Specific aspects of the evolution of antimatter globular clusters domains Authors: Maxim Yu. Khlopov , Orchidea maria Lecian Speaker: Orchidea Maria Lecian Talk delivered at The Fourth Zeldovich virtual meeting, September 7-11, 2020 11 Sep 2020 Abstract: In the Affleck-Dine-Linde scenario of baryosynthesis, there exists a possibility of creation of sufficiently large regions with antibaryon excess . Such regions can evolve in antimatter globular clusters. They appear only in the result of domain evolution and only at the stage of galaxy formation. Their number is demonstrated to increase as a function of time, being determined by the mechanism of antibaryon excess generation and on the properties of the inflaton filed and related ones; the number of clusters depends also on the Hubble parameter at the inflationary stage, as well as on the field initial conditions in the Einstein field Equations. Possible evolution of the domains of the antimatter globular clusters provides observational constraints on the mechanisms of inflation, baryosynthesis and evolution of antibaryon domains in baryon-asymmetrical Universe. http://www.icranet.org/index.php?option=com_content&task=view&id=1254 http://www.icranet.org/zeldovich4

1. Orchidea Maria Lecian
Sapienza University of Rome
ICRA- International Center for Relativistic Astrophysics
Specific aspects of the evolution of antimatter globular clusters
domains
Authors: M.Yu.Khlopov, O.M. Lecian
11th September 2020, Pescara, Italy.
THE FOURTH ZELDOVICH VIRTUAL MEETING
September 7-11, 2020
orchideamaria.lecian@uniroma1.it
lecian@icra.it

2. Abstract
In the Affleck-Dine-Linde scenario of baryosynthesis, there exists a
possibility of creation of sufficiently large regions with antibaryon
excess . Such regions can evolve in antimatter globular clusters.
They appear only in the result of domain evolution and only at the
stage of galaxy formation. Their number is demonstrated to
increase as a function of time, being determined by the mechanism
of antibaryon excess generation and on the properties of the
inflaton filed and related ones; the number of clusters depends also
on the Hubble parameter at the inflationary stage, as well as on
the field initial conditions in the Einstein field Equations. Possible
evolution of the domains of the antimatter globular clusters
provides observational constraints on the mechanisms of inflation,
baryosynthesis and evolution of antibaryon domains in
baryon-asymmetrical Universe.

3. Introduction
In particular inflationary scenarios with nonhomogeneous baryosynthesis, the creation
of antimatter can be obtained from the solution of the matter content of the
Einstein-Field-Equation.
Antimatter is allowed to form antimatter domains, containing a certain number of
antibaryons, whose number, volume and correlation functions can be evaluated.
Theses features are calculated according the statistical distributions which account for
the possibility that
1) a sufficient amount of antimatter (under suitable conditions) can form compact
domains;
2) the time evolution of such domains is allowed by the statistical probabilities that
the antimatter is not annihilated by ordinary matter domains, and/or by the
interaction with external matter (vacuum hypothesis and plasma hypothesis), and/or
by competition interaction among the latter matter presence and vacuum content.
The different statistical distributions are mathematically demonstrated to be
compatible.
The correlation function for the antimatter domains can be evaluated at different ages
of the Universe (under the suitable hypotheses), and can be refined by the use of
statistical estimators, also based on the Bernoulli-distribution comparison, after
considering the Standard Cosmological Principle.

4. Summary
- Symmetry-breaking scenarios:
- ADL scenario.
- Antimatter domains
- Evaluation of the number of antimatter domains:
- Statistical distributions:
- Gaussian distribution;
- Gauss-minus distribution.
- Discrete statistical distributions: Poisson Distribution, Binomial distribution
- Non-central continuous distributions:
- Fisher’s non-central Hypergeometric Distribution;
- Modified Wallenius’ Hypergeometric Distribution;
- Generalized non-central Hypergeometric Distribution;
- Comparison: a theorem
- Correlation function(s) for antimatter domains:
- (Bernoulli distribution-based possibilities)
- Outlook

5. Symmetry-breaking scenario
V (φ1, φ2, χ) = −µ2
1(φ+
1 φ1 + φ+
2 φ2) + λ1[(φ+
1 φ1)2
+ (φ+
2 φ2)2
] + 2λ3(φ+
1
φ1)(φ+
2 φ2)(φ+
1 φ2) + 2λ4(φ+
1 φ2)(φ+
2 φ1) + λ5[(φ+
1 φ2)2
+ h.c.]+
λ6(φ+
1 φ1 + φ+
2 φ2)(φ+
1 φ2 + φ+
2 φ1) − µ2
2χ+
χ + δ(χ+
χ)2
+ 2α(χ+
χ)(φ+
1 φ1 + φ+
2 φ2) + 2β[(φ+
1 χ)(χ+
φ1) + (φ+
2 χ)(χ+
φ2)]
for effective low-energy electroweak SU(2) ⊗ U(1) theory.
A GUT spontaneous CP violation would imply the formation of vacuum structures
separated from the rest of the matter universe by domain walls, whose size is
calculated to grow with the evolution of the Universe. This behavior is calculated not
not affect the evolution of the Universe if the volume energy ˜ρ(V ) density of the walls
for
˜ρ(V ) ∼ σ2
φT4/˜h,
with ˜h the value of the scalar coupling constant.

6. A CP-invariant Lagrangean was chosen of form
L = (∂φ)2
− λ2
(φ2
− χ2
)2
+ ¯ψ(i∂ − m − igγ5φ)ψ
where the vacuum is characterized by < φ >= ση, with σ = ±1.
L = (∂χ)2
−
1
2
m2
χχ2
−4σλ2χη3
−λ2
χ4
+ ¯ψ(iˆ∂−M−i
gm
M
γ5χ−
g2ση
M
χ)ψ.
A CP violation can be achieved after the substitution φ = χ + ση.

7. Characteristical features of antimatter domains
- minimal mass for an antimatter domain to evolve in time without being annihilated
estimated as ≃ 103M⊙
for a temperature below 4000K, with the initial condition of the time t0
corresponding to 9000K;
- antimatter density in the antimatter domains 3 orders of magnitude less than the
baryon density;
- correlation functions can be written after solving the fluid equation for the diffusion
coefficient D(t) via the baryon-to-photon ratio r ≡ nb/n˜γ
∂t
∂t
= D(t) ∂2
t
∂x2 , r(R, t0) = r0, x < 0, r(R, t0) = 0, x > 0,
to evaluate the physical distance covered by atoms after the recombination age until
the present time.

8. Calculation of number of domains
The number of domains ¯N as a function of the number of e-folds ˜N at the time ta
corresponding to ˜Nc reads
¯N( ˜Nc, ˜N0) − ¯N0 ≃ ln
¯N( ˜Nc, ˜N0; ta)
¯N0
≤ ln
(eθ)feff ( ˜Nc)
√
2
1 −
feff ( ˜Nc − ˜N0)
(σ(ta − t0))2
.
In the case the variance should be dependent on the number of e-folds ˜Nι and on the
size of the Universe at the time of Galaxy formation, different estimations can be
conducted, according to the variables which need be outlined.
After considering the definition ˜N ≡ H∆t for the standard deviation,
the following estimation of number of antimatter domain is found
¯N( ˜Nc, ˜N0)− ¯N0 ln
(eθ)feff ( ˜Nc)
√
2


1 − 4π2
f 1 +
gφχ
12πλ
(Hcta − Hιtι)
ln LueHcta −eHιtι
l
2
(ta − tι)2



which depends on the time tι at which the number estimation is formulated, between
the time interval ∆t ≡ (ta − t0), with t0 ≤ tι ≤ ta.

9. Gaussian distribution
The choice for a Gaussian distribution of the number of antibaryons in each domain
leads to the evaluation of the number of antimatter domains as
˜N − ˜N0 ≃ ln
θι
θc
+ ln(1 + θc + θ2
c /2) − ln(1 + θι − θ2
ι /2) ·
· ln
LueHc(tc−t0) − eH0t0
l
(eθ60 − eθι )
θ60 − θ0
χ60 − χ0
[f(ta) − f(t0)] + feff
1
H
3/2
c − H
(3/2)
ι
(tc − tι)

10. Gauss-minus distribution
N(t) − N0 ≡ 4π2
e2θ0
2feff
ln 1
N0l
2
1 − 2 LueNc
ln 1N0l
1 + 1
ln feff
ρc∆tc
ρ0∆t0
(1 − 1
2
( ρc∆tc
ρ0∆t0
)−1) ρ(t) t4 with ρ(t))t4
to be evaluated at the necessary time ι after the time t0.
with a two-parameter Gaussian-modified distribution
P− ∝ 1
2π(σ2− 1
λ2 )
e
−
ak
ς2
e
−
bk
ς2
,
i.e. with a Gaussian-modified dispersion relation
ς2 ≡ (σ2 − 1
λ2 )ς ≡ ln N0
l
+ π − Luec−N0
l
− 1
2
Luec−N0
l
2

11. Poisson distribution
k antibaryon excess regions of some minimal size (containing antibaryons, able to
interact, according to a chosen, modellizable kind of interaction), such that an
antimatter domain can be started to be formed;
˜N(k) − ˜N0(k) ≃ n k
kn
ek
n!
χta
χt0
2
∆feff (t;ta,ti,t0)·
(−2)
4π2 ln LueHc(tc−t0)
−eH0t0
l
k
(t)k−3
the numerical evaluation can be performed by knowing the numerical values of k;
for a particular kj = 0, kj ∈ n, indeed, for the Poisson distribution, as well as for
other probability distributions, an antibaryon not neighbouring with another
antibaryon should be neighbouring with ordinary matter, vacuum or plasma, and the
event does not contribute to the domain formation (within the sum) which defines the
boundary condition for the domain.
More specifically, for ˜F( ˜N(kj) − ˜N0(kj)) = 0, and
k
... + ˜F( ˜N(kj−1) − ˜N0(kj−1)) + ˜F( ˜N(kj) − ˜N0(kj)) + ˜F( ˜N(kj+1) − ˜N0(kj+1)) + ... =
=
k
... + ˜F( ˜N(kj−1) − ˜N0(kj−1)) + ˜F( ˜N(kj+1) − ˜N0(kj+1)) + ...

12. Binomial distribution
˜N(k) − ˜N0(k) ≃
k
1
(k!)(1 − k)!
χta
χt0
2
∆feff (t; ta, ti, t0)
(−2)
4π2
·
· ln
LueHc(tc−t0) − eH0t0
l
k
(t)k−3
k probability of finding an antibaryon-excess region of some minimal size;
pk the probability of finding k − 1 neighbouring antibaryons within an
antibaryon-excess region of some minimal size;
2˜n antibaryon density (at least considered for the neighbouring antibaryon-antibaryon
interaction), expressed by the efficace expression of H for ˜n antibaryons;
n the number of antibaryons surrounding the domain,
i.e. n − k relates the number of antibaryons neighbouring with a baryon; (1 − p)n−k
the probability that an antibaryon (in an antibaryon-excess region of some minimal
size) is neighbouring with an baryon, i.e. at the domain boundary

13. Non-central Fisher’s Hypergeometric Distribution
na = number of antibaryon(s) within one antibaryon cluster;
nb = number of baryon(s) within one cluster;
xmin ≡ max(0, n − na);
xmax ≡ min(n, nb);
N = na + nb;
n ≡ number of considered (neigbouring) antibaryons;
ω ≡ ωa/ωb weight;
example: probability of existence of one antibaryon
P ≡ ωama
ωana+ωbnb
;
variance σ2 = N
N−1/ 1
µ + 1
na−µ + 1
n−µ + 1
µ+mb−n
µ ≡ − 2nanω
(na+n)(1−ω)−N−
√
((na+n)(1−ω)−N)2−4(ω−1)(nanω)
number of domains as a function of the volume V evolving in time
time-dependent volume of the number of domains

14. N(tι) − N(t0) =
1
√
2 t0
tι
x=n
x=n−nb
ma!
x!(ma−x)!
mb!
x!(mb−x)!
ω(ρ)x
x=n
y=n−nb
ma!
y!(ma−y)!
mb!
y!(mb−y)!
ω(ρ)y
·
· ln (ln
ln(˜a)
t
)
(3m2
P l)2
mfeff (8π)2
1 +
gφP l
12πλ
H∆(t) ·
· [
1
˜a
d˜a
dt
1
t
(3Mm2
P l)2
mfeff
1 + gχ
12πλ
Hι∆t
(8π)2
+
− ln ˜a
1
t2
(3Mm2
P l)2
mfeff
1 + gχ
12πλ
Hι∆t
(8π)2
+
−
1
2
ln ˜a
t
(3Mm2
P l)2
mfeff
( gφχMmP l
12πλ
)Hι
1 + gχ
12πλ
Hι∆t(8π)2
]dt ≡

15. ≡
1
√
2 t0
tι
x=n
x=n−nb
ma!
x!(ma−x)!
mb!
x!(mb−x)!
ω(ρ)x
x=n
y=n−nb
ma!
y!(ma−y)!
mb!
y!(mb−y)!
ω(ρ)y
·
· ln
n
mf


−
1
mfeff
( gφχMmP l
12πλ
)Hι
1 + gχ
12πλ
Hι∆t(8π)2
dn
dt
−
n
mfeff
1 + gχ
12πλ
Hι∆t
V 2(t(n))(8π)2
dV
dt


 dt
≡ t0
tι
P(χ) ln χdχ(t)
The Hubble parameter H(tι) can be considered also as HI, as the
FRW scale factor ˜aHI t; in the case of barionic matter with density
ρb, ωρb ≡ 0 for ρb ∼ ˜a−4

16. (modified) Non-central Wallenius’ Hypergeometric
Distribution
N = na + nb;
n ∈ [0, N) ≡ number of considered (neighbouring) antibaryons;
ω ∈ R+;
x ∈ [xmin, xmax];
xmin = max(0, n − mb);
xmax = min(n, ma);
1 − P ≡ ma!
x!(x−ma)!
mb!
(n−x)!(mb−n+x)! 0
1
(1 − qω/D)x(1 − q1/D)n−xdq;
D ≡ ω(ma − x) + mb − n + x;
variance N
N−1
µ(m1−µ)(n−µ)(µ+mb−n)
ma(n−µ)(µ+mb−n)+mbµ(ma−µ);
time-dependent volume of the number of domains:
na = number of antibaryon(s) within one antibaryon cluster
nb = number of baryon(s) within one cluster
xmin ≡ max(0, n − na)
xmax ≡ min(n, nb)
N = na + nb

17. N(tι) − N(t0) =
1
√
2 t0
tι ma!
x!(x − ma)!
mb!
(n − x)!(mb − n + x)!
·
·
0
1
(1 − qω/D
)x
(1 − q1/D
)n−x
dq ·
· ln (ln
ln(˜a)
t
)
(3m2
P l)2
mfeff (8π)2
1 +
gφP l
12πλ
H∆(t) ·
· [
1
˜a
d˜a
dt
1
t
(3Mm2
P l)2
mfeff
1 + gχ
12πλ
Hι∆t
(8π)2
+
− ln ˜a
1
t2
(3Mm2
P l)2
mfeff
1 + gχ
12πλ
Hι∆t
(8π)2
+
−
1
2
ln ˜a
t
(3Mm2
P l)2
mfeff
( gφχMmP l
12πλ
)Hι
1 + gχ
12πλ
Hι∆t(8π)2
]dt ≡

18. N(tι) − N(t0) =
1
√
2 t0
tι ma!
x!(x − ma)!
mb!
(n − x)!(mb − n + x)!
·
·
0
1
(1 − qω/D
)x
(1 − q1/D
)n−x
dq ·
· ln
n
mf


−
1
mfeff
( gφχMmP l
12πλ
)Hι
1 + gχ
12πλ
Hι∆t(8π)2
dn
dt
+
−
n
mfeff
1 + gχ
12πλ
Hι∆t
V 2(t(n))(8π)2
dV
dt
dt

19. Generalized non-central hypergeometric distribution
N = na + nb
n ∈ [0, N)
ω ∈ R+
x ∈ [xmin, xmax]
xmin = max(0, n − mb)
xmax = min(n, ma)
1 − P ≡ ma!
x!(x−ma)!
mb!
(n−x)!(mb−n+x)! 0
1
(1 − qω/D)x(1 − q1/D)n−xdq
D ≡ ω(ma − x) + mb − n + x
- weight parameter ̟ can be interpreted as accounting for the phenomena concerning
the domains and the interaction of the antibaryons within the domains. The
corresponding expression of the probabilities as a function of z ≡ x/ma is obtained as
P(z) = C1I1 + C2I2,
with

20. C1 =
(N − mbma)! mb
ma
!
k
ma
! n−k
ma
!
N − mb − k − 1
ma
!
mb − n − k
ma
!,
C2 = C
mb − n + k
N − mb − k
,
I1 =
0
1
qk/ma (1 − q)(N−mb−k)/ma ln(n−k)/ma (1 − q)−ma/̟
·
· [1 − ln(1 − q)−ma/̟
mb−n+k
ma dq,
I2 =
0
1
1 − q−̟q/ma e−̟q(N−mb−k)/ma q
n−k
ma (1 − q)
mb−n+k−1
ma dq.

21. The number of such domains, according to their number and the number of
antibaryons contained is therefore calculated as
N(tι) − N(t0) =
1
√
2
(C1I1 + C2I2) ·
· ln (ln
ln(˜a)
t
)
(3m2
P l)2
mfeff (8π)2
1 +
gφmP l
12πλ
H∆(t) [
1
˜a
d˜a
dt
1
t
(3Mm2
P l)2
mfeff
1 + gχ
12πλ
Hι∆t
(8π)2
+
− ln ˜a
1
t2
(3Mm2
P l)2
mfeff
1 + gχ
12πλ
Hι∆t
(8π)2
−
1
2
ln ˜a
t
(3Mm2
P l)2
mfeff
( gφχMmP l
12πλ
)Hι
1 + gχ
12πλ
Hι∆t(8π)2
]dt ≡
≡
1
√
2
(C1I1 + C2I2) ·
· ln
n
mf


−
1
mfeff
( gφχMmP l
12πλ
)Hι
1 + gχ
12πλ
Hι∆t(8π)2
dn
dt
−
n
mfeff
1 + gχ
12πλ
Hι∆t
V 2(t(n))(8π)2
dV
dt


 dt.

22. Matter/antimatter symmetric Universe
NB( ¯B) ≃ −
g2f2mθ
8π2
Wi2
θi ±θi/2
∞
dω
sin2 ω
ω
≃
1
2
This way, also the probability should be modified by assuming ρ = const = 1/2, by
substituting the pertinent expression
Pχ ≡ 1 − f21/4
1√
2
+ gφχ
2πλ
√
∆tι −
√
∆t0
σ2
˜N(tta ) − ˜N(tι) =
=
ta
tι
ln
1
2feff
1 +
1
2
gφχ
2πλ
√
t
4π2f2
eff
t2
1
2feff
1
4
t−1/2
dt ≃
1
4
1
2
gφχ
2πλ
1
ta
−
1
tι

23. Comparison
- it is possible to compare the error on the variance of a generaical statistical
distribution with that of a Gaussian statistical distribution, without loss of
information, for the different dependence on time of the GR solutions obtained:
majorize the error by a linear operator S by a linear mapping ¯N weighted by a
parameter φ, which can be also time dependent
|| Sx − φ ¯N ||
in a spherical neighborhood of each antibaryon, placed in ¯x as
X
| Sx − φ ¯N || d¯µ(X)
such that there exists ǫ > 0 such that
inf{ ǫ > 0 : ¯µ{ ¯x ∈ X :| Sx − φ ¯N ||> ǫ} ≤ δ}
where δ of (also time-dependent) Planckian order .
- choice of linearity for the operator S compatible with the first-order approximation i
time for each calculation accomplished;
- competition at the boundary surfaces of the domains with void, plasma,
intergalactical medium, and the production of hypernuclei also possible.

24. Correlation distribution of antimatter domains
N(r) ∝ rD2
1 + ξ(r) ∝ rD2−2
N(tι) − N(t0) =
1
√
2 t0
tι 1
√
2
(C1I1 + C2I2) ·
· ln (ln
ln(˜a)
t
)
(3m2
P l)2
mfeff (8π)2
1 +
gφχMmP l
12πλ
H∆(t) ·
· [
1
˜a
d˜a
dt
1
t
(3Mm2
P l)2
mfeff
1 + gφχ
12πλ
Hι∆t
(8π)2
+
− ln ˜a
1
t2
(3Mm2
P l)2
mfeff
1 + gφχ
12πλ
Hι∆t
(8π)2
+
−
1
2
ln ˜a
t
(3Mm2
P l)2
mfeff
( gφχMmP l
12πλ
)Hι
1 + gφχ
12πλ
Hι∆t(8π)2
]dt ≡

25. ≡
1
√
2 t0
tι 1
√
2
(C1I1 + C2I2) ·
· ln
n
mf


−
1
mfeff
( gφχMmP l
12πλ
)Hι
1 + gφχ
12πλ
Hι∆t(8π)2
dn
dt
−
n
mfeff
1 + gφχ
12πλ
Hι∆t
V 2(t(n))(8π)2
dV
dt


 dt
≡
˜Nι − ˜N0
V 0
r
4πs2
sD2−2
ds
D2 correlation dimension

26. Two-point correlation function:
use of estimators (Hamilton estimator, Davis-and-Peebles estimator, Landy-Szalay
estimator, ...
for generating binomial distributions of pairs of correlated objects.
The distances between pairs of antimatter can be estimated;
- evaluation of the distances for pairs of antimatter domain and matter domain) for
the initial conditions.
A a point of the Poisson simulation can be chosen as
1) an arbitrary point; or
2) the presence of a single antibaryon, according to the presence of antimatter
(antinuclei) in the intergalactic medium, as in [?] for obtaining a correlation within the
corresponding times.
3) pairs of antimatter domains;
4) pairs of antimater domains and antibaryons;
5) pairs characterized by the same interval.
The results of the binomial/Poisson distributions obtained for a simulation of the pairs
of objects defined in the estimators can be compared with the experimental data
referred to, for example,
- primordial non-Gaussianities,
- data available for other times of evolution of the Universe.

27. Thank You for Your attention!

28. References
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(1985) 361;
-A.Vilenkin and L.Ford, Phys.Rev. D26, 1231 (1982); A.D.Linde, Phys.Lett. 116B 335
(1982);
- V. M. Chechetkin, M. y. Khlopov, M. G. Sapozhnikov and Y. B. Zeldovich,
Astrophysical Aspects Of Anti-proton Interaction With He-4 (antimatter In The
Universe), Phys. Lett. 118B (1982) 329;
- V. A. Kuzmin, M. E. Shaposhnikov and I. I. Tkachev, Matter - Antimatter Domains
in the Universe: A Solution of the Vacuum Walls Problem, Phys. Lett. 105B (1981)
167;
- Y. B. Zeldovich, I. Y. Kobzarev and L. B. Okun, Cosmological Consequences of the
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30. Further References
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Konoplich, R. Mignani, S.G. Rubin, A.S. Sakharov Volume: 12,
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