Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 103
Table 23: The reduction of the speed of sound during the electron–positron
annihilation and the corresponding values for the parameter ∆T (Tc = 1 MeV).
cs reduction
∆T
Tc
20% 0.115
15% 0.169
10% 0.276
5% 0.600
Assuming different reductions, in the range 5%–20%, on the value of the speed
of sound we have obtained, from equation (194), the corresponding values for
the parameter ∆T (see Table 23).
In Figure 42 we show the curve for the sound speed as a function of the
temperature with Tc = 1 MeV and with ∆T assuming the values corresponding
to a reduction of 10% and 20% on the sound speed value (Table 23). For
the temperature as a function of time during the electron–positron annihilation
we recover here equation (151). Inserting expression (151) into equation (193)
we obtain, for the sound speed during the electron–positron annihilation, the
following:
c 2 s (t) =
⎡
⎢
⎣3+
∆T
2 T (t)−Tc
∆gT(t)sech ∆T
gep + g ′
T (t)−Tc
ep +∆gtanh ∆T
⎤
⎥
⎦
−1
, (195)
with Tc = 1 MeV. Taking into account that 1 MeV ≈ 7.7 × 10 9 K we obtain
from equation (151) tc ≈ 3.15 s. This corresponds to the instant of time for
which the speed of sound reaches its minimum value. In Figure 43 we represent,
again, the curves of Figure 42 but now as a function of time.
We have considered as a reasonable effective duration for the e + e − annihilation
process the interval for which the sound speed stays below 99% of its
‘background’ value c 2 s,0 =1/3. With the help of equations (193) and (195)
we have obtained this effective width for the values of the parameter ∆T that
we have already considered on Table 23. The obtained results are shown on
Table 24 (in terms of temperature) and on Table 25 (in terms of time).