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One possible way of obtaining a small, non-zero, cosmological constant at the present epoch of the universe is to make the cosmological constant evolve in time due to some physical process. At a phenomenological level this can be done either by just postulating such a variation and explore its consequences or - in a slightly more respectable way - by postulating a scalar field potential as described in section 4. These models, however, cannot explain why a bare cosmological constant [the first term on the right hand side of (7)] is zero. To tackle this issue, one can invoke some field [usually a scalar field] which directly couples to the cosmological constant and decreases its "effective value". We shall now examine two such models.

The key idea is to introduce a field which couples to the trace T = Taa of the energy momentum tensor. If T depends on phi and vanishes at some value phi = phi0, then phi will evolve towards phi = phi0 at which T = 0. This equilibrium solution will have zero cosmological constant [276, 277, 278, 279]. While this idea sounds attractive, there are general arguments as to why it does not work in the simplest context [4].

A related attempt was made by several authors, [276, 280, 281], who coupled the scalar field directly to R which, of course, is proportional to T because of Einstein's equations. Generically, these models have the Lagrangian

Equation 116 (116)

The field equations of this model has flat spacetime solutions at phi = phi0 provided U(phi0) = infty. Unfortunately, the effective gravitational constant in this model evolves as

Equation 117 (117)

and vanishes as U rightarrow infty. Hence these models are not viable.

The difficulty in these models arise because they do not explicitly couple the trace of the Tab of the scalar field itself. Handling this consistently [282] leads to a somewhat different model which we will briefly describe because of its conceptual interest.

Consider a system consisting of the gravitational fields gab, radiation fields, and a scalar field phi which couples to the trace of the energy-momentum tensor of all fields, including its own. The zeroth order action for this system is given by

Equation 118 (118)


Equation 119 (119)
Equation 120 (120)

Here, we have explicitly included the cosmological constant term and eta is a dimensionless number which `switches on' the interaction. In the zeroth order action, T represents the trace of all fields other than phi. Since the radiation field is traceless, the only zeroth-order contribution to T comes from the Lambda term, so that we have T = 4Lambda. The coupling to the trace is through a function f of the scalar field, and one can consider various possibilities for this function. The constant phi0 converts phi to a dimensionless variable, and is introduced for dimensional convenience.

To take into account the back-reaction of the scalar field on itself, we must add to T the contribution Tphi = - phi1 phi1 of the scalar field. If we now add Tphi to T in the interaction term Aint(0) further modifies Tikphi. This again changes Tphi. Thus to arrive at the correct action an infinite iteration will have to be performed and the complete action can be obtained by summing up all the terms. (For a demonstration of this iteration procedure, see [283, 284].) The full action can be found more simply by a consistency argument.

Since the effect of the iteration is to modify the expression for Aphi and ALambda, we consider the following ansatz for the full action:

Equation 121 (121)

Here alpha(phi) and beta(phi) are functions of phi to be determined by the consistency requirement that they represent the effect of the iteration of the interaction term. (Since radiation makes no contribution to T, we expect Arad to remain unchanged.) The energy-momentum tensor for phi and Lambda is now given by

Equation 122 (122)

so that the total trace is Ttot = 4alpha(phi) Lambda - beta(phi) phii phii. The functions alpha(phi) and beta(phi) can now be determined by the consistency requirement

Equation 123 (123)

Using Ttot and comparing terms in the above equation we find that

Equation 124 (124)

Thus the complete action can be written as

Equation 125 (125)

(The same action would have been obtained if one uses the iteration procedure.) The action in (125) leads to the following field equations,

Equation 126 (126)
Equation 127 (127)

Here, square stands for a covariant d'Lambertian, Tiktraceless is the stress tensor of all fields with traceless stress tensor and a prime denotes differentiation with respect to phi.

In the cosmological context, this reduces to

Equation 128 (128)
Equation 129 (129)

It is obvious that the effective cosmological constant can decrease if f increases in an expanding universe.The result can be easily generalized for a scalar field with a potential by replacing Lambda by V(phi). This model is conceptually attractive since it correctly accounts for the coupling of the scalar field with the trace of the stress tensor.

The trouble with this model is two fold: (a) If one uses natural initial conditions and do not fine tune the parameters, then one does not get a viable model. (b) Since the scalar field couples to the trace of all sources, it also couples to dust-like matter and "kills" it, making the universe radiation dominated at present. This reduces the age of the universe and could also create difficulties for structure formation. These problems can be circumvented by invoking a suitable potential V(phi) within this model [285]. However, such an approach takes away the naturalness of the model to certain extent.

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