We assume that the radiation is emitted at some epoch t, with scale factor R, and we detect it at the reference epoch t0. The observed wavelength A0 is related to the emitted wavelength X by equation (20.29), as
Since X = c/v, the observed frequency v0 is related to the emitted frequency v by v/vo = 1/R
This means that
Spectrum of the radiation in the universe as it expands.We plot log intensity vs. log frequency, for three temperatures. 3000 K was the temperature when radiation and matter decoupled. 1500 K is when the scale factor increased by a factor of 2 after decoupling. 750 K is another factor of 2 increase in the scale factor.
Multiplying by hv0 gives us back the energy density spectrum for radiation detected between v0 and v0 + dv0:
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