**4.1. Parametrized equation of state and cosmological observations**

One simple, phenomenological, procedure for comparing observations with
theory is to parameterize the function *w*(*a*) in some
suitable form and determine a finite set of parameters
in this function using the observations. Theoretical models can then be
reduced to a finite set of parameters which can be determined by this
procedure. To illustrate this
approach, and the difficulties in determining the equation of state of
dark energy from the observations, we shall assume that
*w*(*a*) is given by the simple form:
*w*(*a*) = *w*_{0} + *w*_{1}(1 -
*a*); in the *k* = 0 model (which we shall assume for
simplicity), *w*_{0} measures
the current value of the parameter and - *w*_{1} gives its
rate of change at the present epoch. In addition to simplicity, this
parameterization has the advantage of giving finite *w* in the
entire range 0 < *a* < 1.

Figure 12 shows confidence interval contours in
the *w*_{0} - *w*_{1} plane arising from
the full supernova data, obtained by assuming that
_{NR} +
_{} = 1. The three
frames are for
_{NR} =
(0.2, 0.3, 0.4). The following features are
obvious from the figure: (i) The cosmological constant corresponding to
*w*_{0} = - 1, *w*_{1} = 0
is a viable candidate and cannot be excluded. (In fact, different
analysis of many observational results lead to this conclusion
consistently; in other words, at present there is no observational
motivation to assume *w*_{1}
0.) (ii) The result is
sensitive to the value of
_{NR} which
is assumed. This is understandable from equation (44) which shows that
*w*_{X}(*a*) depends on both
*Q*
_{NR} and
*H*(*a*). (We shall discuss this dependence of the results on
_{NR} in
greater detail below). (iii) Note that the axes are not in equal units
in figure 12. The observations can determine
*w*0 with far greater accuracy than *w*_{1}. (iv) The
slanted line again corresponds to
*H*_{0}*d*_{L}(*z* = 0.63) = constant and
shows that the shape of the probability ellipses arises essentially from
this feature.

In summary, the current data definitely supports a negative pressure
component with *w*_{0} < - (1/3)
but is completely consistent with *w*_{1} = 0. If this is
the case, then the cosmological constant is the
simplest candidate for this negative pressure component and there is
very little observational motivation
to study other models with varying *w*(*a*). On the other
hand, the cosmological constant has well known theoretical problems
which could possibly be alleviated in more sophisticated
models with varying *w*(*a*). With this motivation, there has
been extensive amount of work in the
last few years investigating whether improvement in the observational
scenario will allow us to determine whether *w*_{1} is non
zero or not. (For a sample of references, see
[110,
111,
112,
113,
114,
115,
116,
117,
118,
119,
120,
121,
122,
123,
124,
125].)
In the context of supernova based determination of *d*_{L},
it is possible to analyze the situation along the following lines
[79].

Since the supernova observations essentially measure
*d*_{L}(*a*), accuracy in the determination of
*w*_{0} and *w*_{1} from (both the present and
planned future
[126])
supernova observations will crucially depend on how sensitive
*d*_{L} is to the changes in *w*_{0} and
*w*_{1}. A good measure of the sensitivity is provided by
the two parameters

(45) |

Since *d*_{L}(*z*, *w*_{0},
*w*_{1}) can be obtained from theory, the
parameters *A* and *B* can be computed form theory in a straight
forward manner. At any given redshift *z*, we can plot contours of
constant *A* and *B* in the *w*_{0} -
*w*_{1} plane.
Figure (13) shows the result of such an
analysis
[79].
The two frames on the left are at *z* = 1 and the two frames on
the right are at *z* = 3. The top frames give contours of constant
*A* and bottom frame give contours of constant *B*.
From the definition in the equation (45) it is clear that *A* and
*B* can be interpreted as the fractional change in
*d*_{L} for unit change
in *w*_{0}, *w*_{1}. For example, along the
line marked *A* = 0.2 (in the top left frame) *d*_{L}
will change by 20 per cent for unit change in *w*_{0}. It
is clear from the two top frames that for most of the
interesting region in the *w*_{0} - *w*_{1}
plane, changing *w*_{0} by
unity changes *d*_{L} by about 10 per cent or
more. Comparison of
*z* = 1 and *z* = 3 (the two top frames) shows that
the sensitivity is higher at high redshift, as to be expected.
The shaded band across the picture corresponds to the region
in which -1 *w*(*a*)
0 which is of primary
interest in constraining dark energy with negative pressure.
One concludes that determining *w*_{0} from
*d*_{L} fairly accurately will not be too daunting a task.

The situation, however, is quite different as regards
*w*_{1} as illustrated in the bottom two frames. For the
same region of the *w*_{0} - *w*_{1} plane,
*d*_{L} changes only by a few percent when
*w*_{1} changes by unity. That is, *d*_{L} is
much less sensitive to *w*_{1} than to
*w*_{0}. It is going to be significantly
more difficult to determine a value for *w*_{1} from
observations of *d*_{L} in the near future. Comparison of
*z* = 1 and *z* = 3 again shows that the sensitivity is
somewhat better at high redshifts but only marginally.

The situation is made worse by the fact that *d*_{L} also
depends on the parameter
_{NR}. If
varying _{NR}
mimics the variation of *w*_{1} or *w*_{0},
then, one also needs to determine the sensitivity of
*d*_{L} to
_{NR}.
Figure 14 shows contours of constant
*H*_{0} *d*_{L} in the
_{NR} -
*w*_{0} and
_{NR} -
*w*_{1}
planes at two redshifts *z* = 1 and *z* = 3. The two top
frames shows that if one varies the value of
_{NR} in the
allowed range of, say, (0.2, 0.4) one can move along
the curve of constant *d*_{L} and induce fairly large
variation in *w*_{1}. In other words, large changes in
*w*_{1} can be easily compensated by small changes in
_{NR}
while maintaining the same value for *d*_{L} at a given
redshift. This shows that the uncertainty in
_{NR} introduces
further difficulties in determining *w*_{1} accurately from
measurements of *d*_{L}. The two lower frames show that
the situation is better as regards *w*_{0}. The curves are much
less steep and hence varying
_{NR} does not
induce large variations in *w*_{0}. We are once again
led to the conclusion that unambiguous determination of
*w*_{1} from data will be quite difficult. This is
somewhat disturbing since *w*_{1}
0 is a clear
indication of a dark energy component which is evolving.
It appears that observations may not be of great help
in ruling out cosmological constant as the major dark energy
component. (The results given above are based on
[79];
also see
[127]
and references cited therein.)