MODERN COSMOLOGY

Inflationary cosmology 41

The ratio of the tensor and scalar contributions to the variance of microwave
background anisotropies is therefore proportional to the inflationary parameter
:
^2 T
^2 S

 12. 4 ,

inserting the exact coefficient from Starobinsky (1985). If it could be measured,
the gravity-wave contribution to CMB anisotropies would therefore give a
measure of, one of the dimensionless inflation parameters. The less ‘de
Sitter-like’ the inflationary behaviour is, the larger the relative gravitational-wave
contribution is.
Since deviations from exact exponential expansion also manifest themselves
as density fluctuations with spectra that deviate from scale invariance, this
suggests a potential test of inflation. Define thetiltof the fluctuation spectrum
as follows:

tilt≡ 1 −n≡−

dlnδ^2 H

dlnk

.

We then want to express the tilt in terms of parameters of the inflationary potential,
andη. These are of order unity when inflation terminates; andηmust
therefore be evaluated when the observed universe left the horizon, recalling that
we only observe the last 60-odde-foldings of inflation. The way to introduce scale
dependence is to write the condition for a mode of given comoving wavenumber
to cross the de Sitter horizon,

a/k=H−^1.

SinceH is nearly constant during the inflationary evolution, we can replace
d/dlnkby d lna, and use the slow-roll condition to obtain

d

dlnk

=a

d

da

=

φ ̇

H

d

dφ

=−

m^2 P

8 π

V′

V

d

dφ

.

We can now work out the tilt, since the horizon-scale amplitude is

δ^2 H=

H^4

( 2 πφ) ̇^2

=

128 π

3

(

V^3

m^6 PV′^2

)

,

and derivatives ofVcan be expressed in terms of the dimensionless parameters
andη. The tilt of the density perturbation spectrum is thus predicted to be

1 −n= 6 − 2 η.

In section 2.8.5 on CMB anisotropies, we discuss whether this relation is
observationally testable.