200
Inflationary cosmology
than containing of the order of 3.6 x 104 such regions as in section 7.2. The usual
expression (3.117) for the distance to the particle horizon does not hold during
the period of cosmological inflation. Instead,
(' dt'
(7.26)
dH(t) = R(t) lo R(t'}
with R(t) given by (7.2S) if we neglect the period of growth of dH(t) during the
period of radiation-dominated expansion which preceded cosmological inflation.
Thus, during the inflationary period,
dH(t) = H-I(e Ht - 1).
(7.27)
The exponential growth of dH(t) during this period means that, once the universe
has reheated to around the critical temperature after the phase transition has
been completed, the horizon volume is exponentially greater than it was at this
temperature prior to inflation. (Because the energy density in the false vacuum
is of order Tc 4 and the radiation density in the FRW universe is given by (2.22).
reheating to a temperature of order Tc occurs.) Thus, after the subsequent period
of radiation-dominated expansion in the FRW universe, the size of the horizon
volume at t = t, is exponentially greater than in the standard model of cosmology.
It is then easy for one horizon volume at t = t, to contain many times over the
volume which will expand to the presently seeable universe.
Consider next the flatness problem. Let T = Tc be the temperature at
which the low-temperature minimum of the effective potential (the true vacuum)
becomes the absolute minimum. When the phase transition is first order, the
universe will supercool to a temperature T = T, before the phase transition is
completed by tunnelling out of the false vacuum, as described in section 2.9,
and reheating of the universe to a temperature T = T R occurs, with T R '" Tc.
The period when supercooling is occurring is a period of non-adiabatic expansion
which modifies the discussion of the flatness problem given earlier. The flatness
problem was cast in section 7.2 as the unnatural smallness of 10 - 11 = lif H2 R21
at early times, e.g. at the (supersymmetric) grand unification scale. During
inflation, assumed to occur at that scale, H2 is given by (7.24) and is constant.
At the same time, R grows exponentially. In (7.20), 10 - 11 was of order 10- 55 .
Thus. if R2 grows by more than about 55 orders of magnitude during inflation,
we end up with a 'natural' value of 10 - 11 of order I at the supersymmetric
grand unification scale. It is usual to measure inflation in terms of e-folds (one
e-fold being growth of R by a factor of e). In terms of e-folds, what we require to
overcome the flatness problem is around 64 e-folds of inflation 1 . This is sufficient
to solve the horizon problem discussed above.
Finally, turning to the unwanted relics problem, let us consider, for
definiteness, the magnetic monopole problem. In section 3.10, the excessive
1 The WMAP data which suggests that 100- 11 may be two orders of magnitude less than 1 indicates
66 e-folds may be nearer the marlt. This makes little difference and we shall use 64 e-folds throughout.
Inflationary cosmology
than containing of the order of 3.6 x 104 such regions as in section 7.2. The usual
expression (3.117) for the distance to the particle horizon does not hold during
the period of cosmological inflation. Instead,
(' dt'
(7.26)
dH(t) = R(t) lo R(t'}
with R(t) given by (7.2S) if we neglect the period of growth of dH(t) during the
period of radiation-dominated expansion which preceded cosmological inflation.
Thus, during the inflationary period,
dH(t) = H-I(e Ht - 1).
(7.27)
The exponential growth of dH(t) during this period means that, once the universe
has reheated to around the critical temperature after the phase transition has
been completed, the horizon volume is exponentially greater than it was at this
temperature prior to inflation. (Because the energy density in the false vacuum
is of order Tc 4 and the radiation density in the FRW universe is given by (2.22).
reheating to a temperature of order Tc occurs.) Thus, after the subsequent period
of radiation-dominated expansion in the FRW universe, the size of the horizon
volume at t = t, is exponentially greater than in the standard model of cosmology.
It is then easy for one horizon volume at t = t, to contain many times over the
volume which will expand to the presently seeable universe.
Consider next the flatness problem. Let T = Tc be the temperature at
which the low-temperature minimum of the effective potential (the true vacuum)
becomes the absolute minimum. When the phase transition is first order, the
universe will supercool to a temperature T = T, before the phase transition is
completed by tunnelling out of the false vacuum, as described in section 2.9,
and reheating of the universe to a temperature T = T R occurs, with T R '" Tc.
The period when supercooling is occurring is a period of non-adiabatic expansion
which modifies the discussion of the flatness problem given earlier. The flatness
problem was cast in section 7.2 as the unnatural smallness of 10 - 11 = lif H2 R21
at early times, e.g. at the (supersymmetric) grand unification scale. During
inflation, assumed to occur at that scale, H2 is given by (7.24) and is constant.
At the same time, R grows exponentially. In (7.20), 10 - 11 was of order 10- 55 .
Thus. if R2 grows by more than about 55 orders of magnitude during inflation,
we end up with a 'natural' value of 10 - 11 of order I at the supersymmetric
grand unification scale. It is usual to measure inflation in terms of e-folds (one
e-fold being growth of R by a factor of e). In terms of e-folds, what we require to
overcome the flatness problem is around 64 e-folds of inflation 1 . This is sufficient
to solve the horizon problem discussed above.
Finally, turning to the unwanted relics problem, let us consider, for
definiteness, the magnetic monopole problem. In section 3.10, the excessive
1 The WMAP data which suggests that 100- 11 may be two orders of magnitude less than 1 indicates
66 e-folds may be nearer the marlt. This makes little difference and we shall use 64 e-folds throughout.