Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 2
be random and less than one–thousandth of the velocity of light (e.g. d’Inverno,
1993).
Relativistic Cosmology is based on three assumptions: (1) the Cosmological
Principle, (2) Weyl’s postulate and (3) General Relativity2 .
Weyl’s postulate requires that the geodesics of the substractum are orthogonal
to a family of spacelike hypersurfaces. We introduce coordinates
(t, x1 ,x2 ,x3 ) such that these spacelike hypersurfaces are given by constant t
and such that the space coordinates (x1 ,x2 ,x3 ) are constant along the geodesics.
Such coordinates are called comoving coordinates (e.g. d’Inverno, 1993).
Comoving observers are also called fundamental observers.
A flat, homogeneous and isotropic expanding Universe can be described
by the Friedmann–Lemaître–Robertson–Walker (FLRW) metric (e.g. d’Inverno,
1993)
ds 2 = dt 2 − R 2 2 dr
(t)
1 − κr2 + r2 dθ 2 + sin 2 θdφ 2
(1)
where R(t) is the so called scale factor which describes the time dependence
of the geometry (the distance between any pair of galaxies, separated by more
than 100 Mpc, is proportional to R(t)) and κ is a constant which fixes the sign
of the spatial curvature (κ = 0 for Euclidean space, κ> 0 for a closed elliptical
space of finite volume and κ< 0 for an open hyperbolic space). Notice that,
whatever the physics of the expansion, the space–time metric must be of the
FLRW form, because of the isotropy and homogeneity (e.g. Longair, 1998).
Considering the FLRW metric (1), Weyl’s postulate, General Relativity
(with a cosmological constant term Λ) and a comoving coordinate system it
turns out that the field equations lead to two independent equations sometimes
called the Friedmann–Lemaître equations (e.g. Yao et al., 2006; Unsöld &
Bascheck, 2002)
˙R
R
2
¨R Λ
=
R 3
= 8πGρ
3
κ Λ
− +
R2 3
4πG
− (ρ +3p) (3)
3
where we have used relativistic units (c = 1) and a dot denotes differentiation
with respect to cosmic time t. Equation (3) involves a second time derivative
of R and so it can be regarded as an equation of motion, whereas equation (2),
sometimes called Friedmann equation, only involves a first time derivative of R
and so may be considered an integral of motion, i.e., an energy equation.
The addition of a cosmological constant term Λ is equivalent to assume
that matter is not the only source of gravity and there is also an additional
source of gravity in the form of a fluid with pressure pΛ and energy density ρΛ
2 For an introductory text on the Theory of General Relativity see (e.g. Schutz, 1985;
d’Inverno, 1993).
(2)