PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

CHAPTER 1. INTRODUCTION
say, λ P
= λa, associated with any mode of a given wavelength λ, always grows proportional
to the scale factor. In contrast, the Hubble radius, say, d H
= H −1 , which reflects
the size of the horizon in such cases, behaves as a 1/q . Therefore, λ P
/d H
∝ a (q−1)/q , which,
for q < 1, turns larger than unity at a sufficiently early time. In other words, the modes
associated with the perturbations leave the Hubble radius as we go back in time. The
fact that, at early times, the modes of cosmological interest (say, wavelengths larger than
the scale of homogeneity today) are outside the Hubble radius in the hot big bang model
implies that the model cannot provide a causal mechanism for seeding the perturbations.
Inflation, a period of rapid expansion during the stages of the radiation dominated
epoch, solves the horizon problem (and, in fact, other issues too, such as, for example,
the flatness problem) of the standard big bang cosmology in an elegant fashion [6, 7].
Evidently, the horizon problem can be resolved if we can have a phase during the early
stages of radiation domination wherein the modes of cosmological interest are inside the
Hubble radius. It is clear from the above discussion that this can be achieved, provided,
during this period, the physical wavelengths of the modes decrease faster than the Hubble
radius as we go back in time, i.e. if −d(λ P
/d H
)/dt < 0. This condition corresponds to
ä > 0, and it is such an era of accelerated expansion that is referred to as inflation. In most
models of inflation, such as the slow roll scenarios that have drawn constant attention, the
Hubble radius remains approximately constant. As illustrated in Figure 1.2, this property
allows us to bring the modes of cosmological interest inside the Hubble radius at early
times. It can be shown that, in order to ensure that the forward light cone at decoupling
is at least as large as the backward light cone, one requires the universe to expand by
a factor of about 10 28 during inflation 1 . For convenience, the extent of inflation is often
measured in terms of the number of e-folds N, which is defined as the logarithmic ratio
of the scale factor at any given instant to its value at another fixed time. It can, in fact, be
expressed as
N =
∫ t
t ∗
dtH =
[ ]
da a(t)
t ∗
a = ln , (1.4)
a(t ∗ )
∫ t
where t ∗ denotes some fixed time. Typically, one requires about 60–70 e-folds of inflation
to overcome the horizon problem [6, 7]. As we shall soon discuss, the fact that the modes
are inside the Hubble radius during the early phase of inflation allows us to impose well
motivated initial conditions on the perturbations.
1 It should be clarified that the factor of10 28 that we have quoted is an approximate upper limit, and the
actual number depends on the energy scale at which inflation takes place (in this context, see, for instance,
Refs. [32]).
8