need help expanding three-metric - Cosmology

[Gabriel Sampaio]

Hi,

I’m very new to xAct and xPand, and I’d like to describe perturbations around a spatially flat background cosmology in the comoving-synchronous gauge.

In this gauge, the fluid four-velocity is
uμ​=(a,0,0,0),
where a is the scale factor. Perturbations are restricted to the spatial three-metric γij​, so that the line element is
ds2=a2(η)[dη2+γij​dxidxj].

I would like to perturb γij​ to first order in the scalar perturbations ψ and χ:
γij​=(1−2ψ)δij​+(∂i​∂j​−31​∇2)χ.

My goal is to obtain the three-Ricci tensor associated with this spatial metric, expressed in terms of ψ and χ, but I haven’t been able to do so.

Any help would be greatly appreciated.

Thanks!

[Cyril Pitrou]

Hi,

I have made a small notebook which explains how to obtain perturbation equations in a synchronous gauge.

The one you want to use is not the default one in xPand so I also added a section to obtain results with your gauge choice.

If you are interested only in the Ricci of the spatial metric, I think you can just put the Hubble factors to 0 and the scale factor to 1, and this should give you the result.

Regards,

Cyril

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