Calculating the Gravitational wave spectrum with Inflation as a source

[shinobi20]

TL;DR Summary

I am interested in knowing how to calculate the gravitational wave (GW) spectrum with inflation as a source, I have some background in inflation but I am not so familiar about calculating the GW spectrum.

I am interested in knowing how to calculate the gravitational wave (GW) spectrum with inflation as a source, I have some background in inflation but I am not so familiar about calculating the GW spectrum. I am reading a paper (https://arxiv.org/abs/0804.3249) about it, however, a big part of it is still over my head. I have some basic background in Mathematica so I plan to use it to plot the spectrum.

Based on what I read, I must simultaneously calculate some differential equations (dynamical equations of density, Hubble parameter, etc) to produce a value for a certain parameter, in this case, $H$ (Hubble parameter) and then repeat the process for the perturbed value $h_k$.

Another problem that came to mind is that how can I do the iteration in Mathematica, say, since $H$ will evolve through time, and $h_k$ has a differential equation containing $H$, how do I calculate the data points of $h_k$ so as to plot the GW spectrum.

BTW, $\Omega \propto h_k^2$.

Can anyone give me any advice on this? Also does anyone know of any tutorial (whether websites, papers, etc) that can give me more knowledge on this? Any help to point me in the proper direction would greatly help! Thanks!

 

[kimbyd]

I haven't looked at the calculations for this in some time, but the essence of the theory is as follows (from memory):

1) Inflation itself generates isotropic perturbations with comparable amounts of tensor and scalar perturbations.
2) As the universe expands during and after inflation, those perturbations evolve over time. First through super-horizon evolution. When the perturbations become smaller than the cosmological horizon, their evolution changes and they start oscillating.

The difference between the tensor and scalar perturbations is that the tensor perturbations are not amplified by gravitational attraction, so they largely remain the same amplitude they were to start, while the scalar perturbations will have grown significantly.

So in essence, most of what you need in order to do the calculation is already performed for calculating the scalar perturbations. You just need to look up how the scalar perturbation evolution differs from the tensor perturbation evolution.

 

[shinobi20]

I haven't looked at the calculations for this in some time, but the essence of the theory is as follows (from memory):

1) Inflation itself generates isotropic perturbations with comparable amounts of tensor and scalar perturbations.
2) As the universe expands during and after inflation, those perturbations evolve over time. First through super-horizon evolution. When the perturbations become smaller than the cosmological horizon, their evolution changes and they start oscillating.

The difference between the tensor and scalar perturbations is that the tensor perturbations are not amplified by gravitational attraction, so they largely remain the same amplitude they were to start, while the scalar perturbations will have grown significantly.

So in essence, most of what you need in order to do the calculation is already performed for calculating the scalar perturbations. You just need to look up how the scalar perturbation evolution differs from the tensor perturbation evolution.

When the perturbations become smaller than the cosmological horizon, why should their evolution start oscillating? Can that be shown as a solution of some DEs?

 

[kimbyd]

When the perturbations become smaller than the cosmological horizon, why should their evolution start oscillating? Can that be shown as a solution of some DEs?

It's definitely examined rigorously in the actual calculations, but I don't remember the details. I do remember the concept of what is going on: if the wavelength is larger than the cosmological horizon, then it would require communication faster than the speed of light for matter from the peak to communicate with matter on the trough. Such waves do evolve over time, but do so in a rather different way from waves which are smaller than the horizon.