MODERN COSMOLOGY

Inflationary cosmology 37

The comoving horizon size at the end of inflation was therefore

dH(tGUT)aGUT−^1 [c/HGUT][EP/Eγ]EGUT−^1 ,

where the last expression in natural units usesH

√

V/EPEGUT^2 /EP.Fora

GUT energy of 10^15 GeV, this is about 10 m. This is a sobering illustration of the
magnitude of the horizon problem; if we relied on causal processes at the GUT
era to produce homogeneity, then the universe would only be smooth in patches
a few comoving metres across. To solve the problem, we need enoughe-foldings
of inflation to have stretched this GUT-scale horizon to the present horizon size

Nobs=ln

[

3000 h−^1 Mpc

(EP/Eγ)E−GUT^1

]

 60.

By construction, this is enough to solve the horizon problem, and it is also the
number ofe-foldings needed to solve the flatness problem. This is no coincidence,
since we saw earlier that the criterion in this case was

N&

1

2

ln

(

aeq

a^2 GUT

)

.

Now,aeq=ργ/ρ,andρ= 3 H^2
/( 8 πG). In natural units, this translates to
ρ ∼E^2 P(c/H 0 )−^2 ,ora−eq^1 ∼E^2 P(c/H 0 )−^2 /Eγ^4. The expression forNis then
identical to that in the case of the horizon problem: the same number ofe-folds
will always solve both.
Successful inflation in any of these models requires> 60 e-foldings of
the expansion. The implications of this are easily calculated using the slow-roll
equation, which gives the number ofe-foldings betweenφ 1 andφ 2 as

N=

∫

Hdt=−

8 π

m^2 P

∫φ 2

φ 1

V

V′

dφ.

For any potential that is relatively smooth,V′ ∼ V/φ, and so we getN ∼
(φstart/mP)^2 , assuming that inflation terminates at a value ofφrather smaller than
at the start. The criterion for successful inflation is thus that the initial value of
the field exceeds the Planck scale:

φstartmP.

By the same argument, it is easily seen that this is also the criterion needed
to make the slow-roll parametersandη1. To summarize, any model in
which the potential is sufficiently flat that slow-roll inflation can commence will
probably achieve the critical 60e-foldings. Counterexamples can of course be
constructed, but they have to be somewhat special cases.