The External Field Effect and Tests of MOND on Cosmological Scales

The effective force law in galaxies is MOND. At high accelerations, this is the same as Newton’s inverse square law: $g = GM/r^2$. At low accelerations, the deep MOND limit is $g = \sqrt{a_0GM}/r$. The transition between the regimes occurs at $g = a_0 = 1.2 \times 10^{-10}\;\mathrm{m}\,\mathrm{s}^{-2}$.

This is well established in galaxies. The question naturally arises whether this holds on larger scales. A recent paper by Gallardo et al. says no:

On scales from 30 – 230 megaparsecs, we constrain the gravitational acceleration between pairs of halos^$ at separation r to be g ∝ 1/rⁿ with n=2.1±0.3

so Newton’s n = 2, not the n = 1 of the deep MOND limit. Here is their result for the kinematic SZ effect*.

Here the blue line represents Newton’s n = 2 with the boost of dark matter as appropriate for LCDM. This matches the data better than the yellow line with n = 1 representing MOND. But is that the right representation?

Although the bulk of our analysis is model independent, we have also shown the theoretical curves appropriate for MOND. This test is the largest-scale direct test of MOND to date. Our formalism is an excellent approximation to MOND in the low-acceleration regime

This much is good. It’s a nice, general approach to represent 1/r force laws like the deep MOND limit. But is that the right thing to do here?

that said, we have not included the external field effect (EFE) in this analysis. This can modify the acceleration law in the case where the baryonic gravity of an object is less than the gravity of its larger environment, and has been used to explain^# the velocity dispersion of satellite galaxies. However, this effect likely does not affect our analysis.

because the EFE absolutely affects this analysis.

They go on to say words about how the EFE is a thing that applies on small scales (hence the allusion to dwarf galaxies) but they’re looking at large scales so it shouldn’t matter. If only that were true.

The length scale does not matter in MOND. The acceleration scale matters. Is the chunk of the universe over which they’re integrating in the low acceleration regime? Yes. Is the EFE relevant on these scales? Also yes.

Some EFE from large scale structure is unavoidable. Everything feels the tug of everything else. In the deepest depths of the intergalactic medium, the EFE is tiny (maybe 1% of a₀) but also ubiquitous. In the absence of a prominent mass, it dominates. That’s exactly the regime this experiment probes.

The force in the MOND EFE regime looks like a boosted version of Newton: $g \approx (a_0/a_{\mathrm{EFE}})\,GM/r^2$. The boost factor $a_0/a_{\mathrm{EFE}}$ is what we interpret as dark matter: the total mass that we need in excess of what we observe.

So, what Gallardo et al. falsify is a straw man version of MOND in which the force law has n = 1 forever and always. That’s only true in the absence of the EFE, for which the prediction is n = 2 – as observed.

There may be a test in the amplitude of the boost factor. We already know that is in the ballpark that’s required for the cosmic dark matter, $\Omega_m/\Omega_b \approx a_0/a_{\mathrm{EFE}}$, as this is one of the first things I checked when I was surprised to encounter MOND in the previous century. So I don’t see much hope in distinguishing between the two this way.

Reality is more complicated. The amplitude of the EFE throughout intergalactic space depends on the cosmic distribution of mass. This was calculated by Chae et al. (2021), who found it to vary with both distance (from us) and position on the sky:

It’s even worse than this, because most of the baryons are in the IGM. It makes a difference to the amplitude of the EFE how clumpy these are. They’re probably somewhat clustered into filaments and walls, but we don’t really have a great empirical map of that. So: the EFE is definitely there at a level that matters, but precisely what that level might be is rather hard to say. My best guess today is ~2% of a₀, but it could be more, and probably is in places. Could be less in the midst of the deepest voids.

Gallardo et al. do not cite Chae et al. (2021), nor evince awareness^& that there are relevant constraints on the EFE on the scales they probe. Note that the distance range of the figure from Chae et al. (2021) goes out to 150 Mpc, which is where the relevant data of Gallardo et al. are (their last two points are largely irrelevant). So we are talking about very much the same length scales, which does matter to the integration they do. That integration per force averages over any real variations in the EFE.

There is nowhere you can go to completely escape the EFE. A rather profound fact that appears not to be widely appreciated is that there is a minimum cosmic acceleration of order $\sim 10^{-12}\;\mathrm{m}\,\mathrm{s}^{-2}$.

---

^$Saying “halos” here is quite the linguistic bias since there are no dark matter halos in the theory they’re testing.

*There are less obscure ways to do this, but this is what they chose to do and I’m not going to attempt to unravel it here. Ask them.

^#Not just explain, but predict. Repeatedly. Until the same ability to predict kinematics in advance of observation is demonstrated by LCDM, I’m going to remain underwhelmed by post-facto tests that claim to favor LCDM over MOND.

^%If I recall the notation correctly, the actual EFE is $\sqrt{a_0\,e_{\mathrm{N,env}}}$ and in these units $a_0 = 1$ so $\log(e_{\mathrm{N,env}}) = -3$ corresponds to an EFE that is 3% of $a_0$. This plot was made assuming maximal clumping of the IGM baryons which gives an upper limit, so the reality is probably less; see Fig. 6 of Chae et al.

^&I wonder what the refereeing process looked like. I can imagine there being no mention of the EFE initially, with a referee (or perhaps one of the many coauthors) asking if they ought to maybe worry about it and the majority choosing to wave their hands through it. That’s what it looks like.

This is not the first time I’ve encountered the misconception that the EFE can be ignored on cosmic scales. It happened in the submitted version of Aguirre et al. (2001). In that case, I was the referee, and pointed out that the EFE had a profound effect on what they were saying about Lyman-alpha absorbers. To their credit, they listened and corrected it. I think they still kinda low-balled the amplitude of the EFE, but it went from a factor of tens problem to a small problem or maybe not a problem at all.

While on the topic of refereeing, I note that papers that find MOND wanting generally get less scrutiny than those which find that it works. Hardly surprising, once framed that way. I only mention this because there are certain toxic science communicators who rush to social media to denounce the incompetence of the referee any time a MOND-positive paper appears. Funny how they can be sure of the competence of the referee in a process they are not in any way privy to. If you take the time to think about it, you can more likely presume that the MOND paper has been held to a higher, not lower, standard, and weigh its credence appropriately.