Calculating Electron Number Density in Early Universe

[jc09]

Homework Statement

The present number density of electrons in the Universe is the same as that of protons, about
0.2m−3. Consider a time, long before the formation of atoms and CMB, when the scale factor
was one millionth of its present value. What was the number density of electrons then ? Given
that the rest mass of the electron is me = 9.1 × 10−31 kg, were electrons relativistic or nonrelativistic
at that time

Homework Equations

The Attempt at a Solution

I don't know the equation that can link these so I'm stuck, can anyone help?

 

[mark726]

Hello! Thank you for your question. Let's break down the problem and see if we can come up with a solution.

First, we are given the present number density of electrons in the Universe, which is 0.2m^-3. This means that for every cubic meter of space, there are 0.2 electrons.

Next, we are asked to consider a time when the scale factor was one millionth of its present value. This means that the Universe was much smaller at this time, so the number of electrons in a given volume would have been much smaller as well.

To find the number density of electrons at this time, we can use the fact that the total number of electrons in the Universe is conserved. This means that the number density of electrons at any time is proportional to the volume of the Universe at that time.

We can express this relationship as:

ne(t) = ne(0) * (a(t)/a(0))^3

Where ne(t) is the number density of electrons at time t, ne(0) is the present number density of electrons, and a(t) and a(0) are the scale factors at time t and the present time, respectively.

Since we know that the scale factor at this time was one millionth of its present value, we can plug this into the equation to find the number density of electrons at this time:

ne(t) = 0.2m^-3 * (1/10^6)^3 = 0.2 * 10^-18m^-3 = 2 * 10^-19m^-3

Next, we are asked to determine if the electrons were relativistic or non-relativistic at this time. To do this, we can use the fact that the electrons are non-relativistic if their kinetic energy is much smaller than their rest mass energy, and relativistic if their kinetic energy is much larger.

We can express this as:

Kinetic energy = 1/2 * m * v^2

Rest mass energy = m * c^2

Where m is the mass of the electron, v is its velocity, and c is the speed of light.

At this time, the electrons would have been moving very quickly due to the high temperatures and densities of the Universe. To determine if they were relativistic or non-relativistic, we can compare the kinetic energy to the rest mass energy.

If we assume that the velocity of