Physical Foundations of Cosmology

8.2 Perturbations on inflation (slow-roll approximation) 329

wavenumberk, the typical amplitude of fluctuations is of order

δφ(k)∼|δφk|k^3 /^2 ∼

k

ak

∼Hk∼Ha, (8.25)

at the moment of the horizon crossing,Hak∼k(orkηk∼1). During the inflation-
ary stageHa=a ̇increases and a perturbation with a givenkeventually leaves the
horizon. To see whether it will remain large enough after being stretched to galactic
scales, we have to find out how it will behave on supercurvature scales.

8.2.2 The spectrum of generated perturbations

To determine the behavior of long-wavelength perturbations we use the slow-roll
approximation. In Chapter 5 we saw that for the homogeneous mode this approxi-
mation means that in the equation

φ ̈ 0 + 3 Hφ ̇ 0 +V,φ= 0 (8.26)

we can neglect the second derivative with respect to the physical timetand it
simplifies to

3 Hφ ̇ 0 +V,φ 0. (8.27)

To take advantage of the slow-roll approximation for the perturbations, we have to
recast (8.17) and (8.20) in terms of the physical timet:

δφ ̈+ 3 Hδφ ̇−δφ+V,φφδφ−4 ̇φ 0 ̇+ 2 V,φ = 0 , (8.28)

̇+H = 4 πφ ̇ 0 δφ, (8.29)

whereδφ≡δφand we have taken into account that=. First of all we note
that the spatial derivative termδφcan be neglected for long-wavelength inhomo-
geneities. To find the nondecaying slow-roll mode we next omit terms proportional
toδφ ̈and ̇. (After finding the solution of the simplified equations one can check
that the omitted terms are actually negligible.) The equations for the perturbations
become

3 Hδφ ̇+V,φφδφ+ 2 V,φ  0 ,H  4 πφ ̇ 0 δφ. (8.30)

Introducing the new variable

y≡δφ/V,φ

and using (8.27), they further simplify to

3 Hy ̇+ 2 = 0 , H = 4 πVy ̇. (8.31)