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I'm attempting to perform the integration that will yield the sound horizon at recombination: $$ c_s^2 = \frac{c^2}{3}\left[\frac{3}{4}\frac{\rho_{b,0}(1+z)^3}{\rho_{\gamma,0}(1+z)^4} + 1\right]^{-1} $$The present day density of baryons is reasonably easy to calculate as $\rho_b=\Omega_b\times \rho_{crit}$. Using the Planck Collaboration results, I get a present day value of $\rho_b=4.14\times 10^{-25}\space g\space m^{-3}$.

I've seen one post calculate the photon density as $$ \rho_\gamma = \frac{a_B\, T_0^4}{c^2} = 4.64511\times 10^{-31}\;\text{kg}\,\text{m}^{-3}. $$ where $$ a_B = \frac{8\pi^5 k_B^4}{15h^3c^3} = 7.56577\times 10^{-16}\;\text{J}\,\text{m}^{-3}\,\text{K}^{-4} $$ But then the author goes on to calculate the radiation density and suggests plugging the total radiation density value into the formula for sound velocity which doesn't seem right (as the electrons are bound to photons, not all radiation).

What is the right value for $\rho_{\gamma,0}$ for the purpose of calculating the velocity of sound in the pre-recombination fluid?

  • If you are just asking the present value of the photon density then that expression seems correct. I checked it from a book. – seVenVo1d Jan 11 '19 at 20:02
  • Yes, I'm asking for the present day photon density. 2. I'm asking if this is the right value to use in the speed of sound calculation because the post I referenced suggested using the total radiation density. 3. I'm looking for an actual value to double check against the one I'm using.
    • –  Jan 11 '19 at 20:05
  • A reference for the photon "mass" density formula above is https://en.wikipedia.org/wiki/Photon_gas. Look at the second equation there. $U/V$ is the photon energy density. Divide by $c^2$ and you have the density of relativistic mass. This is just a standard blackbody radiation formula. – G. Smith Jan 11 '19 at 20:06
  • @G.Smith - I don't see either a formula or a value for the photon density on the page you referenced. What am I missing? –  Jan 11 '19 at 20:09
  • I edited my comment to explain. – G. Smith Jan 11 '19 at 20:10
  • @G.Smith - Thank you, but I have plenty of formulas. I'd like an answer (an actual value) and then I can work backwards and put the formulas in some context. –  Jan 11 '19 at 20:11
  • The value 4.645e-31 kg/m^3 above is correct. The only number you need to get this, other than the values of fundamental constants, is the CMB temperature $T_0$, which is 2.72548 K, as you can find many places, such as here: https://en.wikipedia.org/wiki/Cosmic_microwave_background – G. Smith Jan 11 '19 at 20:18
  • In other words, the post you cited correctly gave the photon density that you want. – G. Smith Jan 11 '19 at 20:20
  • Whether it is OK to ignore the neutrino density when calculating the velocity of sound in the early universe is not something I can help you with. – G. Smith Jan 11 '19 at 20:24
  • @G.Smith - The post I cited suggested using a value of $\rho_{\gamma}=7.8\times 10^{34}\space g\space m^{-3}$. That's the real reason for this post. Is it a typo? –  Jan 11 '19 at 20:25
  • Yes, it's a typo in the sense that the gamma subscript should be an R. This is the total radiation density (photons+neutrinos) that Pulsar calculated in the link that caverac referenced (https://physics.stackexchange.com/questions/94181/where-is-radiation-density-in-the-planck-2013-results). It is not the photon density, and I don't know whether his formulas for the neutrino density are correct or not. – G. Smith Jan 11 '19 at 20:33
  • The post you linked to doesn't have either of the formulas you gave. They're in the post I just referenced. – G. Smith Jan 11 '19 at 20:35
  • @G.Smith - Further down in the conversation I ask caverac where he got the values. He linked to https://physics.stackexchange.com/questions/94181/where-is-radiation-density-in-the-planck-2013-results, which is where I got the formulas. I agree, it looks like a typo but I've been using that value for a week and it wasn't giving the proper results. –  Jan 11 '19 at 20:37