4 May 1995
PHYSICS LETTERS 6
ELSEVIER Physics Letters B 350 (1995) 13-16
Baryogenesis from long cosmic strings Masoud Mohazzab
Institute for Studies in Theoretical Physics and Mathematics, PO. Box 5746, Tehran 19395, Iran
Received 5 December 1994; revised manuscript received 16 February 1995 Editor: M. Dine
Based on the mechanism of cusp production on long cosmic strings, the baryon asymmetry caused by cusp annihilation has been calculated. The result is compatible with observation and stronger than the results from loops.
Grand Unified Theories (GUTS) are possible frameworks to explain the baryon asymmetry of the universe where the Sakharovs conditions [ I] i.e. 1) baryon number violation, 2) C and CP violation and 3) departure from thermal equilibrium are satisfied. GUTS predict that the universe underwent a series of phase transitions during its early stage of evolution. Cosmic strings are one-dimensional topological de- fects that are generated during the phase transition [ 2,3]. Cosmic strings have to be in the form of loops or infinitely long with the important parameter CL, mass per length of the strings and are very desirable in cosmology. The value Gp N 10m6 is compatible with 1) the scale of GUT symmetry breaking 2) structure formation models and 3) thermal fluctuations of the background radiation (the recent COBE results) 2
The oscillations of cosmic strings typically lead to the formation of cusps. A cusp is a point where two
E-mail address: firstname.lastname@example.org. 2 The South Pole CMB observations at angular scale of less than
lo indicates a non-Gaussian distribution which are consistent with the fluctuations due to topological defects .
segments of the strings overlap and the point reaches the speed of light. There is no topological barrier to conserve cusps from decaying into bursts of Higgs or superheavy particles. The rate of decay and total energy of cusp annihilation has been worked out in  and , respectively. The starting point for the perturbative calculation is to consider, as a toy model, the Mexican hat potential
U(4) = $&(1412 - a2)2 (1.1)
where A0 is a self-coupling constant, q5 is a complex scalar field which gives rise to strings and (+ is the scale of GUT symmetry breaking. 4 is coupled to a scalar field $ via the interaction lagrangian
Lj = -A[&@ (1.2)
where A is the coupling constant of the interaction term.
The dimensionless value that most authors try to derive is the ratio of the net baryon number density, denoted by no - ns, to the entropy density of the universe, S, or 71 = v. The observational value for v based on the measurements of abundances of primordial D, 3He and 7Li is in the range (6-10) x lo-.
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14 M. Moha.zzab/Physics Letters B 350 (1995) 13-16
Cusp annihilation on cosmic strings has long been considered as a mechanism to describe the baryon asymmetry of the universe [ 71. The annihilation of cusps produces superheavy $ particles. Further decay of these particles results in the baryon asymmetry of AB N 10-2-10-3, where AB is the amount of baryon number violation due to CP violation from the decay of superheavy particles [ 81. In [ 71 loops of ordinary cosmic strings are considered and the baryon number production from kink and cusp evaporation has been calculated. The result of [ 71 shows cusp annihilation on loops results in a compatible value for 7. Kink evaporation, on the other hand, cannot describe the observed value of baryon asymmetry. Another mech- anism to be accounted for baryogenesis is the col- lapse of the topological defects such as cosmic string loops or textures [ 91. In the former approach, oscilla- tions of cosmic string loops reduce their radius due to gravitational radiation. The loops shrink to their min- imum size N h-/*c- when they have zero winding number and can decay into superheavy particles. The decay of these superheavy particles accounts for the baryon number asymmetry. The other mechanism of [ 91 is the shrinking of the loops by cusp annihilation. In each period of oscillation of a loop, equivalent to its length, a cusp forms and radiates causing a loss of energy and therefore a decrease of the radius of the loop.
The ratios of baryon number generation of loops due to the decrease of its radius by gravitational radiation and cusp annihilation are, respectively,
rlg.r. = %laxY -39~~~) -314~ 7)3
%.a. = %mxA*( g13*
where q,,ax N O.O3+AB (lo-l3 < AB < lo-*) and NX is the helicity of spin and N is the number of states. y is a constant determined numerically to be y N 100 [ lo]. TF is the temperature when baryon number violating processes freeze out and T, is the critical temperature of string phase transition [ 531.
If the baryon number violating processes fall out of thermal equilibrium at a temperature TF which is be- low the critical temperature T,, then the baryon to en- tropy ratio will be suppressed by only a power of tem- perature. The power is different depending on whether
gravitational radiation or cusp evaporation is the dom- inant decay mechanism. ( 1.3) shows the power 3 for the case the string loops decay gravitationally and i from ( 1.4) when it decays by cusp annihilation3 .
In [ 111 a mechanism has been suggested that pro- duces cusps from colliding traveling waves on long strings. In this work, based on the same mechanism, we calculate the baryon asymmetry caused by long cosmic strings from cusp annihilation.
In the next section, we calculate the amount of baryon number from cusp annihilationon long strings.
2. Baryon number production from long strings
According to a numerical simulation, long strings are 80% of the total cosmic strings formed at the GUT phase transition [ 31. Therefore they may have impor- tant contribution to radiation from cosmic strings.
Long strings do not shrink to a point and the only possibility for them to radiate superheavy particles is through cusp annihilation. Cusps, on long strings, are formed, up to a probability, when two wiggles trav- eling along the string collide. If we assume that the wiggles have random shapes we can find the proba- bility of formation of a cusp is 50% [ II]. The prob- ability of formation of more than one cusp in each collision is also non-zero. Of course it is clear that the shapes of wiggles are not random since the frac- tal dimension of cosmic strings is 1.2 (and not 2 for the random shape) [ 121. Due to this mechanism and the possibility of superheavy particle production from cusp annihilation, long strings could contribute to the baryon number generation.
The scaling distribution of long strings is given by [31
scaling v %. = 7 (2.1)
where v is a constant of the order 100 and t is the cosmological time. nt., is the number of long strings per volume 3. The distribution before the scaling so- lution, however, is different and is given by [ 131
3 I am grateful to R.H. Bmndenberger for this comment.
h4. Mohamb /Physics Letters B 350 (1995) 13-l 6 15
where Y is constant. The distribution of traveling waves on long strings
can roughly be written [ 111 as
K(1, t) = + (2.3)
where 1 is the size (of the order of the wavelength) of the traveling wiggle and cy is a constant. K( 1, r)dZ is the number of small scale wiggles of length 1 in the interval [I, 1 + dl] on a long string of length 1 [ 111.
The total number of cusps per volume will be
where z (t) is the redshift factor at time f and 1 /I represents the frequency of the impacts of the wiggles. The probability of cusp formation on long string upon each collision is denoted by PC.
In the integration (2.4) /tin is a minimum for the size of traveling waves, beyond which gravitational radiation smoothes out the wiggle,
Iti,, N yGpt. (2.5)
Finally the minimum time TV,, is close to the time when cosmic string is formed, i.e. tCr the time when GUT phase transition occurs.
It should be noted that we have neglected the effect of friction and the fact that the traveling waves do not change their shape when they travel along the string [141.
To implement the third Sakharov condition, i.e. out of equilibrium condition, we should have
where I is the decay rate of the superheavy particle and H is the expansion rate of the Universe. Here we consider the decay of superheavy particles and dont consider the inverse decay or collisions. Therefore the minimum time can be approximated by tin = t, N lOm& [ 71, when the inverse decay the Boltzmann factor. t, is given by
is suppressed by
Using the number density (2.2), the total number of cusps at the time t before the scaling solution era will be
nc(t) = 7uP,a 1 1 --
2(yGp)2(Gp)3/2m~j4 t2 tii4
The net baryon number from cusp evaporation is
0 = n,AB (2.8)
The baryon number asymmetry will then be
z -i- = 2(yGp)2(Gp)3/2
1 1 T 1
*j2)2 $r2ges mpt (mptr,)7/4 (2.9)
s is the entropy density, s = &?g*,T3, g*s is the num- ber of states and T is the temperature of the universe at the time t.
By (2.9) it is seen that the baryon asymmetry pro- duced by long strings depends on the era. It is easy to see, however, that the asymmetry produced in the early era is much more important. In fact by using (2.1) instead of (2.2), in the integration (2.4), and performing the integration up to the present time it can be seen that the value of asymmetry is 30 orders of magnitude less than