Is there a kinematic backreaction in cosmology?

I just noticed that a paper has appeared on the arXiv with the confident title There is no kinematic backreaction. Normally one can be skeptical about such bold claims, but this one is written by Nick Kaiser and he’s very rarely wrong…

The article has a very clear abstract:


This is an important point of debate, because the inference that the universe is dominated by dark energy (i.e. some component of the cosmic energy density that violates the strong energy condition) relies on the assumption that the distribution of matter is homogeneous and isotropic (i.e. that the Universe obeys the Cosmological Principle). Added to the assumption that the large-scale dynamics of the Universe are described by the general theory of relativity, this means that we evolution of the cosmos is described by the Friedmann equations. It is by comparison with the Friedmann equations that we can infer the existence of dark energy from the apparent change in the cosmic expansion rate over time.

But the Cosmological Principle can only be true in an approximate sense, on very large scales, as the universe does contain galaxies, clusters and superclusters. It has been a topic of some discussion over the past few years as to whether the formation of cosmic structure may influence the expansion rate by requiring extra terms that do not appear in the Friedmann equations.

Nick Kaiser says `no’. It’s a succinct and nicely argued paper but it is entirely Newtonian. It seems to me that if you accept that his argument is correct then the only way you can maintain that backreaction can be significant is by asserting that it is something intrinsically relativistic that is not covered by a Newtonian argument. Since all the relevant velocities are much less than that of light and the metric perturbations generated by density perturbations are small (~10-5) this seems a hard case to argue.

I’d be interested in receiving backreactions to this paper via the comments box below.


24 Responses to “Is there a kinematic backreaction in cosmology?”

  1. I was convinced that the cosmological constant in non-zero before this was generally accepted, that $H$ was higher than Sandage claimed, that Omega (matter) is much less than 1. I should have made some bets. I’ve always maintained that kinematic backreaction plays no significant role in cosmology, as those who have discussed this with me know. So I will publicly state my opinion here and gladly accept bets. 🙂

  2. I guess that the answer can be “yes” only from “a given place to stand” (Archimedes).

  3. What do you mean when you say “density perturbations are small (~10⁻⁵)”. From what I understand the problem arises when you start to integrate the problem in time. Then you just cannot talk about a snapshot of the Universe. You need to see how the density perturbations grow and curve space-time and how that depends on the inhomogeneities. Then, saying that at some point in time fluctuations are small is not at strong argument at all.

    Anyway, I’m still skeptical about the two options. I find it surprising to find people supporting a position because the believe is true, very unscientific. It seems that we will need to wait for better simulations to answer this.

    This is one of the first steps in that way

    • telescoper Says:

      The correct quotation from my piece is

      ” metric perturbations generated by density perturbations are small (~10-5) ”

      I don’t think simulations will help us much as (a) we can’t do fully relativistic cosmological simulations yet and (b) simulations are not exact anyway, because of resolution and boundary effects.

  4. The title of Nick Kaiser’s paper should really be “There is no global kinematic backreaction in Newtonian cosmology”. That statement is not controversial in the backreaction community, as the result was proven (even more generally) by Buchert and Ehlers over 20 years ago (arXiv:astro-ph/9510056). There is nothing new in arXiv:1703.08809.

    The old Newtonian result that Kaiser restates is irrelevant since the Universe is governed by General Relativity (GR). Any serious cosmology which does away with dark energy via backreaction must not only deal with GR, but also the unsolved problems of GR (see, e.g., arXiv:1612.09309).

    The timescape cosmology (see, e.g., arXiv:1311.3787) tackles such questions in a testable manner, with differences from LCDM that can be distinguished in future by the Euclid satellite. Last December I offered T. Padmanabhan a wager on this – that he did not take it up. If Nick Kaiser or any standard model cosmologist wishes to take up the wager, please get in touch.

  5. Echoing David: The result that the effect of inhomogeneities on the average expansion rate reduces in Newtonian gravity to a boundary term (and is therefore small) is correct. This, however, has been known from 1995, and is repeated in many papers on the subject.

    For example, Thomas Buchert and I say this in the first paragraph of our review:

    In GR, the effect does not reduce to a boundary term, although it remains small in perturbation theory. (See

    Note that smallness of gravitational potentials locally is not enough to guarantee that the effect is small. We in any case know that deviations in the local expansion rate are of order unity (compare collapsing regions, stabilised regions and voids). The question is whether the deviations in the positive direction and the negative direction cancel each other to good accuracy.

    This remains an open question, and there are no reliable quantitative estimates.

  6. This paper is very odd. Just after (8) it says “But since a(t) is arbitrary we may assert that a(t) is such that the RHS of (8) vanishes – i.e. that a(t) is a solution of (1) – in which case the vanishing of the LHS is equivalent to the conventional structure equations (2) and (3).” But by the same logic, one can choose a(t) so that (8) is NOT satisfied. then you do not recover the standard equations of FLRW models.

    Actually he has got the Newtonian discrete dynamics wrong. Gary Gibbons and I got it right here:, see the Theorem and Corollary on page 19.

  7. Reading that seemingly rushed out paper, I was just wondering whether cosmology is entering the fake news and alternative facts era.
    I tried to get this one straight:

  8. Some of the above points are addressed explicitly (and, basically, reinforced) by Buchert in the response he posted on the arXiv today: .

    • telescoper Says:

      Yes, he also posted a link here.

      I’m learning a lot from having posted this paper, as I haven’t really followed the literature on this.

      I still think the argument is largely irrelevant anyway, as it is entirely Newtonian.

      • Peter, the backreaction people who have posted here agree with you that the Newtonian debate is largely irrelevant. What you may have missed is the back story. A few of us have recently been contacted by science writers to comment on the work of Racz et al, who apply a heuristic non-FLRW evolution time-stepping to multi-scale partitioned Newtonian N-body simulations. Following publication in MNRAS Letters they made boldy stated press release leading to various popular news pieces that have or soon will appear, like this one in Science magazine 2 days ago:

        Nick Kaiser comments on this work in his paper. He contends that if one accepts the use of Newtonian N-body simulations, then one must also accept global FLRW evolution, which Racz et al do not. I disagree with Kaiser. While FLRW with dark energy and dark matter happens to work, demanding global average FLRW evolution is not foundationally justified. It presumes an answer to the the fitting problem – on what scale are matter and geometry dynamically coupled in Einstein’s equations? We discussed that in a CQG+ piece that you blogged in January 2016.

        Unlike Newtonian gravity, GR is causal with no action at a distance. While there is an ultimate speed limit set by light, at late epochs the energy density in radiation is negligible and most matter moves with relative speeds much slower than light, making the effective domain for the dominant causal gravitational interactions (from initial density perturbations) very small, of order 2-10 Mpc depending on one’s local environment (void or rich cluster of galaxies), a scale which Ellis and Stoeger call the matter horizon. This justifies the separate universe approximation that Racz et al use.

        As far as I am concerned, Newtonian N-body simulations work as well as they do in LCDM since there is a small scale on which Newtonian gravity is a reasonable approximation, up to issues of recalibration of rulers and clocks. Matter and geometry are decoupled in the standard approach; the cosmological constant is put in just to scale the box. A non-FLRW strong backreaction solution requires a physical principle to explain why the box effectively scales in such a simple way. In the timescape model, I do that by asking what is the largest scale on which the strong equivalence applies, to establish a new principle, the cosmological equivalence principle, for the relativity of asymptotic regions in cosmological averages.

        Intriguingly, in Fig. 5(b) Racz et al appear to reconstruct an effective comoving distance-redshift relation numerically very similar to that of the timescape model fit to the same Planck data, Fig. 8(a). Up to redshift z=6, the timescape model curve in fact tracks between the LCDM curve and Racz et al’s “AvERA analytic” curve, being a little closer to Planck-normalized LCDM than Racz et al’s model.

        Having worked on this for 10 years, with a successful phenomology – but much work still to on mathematical foundations – it is clear that most of the community find the conceptual foundations of GR just too challenging to want to take notice. Although the Racz et al averages of N-body simulations are very heuristic, maybe that is enough pique the interest of some.

  9. A new version of the paper is out:

    The title is improved. There is now more discussion of the literature. Unfortunately a lot of it is misleading. Kaiser presents things that have been studied in the literature for years as either not being addressed or as his original contributions. (E.g. the effect of pressure and the fact that statistical homogeneity and isotropy suppresses boundary terms.)

    The paper also claims to study things that are nor in the paper, like the effect of pressure (only gravity is included). Many more comments could be made.

    Kaiser also doesn’t cite Buchert’s paper (mentioned above), which corrected some of the problems of v1, which I find bad form.

    One can only hope that the referee will not wave the paper through!

  10. Kaiser’s paper has been quickly published. The published version corrects derivation errors that were listed in my comment to his first version on arXiv. I have now posted a revised comment :

  11. What should one do when one notices that someone has published a paper which contains wrong statements, neglects results in the literature, and so on? Should one publish a paper pointing this out, even if it contains no original research? Should journals be obliged to publish such “comments”? A detailed reply is a lot of work, and few will do it if the risk of rejection (due to lack of original research) is high. Sure, one can blog about it, but are there any refereed “comments” journals?

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  18. Just stumbled across this most interesting post and comments. At the risk of making a very naive observation, it seems to me that Peter’s statement of the CP is not quite correct, and the slip may relevant in this sort of study.
    The famous Cosmological Principle is not in fact he assumption that the distribution of matter is homogeneous and isotropic – it is the assumption that the spacetime metric is homogeneous and isotropic on the largest scales (Milne 1932). Of course the two are related, but they are not identical, as I understand it. John Barrow has a very nice paper on this where he shows the constraints set by the CP on matter are much less than what is commonly assumed….I presume all these advanced papers take this into account?

    • It’s obviously not homogeneous and isotropic on small scales, and on very large scales it appears to be (e.g. the CMB). The usual assumption is that the expansion of the universe is governed by the large-scale distribution, and all other motions can be seen as perturbations of this. The back-reaction question is whether the distribution is inhomogeneous enough at intermediate scales to affect (“react back”) on the expansion, thus violating the assumption that a perturbed FLRW metric is a sufficient description in practice.

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