**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:

(46) |

Both these Lagrangians involve one arbitrary function
*V*(). The
first one, *L*_{quin}, which is a natural generalisation of
the Lagrangian for a non-relativistic particle, *L* = (1/2)
^{2}
- *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

(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 -
^{2})^{1/2}.
It has the energy *E* = *m* / (1 -
^{2})^{1/2} and momentum *p* =
*m* / (1 -
^{2})^{1/2}
which are related by *E*^{2} = *p*^{2} +
*m*^{2}. As is well
known, this allows the possibility of having massless particles with finite
energy for which *E*^{2} = *p*^{2}. This is
achieved by taking the limit of *m*
0
and
1, while
keeping the ratio in *E* = *m* / (1 -
^{2})^{1/2}
finite. The momentum acquires a life of its own, unconnected with the
velocity ,
and the energy
is expressed in terms of the momentum (rather than in terms of
) in the
Hamiltonian formulation. We can now
construct a field theory by upgrading *q*(*t*) to a field
. Relativistic
invariance now requires
to depend on
both space and time
[ =
(*t*,
**x**)] and
^{2}
to be replaced by
_{i}
^{i}
. It is also
possible now to treat the mass parameter *m* as a function of
, say,
*V*()
thereby obtaining a field theoretic Lagrangian *L* =
-*V*()(1 -
^{i}
_{i}
)^{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* 0,
_{i}
^{i}
1
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

(48) |

with

(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
and the
pressure *p* and write them in a more suggestive form as

(50) |

where

(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:

(52) |

When 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*
- 1 and
mimic a cosmological constant. When
>>
*V*, the quintessence has
*w* 1 leading to
_{q}
(1 +
*z*)^{6}; the tachyonic field, on the other hand, has
*w* 0 for
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*(), 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
(*a*) =
_{known}(*a*) +
_{}(*a*)
where _{known}(*a*) arises from any known forms of
source (matter, radiation, ...) and
_{}(*a*) is
due to a scalar field. When *w*(*a*) is given, one can
determine the
*V*()
using either (47) or (52). For quintessence,
(47) along with (43) gives

(53) |

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

(54) |

Given *Q*(*a*), *w*(*a*) these equations implicitly
determine
*V*().
We have already seen that, for any cosmological evolution specified
by the functions *H*(*a*) and
_{k}(*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]

(55) | |

(56) |

where *Q*(*a*)
[8 *G*
_{known}(*a*) / 3*H*^{2}(*a*)].
We shall now discuss some examples of this result:

- Consider a universe in which observations suggest that
*H*^{2}(*a*) =*H*_{0}^{2}*a*^{-3}. Such a universe could be populated by non relativistic matter with density parameter_{NR}= = 1. On the other hand, such a universe could be populated entirely by a scalar field with a potential*V*() =*V*_{0}exp[- (16*G*/3)^{1/2}]. One can also have a linear combination of non relativistic matter and scalar field with the potential having a generic form*V*() =*A*exp[-*B*]. - Power law expansion of the universe can be generated by
a quintessence model with
*V*() =^{-}. In this case, the energy density of the scalar field varies as_{}*t*^{-2 / (2 + )}; if the background density_{bg}varies as_{bg}*t*^{-2}, the ratio of the two energy densities changes as (_{}/_{bg}=*t*^{4 / (2 + )}). Obviously, the scalar field density can dominate over the background at late times for > 0. - A different class of models arise if the potential is taken
to be exponential with, say,
*V*() exp(- /*M*_{Pl}). When*k*= 0, both_{}and_{bg}scale in the same manner leading to(57) where

*w*_{bg}refers to the background parameter value. In this case, the dark energy density is said to "track" the background energy density. While this could be a model for dark matter, there are strong constraints on the total energy density of the universe at the epoch of nucleosynthesis. This requires_{}0.2 requiring dark energy to be sub dominant at all epochs. - Many other forms of
*H*(*a*) can be reproduced by a combination of non-relativistic matter and a suitable form of scalar field with a potential*V*(). As a final example [68], suppose*H*^{2}(*a*) =*H*_{0}^{2}[_{NR}*a*^{-3}+ (1 -_{NR})*a*^{-n}]. This can arise, if the universe is populated with non-relativistic matter with density parameter_{NR}and a scalar field with the potential, determined using equations (55), (56). We get(58) where

(59) and is a constant.

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

(60) | |

(61) |

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

As an example, consider a universe with power law expansion
*a* = *t*^{n}. If it is populated only by a tachyonic
scalar field, then *Q* = 0; further,
(*aH'*/*H*) in equation (60) is a constant
making a
constant. The complete solution is given by

(62) |

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

(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
*t*^{-}
[170,
171,
172,
173,
174,
175,
68]
or
*a*^{-},
[176,
177,
178,
179,
180,
181,
182,
183,
184,
185,
186,
187,
188,
189,
190,
191],
exponential decay
exp(- *a*)
[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*().
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 *d*_{L} or angular diameter
distance *d*_{A}. 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.