Physical Foundations of Cosmology

8.5 Self-reproduction of the universe 353

quantum fluctuations

H−^1

∆φq ∼H

Fig. 8.7.

the size of this domain grows toH−^1 exp(Ht)∼eH−^1 and hence gives rise to

e^3 20 new domains of sizeH−^1. Now consider the averaged value of the scalar

field in each of these new domains. During a Hubble time the classical scalar field

decreases by an amount

φcl−

V,φ

3 H

tH∼−φ−^1. (8.130)

Simultaneously, quantum fluctuations stretched from sub-Hubble scales begin to

contribute to the mean value of the scalar field in each domain of sizeH−^1. Quantum

fluctuations with wavelength of orderH−^1 and amplitudeφq∼H∼mφare

superimposed on the classical field; in half of the regions they decrease the value

of the scalar field still further (Figure 8.7), while in the other half they increase the

field value. The overall change ofφin these latter domains is about

φtot=φcl+φq∼−φ−^1 +mφ. (8.131)

It is clear that ifφ>m−^1 /^2 , the field grows and inflation always produces regions

where the scalar field exceeds its “initial” value. In Figure 8.8 we sketch the typical

trajectories describing the evolution of the scalar field within a typical Hubble

domain. Forφm−^1 /^2 , quantum fluctuations only slightly disturb the classical

slow-roll trajectory, while forφm−^1 /^2 they dominate and induce a “random

walk.” Because each domain of sizeH−^1 in turn produces other domains at an

exponential rate, the physical volume of space where the scalar field is larger than its

initial value grows exponentially. Thus, inflation continues forever and the universe

is said to be “self-reproducing.” In those regions where the field drops below the