Questions about the xPand package

[François Rose]

Hi,

I am using xPand for my research in cosmology. I have the following questions:

1.- For a FRW Flat metric the background metric is given, by default in xPand, as

ds^2 = a(t)^2(-d\tau^2+\delta_{ij} dx^i dx^j )

in conformal time.  The question is, is there a, somewhat easy, way introduce a metric in the background as follows

ds^2 = a(t)^2(-N(t)^2 d\tau^2+\delta_{ij} dx^i dx^j )

In which   N(t) is the lapse function? I am aware that the lapse function is introduced into the package, but it is not clear to me how can it be implemented in the background metric as I mentioned above. 

2.- I would like to know if the gauge "ScalarFieldComovingGauge" is what is sometimes know in the literature as the unitary gauge. That is the gauge in which the scalar field perturbation vanishes?

3.- The first order metric perturbation in the "ScalarFieldComovingGauge" is given by

ds^2 = -(1+2\phi)dt^2 +2a\partial_iB dt dx^i +a(t)^2(h_{ij} (1-2\psi)dx^i dx^j )

if the vector and tensor metric perturbations are set to vanish. I would like to know if there is a way to rewrite it in the following way

ds^2 = -(1+2\phi)dt^2 +2\partial_iB dt dx^i +a(t)^2(h_{ij} (1+2\psi)dx^i dx^j ),

that is with no scale factor in the non-diagonal terms and with a plus sign in front of the \psi scalar perturbation of the metric.

Regards,

François

[Cyril Pitrou]

Hi,

I am using xPand for my research in cosmology. I have the following questions:

Thanks a lot.
 

1.- For a FRW Flat metric the background metric is given, by default in xPand, as

ds^2 = a(t)^2(-d\tau^2+\delta_{ij} dx^i dx^j )

in conformal time.  The question is, is there a, somewhat easy, way introduce a metric in the background as follows

ds^2 = a(t)^2(-N(t)^2 d\tau^2+\delta_{ij} dx^i dx^j )

No. xPand is based on a geometrical splitting of the background (and a conformal transformation to handle the scale factor). So what is defined is the normalized time like vector, say n^\mu, which corresponds roughly to d\tau or N(t) d\tau.
What appears as a time derivative is a Lie derivative in the direction of this timelike vector.

What is maybe possible to do, is to massage the final results with some rules to transform time derivatives. This is what is already done when you set the option to express the results in terms of cosmic time (N(t)=1/a(t)).
Indeed what you have is to use d\tau = N() d\tau' so any time derivative wrt to tau is expressed as a time derivative wrt to tau'. But nothing is built in for that.
 

In which   N(t) is the lapse function? I am aware that the lapse function is introduced into the package, but it is not clear to me how can it be implemented in the background metric as I mentioned above. 

2.- I would like to know if the gauge "ScalarFieldComovingGauge" is what is sometimes know in the literature as the unitary gauge. That is the gauge in which the scalar field perturbation vanishes?

Yes. The name is bad. It should be named Unitary gauge.

 

3.- The first order metric perturbation in the "ScalarFieldComovingGauge" is given by

ds^2 = -(1+2\phi)dt^2 +2a\partial_iB dt dx^i +a(t)^2(h_{ij} (1-2\psi)dx^i dx^j )

This is True in cosmic time yes.
 

if the vector and tensor metric perturbations are set to vanish. I would like to know if there is a way to rewrite it in the following way

ds^2 = -(1+2\phi)dt^2 +2\partial_iB dt dx^i +a(t)^2(h_{ij} (1+2\psi)dx^i dx^j ),

that is with no scale factor in the non-diagonal terms and with a plus sign in front of the \psi scalar perturbation of the metric.

For the sign, there are three ways.
First, you can hack directly the code of xPand where the perturbation of the metric is defined.

Or second in your final results you can use a replacement rule psi->-psi.

Or third, you use the method of xPand where you define by yourself the set of rules for the perturbation of the metric which are given to the function SplitPerturbation when handling expression, instead of using the function ToxPand which does everything without control of the user. In the documentation, I think there are some examples where this more low level method is used. If you do not see how to do it I can help you next week.

For the absence of scale factor in front of the g_0i term this again can be handle by this low level method where you define by yourself the perturbation of the metric. You would just need to divide by the scale factor the correspond part of the perturbation. Or again alternatively you can massage the final results since it amounts to the replacement B-> B/a.

I attach a notebook where I use the method where I redefine the needed rule and use it in SplitPerturbations.

Ask me if you have further questions!

Cyril

 

Regards,

François

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