de Sitter Space and Cosmology

By Mark Trodden | April 15, 2012 6:08 am

A standard topic in an introductory General Relativity (GR) course is the study of maximally symmetric solutions. These are flat (Minkowski) spacetime, de Sitter spacetime (obtained when the cosmological constant is positive) and Anti-de Sitter spacetime (when the cosmological constant is negative). While this last space has been of great interest in physics during the last fifteen years due to its central role in the correspondence between gauge theories and gravity, it is de Sitter space with which I’ll be concerned here.

The idea of cosmological inflation is our best developed idea of how the physics of the early universe might lead to the observed universe today. This idea has been widely discussed in popular books and beyond, and in this context, many students have heard the loose description that inflation occurs when the universe is in an almost de Sitter state, and undergoes exponentially rapid expansion. There is nothing wrong with this explanation, but one consequence of accepting it before having a thorough grounding in GR is that it seems to imply that de Sitter space is a solution to GR that undergoes a rapid change over time. This leads to a few confused looks when I get to maximally symmetric spaces in my course.

You see, maximal symmetry means that you should be able to look at the space at different places and at different times and the metric should be just the same. So how are we to square that with the idea of an exponentially growing universe? Well, it all comes down to coordinate choices and the crucial existence of other matter in the universe.

Pure de Sitter space – the solution to the Einstein equations with a positive cosmological constant and no other matter sources – is, indeed, a maximally symmetric space. There exist a number of particularly useful coordinate choices for this space. In some cases, these consist of picking a useful time choice, and thus defining a family of spacelike surfaces (the spatial part of the spacetime at a constant value of this time choice). This is referred to as a slicing of the space, and it is, actually, possible to slice the space in three different ways that correspond to cosmologically expanding spaces with flat, positively-curved and negatively curved spatial parts, respectively. These are the ways of describing de Sitter space that are useful when considering inflation. However, there also exists a choice of coordinates in which the metric does not depend on time at all, and the mere existence of such a choice is enough to tell us that there is no fundamental sense in which this is an expanding cosmological spacetime. In fact, from what I just wrote, you might have a related question: even in the cosmological coordinates, what decides if the universe is flat, positively, or negatively curved?

In the case of pure de Sitter space there is no answer to these questions. All the coordinate choices are equally allowed of course, and so we might as well look at the static coordinates, and there is no cosmology here. However, importantly, in cosmology we are never interested in pure de Sitter space. We know that there is other matter in the universe. This may be either in the form of particles like us, or, in the case of inflation, the background field that causes inflation in the first place – the inflaton. These types of matter mean that the behavior of the metric is at best almost de Sitter – the difference from pure de Sitter being that, crucially, there are only certain coordinate systems in which the regular matter is homogeneous and isotropic, whereas for a cosmological constant this is true in all coordinate systems. Thus an almost de Sitter space has less symmetry than pure de Sitter. One is free to transform coordinates as much as one likes, but there will no longer be any choices in which the metric is static!

Of course, we find it most convenient to discuss cosmology in the (Friedmann, Robertson-Walker) coordinates that exploit the natural homogeneity and isotropy of the relevant matter sources. This picks out a slicing of the spacetime, and in this slicing, when the universe is almost de Sitter, the universe does expand almost exponentially rapidly – inflation! This also decides among the flat, positively and negatively curved options for the spatial part of the metric.

So it matters that inflation is “quasi-de Sitter”. It is this that gives sense to statements about inflation beginning, ending, and even operating in the way we usually describe. de Sitter space is beautiful symmetric and rich, but out real universe is somewhat messier, even at its earliest times.

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  • Andy

    Great post, Mark! So, just like in dS, is it then also possible to choose coordinates in AdS space such that the metric does not depend on time at all? In that regard, when people say that negative CC leads to a Big Crunch, they are not talking about pure AdS space, are they?

  • James

    I agree, great post. I’ve been wondering about this for a while. Do you have any links to proper formal expositions of this issue (for dS and AdS)?

  • Torbjörn Larsson, OM

    I had never thought of that. (But I haven’t studied GR either.) So does this imply that if a universe is expanding, it is certain to have interesting fields (particles and/or inflatons)?

    And that we could never have a non-expanding universe in our past, since that would have been stuck for ever? I dunno if quantum fluctuations would take care of that, I thought the SN1987 photon results told us the metric doesn’t fluctuate.

  • de Stander

    I see what you are driving at, but I’m not really convinced. There’s no global timelike Killing field in dS, and that sounds like a good definition of “time dependent”. As for static coordinates: one could argue that all you
    are doing there is chopping out a [non-geodesically complete] chunk of dS, and then adopting coordinates specially tailored to cancel out the time-dependence — the sort of thing that is often possible on incomplete chunks of spacetimes.

  • Phillip Helbig

    “In that regard, when people say that negative CC leads to a Big Crunch, they are not talking about pure AdS space, are they?”

    A negative cosmological constant will cause the universe to collapse, at least in the standard coordinates used in cosmology. If there is no matter, then it will also collapse, but in this case perhaps you have more coordinate freedom.

  • ossicle

    Oof, Mark’s posts are extremely frustrating for me, I just don’t have enough background to follow what he’s talking about, and he gets all these rave reviews from commenters about how incisive and useful they are! :p Happy for all y’all, though…

  • Mark Trodden

    James #3: Many good GR books discuss this. I don’t have Sean’s in front of me but think he discusses it a little. The various coordinates are certainly discussed in Hawking and Ellis, for example.

    de Stander #5: I think that’s a fair point. But the manifolds relevant to cosmology are typically geodesically incomplete, and we still need matter to choose among the slicings.

    ossicle #7: Sorry about that. They’re not all like this, but the last couple have been.

  • matt

    more of these posts please !

  • James

    As far as I recall, Sean’s book mentions the issue, but doesn’t go into a great deal of detail. I think other GR books are similar. I suppose a bit of arXiv-trawling is in order.


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About Mark Trodden

Mark Trodden holds the Fay R. and Eugene L. Langberg Endowed Chair in Physics and is co-director of the Center for Particle Cosmology at the University of Pennsylvania. He is a theoretical physicist working on particle physics and gravity— in particular on the roles they play in the evolution and structure of the universe. When asked for a short phrase to describe his research area, he says he is a particle cosmologist.


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