On the functional form of the radial acceleration relation

TL;DR

The study applies Exhaustive Symbolic Regression (ESR) to the SPARC radial acceleration relation (RAR) to derive analytic mappings between baryonic and total accelerations without strong priors and to test MOND-like limiting behavior. By exhaustively generating functions up to complexity 9 and using minimum description length (MDL) as a model selector, ESR identifies functions that describe the data more accurately and more simply than traditional MOND interpolating functions. The results indicate that, although many top ESR functions approximate $g_\text{obs} \propto g_\text{bar}$ at high accelerations, the coefficient is not universally unity and the deep-MOND limit is not robustly supported by SPARC; mock MOND data show ESR may not reliably recover the generating function in finite dynamical range. Consequently, SPARC data alone are insufficient to decisively confirm or rule out law-like gravity, and stronger constraints will require expanded dynamical range and more sophisticated treatment of uncertainties and covariances.

Abstract

We apply a new method for learning equations from data -- Exhaustive Symbolic Regression (ESR) -- to late-type galaxy dynamics as encapsulated in the radial acceleration relation (RAR). Relating the centripetal acceleration due to baryons, $g_\text{bar}$, to the total dynamical acceleration, $g_\text{obs}$, the RAR has been claimed to manifest a new law of nature due to its regularity and tightness, in agreement with Modified Newtonian Dynamics (MOND). Fits to this relation have been restricted by prior expectations to particular functional forms, while ESR affords an exhaustive and nearly prior-free search through functional parameter space to identify the equations optimally trading accuracy with simplicity. Working with the SPARC data, we find the best functions typically satisfy $g_\text{obs} \propto g_\text{bar}$ at high $g_\text{bar}$, although the coefficient of proportionality is not clearly unity and the deep-MOND limit $g_\text{obs} \propto \sqrt{g_\text{bar}}$ as $g_\text{bar} \to 0$ is little evident at all. By generating mock data according to MOND with or without the external field effect, we find that symbolic regression would not be expected to identify the generating function or reconstruct successfully the asymptotic slopes. We conclude that the limited dynamical range and significant uncertainties of the SPARC RAR preclude a definitive statement of its functional form, and hence that this data alone can neither demonstrate nor rule out law-like gravitational behaviour.

Figures (4)

• Figure 1: The Simple, Standard and RAR IFs, a double power law, and Simple IF with a global external field strength in AQUAL, overlaid on the SPARC data (blue points). The parameters are set to their maximum-likelihood values shown in Table \ref{['tab:real_results']}. The dashed black line shows the one-to-one relation (Newtonian limit) and the cross in the lower right shows the average uncertainty size.
• Figure 2: The top 20 functions found by ESR overlaid on the SPARC data (blue points), colour-coded by their relative probability in the full function list. The top panel fits the SPARC data, the middle panel mock data generated by the RAR IF, and the bottom panel mock data generated by the Simple IF with universal external field strength $g_\text{ex}=1.2\times10^{-12}$ m s$^{-2}$. The mock datasets are $10$ times larger than SPARC, although this is factored out in the description length calculation.
• Figure 3: The logarithmic slopes $s \equiv \frac{d\log(g_\text{obs})}{d\log(g_\text{bar})}$ of the top 10 ESR functions on each dataset, for comparison with the low- and high-$g_\text{bar}$ MONDian expectations 1/2 and 1 respectively (blue and red vertical dashed lines). The blue and red points are the limiting slopes $s_- \equiv \lim_{g_\text{bar}\to0^+} s$ and $s_+ \equiv \lim_{g_\text{bar}\to\infty} s$, while cyan and magenta indicate the slopes at the minimum and maximum $g_\text{bar}$ of the SPARC data (0.0083 and 65.4). In case a slope depends on a parameter value we show the 95% confidence interval as a bar (often very thin), obtained from an MCMC fit. Arrowheads indicate points or bars beyond the range of the plot.
• Figure 4: The Pareto fronts identified by ESR for the SPARC, RAR IF mock and Simple IF + EFE mock datasets, for both $\log(\mathcal{L})$ (blue) and total description length $L(D)$ (red). The quantities plotted have the minimum values subtracted so that the best results appear at 0. Also shown are the results of the RAR, Simple and Standard IFs, Simple IF + EFE and double power law fits. ESR significantly outperforms these "by eye" guesses, even for mock data generated from them. Short diagonal lines on the $x$-axis indicate breaks. In the left and right panels both red and blue points for the Standard IF at complexity 14 lie above the top of the plot.