The Friedmann models 23The ‘flat’ universe withk=0 arises for a particular critical density. We
are therefore led to define a density parameter as the ratio of density to critical
density:
≡ρ
ρc=
8 πGρ
3 H^2.
Sinceρ and H change with time, this defines an epoch-dependent density
parameter. The current value of the parameter should strictly be denoted by 0.
Because this is such a common symbol, we shall keep the formulae uncluttered
by normally dropping the subscript; the density parameter at other epochs will be
denoted by(z). The critical density therefore just depends on the rate at which
the universe is expanding. If we now also define a dimensionless (current) Hubble
parameter as
h≡H 0
100 km s−^1 Mpc−^1,
then the current density of the universe may be expressed as
ρ 0 = 1. 88 × 10 −^26 h^2 kg m−^3
= 2. 78 × 1011 h^2 M Mpc−^3.A powerful approximate model for the energy content of the universe is to
divide it into pressureless matter (ρ∝R−^3 ), radiation (ρ∝R−^4 ) and vacuum
energy (ρconstant). The first two relations just say that the number density
of particles is diluted by the expansion, with photons also having their energy
reduced by the redshift; the third relation applies for Einstein’s cosmological
constant. In terms of observables, this means that the density is written as
8 πGρ
3=H 02 (v+ma−^3 +ra−^4 )(introducing the normalized scale factora=R/R 0 ). For some purposes, this
separation is unnecessary, since the Friedmann equation treats all contributions to
the density parameter equally:
kc^2
H^2 R^2=m(a)+r(a)+v(a)− 1.Thus, a flatk = 0 universe requires
∑
i = 1 at all times, whatever the
form of the contributions to the density, even if the equation of state cannot be
decomposed in this simple way.
Lastly, it is often necessary to know the present value of the scale factor,
which may be read directly from the Friedmann equation:
R 0 =c
H 0[(− 1 )/k]−^1 /^2.The Hubble constant thus sets thecurvature length, which becomes infinitely
large asapproaches unity from either direction.