Inflationary cosmology 35If the field is constant both spatially and temporally, the equation of state is then
p=−ρ, as required if the scalar field is to act as a cosmological constant; note
that derivatives of the field spoil this identification.
Treating the field classically (i.e. considering the expectation value〈φ〉,we
get from energy–momentum conservation (T;μνν =0) the equation of motion
φ ̈+ 3 Hφ ̇−∇^2 φ+dV/dφ= 0.This can also be derived more easily by the direct route of writing down the action
S=
∫
L√
−gd^4 xand applying the Euler–Lagrange equation that arises from a
stationary action (
√
−g=R^3 (t)for an FRW model, which is the origin of the
Hubble drag term 3Hφ ̇).
The solution of the equation of motion becomes tractable if we both ignore
spatial inhomogeneities inφand make theslow-rolling approximationthat|φ ̈|
is negligible in comparison with| 3 Hφ ̇|and|dV/dφ|. Both these steps are
required in order that inflation can happen; we have shown earlier that the vacuum
equation of state only holds if in some senseφchanges slowly both spatially and
temporally. Suppose there are characteristic temporal and spatial scalesTand
Xfor the scalar field; the conditions for inflation are that the negative-pressure
equation of state fromV(φ)must dominate the normal-pressure effects of time
and space derivatives:
Vφ^2 /T^2 , Vφ^2 /X^2 ,hence|dV/dφ|∼V/φmust beφ/T^2 ∼φ ̈.Theφ ̈term can therefore be
neglected in the equation of motion, which then takes the slow-rolling form for
homogeneous fields:
3 Hφ ̇=−dV/dφ.
The conditions for inflation can be cast into useful dimensionless forms. The
basic conditionVφ ̇^2 can now be rewritten using the slow-roll relation as
≡
m^2 P
16 π(V′/V)^2 1.
Also, we can differentiate this expression to obtain the criterionV′′V′/mP.
Using slow-roll once more gives 3Hφ/ ̇ mPfor the right-hand side, which is in
turn 3 H
√
V/mPbecauseφ ̇^2 V, giving finallyη≡m^2 P
8 π(V′′/V) 1
(recall that for de Sitter spaceH=
√
8 πGV(φ)/ 3 ∼√
V/mPin natural units).
These two criteria make perfect intuitive sense: the potential must be flat in the
sense of having small derivatives if the field is to roll slowly enough for inflation
to be possible.