Physical Foundations of Cosmology

1.3 From Newtonian to relativistic cosmology 23
If the pressurepwithin an expanding sphere of volumeVis significant, then
the total energy,E=εV, is no longer conserved because the pressure does work,
−pdV.According to the first law of thermodynamics, this work must be equal to
the change in the total energy:

dE=−pdV. (1.63)

SinceV∝a^3 , we can rewrite this conservation law as

dε=−3(ε+p)dlna (1.64)

or, equivalently,

ε ̇=− 3 H(ε+p). (1.65)

This relation is the new version of (1.12) and it turns out to be the energy conser-
vation equation,T0;αα=0, in an isotropic, homogeneous universe.
The acceleration equation is also modified for matter with nonnegligible pressure
since, according to General Relativity, the strength of the gravitational field depends
not only on the energy density but also on the pressure. Equation (1.15) becomes
the first Friedmann equation:

a ̈=−

4 π

3

G(ε+ 3 p)a. (1.66)

The real justification for the form of the pressure contribution is that the accelera-
tion equation (1.66) follows from any diagonal spatial component of the Einstein
equations. Multiplying (1.66) bya ̇, using (1.65) to expresspin terms ofε,ε ̇and
H, and integrating, we obtain the second Friedmann equation:

H^2 +

k

a^2

=

8 πG

3

ε. (1.67)

This looks like the Newtonian equation (1.18) withk=− 2 E, though (1.67) applies
for an arbitrary equation of state. However,kis not simply a constant of integra-
tion: the 0−0 Einstein equation tells us that it is exactly the curvature introduced
before, that is,k=±1or0.Fork=±1, the magnitude of the scale factorahas a
geometrical interpretation as the radius of curvature.
Thus, in General Relativity, the value of cosmological parameter, ≡ε/εcr,
determines the geometry. If

1, the universe is closed and has the geome-
try of a three-dimensional sphere(k=+ 1 ); =1 corresponds to a flat universe
(k= 0 ); and in the case of
<1, the universe is open and has hyperbolic geometry
(k=−1).
The combination of (1.67) and either the conservation law (1.65) or the accel-
eration equation (1.66 ), supplemented by the equation of statep=p(ε), forms a
complete system of equations that determines the two unknown functionsa(t) and