 Cosmological surface-brightness dimming

For freely propagating radiation with the usual distance duality relation, a fixed emitted bolometric surface brightness is observed smaller by $(1+z{)}^4$. One factor each comes from photon energy and arrival rate, and two from the relation of projected angular area to luminosity distance. In a fixed observed photometric passband, spectral shifts add a band-dependent correction; the bolometric power is not automatically the monochromatic one. A distortion created by scattering the local cosmic microwave background has a different redshift comparison, as in the thermal Sunyaev-Zeldovich effect.