Calculating Redshift of Universe Domination

[Radiohannah]

Homework Statement

Hey :-)

I want to calculate the redshift at which the Universe became vacuum dominated, with given values for the energy densities of Omega_{m} and Omega_{Lambda}.

Homework Equations

I know that the scale factors

rho_{m} is proportional to a^{-3}
rho_{r} is proportional to a^{-4}
rho_{v} is proportional to a^{0}


I know that the relationship between the scale factor and the redshift is

(1+z) = a(t) / a_{0}(t)...(*)

and

H^{2} = h_{0}^2 [ Omega_{m}/a^{3} + Omega_{r}/a^{4} + Omega_{Lambda} + Omega_{k}/a^{2} ]

The Attempt at a Solution

I know that at early enough times the Universe was radiation dominated. I have plotted log rho vs log a for each contributor, and can see that the Universe was originally radiation dominated, then matter dominated, then vacuum dominated. I thought that if I calculated the point at which the matter and vacuum slopes intersected, I could solve for a, and then use equation (*), but I wouldn't know what to use for a_{0}. And if I did equate a^{-3} to a^{0}, I just get 1. And that wouldn't be using the given values for Omega_{Lambda} or Omega_{m}.

I'm not very confident in my approach and any advice would be warmly appreciated!

Thanks

 

[mark726]

for your post! Your approach is on the right track, but there are a few things you need to consider. First, the scale factor a in the equation (1+z)=a(t)/a0(t) is not the same as the scale factor used in the equation for energy densities. The scale factor in the energy density equation is the present scale factor, a0, while the scale factor in the redshift equation is the scale factor at a specific time in the past, a(t).

To calculate the redshift at which the Universe became vacuum dominated, you will need to solve for the scale factor a at the time when the matter and vacuum energy densities are equal. This can be done by setting the matter and vacuum energy densities equal to each other and solving for a. Once you have a, you can plug it into the redshift equation to calculate the redshift.

In order to use the given values for Omega_m and Omega_Lambda, you will need to first convert them to energy density values using the equations given. Once you have the energy density values, you can use them in the equation for H^2 to solve for a. Then, plug in the value of a into the redshift equation to calculate the redshift at which the Universe became vacuum dominated.

I hope this helps and good luck with your calculations!