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Help me derive the relativistic equation of motion for the Universe

  1. Feb 21, 2014 #1
    Guys,

    My calculus is really rusty and I need help solving this equation using a time derivative (denoted with a dot) in order to get the relativistic equation of motion for the Universe

    The equation is:

    adot^2 = Λc^2a^2/3

    where adot is the time derivative of the scale factor, lambda is a cosmological constant, and c is the speed of light.

    The answer should be:

    a(t) = a(t0)e^Ht

    where H is Hubble's constant = sqrt(c^2Λ/3)

    But I don't know how to get there. Any help is appreciated!
     
  2. jcsd
  3. Feb 21, 2014 #2

    tiny-tim

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    Hi QuantumX! Welcome to PF! :smile:

    (try using the X2 button just above the Reply box :wink:)
    So da/dt = √(Λc2/3)a = Ha :smile:
     
  4. Feb 21, 2014 #3
    Thanks Tiny-Tim,

    So then the first equation is just the relationship between the scale factor and the Hubble constant...

    So how do I then get a(t) = a(t0)e^Ht ? (that is for t>t0). That's what i need to ultimately arrive at. It's supposed to showcase that the Universe is accelerating in its expansion.
     
  5. Feb 22, 2014 #4

    tiny-tim

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    da/dt = Ha

    so dt/da = 1/Ha, so t = loga/H + constant

    so loga = Ht + constant

    so a = eHt + constant = econstanteHt

    and the econstant is a(to)
     
  6. Feb 24, 2014 #5
    Thanks!
     
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