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Age of Universe (Liddles Modern Cosmology)

  1. Nov 18, 2011 #1
    I am currently reading Liddles Introduction to Modern Cosmology (2nd Ed) and having trouble with problem 8.4, about the age of the universe with a cosmological constant.
    The question asks to derive the formula for the age by first writing the Fridemann equation in such a model as
    [itex]\frac{da}{dt}[/itex] = [itex]H^{2}_{0}[/itex] [itex][Ω_{0}a^{-1}[/itex] + [itex]\left(1-Ω_{0})a^{-2}\right][/itex]

    But i'm not sure how this step is found, so can't proceed any further. I would appreciate any help to get me started.
  2. jcsd
  3. Nov 21, 2011 #2


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    Welcome to PF hawker3! :smile:

    I'd be happy to help you, but in order to do so, you need to tell me what form of the Friedmann equation you are starting with. Then I can give you suggestions on what manipulations to perform in order to get it into the form given above.
  4. Nov 21, 2011 #3


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    I'm having trouble finding a set of conditions in which the formula is correct. Look at the matter-only case for example: a [itex]\propto[/itex]t2/3. Then the left side is [itex]\propto[/itex] t-1/3 whereas the 2 right side terms are [itex]\propto[/itex] t-2/3 and t-4/3.
  5. Nov 21, 2011 #4


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    I think that the relation a ~ t2/3 is specific to the "critical" or "Einstein-de Sitter" universe which is not merely matter-dominated, but also happens to have a critical density of matter so that Ω0 = Ωm = 1. Hence (1-Ω0) = 0.

    Even so, the equation is also a bit off. At the risk of giving too much away, it should be (da/dt)2 on the left-hand side. You can see that this makes things work out for the critical case.

    The a-2 dependence on the (1-Ω0) term also seems wrong to me. If you assume that there is no spatial curvature, and that the matter shortfall (1-Ω0) is made up of "something" that has a constant energy density (e.g. dark energy), then this term should have an a2 dependence. If you assume that there is nothing else aside from matter, then (1 - Ω0) is the curvature term, and it should have an a0 dependence (i.e. no dependence on a at all).
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