Physical Foundations of Cosmology

5.3 How can gravity become “repulsive”? 233

rate wasmuch smallerthan the rate of expansion today, that is,a ̇i/a ̇ 0 1. More
precisely, the CMB observations require that the variation of the energy density
on the present horizon scale does not exceed 10−^5. The traces of an initial large
inhomogeneity will be sufficiently diluted only ifa ̇i/a ̇ 0 < 10 −^5. Rewriting (5.8) as

0 = 1 +( (^) i− 1 )

(

a ̇i

a ̇ 0

) 2

, (5.14)

we see that if| (^) i− 1 |∼O( 1 )then

0 = 1 (5.15)
to very high accuracy. This importantrobust predictionof inflation has a kinemat-
ical origin and it states that thetotalenergy density of all components of matter,
irrespective of their origin, must be equal to the critical energy density today.
We will see later that amplified quantum fluctuations lead to tiny corrections to

0 =1, which are of order 10−^5. It is worth noting that, in contrast to a deceler-
ating universe where (t)→1ast→0, in an accelerating universe (t)→1as
t→∞, that is, =1isitsfutureattractor.
Problem 5.3Why does the consideration above fail for (^) i=0?

5.3 How can gravity become “repulsive”?

To answer this question we recall the Friedmann equation (1.66):

a ̈=−

4 π

3

G(ε+ 3 p)a. (5.16)

Obviously, if the strong energy dominance condition,ε+ 3 p>0, is satisfied, then
a ̈<0 and gravity decelerates the expansion. The universe can undergo a stage
of accelerated expansion witha ̈>0 only if this condition is violated, that is, if
ε+ 3 p<0. One particular example of “matter” with a broken energy dominance
condition is a positive cosmological constant, for whichpV=−εVandε+ 3 p=
− 2 εV<0. In this case the solution of Einstein’s equations is a de Sitter universe−
discussed in detail in Sections 1.3.6 and 2.3. FortH−^1 , the de Sitter universe
expands exponentially quickly,a∝exp(Ht), and the rate of expansion grows as
the scale factor.Theexactde Sitter solution fails to satisfy all necessary conditions
for successful inflation: namely, it does not possess a smooth graceful exit into the
Friedmann stage. Therefore, in realistic inflationary models, it can be utilized only
as a zero order approximation. To have a graceful exit from inflation we must allow
the Hubble parameter to vary in time.