**8.4. Reconstructing the effective potential
V()
for a time-dependent
-term**

In view of the large number of models capable of
predicting a small cosmological constant at the present epoch, it is
necessary to ask whether cosmological observations themselves may be
used to determine model parameters uniquely.
The answer to this question
is (fortunately) in the affirmative, at least for the class of minimally
coupled scalar field models discussed earlier.
This is easily demonstrated by considering Equations (114)
and (98) which
express *V*() and
in terms of the Hubble
parameter *H*
and its first derivative *dH*/*dz*. Consequently one can
determine the
form of the potential *V*(*z*) (mimicking the
-term)
if the Hubble parameter *H*(*z*) is
known from observations. There are two independent methods for
determining *H*(*z*).
The first is related to the luminosity distance *d*_{L},
discussed in section 4.2
[186,
101].
From (28) we easily find

(119) |

Thus the luminosity distance *d*_{L}(*z*) determines
the Hubble parameter
*H*(*z*) uniquely ! Now (98, 114) can be used to reconstruct the
form of the potential *V*(*z*) (or
*V*()) and the equation
of state
*w*_{}(*z*)
in a model independent manner. (However, what is required
for an unambiguous determination of
*V*() is the present matter
density _{m}
[186].)
Formula (119) can also be used for an unambiguous determination
of *H*(*z*) from the angular-size distance
*d*_{A}(*z*) introduced in
section 4.5, if we
use the relation *d*_{A}(*z*) = *d*_{L}(1 +
*z*)^{-2}
[187].

Another means of
determining *H*(*z*) is associated with the growth of linear
density fluctuations responsible for the formation of large scale structure
[185,
186].
The growth of linearized perturbations in a collisionless medium has the
well known form

(120) |

where the value of *H* is determined from (97).
(On scales << 200h^{-1} Mpc the
-field is practically
unclustered and can be treated as a smooth component if
|*m*_{}^{2}|
|*d*^{2}*V* /
*d*^{2}|
img src="../../New_Gifs/ltapprox.gif" alt="ltapprox">
*H*_{0}^{2}).
Although it is not possible to solve (120) analytically for an arbitrary
potential *V*(), the
inverse problem of determining *H* once
is known is exactly
solvable ! We demonstrate this by first performing
a change of variables
*t*
*a*, *d* / *dt*
*aHd* /
*da* which
reduces (120) to a first order linear differential equation
for *H*^{2}:

(121) |

Equation (121) has the exact solution

(122) |

where ' =
*d* /
*dz*. Setting *z* = 0 in this expression, we
arrive at a very interesting relationship between
_{0} and
(*z*):

(123) |

Substituting this relationship in (122) we finally obtain

(124) |

Clearly knowing and
' we can determine
*H*(*z*) and hence *V*(*z*).
For very large *z*, *z* >> 1, observational difficulties
make it unlikely that
will be known to great accuracy, at least
in the near future. However in this regime the flat matter dominated
solution
(1 +
*z*)^{-1} provides a very good approximation
since _{m}
1 for
*z* >> 1.

It should be pointed out that the above method of reconstructing the
-term potential
from observations
is complementary to that used to reconstruct the inflaton potential
[126].
Whereas the luminosity distance *d*_{L} or the growth rate
of the linearized density contrast
(*z*) can be used to
reconstruct *V*(),
the inflaton potential is reconstructed on the basis of
the primordial amplitude and spectrum of relic density perturbations and
gravity waves created during inflation (also see
[141]).