Tensor perturbation around FRW metric.

[Jonghee Kang]

Hello, I am quite new to xTensor package and have a few questions.

I am trying to calculate Ricci and Riemann Tensors of tensor perturbations around FRW.

So, the metric that I am using is g_{00} = -1, g_{i0}=g_{0i}=0, g_{ij}=a(t)^2*exp(\gamma_{ij}) where \gamma_{ij} is traceless and transverse.

Since I want the expressions up to quadratic order in \gamma, let me write g_{ij}=a(t)^2*(\delta_{ij} + \gamma_{ij} + 1/2 \gamma_{il}\gamma_{lj} )

In my notebook, first section defines proper packages and second section defines manifolds, metrics and basis. 

Please notice that I define two manifolds ; one is 3-dim spatial one and the other is 1-dim time one. And then, I merge these two into 3+1 dim manifold.

In third section, I used "xCoba" to assign specific FRW background metric to my "metricg". By doing this and using 'xCoba' further, 

I could calculate RicciCd and RicciScalarCd of unperturbed FRW background metric.

Now, it is time to consider perturbation. First, I defined a tensor on spatial manifold and make it traceless and transverse using

AutomaticRules[\[Gamma], MakeRule[{\[Gamma][i, -i], 0}, PatternIndices -> All, MetricOn -> All]]

AutomaticRules[\[Gamma], MakeRule[{PD[-i]@\[Gamma][i, -j], 0}, PatternIndices -> All,  MetricOn -> All]]

Here is my first question. are these two rules correct ones to make traceless and transverse?

Let me go on. After this, I defined the metric perturbation around 'metricg' which is set to FRW background metric using 'xCoba'.

By definition, perturbation is a tensor in 3+1 dim manifold. What I am doing is to consider only spatial tensor perturbation, so I make a few rules which are following

Flatten[{MakeRule[{pert[LI[1], -i, -j], s[t[]]^2* \[Gamma][-i, -j]}, MetricOn -> None], 

  MakeRule[{pert[LI[1], i, j], \[Gamma][i, j]/s[t[]]^2},  MetricOn -> None]}]

Tensor1 = pert[-a_?Tangenttime`Q, -b_?Tangenttime`Q] :> 0

Tensor2 = pert[-a_?Tangenttime`Q, -j_?TangentM3`Q] :> 0

Tensor3 = pert[-j_?TangentM3`Q, -a_?Tangenttime`Q] :> 0

Tensor4 = pert[a_?Tangenttime`Q, b_?Tangenttime`Q] :> 0

Tensor5 = pert[a_?Tangenttime`Q, j_?TangentM3`Q] :> 0

Tensor6 = pert[j_?TangentM3`Q, a_?Tangenttime`Q] :> 0

Here is my second question. are these rules okay?

Now, I calculated the perturbation of Ricci tensor using "Perturbed[RicciCd[-a, -b], 2] // ExpandPerturbation //  CovDToChristoffel // ChristoffelToMetric".

But, it gives just abstract expression in terms of "metricg" and "pert". What I want is the expression in terms of a[t] which is a scale factor and \gamma_{ij} which is a tensor perturbation. How can I do this?

Thanks in advance.

[Leo Stein]

Hi Jonghee,
For cosmological perturbation theory, a huge amount of work has already been done by Cyril Pitrou, Xavier Roy, and Obinna Umeh in the xPand package. I would recommend using their package so that you don't have to re-build the same tools that they've already built!
The xPand home page is here: http://www2.iap.fr/users/pitrou/xpand.htm
If you are so inclined, the latest version of their code can also be found on github, as part of xAct-contrib: https://github.com/xAct-contrib/xPand

Best

Leo

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[Jonghee Kang]

Thank you for your answer.

I have already looked into xPand package. But, to me, it seems that xPand is good at dealing with scalar perturbation.

What I am doing now is to ignore all scalar and vector perturbations and include only traceless and transverse tensor perturbations in the language of SVT decomposition.

So, differentiating time and spatial coordinate is crucial and I should replace a generic metric perturbation h_{ab} into \gamma_{ij}. 

I still don't know how to do this. 

[Leo Stein]

Hi Jonghee,

xPand does know how to do scalar, vector, and tensor perturbations (in fact it can handle tensors with an arbitrary number of indices). Look at some of the notebooks in xPand/Examples/ to see the SVT decomposition arise. In particular, the metric tensor perturbations are named Eth[LI[p], LI[q], a, b] for the qth time derivative of the pth order perturbation with indices (a,b). For details of their convention, see Sec. IIIB, Eqs. (37-39) of the xPand paper (http://arxiv.org/abs/1302.6174).

Best,

Leo

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[Jonghee Kang]

Okay, now I see your point and I use xPand to compute something. 

However, it gives almost correct but slightly wrong answer, which I don't understand.

In the attached notebook, I compute the perturbation of Ricci scalar. Second last line of this notebook is the result of this.

We know that quadratic Ricci scalar for tensor mode contains \dot{E_{ab}} \dot{E^{ab}} for time derivative.

However, at the first glance, I am not sure whether we would end up with this term. 

I tried applying CovDToChristoffel and ChristoffelToMetric, but it just made the result more formidable.

Is there any better way to simplify the result in a tractable way in terms of only ordinary derivatives, scale factor and E_{ab} ?

I really appreciate your help so far.

[Jonghee Kang]

Oops, sorry about the multiple emails too... I forgot to attach my notebook. Here it is. 

[Cyril Pitrou]

Hi Jonghee,

I have looked at your notebook, and I do see terms which are quadratic in the derivative of the tensor.
I have modified your notebook (see attached), so as to consider perturbation which are only of the type you mentionned (that is only the spatial part is perturbed, and it is of the form Exp[ 2 E_ij ] where E_ij is traceless and transverse).

The usual gauge choices in xPand do not consider this exponential form so I had to build the rule and not use SplitMetric, but this is very easy.

You can then add vectors and scalars if necessary like in the Synchronous gauge.

In conformal time the time derivatives are written with a prime and in cosmic time with a dot.

Please do not hesiatte to ask further questions.

Best,
Cyril

On 21 October 2014 04:04, Jonghee Kang <jghe...@gmail.com> wrote:

Oops, sorry about the multiple emails too... I forgot to attach my notebook. Here it is. 

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[Jonghee Kang]

Hi Cyril,

First of all, thank you so much.

I have a further question. Eth tensor you used is transverse with respect to covariant derivative cd.

However, in my case, it should be transverse with respect to ordinary derivative. 

At this point, actually, I am quite confused about carrying or not carrying the scale factor a(t). 

We have the induced metric 'h'. Is it just flat space one or FRW space one?

If it is flat space one, the covariant derivative cd is identical to ordinary partial derivative. But, in this case, the final answer does not look like the desired Ricci Scalar.

I know that Ricci scalar should be " 12H^2 + 6 \dot{H} + \dot{E_{ab}} \dot{E^{ab}} - 1/a^2 \partial_c E_{ab} \partial^c E^{ab} ".