A simple generalization of the scalar field models is a collection of several scalar fields [119]. These
have kinetic energy
and potential energy given by a function
of all the
. If the
are
thought of as defining a mapping with values in
endowed with the Euclidean metric then it is easy to
see a further generalization. Simply replace
by a Riemannian manifold
and use the metric
to define a kinetic energy as in a wave map or nonlinear
-model. The unknown in the equation is
then a mapping
from spacetime to
and the potential is a function on
. A more concrete
description of
can be obtained by using its components
in a local coordinate chart on
. One
type of model is called assisted inflation and has a potential which is the sum of exponentials
of scalar fields. The name comes from the fact that even if each of these exponentials alone
decays too fast to produce inflation they can assist each other so as to produce inflation in
combination.
A more radical generalization is to consider a scalar field with Lagrangian where
. This is known as
-essence [28]. In quintessence models the equation of motion of
the scalar field is always hyperbolic so that the Einstein-matter equations have a well-posed initial value
problem. Under the assumption that the potential is non-negative the dominant energy condition is always
satisfied. These properties need not hold in
-essence models unless the function
is restricted. In fact
there is a motivation for considering models in which the dominant energy condition is violated. The value
of
in our universe can in principle be determined by observation. It is not far from
and if it happened to be less than
(which is consistent with the observations) then
the dominant energy condition would be violated. It would be desirable to determine general
conditions on
which guarantee well-posedness and/or the dominant energy condition. In
-essence the equations of motion are in general quasilinear and not semilinear as they are
in the case of quintessence. This may lead to the spontaneous formation of singularities in
the matter field. It would be interesting to know under what conditions on
this can be
avoided.
Partial answers to the questions just raised can be found in [152, 151, 138]. An interesting class of
models which seem to be relatively well-behaved are the tachyon models where
for some non-negative potential
. Despite their name they have characteristics which lie inside the light
cone. Specialising further to
gives a model equivalent to an exotic fluid, the Chaplygin
gas. More information about the equivalence between different matter models can be found
in [305].
When the dominant energy condition is violated new phenomena can occur. It is possible for an
expanding cosmological model to end after finite proper time, something known as the big rip since before
the final time all physical systems are ripped apart [74]. As this final time is approached the
mean curvature tends to infinity, as does the energy density. This kind of behaviour can be seen
explicitly for a fluid with and
. It is not clear that it is reasonable to consider
such a fluid, but similar things could happen for other matter fields violating the dominant
energy condition. It seems that there is no overview in the literature of what matter models are
concerned.
To end this section we list without further comment some other exotic models which have been
considered. There is the curvature-coupled scalar field (where there are some mathematical results [57]) and
theories where the Einstein-Hilbert Lagrangian is replaced by some other function of the curvature.
There are also models, different from Einstein gravity, which are motivated by loop quantum
gravity [63] and brane-world theories [238] where the form of the Hamiltonian constraint is
modified.
![]() |
http://www.livingreviews.org/lrr-2005-6 |
© Max Planck Society and the author(s)
Problems/comments to |