# Find the Age of the Universe when Matter and Radiation densities were equal

#### ChrisJ

1. Homework Statement
Currently, the density of matter $\rho_0^M$ and of radiation $\rho_0^R$ have values of approximately $10^{-29}$ g/cm^3 and $10^{-33}$ g/cm^3 respectively.

Estimate the ratio of the cosmic scale factors $a_{eq}$ (scale factor at equality) and $a_0$ (scale factor now)

Hence obtain a rough estimate of the age of the universe at the time of matter-radiation equality. (Assume $t_0 = 10^{10}$ years)

2. Homework Equations
$\rho = \rho_0 (\frac{a_0}{a})^{3(1+w)}$ ??

3. The Attempt at a Solution
I may have found the ratio of scale factors, but I am almost certain it is incorrect, but whether it is or not I am a bit stuck on how to then get the estimate for the age of the universe .

I have done this so far:

$$\rho_{eq}^M = \rho_0^{M} (\frac{a_0}{a_{eq}})^{3} \\ \implies \frac{a_0}{a_{eq}}) = \sqrt[3]{(\frac{\rho_{eq}^R}{ \rho_0^{R}})} \\ \rho_{eq}^R = \rho_0^{R} (\frac{a_0}{a_{eq}})^{4} \\ \implies \frac{a_0}{a_{eq}}) = \sqrt[4]{(\frac{\rho_{eq}^R}{ \rho_0^{R}})} \\ \implies \sqrt[4]{\frac{\rho_{eq}^R}{ \rho_0^{R}}} = \sqrt[3]{\frac{\rho_{eq}^M}{ \rho_0^{M}}} \\ \therefore \frac{\sqrt[3]{\rho_0^M}}{\sqrt[4]{\rho_0^R}} = \frac{a_0}{a_{eq}} \\ \frac{\sqrt[3]{10^{-29}}}{\sqrt[4]{10^{-33}}} = 0.0383$$
Any help/advice is much appreciated :)

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#### Orodruin

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You are not solving your system of equations correctly. (You can see this simply from the fact that your final result is not dimensionally consistent.) For some reason you seem to be eliminating the very thing you should be computing.

#### ChrisJ

You are not solving your system of equations correctly. (You can see this simply from the fact that your final result is not dimensionally consistent.) For some reason you seem to be eliminating the very thing you should be computing.
Yeah I knew it was wrong, but earlier I just couldnt see a way to make progress with it. After having a long break and taking another look at it now, I managed to get the ratio of the scale factors straight away (at least I think so)! I cant beleive how I could have got it so wrong! Taking a break an coming back to a question, really does help haha.

Right now I have just done:
$$\rho_{eq}^M = \rho_0^{M} (\frac{a_0}{a_{eq}})^{3} \\ \rho_{eq}^R = \rho_0^{R} (\frac{a_0}{a_{eq}})^{4} \\ \rho_{eq}^M = \rho_{eq}^R \\ \therefore \rho_0^{M} (\frac{a_0}{a_{eq}})^{3} =\rho_0^{R} (\frac{a_0}{a_{eq}})^{4} \\ \frac{\rho_0^{M}}{\rho_0^{R}} = \frac{a_{eq}^3}{a_0^3} \times \frac{a_0^4}{a_{eq}^4} \\ \frac{\rho_0^{M}}{\rho_0^{R}} = \frac{a_0}{a_{eq}} \\ \therefore \frac{a_{eq}}{a_0} = \frac{\rho_0^R}{\rho_0^{M}} = \frac{10^{-33}}{10^{-29}} = \frac{1}{10000}$$

But how can I get an estimate of the age of the universe @ matter-radiation equivalency? I know I could relate it to the redshift, but I need the answer in units of time (years), it does say only a rough estimate is needed, not an exact answer.

Thanks :)

#### ChrisJ

After going through my notes, I found this equation.

$$\frac{a}{a_0} = t^{\frac{2}{3(1+w)}}$$

The only thing is, for matter dominated w=0 and for radiation w=1/3, but what about when matter and radiation are equal?

If I use w=0, then the age would be $t=10000^{\frac{3}{2}}=1$Myr which is more than a order of magnitude off, as from what I have read, the age was around 60000 years. Plus I have not used the "Assume $t_0=10^{10}$ years"

#### Orodruin

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Again your equation is dimensionally inconsistent. It is not an equality but a proportionality.

#### ChrisJ

Again your equation is dimensionally inconsistent. It is not an equality but a proportionality.
Ok now I am confused, the first time I knew I was wrong, but this time I have to admit I thought I had the estimate of the ratio of scale factors correct. Do you mean the equation in the relevant equations section? If so, then I didn’t know, its in the lecture notes/slides as an equality.

Thanks,

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#### Orodruin

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You have the ratio of two scale factors (dimensionless) equal age^[1/(1+w)] (dimension time^[1/(1+w)]).

#### ChrisJ

You have the ratio of two scale factors (dimensionless) equal age^[1/(1+w)] (dimension time^[1/(1+w)]).
Oh sorry, you are talking about the one with time in it. I thought you were referring to my calculation of the ratio of the scale factors. At least is that bit now correct?

#### Orodruin

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Oh sorry, you are talking about the one with time in it. I thought you were referring to my calculation of the ratio of the scale factors. At least is that bit now correct?
Yes. The time inference is not.

#### ChrisJ

Yes. The time inference is not.
Ok thanks. Is the final answer, seeing as the question is only asking for a rough estimate, then simply just 10,000 years? If so, I don't know what I was supposed to do with the information that $t_0~10^{10}$ years.

#### Orodruin

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Ok thanks. Is the final answer, seeing as the question is only asking for a rough estimate, then simply just 10,000 years? If so, I don't know what I was supposed to do with the information that $t_0~10^{10}$ years.
No, as I said, the equation you used to deduce this is dimensionally inconsistent.

#### ChrisJ

No, as I said, the equation you used to deduce this is dimensionally inconsistent.
I thought you said that I had calculated the ratio of scale factors correct as being 1/10000?

EDIT: I was not refering to the equation with time in at all, but never mind, it obviously isnt, as otherwise you would have said.

Do I need to use $a(t)=(\frac{t}{t_0})^{\frac{2}{3(1+w)}}$ ?

#### Orodruin

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EDIT: I was not refering to the equation with time in at all, but never mind, it obviously isnt, as otherwise you would have said.
Yes you were implicitly doing this by stating 10000 years. You cannot get a dimensionful number out of a single dimensionless one.

Yes, you need to relate the scale factors to the time today.

#### ChrisJ

Yes you were implicitly doing this by stating 10000 years. You cannot get a dimensionful number out of a single dimensionless one.

Yes, you need to relate the scale factors to the time today.
Ok thanks.

I dont know what value of w to use for when matter and radiation were equal, but if I use 0 then I still end up with 10,000 years. I've done..
$$a_{eq} = (\frac{t_{eq}}{t_0})^{\frac{2}{3}} = t_{eq}^{\frac{2}{3}} t_{0}^{\frac{-2}{3}} \\ a_{0} = (\frac{t_{0}}{t_0})^{\frac{2}{3}}= 1 \\$$
From before, $\frac{a_{eq}}{a_{0}} = 10^{-4}$
$$\implies t_{eq} = (10^{-4} t_{0}^{\frac{2}{3}})^\frac{3}{2}=(10^{-4})^\frac{3}{2} t_0 \\ t_{eq} = (10^{-4})^\frac{3}{2} 10^{10} = 10000 yrs$$
If I do the same but use w=1/3 then I end up with 100 years. But 10,000 is a closer estimate to the real value.

"Find the Age of the Universe when Matter and Radiation densities were equal"

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