A Cyclic Universe 169Universe, we also have a precise theoretical specification for how much matter there
should be. This links dark matter to inflation.
We have already noted that the scalar inflaton field produced a spectrum of frozen
density and radiation perturbations beyond the horizon, which moved into sight when
the expansion of the Universe decelerated. In the post-inflationary epoch when the
Friedmann expansion takes over we can distinguish between two types of pertur-
bations,adiabatic fluctuations, also calledcurvature perturbations,andisocurvature
fluctuations, also calledisothermalperturbations. In the first case, the perturbations
in the local number density,훿m≡훿휌m∕휌m, of each species of matter—baryons, leptons,
neutrinos, dark matter—is the same. In particular, these perturbations are coupled to
those of radiation,훿r≡훿휌r∕휌r,sothat4훿m= 3 훿r[from Equation (6.39)]. By the princi-
ple of covariance, perturbations in the energy-momentum tensor imply simultaneous
perturbations in energy density and pressure, and by the equivalence principle, vari-
ations in the energy-momentum tensor are equivalent to variations in the curvature.
Curvature perturbations can have been produced early as irregularities in the met-
ric, and they can then have been blown up by inflation far beyond the Hubble radius.
Thus adiabatic perturbations are a natural consequence of cosmic inflation. In con-
trast, inflation does not predict any isocurvature perturbations.
Let us write the power spectrum of density perturbations in the form
푃(푘)∝푘푛s, (7.60)where푛sis thescalar spectral index. Inflationary models predict that the primordial
fluctuations have an equal amplitude on all scales, an almost scale-invariant power
spectrum as the matter fluctuations cross the Hubble radius, and are Gaussian. This is
the Harrison–Zel’dovich spectrum for which푛s=1(푛s=0 would correspond to white
noise).
A further prediction of inflationary models is that tensor fluctuations in the space-
time metric, satisfying a massless Klein–Gordon equation, have a nearly scale-
invariant spectrum of the form in Equation (7.59) withtensor spectral index푛t,just
like scalar density perturbations, but independently of them. The ratio푟=푛t∕푛sis now
eagerly measured by several teams.
The above predictions are generic for a majority if inflation models which differ in
details. Inflation as such cannot be either proved or disproved, but specific theories
can be and will be ruled out by these observations.
7.5 A Cyclic Universe
As we have seen, ‘consensus’ inflation by a single inflaton field solves the problems
described in Section 7.1. But in the minds of some people it does so at a very high
price. It does not explain the beginning of space and time, it does not predict the
future of the Universe, or it sweeps these fundamental questions under the carpet of
theanthropic principle. It invokes several unproven ingredients, such as a scalar field
and a scalar potential, suitably chosen for the field to slow-roll down the potential
while its kinetic energy is negligible, and such that it comes to a graceful exit where