← Cosmology

Cosmic Microwave Background Radiation

Friday, April 22, 2022

Universal Cooling

Since the universe is expanding, it must also be cooling (as a gas does when it expands). Going far enough back in time, we should see a universe so hot and dense that stable matter could not form. Eventually these particles would decay into stable matter and the photons involved could escape the hot "gas" of particles and photons of the early universe.

Blackbody Radiation of the Universe

The wavelength spectrum of photons from the early universe resemble a blackbody of temperature TT. Since these wavelengths expand as the universe expands, the temperature today should be around 510 K5-10~\text{K}, with photons of an energy kTkT on the order of 103 eV10^{-3}~\text{eV} (microwaves).

Relating the number of photons dNdN in the energy interval dEdE around EE as the number per unit volume, we get

N(E) dEV=8πE2(hc)31eE/kT1\frac{N\left(E\right)~dE}{V}=\frac{8\pi E^2}{\left(hc\right)^3}\frac{1}{e^{E/kT}-1}

Finding the total number of photons, we integrate over energy:

NV=1V0N(E) dE=8π(hc)30E2 dEeE/kT1=8π(hc)3(kT)30x2 dxex1\frac{N}{V}=\frac{1}{V}\int_0^\infty N\left(E\right)~dE=\frac{8\pi}{\left(hc\right)^3}\int_0^\infty \frac{E^2~dE}{e^{E/kT}-1}=\frac{8\pi}{\left(hc\right)^3}\left(kT\right)^3\int_0^\infty \frac{x^2~dx}{e^x-1}

Solving for the standard integral, we find the number density to be around

N/V=(2.03×107 photons/m3K3)T3N/V=\left(2.03\times10^7~\text{photons}/\text{m}^3\cdot\text{K}^3\right)T^3

Using the energy density formula, we find

U=8π5k415(hc)3T4=(4.72×103 eV/m3K4)T4U=\frac{8\pi^5k^4}{15\left(hc\right)^3}T^4=\left(4.72\times 10^3~\text{eV}/\text{m}^3\cdot\text{K}^4\right)T^4

The average energy per photon is therefore

Em=UN/V=(2.33×104 eV/K)TE_\text{m}=\frac{U}{N/V}=\left(2.33\times 10^{-4}~\text{eV}/\text{K}\right)T

Measuring the CMB

Using microwave antennae and space probes designed to measure the cosmic microwave background radiation, the temperature is found to be around T=2.725 KT=2.725~\text{K}. It is also measured to be constant in all directions, filling the entire universe just after the Big Bang.