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Friedmann equation - show Big Bang happened given conditions

  1. Mar 23, 2017 #1
    1. The problem statement, all variables and given/known data

    Use Friedmann equations to show that if ##\dot{a} > 0##, ##k<0## and ##\rho>0## then there exists a ##t*## in the past where ##a(t*)=0##

    2. Relevant equations

    Friedmann :

    ##(\frac{\dot{a}}{a})^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2}##


    3. The attempt at a solution

    re-arrange as:

    ##\dot{a}^2=1+\frac{8\pi G}{3}a^2\rho##

    where I have used ##k<0## can be/is standard to set to ##k=-1##

    Then since ##\rho>0##, and ##\dot{a}##>0 implies I should take the positive square root of this , this implies that ##\dot{a}>1## for all time.

    Now I do not follow the next part of my solution which says:

    Thus ##a(t)>0## is decreasing at a rate that is always greater than ##1## and there was necessarily a finite time ##t*## in the past where ##a(t*)=0##

    How have we concluded a decrease, do we not need ##\ddot{a}## to make this conclusion? Can someone please explain where this comes from?

    Many thanks in advance
     
  2. jcsd
  3. Mar 23, 2017 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    I don't think you can just set k=-1, although it doesn't change the conclusion here. It is not necessary to do that, all you need is k<0 in the following argument.

    If the first derivative is positive, then the original function decreases if we go back in time. This does not depend on the second derivative.
     
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