Physical Foundations of Cosmology

8.2 Perturbations on inflation (slow-roll approximation) 333

corresponding to gravitational waves can be estimated on dimensional grounds
ash∼lPl/L, wherelPl∼ 10 −^33 cm. It is incredibly small: on galactic scales
L∼ 1025 cm, soh∼ 10 −^58. Scalar metric perturbations due to vacuum fluctua-
tions of the scalar field are even smaller. Thus the only way to get the required
amplitude ∼ 10 −^5 on large scales from initial quantum fluctuations is by stretch-
ing the very short-wavelength fluctuations. During this stretching, the mode must
not lose its amplitude. Let us consider a scalar field perturbation, which determines
the metric fluctuations, and find out what generally happens to its amplitude when
the spatial size of the perturbation is stretched. As we have seen, the amplitude
decays in inverse proportion to the spatial size until the perturbation starts to “feel”
the curvature of the universe. This happens when its size begins to exceed the
curvature scaleH−^1. Therefore, if during expansion the perturbation size always
remains smaller than the curvature scale, then its amplitude continuously decreases;
it “arrives” at large scales with negligible vacuum amplitude. In a decelerating uni-
verse, the curvature scaleH−^1 =a/a ̇grows faster than the physical wavelength
of the perturbation

(

λph∝a

)

becausea ̇is decreasing (see Figure 8.3). Hence, if
a perturbation is initially inside the horizon, it remains there and decays. Perturba-
tions on those scales which were initially a little larger than the Hubble radius will
soon enter the horizon and also decay. Thus, in a decelerating expanding universe,
quantum fluctuations can never be significantly amplified to become relevant for
large-scale structure.

ai a

Hi−^1

H

−^1 (

no inflation

)

H−^1 (inflation)

λph

∼

a

Fig. 8.3.