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

In summary: Ok thanks.I don't 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
  • #1
ChrisJ
70
3

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)[/B]

Homework Equations


##\rho = \rho_0 (\frac{a_0}{a})^{3(1+w)}## ??[/B]

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:[/B]
[tex]
\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
[/tex]
Any help/advice is much appreciated :)
 
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  • #2
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.
 
  • #3
Orodruin said:
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 couldn't 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 can't believe 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:
[tex]
\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}
[/tex]

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 :)
 
  • #4
After going through my notes, I found this equation.

[tex]
\frac{a}{a_0} = t^{\frac{2}{3(1+w)}}
[/tex]

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"
 
  • #5
Again your equation is dimensionally inconsistent. It is not an equality but a proportionality.
 
  • #6
Orodruin said:
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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  • #7
edit: made mistake
 
Last edited:
  • #8
You have the ratio of two scale factors (dimensionless) equal age^[1/(1+w)] (dimension time^[1/(1+w)]).
 
  • #9
Orodruin said:
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?
 
  • #10
ChrisJ said:
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.
 
  • #11
Orodruin said:
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.
 
  • #12
ChrisJ said:
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.
 
  • #13
Orodruin said:
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 referring 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)}}## ?
 
  • #14
ChrisJ said:
EDIT: I was not referring 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.
 
  • #15
Orodruin 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.

Ok thanks.

I don't 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..
[tex]
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 \\
[/tex]
From before, ##\frac{a_{eq}}{a_{0}} = 10^{-4}##
[tex]
\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
[/tex]
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.
 

1. What is the significance of finding the age of the universe when matter and radiation densities were equal?

Finding the age of the universe when matter and radiation densities were equal is crucial in understanding the evolution of the universe. This event, known as the matter-radiation equality, marks an important transition in the history of the universe and provides valuable insights into the early stages of its formation.

2. How is the age of the universe when matter and radiation densities were equal calculated?

The age of the universe when matter and radiation densities were equal is calculated using the Friedmann equation, which relates the expansion rate of the universe to its density. By equating the density of matter and radiation at a certain point in time, we can determine the age of the universe at that moment.

3. What is the current estimated age of the universe when matter and radiation densities were equal?

According to current estimates, the age of the universe when matter and radiation densities were equal is approximately 380,000 years after the Big Bang. This is known as the cosmic microwave background (CMB) era, as it is the time when the universe became transparent to light.

4. How does the age of the universe when matter and radiation densities were equal affect our understanding of the universe?

The age of the universe when matter and radiation densities were equal provides important constraints on theories of the universe's formation and evolution. It allows us to better understand the distribution of matter and energy in the universe and the processes that shaped its development.

5. What implications does the age of the universe when matter and radiation densities were equal have for future research?

Studying the age of the universe when matter and radiation densities were equal can help us make predictions about the future of the universe, such as its ultimate fate and the potential existence of dark energy. It also provides a starting point for further research into the early stages of the universe and its fundamental properties.

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