Neutrino cosmology problem

Question

Assume there are one billion neutrinos for every proton. Using the limits on baryon density given by big bang nucleosynthesis, and assuming a Hubble constant of 70 km s−1 Mpc−1, how massive would the neutrino have to be, compared to the proton, to provide the balance of the closure density, ΩM = 1?

Answer 1

Apply Given, Asked, Picture, Solve GAPS strategy

Given: Hubble Constant H₀ = 70 kms^-1 Mpc^-1, balance of closure density Ω_M = 1,

           1 billion = 1 x 10^9 neutrino for every proton

Asked: mass ratio neutrino mass/proton mass = Mn/Mp

Assume: Volume of universe best current estimate Vu = 3.566 x 10^80 m^3

              Proton mass Mp = 1.673 x 10 ^-27 kg

              Density parameter best estimate Ω_M0 = 0.27 ± 0.04 = 0.31 ≅ 0.3

Assumption conditions: 1. Universe filled with Hydrogen atoms, abundant

                                      2. Universe lack dark matter and dark energy

First calculate number of protons for mass estimate of the universe at current Ω_M0

ρ_c critical density parameter = 3H₀^2/8πG

= 9.2 x 10^-27 kg/m^3 for H₀ = 70 kms^-1 Mpc^-1

Ω_M0 = ρ_m0/ρ_c, ρ_m0 = 0.3 x 9.2 x 10^-27 = 2.76 x 10^-27

Mass of universe filled with protons Mu = ρ_m0 x Vu

                                                               = 2.76 x 10^-27 x 3.566 x 10^80

                                                               = 9.843 x 10^53 kg

Mu = Proton mass x number of protons

no. of protons N = 9.843 x 10^53/1.673 x 10^-27 = 5.88 x 10^80

no. of neutrinos = 5.88 x 10^80 x 10^9 = 5.88 x 10^89

Next calculate matter density ρ_m, given Ω_M = 1,

Ω_M = ρ_m/ρ_c, Matter density ρ_m= ρ_c= 9.2 x 10^-27 kg/m^3

Matter density ρ_m = (mass proton filled universe + mass neutrinos)/Universe volume

9.2 x 10^-27 = (9.843 x 10^53 + 5.88 x 10^89 x Mn)/3.566 x 10^80 Solve for Mn

9.2 x 10^-27 x 3.566 x 10^80 = 9.843 x 10^53+ 5.88 x 10^89 x Mn

3.281 x 10^54 – 0.984 x 10^54 = 5.88 x 10^89 x Mn

Mn = 2.297 x 10^54/5.88 x 10^89 = 0.391 x 10^-35

Ratio Neutrino mass/proton mass = Mn/Mp

= 0.391 x 10^-35/1.673 x 10 ^-27 = 0.233 x 10^-8

I hope this helps.

Sincerely,

Shailesh (Sky) Kadakia, Expert tutor

Physics, Astronomy, Cosmology & All Math

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