New inflation
203
with the covariant derivative defined by
D'AV'" = a'AV'" + rrpVp
(7.30)
for any 4·vector V"'. Assuming a homogeneous field 41, so that the spatial
gradients are zero,
V4J = 0,
(7.31)
the Euler-Lagrange equation reduces to
~ + r:04> + V'(4J) = o.
(7.32)
With the coefficients of affine connection for the Robertson-Walker metric as in
section 1.2, the explicit equation of motion is
~ + 3H4> + V'(tP) = 0
(7.33)
where H is the Hubble 'constant' (exercise 1). If there is a range of values of
41 for which slow roll occurs, then in that region the motion is dominated by the
'frictional' teon 3H4> and the ~ teon is neglected. Then the equation of motion
simplifies to
4> = _ V'(tP) .
(7.34)
3H
We derive the conditions that V(tP) must satisfy to obtain
13:4>1 «
(7.35)
1.
The double time derivative ~ may be estimated from (7.34) as
..
1 , . 1
2 •
41 = - 3H V (4J)tP + 3 H - HV(4J).
(7.36)
An estimate of if is now required. When the vacuum energy density Pv
dominates over the radiation density and the curvature teons, the Friedmann
equation gives
2
81rGN
81r_2
H = --Pv = -m p Pv·
(7.37)
3
3
Also, the energy-momentum tensor for the scalar field is given by
a.c atP
T. - - - - - 8 .c
(7.38)
"''' - a(a"'tP) oxv
"'''
= a",4Ja"tP - !g",vOl.tPo'AtP + g",v V(4J).
(7.39)
For a homogeneous field 41, the vacuum energy density pv is given by
1 ·2
pv = Too = ' 1.41 + V(4J).
(7.40)