Physical Foundations of Cosmology

8.3 Quantum cosmological perturbations 347

by the power law,δ 2 (k)∝knS−^1 , and thus characterize it by the spectral indexnS.
A flat spectrum corresponds tonS=1.
The expression for the spectral index follows from (8.106):

nS− 1 ≡

dlnδ 2

dlnk

− 3

(

1 +

p

ε

)

−

1

H

(

ln

(

1 +

p

ε

))•

−

(lncs)

H

, (8.108)

where the quantities on the right hand side must be calculated at the time of horizon
crossing. In deriving this formula we have taken into account thatdlnkdlnak.
This relation follows from the condition determining horizon crossing,cskHak,
if we neglect the change incsandH. All terms on the right hand side of (8.108) are
negative for a generic inflationary scenario. Therefore, inflation does not predict
a flat spectrum, as is quite often mistakenly stated. Instead, it predicts a red-tilted
spectrum:nS<1 so that the amplitude grows slightly towards the larger scales. The
physical reason for this tilt is the necessity for a smooth graceful exit. To obtain
an estimate for the tilt we note that the galactic scales cross the horizon around
50–60 e-folds before the end of inflation. At this time( 1 +p/ε)is larger than 10−^2.
The second term in (8.108) is about the same order of magnitude and the spectral
index can thus be estimated asnS 0 .96. The concrete value ofnSdepends on a
particular inflationary scenario. Even without knowing this scenario, however, one
could expect thatnS≤ 0 .97. By inspection of the variety of scenarios, one infers
that it is rather difficult to get a very large deviation from the flat spectrum and that
it is likelynS> 0 .92.

Problem 8.14Consider inflation in a model with potentialVand verify that

nS− 1 −

3

8 π

(

V,φ

V

) 2

+

1

4 π

V,φφ

V

. (8.109)

Check that for the power-law potential,V∝φn,

nS− 1 −

n(n+ 2 )

8 πφk^2 Ha

−

n+ 2

2 N

, (8.110)

whereNis the number of e-folds before the end of inflation when the corresponding
perturbation crosses the horizon. In the case of a massive scalar field,n=2, and
nS 0 .96 on galactic scales for whichN50. For the quartic potentialn=4 and
nS 0 .94. How much does the spectral index “run” when the scale changes by one
decade?

How do quantum fluctuations become classical?When we look at the sky we see
the galaxies in certain positions. If these galaxies originated from initial quantum
fluctuations, a natural question arises: how does a galaxy, e.g. Andromeda, find it-
self at aparticularplace if the initial vacuum state was translational-invariant with