160 Cosmic Inflation
If we make the further approximation that the friction term in Equation (7.24)
dominates,휙≪̈ 3 퐻휙̇, the equation of motion for scalar particles is approximately
3 퐻휙̇+푉′(휙)≃ 0. (7.33)Slow rolling is characterised by the two parameters휂≡
푚^2 Planck
16 휋(
푉′′
푉
)
≪ 1 ,휖≡
푚^2 Planck
16 휋(
푉′
푉
) 2
≪ 1 , (7.34)
where푚^2 Planck=푐^2 ∕퐺. Single-field inflation occurs when the Universe is dominated by
the inflaton field휙and obeys the slow-roll conditions in Equation (7.32). Inflationary
models assume that there is a moment when this domination starts and subsequently
drives the Universe into a de Sitter-like exponential expansion in which푇≃0.Alan
Guthin 1981 [2] named this aninflationary universe.
Graceful Exit. Clearly the cosmic inflation cannot go on forever if we want to arrive
at our present slowly expanding Friedmann–Lemaitre universe. Thus there must be a
mechanism to halt the exponential expansion, agraceful exit. The freedom we have to
arrange this is in the choice of the potential function and its temperature-dependence,
푉(휙,푇). Inflation ends when^12 휙̇^2 dominates over푉(휙)in Friedmann’s Equation (7.25)
when the inflaton field arrives at the minimum휙=0 of the potential in Figure 7.2.
The timescale for inflation is
퐻=√
8 휋퐺
3
⟨푉 0 ⟩∝
√
ℏ푐
푀P
≃( 10 −^34 s)−^1. (7.35)One may expect that휙should oscillate near this minimum, but in a rapidly expand-
ing universe, the inflaton field approaches the minimum very slowly, like a ball in a
viscous medium, the viscosity푉′(휑)being proportional to the speed of expansion.
In the expansion the scale factor푎grows so large that the third inequality follows.
Equations (7.25) and (7.24) then simplify to
퐻^2 =8 휋
3 푀P^2
푉(휙) (7.36)
and
3 퐻̇휑=−푉′(휑). (7.37)
There are many ways to go beyond single-field slow-roll. So far we have described
the minimally coupled action which implies that there is no direct coupling
between the inflaton and the metric. We could also entertain the possibility that
the Einstein-Hilbert action is modified at high energy with푓(푅) terms or with
noncanonical terms퐹(푅,휙)as in Equation (5.85), or that more than one field is
relevant during inflation. These subjects are advanced and beyond the scope of the
present monograph.
Entropy. Suppose that there is a symmetry breaking phase transition from a hot
퐺GUT-symmetric phase dominated by the scalar field휙to a cooler퐺s-symmetric phase.