Deriving fate of the universe from Friedman's equations

[andrewkirk]

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).

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.

 

[George Jones]

A diagram and analysis for this is given, e.g., in section 15.4 of the book by Hobson, Efstathiou, and Lasenby.

 

[andrewkirk]

Thank you George. I managed to track down a copy of that book. When I did and read the relevant parts of Hobson et al, I discover that I have been led astray, yet again, by the aggravating Bernard Schutz, who claimed in the Cosmology chapter of his 'A first course in general relativity' that only one independent equation can be derived from the Einstein field equation in the FLRW universe (he waves a hand at the Bianchi identities as justification but provides no working). So I just used one (the 00 equation), and found I didn't have enough information. I could only derive one of the two Friedman equations.

Hobson et al use two equations from the Einstein tensor equation - the 00 equation and the 22 equation - and from that are able to derive both the Friedman equations. Because the extra equation contains an explicit formula for $\ddot{R}$, it enables one to answer questions such as whether an expanding universe rebounds (as opposed to asymptotically approaching stasis) in certain circumstances.

This is by no means the first crucial error I have found in Schutz, and I am feeling very close to consigning it to the flames. But if I did that, I'd lose all the notes I've scribbled in the margins.

 

[George Jones]

I discover that I have been led astray, yet again, by the aggravating Bernard Schutz

Books are written by humans, and thus almost all books (even the good ones) contain errors. For example

Unfortunately, there is some subtlety here, and this subtlety seems to have confused Hobson, Efstathiou, and Lasenby (HEL). Most of the subtlety has to do with Woodhouse's "second fundamental confusion of calculus."

By HEL's own definition on page 248,

... fix the other coordinates at their values at P and consider an infinitesimal variation $dx^\mu$ in the coordinate of interest. If the corresponding change in the interval $ds^2$ is positive, zero or negative, then $x^\mu$ is timelike, null or spacelike respectively.

$p$ in Eddington-FinkelStein coordinates $\left(p,r,\theta,\phi \right)$ is a timelike coordinate, not a null coordinate. To see this, apply HEL's prescription on page 248 to equation (11.6). Varing $p$ while holding $r$, $\theta$, and $\phi$ constant gives $dr = d\theta = d\phi = 0$ and

$$ds^2 = \left( 1 - \frac{2M}{r} \right) dp^2.$$

Hence, (when $r > 2M$) $ds^2$ is positive, and $p$ is a timelike coordinate.

HEL are thinking of $p$ in Kruskal coordinates $\left(p,q,\theta,\phi \right).$. In this case, applying the page 248 prescription to equation (11.16) gives that $p$ is a null coordinate. Do you see why?

What type of coordinate is $r$ in Eddington-FinkelStein coordinates $\left(p,r,\theta,\phi \right)$?

By now, you should be thoroughly confused! How can the "same" $p$ be timelike in one set of coordinates and null in another set of coordinates? If you want, I am willing to spend some time explaining in detail what is going on here, and what Woodhouse's "second fundamental confusion of calculus" is.