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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 G 3 c 2 ( + 3 P ) + 3 . (1) Initially only a density of non-relativistic matter is present. Non-relativistic matter has P = 0, so a cosmological constant with = 4 G 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, nr , and relativistic matter, r , must still equal c 2 . From the equation of state for relativistic matter, e.g. Ryden equation (4.56), P r = r / 3. Thus, Equation 1 becomes: a a =- 4 G 3 c 2 ( nr + r + 3( r / 3)) + 3 (2) =- 4 G 3 c 2 ( c 2 + r ) + 4 G 3 (3) =- 4 G 3 c 2 ( 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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- Fall '08