Solution

The given formula is the integral of the Planck photon distribution. Set

x = \frac{h ν}{k _{B} T}, d ν = \frac{k _{B} T}{h} d x .

Then

n = \frac{8 π}{c ^{3}} (\frac{k _{B}}{h})^{3} T^{3} \int_{0}^{\infty} \frac{x ^{2}}{e ^{x} - 1} d x = α T^{3},

where

α = \frac{8 π}{c ^{3}} (\frac{k _{B}}{h})^{3} \int_{0}^{\infty} \frac{x ^{2}}{e ^{x} - 1} d x .

Each photon of frequency

ν

has energy

h ν

, so the energy density is

ρ = \frac{8 πh}{c ^{3}} \int_{0}^{\infty} \frac{ν ^{3}}{e ^{h ν / (k_{B} T)} - 1} d ν = \frac{8 π k _{B}^{4}}{c ^{3} h ^{3}} T^{4} \int_{0}^{\infty} \frac{x ^{3}}{e ^{x} - 1} d x = ξ T^{4} .

After decoupling, cosmological redshift gives

ν (t) = \frac{a ( t _{dec} )}{a ( t )} ν_{dec}, ν_{dec} = \frac{a ( t )}{a ( t _{dec} )} ν (t) .

The occupation number is conserved along the freely propagating photon trajectory. Its exponential argument at decoupling becomes

\frac{h ν _{dec}}{k _{B} T _{dec}} = \frac{h ν ( t )}{k _{B} ( a ( t _{dec} ) T _{dec} / a ( t ) )} .

Thus the spectrum still has the Planck form if

T (t) = \frac{a ( t _{dec} )}{a ( t )} T_{dec} .

Frequency and temperature redshift by the same factor, which is the redshift preservation of a thermal photon spectrum.

Solved by gpt-5.6-sol high.

 Ancestors (10)