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Closed linear cosmology implies G M / R = c^2?

  1. Jul 30, 2013 #1
    I have a question about a linear FRW cosmology with [itex]k=+1[/itex].

    Assuming zero cosmological constant the first Friedmann equation can be written:

    $$\left(\frac{\dot R}{R}\right)^2 + \frac{kc^2}{R^2}=\frac{8\pi G}{3}\rho$$

    where scalar curvature [itex]k=-1[/itex] (open),0 (flat) or +1 (closed) and [itex]R(t)[/itex] is the radius of curvature of space.

    Now I assume a linear cosmology so that:

    $$R = c t.$$

    Thus the Friedmann equation becomes:

    $$\frac{c^2}{R^2} + \frac{kc^2}{R^2}=\frac{8\pi G}{3}\rho.$$

    If [itex]k=-1[/itex] then [itex]\rho=0[/itex]. This is the empty Milne Universe with negative spatial curvature.

    If [itex]k=0[/itex] then we have a flat Universe with:

    $$\frac{c^2}{R^2} = \frac{8 \pi G}{3} \rho.$$

    Because the Universe is flat I can write an expression for the total mass [itex]M[/itex] in a sphere of radius [itex]R[/itex]:

    $$M = \frac{4}{3} \pi R^3 \rho\ \ \ \ \ \ \ \ \ \ \ \ (1)$$

    so that

    $$\frac{GM}{R}=\frac{c^2}{2}.$$

    Now I want to look at the case with [itex]k=+1[/itex]:

    $$\frac{c^2}{R^2} + \frac{c^2}{R^2}=\frac{8\pi G}{3}\rho.$$

    One could say that this case must have a positive spatial curvature and thus the equation (1) linking mass, density and volume of a sphere is not valid.

    But I can define [itex]\rho^\prime = \rho/2[/itex] so that I get the equation:

    $$\frac{c^2}{R^2} = \frac{8 \pi G}{3} \rho^\prime.$$

    This is exactly the same as the flat Friedmann equation but with half the density. Therefore equation (1) is still valid. Thus a sphere of radius [itex]R[/itex] will have a mass [itex]M/2[/itex] so that, in this case, one would get the equation:

    $$\frac{GM}{R}=c^2.$$

    Is this correct?

    I ask because the linear model with [itex]k=+1[/itex] is unique in that the curvature term and the density term have exactly the same [itex]1/R^2[/itex] dependence. I thought therefore that maybe the curvature can be included with the density leaving an effectively flat cosmology.
     
  2. jcsd
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