I have been trying to derive the different ultimate 'fates' of the universe from Friedman's equations, for the different possible values of k (shape parameter) and ##\Lambda## (cosmological constant).(adsbygoogle = window.adsbygoogle || []).push({});

There are nine possibilities, according to whether each of k and ##\Lambda## are negative, zero or positive.

The equation I am using is the matter-dominated Friedman equation:##\dot{R}^2 = \frac{8\pi A}{3R} - k + \frac{\Lambda}{3} R^2##, where A is a positive constant, related to the density of mass-energy at a certain time.

The cases where ##\Lambda=0## can be easily done using effective potential techniques, leading to the conclusions that:

- if k = 0 (flat universe), the universe expands without limit, with the expansion rate asymptotically approaching zero

- if k < 0 (hyperbolic universe), the universe expands without limit, with the expansion rate asymptotically approaching a fixed positive rate.

- if k > 0 (elliptic universe), the universe will eventually stop expanding, turn around and collapse.

The main fate (call it F1) that I have been unable to rule out in some cases is the possibility that the scale factor will increase at an ever-slowing rate, so that it asymptotically approaches some maximum value ##R_{max}##. In that case, the rebound and subsequent collapse would not happen, because the point of rebound is never reached.

Another conceivable fate (call it F2) would be that ##R_{max}## is reached and the scale factor freezes there, in a very unstable equilibrium.

For ##\Lambda>0##, I can demonstrate that the expansion continues without limit, and the expansion rate also eventually increases without limit, unless ##k>0##. I have been unable to rule out F1 or F2 for that case.

For ##\Lambda<0##, I have not managed to prove the final outcome for any of the three shapes. For all three cases, I can demonstrate that the scale factor R is bounded above, so that rules out unlimited expansion, but I have been unable to rule out F1 or F2.

The techniques I have tried have involved a combination of differentiating the above equation with respect to t, and also dropping selected components and then using inequalities. THis allowed me to reach the conclusions I have so far, but not all the cases are solved.

Of course, for any given values of R, k, A and ##\Lambda##, one can numerically solve the equation, but I was hoping for a general solution that expresses the ultimate fate based on simple inequalities of those numbers.

I would be grateful for any suggestions.

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# Deriving fate of the universe from Friedman's equations

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