Find the number density of photons:
Determine the number density of black body (CMB) photons by the relation
Here, is the energy density of photons, is Boltzmann constant and is the temperature of CMB.
The energy density of CMB photons is related to the temperature of CMB as
Here, is the radiation constant.
Substitute in the number density equation,
Substitute for, forand for
Find the number density of baryons:
Determine the number density of baryons in the present day universe by the relation,
Here, is the density of baryons in the present day universe and is the mass of hydrogen atom.
Substitute for and forto find
Find the ratio:
Take the ratio of number density of baryons to the number density of black body photons in the present day universe
This is same as the ratio of number of baryons to the number of photons in today’s universe.
So the ratio of number of baryons to the number of photons in today’s universe is.
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