Slow Roll Parameters: Deriving and Computing

[AlphaNumeric]

Not sure if this is a diff geom. question or more appropriate for the strings forum or even relativity or cosmology.

I'm doing work involved in inflationary models for compact spaces and the two important quantities are the slow roll parameters $$\epsilon$$ and $$\eta$$. Previously I've been using the definitions described by Quevado et al. in this paper (Equations 2.12 to 2.16) but after a discussion about such things with someone far more knowledgeable about this whole area than me, I've been informed such algebraic expressions might not be true.

How would I go about deriving slow roll parameters for a potential surface (typically in complex fields)? I've checked the references of that paper but they just state the formula. Seems to be a normalised "rate of change" (ie epsilon) and "how sharp is the turning point in the potential?" (ie largest negative eigenvalue, eta) but often there's disagreement in how to compute things like the Hessian depending on the sign of the potential etc.

I just don't want to spend a month doing work on volumes of parameter space which lead to viable inflation only to find I'm using the wrong definition!

If this is more appropriate for one of the more physics based forums rather than this forum can a mod please move it

 

[Haelfix]

Im not really sure what his objection is to that statement, afaik its pretty standard, at least in the textbook inflationary models usually studied. Perhaps an expert on inflationary model building can chime in (prolly in the cosmology forum)

Physically I view the slow roll approximation as assuming the magnitude of the second time derivative in phi (your inflaton field) is irrelevant w.r.t to drag terms (usually constant * first derivative in phi) as well as dV/dphi. Or that the potential must be sufficiently flat enough (eg small derivatives) so that the field rolls slowly enough for inflation to occur.