How Is the Scale Factor R(t)/R(0) Calculated in Cosmology?

[mrjaffa]

Homework Statement

The energy density of the universe for radiation, matter and cosmological constant have changed over the years and there was a time (t), when it was equal for matter and radiation.

Assuming the universe is 13.7 billion years old, estimate R(t) / R(0) where R(0) is the scale factor for now.

The question says to use information given in the textbook. In the textbook, we are told that R(t) was approximately around a few time 10⁴ years old. We are also given a graph where it plots energy densities over scale factor (R(t)/R(0)). So on the graph, it shows the point where energy density for matter and radiation was equal : R(t) / R(0) = 10^-4 and it also says it was this in the text.

Homework Equations

R(t) / R(0) where R(0) is now which I presume is 1.

The Attempt at a Solution

So... I'm already given the answer in the textbook right? It's 10^-4. I'm getting confused I think as Rt is essentially the same as Rt / R0 as R0 is 1. So I don't see the point in Rt / R0 if you're just divinding by 1.

It doesn't however show us how it calculated Rt to get 10^-4. R is a scale factor, essentially the distance between two cosmic coordinates. So R0 is this scale now which equals 1 and Rt is billion of years ago when the universe was much much smaller so 10^-4 makes sense. But how do they get this number? I thought as we were told the age of the universe now and Rt to be around 30000 years old, it was just 30000 / 13 billion but that doesn't give me 10^-4.

Thanks for any help guys.

 

[wabbit]

R(0) is probably not normalized to 1 in your book, so R(t)/R(0) is the relevant ratio - but you can certainly normalize to R(0)=1 and use just R(t) instead.
As to the relation with when matter and energy densities were equal, you need to look at how energy density and matter density scale with R, and how their ratio does.

 

[mrjaffa]

Thanks for your reply.

I've been told that we just the value in the book of 10^-4 for R(t).