hw4a-sol - Ph 444 Solutions for Problem Set 4 1(Ryden 4.2...

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Unformatted text preview: Ph 444 Solutions for Problem Set 4 1. (Ryden 4.2) The acceleration of the universe is governed by equation (4.64) from Ryden: ¨ a a =- 4 πG 3 c 2 ( ǫ + 3 P ) + Λ 3 . (1) Initially only a density ρ of non-relativistic matter is present. Non-relativistic matter has P = 0, so a cosmological constant with Λ = 4 πGρ produces zero acceleration. If some of the non-relativistic matter is converted to photons, say by stars or a decaying particle, then by energy conservation the energy density in non-relativistic matter, ǫ nr , and relativistic matter, ǫ r , must still equal ρc 2 . From the equation of state for relativistic matter, e.g. Ryden equation (4.56), P r = ǫ r / 3. Thus, Equation 1 becomes: ¨ a a =- 4 πG 3 c 2 ( ǫ nr + ǫ r + 3( ǫ r / 3)) + Λ 3 (2) =- 4 πG 3 c 2 ( ρc 2 + ǫ r ) + 4 πGρ 3 (3) =- 4 πG 3 c 2 ( ǫ r ) . (4) Thus, the universe would start to collapse because of the pressure associated with the radiation. That increased pressure initiates collapse is counter-intuitive, but is consistent with pressure contributing to the gravitational acceleration in General Relativity. 2. (Ryden 5.2) The redshift is given by 1 + z = a /a e . Taking the derivative of z with respect to the time of observation, t...
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hw4a-sol - Ph 444 Solutions for Problem Set 4 1(Ryden 4.2...

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