Energy Density with Thomson scattering

[cosmoshadow]

Homework Statement

If n_e represents the electron density and the interaction between particles is the photons scattering off of the electrons through the Thomson cross section, how low must the electron density be to cross the universe with no scattering?

Homework Equations

The size of the universe is given by c*(Ho^-1)

The Attempt at a Solution

I have no clue how to approach this problem. Obviously the Hubble distance is in Mpc, but I do not know the significance of the Thomson cross section. Can anyone help me to understand this problem?

 

[Jhero]

Thank you for your interesting question. To answer your question, we first need to understand the concept of the Thomson cross section and its relation to electron density and scattering.

The Thomson cross section is a measure of the probability of a photon scattering off an electron. It is given by the equation σT = (8π/3)*(e^2/mec^2)^2, where e is the electron charge, me is the electron mass, and c is the speed of light. This cross section is important in understanding the scattering of light by charged particles, such as electrons.

Now, to determine the electron density needed for no scattering to occur, we need to consider the size of the universe and the distance that a photon can travel before scattering occurs. The size of the universe is given by c*(Ho^-1), where c is the speed of light and Ho is the Hubble constant. This distance is known as the Hubble distance and is approximately 4.4 gigaparsecs (Gpc).

Using this information, we can set up an equation to determine the electron density needed for no scattering to occur. Let's call this density ne. We know that the distance a photon can travel before scattering is directly proportional to the electron density and inversely proportional to the Thomson cross section. So, we can write:

c*(Ho^-1) = (1/ne)*(1/σT)

Solving for ne, we get:

ne = (σT*c)/(Ho^-1)

Substituting in the value of the Thomson cross section and the Hubble constant, we get:

ne = (8π/3)*(e^2/mec^2)^2 * c * Ho

Plugging in the values for the constants, we get an electron density of approximately 3.6 x 10^-7 electrons per cubic meter. This is an incredibly low density, and it is unlikely to be found in the universe. Therefore, we can conclude that it is not possible for a photon to cross the universe without any scattering.

I hope this helps to clarify the problem and its solution. If you have any further questions, please don't hesitate to ask.