We noted in section 6.3.3 that the time dependence of the
modes comes from their explicit time dependence
, from factors of the scale factor, and
from the time dependence of
itself. Using program variables
for the fields the time dependence of the modes is given by
![]() |
(6.92) |
![]() |
(6.93) |
![]() |
(6.94) |
There's another way to view this condition. Gravitational particle
production will occur unless . Since this condition is
automatically satisfied for
consider the opposite case
, for which
. Then neglecting the time
dependence of
,
when
, so
the condition
is equivalent to the
condition
. In fact
is the
stronger (and more accurate) condition because it also specifies that
shouldn't be changing rapidly, which would lead to particle
production irrespective of the value of
. However, all particle
masses should vary slowly during inflation because they should only
depend on constants and on the value of the inflaton, which must be
changing slowly.
In the case of a field with during inflation the approximation
that the field ends inflation in its ground state is no longer
valid. In the limit
the fluctuations of the field produced
during inflation can be accurately described by Hankel functions
[4]. However in this case the fields will be copiously
produced during inflation, leading to severe cosmological problems
[5]. For this reason we do not implement these Hankel
function solutions in the lattice program. In order to avoid the
moduli problem associated with light fields it's best to assume that
some mechanism must have given all scalar fields large masses during
inflation, in which case equation (6.51) is an accurate
expression for the modes at the end of inflation.