system
Given the following post, respond as if you're commenting on Reddit:
human
Is energy conserved in the redshift of cosmic microwave background (CMB) radiation?

If I understood correctly, the CMB originated at a time when the universe was much hotter and denser than today, and originally consisted of photons of much shorter wavelength than today. As the universe expanded and cooled, the photons comprising this background radiation did too, and at present it appears to us in the microwave range of wavelengths.

I don't understand how the law of conservation of energy holds for the total amount of energy carried by the CMB. The number of photons comprising the CMB can't increase. It can only remain the same or decrease (some photons will get absorbed by matter), and their wavelength is continually increasing. The total amount of energy carried by the CMB should therefore decrease (photons will longer wavelengths have less energy). If energy is conserved, where did this energy go?
gpt
Yes, the CMB loses energy. There are two perspectives on this:

First, because time translation is not a symmetry of an expanding universe (no timelike Killing field), we may take the perspective that energy just isn't conserved at cosmological scales.

Second, we can quantify how much energy gets lost, eg using Noether's *second* theorem. If you have your observers follow the Hubble flow (energy depends on the reference frame), this gets you the first Friedmann equation. In case of a spatially flat universe (and using units where c=1,G=1), the changing energy densities of matter, radiation and dark energy will sum to 3H²/8π. If we interpret this term as the energy density of the gravitational field, total energy will exactly balance and hence be conserved.