Final answer:
The energy of photon radiation as the universe expands is influenced by the scale factor, R, and changes according to the expression E = E_0 / R. This is due to the volume expansion which dilutes the photon density and redshifting which stretches the photon's wavelength, reducing its energy, leading to an overall energy density scaling as 1 / R^4.
Step-by-step explanation:
The energy of a photon radiation as the universe expands changes with the scale factor, R. The expression for this evolution of energy is E = E_0 / R, where E is the energy of the photon as observed, E_0 is the energy at the time of emission, and R is the scale factor of the universe. There are two components to this dependence on R. First, there is the reduction in energy due to the redshifting of photons as space expands, which corresponds to the wavelength of the photon stretching—this represents the photon's energy loss over cosmological distances. Second, there is the decrease in the density of photons due to the increase in volume associated with the expansion of the universe.The simple volume expansion speaks to the increase of space itself, causing the photons to be more spread out and therefore making their energy density decrease as the scale factor increases. This can be represented as a proportional relationship where the density of radiation energy goes as 1 / R^3, a direct consequence of the volume's dependency on the scale factor (since volume scales as R^3). Moreover, since the energy of each photon is redshifted and decreases with the expansion of space, the energy density of radiation actually scales as 1 / R^4.To summarize, as the scale factor grows, the energy density of photon radiation decreases both due to the expansion of space (simple volume expansion) and the redshifting of photons (additional consideration).