Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 30
Log 10Rt
0
-10
-20
-30
-40
-50
-60
Inflation EW QCD
-40 -30 -20 -10 0 10
Log 10t1s
matter
Figure 7: The scale factor as a function of time. The gray regions corresppond to
the inflationary period, the EW and QCD transitions and the matter–dominated
era. In blue (right side) we have the dark energy dominated era. The other
regions (in white) correspond to radiation–dominated periods.
Another observable quantity inherent in the CMB is the variation in temperature
(or intensity) from one part of the microwave sky to another. Since the first
detection of these anisotropies by the COBE satellite in 1992, there has been
intense activity to map the sky at increasing levels of sensitivity and angular
resolution. Observations have shown us that the CMB contains anisotropies at
the 10 −5 level (e.g. Yao et al., 2006)
∆T
T
∼ 10−5
(90)
over a wide range of angular scales. Density fluctuations over the plasma in thermal
equilibrium gave rise to temperature fluctuations (denser regions were hotter).
Hence, the temperature anisotropies in the CMB bring us direct evidence
of the density contrast at recombination. This small temperature anisotropy,
whose existence is predicted by cosmological models, provides the clue to the
origin of structure and is an important confirmation of theories of the early
Universe (e.g. Boyanovsky et al., 2006).
These anisotropies are usually expressed by using a spherical harmonic expansion
of the CMB sky (e.g. Yao et al., 2006)
T (θ, φ) =
almYlm(θ, φ) (91)
l,m
where Ylm(θ, φ) is the so–called spherical harmonic function of degree l and
order m 10 .
10 Ylm(θ, φ) represents the angular part of the solution of Laplace’s equation
(∇ 2 f(r, θ, φ) = 0). The degree l and order m are integers such that l ≥ 0 and |m| ≤ l.
The coefficients alm are constants. The expansion is exact as long as l goes to infinity.