Hamster. Not to scale. Img src: Petopedia. |

Dark matter, to remind you, are hypothetical clouds of particles that hover around galaxies. We can’t see them because they neither emit nor reflect light, but we do notice their gravitational pull because it affects the motion of the matter that we *can* observe. Modified gravity, on the other hand, posits that normal matter is all there is, but the laws of gravity don’t work as Einstein taught us.

In a 2016 paper, McGaugh, Lelli, and Schomberg analyzed data from a set of about 150 disk galaxies. They identified the best-fitting acceleration scale for each of them and found that the distribution is clearly peaked around a mean-value:

Histogram of best-fitting acceleration scale. Blue: Only high quality data. Via Stacy McGaugh. |

McGaugh *et al* conclude that the data contains evidence for a universal acceleration scale, which is strong support for modified gravity.

Then, a month ago, Nature Astronomy published a paper titled “Absence of a fundamental acceleration scale in galaxies“ by Rodrigues et al (arXiv-version here). The authors claim to have ruled out modified gravity with at least 5 σ, ie with high certainty.

That’s pretty amazing given that two months ago modified gravity worked just fine for galaxies. It’s even more amazing once you notice that they ruled out modified gravity using the same data from which McGaugh *et al* extracted the universal acceleration scale that’s evidence for modified gravity.

Here is the key figure from the Rodrigues *et al* paper:

Figure 1 from Rodrigues et al |

A first observation is that the two studies don’t use the same data analysis. The main difference is the priors for the distribution of the parameters which are the acceleration scale of modified gravity and the stellar mass-to-light ratio. Where McGaugh *et al* use Gaussian priors, Rodrigues *et al* use flat priors over a finite bin. The prior is the assumption you make for what the likely distribution of a parameter is, which you then feed into your model to find the best-fit parameters. A bad prior can give you misleading results.

Back to the galaxies. As we’ve seen, if you start with an unmotivated prior, you can end up with a “best fit” (the 300 pound hamster) that’s unlikely for reasons your software didn’t account for. At the very least, therefore, you should check that whatever the resulting best-fit distribution of your parameters is doesn’t contradict other data. The Rodrigues *et al* analysis hence raises the concern that their best-fit distribution for the stellar mass-to-light ratio doesn’t match commonly observed distributions. The McGaugh paper on the other hand starts with a Gaussian prior, which is a reasonable expectation, and hence their analysis makes more physical sense.

Let me tell you a story to illustrate what’s going on. Suppose you are Isaac Newton and an apple just banged on your head. “Eureka,” you shout and postulate that the gravitational potential fulfils the Poisson-equation.^{*} Smart as you are, you assume that the Earth is approximately a homogeneous sphere, solve the equation and find an inverse-square law. It contains one free parameter which you modestly call “Newton’s constant.”

Back to 2018 and modified gravity. Same difference. In the Rodrigues *et al *paper, the authors rule out that modified gravity’s one-parameter law fits all disk galaxies in the sample. This shouldn’t come as much of a surprise. Galaxies aren’t disks with bulges any more than the Earth is a homogeneous sphere. It’s such a crude oversimplification it’s remarkable it works at all.

Without the comparison to particle dark matter, therefore, the only thing I learn from the Rodrigues *et al* paper is that a non-universal acceleration scale fits the data better than a universal one. And that I could have told you without even looking at the data.

^{*}Dude, I know that Newton isn’t Archimedes. I’m telling a story not giving a history lesson.