How to calculate CBR lambda vs age of universe?

[Fire Tech]

hello,

I'm wondering how to calculate, even approximately for the relatively mature universe, what the CBR wavelength was as a function of age of the universe. I get that the CBR began during recombination, though have read different values for the black body temperature at that recombination time from H ionization value 13.6 eV, to about 3000K. Those give starting wavelength for CBR of 91nm and 965nm respectively, though have read it started from 400nm to 800nm elsewhere. Currently, the CBR is 1.8mm lambda. But, what was it a billion years ago, or 5 billion years ago? Does it just vary linearly with age of universe such that the start was close to zero, so as a good (simple) approximation I could just take the current wavelength 1.83mm * (12.7 / 13.7) = 1.69 mm? for the CBR value a billion years ago (12.7 and 13.7 being the age of universe a billion years ago, or today, respectively.

Thanks,
rt

 

[Some Slacker]

hello,

I'm wondering how to calculate, even approximately for the relatively mature universe, what the CBR wavelength was as a function of age of the universe. I get that the CBR began during recombination, though have read different values for the black body temperature at that recombination time from H ionization value 13.6 eV, to about 3000K. Those give starting wavelength for CBR of 91nm and 965nm respectively, though have read it started from 400nm to 800nm elsewhere. Currently, the CBR is 1.8mm lambda. But, what was it a billion years ago, or 5 billion years ago? Does it just vary linearly with age of universe such that the start was close to zero, so as a good (simple) approximation I could just take the current wavelength 1.83mm * (12.7 / 13.7) = 1.69 mm? for the CBR value a billion years ago (12.7 and 13.7 being the age of universe a billion years ago, or today, respectively.

Thanks,
rt

I think you have a good starting point, better than I would have come up with for sure! But I don't see your calculation take into consideration the expansion of space and its apparent changes, I'm no scientist but it seems this would have to factor in somewhere...

 

[George Jones]

Wavelength is proportional to the scale factor of the universe, and the scale factor does vary linearly with time.

[EDIT]"does vary" should be "does not vary"[/EDIT]

 

[mfb]

[EDIT]"does vary" should be "does not vary"[/EDIT]

However, it is a good approximation for the time after the CMB was created.

Edit: Ok, I underestimated gravity.

though have read different values for the black body temperature at that recombination time from H ionization value 13.6 eV, to about 3000K

To get long mean free path lengths (as you want photons which survive), the temperature has to be below these 13.6eV by some significant factor.

 

[Ich]

However, it is a good approximation for the time after the CMB was created.

Not really. The CMB has a redshift of 1089 or so, but the universe is some 36000 times older now.
From the redshift you can deduce an original temperature of 2967 K.

 

[George Jones]

In terms of actual math, the functional dependence of the scale factor on time is close to

\sinh^{\frac{2}{3}}t.

 

[Ich]

In terms of actual math, the functional dependence of the scale factor on time is close to

\sinh^{\frac{2}{3}}t.

Right, but the formula doesn't work without the correct scaling of t in the sinh:
\sinh^{\frac{2}{3}}\left( 1.5 H_0 \sqrt{\Omega_{\Lambda}}t \right).

 

[Fire Tech]

Thanks for the equations and please forgive my ignorance, but I don't know what values to plug in for the variables / constants in the sinh equation...

Could you let me know what the CBR wavelength would have been, 1, 3, and 6 billion years ago and please show the math so I can then plug in other numbers on my own?

thanks,

rt

 

[Ich]

The formula ist the last one in the German wikipedia. I hope someone can provide an English reference.

\Omega_{\Lambda} is the fraction of Dark Energy of the total Energy of the universe. It is ~0.73.
H_0 is the Hubble parameter, about 1/(13.7 bn years).
You plug in the time now (also 13.7 bn years) and get a number (the scale factor).
You plug in another time, say, 1 bn years ago = 12.7 bn years and get another number.
The ratio of these numbers is also the ratio of the respective wavelengths.

The formula is still an approximation. For the correct numbers according to the LCDM model, use Ned Wright's cosmology calculator.