# Calculating the Gravitational wave spectrum with Inflation as a source

• A
• shinobi20
In summary, the calculation of the gravitational wave (GW) spectrum with inflation as a source involves simultaneously calculating differential equations for density, Hubble parameter, and other parameters. The evolution of the perturbations over time is key, with the scalar perturbations growing significantly while the tensor perturbations remain relatively constant. The evolution of the perturbations changes when they become smaller than the cosmological horizon, resulting in oscillations. This can be shown as a solution of differential equations, which are rigorously examined in the actual calculations. Further information on the scalar and tensor perturbation evolution can be found through research and tutorials.
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!

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.

kimbyd said:
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?

shinobi20 said:
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.

## 1. What is the significance of calculating the gravitational wave spectrum with inflation as a source?

The gravitational wave spectrum refers to the range of frequencies at which gravitational waves are emitted. Inflation, a period of rapid expansion in the early universe, is believed to be a source of gravitational waves. By calculating the gravitational wave spectrum with inflation as a source, we can gain a better understanding of the early universe and potentially confirm the theory of inflation.

## 2. How is the gravitational wave spectrum with inflation as a source calculated?

The calculation of the gravitational wave spectrum with inflation as a source involves using mathematical models and equations to predict the expected frequency and amplitude of gravitational waves produced during inflation. This requires knowledge of the properties of inflation, such as its energy density and duration, as well as the properties of gravity.

## 3. What are some potential applications of understanding the gravitational wave spectrum with inflation as a source?

Understanding the gravitational wave spectrum with inflation as a source can have various applications in the field of cosmology. It can help us better understand the early universe and the processes that led to its formation. It can also provide insights into the nature of gravity and potentially lead to the development of new theories or models.

## 4. What challenges are involved in calculating the gravitational wave spectrum with inflation as a source?

One of the main challenges in calculating the gravitational wave spectrum with inflation as a source is the lack of direct observational evidence. Inflation is a theoretical concept, and while there is some evidence to support it, it has not been directly observed. This means that the calculations rely heavily on mathematical models and assumptions, which may not accurately reflect the true nature of inflation.

## 5. How does the calculation of the gravitational wave spectrum with inflation as a source contribute to our understanding of the universe?

The calculation of the gravitational wave spectrum with inflation as a source is an important step towards understanding the early universe and the fundamental laws of physics. It can help us test and refine our theories and models, and potentially uncover new insights into the nature of the universe and its origins. Additionally, it can contribute to the development of new technologies and advancements in the field of cosmology.

Replies
2
Views
1K
Replies
1
Views
1K
Replies
4
Views
2K
Replies
1
Views
2K
Replies
3
Views
2K
Replies
11
Views
2K
Replies
8
Views
2K
Replies
55
Views
4K
Replies
2
Views
2K
Replies
13
Views
3K