Modified Gravity and the Radial Acceleration Relation, Again

Have I recently mentioned that I am now proud owner of my personal modified gravity theory? I have called it “Covariant Emergent Gravity.” Though frankly I’m not sure what’s emergent about it; the word came down the family tree of theories from Erik Verlinde’s paper. Maybe I had better named it Gravity McGravace, which is about equally descriptive.

It was an accident I even wrote a paper about this. I was supposed to be working on something entirely different – an FQXi project on space-time defects – and thought that maybe Verlinde’s long-range entanglement might make for non-local links. It didn’t. But papers must be written, so I typed up my notes on how to blend Verlinde’s idea together with good, old, general relativity.

Then I tried to forget about the whole thing. Because really there are enough models of modified gravity already. Also, I’m too fucking original to clean up somebody else’s math. Besides, every time I hear the name “Verlinde” it reminds me that I once confused Erik Verlinde with his brother Herman, even though I perfectly know they’re identical twins. It’s a memory I’d rather leave buried in the depths of my prefrontal cortex.

But next thing I know I have a student who wants to work on modified gravity. He’s a smart young man. Indeed, I now think he is a genius. See, while I kept blathering about the awesomeness of McGaugh et al’s recent data on the radial acceleration relation, he had the brilliant idea of plotting the prediction from my model over the data.

Figure 1 from arXiv:1803.08683

Eh, I thought, look at this. (Deep thoughts are overrated.)

The blue squares in this figure are the data points from the McGaugh et al paper. The data come from galactic rotation curves of 156 galaxies, spanning several orders of magnitude. The horizontal axis (g_B) shows the acceleration that you would expect from the “normal” (baryonic) mass. The vertical axis (g_tot) shows the actually observed (total) acceleration. The black dotted line is normal gravity without dark matter. The red curve is the prediction from my model; 1σ-error in pink. For details, see paper.

As the data show, the observed acceleration is higher than what the normal (Newtonian limit of ) general relativity predicts, especially at low accelerations. Physicists usually chalk this mismatch up to dark matter. But we have known for some decades that Milgrom’s Modified Newtonian Dynamics (MOND) does a better job explaining the regularity of this relation, in the sense that MOND requires less fumbling to fit the data.

However, while MOND does a good job explaining the observations, it has the unappealing property of requiring an “interpolation function”. This function is necessary to get a smooth transition from the regime in which gravity is modified (at low acceleration) to the normal gravity regime, which must be reproduced at high acceleration to fit observations in the solar system. In the literature one can find various choices for this interpolation function.

Besides the function, MOND also has a free constant that is the acceleration scale at which the transition happens. At accelerations below this scale, MOND effects become relevant. Turns out this constant is to good approximation the square root of the cosmological constant. No one really knows why that is so, but a few people have put forward ideas where this relation might come from. One of them is Erik Verlinde.

Verlinde extracts the value of this constant from the size of the cosmological horizon. Something about an insertion of mass into de-Sitter space changing the volume entropy and giving rise to a displacement vector that has something to do with the Newtonian potential. Among us, I think this is nonsense. But then, what do I know. Maybe Verlinde is the next Einstein and I’m just too dumb to understand his great revelations. And in any case, his argument fixes the free constant.

Then my student convinced me that if you buy what I wrote in my last year’s paper, Covariant Emergent Gravity doesn’t need an interpolation function. Instead, it gives rise to a particular interpolation function. So then, we were left with a particular function without free parameters.

If you have never worked in theory-development, you have no idea how hair-raisingly terrible a no-parameter model is. It either fits or it doesn’t. There’s no space for fudging here. It’s all or nothing, win or lose.

We plotted, we won. Or rather, Verlinde won. It’s our function with his parameter that you see plotted in the above figure. Fits straight onto the data.

I’m not sure what to make out of this. The derivation is so ridiculously simple that Kindergarten math will do it. I’m almost annoyed I didn’t have to spend some weeks cracking non-linear partial differential equations because then at least I’d feel like I did something. Now I feel like the proverbial blind chick that found a grain.

But well, as scientists like to say, more work is needed. We’re still scratching our heads over the gravitational lensing. Also the relation to Khoury et al’s superfluid approach has remained murky.

So stay tuned, more is to come.