Old inflation
199
7.3 Old inflation
A solution to all three problems discussed in the last section is for there to have
been a period of very rapid expansion of the universe (cosmological inflation)
during which the scale factor of the universe grew by a large amount. We shall
discuss shortly how much expansion is sufficient to solve these problems. A
simple mechanism to produce the required expansion is for the universe to have
supercooled in a false vacuum prior to undergoing a first-order phase transition to
the true vacuum in the way described in section 2.9. In that case, a large positive
vacuum energy, constant until the phase transition is completed, can drive a period
of expansion in a de Sitter universe. After some supercooling has occurred, the
vacuum energy density in the Friedmann equation will dominate the radiation
energy density and the curvature term, and the Friedmann equation simplifies to
2
R2 81fGN
(7.22)
H = R2 = - 3 - V
where V is the vacuum energy density in the false vacuum. This is equivalent to
the de Sitter equation with cosmological constant
A = 81fGNV = 81fmp2V.
(7.23)
Moreover, the Hubble constant during the inflationary era has a constant value
given by
H2 = lA.
(7.24)
During cosmological inflation, the scale factor of the universe grows
exponentially
R(t) ex exp (~t) = eH'.
(7.25)
The exponential expansion means that by the time the transition to the true
vacuum occurs, the scale factor of the universe may have increased by many
orders of magnitude. The phase transition will be completed by the formation
of bubbles of the true vacuum, as discussed in section 2.9. Once formed, the
bubbles will tend to coalesce and the energy stored in the walls of the bubbles
will be released resulting in the universe reheating. Thereafter, the universe
will evolve as a (in the first instance) radiation-dominated Friedmann-RobertsonWalker (FRW) universe. However, the initial conditions for the evolution of the
FRW universe will have been drastically modified by the period of inflation. If
sufficient inflation has occurred, the various problems discussed in the previous
section will be solved. We now estimate how much inflation is required for this
purpose.
Consider first the horizon problem. This problem will be resolved if the
presently observable universe lies in a single region which was causally connected
at the time of decoupling of photons from matter (the recombination time), rather
199
7.3 Old inflation
A solution to all three problems discussed in the last section is for there to have
been a period of very rapid expansion of the universe (cosmological inflation)
during which the scale factor of the universe grew by a large amount. We shall
discuss shortly how much expansion is sufficient to solve these problems. A
simple mechanism to produce the required expansion is for the universe to have
supercooled in a false vacuum prior to undergoing a first-order phase transition to
the true vacuum in the way described in section 2.9. In that case, a large positive
vacuum energy, constant until the phase transition is completed, can drive a period
of expansion in a de Sitter universe. After some supercooling has occurred, the
vacuum energy density in the Friedmann equation will dominate the radiation
energy density and the curvature term, and the Friedmann equation simplifies to
2
R2 81fGN
(7.22)
H = R2 = - 3 - V
where V is the vacuum energy density in the false vacuum. This is equivalent to
the de Sitter equation with cosmological constant
A = 81fGNV = 81fmp2V.
(7.23)
Moreover, the Hubble constant during the inflationary era has a constant value
given by
H2 = lA.
(7.24)
During cosmological inflation, the scale factor of the universe grows
exponentially
R(t) ex exp (~t) = eH'.
(7.25)
The exponential expansion means that by the time the transition to the true
vacuum occurs, the scale factor of the universe may have increased by many
orders of magnitude. The phase transition will be completed by the formation
of bubbles of the true vacuum, as discussed in section 2.9. Once formed, the
bubbles will tend to coalesce and the energy stored in the walls of the bubbles
will be released resulting in the universe reheating. Thereafter, the universe
will evolve as a (in the first instance) radiation-dominated Friedmann-RobertsonWalker (FRW) universe. However, the initial conditions for the evolution of the
FRW universe will have been drastically modified by the period of inflation. If
sufficient inflation has occurred, the various problems discussed in the previous
section will be solved. We now estimate how much inflation is required for this
purpose.
Consider first the horizon problem. This problem will be resolved if the
presently observable universe lies in a single region which was causally connected
at the time of decoupling of photons from matter (the recombination time), rather