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hv0fkRT .

In addition, photons emitted in the frequency range dv will be observed in the frequency range dvo, given by dv0 = R dv (21.4)

The photons emitted between v and v + dv are now observed between v0 and v0 + dv0. Also, all volumes increase by a factor of 1/R3. Combining all of these, we now have the number of observed photons per unit volume with frequencies between v0 and v0 + dv0 as

This looks just like a blackbody spectrum at temperature T0 = RT. Therefore, the radiation will still have a blackbody spectrum, but will appear cooler by a factor of R. In Fig. 21.2, we see a few examples of how the spectrum evolves at different temperatures.

The redshift of the background radiation has an interesting consequence on the evolution of the universe. This is illustrated in Fig. 21.3. The energy density of the matter in the universe is pmatc2, where pmat is the density of matter. We have already seen in Chapter 20 that the density is proportional to 1/R3. The number density of photons in the universe is also proportional to 1/R3. However, the redshift means that the energy per photon is proportional to 1/R, so the energy density of radiation in the universe is proportional to 1/R4. This means that the energy density of radiation (integrated over

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