Our knowledge of the fundamental physics is insufficient to show which potential for the scalar field is most relevant for physics. It therefore makes sense to study the dynamics for large classes of potentials. A useful way of organizing the possibilities uses the ‘rolling’ picture. In a spatially homogeneous spacetime the scalar field satisfies
This resembles the equation of motion of a ball rolling on the graph of the potential The simplest case is where the potential has a strictly positive minimum. In [303] it was proved under
some technical assumptions that a direct analogue of Wald’s theorem holds. The late time behaviour of the
geometry closely resembles that for a cosmological constant. The value of this effective cosmological
constant is , where
is the value where
has its minimum. The asymptotic
behaviour of the matter fields was determined in the case of collisionless matter and perfect fluids with a
linear equation of state.
Another important case is where is everywhere positive and decreasing and tends to zero as
. The ‘rolling’ picture suggests that
should tend to infinity as
. Under suitable
technical assumptions this is true and information can be obtained concerning the asymptotics. The
exponential potential is a borderline case. An important assumption is that
or, more
generally
. Intuitively this says that the potential falls off no faster at infinity than
an exponential potential which gives rise to power-law inflation. A theorem in [306
] where
this assumption is made in a set-up like that in Wald’s theorem shows that there is always
accelerated expansion for
sufficiently large. If it is further assumed that
as
then it is possible to say a lot more. It is found that, if
is the tracefree part of the
second fundamental form,
is the spatial scalar curvature, and
is the energy density of
matter other than the scalar field then
,
, and
tend to
zero as
. In the limit
the solution is approximated by one which is isotropic
and spatially flat and contains no matter other than the scalar field. This kind of situation is
sometimes called intermediate inflation since the potential is intermediate between a constant
(corresponding to a cosmological constant) and an exponential (corresponding to power-law
inflation).
If as
and
is bounded for large
then it is possible to get further
information. This is related to the ‘slow-roll approximation’. The intuitive idea is that if the slope of the
graph of
is not too steep the ball will roll slowly and certain quantities will change gradually. It can be
proved that asymptotically the term with second order derivatives in Equation (4
) can be neglected and
that the late-time behaviour is described approximately by the resulting first order equation.
In fact this can be further simplified to give the equation
for
alone.
This asymptotic description is not only interesting in itself; it gives a powerful method for
determining the late time asymptotics when a specific potential has been chosen. For more details
see [306].
Both models with a positive cosmological constant and models with a scalar field with exponential
potential are called inflationary because the rate of expansion is increasing with time. There is also
another kind of inflationary behaviour that arises in the presence of a scalar field with power law
potential like or
. In that case the inflationary property concerns the behaviour of the
model at intermediate times rather than at late times. The picture is that at late times the
universe resembles a dust model without cosmological constant. This is known as reheating. The
dynamics have been analysed heuristically by Belinskii et al. [46]. Part of their conclusions
have been proved rigorously in [302]. Calculations analogous to those leading to a proof of
isotropization in the case of a positive cosmological constant or an exponential potential have
been done for a power law potential in [249]. In that case, the conclusion cannot apply to late
time behaviour. Instead, some estimates are obtained for the expansion rate at intermediate
times.
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