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7.5 Other models for cosmic acceleration

Nonlinear minimally coupled scalar fields were originally applied to the very early universe which is why the name inflation is often attached to them. Now many of these models have been recycled to model cosmic acceleration at later epochs up to the present day. In the latter context the name quintessence is used. More generally an exotic matter field violating the strong energy condition and leading to cosmic acceleration is often referred to as dark energy. In this section some models will be considered which go beyond the cosmological constant and ordinary quintessence. There is such a proliferation of models in the literature that the list considered here is far from complete.

A simple generalization of the scalar field models is a collection of several scalar fields fi [119]. These have kinetic energy sum (fi)2 i and potential energy given by a function V of all the fi. If the fi are thought of as defining a mapping with values in Rn endowed with the Euclidean metric then it is easy to see a further generalization. Simply replace n R by a Riemannian manifold (N, h) and use the metric h to define a kinetic energy as in a wave map or nonlinear s-model. The unknown in the equation is then a mapping f from spacetime to N and the potential is a function on N. A more concrete description of f can be obtained by using its components fi in a local coordinate chart on N. 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 p(f, X) where 1 a X = - 2 \~/ f\ ~/ af. This is known as k-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 k-essence models unless the function p is restricted. In fact there is a motivation for considering models in which the dominant energy condition is violated. The value of w = p/r in our universe can in principle be determined by observation. It is not far from - 1 and if it happened to be less than - 1 (which is consistent with the observations) then the dominant energy condition would be violated. It would be desirable to determine general conditions on p which guarantee well-posedness and/or the dominant energy condition. In k-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 p this can be avoided.

Partial answers to the questions just raised can be found in [152151138]. An interesting class of models which seem to be relatively well-behaved are the tachyon models where V~ -------- p(f,X) = - V (f) 1- 2X for some non-negative potential V. Despite their name they have characteristics which lie inside the light cone. Specialising further to V (f) = 1 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 p = wr and w < - 1. 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 [57Jump To The Next Citation Point]) 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.

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