Archive of Astronomy Questions and Answers

How is it possible for the energy in the cosmic background radiation to remain constant as the universe expands?

Because the cosmological expansion is adiabatic. This means that the following holds:

                   3
  n(photons/cc) * R    = constant

for every epoch, where n is the total number of photons that make up the Planck distribution of the cosmic background radiation field. This is constant because there are no large-scale processes that can create or destroy photons in the background radiation field. Now, to define a total energy, we can multiply by the energy per photon, e = h x frequency. But because of the expansion of the universe the frequency of each photon varies as

  wavelength = wavelength(0) * R

so                       
  
freq = freq(0) / R

This is because space is stretching the wavelengths more and more as time goes on and R increases. If you now compute

               constant
    n     =   -------- 
                 3
                R    

you get

  E = e * N

            -4
  E = E(0) R      ergs/cc

Since we are still talking about Planck spectra, if you integrate the spectrum over frequency, you get the Stephan-Boltzman Law which says the total energy is proportional to the fourth power of the temperature. But from general relativity,


   KT / h*frequency  = constant 

and so


T = T(0) / R           (  alternately, T = T(0) x (1 + z) )

The net result is that both sides of the above equation are proportional to the fourth power of the scale factor at a given epoch. This means that the total energy of the CMBR defined in this way is independent of the epoch and scale factor. This is what we mean by 'adiabatic expansion' since the total energy of the system remains constant.