Physical Foundations of Cosmology

8.4 Gravitational waves from inflation 351

and hence the spectrum of the gravitational waves is also slightly tilted to the
red. (Note that the tensor and scalar spectral indices are defined differently−see
(8.108).) The ratio of tensor to scalar power spectrum amplitudes on supercurvature
scales during the post-inflationary, radiation-dominated epoch is

δh^2

δ^2

 27

[

cs

(

1 +

p

ε

)]

kHa

. (8.126)

For a canonical scalar field(cs= 1 ), this ratio is between 0.2 and 0.3. However, in
kinflation, wherecs1, it can be strongly suppressed. Thus, at least in principle,
kinflation is phenomenologically distinguishable from inflation based on a scalar
field potential.

Post-inflationary epochWe found in Section 7.3.2 that the amplitude of a grav-
itational wave stays constant on supercurvature scales irrespective of changes in
the equation of state. When the gravitational wave re-enters the horizon, however,
its amplitude begins to decay in inverse proportion to the scale factor. Hence the
spectrum remains unchanged only on large scales and is altered within the Hubble
horizon. Neglecting the dark energy component, we can express the Hubble con-
stant at earlier times in terms of its present valueH 0 as

H(a)H 0

{

(a 0 /a)^3 /^2 , z<zeq,

z−eq^1 /^2 (a 0 /a)^2 , z>zeq,

(8.127)

wherezeqis the redshift at matter–radiation equality. For a gravitational wave with
a comoving wavenumberk, the value of the scale factor at horizon crossing,ak,
is determined from the conditionkH(ak)ak.After that the amplitude decreases
by a factorak/a 0 and we obtain the following spectrum at the present time:

δh∼H

⎧

⎪⎨

⎪⎩

z−eq^1 /^2

(

λphH 0

)

,λph<H 0 −^1 z−eq^1 /^2 ,

(

λphH 0

) 2

, H 0 −^1 >λph>H 0 −^1 z−eq^1 /^2 ,

1 ,λph>H 0 −^1 ,

(8.128)

whereHis the value of the Hubble constant during inflation andλph∼a 0 /k
is the physical wavelength. This spectrum is sketched in Figure 8.6. On scales of
several light years the typical amplitude of the primordial gravitational waves can
be estimated as roughly 10−^17 for a realistic model of inflation. This amplitude
drops linearly towards smaller scales and so the prospects of direct detection of the
primordial gravitational background are not very promising. However, as we shall
see in the next chapter, these gravitational waves influence the CMB temperature
fluctuations and therefore may be detectable indirectly.