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# hw3-sol - Ph 444 Solutions for Problem Set 3 2 1 The...

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Ph 444 Solutions for Problem Set 3 1. The critical energy density is ǫ c = 3 H 2 0 / (8 πGc 2 ) and so the critical density is ρ c = 3 H 2 0 / (8 πG ). The uncertainty in the critical density is σ ρ c = ∂ρ c ∂H 0 σ H 0 = ∂H 0 parenleftBigg 3 H 2 0 8 πG parenrightBigg σ H 0 = parenleftbigg 6 H 0 8 πG parenrightbigg σ H 0 (1) σ ρ c ρ c = 2 σ H 0 H 0 (2) I find that fractional uncertainties are usually more meaningful than the uncertainty itself. The formulae are also often simpler. Plugging in the value of H 0 gives ρ c = 3(70 . 6 km s 1 Mpc 1 ) 2 (1 Mpc / 3 . 086 × 10 19 km) 2 8 π (6 . 673 × 10 11 m 3 kg 1 s 1 ) = 9 . 362 × 10 27 kg m 3 . (3) For use in problem 2, convert the units to M Mpc 3 : ρ c = (3 . 086 × 10 22 m / Mpc) 3 1 . 989 × 10 30 kg /M (9 . 362 × 10 27 kg m 3 ) = 1 . 38 × 10 11 M Mpc 3 . (4) The fractional uncertainty is σ ρ c ρ c = 2 parenleftBigg 1 . 8 km s 1 Mpc 1 70 . 6 km s 1 Mpc 1 parenrightBigg = 2(0 . 025) = 0 . 051 . (5) Thus, the critical density is ρ c = (9 . 362 ± 0 . 48) × 10 27 kg m 3 = (1 . 38 ± 0 . 07) × 10 11 M Mpc 3 . (6) 2. If j is the luminosity density of galaxies, then the average M/L per galaxy needed for galaxies to produce the critical density is ( M/L ) c = ρ c /j . The propagation of errors is slightly tricky in this case as both j and ρ c depend on H 0 and so errors in these quantities are not independent. Adding uncertainties in quadrature requires that the uncertainties be independent. The solution is to express ( M/L ) c explicitly as a function of H 0 .

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hw3-sol - Ph 444 Solutions for Problem Set 3 2 1 The...

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