Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 31
Theoretical models generally predict that the alm modes are Gaussian random
fields. Tests show that this is an extremely good simplifying approximation,
with only some relatively weak indications of non–Gaussianity or statistical
anisotropy at large scales. With the assumption of Gaussian statistics, and if
there is no preferred axis, then it is the variance of the temperature field which
carries the cosmological information, rather than the values of the individual
alm coefficients. In other words, the power spectrum in l fully characterizes the
anisotropies (e.g. Yao et al., 2006).
On small sections of the sky where its curvature can be neglected, the spherical
harmonic analysis becomes ordinary Fourier analysis in two dimensions and
l becomes the Fourier wavenumber. Since the angular wavelength θ =2π/l,
larger multipole moments correspond to smaller angular scales, with l ∼ 10 2
representing degree scale separations. In this limit the power spectrum is usually
displayed as (e.g. Hu & Dodelson, 2002)
∆T
T
2
= l(l + 1)
where (e.g. Yao et al., 2006)
2π Cl (92)
Cl ≡ 〈|alm| 2 〉. (93)
The CMB mean temperature of 2.725 K (cf. equation 89) can be regarded
as the monopole component (a00) of CMB maps. Since all mapping experiments
involve difference measurements, they are insensitive to this average level.
Monopole measurements can only be made with absolute temperature devices,
such as the Far–InfraRed Absolute Spectrophotometer (FIRAS) instrument on
the COBE satellite. Such measurements of the spectrum are consistent with a
blackbody distribution over more than three decades in frequency (e.g. Yao et
al., 2006).
The largest anisotropy is in the l =1dipole first spherical harmonic, with
amplitude 3.346 ± 0.017 mK. The dipole is interpreted to be the result of the
Doppler shift caused by the solar system motion relative to the nearly isotropic
blackbody field, as confirmed by measurements of the radial velocities of local
galaxies (e.g. Yao et al., 2006).
Excess variance in CMB maps at higher multipoles (l ≥ 2) is interpreted as
being the result of perturbations in the density of the early Universe, manifesting
themselves at the epoch of the last scattering of the CMB photons. In the hot
Big Bang picture, this happens at a redshift z 1090, with little dependence
on the details of the model (e.g. Yao et al., 2006).
In Figure 8 we show the theoretical CMB anisotropy power spectrum (according
to the standard ΛCDM model). Notice that the physics underlying the
Cl’s can be separated into four main regions: the ISW Rise (l 2), the Sachs–
Wolfe plateau (l 100), the acoustic peaks (100 l 1000) and the damping
tail (l 1000).
The horizon scale at photon decoupling corresponds to l ≈ 100. Anisotropies
at larger scales (l