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4.2. Theoretical models with time dependent dark energy: cosmic degeneracy

The approach in the last section was purely phenomenological and one might like to construct some physical model which leads to varying w(a). It turns out that this is fairly easy, and - in fact - it is possible to construct models which will accommodate virtually any form of evolution. We shall now discuss some examples.

A simple form of the source with variable w are scalar fields with Lagrangians of different forms, of which we will discuss two possibilities:

Equation 46 (46)

Both these Lagrangians involve one arbitrary function V(phi). The first one, Lquin, which is a natural generalisation of the Lagrangian for a non-relativistic particle, L = (1/2) qdot2 - V(q), is usually called quintessence (for a sample of models, see [128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138]). When it acts as a source in Friedman universe, it is characterized by a time dependent w(t) with

Equation 47 (47)

The structure of the second scalar field can be understood by a simple analogy from special relativity. A relativistic particle with (one dimensional) position q(t) and mass m is described by the Lagrangian L = - m(1 - qdot2)1/2. It has the energy E = m / (1 - qdot2)1/2 and momentum p = m qdot / (1 - qdot2)1/2 which are related by E2 = p2 + m2. As is well known, this allows the possibility of having massless particles with finite energy for which E2 = p2. This is achieved by taking the limit of m rightarrow 0 and qdot rightarrow1, while keeping the ratio in E = m / (1 - qdot2)1/2 finite. The momentum acquires a life of its own, unconnected with the velocity qdot, and the energy is expressed in terms of the momentum (rather than in terms of qdot) in the Hamiltonian formulation. We can now construct a field theory by upgrading q(t) to a field phi. Relativistic invariance now requires phi to depend on both space and time [phi = phi(t, x)] and qdot2 to be replaced by partiali phi partiali phi. It is also possible now to treat the mass parameter m as a function of phi, say, V(phi) thereby obtaining a field theoretic Lagrangian L = -V(phi)(1 - partiali phi partiali phi)1/2. The Hamiltonian structure of this theory is algebraically very similar to the special relativistic example we started with. In particular, the theory allows solutions in which V rightarrow 0, partiali phi partiali phi rightarrow1 simultaneously, keeping the energy (density) finite. Such solutions will have finite momentum density (analogous to a massless particle with finite momentum p) and energy density. Since the solutions can now depend on both space and time (unlike the special relativistic example in which q depended only on time), the momentum density can be an arbitrary function of the spatial coordinate. This provides a rich gamut of possibilities in the context of cosmology. [139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165], This form of scalar field arises in string theories [166] and - for technical reasons - is called a tachyonic scalar field. (The structure of this Lagrangian is similar to those analyzed in a wide class of models called K-essence; see for example, [159]. We will not discuss K-essence models in this review.)

The stress tensor for the tachyonic scalar field can be written in a perfect fluid form

Equation 48 (48)


Equation 49 (49)

The remarkable feature of this stress tensor is that it could be considered as the sum of a pressure less dust component and a cosmological constant [164] To show this explicitly, we break up the density rho and the pressure p and write them in a more suggestive form as

Equation 50 (50)


Equation 51 (51)

This means that the stress tensor can be thought of as made up of two components - one behaving like a pressure-less fluid, while the other having a negative pressure. In the cosmological context, the tachyonic field is described by:

Equation 52 (52)

When phidot is small (compared to V in the case of quintessence or compared to unity in the case of tachyonic field), both these sources have w rightarrow - 1 and mimic a cosmological constant. When phidot >> V, the quintessence has w approx 1 leading to rhoq propto (1 + z)6; the tachyonic field, on the other hand, has w approx 0 for phidot rightarrow 1 and behaves like non-relativistic matter. In both the cases, -1 < w < 1, though it is possible to construct more complicated scalar field Lagrangians with even w < - 1. (See for example, [167]; for some other alternatives to scalar field models, see for example, [168].)

Since the quintessence field (or the tachyonic field) has an undetermined free function V(phi), it is possible to choose this function in order to produce a given H(a). To see this explicitly, let us assume that the universe has two forms of energy density with rho(a) = rhoknown(a) + rhophi(a) where rhoknown(a) arises from any known forms of source (matter, radiation, ...) and rhophi(a) is due to a scalar field. When w(a) is given, one can determine the V(phi) using either (47) or (52). For quintessence, (47) along with (43) gives

Equation 53 (53)

For tachyonic scalar field, (52) along with (43) gives

Equation 54 (54)

Given Q(a), w(a) these equations implicitly determine V(phi). We have already seen that, for any cosmological evolution specified by the functions H(a) and rhok(a), one can determine w(a); see equation (44). Combining (44) with either (53) or (54), one can completely solve the problem.

Let us first consider quintessence. Here, using (44) to express w in terms of H and Q, the potential is given implicitly by the form [169, 165]

Equation 55 (55)
Equation 56 (56)

where Q(a) ident [8pi G rhoknown(a) / 3H2(a)]. We shall now discuss some examples of this result:

Similar results exists for the tachyonic scalar field as well [165]. For example, given any H(t), one can construct a tachyonic potential V(phi) so that the scalar field is the source for the cosmology. The equations determining V(phi) are now given by:

Equation 60 (60)
Equation 61 (61)

Equations (60) and (61) completely solve the problem. Given any H(t), these equations determine V(t) and phi(t) and thus the potential V(phi).

As an example, consider a universe with power law expansion a = tn. If it is populated only by a tachyonic scalar field, then Q = 0; further, (aH'/H) in equation (60) is a constant making phidot a constant. The complete solution is given by

Equation 62 (62)

where n > (2/3). Combining the two, we find the potential to be

Equation 63 (63)

For such a potential, it is possible to have arbitrarily rapid expansion with large n. (For the cosmological model, based on this potential, see [158].)

A wide variety of phenomenological models with time dependent cosmological constant have been considered in the literature. They involve power law decay of cosmological constant like Lambda propto t-alpha [170, 171, 172, 173, 174, 175, 68] or Lambda propto a-alpha, [176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191], exponential decay Lambda propto exp(- alphaa) [192] and more complicated models (for a summary, see [68]). Virtually all these models can be reverse engineered and mapped to a scalar field model with a suitable V(phi). Unfortunately, all these models lack predictive power or clear particle physics motivation.

This discussion also illustrates that even when w(a) is known, it is not possible to proceed further and determine the nature of the source. The explicit examples given above shows that there are at least two different forms of scalar field Lagrangians (corresponding to the quintessence or the tachyonic field) which could lead to the same w(a). A theoretical physicist, who would like to know which of these two scalar fields exist in the universe, may have to be content with knowing w(a). The accuracy of the determination of w(a) depends on the prior assumptions made in determining Q, as well as on the observational accuracy with which the quantities H(a) can be measured. Direct observations usually give the luminosity distance dL or angular diameter distance dA. To obtain H(a) from either of these, one needs to calculate a derivative [see, for example, (17)] which further limits the accuracy significantly. As we saw in the last section, this is not easy.

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