3. Adiabatic approximation
In this section we briefly comment on some results about the time-dependent mass term (16) of Eq. (13). The detailed analysis that justifies our assertions is carried out in Buchert et al. 1997.
For equations of state of the form , the time-dependent mass term (16) is identically zero for flat FRW backgrounds, but evolves with time in closed and open FRW backgrounds, keeping a constant sign during the epoch of interest: in open backgrounds and when the background is closed.
There are two time scales in our problem: an intrinsic time scale associated with the dynamical evolution prescribed by Eq. (13), which is determined by the values of the coefficients in it, and an extrinsic time scale associated with the background expansion, which is determined by the parameters defining the background cosmology. A numerical study of the function for the physically interesting range of parameters, reveals that its relative variation over these two time scales is small, which justifies an "adiabatic" approximation: we can assume to be a constant, rather than a function of time, when performing the Renormalization Group analysis in the next section. This assumption greatly simplifies the analysis, and the presence of a non-zero mass term in the KPZ equation results in a much richer renormalization group trajectory flow and fixed point structure than that corresponding to the massless case, as we will see below.
© European Southern Observatory (ESO) 1999
Online publication: March 10, 1999