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Problem for simulation on recollapse model

  1. Jan 4, 2010 #1

    I post another topic because my first main problem about the modelisation of scale factor by solving numerically friedmann equations is resolved.
    However, It remains one problem, that of the recollapse when integration starts at a early age of the universe.

    i have reformulated the differential system through the equation :

    [tex]H^2(a) = \bigg(\frac{1}{a} \frac{da}{dt}\bigg)^{2} = H_0^2 \left( \frac{\Omega_m}{a^3} + \frac{\Omega_k}{a^2} + \Omega_\Lambda \right)

    which gives equation n°1 :

    \left( \frac{da}{dt} \right)^2 = H_0^2 \left( \frac{\Omega_m}{a} + \Omega_k + \Omega_\Lambda a^{2} \right)

    then we have :

    [tex] a = \Omega_{m} \bigg( \frac{1}{H_{0}^{2}} \bigg(\frac{da}{dt}\bigg)^{2} - \Omega_{k} - \Omega_{\Lambda} a^{2} \bigg)^{-1} [/tex] ​

    So, the differential equation can be written in this way :


    a''=\bigg(\frac{-4 \pi G}{3c^{2}}(\frac{\rho_{0} c^{2}}{a^{3}}+3 p)+\frac{\Lambda}{3}\bigg) \Omega_{m} \bigg( \frac{1}{H_{0}^{2}} \bigg(\frac{da}{dt}\bigg)^{2} - \Omega_{k} - \Omega_{\Lambda} a^{2} \bigg)^{-1} [/tex] ​

    Finally, i managed to get the curve representing the "recollapse" with [tex]\Omega_{m} > 1 [/tex], [tex] a_{0} = 1 [/tex], and [tex] a'_{0} = H_{0} [/tex] and [tex] \Omega_{\Lambda}=0 [/tex] ( https://www.physicsforums.com/attachment.php?attachmentid=22861&d=1262647111") . The integration begins at [tex] t_{0}=13.7 Gyr [/tex]. The results are validated by the value of the maximum of scale factor before the recollapse :

    [tex] a_{max} = a ( - \frac{\Omega_{m}}{\Omega_{k}} )[/tex]

    [tex] t_{max} = - \frac{5}{-4} = 1.25 [/tex] on the figure 5 ​

    I would like to get the first part of this curve ( for [tex] t0 << 13.7 Gyr [/tex] ), ie from different initial conditions. I tried with these conditions :

    [tex] a_{0} = 0.001 [/tex], and [tex] a'_{0} =H_{0} (5*10^{3}-4)^{1/2} = 70.6824 H_{0} [/tex]

    respecting so equation n°1 for the value of [tex] a'_{0} [/tex]. For this case, the Matlab solver begins integration at [tex] t_{0}=380.000 years [/tex], so a redshift z=1000.

    I get the https://www.physicsforums.com/attachment.php?attachmentid=22862&d=1262647111". There's well the recollapse design but the maximum is not equal to 1.25 (much more little, about 0.09 ) and the big crunch occurs fastier than on figure 5.

    What's the problem, How it is done ? Why i don't get the same recollapse as in figure 5, i mean the same maximum and time "t_final" of big crunch ( [tex] a(t_{final})=0 [/tex].

    Thanks a lot.
    Last edited by a moderator: Apr 24, 2017
  2. jcsd
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