Today I attended a cosmology discussion group where the paper being considered was by Jean-Luc Lehners and Jerome Quintin and was entitled A small Universe. Here is the abstract:

Many cosmological models assume or imply that the total size of the universe is very large, perhaps even infinite. Here we argue instead that the universe might be comparatively small, in fact not much larger than the currently observed size. A concrete implementation of this idea is provided by the no-boundary proposal, in combination with a plateau-shaped inflationary potential. In this model, opposing effects of the weighting of the wave function and of the criterion of allowability of the geometries conspire to favour small universes. We point out that a small size of the universe also fits well with swampland conjectures, and we comment on the relation with the dark dimension scenario.

arXiv:2309.03272

This paper is based on rather speculative arguments. I don’t have anything against those, but the discussion of this particular case reminded me that the idea that the Universe might be much smaller than we think is one that has come and gone many times during my lifetime. The point is that Einstein’s equations of general relativity are local in that they relate the geometric properties of space-time at specific coordinate position to the energy and momentum at the same position. When we make cosmological models based on these equations we usually assume a great deal of symmetry, i.e. that defined in a certain way the spatial sections which form surfaces of simultaneity have the same curvature everywhere, regardless of spatial position. The standard cosmology takes this curvature to be zero, in fact, so the spatial sections are Euclidean (flat), though the curvature could be positive or negative.

Usually when we assume the universe is flat we also assume that it is infinite, but it is possible in principle that a flat universe could be finite, for example in the case of a cube with opposite faces identified so that it has a sort of toroidal symmetry that has no physical edge. The size of the notional cube defines a topological scale which is independent of Einstein’s equations. That’ just a simple example: the topology does not have to be based on a cube; it could be, for example, a rhombic dodecahedron…

Likewise when we talk about a universe with negative spatial curvature we also assume it to be infinite, but that doesn’t have to be the case either: there are spaces with negative spatial curvature which are finite. A manifestation of this idea that I remember from way back in 1999 was a paper by Neil Cornish and David Spergel entitled A small universe after all?

Observing a small universe from the inside produces many interesting effects like a sort of cosmic hall of mirrors. For instance, if you can see far enough you will see the back of your own head. More realistically, the observed large-scale structure of the universe would repeat, and there would be correlated features in the cosmic microwave background. The idea is therefore amenable to observational test; the absence of any of the predicted correlations places constrains the topological scale to be comparable to the size of the observable universe or larger. Of course if it’s infinitely large then the small universe is not small at all…

(For a bit of gratuitous self-promotion, I refer you to a paper I wrote with Graca Rocha and Patrick Dineen back in 2004 using the WMAP observations of the CMB to constrain compact topologies. Given the wealth of new data we have acquired since then I’m sure the constraints are even stronger now.)

Anyway, my point is that speculative ideas are all very well but they don’t mean much if you can’t test them. This one at least has the virtue of making testable predictions.


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