The original photon would be redshifted to an energy of 2.481233 eV, it would have emitted new photons with a total energy of 0.0001875954 eV.
If that were one photon, it would have a wavelength of 6609127.569226111 nm.
"The CMB has a thermal black body spectrum at a temperature of 2.72548±0.00057 K. The spectral radiance dEν/dν peaks at 160.23 GHz, in the microwave range of frequencies, corresponding to a photon energy of about 6.626 ⋅ 10−4 eV"
The conjecture is that photons redshift, and the deducted energy produces a new photon, which are detected as the CMB.
How often is a new photon produced this way?
Let's say a photon is emitted with wavelength of 499.65 nm, which is energy of 2.481420854598219 eV.
It's going to travel 1 million light years, and using a value of 74 km/s/Mpc for Hubble's constant, convert that ly/y/Mly and that is 0.0000756.
After 1 million years then, it's speed would be 0.9999244 c.
c = freq_emit wavelength c 0.9999244 = freq_new * wavelength
freq_emit wavelength = (freq_new wavelength) / 0.9999244
0.9999244 freq_emit wavelength = freq_new * wavelength
0.9999244 * freq_emit = freq_new
E=hf
0.9999244 freq_emit h = freq_new * h
0.9999244 * E_emit = E_new
E_new = 0.9999244 * 2.481420854598219
The original photon would be redshifted to an energy of 2.481233 eV, it would have emitted new photons with a total energy of 0.0001875954 eV.
If that were one photon, it would have a wavelength of 6609127.569226111 nm.
"The CMB has a thermal black body spectrum at a temperature of 2.72548±0.00057 K. The spectral radiance dEν/dν peaks at 160.23 GHz, in the microwave range of frequencies, corresponding to a photon energy of about 6.626 ⋅ 10−4 eV"
https://en.wikipedia.org/wiki/Cosmic...ave_background
A CMB photon has energy 0.0006626 eV, and wavelength 1871177.075 nm.
So... assuming a fresh local CMB photon has that energy, how many millions of years would it take a photon to pop out another photon at that energy?
0.0006626 eV / 0.0001875954 eV My-1 = 3.532069549679789 My
So... according to all this, a photon in the visible spectrum should pop out a CMB photon every 3.5 million years.