Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 22
Table 2: The best fit values for the ΛCDM model free parameters according to
the WMAP data (Spergel et al., 2007).
Parameter Value Description
h 0.734 +0.028
−0.038
100Ωbh 2 2.233 +0.072
−0.091
Ωmh 2 0.1268 +0.0072
−0.0095
τ 0.088 +0.043
−0.054
As
ns
0.801 +0.043
−0.054
0.951 +0.015
−0.019
Normalized Hubble constant
Baryon density
Total matter density
Optical depth to reionization
Scalar fluctuation amplitude at k =0.002 Mpc −1
Scalar spectral index at k =0.002 Mpc −1
Ωb ≈ 0.042. (53)
Notice that the dark energy density ΩΛ (cf. equation 35) is not a free parameter
because, since the ΛCDM model assumes a flat Universe (Ω = 1), we have,
according to equation (36)
ΩΛ =1− Ωm ≈ 0.76. (54)
Other derived parameters are the age of the Universe t0 (Section 1.6) and the
critical density ρ0 (cf. equation 32). Inserting the value of H0 into equation
(32) one obtains
ρc = ρ0 ≈ 1.013 × 10 −26 kgm −3 . (55)
The cosmological constant, which is also a derived parameter, is given by (see
equations 32 and 35)
Λ= 8πG
c 2 ρΛ = 8πG
c 2 ΩΛρc = 3H2 0 ΩΛ
c 2 . (56)
Inserting the obtained values for H0 (equation 51) and ΩΛ (equation 54) into
equation (56) one obtains 8
Λ ≈ 1.44 × 10 −52 m −2 . (57)
8 When one calculates the theoretical value of Λ one ends up with a value about 120 orders
of magnitude larger than the experimentally measured one. This has been called the worst
mismatch between theory and experiment in the whole of science (e.g. Weinberg, 2000).