When astronomers look at distant Type-Ia supernovae, they find them to be dimmer than expected. The most popular interpretation of this result is that it’s caused by the expansion of the Universe getting faster – accelerating – and so the supernovae are further away than we first thought. This conclusion is only valid if you assume that the cosmos is filled with homogeneously-distributed matter. In practise, we know that that’s not the way things are (we see galaxy clusters surrounded by large voids; certainly not homogeneous). But, if you can smooth over all of the lumps and bumps on sufficiently large scales, you should get a homogeneous model out. The model that describes this situation is called a Friedmann-Lemaitre-Robertson-Walker (FLRW) model, and when you fit that to the supernova data, you find that about 70% of the energy density of the Universe has to be made up of some sort of mysterious “dark energy“. This is weird. We don’t know what it is.

Much of what we do in cosmology rests on assuming that this smoothing procedure is valid, but actually, we aren’t quite sure that it is yet*. It might be the case that the Universe isn’t homogeneous on large scales, for example, or that the smoothing procedure introduces corrections into the equations for the FLRW model (so-called “backreaction” effects). So, to find out whether the smoothing procedure is sensible or not, people look at inhomogeneous models of the Universe. By necessity, these are gross simplifications; working with inhomogeneous general relativistic models is hard, and you tend only to be able to solve the relevant equations for simple models with lots of symmetry. The real Universe isn’t so symmetric.

Once you move away from the simplicity of the homogeneous FLRW models, lots of things change. In particular, the choice of model strongly affects how you interpret your observations. Over the past few weeks (OK, months…), we’ve been looking at the way acceleration is defined in inhomogeneous spacetimes, to see how conclusions about the existence of dark energy are affected by throwing inhomogeneity into the mix.

We pick out four possible definitions of acceleration as being particularly relevant to current ideas on the origin of the apparent accelerating expansion:

- The local acceleration of the expansion of a spatial volume, governed by the Raychoudhuri equation;
- The acceleration inferred by an observer fitting an FLRW distance-redshift relation to the low-z part of their observed Hubble diagram;
- The acceleration inferred by reconstructing only the Hubble diagram locally (i.e. exactly at z=0), as described by Kristian and Sachs;
- The acceleration of the effective scale factor of volumes that have been spatially averaged according to the Buchert scalar averaging procedure.

For an exactly homogeneous and isotropic FLRW model, all of these definitions converge to give the same measure of acceleration. In the general case, however, they can all be quite different, to the point where some of them can show strong acceleration, whilst others are decelerating. This is shown in the plot above, for a “spherical collapse” model. This is an inhomogeneous model made up of a bunch of disconnected FLRW regions with different expansion rates and densities. For definitions (2) and (4), we see acceleration, but for (1) and (3) we see deceleration. Yikes. The upshot is that using supernovae might not be the best idea if you want to find out if the apparent acceleration is caused by a cosmological constant or not. It also has some interesting things to say about the relationship between what we actually observe, and what happens when you work with a smoothed-out model (the “fitting problem”).

We’re close to releasing a paper on this, so I’ll have more on it soon.

** When I say “we”, I actually mean “a small subset of the cosmology community”. I think it’s fair to say that many cosmologists feel that the issue is settled, or are not even aware of it. My personal feeling is that it’s not settled, and there’s still much work to be done before it is. But hey, everyone thinks that their own area of research is important, right?*

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