- Email: [email protected]
Evolution and observational signature of diffused antiworld
Astroparticle Physics 12 Ž2000. 367–372 www.elsevier.nlrlocaterastropart
Evolution and observational signature of diffused antiworld M.Yu. Khlopov a
a,b,c,d,)
, R.V. Konoplich a,b,c,d , R. Mignani a , S.G. Rubin A.S. Sakharov b,d
b,d
,
Dipartimento di Fisica ‘‘E.Amaldi’’, III UniÕersita’ di Roma ‘‘Roma Tre’’, and INFN, Sezione di Roma III, Rome, Italy b Center for CosmoParticle Physics ‘‘Cosmion’’, Moscow, Russian Federation c Institute of Applied Mathematics, Moscow, Russian Federation d Moscow Engineering Physics Institute (Technical UniÕersity), Moscow, Russian Federation Received 12 November 1998; received in revised form 27 April 1999; accepted 21 June 1999
Abstract The existence of macroscopic regions with antibaryon excess in the baryon asymmetric Universe with general baryon excess is the possible consequence of practically all models of baryosynthesis. Diffusion of matter and antimatter to the border of antimatter domains defines the minimal scale of the antimatter domains surviving to the present time. A model of diffused antiworld is considered, in which the density within the surviving antimatter domains is too low to form gravitationally bound objects. The possibility to test this model by measurements of cosmic gamma ray fluxes is discussed. The expected gamma ray flux is found to be acceptable for modern cosmic gamma ray detectors and for those planned for the near future. q 2000 Elsevier Science B.V. All rights reserved.
It was recently stated w1x that the existence of antimatter domains with scale up to 1000 Mpc in a baryon-symmetric Universe cannot escape contradiction with the observed gamma background. This argument does not exclude the case when the Universe is globally baryon asymmetric and the antibaryons within antimatter domains contribute a small fraction of the total baryon charge. On the other hand, it was shown w2–4x that the existence of antimatter domains in the baryon dominated Universe is a profound signature for the origin and evolution of baryon matter and its inhomogeneity. Depending on its parameters, the mechanism of in-
)
Corresponding author. E-mail address: [email protected] ŽM.Yu. Khlopov.
homogeneous baryosynthesis can lead to both highand low-density antibaryon domains. According to w5x, high-density domains can evolve into antimatter stellar objects, so that a globular cluster of antimatter stars can exist in our Galaxy, what may be tested in the cosmic searches for antimatter planned for the near future. On the other hand, as it was mentioned in Ref. w5x, low-density antibaryon domains cannot evolve into gravitationally bounded objects. The case of such ‘‘diffused antiworld’’ is considered in the present paper. There are different ways for antimatter domain formation. A simple modification of the baryogenesis scenario will lead to formation of domains with different signs of baryon asymmetry. To achieve this, the condition should be satisfied that C- and CP-violation must have different signs in different spatial
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368
M.Yu. KhlopoÕ et al.r Astroparticle Physics 12 (2000) 367–372
regions. For instance, this can occur if there exist two sources of CP violation, a spontaneous and an explicit one. Spontaneous symmetry breaking could naturally proceed at the second order phase transition which started after the inflationary stage w6x. One would expect that CP-odd amplitudes are not universal, space-independent quantities, but have different signs and values in different space points: f s s f s Ž x .. If the amplitude of spontaneous symmetry breaking is small in comparison with the explicit one, f s - f e , there would be relatively small fluctuations in the baryonic number density around the uniform baryonic background. In the opposite case, f e - f s , the amplitude of fluctuations is large with respect to the baryonic background density. One can see that if the domains with high antibaryonic density are formed in the second-order phase transition, their elementary size l i at the moment of formation is determined by l i , 1rlTc w7x, where Tc is the Ginzburg critical temperature at which the phase transition takes place and l is the self-interaction coupling constant of the field which breaks CP symmetry. In this case different elementary domains would expand together with the Universe and now their size would reach the value l 0 , l i ŽTcrT0 . s 1rlT0 , 10y5 cmrl, where T0 is the present temperature of the background radiation. Another possible mechanism for generation of large fluctuations of baryon asymmetry in space is based on the model of baryogenesis with baryonic charged condensate. At first, the baryon asymmetry is accumulated in the form of a condensate of the scalar superpartner of the colorless and electrically neutral combination of quark and lepton fields ² X : w8,9x. This condensate might be formed during the inflationary stage if baryonic and leptonic charges were not conserved and the potential UŽ X . had a flat direction. The subsequent decay of this condensate would result in a considerable baryon asymmetry. Depending upon the initial conditions in the different space points, it is possible to produce baryons or antibaryons in the decay of the condensate ² X : w10x. The characteristic size of an elementary domain with a definite sign of baryonic charge was estimated in Ref. w10x. At the end of inflation it is equal to r2 . l i , Hy1 expŽ ly1 , where H is the Hubble paramX eter at the end of inflation and l X is the constant of self-interaction of the field X.
In any case all the above-mentioned mechanisms are able to generate domains of antimatter of any baryon density and on any scale. Such domains may consist of many elementary domains of small scale, as in the case of the standard phase transition, or it can appear like one moderately inflated elementary domain as in the case of the phase transition triggered by inflation. Let us consider the case of low-density antimatter regions, corresponding to the model of diffused antiworld. When the density of antimatter r b within a domain is 3 orders of magnitude less than the baryon density r b Žwhich we assume in the further discussion corresponding to V b h 2 s 0.1, where h is the Hubble parameter normalized to 100 km sy1 Mpcy1 ., the cosmological nucleosynthesis in the period t , 1 % 10 s results in a nontrivial chemical composition. For smaller densities of antibaryons within domains no antinuclei are formed. Indeed, the reaction nqp™dqg
Ž 1.
is a crucial one for antimatter nucleosynthesis. Below 1 MeV the cross section for the reaction Ž1. practically does not depend on temperature and is given by
² s Õ :f 4.55 P 10y2 0 cm3rc .
Ž 2.
The antideuterium production in the reaction Ž1. is effective only if the reaction rate exceeds the expansion rate of the Universe, n b Ž T . ² s Õ :) H .
Ž 3.
It is convenient to rewrite Ž3. in terms of temperature and present baryon density, 850 N Ž T .
y1r2
T
ž /ž MeV
nb y7
10
cmy3
/b
R1 ,
Ž 4.
where N ŽT . is an effective number of degrees of freedom, b s r brr b . It can be easily seen from Ž4., that if r b is as small as 10y4r b , the reaction Ž1. is not effective below few MeV where the antideuterium state could be formed. A similar situation occurs in the case of recombination at Z , 1500. According to the theory of recombination w11x the validity of the Saha equation assumes that the equilibrium condition is fulfilled,
M.Yu. KhlopoÕ et al.r Astroparticle Physics 12 (2000) 367–372
corresponding to the recombination rate exceeding the expansion rate of the expansion of the Universe. One can easily find with the use of recombination coefficient w1x to all states but the ground state, eV
² s Õ :rec s 1.14 P 10y1 3 q0.814 P
ž / ž /
1r2
T
T
eV
1 y 2.2 P log
T
ž / eV
1r3
cm3rs ,
Ž 5.
that the equilibrium condition following from Ž3., Ž5., Z
ž
nb y7
10
cmy3
/b
R1 ,
diffused antiworld. Let us assume that the Universe contains regions of very small antibaryon excess density. At temperatures above several MeV these regions cannot be strongly affected by the diffusion of surrounding particles, because their mean free path length is small enough. Therefore, we will consider the evolution of these antibaryon domains at temperatures 4 P 10 3 K - T - 10 9 K, when the Universe is radiation-dominated and contains mainly electrons, protons, photons and neutrinos. If the size of the antibaryon region is much greater than the mean free path of surrounding particles we can solve a one-dimensional problem assuming that the ‘‘initial’’ baryon density at T s 10 9 K is given by
Ž 6.
is violated at the recombination temperature corresponding to Z , 1500 for b Q 10y4 . The similar estimation can be obtained with the use of the recombination coefficients w11x for 2P and 2S Hydrogen states. Therefore, at the density r brr b - 10y4 , owing to the low antimatter density inside the domain, no recombination takes place at Z , 1500 and the antimatter domain remains ionized after recombination in regions of baryonic matter. At such low densities, antimatter within a domain represents the radiation dominated positron–antiproton plasma rg ) r b for which the Jeans scale l J s l hr '3 is of the order of cosmological horizon l h which prevents development of gravitational instability. Radiation dominated antimatter plasma behaves as the component with the relativistic equation of state and cannot participate to the process of growth of density fluctuations, triggered by the dark matter. Note that at very small scales l - l T ; Ž ng s T .y1 ; 3 P 10 21 Ž1 q z .y3 cm the radiation pressure is ineffective which may cause the growth of density fluctuations induced by cold dark matter. As it is shown below such scale is smaller than the survival scale, so that this effect is not essential for individual small scale domains. Within large domains of diffused antimatter growth of small scale inhomogeneities, triggered by cold dark matter, leads to an antibaryon graining small scale, which as we show below increases the effect of annihilation. Let us give a quantitative estimation of the surviving size and the observational effects of domains of a
369
n b Ž R ,t 0 . s
½
n0 , 0,
x-0 , x)0 .
Ž 7.
In this case the diffusion equation for baryons is E nb E 2 nb s DŽ t . y a nb , Ž 8. Et E x2 where DŽ t . is a diffusion coefficient. The last term in Ž8. takes into account the expansion of the Universe. Note that, since the antibaryon component is very small, we neglected it in the diffusion equation. Let us introduce a new variable r defined as the baryon to photon ratio, r s n brng . Since the evolution of the photon density is given by the equation E ngrE t s ya ng , we can rewrite Ž8. in terms of r as Er E 2r s DŽ t . 2 , Ž 9. Et Ex where r Ž R ,t 0 . s
½
r0 , 0,
x-0 , x)0 .
We note that in the problem under consideration photons are distributed uniformly in baryon domains as well as in the antibaryon ones. Due to electroneutrality of the media, the motion of protons is strongly related with the motion of electrons. The last ones interact with photons, and this interaction defines the diffusion coefficient DŽ t . for protons. According to w12x this coefficient Žin units where the Boltzmann constant k b s 1. is given by 3Tg c DŽ t . f f 0.61 P 10 32 Zy3 cm2rs , Ž 10 . 2 rg s T
M.Yu. KhlopoÕ et al.r Astroparticle Physics 12 (2000) 367–372
370
where Tg and rg are the temperature and the energy density of the radiation, respectively, c is the velocity of light, s T is the Thomson cross section, Z is the red shift which is related with the time t at this stage by t f 2.6 P 10 19 srZ 2 convenient to extract the t-dependence and to write DŽ t . in the following form:
Solving Ž9. with the diffusion coefficient Ž11., we find
dominated stage, remain practically unaffected by the diffusion of the ordinary matter. Note that according to Ž13. in the limit t 4 t 0 the motion of the boundary is determined by the final time t and does not depend on t 0 . This is due to the fact that at the end of the radiation-dominated stage the particle number density drops significantly and as a consequence the mean free-path length increases, thus giving an important contribution to the boundary motion. Below 4000 K atoms are formed in baryon domains. Since the antibaryon density is assumed to be small enough Ž r brr b - 10y4 . in our approach, we can consider the flow of hydrogen atoms into antibaryon regions as a motion of free-streaming atoms. The physical distance traveled by the atoms after recombination until the present time t p is given by
r Ž R ,t .
d f ap
D Ž t . s DŽ t0 .
3r2
t
ž / t0
.
Ž 11 .
At the initial time t 0 , corresponding to the temperature T0 s 10 9 K, DŽ t 0 . is given according to Ž10. by D Ž t 0 . f 1.24 P 10 6 cm2rs .
s
r0
Ž 12 .
°
~1 y F ( 2¢
5 8
x
(D t
0 0
Ž trt 0 .
5r2
y1
¶• ß, Ž 13 .
where F is the error function. Assuming t 4 t 0 , we get r Ž R ,t . f
r0 2
½ ( 1yF
x
8
'Dt
5
,
Ž 14 .
or, in terms of temperature, r Ž R ,t . f
r0 2
½
y4
1 y F 0.527 P 10
T
ž / T0
5r2
x cm
5
.
Ž 15 .
ÕŽ t. dt
rec
aŽ t .
,
Ž 17 .
where Õ Ž t . is the average velocity of atoms and aŽ t . is the scale factor of the Universe. It is convenient to set a s ya p , where y s 1rŽ1 q Z . s TprT, and to rewrite Ž17. as df
5
tp
Ht
ap
Õ Ž a. d a
rec
aa˙
Ha
s
ÕŽ y. d y
1
Ha
recra p
yy˙
.
Ž 18 .
At the instant of recombination the velocity of atoms is given by the thermal value Õrec f c Trecrm , where m is the mass of the atom. After 4000 K, the typical velocity of atoms Õ f prm is redshifted down as 1ra ; T. Therefore, below 4000 K we have Õ f cŽTprm. mrT Ž1ry .. Taking the equation
(
'
2 Ž y˙ . s
8p 3
Gr Ž y . y 2
Ž 19 .
If we choose T s 4000 K at the end of the radiation-dominated epoch, then r0 x r Ž R ,t . < 4000 K f 1yF 0.524 . Ž 16 . 2 pc
Žwhere G is Newton’s constant. and substituting for the density the expression r Ž y . s r 0ry 3, in the case of a matter-dominated Universe we find
It follows from Ž16. that the average displacement of the boundary between baryon and antibaryon domains is about D x , 0.2 pc. Therefore, the primordial antibaryon regions of low density, which grow up to 1 pc or more at the end of the radiation-
df
½
5
s
3
cTp
m
m
Trec
( ( ( ( 8p Gr 0 3
m
8p Gr 0
Trec
.
dy
1
HT rT p
rec
y 3r2
Ž 20 .
M.Yu. KhlopoÕ et al.r Astroparticle Physics 12 (2000) 367–372
Substituting r 0 s rc , where rc s 1.88 P 10y2 9 h 2 s grcm3 is the critical density of the Universe, we obtain d ; 3rh kpc. In the wide range of the Hubble constant h between 0.4 and 1 the free-streaming length of atoms will be of the order of several kpc. Therefore, antibaryon regions of the same size will be filled at present by hydrogen atoms. A key observation to test the model of diffused antiworld could be the search for gamma rays from a boundary annihilation of antimatter and hydrogen atoms. Let us consider first the possibility of annihilation in antimatter domains filled with hydrogen atoms Žthe annihilation of matter–antimatter domains in baryon symmetric Universe was considered in Ref. w1,13x.. The equation for the number density of antiprotons which takes into account both the annihilation and the expansion of the Universe is given by d nb dt
s y² s Õ : n b n b y a n b .
Ž 21 .
In the limit n b 4 n b we can neglect the variation of n b due to annihilation. Then, introducing r s n brng , r s n brng , and solving Ž21., we find that at present time in antimatter domains filled with hydrogen atoms, r p s rrec exp y
tp
Ht
² s Õ :ng r d t .
Ž 22 .
Since according to w14x at energies below 10 eV the cross section of pH annihilation is given by ² s Õ : f 2.7 P 10y9 cm3rs ,
Ž 23 .
we find that the integral in Ž22. is much greater than unity. So, no gamma radiation occurs at present from such regions because practically all antiprotons have been already annihilated. Therefore, the radiation comes only from the narrow region of the boundary between matter and antimatter domains, where antibaryons have not annihilated yet. The width of this region is d ; ÕD t, where D t is defined from the condition tp
Ht yD t² s Õ : n p
b
d t f ² s Õ: nb D t ; 1 ,
and the velocity Õ is given by
Ž 24 .
T
Õfc
m
(
3m
.
Trec
Ž 25 .
Substituting all numerical values, we find for the width of the region,
(
dfc
3
f 0.86
Tp Tp
1
Trec m ² s Õ : n b 10y7 cmy3
ž
nb
/
pc .
Ž 26 .
The gamma flux near Earth in this case is given by dF f dv dV
d n d Ng
V
,
d t d v 4p rA2
Ž 27 .
where d nrdt s ² s Õ : n b n b is the rate of annihilation per unit volume and per unit time; d Ngrd v is the differential cross section for an inclusive gamma production; V s 4p R 2 d is the volume of the annihilating boundary of the diffused antiworld at present; R is the size of the diffused antiworld region; rA is the distance between Earth and the diffused antiworld region. Integrating Ž27. over photon energy, we obtain dF dV
rec
371
f²s
R
2
ž /
Õ :b n 2b ² Ng : d
,
rA
Ž 28 .
where ² Ng : is the average number of photons per one act of annihilation Žtypically ² Ng : f 4., b s n brn b < 1. Finally, taking into account Ž24., we get dF dV
f b n b² Ng :
R
ž /( rA
f 2.6 P 10y4b
=
R
ž / rA
2
ž
c
3
Tp Tp Trec m
nb 10y7 cmy3
/
2
cmy2 sy1 sry1 .
Ž 29 .
The absence of absorption bands in the spectra of long-distance quasars indicates that the temperature of the intergalactic gas is high enough and that this
M.Yu. KhlopoÕ et al.r Astroparticle Physics 12 (2000) 367–372
372
gas is ionized. If this is the case we have to change the cross section in Ž23. by the corresponding cross section for pp annihilation w14x, ² s Õ : f 6.5 P 10y1 7 Ž Õrc . cm3rs .
Ž 30 .
However, for pp annihilation the integral in Ž22. will be much greater than unity, similarly to the pH case. Therefore, expression Ž29. for the photon flux remains valid for pp annihilation because this flux does not depend explicitly on ² s Õ :. Up to now, no evidence for a major anisotropy in the gamma background has been observed. Below 1 GeV at high latitudes the diffused photon background is given by w15x
Fdif dV
f 10y6 cmy2 sy1 sry1 .
Ž 31 .
It was shown above that a diffused antiworld can exist if b Q 10y4 . Therefore, it follows from Ž29. that the possible gamma flux from the annihilation at the boundary of two worlds should be about or less than 10y8 cmy2 sy1 sry1 . This means that such diffused antiworlds could exist, in particular not far from our Galaxy, successfully avoiding to be detected. However, the development of modern detectors of gamma rays, like EGRET and AMS with the flux sensitivity up to 10y8 cmy2 sy1 sry1 , gives a hope to detect diffused antiworlds. In this article we have found that the minimal scale of an antibaryon domain, which was not destroyed by annihilation, must be larger then 1 kpc at present. Independently of the conditions of creation of such an antiworld domain, the density within it will not increase due to gravitational instability, provided that two conditions are satisfied. Namely, this domain must be located in the void and the density of antimatter must be less than the critical density. It means that the space structure and geometrical form of the annihilation boundary of such a domain is conserved. In the case when the antiworld region of scale R consists of many antimatter domains of minimal annihilation scale d, the annihilation in such a region takes place at the border of each little domain. As a result the annihilation occurs in the
whole volume of this region and, as a consequence, the gamma flux increases, owing both to the volume effect and the possibility for the antimatter density within a small domain to be as high as b f 1. This picture of volume annihilation in antiworld regions can be tested in searches for the angular variation of gamma flux in the angular range a ; RrrA . Acknowledgements This work was supported in part by the scientific and educational center ‘‘Cosmion’’ and performed within the framework of the International projects Astrodamus, Cosmion-ETHZ and Eurocos-AMS. M.K. and R.K. are grateful to I Rome University ‘‘La Sapienza’’ and III Rome University ‘‘Roma Tre’’ for hospitality and support. We thank B. Kerbikov, A. Kudrjavtsev, and A. Sudarikov for interesting discussions and suggestions.
References w1x A.G. Cohen, A. De Rujula, S.L. Glashow, Astrophys. J. 495 Ž1998. 539. w2x V.M. Chechetkin, M.Yu. Khlopov, M.G. Sapozhnikov, Ya.B. Zeldovich, Phys. Lett. 118B Ž1982. 329. w3x V.M. Chechetkin, M.Yu. Khlopov, M.G. Sapozhnikov, Nuovo Cimento 5 Ž1982. 1. w4x M.Yu. Khlopov, Cosmion’94, Proc. 1st Int. Conf. Cosmoparticle Physics, M.Yu. Khlopov, M.E. Prokhorov, A.A. Starobinsky, J. Tran Thanh Van, eds. ŽEditions Frontieres, 1996. p. 67. w5x M.Yu. Khlopov, Gravitation & Cosmology 4 Ž1998. 1. w6x T.D. Lee, Phys. Rev. D 8 Ž1973. 1226. w7x A.D. Dolgov, Preprint TAC-1996-010 Ž1996.; hepphr9605280 Ž1996.. w8x I. Affleck, M. Dine, Nucl. Phys. B 249 Ž1985. 361. w9x A.D. Linde, Phys. Lett. 160B Ž1985. 243. w10x A.D. Dolgov, Phys. Rep. 222 Ž1992. 311. w11x Ya.B. Zeldovich, V.G. Kurt, R.A. Sunyaev, ZhETF 55 Ž1968. 278. w12x Ya.B. Zeldovich, I.D. Novikov, The Structure and Evolution of the Universe ŽMoscow, Nauka, 1971.. w13x F.W. Stecker, Nucl. Phys. B 252 Ž1985. 25. w14x D.L. Morgan, V.W. Hudges, Phys. Rev. D 2 Ž1970. 1389. w15x M. Pohl, astro-phr9807267 Ž1998..
Evolution and observational signature of diffused antiworld M.Yu. Khlopov a
a,b,c,d,)
, R.V. Konoplich a,b,c,d , R. Mignani a , S.G. Rubin A.S. Sakharov b,d
b,d
,
Dipartimento di Fisica ‘‘E.Amaldi’’, III UniÕersita’ di Roma ‘‘Roma Tre’’, and INFN, Sezione di Roma III, Rome, Italy b Center for CosmoParticle Physics ‘‘Cosmion’’, Moscow, Russian Federation c Institute of Applied Mathematics, Moscow, Russian Federation d Moscow Engineering Physics Institute (Technical UniÕersity), Moscow, Russian Federation Received 12 November 1998; received in revised form 27 April 1999; accepted 21 June 1999
Abstract The existence of macroscopic regions with antibaryon excess in the baryon asymmetric Universe with general baryon excess is the possible consequence of practically all models of baryosynthesis. Diffusion of matter and antimatter to the border of antimatter domains defines the minimal scale of the antimatter domains surviving to the present time. A model of diffused antiworld is considered, in which the density within the surviving antimatter domains is too low to form gravitationally bound objects. The possibility to test this model by measurements of cosmic gamma ray fluxes is discussed. The expected gamma ray flux is found to be acceptable for modern cosmic gamma ray detectors and for those planned for the near future. q 2000 Elsevier Science B.V. All rights reserved.
It was recently stated w1x that the existence of antimatter domains with scale up to 1000 Mpc in a baryon-symmetric Universe cannot escape contradiction with the observed gamma background. This argument does not exclude the case when the Universe is globally baryon asymmetric and the antibaryons within antimatter domains contribute a small fraction of the total baryon charge. On the other hand, it was shown w2–4x that the existence of antimatter domains in the baryon dominated Universe is a profound signature for the origin and evolution of baryon matter and its inhomogeneity. Depending on its parameters, the mechanism of in-
)
Corresponding author. E-mail address: [email protected] ŽM.Yu. Khlopov.
homogeneous baryosynthesis can lead to both highand low-density antibaryon domains. According to w5x, high-density domains can evolve into antimatter stellar objects, so that a globular cluster of antimatter stars can exist in our Galaxy, what may be tested in the cosmic searches for antimatter planned for the near future. On the other hand, as it was mentioned in Ref. w5x, low-density antibaryon domains cannot evolve into gravitationally bounded objects. The case of such ‘‘diffused antiworld’’ is considered in the present paper. There are different ways for antimatter domain formation. A simple modification of the baryogenesis scenario will lead to formation of domains with different signs of baryon asymmetry. To achieve this, the condition should be satisfied that C- and CP-violation must have different signs in different spatial
0927-6505r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 6 5 0 5 Ž 9 9 . 0 0 0 9 9 - 7
368
M.Yu. KhlopoÕ et al.r Astroparticle Physics 12 (2000) 367–372
regions. For instance, this can occur if there exist two sources of CP violation, a spontaneous and an explicit one. Spontaneous symmetry breaking could naturally proceed at the second order phase transition which started after the inflationary stage w6x. One would expect that CP-odd amplitudes are not universal, space-independent quantities, but have different signs and values in different space points: f s s f s Ž x .. If the amplitude of spontaneous symmetry breaking is small in comparison with the explicit one, f s - f e , there would be relatively small fluctuations in the baryonic number density around the uniform baryonic background. In the opposite case, f e - f s , the amplitude of fluctuations is large with respect to the baryonic background density. One can see that if the domains with high antibaryonic density are formed in the second-order phase transition, their elementary size l i at the moment of formation is determined by l i , 1rlTc w7x, where Tc is the Ginzburg critical temperature at which the phase transition takes place and l is the self-interaction coupling constant of the field which breaks CP symmetry. In this case different elementary domains would expand together with the Universe and now their size would reach the value l 0 , l i ŽTcrT0 . s 1rlT0 , 10y5 cmrl, where T0 is the present temperature of the background radiation. Another possible mechanism for generation of large fluctuations of baryon asymmetry in space is based on the model of baryogenesis with baryonic charged condensate. At first, the baryon asymmetry is accumulated in the form of a condensate of the scalar superpartner of the colorless and electrically neutral combination of quark and lepton fields ² X : w8,9x. This condensate might be formed during the inflationary stage if baryonic and leptonic charges were not conserved and the potential UŽ X . had a flat direction. The subsequent decay of this condensate would result in a considerable baryon asymmetry. Depending upon the initial conditions in the different space points, it is possible to produce baryons or antibaryons in the decay of the condensate ² X : w10x. The characteristic size of an elementary domain with a definite sign of baryonic charge was estimated in Ref. w10x. At the end of inflation it is equal to r2 . l i , Hy1 expŽ ly1 , where H is the Hubble paramX eter at the end of inflation and l X is the constant of self-interaction of the field X.
In any case all the above-mentioned mechanisms are able to generate domains of antimatter of any baryon density and on any scale. Such domains may consist of many elementary domains of small scale, as in the case of the standard phase transition, or it can appear like one moderately inflated elementary domain as in the case of the phase transition triggered by inflation. Let us consider the case of low-density antimatter regions, corresponding to the model of diffused antiworld. When the density of antimatter r b within a domain is 3 orders of magnitude less than the baryon density r b Žwhich we assume in the further discussion corresponding to V b h 2 s 0.1, where h is the Hubble parameter normalized to 100 km sy1 Mpcy1 ., the cosmological nucleosynthesis in the period t , 1 % 10 s results in a nontrivial chemical composition. For smaller densities of antibaryons within domains no antinuclei are formed. Indeed, the reaction nqp™dqg
Ž 1.
is a crucial one for antimatter nucleosynthesis. Below 1 MeV the cross section for the reaction Ž1. practically does not depend on temperature and is given by
² s Õ :f 4.55 P 10y2 0 cm3rc .
Ž 2.
The antideuterium production in the reaction Ž1. is effective only if the reaction rate exceeds the expansion rate of the Universe, n b Ž T . ² s Õ :) H .
Ž 3.
It is convenient to rewrite Ž3. in terms of temperature and present baryon density, 850 N Ž T .
y1r2
T
ž /ž MeV
nb y7
10
cmy3
/b
R1 ,
Ž 4.
where N ŽT . is an effective number of degrees of freedom, b s r brr b . It can be easily seen from Ž4., that if r b is as small as 10y4r b , the reaction Ž1. is not effective below few MeV where the antideuterium state could be formed. A similar situation occurs in the case of recombination at Z , 1500. According to the theory of recombination w11x the validity of the Saha equation assumes that the equilibrium condition is fulfilled,
M.Yu. KhlopoÕ et al.r Astroparticle Physics 12 (2000) 367–372
corresponding to the recombination rate exceeding the expansion rate of the expansion of the Universe. One can easily find with the use of recombination coefficient w1x to all states but the ground state, eV
² s Õ :rec s 1.14 P 10y1 3 q0.814 P
ž / ž /
1r2
T
T
eV
1 y 2.2 P log
T
ž / eV
1r3
cm3rs ,
Ž 5.
that the equilibrium condition following from Ž3., Ž5., Z
ž
nb y7
10
cmy3
/b
R1 ,
diffused antiworld. Let us assume that the Universe contains regions of very small antibaryon excess density. At temperatures above several MeV these regions cannot be strongly affected by the diffusion of surrounding particles, because their mean free path length is small enough. Therefore, we will consider the evolution of these antibaryon domains at temperatures 4 P 10 3 K - T - 10 9 K, when the Universe is radiation-dominated and contains mainly electrons, protons, photons and neutrinos. If the size of the antibaryon region is much greater than the mean free path of surrounding particles we can solve a one-dimensional problem assuming that the ‘‘initial’’ baryon density at T s 10 9 K is given by
Ž 6.
is violated at the recombination temperature corresponding to Z , 1500 for b Q 10y4 . The similar estimation can be obtained with the use of the recombination coefficients w11x for 2P and 2S Hydrogen states. Therefore, at the density r brr b - 10y4 , owing to the low antimatter density inside the domain, no recombination takes place at Z , 1500 and the antimatter domain remains ionized after recombination in regions of baryonic matter. At such low densities, antimatter within a domain represents the radiation dominated positron–antiproton plasma rg ) r b for which the Jeans scale l J s l hr '3 is of the order of cosmological horizon l h which prevents development of gravitational instability. Radiation dominated antimatter plasma behaves as the component with the relativistic equation of state and cannot participate to the process of growth of density fluctuations, triggered by the dark matter. Note that at very small scales l - l T ; Ž ng s T .y1 ; 3 P 10 21 Ž1 q z .y3 cm the radiation pressure is ineffective which may cause the growth of density fluctuations induced by cold dark matter. As it is shown below such scale is smaller than the survival scale, so that this effect is not essential for individual small scale domains. Within large domains of diffused antimatter growth of small scale inhomogeneities, triggered by cold dark matter, leads to an antibaryon graining small scale, which as we show below increases the effect of annihilation. Let us give a quantitative estimation of the surviving size and the observational effects of domains of a
369
n b Ž R ,t 0 . s
½
n0 , 0,
x-0 , x)0 .
Ž 7.
In this case the diffusion equation for baryons is E nb E 2 nb s DŽ t . y a nb , Ž 8. Et E x2 where DŽ t . is a diffusion coefficient. The last term in Ž8. takes into account the expansion of the Universe. Note that, since the antibaryon component is very small, we neglected it in the diffusion equation. Let us introduce a new variable r defined as the baryon to photon ratio, r s n brng . Since the evolution of the photon density is given by the equation E ngrE t s ya ng , we can rewrite Ž8. in terms of r as Er E 2r s DŽ t . 2 , Ž 9. Et Ex where r Ž R ,t 0 . s
½
r0 , 0,
x-0 , x)0 .
We note that in the problem under consideration photons are distributed uniformly in baryon domains as well as in the antibaryon ones. Due to electroneutrality of the media, the motion of protons is strongly related with the motion of electrons. The last ones interact with photons, and this interaction defines the diffusion coefficient DŽ t . for protons. According to w12x this coefficient Žin units where the Boltzmann constant k b s 1. is given by 3Tg c DŽ t . f f 0.61 P 10 32 Zy3 cm2rs , Ž 10 . 2 rg s T
M.Yu. KhlopoÕ et al.r Astroparticle Physics 12 (2000) 367–372
370
where Tg and rg are the temperature and the energy density of the radiation, respectively, c is the velocity of light, s T is the Thomson cross section, Z is the red shift which is related with the time t at this stage by t f 2.6 P 10 19 srZ 2 convenient to extract the t-dependence and to write DŽ t . in the following form:
Solving Ž9. with the diffusion coefficient Ž11., we find
dominated stage, remain practically unaffected by the diffusion of the ordinary matter. Note that according to Ž13. in the limit t 4 t 0 the motion of the boundary is determined by the final time t and does not depend on t 0 . This is due to the fact that at the end of the radiation-dominated stage the particle number density drops significantly and as a consequence the mean free-path length increases, thus giving an important contribution to the boundary motion. Below 4000 K atoms are formed in baryon domains. Since the antibaryon density is assumed to be small enough Ž r brr b - 10y4 . in our approach, we can consider the flow of hydrogen atoms into antibaryon regions as a motion of free-streaming atoms. The physical distance traveled by the atoms after recombination until the present time t p is given by
r Ž R ,t .
d f ap
D Ž t . s DŽ t0 .
3r2
t
ž / t0
.
Ž 11 .
At the initial time t 0 , corresponding to the temperature T0 s 10 9 K, DŽ t 0 . is given according to Ž10. by D Ž t 0 . f 1.24 P 10 6 cm2rs .
s
r0
Ž 12 .
°
~1 y F ( 2¢
5 8
x
(D t
0 0
Ž trt 0 .
5r2
y1
¶• ß, Ž 13 .
where F is the error function. Assuming t 4 t 0 , we get r Ž R ,t . f
r0 2
½ ( 1yF
x
8
'Dt
5
,
Ž 14 .
or, in terms of temperature, r Ž R ,t . f
r0 2
½
y4
1 y F 0.527 P 10
T
ž / T0
5r2
x cm
5
.
Ž 15 .
ÕŽ t. dt
rec
aŽ t .
,
Ž 17 .
where Õ Ž t . is the average velocity of atoms and aŽ t . is the scale factor of the Universe. It is convenient to set a s ya p , where y s 1rŽ1 q Z . s TprT, and to rewrite Ž17. as df
5
tp
Ht
ap
Õ Ž a. d a
rec
aa˙
Ha
s
ÕŽ y. d y
1
Ha
recra p
yy˙
.
Ž 18 .
At the instant of recombination the velocity of atoms is given by the thermal value Õrec f c Trecrm , where m is the mass of the atom. After 4000 K, the typical velocity of atoms Õ f prm is redshifted down as 1ra ; T. Therefore, below 4000 K we have Õ f cŽTprm. mrT Ž1ry .. Taking the equation
(
'
2 Ž y˙ . s
8p 3
Gr Ž y . y 2
Ž 19 .
If we choose T s 4000 K at the end of the radiation-dominated epoch, then r0 x r Ž R ,t . < 4000 K f 1yF 0.524 . Ž 16 . 2 pc
Žwhere G is Newton’s constant. and substituting for the density the expression r Ž y . s r 0ry 3, in the case of a matter-dominated Universe we find
It follows from Ž16. that the average displacement of the boundary between baryon and antibaryon domains is about D x , 0.2 pc. Therefore, the primordial antibaryon regions of low density, which grow up to 1 pc or more at the end of the radiation-
df
½
5
s
3
cTp
m
m
Trec
( ( ( ( 8p Gr 0 3
m
8p Gr 0
Trec
.
dy
1
HT rT p
rec
y 3r2
Ž 20 .
M.Yu. KhlopoÕ et al.r Astroparticle Physics 12 (2000) 367–372
Substituting r 0 s rc , where rc s 1.88 P 10y2 9 h 2 s grcm3 is the critical density of the Universe, we obtain d ; 3rh kpc. In the wide range of the Hubble constant h between 0.4 and 1 the free-streaming length of atoms will be of the order of several kpc. Therefore, antibaryon regions of the same size will be filled at present by hydrogen atoms. A key observation to test the model of diffused antiworld could be the search for gamma rays from a boundary annihilation of antimatter and hydrogen atoms. Let us consider first the possibility of annihilation in antimatter domains filled with hydrogen atoms Žthe annihilation of matter–antimatter domains in baryon symmetric Universe was considered in Ref. w1,13x.. The equation for the number density of antiprotons which takes into account both the annihilation and the expansion of the Universe is given by d nb dt
s y² s Õ : n b n b y a n b .
Ž 21 .
In the limit n b 4 n b we can neglect the variation of n b due to annihilation. Then, introducing r s n brng , r s n brng , and solving Ž21., we find that at present time in antimatter domains filled with hydrogen atoms, r p s rrec exp y
tp
Ht
² s Õ :ng r d t .
Ž 22 .
Since according to w14x at energies below 10 eV the cross section of pH annihilation is given by ² s Õ : f 2.7 P 10y9 cm3rs ,
Ž 23 .
we find that the integral in Ž22. is much greater than unity. So, no gamma radiation occurs at present from such regions because practically all antiprotons have been already annihilated. Therefore, the radiation comes only from the narrow region of the boundary between matter and antimatter domains, where antibaryons have not annihilated yet. The width of this region is d ; ÕD t, where D t is defined from the condition tp
Ht yD t² s Õ : n p
b
d t f ² s Õ: nb D t ; 1 ,
and the velocity Õ is given by
Ž 24 .
T
Õfc
m
(
3m
.
Trec
Ž 25 .
Substituting all numerical values, we find for the width of the region,
(
dfc
3
f 0.86
Tp Tp
1
Trec m ² s Õ : n b 10y7 cmy3
ž
nb
/
pc .
Ž 26 .
The gamma flux near Earth in this case is given by dF f dv dV
d n d Ng
V
,
d t d v 4p rA2
Ž 27 .
where d nrdt s ² s Õ : n b n b is the rate of annihilation per unit volume and per unit time; d Ngrd v is the differential cross section for an inclusive gamma production; V s 4p R 2 d is the volume of the annihilating boundary of the diffused antiworld at present; R is the size of the diffused antiworld region; rA is the distance between Earth and the diffused antiworld region. Integrating Ž27. over photon energy, we obtain dF dV
rec
371
f²s
R
2
ž /
Õ :b n 2b ² Ng : d
,
rA
Ž 28 .
where ² Ng : is the average number of photons per one act of annihilation Žtypically ² Ng : f 4., b s n brn b < 1. Finally, taking into account Ž24., we get dF dV
f b n b² Ng :
R
ž /( rA
f 2.6 P 10y4b
=
R
ž / rA
2
ž
c
3
Tp Tp Trec m
nb 10y7 cmy3
/
2
cmy2 sy1 sry1 .
Ž 29 .
The absence of absorption bands in the spectra of long-distance quasars indicates that the temperature of the intergalactic gas is high enough and that this
M.Yu. KhlopoÕ et al.r Astroparticle Physics 12 (2000) 367–372
372
gas is ionized. If this is the case we have to change the cross section in Ž23. by the corresponding cross section for pp annihilation w14x, ² s Õ : f 6.5 P 10y1 7 Ž Õrc . cm3rs .
Ž 30 .
However, for pp annihilation the integral in Ž22. will be much greater than unity, similarly to the pH case. Therefore, expression Ž29. for the photon flux remains valid for pp annihilation because this flux does not depend explicitly on ² s Õ :. Up to now, no evidence for a major anisotropy in the gamma background has been observed. Below 1 GeV at high latitudes the diffused photon background is given by w15x
Fdif dV
f 10y6 cmy2 sy1 sry1 .
Ž 31 .
It was shown above that a diffused antiworld can exist if b Q 10y4 . Therefore, it follows from Ž29. that the possible gamma flux from the annihilation at the boundary of two worlds should be about or less than 10y8 cmy2 sy1 sry1 . This means that such diffused antiworlds could exist, in particular not far from our Galaxy, successfully avoiding to be detected. However, the development of modern detectors of gamma rays, like EGRET and AMS with the flux sensitivity up to 10y8 cmy2 sy1 sry1 , gives a hope to detect diffused antiworlds. In this article we have found that the minimal scale of an antibaryon domain, which was not destroyed by annihilation, must be larger then 1 kpc at present. Independently of the conditions of creation of such an antiworld domain, the density within it will not increase due to gravitational instability, provided that two conditions are satisfied. Namely, this domain must be located in the void and the density of antimatter must be less than the critical density. It means that the space structure and geometrical form of the annihilation boundary of such a domain is conserved. In the case when the antiworld region of scale R consists of many antimatter domains of minimal annihilation scale d, the annihilation in such a region takes place at the border of each little domain. As a result the annihilation occurs in the
whole volume of this region and, as a consequence, the gamma flux increases, owing both to the volume effect and the possibility for the antimatter density within a small domain to be as high as b f 1. This picture of volume annihilation in antiworld regions can be tested in searches for the angular variation of gamma flux in the angular range a ; RrrA . Acknowledgements This work was supported in part by the scientific and educational center ‘‘Cosmion’’ and performed within the framework of the International projects Astrodamus, Cosmion-ETHZ and Eurocos-AMS. M.K. and R.K. are grateful to I Rome University ‘‘La Sapienza’’ and III Rome University ‘‘Roma Tre’’ for hospitality and support. We thank B. Kerbikov, A. Kudrjavtsev, and A. Sudarikov for interesting discussions and suggestions.
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