82 Cosmological Models
component푇 00 and the space–space component푇 11 are then
푇 00 =휌푐^2 ,푇 11 =
푝푎^2
1 −푘휎^2
, (5.1)
taking푔 00 and푔 11 from Equation (2.33). We will not need푇 22 or푇 33 because they just
duplicate the results without adding new dynamical information. In what follows we
shall denote mass density by휌and energy density by휌푐^2. Occasionally, we shall use
휌m푐^2 to denote specifically the energy density in all kinds of matter: baryonic, leptonic
and unspecified dark matter. Similarly we use휌r푐^2 or휀to specify the energy density
in radiation.
On the left-hand side of the Einstein Equations (3.29) we need퐺 00 and퐺 11 to equate
with푇 00 and푇 11 , respectively. We have all the tools to do it: the metric components푔휇휈
are inserted into Equation (3.13) for the affine connection, and subsequently we calcu-
late the components of the Riemann tensor from the expression (3.16) using the metric
components and the affine connections. This lets us find the Ricci tensor components
that we need,푅 00 and푅 11 , and the Ricci scalar from Equations (3.17) and (3.18),
respectively. All this would require several pages to work out (see, e.g., [1, 2]), so I only
give the result:
퐺 00 = 3 (ca)−^2(
푎̇^2 +푘푐^2
)
, (5.2)
퐺 11 =−푐−^2
(
2 푎̈푎+푎̇^2 +푘푐^2
)(
1 −푘휎^2
)− 1
. (5.3)
Here푎is the cosmic scale factor푎(푡)at time푡. Substituting Equations (4.1)–(4.3) into
the Einstein Equations (3.29) we obtain two distinct dynamical relations for푎(푡):
푎̇^2 +푘푐^2
푎^2=^8 휋퐺
3
휌, (5.4)
2 푎̈
푎
+푎̇
(^2) +푘푐 2
푎^2
=−^8 휋퐺
푐^2
푝. (5.5)
These equations were derived in 1922 by Friedmann, seven years before Hubble’s
discovery, at a time when even Einstein did not believe in his own equations because
they did not allow the Universe to be static. Friedmann’s equations did not gain gen-
eral recognition until after his death, when they were confirmed by an independent
derivation (in 1927) by Georges Lemaitre (1894–1966). For now they will constitute
the tools for our further investigations.
The expansion (or contraction) of the Universe is inherent to Friedmann’s equa-
tions. Equation (5.4) shows that the rate of expansion,푎̇, increases with the mass den-
sity휌in the Universe, and Equation (5.5) shows that it may accelerate. Subtracting
Equation (5.4) from Equation (5.5) we obtain
2 푎̈
푎=−
8 휋퐺
3 푐^2
(휌푐^2 + 3 푝), (5.6)
which shows that the acceleration decreases with increasing pressure and energy den-
sity, whether mass or radiation energy. Thus it is more appropriate to talk about the
decelerationof the expansion. Equation 5.6 is also called theRaychauduri equation.