We have in MOND a formula that has had repeated predictive successes. Many of these have been true a priori predictions, like the absolute nature of the baryonic Tully-Fisher relation, the large mass discrepancies evinced by low surface brightness galaxies, and the velocity dispersions of  many individual dwarf Spheroidal galaxies like Crater 2. I don’t see how these can be an accident. But what we lack is an underlying theoretical basis for the observed MONDian phenomenology: Why does this happen?

One apparently promising idea is the emergent gravity hypothesis by Erik Verlinde. Gravity is not a fundamental force so much as a consequence of microscopic entanglement. This manifests on scales comparable to the Hubble horizon, in particular, with an acceleration of order the speed of light times the Hubble expansion rate. This is  close to the acceleration scale of MOND.

An early test of emergent gravity was provided by weak gravitational lensing. It does remarkably well at predicting the observed lensing signal with no free parameters. This is promising – perhaps we finally have a physical basis for MOND. Indeed, as Milgrom points out, the equivalent success was already known in MOND.

The weak lensing signal for galaxies stacked in several mass bins from Brouwer et al. (2016). The straight line predicted by emergent gravity (EG) is a good fit to the data with no free parameters to adjust.

Weak lensing occurs deep in the MOND regime, at very low accelerations far from the lensing galaxy. In that regard, the results of Brouwer et al. can be seen as an extension of the radial acceleration relation to much lower accelerations. In this limit, it is fair to treat galaxies as point masses – hence the similarity of the solid and dashed lines in the figure above.

For rotation curves, it is not fair to approximate galaxies as point masses. Rotation curves are observed in the midst of the stellar and gaseous mass distribution. One must take account of the detailed distribution of baryons to treat the problem properly. This is something MOND is very successful at.

Emergent gravity converges to the same limit as MOND in the point mass case, which holds for any mass distribution once you get far enough away. It is not identical for finite mass distributions. When one solves the equation of emergent gravity for a finite mass distribution, you get one term that looks like MOND, which gives the success noted above. But you also get an additional term that depends on the gradient of the mass distribution, dM/dr.

The additional term that emerges for extended mass distributions in emergent gravity lead to different predictions than MOND. This is good, in that it makes the theories distinguishable. This is bad, in that MOND already provides good fits to rotation curves. Additional terms are likely to mess that up.

And so it does. Two independent studies recently come to this conclusion: one including myself (Lelli et al. 2017) and another by Hees et al. (2017). The studies differ in their approach. We show that the additional term in emergent gravity leads to the prediction of a larger mass discrepancy than in MOND, driving one to abnormally low stellar mass-to-light ratios and degrading the empirical radial acceleration relation. Hees et al. make detailed rotation curve fits, showing that the dM/dr term over-amplifies bumps & wiggles in the rotation curve. It has always been intriguing that MOND gets these right: this is a non-trivial success to reproduce.

Rotation curves predicted for various exponential disks (top row) by Newton (dotted line), MOND (red dashed line), and emergent gravity (solid line). Note that emergent gravity predicts more of a hump at the peak of the rotation curve, leading to the hook in the corresponding radial acceleration relation (lower panels). From Lelli et al. (2017).

The situation looks bad for emergent gravity. One caveat is that at present we only have solutions for emergent gravity in the case of a spherical cow. Conceivably a better treatment of the geometry would change the result, but it won’t eliminate the dM/dr term. So this seems unlikely to help with the fundamental problem: this term needs not to exist.

Perhaps emergent gravity is a clue to what ultimately is going on – a single step in the right direction. Or perhaps the similarity to MOND is misleading. For now, the search for a satisfactory explanation for the observed phenomenology continues.

10 thoughts on “Emergent Gravity hits a pothole

  1. Is the dM/dr term something that is unavoidable in emergent gravity theories, or is it particular to Dr. Verlinde’s theory?


  2. The dM/dr term is particular to Verlinde’s theory as it now stands. It is much harder to say whether it is generic to all possible emergent gravity theories. It does arise in a pretty natural way there, so may be hard to get rid of. But theorists are very clever, so one can never say never. Perhaps there could be some screening mechanism. But rotation curve fits do not just fall out in the satisfactory way the lensing result did.


    1. “… all possible emergent gravity theories …” My guess is that given any plausible empirical data there exists some emergent gravity theory that can adequately approximate such data.
      “According to Verlinde (2016), “The central idea of this paper is that the volume law contribution to the entanglement entropy, associated with the positive dark energy, turns the otherwise “stiff” geometry of spacetime into an elastic medium. We find that the elastic response of this ‘dark energy’ medium takes the form of an extra ‘dark’ gravitational force that appears to be due to ‘dark matter’.”
      https://arxiv.org/pdf/1611.02269.pdf “Emergent Gravity and the Dark Universe”
      According to Motl (with omission of much negative response), “… Erik Verlinde’s paper could be a sketch of something that is on the right track. …”
      It might be possible to fix Verlinde’s theory by starting with some kind of emergent function that matches MOND and then constructing the emergent elastic medium from the MOND-compatible emergent function.


  3. Hi Stacey! Thank you a lot for sharing your insights about our fantastic universe! I was wondering if you heard about the Hajdukovic hypothesis (https://arxiv.org/pdf/1106.0847.pdf), namely the one that starts assuming that the gravitational interactions of antimatter and matter are repulsive by nature ( thus killing the WEP)? In this case it follows that the quantum vacuum is a gravitational dipolar medium that can exhibit anti-screening effects at large scales. It seems that it reproduces the MOND phenomenology. Did you have some time to delve into it and if yes, what is your opinion about it?


  4. I have heard the possibility of matter and antimatter behaving differently in this context, but have no well-formed opinion as to how it applies to MOND. It seems unlikely to me to matter. But we live in an implausible universe.


  5. Thanks for you answer! I guess you’re being quite busy ATM with all the “buzz” around dark matter recently. I’ll keep following your work with great interest. FYI I’ve been reading your MOND pages since 2007, the year of the “bullet cluster” paper if my memory serves. Your analysis are always a fresh breath in the tornado of “epicyclical” science we have to currently endure. Hopefully science will prevail and we will checkmate the new inquisitors ( I think we can fairly call them that). But please, take a lil’ time to read the paper in the link, because it seems that you ignore the ideas in it. They are relevant to MOND, I promise!


  6. I can only read so many million papers at a time. This one sounds related to the gravitationally polarized dark matter of Blanchett, and the vacuum inertia of Milgrom. These are promising leads, but I think a lot of work has to be done to sort out which commonalities may be important. If I can figure that out I’ll let the world know.


  7. Blanchet found that MOND phenomology was somehow reminiscent of some sort of polarization effect due to a gravitational dipolar medium. But to have to introduce a new kind of “di-gravitational dark matter” is not satisfying IMHO as it kills the purpose of a modified theory of gravity like MOND without DM. Hajdukovic thinks that this gravitationaly polarizable medium could very well simply be the quantum vacuum if and only if matter and antimatter are mutually repulsive with respect to gravitational forces. There is an interesting numeric coincidence that seems to appear in this context: the universal MOND parameter a0 seems to be of the order of the gravitational field created by a pion at a distance equal to its Compton wavelength. This makes sense: only when the gravitational acceleration becomes less than the mutual gravitational acceleration of virtual pions , the anti-screening effect due to the QCD vacuum ( dominated by virtual pions) start to kick in, because only then saturation of gravitational dipoles with pion mass is achieved. This point to a new anti-gravity paradigm IMHO, where gravitational effects at large scales could very well be dominated by the gravitational polarization of the QCD vacuum, as postulated by Hajdukovic.


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