Problems deriving the action for cosmological perturbations using xPand

[Paul Mezgolits]

Hello,

I have been trying to derive the actions for the perturbations from the Einstein-Hilbert action but encounter some problems. I have tried two different approaches one produces the correct results the other dose not.

The approach that works is the following: I set up xPand to do Perturbation around Minkowski spacetime and perturb the action to second order. then i do partial integration to get the desiered form. Then i do the conformal transformation from Minkoswki to a Flat FRW spacetime by hand (which for the case at hand amounts to multiplying the previously obtained term by a^2). This gives the correct action.

However when I set up xPand to perturb around an FRW spacetime I get (after partial integration) the desiered expression + a term I can't seem to get rid of.

I have included a notebook where i have done both approaches and commented how i have done it.
i would appriciate any comments and advice on why the two approaches do not produce the same result.

thanks for your time and effort
paul

[Cyril Pitrou]

Hi Paul,

In fact both results are correct. By not perturbing the Lagrangian of matter, you assume implicitely that H^2+2H'=0.

If you consider the lagrangian of a Scalar Field for instance, then there is a term coming from the variation of the determinant of the metric times the Lagrangian. So this gives an extra E_ij E^ij P, where P is the pressure of the field.

Then the identity you should use is then something  like H^2+2H'+P=0, and again the extra term disappears.

Another solution is to consider a parameterzation of the perturbation of the form   Exp[E_ij]. That way since E_ij is traceless, this perturbation conserves the determinant of the metric, and you never have to care about the Lagrangian of matter. This is what has been done by Jonghee Kang.

I attach a version of his notebook which does that.
The advantage on top of that is that the integrations by parts on the perturbed action are not even needed with this parameterization.

Best,
Cyril

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[Paul Mezgolits]

Hi Cyril,

Thank you for answering my question. Unfortunately this is a very busy weekend and i have not had time to think about your suggestions in great detail yet.
As a first reaction let me state that i think you are probably right, i was however under the impression that i should be able the derive what i have in my original post considered to be the correct expression on a flat FRW spacetime as long as i do not use the equations of motion. i have come to this conclusion after reading parts of Mukhanov, Feldman und Brandenbergers review article on cosmological perturbation theory. It is possible that i have read this too superficially though.

i hope to have some time to think about this in greater detail on Monday.

greetings
Paul

[Paul Mezgolits]

I'm sorry it took me so long to get back to this. Christmas got in the way.

I have finally gotten around to trying the suggested approach in the simplest case i could think of: perturbation theory around a dS background with a cosmological constant in the action. Using the equations of motion the terms i had trouble with vanish and only the result I expected remains.

I have attached a Notebook outlining what i have done, in case people are interested in the resolution of my problem.

I would once more like to thank for your help!

Greetings
Paul