Renzo's rule revisited: A statistical study of galaxies' baryon - dark matter coupling

TL;DR

This work tests Renzo's rule—the proposed one-to-one correspondence between features in a galaxy's luminous and total rotation curves—using a systematic statistical framework. By extracting smooth RC trends with Gaussian Process Regression, identifying small-scale features, and evaluating their correlations via Pearson coefficients and dynamic time warping against MOND and LCDM predictions (including SHAM-derived halos), the authors analyze multiple datasets (NGC 1560, SPARC, LITTLE THINGS) and simulated galaxies. Across these data, they find only limited, dataset-dependent hints of Renzo's rule (notably in NGC 1560) and, on average, no robust evidence that features in $V_{ ext{obs}}$ reflect those in $V_{ ext{bar}}$ beyond what LCDM or MOND would predict; SPARC in particular shows an excess of features in $V_{ ext{obs}}$ lacking baryonic counterparts. Mock data reveal that distinguishing MOND from LCDM hinges on high-quality baryonic features and low, well-characterized uncertainties, highlighting current data limitations as the main barrier to a decisive test of Renzo's rule.

Abstract

We present a systematic statistical analysis of an informal astrophysical phenomenon known as Renzo's rule (or Sancisi's law), which states that "for any feature in a galaxy's luminosity profile, there is a corresponding feature in the rotation curve, and vice versa." This is often posed as a challenge for the standard LCDM model while supporting alternative theories such as MOND. Indeed, we identify clear features in the dwarf spiral NGC 1560 -- a prime example for Renzo's rule -- and find correlation statistics which support Renzo's rule with a slight preference for MOND over LCDM halo fits. However, a broader analysis on galaxies in the SPARC database reveals an excess of features in rotation curves that lack clear baryonic counterparts, with correlation statistics deviating up to $3σ$ on average from that predicted by both MOND and LCDM haloes, challenging the validity of Renzo's rule. Thus we do not find clear evidence for Renzo's rule in present galaxy data overall. We additionally perform mock tests, which show that a definitive test of Renzo's rule is primarily limited by the lack of clearly resolved baryonic features in current galaxy data.

Figures (15)

• Figure 1: The two different sets of RCs used for the dwarf spiral galaxy NGC 1560. In both plots, the total observed RC ($V_{\text{obs}}$; error bars), its gaseous component ($V_{\text{gas}}$; dotted line) and its stellar disc component ($V_{\text{disc}}$; dashed line) are given; the total baryonic contribution ($V_{\text{bar}}$; dash-dot line) is calculated using equation \ref{['eqn:Vbar']}, assuming uniform mass-to-light ratios of $\Upsilon_{\text{disc}} = 0.4$ and $\Upsilon_\text{disc} = 1.43$, respectively. Observe the features at around 4-6 kpc in both dataset, especially the large "kink" in Sanders's RCs, which is visible in both $V_{\text{obs}}$ and $V_{\text{bar}}$, thus displaying a clear example of Renzo's rule.
• Figure 2: We demonstrate how DTW aligns $\delta$$V_{\text{obs}}$ to $\delta$$V_{\text{bar}}$ in Sanders's NGC 1560 data. Here, the grey lines depict the optimal mapping $\pi\in\mathcal{M}$ which minimizes the Euclidean separation of $\delta$$V_{\text{obs}}$ and $\delta$$V_{\text{bar}}$ to give the (normalized) DTW alignment cost, DTW$(\delta$$V_{\text{obs}}$$, \delta$$V_{\text{bar}}$$)$, as defined in equation \ref{['eqn:dtw']}. Note that the two sets of residuals are shifted apart vertically to aid visualization.
• Figure 3: Statistics of $V_{\text{obs}}$, $V_{\text{MOND}}$ and $V_{\Lambda\text{CDM}}$ relative to $V_{\text{bar}}$ for the two versions of NGC 1560; (left) RCs obtained from Sanders_2007 using Plot Digitizer, and (right) RCs from Gentile_2010 (S. McGaugh, priv. comm.). $V_{\text{MOND}}$ and $V_{\Lambda\text{CDM}}$ are generated with MCMC fits. Each plot contains three panels showing, from top to bottom: the various data and fits, residuals from the GPR (with kernel length-scale $l=4.5$ kpc, and fitted with points within the 4.2-6.2 kpc window removed), and the Pearson coefficients as a function of maximum radii, where the bands outline the $1\sigma$-confidence range around the displayed means (solid lines) as obtained from the 1000 realizations of each RC. The dashed vertical lines indicate the boundaries of the 4.2-6.2 kpc window, within which features are identified in both $V_{\text{obs}}$ and $V_{\text{bar}}$ of Sanders's RCs, but not in that of Gentile et al. -- note that in the original GPR, i.e., without removing points in the 4.2-6.2 kpc window, the now-visible dip in Gentile et al. is part of a larger 'wiggle' at around 4.5-7.5 kpc, which does not achieve a signal-to-noise ratio of 2.0 to trigger the feature identification algorithm; of course, with the new, refitted GPR, the large dip is identified by Algorithm \ref{['alg:ft_id']}, but it is technically not a feature since points are altered for the new GPR (which may have introduced bias in its residuals). Error bars for MOND and $\Lambda$CDM, which are of the same size as that in $V_{\text{obs}}$, are suppressed to avoid clutter. Note that we only show the correlation coefficients at $\geq5$ data points since, before that, $\rho_p$ simply fluctuates unpredictably between -1 and 1 due to pure noise.
• Figure 4: A summary of all correlation statistics on NGC 1560 --- Pearson coefficient (left) and DTW cost (right; here the x-axis is reversed such that stronger correlations appear towards the right in both plots). The four datasets involved, ordered from top to bottom, are (i) the complete RCs in Sanders_2007 from plot digitization, (ii) the complete RCs in Gentile_2010 from priv. comm. with S. McGaugh, (iii) Sanders's RCs restricted to 4.2-6.2 kpc, where a feature is identified in both $V_{\text{bar}}$ and $V_{\text{obs}}$, and (iv) Gentile's RCs, similarly restricted to 4.2-6.2 kpc, but where no features were identified. Our results indicate that in most cases, especially in the RCs from Gentile et al., features in $V_{\text{obs}}$ and $V_{\text{bar}}$ of correlate stronger than what is expected from $\Lambda$CDM (specifically that from a smooth NFW halo) and even MOND. However, in many cases (especially for DTW), the same statistics fail to separate clearly the expectations from MOND and $\Lambda$CDM, likely due to large and strongly correlated uncertainties the data, which over-scatter the corresponding Monte-Carlo samples, weakening the expected correlations. This suggests that a more detailed study of the dwarf spiral is required to better understand the origin and strong correlations of its features.
• Figure 5: We display four SPARC galaxies where at least one feature is identified in $V_{\text{obs}}$ (black error bars), but not in $V_{\text{bar}}$ (red error bars). The blue and orange solid lines show the GPR fits for $V_{\text{obs}}$ and $V_{\text{bar}}$ respectively, relative to which features are identified. While there are interesting cases where prominent features are visible in $V_{\text{obs}}$ without similar fluctuations in $V_{\text{bar}}$, most notably in UGC 6787 (as shown on the bottom-right panel), strongly correlated (and possibly overestimated) a priori uncertainties in galaxy properties such as distances, mass-to-light ratios, etc., also contributed heavily to the lack of identifiable features in $V_{\text{bar}}$; these uncertainties are treated as uncorrelated by Algorithm \ref{['alg:ft_id']}, making it practically impossible for fluctuations in $V_{\text{bar}}$ to achieve the threshold signal-to-noise ratio of $T=2.0$.