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8.1. Overview

Having, under the pressure of observational evidence and an aesthetical desire to keep the inflationary scenario of the early universe as simple as possible, admitted the existence of a constant positive Lambda-term, it is natural to take a step beyond Einstein's original hypothesis and consider the additional possibility that the Lambda-term is not an exact constant, but rather, describes a new dynamical degree of freedom (perhaps even a new form of matter). Really, neither observational data, nor inflationary considerations tell us that a "cosmological constant" is constant (though, as discussed above, it should change sufficiently slowly with time, in particular, slower than the Ricci tensor). In fact the effective Lambda-term which appears in the inflationary scenario of the early universe is never an exact constant and rarely even an approximate constant of motion. (A recent analysis of observational data in the light of a time dependent Lambda may be found in [203].)

To quantitatively describe this new degree of freedom (or a new form of matter), some phenomenological models of a dynamical Lambda-term have to be introduced. The word "phenomenological" means that no attempt to derive these models from an underlying quantum field theory is being made, in contrast to examples discussed in previous sections. Historically, many phenomenological Lambda-models were proposed since 1986 (not counting the "C-field" of Hoyle and Narlikar (1962) which was perhaps the earliest, though unsuccessful, attempt to introduce a dynamical Lambda-term in cosmology). Depending upon their level of "fundamentality", these phenomenological methods may be classified into 3 main groups:

1) Kinematic models.
Here Lambda is simply assumed to be a function of either the cosmic time t or the scale factor a(t) of the FRW cosmological model.

2) Hydrodynamic models.
Here a Lambda-term is described by a barotropic fluid with some equation of state pLambda(rhoLambda) (dissipative terms may also be present).

3) Field-theoretic models.
The Lambda-term is assumed to be a new physical classical field (which we shall call a lambda-field) with some phenomenological Lagrangian.

Of course, models from the last group are in a sense also the most fundamental. In particular, they may be used in a non-FRW setting. Additionally, their quantization is straightforward. However, if we restrict ourselves to a FRW model with small perturbations, the three different way of describing a Lambda-term could lead to converging results. Note also, that it was recently proposed to call a dynamical Lambda-term "quintessence" [24] (irrespective of the specifities of modelling, so this notion is wider than the notion of a lambda-field), though we don't consider it to be obligatory.

In the case of field-theoretic models, the most simple and natural is the model of a scalar field phi with some self-interaction potential V(phi), minimal coupling to gravity and no (or very weak) coupling to other known physical fields. The latter requirement follows not only from simplicity, but also from observational evidence (see [27] for a recent analysis of upper bounds on coupling of phi to the electromagnetic field). The assumption of minimal coupling to gravity may be relaxed, but only slightly (see [33] for constraints on the coefficient xi in case of the xi Rphi2/2 coupling). Since the minimally coupled scalar field model has proven to be extremely successfuly in the case of the inflationary scenario, one might be tempted to use it for the description of a Lambda-term. As a result, an overwelming part of recent theoretical activity has focussed on the scalar field model (more appropriately, on this class of models differing between themselves by the form of the scalar field potential V(phi)).

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