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Homework Help: Cosmological epoch of matter-radiation equality

  1. Nov 12, 2012 #1
    1. The problem statement, all variables and given/known data
    If Ω_0m=0.25 and Ω_0R=7.4*10^-5 calculate the redshift when the two densities Ω_m and Ω_R are equal.

    Relevant Equations
    [itex]\Omega = \frac{rho}{rho_{crit}}[/itex]
    [itex]\rho_{0,crit} = \frac{3 H_{0}^{2}}{8 \pi G}[/itex]

    The attempt at a solution

    convert matter density: [itex]\epsilon_{0,m} = \rho_{0,m} c^{2} = \Omega_{m,0} \rho_{crit,0} c^{2}[/itex]

    sub in for critical density: [itex]\epsilon_{0,m} = \Omega_{m,0} \frac{3 H_{0}^{2}}{8 \pi G} c^{2}[/itex]

    calculate ratio of matter to radiation: [itex]\frac{\epsilon_{R}}{\epsilon{M}} = \frac{\Omega_{R,0}}{\Omega{M,0}} \frac{8 \pi G c^{2}}{3 H_{0}^{2}}[/itex]

    and as [itex]\epsilon_{R} \propto 1/a^{4}[/itex] and [itex]\epsilon_{M} \propto 1/a^{3}[/itex] and ρ0/a^3 = ρ

    [itex]\frac{\epsilon_{R}}{\epsilon{M}} = \frac{\epsilon_{0,R}}{\epsilon{0,M}} 1/a[/itex]

    put [itex]\frac{\epsilon_{R}}{\epsilon{M}} = 1[/itex] so

    [itex]1 = \frac{\Omega_{M,0}}{\Omega{R,0}} \frac{3 H_{0}^{2}}{8 \pi G c^{2}} (1+z)[/itex]

    This comes out as 1+z = 3.215*10^-6
    and so gives a negative redshift!

    Now I have either done something terribly wrong or the Omegas given are for an arbitrary universe in which the equality epoch has yet to occur!

  2. jcsd
  3. Nov 12, 2012 #2
    Sudden realisation (maybe):

    Is it because
    (ϵR)(ϵM) = (ϵ_0,R) (ϵ_0,M) a

    (ϵR)(ϵM) = (ϵ_0,R) (ϵ_0,M) 1/a??

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