Next Contents Previous

8.4. Reconstructing the effective potential V(phi) for a time-dependent Lambda-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(phi) and phi 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 Lambda-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 dL, discussed in section 4.2 [186, 101]. From (28) we easily find

Equation 119 (119)

Thus the luminosity distance dL(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(phi)) and the equation of state wphi(z) in a model independent manner. (However, what is required for an unambiguous determination of V(phi) is the present matter density Omegam [186].) Formula (119) can also be used for an unambiguous determination of H(z) from the angular-size distance dA(z) introduced in section 4.5, if we use the relation dA(z) = dL(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

Equation 120 (120)

where the value of H is determined from (97). (On scales << 200h-1 Mpc the Lambda-field is practically unclustered and can be treated as a smooth component if |mphi2| ident |d2V / dphi2| img src="../../New_Gifs/ltapprox.gif" alt="ltapprox"> H02). Although it is not possible to solve (120) analytically for an arbitrary potential V(phi), the inverse problem of determining H once delta is known is exactly solvable ! We demonstrate this by first performing a change of variables t rightarrow a, d / dt rightarrow aHd / da which reduces (120) to a first order linear differential equation for H2:

Equation 121 (121)

Equation (121) has the exact solution

Equation 122 (122)

where delta' = ddelta / dz. Setting z = 0 in this expression, we arrive at a very interesting relationship between Omega0 and delta(z):

Equation 123 (123)

Substituting this relationship in (122) we finally obtain

Equation 124 (124)

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

It should be pointed out that the above method of reconstructing the Lambda-term potential from observations is complementary to that used to reconstruct the inflaton potential [126]. Whereas the luminosity distance dL or the growth rate of the linearized density contrast delta(z) can be used to reconstruct V(phi), 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]).

Next Contents Previous