Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 106
5 Fluctuations
It was already realized many years ago that a spectrum of primordial fluctuations
can lead to the formation of PBHs (e.g. Carr & Hawking, 1974; Carr, 1975;
Novikov et al., 1979). What was initially considered was a spectrum of classical
fluctuations instead of a spectrum of quantum fluctuations. We have now in
cosmology a paradigm based on the existence of Inflation (Section 1.3) which
allows us to consider the quantum origin of the fluctuations (Polarski, 2001).
During inflation fluctuations of quantum origin, of the inflaton (i.e. the scalar
field driving inflation) are produced. These fluctuations are then stretched to
scales much larger than the Hubble radius RH (equation 28) at the time when
they were produced.
Once a physical wavelength becomes larger than the Hubble radius, it is
causally disconnected from physical processes. The inflationary era is followed
by a radiation–dominated and matter stages where the acceleration of the scale
factor becomes negative (see Sections 1.1 and 1.2). With a negative acceleration
of the scale factor, the Hubble radius grows faster than the scale factor, and
wavelengths that were outside, can re–enter the Hubble radius. This is the main
concept behind the inflationary paradigm for the generation of temperature
fluctuations as well as for providing the seeds for LSS formation (e.g. Boyanovsky
et al., 2006). In fact, with this mechanism we can explain all the inhomogeneities
we see today even on the largest cosmological scales as well as the production
of PBHs (Polarski, 2001).
WMAP has provided perhaps the most striking validation of inflation as a
mechanism for generating superhorizon fluctuations, through the measurement
of an anticorrelation peak in the temperature–polarization angular power spectrum
at l ∼ 150 corresponding to superhorizon scales (e.g. Boyanovsky et al.,
2006, Section 1.7).
5.1 The quantum–to–classical transition
Although there is a great diversity of inflationary models (Section 1.3), they
generically predict a gaussian and nearly scale invariant spectrum of primordial
fluctuations which is an excellent fit to the highly precise wealth of data provided
by the WMAP (e.g. Boyanovsky et al., 2006).
The inhomogeneities that we observe today do not display any property
typical of their quantum origin. On the large cosmological scales probed by
the observations, the fluctuations appear to us as random classical quantities.
This means that there was, at some time in the past, a quantum–to–classical
transition (Polarski, 2001).
Each field mode can be split into two linearly independent solutions: the
growing mode and the decaying mode. At reentrance inside the Hubble radius,
during the radiation–dominated or the matter–dominated stage, the decaying
mode is usually vanishingly small, and can, therefore, be safely neglected. As
a result, the field mode behaves like a stochastic classical quantity (for more
details see Polarski (2001) and Polarski & Starobinsky (1996)).