Cosmological Backreaction

In the last few weeks a disagreement has surfaced at the arXiv. The disagreement concerns whether backreaction is important in cosmology.

To summarise my take on the whole thing, it seems to me that the two sides of this disagreement are, to a large extent, talking past each other. I don't doubt that there is genuine disagreement where definitions overlap, but, at least to my present understanding, much of the disagreement actually just lies in what should be considered "backreaction". There seems to be a secondary, though related, disagreement concerning whether one should start with observations and use them to methodically construct a model of the universe, or instead start with a model of the universe and then see whether it fits the data. The side that favours first constructing the model would say that a model without any backreaction is entirely self-consistent and fits the data well enough not to be concerned. To the other side this still doesn't prove that backreaction must be negligible.

But OK, what is cosmological backreaction?

Backreaction itself is quite a common term in physical sciences.

In a surprising proportion of calculations about nature we would normally analyse some sort of interesting object, existing within some external system, but in a scenario where the behaviour of the object has no measurable influence on the overall system. Then, calculating predictions essentially amounts to two independent steps: firstly, calculating what the background system is doing, and then calculating how the interesting object will react to that.

However, this type of scenario isn't always accurate. When it isn't, the background system could be described as "backreacting" to the object's behaviour.

Backreaction effects often make calculations much more difficult. Essentially, you can't determine what the object will do until you know what the background is doing, but with backreaction you don't know what the system is doing until you know what the object is doing.

With cosmological backreaction the interesting objects are the structures in the universe. These are the things we can observe and are the things we can then use to learn about the universe as a whole. If backreaction doesn't exist, then we can happily calculate what we expect for the average behaviour of the universe and see whether the structures we see match that prediction. If backreaction does exist, we can't, at least not so easily.

Well then, is it important?

Most of the cosmology community would, with varying degrees of confidence, predict that, up to the level of accuracy we have currently measured the universe, the formation of structures does not affect the average behaviour of the universe. The reasons why this belief is prevalent might vary person to person. To me, by far the most convincing one is that there is a modelfor the average behaviour of the universe that fits observations very well  and assumes any backreaction is small enough to be ignored. This model is the FLRW metric with cold dark matter and a cosmological constant.

This isn't a particularly satisfying reason though. The behaviour of the universe, on the scales relevant to the formation of structures, and larger, is described by general relativity. This is a complete, deterministic theory. Surely, one can just calculate how big the backreaction is and know whether it is big or not?

It turns out this isn't so simple and is why there can be arguments about how big/relevant the effect can be. The reason for this is the following:

* In general relativity there is a set of equations (Einstein's equations) that describe what gravity is like given what matter there is and what the matter is doing.
* Einstein's equations are non-linear - i.e., very loosely, if you double the amount of matter you don't just double the "amount" of gravity.
* This non-linearity means that averages do not commute. What this means is that even if we know what the average distribution of matter in the universe is, this doesn't mean that we can naively use Einstein's equations to determine what the average gravitational degrees of freedom (i.e. the metric) are.
* The FLRW metric that describes the average gravitational behaviour of the universe in the standard cosmological model requires a distribution of matter that is homogeneous and isotropic. That is, the same everywhere and with no special direction.

It might very well be the case that, on average, the universe is both homogeneous and isotropic. However, what makes the backreaction calculation incredibly difficult is that, on the scales where structures exist, the universe is very, very far from either.

If the no-backreaction model works, why do people care?

If we don't know how big it is, backreaction could in principle show up in measurements at any time.

If tomorrow a significant anomaly shows up that doesn't go away and becomes more and more significant as similar measurements are made then everybody with their own pet dark matter or dark energy model would jump on the anomaly. Some of these pet models would fit the anomaly well. If that "anomaly" was just a consequence of backreaction we could then be faced with a situation where some new modified gravity, or dark matter, model becomes crowned when all we've done is measure a subtle effect of general relativity.

In fact, some people would argue that this has already happened. In 1999 such an anomaly was measured. Supernovae seemed dimmer than they should be. The missing thing that was need to explain this was labelled "dark energy". The model that has now become the standard cosmological model introduced a cosmological constant to the gravitational side of Einstein's equations. It so happened that this model was simple and has survived and fit the data well. But, at least for a while, there was a lot of speculation that the apparent acceleration could be due to backreaction.

There is still some speculation that dark energy might just be backreaction but that particular possibility seems very unlikely, at least to me, in 2015. Having said that, it hasn't been absolutely proven to be incorrect and just because right now I would require pretty long odds before betting on it, doesn't mean future evidence might show it to be true.

Or someone might conclusively rule it out tomorrow.

I'll try to elaborate on this (all) some more in future posts...

Twitter: @just_shaun