Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 102
4 The electron–positron annihilation
During a first–order cosmological phase transition the Universe experiences a
drastic reduction on the sound speed. In fact the speed of sound vanishes for a
while (see e.g. QCD phase transition, Section 2). Less dramatic reductions may
also occur during higher–order phase transitions or particle annihilation periods
in the early Universe (e.g. Jedamzik & Niemeyer, 1999; Kämpfer, 2000).
This is the case, for example, of the electron–positron (e + e− ) annihilation
process which becomes predominant as soon as the radiation temperature drops
below the mass of the electron (∼ 1 MeV). A reduction in the sound speed value
of order 10 − 20% for a few Hubble times does occur during the cosmic e + e− annihilation. There is the possibility of an enhancement in PBH formation
on the e + e− annihilation horizon mass scale of approximately, M ∼ 105M⊙ (Jedamzik, 1997).
The neutrino degrees of freedom are not affected at all by this process. As a
result, we have the disappearance of four fermionic degrees of freedom, i.e., the
ones corresponding to electrons and positrons (e.g. Zimdahl & Pavón, 2001).
Thus, we have (cf. Table 13) ∆g = 10.75 − 7.25 = 3.5.
Below temperatures of ∼ 1 MeV the three neutrino flavours are decoupled
chemically and kinetically from the plasma and the entropy of the relativistic
electrons is transfered to the photon entropy, but not to the neutrino entropy
when electrons and positrons annihilate. This leads to an increase of the photon
temperature relative to the neutrino temperature by (e.g. Schwarz, 2003)
1/3 Tν 4
= . (192)
11
Tγ
There are other annihilation processes that could lead to an equivalent reduction
on the speed of sound (e.g. muon annihilation). In this work we concentrate on
the electron–positron annihilation process and on its eventual consequences in
the context of PBH production.
We adopt for the profile of the sound speed during the e + e − annihilation
process, for simplicity, an expression similar to (143) or (178). Thus, we consider
c 2 s (T )=
3+
∆gTsech 2 T −Tc
∆T
∆T gep + g ′ ep +∆gtanh T −Tc
∆T
−1
(193)
where g ′ ep = 10.75 and gep =7.25. We also consider a critical temperature
Tc = 1 MeV. The parameter ∆T must be determined in order to achieve results:
reductions of order 10 − 20% on the sound speed must take place.
The minimum value for the sound speed is attained for T ≈ Tc. Considering
T = Tc in equation (193) we obtain the following expression giving an approximate
value for the minimum speed of sound during the e + e− annihilation
process:
c 2 s,min ≈
3+ g′ ep − gep
+ gep)
∆T
Tc (g′ ep
−1
. (194)