A strong falsification of the universal radial acceleration relation in galaxies

In the past few decades, many studies revealed that there exist some apparent universal relations which can describe the dynamical properties in galaxies. In particular, the radial acceleration relation (RAR) is one of the most popular relations discovered recently which can be regarded as a universal law to connect the dynamical radial acceleration with the baryonic acceleration in galaxies. This has revealed an unexpected close connection between dark matter and baryonic matter in galaxies. In this article, by following the recent robust Galactic rotation curve analyses, we derive the Galactic RAR (GRAR) and show for the first time that it deviates from the alleged universal RAR at more than $5\sigma$. This provides a strong evidence to falsify the universal nature of RAR in galaxies claimed in past studies.


Introduction
It has been argued that there exist some apparent universal relations which can connect the dynamical properties between dark matter and baryonic matter in galaxies.For example, the Tully-Fisher relation (Tully & Fisher 1977) and Faber-Jackson relation (Faber & Jackson 1976) show that the luminosity of galaxies is somewhat correlated with the stellar velocities in galaxies.As the luminosity and stellar velocities are determined by baryonic content and dark matter content respectively, these relations have revealed some potential correlations between dark matter and baryonic matter, which is not expected based on the standard Lambda Cold Dark Matter model.Therefore, investigating in these universal relations is utterly important to understand the potential interplay between dark matter and baryons.
Recently, there is one intriguing relation discovered called the radial acceleration relation (RAR) (McGaugh, Lelli & Schombert 2016;Lelli et al. 2017;Mistele et al. 2023), which connects the dynamical radial acceleration with baryonic acceleration in galaxies.By analyzing the data of 153 rotating galaxies in the SPARC sample, the dynamical radial acceleration a dyn and baryonic acceleration a bar follow a universal analytic relation with small scatters (McGaugh, Lelli & Schombert 2016): where a 0 = 1.20 ± 0.02 (random) ±0.24 (systematic) ×10 −10 m/s 2 .This RAR is tantamount to a natural law for rotating galaxies as claimed in McGaugh, Lelli & Schombert (2016).Furthermore, a more recent study combining kinematic and lensing data shows that the applicability of RAR can even extend to different types of galaxies, including early-and late-type galaxies, over a large dynamic range (Mistele et al. 2023), which suggests the existence of a consistent and universal relation manifested in all galaxies.
Apart from the unexpected close relation between dark matter and baryonic matter, this relation also suggests the existence of a universal acceleration scale a 0 , which is consistent with the theory of Modified Newtonian Dynamics (MOND) (Li et al. 2018).Also, the universal analytic relation in Eq. ( 1) is consistent with one of the interpolating functions suggested in MOND (Dutton et al. 2019).Therefore, the RAR becomes one important indicator to challenge the dark matter model (Li et al. 2022).Nevertheless, it should be noted that some recent simulations using the Lambda Cold Dark Matter model can still reproduce the RAR (Stone & Courteau 2019;Paranjape & Sheth 2021).
On the other hand, there are some recent studies challenging the claimed universality of RAR.For example, the RAR shown in galaxies is not consistent with the data of galaxy clusters (Chan & Del Popolo 2020;Chan & Law 2022).Also, whether the RAR can be applied in elliptical galaxies is controversial (Chae et al. 2020;Chan, Desai & Del Popolo 2022;Dabringhausen & Kroupa 2023).Generally speaking, verifying or falsifying the RAR requires very high quality measurements of rotation velocity in galaxies.Too large uncertainty in velocity measurements would make the judgment of the existence of a universal RAR very difficult (Desmond, Bartlett & Ferreira 2023).Fortunately, some recent studies using the data of Gaia have produced accurate rotation curves in our Galaxy.The percentage errors in rotation velocities can be smaller than 5% (Eilers et al. 2019;Ou et al. 2023;Jiao et al. 2023).These data are very useful in examining our understanding about dynamics in galaxies, especially in the low acceleration regime (Chan & Law 2023).In this article, by using two recent robust analyses of Galactic rotation curve (GRC), we derive the Galactic RAR (GRAR) and show that it deviates significantly from the universal RAR derived from the SPARC sample.This significantly challenges the universal nature of RAR in galaxies claimed in past studies.

Modeling the radial acceleration
Recently, some studies have analyzed the new GRC data obtained by the Gaia Collaboration (Eilers et al. 2019;Ou et al. 2023;Jiao et al. 2023).The Gaia DR 3 has given much better improved parallaxes and proper motions measurement than that in previous measurements (Jiao et al. 2023).Generally speaking, the GRCs obtained by different studies are consistent with each other.In the followings, we mainly follow two most recent robust analyses of the GRC data (Ou et al. 2023;Jiao et al. 2023) and obtain the dynamical radial acceleration of our Galaxy.
The dynamical radial acceleration is given by where V c is the stellar rotation velocity at the distance R from the Galactic Center.The values of V c at different R can be found in Ou et al. (2023); Jiao et al. (2023), in which the data in Ou et al. (2023) cover a larger range of R = 6 − 27 kpc.The dynamical radial acceleration can be directly calculated from these data.
For the baryonic radial acceleration, we need to adopt a baryonic model to calculate the value.We follow the benchmark baryonic model outlined in Misiriotis et al. (2006); de Salas et al. (2019), in which the model parameters are determined by direct baryonic observations (Misiriotis et al. 2006).This model has been adopted by many recent studies (Ou et al. 2023;Jiao et al. 2023).The baryonic matter consists of two major components: bulge and disks.The bulge component is characterized by the Hernquist potential ( de Salas et al. 2019): The values of the constant parameters r b and M b are shown in Table 1.The corresponding velocity contribution is given by For the disks component, it consists of a stellar disk, two dust disks (cold and warm dust), and two gas disks (H 2 and HI gas).All of the surface mass density in the disk components can be represented by an exponential function (Misiriotis et al. 2006): where Σ 0 = M d /(2πR 2 0 ) is the central surface density.The corresponding scale disk radius R 0 and disk mass M d for each disk component can be found in Table 1.The RC contribution for each disk is given by (Freeman 1970) where I n and K n are the modified Bessel functions of the n th kind.
The baryonic RC contributed by the bulge and disks components is given by Therefore, we can get the baryonic radial acceleration:

Data analysis
By using two samples of GRCs obtained in Ou et al. (2023); Jiao et al. (2023), we plot the values of a dyn against a bar in Fig. 1, which represent the GRARs.We can see that both GRARs are consistent with each other.However, when comparing the GRARs with the RAR obtained by using SPARC data (see the brown error bars in Fig. 1), we can see some systematic deviation between them.The two regions of the GRARs lie near the edge of the 1σ margins of the SPARC RAR across log a bar = −10.75 to log a bar = −9.75.When log a bar < −10.75, a significant drop in a dyn can be seen when a bar decreases.This behavior is mainly related to the GRC decline at R > 20 kpc (Ou et al. 2023;Jiao et al. 2023).However, the SPARC RAR does not show any significant drop in a dyn when log a bar ∼ −10 (McGaugh, Lelli & Schombert 2016).If we look at the entire range of the latest study of the RAR, the drop in a dyn occurs at log a bar < −14, but not at log a bar < −10.75 (Mistele et al. 2023).Nevertheless, since the uncertainty of the SPARC RAR is relatively large, such a deviation between the regions of the GRARs and SPARC RAR is still statistically allowed.
Nevertheless, past studies claimed that there exists a universal analytic form of RAR like Eq. ( 1) satisfying the SPARC data (McGaugh, Lelli & Schombert 2016;Mistele et al. 2023).The best-fit acceleration scale is a 0 = (1.20±0.26)×10−10 m/s 2 (McGaugh, Lelli & Schombert 2016).The blue dashed line in Fig. 1 indicates the best-fit analytic RAR with a 0 = 1.2×10 −10 m/s 2 .If it represents the universal form of RAR, it significantly deviates from the GRAR.Following the same form of Eq. ( 1), we can get the range of a 0 which can best fit with the GRAR.The best-fit acceleration scale for the RC samples in Jiao et al. (2023) and Ou et al. (2023) are a 0 = 1.67 × 10 −10 m/s 2 (χ 2 = 30.2) and a 0 = 1.88 × 10 −10 m/s 2 (χ 2 = 403) respectively.However, these best-fit values are ruled out by the corresponding GRC data at more than 2σ and 5σ respectively, which means that these so-called best-fit scenarios are indeed very poor fits.For the RC sample in Jiao et al. (2023), the 5σ range is a 0 = 1.67 +0.25  −0.22 × 10 −10 m/s 2 , which lies outside the best-fit range of a 0 in the SPARC RAR.Therefore, both GRCs show discrepancies with the universal form of RAR at more than 5σ.
As the analysis in Ou et al. (2023) has provided a more complete set of GRC sample with R = 6 − 27 kpc, we further investigate the discrepancy statistically by the χ 2 function using this sample.The corresponding statistical framework and the probability conversion can be found in Boudaud et al. (2015).In Fig. 2, we plot the χ 2 value against the acceleration scale a 0 by assuming that Eq. ( 1) is the universal analytic form of RAR.We can see that the χ 2 value depends sensitively on a 0 .The best-fit a 0 = 1.88 × 10 −10 m/s 2 is ruled out far beyond 10σ.For the range of a 0 = (1.20 ± 0.26) × 10 −10 m/s 2 derived from the SPARC sample, the χ 2 values are larger than 7000.Therefore, if Eq. ( 1) is the best-fit analytic form satisfying the SPARC data, it would not satisfy the GRAR derived from Ou et al. (2023) for any a 0 .In other words, it is quite likely that there is no universal RAR satisfying both GRC and SPARC data simultaneously.

Discussion
Strictly speaking, this is not the first time to get the GRAR.A few recent studies have indeed considered or generated the GRAR by using the old data of the GRC (Islam & Dutta 2020;Oman et al. 2020).However, the major objective of using the GRAR in these studies is to differentiate different modified gravity theories or constrain the slope of the RAR, but not testing the existence of a universal RAR in galaxies.Moreover, some other studies claimed that the data of GRC is consistent with the RAR (McGaugh 2016(McGaugh , 2019)).However, the data of GRC used in these studies are not the updated one.In our study, we follow the data of the latest Gaia DR 3 which has given much better improved parallaxes and proper motions measurement than that in previous measurements (Jiao et al. 2023).Therefore, this is the first time to examine whether there exists a universal and consistent RAR simultaneously satisfying the latest high quality GRC data from Gaia DR 3 and the data in SPARC galaxies.We have shown that the latest GRC data from two different samples show a significant deviation from the claimed universal SPARC RAR.For a more complete sample of RC in Ou et al. (2023) (with a larger range of R), the best-fit a 0 is ruled out at more than 10σ.Although the GRAR considered contains the data of one galaxy (i.e.our Galaxy) only, the GRC data used are robust and contain extremely small systematic and observational uncertainties (Ou et al. 2023;Jiao et al. 2023).Therefore, the resultant GRARs obtained are very reliable and we can arrive at a strong conclusion.If our Galaxy is not a special one, this implies that either there is no universal form of RAR or the so-called acceleration scale a 0 is not a universal constant.
The existence of a universal RAR is a strong indicator of the correlation between dark matter and baryons.This provides a strong evidence to support modified gravity theories rather than the postulation of dark matter (McGaugh, Lelli & Schombert 2016;Li et al. 2022).If there is no universal RAR in galaxies, such a potential strong correlation between baryonic matter and dark matter may not exist.This also gives negative impact to any modified gravity theories which predict the association between dynamical mass and baryonic mass.In fact, some previous studies have already challenged that there is no universal form of RAR or there is no universal acceleration scale.For example, the RARs obtained by the data of galaxy clusters and elliptical galaxies are significantly different from the SPARC RAR (Chan & Del Popolo 2020;Chan, Desai & Del Popolo 2022).Also, the best-fit values of a 0 are much larger than that obtained by SPARC data (Chan & Del Popolo 2020).Even using the SPARC sample, some studies argue that the universal acceleration scale does not exist (Rodrigues et al. 2018).Although these results are still controversial and inconclusive, our results add strong evidence to support falsifying the existence of a universal RAR, even in galaxies.
One may argue that the baryonic model used might not be accurate enough so that the discrepancy between the GRAR and the SPARC RAR is not justified.In fact, there are some other baryonic models proposed which might give different baryonic mass distributions (de Jong et al. 2010;Calchi Novati & Mancini 2011).Different baryonic models might follow similar functional forms of baryonic density distributions but with slightly different parameters.However, as shown in Fig. 1, the discrepancy originates from the fact that the baryonic mass in our Galaxy is smaller than that predicted by the SPARC RAR.In other words, any baryonic model which can give a larger baryonic mass might be able to alleviate the discrepancy.Based on the comprehensive analysis in Jiao et al. (2023) considering several most representative baryonic models, the baryonic model we used (called B2 model in Jiao et al. (2023)) has already given the largest amount of baryonic mass.The other baryonic models give 1%-10% smaller in baryonic mass (see Table 4 in Jiao et al. (2023)).Therefore, our baryonic model already provides the most optimal calculations for the GRAR to match the SPARC RAR.Roughly speaking, in order to match the best-fit SPARC RAR, the baryonic mass has to be larger by more than 30%.In other words, appealing to the variation of baryonic models does not help in alleviating the discrepancy.
As shown in Fig. 1 that the discrepancy between GRAR and SPARC RAR becomes much more significant in the small a bar regime, we anticipate that more data at large R (i.e.small a bar ) can further verify our conclusion and test for dark matter and modified gravity theories (Mercado et al. 2023).In fact, there are some GRC data obtained for R > 30 kpc using different methods (Vasiliev, Belokurov & Erkal 2021;Wang, Hammer & Yang 2022).However, the involved uncertainties are still quite large so that these data are not good enough for performing accurate analysis.To conclude, our analysis shows that either there is no universal form of RAR or the universal acceleration scale does not exist.
Fig. 1.-The black and red dots with error bars represent the GRAR derived from the GRC samples in Jiao et al. (2023) and Ou et al. (2023) respectively.The brown error bars indicate the uncertainties of the SPARC RAR obtained in McGaugh,Lelli & Schombert (2016).The blue dashed line represents the best-fit universal RAR using Eq.(1) with a 0 = 1.2 × 10 −10 m/s 2 .The green dotted line represents the best-fit GRAR using Eq.(1) with a 0 = 1.88 × 10 −10 m/s 2 .

Fig. 2 .
Fig. 2.-The black solid line represents the χ 2 values against a 0 .The green and red dashed lines indicate the upper limits of χ 2 ruling out Eq. (1) to account for the GRAR derived from the data in Ou et al. (2023) at 5σ and 10σ respectively.The region bounded by the blue dotted lines represents the best-fit range of a 0 derived from the SPARC sample (McGaugh, Lelli & Schombert 2016).
. Component r b or R 0 (kpc) M b or M d (M ⊙ )