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 [tex]\epsilon[/tex] and [tex]\eta[/tex]. 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.

 

[tacman]

Thank you for sharing your question and concerns about the derivation and computation of slow roll parameters in the context of inflationary models for compact spaces. This is definitely a topic that falls within the realm of differential geometry and is relevant to the study of strings, relativity, and cosmology.

Firstly, it is important to note that there is no one "correct" definition of slow roll parameters, as different researchers may use slightly different definitions depending on their specific context and goals. However, there are general guidelines and principles that can help guide the derivation and computation of these parameters.

One approach to deriving slow roll parameters is to start with the definition of the potential surface in complex fields and use mathematical techniques such as the Hessian (as you mentioned) and other tools from differential geometry to analyze the behavior of the potential at different points. This can help identify the rate of change of the potential and the sharpness of the turning points, which are key components of the slow roll parameters.

Another approach is to use the equations of motion for the fields and the Friedmann equations to derive the slow roll parameters. This approach is more common in the context of inflationary models and can provide a deeper understanding of the physical implications of these parameters.

In either case, it is important to carefully consider the assumptions and approximations made in the derivation and to compare your results with those of other researchers in the field. This can help ensure that your definitions and computations are in line with current understanding and can also help identify any discrepancies or disagreements.

In summary, the derivation and computation of slow roll parameters involves a combination of mathematical techniques from differential geometry and physical principles from inflationary models. It is always a good idea to consult with experts in the field and to carefully compare your results with those of others to ensure accuracy and consistency. I hope this helps and good luck with your research!