36 An introduction to the physics of cosmology
Similar arguments can be made for the spatial parts. However, they are less
critical: what matters is the value of∇φ=∇comovingφ/R.SinceRincreases
exponentially, these perturbations are damped away: assumingVis large enough
for inflation to start in the first place, inhomogeneities rapidly become negligible.
This ‘stretching’ of field gradients as we increase the cosmological horizon
beyond the value predicted in classical cosmology also solves a related problem
that was historically important in motivating the invention of inflation—the
monopole problem. Monopoles are point-like topological defects that would be
expected to arise in any phase transition at around the GUT scale (t∼ 10 −^35 s).
If they form at approximately one per horizon volume at this time, then it follows
that the present universe would contain1 in monopoles. This unpleasant
conclusion is avoided if the horizon can be made much larger than the classical
one at the end of inflation; the GUT fields have then been aligned over a vast
scale, so that topological-defect formation becomes extremely rare.
2.5.2 Ending inflation
Although spatial derivatives of the scalar field can thus be neglected, the same is
not always true for time derivatives. Although they may be negligible initially,
the relative importance of time derivatives increases asφrolls down the potential
andVapproaches zero (leaving aside the subtle question of how we know that the
minimum is indeed at zero energy). Even if the potential does not steepen, sooner
or later we will have1or|η|1 and the inflationary phase will cease.
Instead of rolling slowly ‘downhill’, the field will oscillate about the bottom of
the potential, with the oscillations becoming damped by the 3Hφ ̇friction term.
Eventually, we will be left with a stationary field that either continues to inflate
without end, ifV(φ= 0 )>0, or which simply has zero density. This would be
a most boring universe to inhabit, but fortunately there is a more realistic way in
which inflation can end. We have neglected so far the couplings of the scalar field
to matter fields. Such couplings will cause the rapid oscillatory phase to produce
particles, leading toreheating. Thus, even if the minimum ofV(φ)is atV=0,
the universe is left containing roughly the same energy density as it started with,
but now in the form of normal matter and radiation—-which starts the usual FRW
phase, albeit with the desired special ‘initial’ conditions.
As well as being of interest for completing the picture of inflation, it is
essential to realize that these closing stages of inflation are theonlyones of
observational relevance. Inflation might well continue for a huge number ofe-
foldings, all but the last few satisfying,η1. However, the scales that left the
de Sitter horizon at these early times are now vastly greater than our observable
horizon,c/H 0 , which exceeds the de Sitter horizon by only a finite factor. If
inflation was terminated by reheating to the GUT temperature, then the expansion
factor required to reach the present epoch is
aGUT−^1 EGUT/Eγ.