Physical Foundations of Cosmology

234 Inflation I: homogeneous limit

Let us now determine the general conditions which must be satisfied in a suc-
cessful inflationary model. Because

a ̈

a

=H^2 +H ̇, (5.17)

anda ̈should become negative during a graceful exit, the derivative of the Hubble
constant,H ̇, must obviously be negative. The ratio|H ̇|/H^2 grows toward the end of
inflation and the graceful exit takes place when|H ̇|becomes of orderH^2. Assuming
thatH^2 changes faster thanH ̇, that is,|H ̈|< 2 HH ̇, we obtain the following generic
estimate for the duration of inflation:

tf∼Hi/|H ̇i|, (5.18)

whereHiandH ̇irefer to the beginning of inflation. Att∼tf the expression on
the right hand side in (5.17) changes sign and the universe begins to decelerate.
Inflation should last long enough to stretch a small domain to the scale of the
observable universe. Rewriting the conditiona ̇i/a ̇ 0 < 10 −^5 as

a ̇i

a ̇f

a ̇f

a ̇ 0

=

ai

af

Hi

Hf

a ̇f

a ̇ 0

< 10 −^5 ,

and taking into account thata ̇f/a ̇ 0 should be larger than 10^28 , we conclude that
inflation is successful only if

af

ai

> 1033

Hi

Hf

.

Let us assume that|H ̇i|Hi^2 and neglect the change of the Hubble parameter.
Then the ratio of the scale factors can be roughly estimated as

af/ai∼exp

(

Hitf

)

∼exp (Hi^2 /|H ̇i|)> 1033. (5.19)

Hence inflation can solve the initial conditions problem only iftf> 75 Hi−^1 , that
is, it lastslongerthan 75 Hubble times (e-folds). Rewritten in terms of the initial
values of the Hubble parameter and its derivative, this condition takes the form

|H ̇i|

Hi^2

<

1

75

. (5.20)

Using the Friedmann equations (1.67) and (1.68) withk=0, we can reformulate
it in terms of the bounds on the initial equation of state

(ε+p)i

εi

< 10 −^2. (5.21)