MODERN COSMOLOGY

Inflationary cosmology 33

times. To solve the horizon problem and allow causal contact over the whole
of the region observed at last scattering requires a universe that expands ‘faster
than light’ neart=0:R∝tα, withα>1. If such a phase had existed, the
integral for the comoving horizon would have diverged, and there would be no
difficulty in understanding the overall homogeneity of the universe—this could
then be established by causal processes. Indeed, it is tempting to assert that the
observed homogeneityprovesthat such causal contact must once have occurred.
What condition does this place on the equation of state? In the integral forrH,
we can replace dtby dR/R ̇, which the Friedmann equation says is proportional
to dR/

√

ρR^2 at early times. Thus, the horizon diverges provided the equation of
state is such thatρR^2 vanishes or is finite asR→0. For a perfect fluid with
p≡(− 1 )as the relation between pressure and energy density, we have the
adiabatic dependencep∝R−^3 , and the same dependence forρif the rest-mass
density is negligible. A period of inflation therefore needs

< 2 / 3 ⇒ρc^2 + 3 p< 0.

Such a criterion can also solve the flatness problem. Consider the Friedmann
equation,

R ̇^2 =^8 πGρR

2

3

−kc^2.

As we have seen, the density term on the right-hand side must exceed the
curvature term by a factor of at least 10^60 at the Planck time, and yet a more
natural initial condition might be to have the matter and curvature terms being of
comparable order of magnitude. However, an inflationary phase in whichρR^2
increases as the universe expands can clearly make the curvature term relatively
as small as required, provided inflation persists for sufficiently long.
We have seen that inflation will require an equation of state with negative
pressure, and the only familiar example of this is thep =−ρc^2 relation that
applies for vacuum energy; in other words, we are led to consider inflation as
happening in a universe dominated by a cosmological constant. As usual, any
initial expansion will redshift away matter and radiation contributions to the
density, leading to increasing dominance by the vacuum term. If the radiation
and vacuum densities are initially of comparable magnitude, we quickly reach a
state where the vacuum term dominates. The Friedmann equation in the vacuum-
dominated case has three solutions:

R∝

{sinhHt (k=− 1 )

coshHt (k=+ 1 )

expHt (k= 0 ),

where H =

√

c^2 / 3 =

√

8 πGρvac/3; all solutions evolve towards the
exponentialk = 0 solution, known asde Sitter spacetime. Note thatH is
not the Hubble parameter at an arbitrary time (unlessk =0), but it becomes