162 Cosmic Inflation
As long as there were no decays or annihilations of massive particles, and all other
interactions conserve the total number of photons푛훾in a comoving volume is a con-
stant. Since the entropy per photon is푠∝푇^3 the total entropy푆퐻due to cosmic
microwave photons of temperature푇푅within our current horizon푑퐻is
푆퐻≈푇푅^3 푑퐻^3 ≈
(
푇푅
퐻 0
) 3
≈ 1088. (7.39)
the entropy per particle is suddenly increased by the very large factor
푍^3 =
(
푇R
푇cool) 3
, (7.40)
where the ratio푇R∕푇coolis of the order of magnitude of 10^29 .Thisisaverynonadiabatic
process.
Using estimates of the primordial density fluctuations휕휌∕휌≈ 10 −^5 the energy scale
is of the order of 10−^4 푀푃, so that the number of e-folds of the inflation is푁≈ 60
or larger. There is no upper bound to푁, so it could in fact be infinite, calledeternal
inflation.
At the end of inflation the Universe is a hot bubble of particles and radiation in
thermal equilibrium. The energy density term in Friedmann’s equations has become
dominant, and the Universe henceforth follows a Friedmann–Lemaitre type evolution
as described in Chapters 5 and 6.
The flatness problem is now solved if the part of the Universe which became our
Universe was originally homogeneous and has expanded by the de Sitter scale factor
[Equation (5.59)]
푎=e퐻휏≃ 1029 , (7.41)or퐻휏≃65. Superimposed on the homogeneity of the pre-inflationary universe there
were small perturbations in the field휑or in the vacuum energy. At the end of inflation
these give rise to density perturbations which are the seeds of later mass structures
and which can easily explain 10^90 particles in the Universe.
It follows from Equations (7.33) and (7.36) that the duration of the inflation was
휏≃ 65 × 10 −^34 s. (7.42)Then also the horizon problem is solved, since the initial particle horizon has been
blown up by a factor of 10^29 to a size vastly larger than our present Universe. [Note that
the realistic particle horizon is not infinite as one would obtain from Equation (7.9),
because the lower limit of the integral is small but nonzero.] Consequently, all the
large-scale structures seen today have their common origin in a microscopic part of
the Universe long before the last scattering of radiation.
The development of this scenario is similar to Linde’s scenario shown in Figure 7.3,
except that the vertical scale here grows ‘only’ to 10^29.
When our bubble of space-time nucleated, it was separated from the surrounding
supercooled hot phase by domain walls. When the phase transition finally occurred
the enormous amounts of latent heat was released to these walls. The mechanism
whereby this heat was transferred to particles in the bubbles was by the collision of