Inflaton and the boltzmann equation's density/velocity sources

[twoform]

Hi,

I'm looking for a resource which explains how the inflaton (and metric) perturbations are related to the CMB; in particular a lot of common sources (Dodelson in particular and some reviews I found on arxiv) simply state that the photon-baryon fluid has a 'monopole' (density perturbation) and a 'dipole' (velocity perturbation) term. The first is related to the CMB via dT/T = 1/3 \phi according to sachs-wolfe, but I can't find anyone who shows how to get the velocity perturbation term from the inflaton/metric perturbations.

Thanks for any help,
Dan

 

[AndyW]

Hi Dan,

The relationship between the inflaton and metric perturbations and their effects on the CMB can be quite complex and involves a lot of mathematical calculations. However, I can provide you with some key concepts and resources that can help you understand this relationship better.

Firstly, it is important to understand that the inflaton is a hypothetical scalar field that is thought to have driven the rapid expansion of the early universe during a period known as inflation. This inflationary period is believed to have left behind small perturbations in the inflaton field, which in turn affect the metric of the universe.

These perturbations in the inflaton field and the metric can be quantified using a mathematical framework known as cosmological perturbation theory. This theory allows us to calculate how the perturbations in the inflaton field and the metric evolve over time and how they affect the CMB.

One key concept to understand is that the inflaton perturbations and the metric perturbations are not directly related to the CMB. Instead, they affect the CMB through their impact on the photon-baryon fluid in the early universe. As you mentioned, the monopole and dipole terms in the photon-baryon fluid are related to the density and velocity perturbations, respectively.

To understand how the velocity perturbation term is related to the inflaton and metric perturbations, we need to look at the equations that govern the evolution of the photon-baryon fluid. These equations take into account the effects of gravity, which is influenced by the inflaton and metric perturbations. Through these equations, we can see how the velocity perturbations are affected by the inflaton and metric perturbations.

I would recommend looking into resources on cosmological perturbation theory, such as the book "Cosmological Perturbation Theory" by Valery Rubakov, for a more detailed explanation and mathematical derivation of this relationship. Additionally, there are many research papers available on arXiv that discuss this topic in depth.

I hope this helps guide you in your understanding of the relationship between inflaton and metric perturbations and the CMB. Best of luck in your research!