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. 

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