The observations which suggest the existence of non-zero cosmological constant - discussed in the last section - raises serious theoretical problems which we mentioned in section 1.1. These difficulties have led people to consider the possibility that the dark energy in the universe is not just a cosmological constant but is of more complicated nature, evolving with time. Its value today can then be more naturally set by the current expansion rate rather than predetermined earlier on - thereby providing a solution to the cosmological constant problems.

Though a host of models have been constructed based on this hope, none
of them provides a satisfactory solution to the problems of
fine-tuning.
Moreover, all of them involve an evolving equation of state parameter
*w*_{X}(*a*) for the unknown ("X") dark
energy component, thereby taking away all predictive power from
cosmology
[165].
Ultimately, however, this
issue needs to settled observationally by checking whether
*w*_{X}(*a*) is a constant [equal to -1,
for the cosmological constant] at all epochs
or whether it is indeed varying with *a*. We shall now discuss
several observational and theoretical issues connected with this theme.

While the complete knowledge of the *T*_{b}^{a}
[that is, the knowledge of the right hand side
of (20)] uniquely determines *H*(*a*), the converse is not
true. If we know only the function
*H*(*a*), it is not possible to determine the nature of the
energy density which is present in the universe. We have already seen
that geometrical measurements
can *only* provide, at best, the functional form of
*H*(*a*). It follows that purely geometrical measurements of
the Friedmann universe will never allow us to determine the material
content of the universe.

[The only exception to this rule is when we assume that
*each of the components in the universe has constant
w_{i}*. This is fairly strong assumption and, in fact,
will allow us to determine the
components of the universe from the knowledge of the function

(41) |

with a term having *w*_{i} = - (1/3) added to the sum.
Let
3(1 + *w*)
and
() denote the fraction of the
critical density contributed by matter with
*w* = (/3) -
1. (For discrete values of *w*_{i} and
_{i},
the function
() will be a sum of Dirac
delta functions.) In the continuum
limit, equation (41) can be rewritten as

(42) |

where (*a*/*a*_{0}) = exp(*q*). The function
()
is assumed to have finite support (or decrease fast enough) for the
expression on the right hand side to converge.
If the observations determine the function *H*(*a*), then the
left hand side can be expressed as a function of *q*. An inverse
Laplace transform of this equation will then determine the form of
()
thereby determining the composition of the universe, as long as all
matter can be described by an equation of state of the form
*p*_{i} = *w*_{i}
_{i}
with *w*_{i} = constant for all *i* = 1,...., *N*.]

More realistically one is interested in models which has a complicated
form of *w*_{X}(*a*)
for which the above analysis is not applicable.
Let us divide the source energy density into two components:
_{k}(*a*), which is known from
independent observations and a component
_{X}(*a*) which is not known. From (20), it
follows that

(43) |

Taking a derivative of
ln_{X}(*a*) and using (19), it is easy to obtain
the relation

(44) |

If geometrical observations of the universe give us *H*(*a*)
and other observations give us
_{k}(*a*) then
one can determine *Q* and thus
*w*_{X}(*a*). While this is possible, in principle the
uncertainties in measuring both *H* and *Q* makes this a
nearly impossible route to follow
in practice. In particular, one is interested in knowing whether
*w* evolves with time or a constant and this turns out to be a very
difficult task observationally. We shall now briefly discuss some of the
issues.