# Integration of Planck's law to find photon density in space

## Homework Statement

The universe is filled with EM radiation emanating from the Big Bang. This radiation was initially unimaginably hot but, as the universe has expanded, it has cooled to 3K. The distribution of the energy density of these photons in frequency (or wavelength) is given by the Planck formula. At this temperature, what is the wavelength of the photons at the peak of the Planck distribution? You can use the Planck formula to figure out how much total energy there is per unit volume by integrating over frequency. By dividing by hv and integrating over frequency, you can figure out how many photons there are per unit volume. Use your result to estimate how many Big Bang relic protons there are per cubic meter of intergalactic space. For comparison, the average density of hydrogen atoms in the universe is just one per cubic meter, you will find that there are a lot more protons.

## Homework Equations

from Wien displacement law:
λ_max*T = (hc)/4.9651*K
Planck's law:
du = (8*pi/c^3)[hv/(exp(hv/kT)-1)]v^2*dv
where h = Planck's constant, k = Boltzmann's constant; c = speed of light,
v = frequency

## The Attempt at a Solution

integrate Planck's law from 0 to infinity
replace hv/kT with x
and integrate from 0 to infinity
[(k^3*T^3)/(h^2)]*[8*pi/c^3][x^3/(exp(x)-1)]dx]
U = [(k^3*T^3)/(h^2)]*[8*pi/c^3](gamma(4)zeta(4))
= 9.76e-25 J/m^3
Use λ_max*T = (hc)/4.9651*K at T = 3K to find
λ_max = 9.68e-4m
v_max = c/λ_max = 3.1e11 Hz
photon density = U/hv = U/h*v_max (which is the highest distribution frequency for v)
= 0.0047 protons/m^3
But this isn't much larger than the number of hydrogen atoms per cubic meter, as the question says.

Hi!

How did you get:

du = (8*pi/c^3)[hv/(exp(hv/kT)-1)]v^2*dv

?

This is very urgent please :)