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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 nonrelativistic matter is present. Nonrelativistic matter has P = 0, so a cosmological constant with Λ = 4 πGρ produces zero acceleration. If some of the nonrelativistic matter is converted to photons, say by stars or a decaying particle, then by energy conservation the energy density in nonrelativistic 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 counterintuitive, 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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 Fall '08
 Staff
 Acceleration, General Relativity, Big Bang, Redshift, Hubble's law, Physical cosmology

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