Defining a scalar as a function of just time while working with xpand

[jo...@iisertvm.ac.in]

Hi
        I am trying to work with xpand. Here I am wondering how I'll define a Scalar as a function of only time. I am trying to use constant scalar field gauge in calculating      osmological perturbations in f(R) gravity. So I feel defining scalar field as a function of only time would be extremely useful. Please suggest me how it could be done.

Thanking Jose

[Cyril Pitrou]

Hi Jose!

If you look at the doc of xPand, it is not extremely explicit but you can define a projected tensor with DefProjectedTensor. In that function, there is an option to specify that the tensor lives on the background only. When this opton is specified, the tensor has only time derivatives because by construction, the background is homogeneous.

I rarely use DefProjectedTensor directly in examples because many tensor are kind of predefined by DefMatterFields and DefMetricFields which internaly call DefProjectedTensor to define the usual metric and matter fields.

But in the example 7 (see Example folder), I use directly DefProjectedTensor to define the various tensors that are used in the 1+3 splitting.

You have to look in the documentation by evaluating
?DefProjectedTensor

to see how this function works.

If you define a scalar filed Phi as
DefProjectedTensor[Phi[], h, PrintAs -> "Phi",  SpaceTimesOfDefinition -> {"Background"}]
then it should have only time derivative and no perturbations.

That is \Phi[LI[1],LI[0]] = 0 (it is the first perturbation) but \Phi[LI[0],LI[1]] is not 0 because it is the first time derivative on the background.

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Alternatively you could work with the predefined fields which are defined by DefMatterFields. In those there is a scalar field, (phi[]). You can look at the examples on Friedmann-Lemaitre, example 4 in the Example fodler, as there is a section with a scalar field which is used in a Klein-Gordon equation.

You could use that predefined scalar field, and add the rule that this scalar field is 0 when perturbed.
You add something like

phi[LI[n_],LI[m_]:=/;n>0;

and that would kill all the perturbations of the scalar field.
Do not hesitate to ask me more if you need, and if xPand is useful to you you need to cite it in the acknowledgements of your paper.

Best,

Cyril

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