Blog #35: WS11.1, #3

3. Baryon-to-photon ratio of our universe. 

a) Despite the fact that the CMB has a very low temperature, the number of photons is enormous. Let us estimate what that number is. Each photon has energy \(h\nu\). From the Planck spectrum, figure out the number density, \(n_{\nu}\), of the photon per frequency interval \(d\nu\). Integrate over \(d\nu\) to get an expression for total number density of photon given temperature T. Keep all factors and use the fact that \(\int^{\infty}_0\frac{x^2}{e^x-1}dx\approx2.4\). 

The number density is the energy density (\(u_{\nu}d\nu\)) divided by the energy per photon (\(h\nu\).
\(n_{\nu}=\int^{\infty}_0\frac{8\pi h\nu^3}{c^3}\frac{1}{e^{\frac{h_P\nu}{k_BT}}-1}\frac{1}{h\nu}d\nu=\frac{8\pi}{c^3}\int^{\infty}_0\frac{\nu^2}{e^{\frac{h_P\nu}{k_BT}}-1}d\nu\)
By doing u-substitution and using the fact provided above, we can finish the integration.
\(u=\frac{h_P\nu}{k_BT}\)
\(n_{\nu}=\frac{8\pi}{c^3}(\frac{k_BT}{h_P})^3\int^{\infty}_0\frac{u^2}{e^u-1}du=\frac{8\pi}{c^3}(\frac{k_BT}{h_P})^3 2.4\)

b) Using the following values for the constants: \(k_B=1.38\times10^{-16}erg/K\), \(c=3.00\times10^{10}cm/s\), \(h_P=6.62\times10^{-27}erg\cdot s\), and use the temperature of the CMB today that you have computed to calculate the number density of photons in our universe today. 

\(\frac{8\pi}{(3\times10^{10}cm/s)^3}[\frac{(1.38\times10^{-16}erg/K)(2.72K)}{6.62\times10^{-27}erg\cdot s}]^32.4=407/cm^3\)

c) Let us calculate the average baryon number density today. In general, baryons refer to protons or neutrons. The present-day density (matter + radiation + dark energy) of our universe is \(9.2\times10^{-30}g/cm^3\). The baryon density is about 4% of it. The masses of proton and neutron are very similar (\(\approx1.7\times10^{-24}g\)). What is the number density of baryons? 

\(\frac{0.04(9.2\times10^{-30}g)}{1cm^3}\frac{1 baryon}{1.7\times10^{-24}g}=2.16\times10^{-7} baryons/cm^3\)

d) Divide the above two numbers to get the baryon-to-photon ratio. As you can see, our universe contains many more photons than baryons. 

\(\frac{2.16\times10^{-7}baryons/cm^3}{407photons/cm^3}=5.32\times10^{-10}baryons/photon\)