antisupernovae.

It is in fact surprising that Dirac was not the first person to talk about
antimatter. In 1898, 30 years before Dirac and one year after discovery of
electron (J.J. Thomson, 1897) Arthur Schuster (another British physicist)
conjectured that there might be other sign electricity, antimatter, and
supposed that there might be entire solar systems, made of antimatter,
indistinguishable from ours. Schuster made fantastic wild guess that matter
and antimatter are capable to annihilate and produce vast energy.

Schuster believed that they were gravitationally repulsive having negative
mass. Two such objects on close contact should have vanishing mass!?

Quoting his paper schuster : “When the year’s work is over and all sense of
responsibility has left us, who has not occasionally set his fancy free to
dream about the unknown, perhaps the unknowable?’… Astronomy, the oldest and
yet most juvenile of the sciences, may still have some surprises in store. May
antimatter be commended to its case”.

According to the classical scenario of the generation of the cosmological
baryon asymmetry, proposed by A.D. Sakharov sakharov  , the baryon excess in
the universe is on the average homogeneous, having the same sign determined by
the sign of symmetry breaking between particles and antiparticles. However,
there are plenty of mechanisms leading to space varying amplitude of C and CP
violations with possible sigh changing. If this is the case, the sign of
baryon asymmetry could also be different leading to formation of matter and
antimatter domains in the universe. Possible models of C and CP violation in
cosmology that possess this property are reviewed in AD-CP .

###  XIII.2 Matter and antimatter in the Universe

To the best of my knowledge the first papers on cosmological antimatter were
published by F. Stecker in 1971 stecker-1 , independently on the three year
earlier papers by Konstantinov et al konstan-1 ; konstan-2  on search of
antimatter in the Galaxy, see below subsection XIII.3. Further development of
the idea of matter-antimatter domain structure of the universe was presented
in ref. stecker-2 .

The analysis, performed in ref. C-AdR-G , permits to conclude that
matter–antimatter symmetric universe or close to that, is excluded by the
observed cosmic diffuse gamma-ray background and a distortion of the cosmic
microwave background.

However, there still remains some space for the fraction of cosmological
antimatter, but considerably restricted. It is argued by G. Sgteigman in ref.
GS-1  that the nearest anti-galaxy should be out of our galaxy cluster and
thus could not be closer than at $\sim 10$ Mpc. In a subsequent paper by the
same author GS-2  it is argued that the fraction of antimatter in Bullet
Cluster should be below $3\times 10^{-6}$.

Summary of the situation of the year 2002 was presented in two keynote
lectures at 14th Rencontres de Blois on Matter - Anti-matter Asymmetry
stecker-Blois ; dolgov-Blois .

###  XIII.3 Antimatter in Milky Way

The search of antimatter in the Milky Way was intiated by Konstantinov with
coworkers in 1968 konstan-1 ; konstan-2 . This activity was strongly
criticised by Ya.B. Zeldovich despite very friendly relations between the two.
In agreement with canonic faith no antimatter may exist in our Galaxy and that
explains negative attitude of Zeldovich to Konstantinov activity. Until
recently there was no reason to suspect that any noticeable amount of
antimatter might be in the Galaxy. The predictions of refs. DS ; DKK  were not
taken seriously. Now there are a lot of data indicating that Milky Way
contains siginificant amount of antimatter of different kinds: positrons,
antinuclei, and possibly antistars. The observations do not violate the
existing bounds on galactic antimatter. According to the predictions of papers
DS ; DKK  antimatter objects could be not only in the Galaxy but in its halo
as well.

According to ref. ballmoos-bound , the analysis of the intensity of gamma rays
created by the Bondi accretion of interstellar gas to the surface of an
antistar would allows to put a limit on the relative density of antistars in
the Solar neighbourhood: $N_{\bar{*}}/N_{*}<4\cdot 10^{-5}$ inside 150 pc from
the Sun.

The bounds on galactic antimatter are analysed in refs. bd ; db ; bdp . The
limits on the density of galactic antistars are rather loose, because the
annihilation proceeds only on the surface of antistars, i.e on the objects
with short mean free path of protons, so the extra luminosity created by
matter-antimatter annihilation is relatively low.
Anti-evidence: cosmic positrons.
Existence of rich populations of positrons in the Galaxy was noticed long ago
through the observations of 511 keV gamma ray line (see anti-e1 ; anti-e2 ;
anti-e3  and references therein) with the flux

$\displaystyle{{\Phi_{511\;{\rm keV}}={1.07\pm 0.03\cdot 10^{-3}}\;{\rm
photons\;cm^{-2}\,s^{-1}}.}}$ (5)

The width of the line is about 3 keV. The emission mostly goes from the
Galactic bulge and at much lower level from the disk.This unambiguously
indicates the frequent annihilation of nonrelativistic $e^{+}e^{-}$ pairs in
the Galactic bulge with the rate anti-e1

$\displaystyle\dot{N}_{ee}^{\text{bulge}}\sim 10^{43}\text{ s}^{-1}.$ (6)

Note that one of the brightest X-ray sources in the region around the Galactic
Center got the name Great Annihilator gr-ann . Possibly it is a microquasar
first detected in soft X-rays by the Einstein Observatory Ein-observ  and
later detected in hard X-rays by the space observatory “Granat” granat .

There is no commonly accepted point of view on the origin of the cosmic
positrons. The conventional hypothesis that positrons are created in strong
magnetic fields of pulsars is at odds with the AMS data AMS-19 . However, this
conclusion is questioned in ref. kachel  where it is shown that these features
could be consistently explained by a nearby source which was active $\sim 2$
Myr ago and has injected $(1-2)\times 10^{50}$ erg in cosmic rays.

A competing option is that positrons are created by the Schwinger process at
the horizon of small black holes with masses $\gtrsim 10^{20}$ g . This
mechanism was suggested in ref. BDP  and discussed in more detail in ref. AD-
AR .

One more possibility that is closer to the spirit of this talk is that
positrons are primordial, produced in the early universe in relatively small
antimatter domains DS ; DKK . Possible observation of the unexpectedly high
flux of antinuclei anti-nuc-AMS-1 ; anti-nuc-AMS-2  and antistars in the
Galaxy anti-stars  strongly supports this hypothesis, in particular, by ref.
bpbbd , where it is advocated that antihelium cosmic rays are created by
antistars.
Anti-evidence: cosmic antinuclei.

In 2018 AMS-02 announced possible observation of six $\bm{\overline{He}^{3}}$
and two $\bm{\overline{He}^{4}}$ choutko ; ting-CERN . Accumulated by 2022
data contains some more events: 7 $\overline{D}$ (at energies $\lesssim 15$
GeV) and 9 $\overline{He}^{4}$ at ($E\sim 50$ GeV). These numbers correspond
roughly speaking to $\bm{\overline{He}/He\sim 10^{-9}}$. This number is much
larger than the expected number of $\overline{He}^{4}$, if it were created in
cosmic ray collisions. It is possible that the total flux of anti-helium is
even much higher because low energy $\bm{\overline{He}}$ may escape
registration in AMS.

The probability of the secondary creation of different antinuclei was
estimated in ref. cosm-anti-nuc . According to this work, anti-deuterium could
be most efficiently produced in the collisions ${\bar{p}\,p}$ or
${\bar{p}\,He}$ that can create the flux ${\sim
10^{-7}/m^{2}/s^{-1}}$/steradian/GeV/neutron), i.e. 5 orders of magnitude
below the observed flux of antiprotons. Antihelium could be created in the
similar reactions and the fluxes of ${\overline{He}^{3}}$ and
${\overline{He}^{4}}$, that could be created in cosmic rays would respectively
be 4 and 8 orders of magnitude smaller than the flux of the secondary created
anti-D.

According to the works DS ; DKK , antinuclei should be primordial i.e. created
in the very early universe during big bang nucleosynthesis (BBN) inside
antimatter bubbles with high baryon density. However, the standard anti-BBN
surely does not help, since normally BBN gives 75% of hydrogen, 25% of
helium-4, and a minor fraction of deuterium, at the level a few times
$10^{-5}$, in a huge contrast to the observed ratio of anti-deuterium to anti-
helium which is of order unity. The same problem exists for the ratio of
$\overline{He}^{3}$ to $\overline{He}^{4}$, that is also of order unity
instead of the standard $\sim 3\times 10^{-5}$.

If we assume that in the model of DS ; DKK  the abundances of anti-D and anti-
He are determined by normal BBN with large baryon-to-photon ratio $\beta\sim
1$, the problem would be even more pronounced, because amount of deuterium and
helium-3 would be negligibly small, even much less than $10^{-5}$. On the
other hand in our scenario formation of primordial elements takes place inside
non-expanding compact stellar-like objects with practically fixed temperature.
If the temperature is sufficiently high, this so called BBN may stop with
almost equal abundances of D and He. One can see that looking at abundances of
light elements at a function of temperature. If it is so, antistars may have
equal amount of $\overline{D}$ and $\overline{He}$
Anti-evidence: antistars in the Galaxy.
A striking announcement of the possible discovery of anti-stars in the Galaxy
was made in ref. antistars  The catalog 14 antistar candidates was identifyed,
not associated with any objects belonging to established gamma-ray source
classes and with a spectrum compatible with baryon-antibaryon annihilation.
There results are illustrated in Fig. 9.

Figure 9:  Positions and energy flux in the 100 MeV - 100 GeV range of
antistar candidates selected in 4FGL-DR2. Galactic coordinates. The background
image shows the Fermi 5-year all-sky photon counts above 1 GeV

In ref. bbbdp  a supplimentary method of antistar identification was proposed.
In astrophysically plausible cases of the interaction of neutral atmospheres
or winds from antistars with ionised interstellar gas, the hadronic
annihilation will be preceded by the formation of excited $\bm{p\bar{p}}$ and
$He{\bar{p}}$ atoms. These atoms rapidly cascade down to low levels prior to
annihilation giving rise to a series of narrow lines which can be associated
with the hadronic annihilation gamma-ray emission. The most significant are L
(3p-2p) 1.73 keV line (yield more than 90%) from ${p\bar{p}}$ atoms, and M
(4-3) 4.86 keV (yield $\sim 60$%) and L (3-2) 11.13 keV (yield about 25%)
lines from $He^{4}\bar{p}$ atoms. These lines can be probed in dedicated
observations by forthcoming sensitive X-ray spectroscopic missions XRISM and
Athena and in wide-field X-ray surveys like SRG/eROSITA all-sky survey.

Connection between antistars and the observed fluxes of antinuclei was studied
in recent paper bpbbd . A minor population of antistars in galaxies has been
predicted by some of non-standard models of baryogenesis and nucleosynthesis
in the early Universe, and their presence is not yet excluded by the currently
available observations. Detection of an unusually high abundance of antinuclei
in cosmic rays can probe the baryogenesis scenarios in the early Universe.

It is shown that the flux of antihelium cosmic rays reported by the AMS-02
experiment can be explained by Galactic anti-nova outbursts, thermonuclear
anti-SN Ia explosions, a collection of flaring antistars, or an extragalactic
source with abundances not violating existing gamma-ray and microlensing
constraints on the antistar population.

##  XIV Mechanism of creation of PBH and antimatter

The mechanism of PBH and galactic antimatter creation DS ; DKK  is essentially
based on the scenario of supersymmetry (SUSY) motivated baryogenesis, proposed
by Affleck and Dine (AD) AD . SUSY generically predicts existence of scalars,
$\chi$, with non-zero baryonic number, ${B\neq 0}$. Another prediction of high
energy SUSY models is an existence of flat directions in the $\chi$-potential,
either quartic (self-interaction):

$\displaystyle{{U_{\lambda}(\chi)=\lambda|\chi|^{4}\left(1-\cos
4\theta\right)}}$ (7)

or quadratic, i.e. the mass term,
$\bm{U_{m}=m^{2}\chi^{2}+m^{*\,2}\chi^{*\,2}}$:

(8)

where ${\chi=|\chi|\exp(i\theta)}$ and $\bm{m=|m|e^{\alpha}}$. If ${\alpha\neq
0}$, C and CP are broken. Potential energy does not rise along these flat
directions.

At the inflationary epoch the average value of $\chi^{2}$ linearly rises with
time vil ; adl ; aas  see also ad-dp . In other words, $\chi$ bosons may
condense along flat directions of the quartic potential, when and if its mass
was smaller than the inflationary Hubble parameter.

In GUT SUSY baryonic number is naturally non-conserved, because of generic
non-invariance of ${U(\chi)}$ w.r.t. phase rotation
$\chi\rightarrow\chi\exp(i\theta)$.

After inflation ${\chi}$ was far away from origin due to rising quantum
fluctuations and, when inflation ends, it started to evolve down to the
equilibrium point, ${\chi=0}$, according to equation of motion that formally
coincides with the equation of motion of a point-like particle in Newtonian
mechanics:

$\displaystyle{{\ddot{\chi}+3H\dot{\chi}+U^{\prime}(\chi)=0.}}$ (9)

The baryonic number of $\chi$:

$\displaystyle{{B_{\chi}=\dot{\theta}|\chi|^{2}}}$ (10)

is analogous to mechanical angular momentum in complex plane
$[Re\chi,Im\chi]$. After ${{\chi}}$ decays, the accumulated baryonic number of
$\chi$ is transferred into baryonic number of quarks in B-conserving process.
AD baryogenesis could lead to baryon asymmetry of order of unity, much larger
than the observed asymmetry ${\sim 10^{-9}}$.

If ${m\neq 0}$ and the flat direction of quadratic and quartic valleys are
different, the angular momentum, B, would be generated by the ”rotation”
induced by the motion of $\chi$ from quartic flat direction to the quadratic
one. In other words, the field $\chi$ would acquire non-zero baryonic number,
generically very large .

If CP-odd phase $\bm{\alpha}$ is non-vanishing, both baryonic and antibaryonic
domains might be formed with possible dominance of either of them. Matter and
antimatter objects may exist but possibly global ${B\neq 0}$.

An essential development proposed in works DS ; DKK  was an introduction of
the new interaction between the Affleck-Dine field and the inflaton $\Phi$,
the first term in the equation below:

$\displaystyle
U={g|\chi|^{2}(\Phi-\Phi_{1})^{2}}+\lambda|\chi|^{4}\,\ln\left(\frac{|\chi|^{2}}{\sigma^{2}}\right)+\lambda_{1}(\chi^{4}+h.c.)+(m^{2}\chi^{2}+h.c.).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(11)

This coupling between $\chi$ and the inflaton is the general renormalizable
interaction of two scalar field. The only tuning is the assumption that $\Phi$
reaches the value $\Phi_{1}$ during inflation significantly before it ends,
with the remaining number of e-foldings about 30-40.

The window to the flat directions is open, near ${\Phi=\Phi_{1}}$. At that
period the field ${\chi}$ could rise to large values, according to the quantum
diffusion equation derived by Starobinsky, generalised to a complex field
$\chi$.

If the window to flat direction, when $\bm{\Phi\approx\Phi_{1}}$ is open only
during a short period, cosmologically small but possibly astronomically large
bubbles with high baryon-to-photon ratio ${\beta}$ could be created, occupying
a tiny fraction of the total universe volume, while the rest of the universe
has the observed $\beta\approx 6\cdot 10^{-10}$, created by the normal small
$\chi$. The fundament of PBH creation has been build at inflation by making
large isocurvature fluctuations at relatively small scales, with practically
vanishing density perturbations.

The initial isocurvature perturbations are contained in large baryonic number
of massless quarks in rather small bubbles. We call them high baryonic
bubbles, HBBs. Density perturbations were generated rather late after QCD
phase transition, at temperatures around 100 MeV, when massless quarks turned
into massive baryons. The resulting high density contrast could lead to
creation of PBHs. The mechanism is very much different from any other
described in the literature models of PBH formation.

The emerging universe looks like a piece of Swiss cheese, where holes are high
baryonic density objects occupying a minor fraction of the universe volume.

Outcome of the DS/DKK mechanism:

  * •

PBHs with log-normal mass spectrum - confirmed by the observations!

  * •

Compact stellar-like objects, similar to cores of red giants.

  * •

Disperse hydrogen and helium clouds with (much) higher than average
$\bm{n_{B}}$ density.

  * •

Strange stars with unusual chemistry and velocity.

  * •

${\beta}$ may be negative leading to creation of (compact?) antistars which
could survive annihilation despite being submerged into the homogeneous
baryonic background.

  * •

Extremely old stars could exist and indeed they are observed, even, ”older
than universe star” is found; its prehistoric age is mimicked by the unusual
initial chemistry.

The mechanism of PBH creation pretty well verified by the data on the BH mass
spectrum and on existence of antimatter in the Galaxy, especially of
antistars. So we may expect that it indeed solves the problems created by HST
and JWST.

Thus we may conclude that canonical $\Lambda$CDM cosmology is saved by PBHs.
Antimatter in our backyard is predicted and found.

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