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de Sitter spaces in NKS/Mathematica?

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Posted by: Tony Smith

I've been taking a rushed, self-directed course in inflation cosmology courtesy of the web and a Dirac medallist, which led to me needing to know more about de Sitter spaces, so I aggravated my shoulder checking the indexes of the NKS and Mathematica™ books and found no reference to de Sitter in either.

My gut feeling is that de Sitter is at a level of abstraction that Mathematica does not target, but I really need somebody who has a clue to fill the gaps.


Posted by: Jason Cawley

de Sitter space is essentially the case or solution of Einstein's equations with positive cosmological constant and no matter (pure vaccum). It results in a hyperbolic geometry or ongoing exponential expansion of the (empty and purely symmetric) universe. As such it was one of the precursors of inflation theories. You get phenomena later associated with inflation theories, like cosmological horizons (regions of the space casually disconnected from each other by the speed of their recession) etc.

Consideration of it initially arose because it was one of the cases of GR simple enough to be easily solved, and continued because there was empirical evidence for cosmological expansion that had to be accounted for etc. More strictly, when that is all you have one speaks of a de Sitter universe, while the mathematical object with the same geometry is de Sitter space. (Which may still be a useful first approximation for thinking about universe expansion, while obviously our actual universe is not devoid of matter etc.)

You can look for more on the scienceworld companion to mathworld, but I don't know how illuminating you'd find it. The only thing approaching a discussion of it in the NKS book is the cosmology note on page 1055 - somewhat dated in that it speaks of the inflation scalar field as a Higgs field, as Guth originally suggested but has since been given up by inflation theorists. The note does at least relate it to Wolfram's own expectations for a dynamic network model.

As for Mathematica, I am sure plenty of people investigate Einstein equations with it, but de Sitter in particular is too simple to need it, really. All that happens is the empty universe uniformly expands at an exponential rate given by a single parameter.

I hope this helps.




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