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3. Matter-Radiation Equality In this problem, we will estimate the redshift at which the matter and radiation density of the universe were equal. The important thing to note here is that the mass-energy density of radiation, $ ho_r$, evolves as $ ho_r = ho_{r,0} imes (1 + z)^4$, where $ ho_{r,0}$ represents the present-day radiation density. The redshift dependence of $ ho_r$ takes this form because we need to account for both the evolving density of photons and their evolving photon energies with redshift. You have already calculated how the matter density, $ ho_m$, evolves with redshift. (a) In a universe where the current value of $Omega_m$ is $Omega_{m,0} = 0.3$, what is the present-day matter density, $ ho_{m,0}$? Express your answer in kg/m³. (b) Given the present-day radiation density of $ ho_{r,0} = 7.8 imes 10^{-31}$ kg/m³ (where energy density has been expressed as the equivalent mass density via a factor of 1/c², and represents the contributions of both cosmic microwave background radiation and neutrinos), solve for the redshift at which matter and radiation density were equal.

          3. Matter-Radiation Equality
In this problem, we will estimate the redshift at which the matter and radiation density of
the universe were equal. The important thing to note here is that the mass-energy density of
radiation, $
ho_r$, evolves as $
ho_r = 
ho_{r,0} 	imes (1 + z)^4$, where $
ho_{r,0}$ represents the present-day radiation
density. The redshift dependence of $
ho_r$ takes this form because we need to account for both the
evolving density of photons and their evolving photon energies with redshift. You have already
calculated how the matter density, $
ho_m$, evolves with redshift.
(a) In a universe where the current value of $Omega_m$ is $Omega_{m,0} = 0.3$, what is the present-day matter
density, $
ho_{m,0}$? Express your answer in kg/m³.
(b) Given the present-day radiation density of $
ho_{r,0} = 7.8 	imes 10^{-31}$ kg/m³ (where energy density
has been expressed as the equivalent mass density via a factor of 1/c², and represents the
contributions of both cosmic microwave background radiation and neutrinos), solve for the
redshift at which matter and radiation density were equal.

3. Matter-Radiation Equality
In this problem, we will estimate the redshift at which the matter and radiation density of
the universe were equal. The important thing to note here is that the mass-energy density of
radiation, hor, evolves as hor =
hor,0 imes (1 + z)^4, where hor,0 represents the present-day radiation
density. The redshift dependence of hor takes this form because we need to account for both the
evolving density of photons and their evolving photon energies with redshift. You have already
calculated how the matter density, hom, evolves with redshift.
(a) In a universe where the current value of Omegam is Omegam,0 = 0.3, what is the present-day matter
density, hom,0? Express your answer in kg/m³.
(b) Given the present-day radiation density of hor,0 = 7.8 imes 10^-31 kg/m³ (where energy density
has been expressed as the equivalent mass density via a factor of 1/c², and represents the
contributions of both cosmic microwave background radiation and neutrinos), solve for the
redshift at which matter and radiation density were equal.

Added by Juana H.

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