Wednesday, July 1, 2015

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


  1. Because I didn't get around to it in the bulk of the post, I'll clarify my thoughts on the disagreement. Green and Wald, who claim to have proven that backreaction isn't important explicitly only consider scenarios where the true metric is close to some background metric, everywhere, for all time. Although they do allow the metric's derivatives to be large they explicitly require the metric itself to be close to the background. In most scenarios where backreaction (at least how I would define it) is important, my best understanding is that this won't be true. Green and Wald explicitly acknowledge this in all their recent papers and claim that they don't consider such a scenario to be suitably described as "backreaction".

  2. "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."

    I agree. Back when the observers said "supernovae are fainter than they should be in the Einstein-de Sitter universe" then, yes, it could have been backreaction, or grey dust, or calibration errors, or any number of things. Now the data are much better. Not only can they be fit with 1920s cosmological models, but the allowed region of parameter space is consistent with that from completely independent cosmological tests. This would require a very finely tuned conspiracy for all tests to give the "wrong" answer in precisely the same way.

  3. "This model is the FLRW metric with cold dark matter and a cosmological constant." The cosmological constant seems very good on the evidence, but Kroupa says that the Lambda-CDM model has been ruled out and that galactic dynamics is Milgromian. It seems to me that Kroupa is definitely correct. If cold dark matter particles exist, then why are there no dark matter haloes in our own solar system?
    “I came to the subject a True Believer in dark matter, but it was MOND that nailed the predictions for the LSB galaxies that I was studying (McGaugh & de Blok, 1998), not any flavor of dark matter. So what I am supposed to conclude? …” — McGaugh
    “The currently (2010) widely accepted/believed description of the birth and evolution of the universe and of its contents is "Lambda Cold Dark Matter Concordance Cosmological Model" (LCDM CCM) … My own research was very much confined to the early version of the LCDM CCM (mid-1990's) when I began performing numerical experiments on the satellite galaxies of the Milky Way. I was quite happy with the CCM, as everyone else, and did not bother with the fundamental issues raised by some. With time, however, it became apparent that the LCDM CCM accounts poorly for the properties of the satellite galaxies and their distribution about the Milky Way. Warm dark matter models fared no better.” — Kroupa

  4. Just because Kroupa (no, he is not the new Kepler) says so doesn't make it so, and neither does your posting similar comments wherever a blog post touches this topic (at least they are not grossly off-topic).

    Why no dark matter haloes in our own solar system? Is this a serious question? Check out the size of even a small halo.

    I'm sure Stacy has learned some things since 1998. Yes, MOND seems to work better in some regimes, which, probably not coincidentally, are regimes where simulations are very hard, because small-scale "gastrophyics" and what not are involved. However, there are other regimes where LCDM works better. Touting one and ignoring the other doesn't help the debate. Even if his LSB galaxies work better with MOND, one has to come up with some explanation what the alternative to LCDM is where it does work, and MOND doesn't.

    The same comments apply to Kroupa, though perhaps he hasn't learned as much since 1998. Debate is good, and Kroupa even debated Simon White. But the it seems that MOND, while intriguing, is not the last word, and LCDM won't just be thrown out the window. Probably, a solution which explains why MOND works, whether or not such a solution involves some sort of MOND, would probably leave LCDM untouched in other areas.

  5. My guess is that, 10 years from now, a popular equation will be: dark-matter-particles = phlogiston. My guess is that Gravity Probe B discovered the Fernández-Rañada-Milgrom effect — can anyone point out a previous case in which 4 (out of 4) scientific instruments malfunctioned in a remarkably predictable manner? I say that that MOND has been outrageously ignored — for example:
    "Spacetime, Spin and Gravity Probe B" by James M. Overduin, 2015
    According to the research of Milgrom, McGaugh, Kroupa, Pawlowski, and several others, there are 2 possibilities:
    (1) Newtonian-Einsteinian gravitational theory is 100% correct but appears to be significantly wrong for some unknown reason.
    (2) Newtonian-Einsteinian gravitational theory really is slightly wrong.

  6. Shaun: you are perfectly correctly that Green and Wald are using a different definition of backreaction, and in arxiv:1506.06452 they state that their formalism was "never intended or claimed to apply" to various approaches to backreaction that constitute much of the literature.

    Unfortunately, time-challenged non-experts only read titles and abstracts, rather than studying the fine print of the content of papers peripheral to their own research. In fact, Green and Wald's mathematical results are inapplicable to much of the literature, because the averaged cosmic variables in many backreaction schemes are not an exact solution of Einstein's equations, but rather of equations with additional backreaction terms on any coarse-grained scale, and not only in an ultra-local limit.

    In their past work, Green and Wald have strongly criticized other approaches while arguing that their own framework should provide an accurate description of the actual Universe, leaving it unclear to what extent their mathematical results are limited by restrictive assumptions which may not apply to the actual physical problem.

    In Green and Wald's framework backreaction can only contribute a radiation-like trace-free term to the effective stress-energy tensor in the average equations. We show that such a result is not a feature of general averaging schemes. Furthermore, even if an average background FLRW metric is assumed, Green and Wald's proof of the trace-free property relies on specific technical assumptions when taking weak-limits, one of them implying that the scheme does not actually provide an averaging operation as it is generally understood. If one takes a more general mathematical framework in dealing with such limits, specifically by emphasizing mathematical and physical control on the properties of curvature and matter fluctuations, then their result does not follow.

    In a new paper, arxiv:1601.06789, they refer to their non-peer reviewed note arxiv:1506.06452 as giving a "refutation" of our arguments. Our referees, one of whom effectively also wrote a report on arxiv:1506.06452 disagreed. To be clear we have not claimed that they make an error of logic, but rather that their framework is too narrow to capture the essential physics of the averaging and backreaction problems.

    We see no point in continuously going over these arguments. We wish to move things on, as there is much exciting work to be done, and the point of our article was to dispell a few myths so people are not discouraged to work in this field. Thus when invited to write a piece for CQG+, we have focused on the challenges of the future:

    1. Thanks for the comment David. I think your words below summarise the situation to me quite well. I'm confused as to why they're still pushing this results as so general.

      "To be clear we have not claimed that they make an error of logic, but rather that their framework is too narrow to capture the essential physics of the averaging and backreaction problems."


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