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mathematical cosmology: general interpretation of alpha
In arXiv:astro-ph/0404319, Lake introduces a quantity called alpha and
points out that it essentially measures the ratio of the value of the cosmological constant to that in Einstein's static universe (or those asymptotic to it). This works only for k=+1 and lambda0. For k=+1, there is another interpretation as well: it is proportional to the product of the mass of the universe and the value of the cosmological constant. (This conclusion is present, but not very explicit, at the URL mentioned in reference [12] in Lake's paper.) The interesting thing about alpha is that it is a combination of parameters each of which, in general, is not constant during the evolution of the universe, but the combination is constant. If one thinks of the cosmological parameters Omega and lambda as constituting a dynamical system, this is a constant of motion. What about other cases, i.e. k=-1? Is there a physical interpretation for this quantity? (See reference [14] in Lake's paper.) Lake's paper is rather terse, but contains all the necessary information. Some background, more than enough, is provided by the references. In particular, [8] and [12] should be read by everyone even remotely interested in such topics. (The notation is not uniform. Sometimes it is just a simple change of variables or different coefficients depending on the system of units used (so the same thing denoted by different (combinations of) symbols). It becomes more confusing when two people use the same symbol to denote different things.) If the universe collapses in the future, one can calculate the maximum size of the scale factor from the values of the cosmological parameters at any time. Obviously, along a trajectory in the Omega-lambda plane representing the evolution of a cosmological model, this maximum value is constant. However, there seems to be no simple relation between this "constant of motion" and alpha. Or is there? Note that there are models with k=+1 which collapse in the future, so in these cases both constants of motion are present, though again without any obvious connection. It is no problem to calculate all the interesting quantities. The question is whether there is any relation between these two constants of motion and whether either or both can be generalized to the case k=-1. Even better would be a generalization including the special cases lambda=0 and Omega=0 (for which alpha=0) and k=0 (for which alpha is infinite). |
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