PHYSICAL REVIEW X 9, 031020 (2019)

Reconciling the Diversity and Uniformity of Galactic Rotation Curves with Self-Interacting Dark Matter

† Tao Ren,1 Anna Kwa,2 Manoj Kaplinghat,2,* and Hai-Bo Yu1, 1Department of Physics and Astronomy, University of California, Riverside, California 92521, USA 2Department of Physics and Astronomy, University of California, Irvine, California 92697, USA

(Received 23 October 2018; revised manuscript received 25 April 2019; published 7 August 2019)

Galactic rotation curves exhibit diverse behavior in the inner regions while obeying an organizing principle; i.e., they can be approximately described by a radial acceleration relation or the modified Newtonian dynamics phenomenology. We analyze the rotation curve data from the SPARC sample and explicitly demonstrate that both the diversity and uniformity are naturally reproduced in a hierarchical structure formation model with the addition of dark matter self-interactions. The required concentrations of the dark matter halos are fully consistent with the concentration-mass relation predicted by the Planck cosmological model. The inferred stellar mass-to-light (3.6 μm) ratios scatter around 0.5 M⊙=L⊙,as expected from population synthesis models, leading to a tight radial acceleration relation and a baryonic Tully-Fisher relation. The inferred stellar-halo mass relation is consistent with the expectations from abundance matching. These results provide compelling arguments in favor of the idea that the inner halos of galaxies are thermalized due to dark matter self-interactions.

DOI: 10.1103/PhysRevX.9.031020 Subject Areas: Astrophysics, Cosmology

ϒ ¼ 0 5 I. INTRODUCTION constant stellar mass-to-light ratio ⋆;disk . M⊙=L⊙ and ϒ⋆ ¼ 0.7 M⊙=L⊙ in the 3.6-μm band. The scatter Galactic rotation curves of spiral galaxies show a variety ;bulge of behaviors in the inner parts, even across systems with in this radial acceleration relation (RAR) is around similar halo and stellar masses, which lacks a self-con- 0.1 dex, and the tightness of this relation has been sistent explanation in the standard cold dark matter (CDM) interpreted as a signature of MOND [19]. model [1–13]. Along with this diversity, a long-standing It has long been argued that the acceleration scale observation is that many rotation curves can be understood (including the cH0 dependence) can emerge from hierar- in terms of modified Newtonian dynamics (MOND) chical structure formation predicted in CDM [20,21]. phenomenology [14,15] (see Ref. [16] for a review); Recent hydrodynamical simulations of galaxy formation i.e., there exists a characteristic gravitational acceleration with CDM have clearly shown that a RAR emerges −10 2 [22–24]. However, these simulated galaxies do not re- scale, g† ≈ 10 m=s ∼ cH0=7, with H0 being the present Hubble expansion rate, below which the observed present the full range of diversity in the SPARC data set, pffiffiffiffiffiffiffiffiffiffiffiffi and they cannot yet explain the rotation curves of low- and acceleration can be approximated as g†g , with g bar bar high-surface-brightness galaxies simultaneously. being the baryonic acceleration (a.k.a. Milgrom’slaw). In this paper, we show that self-interacting dark matter More recently, McGaugh et al. [17] analyzed the Spitzer (SIDM) provides a unified way to understand the diverse Photometry and Accurate Rotation Curves (SPARC) data rotation curves of spiral galaxies while reproducing set [18] and showed that there is a tight relation between the RAR with a small scatter. We analyze the SPARC the total gravitational acceleration at any radius and the data set based on the SIDM halo model proposed in acceleration contributed by the baryons, assuming a Refs. [25,26] and demonstrate three key observations leading to this result. σ ∼ 1 2 *Corresponding author. (i) For cross section per unit mass =m cm =g, [email protected] dark matter self-interactions thermalize the inner † Corresponding author. regions at distances less than about 10% of the virial [email protected] radius of galactic halos, while the outer regions remain unchanged. Thus, SIDM inherits essential Published by the American Physical Society under the terms of features of the ΛCDM hierarchical structure for- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to mation model such as the halo concentration-mass the author(s) and the published article’s title, journal citation, relation, which sets the characteristic acceleration and DOI. scale of halos.

2160-3308=19=9(3)=031020(12) 031020-1 Published by the American Physical Society REN, KWA, KAPLINGHAT, and YU PHYS. REV. X 9, 031020 (2019)

(ii) In the inner halo, thermalization ties dark matter and We match this isothermal profile to a Navarro-Frenk- baryon distributions together [25,27,28], and the White (NFW) form [33,34] at r1, where a dark matter SIDM halo can naturally accommodate the diverse particle has scattered Oð1Þ timesovertheageofthe range of “cored” and “cusped” central density pro- galaxy, assuming continuity in both the density and the files, depending on how the baryons are distributed. enclosed mass at r1.Inthisway,theisothermal ρ σ Combined with the scatter in the concentration-mass parameters ( 0, v0) directly map onto the NFW param- ρ relation, this provides the diversity required to ex- eters (rs, s)or(rmax, Vmax). This model provides an plain the rotation curves [26,29,30]. approximate way to calculate the SIDM distribution in a (iii) For the same σ=m that addresses the diversity halo if its CDM counterpart is known, and vice versa. It problem, the baryon content of the galaxies and correctly predicts the halo central density and its the mass model of their host halos also lead to the scalings with the outer halo properties, stellar profiles, RAR with a scatter as small as the one in Ref. [17].In andcrosssection,asconfirmedinbothisolatedand ϒ our SIDM fits, the inferred stellar ⋆;disk values for cosmological N-body simulations with and without individual galaxies have a distribution peaked toward baryons; see, e.g., Refs. [26,28,30,35,36].Seethe 0.5 M⊙=L⊙, as expected from stellar population Appendix and the Supplemental Material [32] for a synthesis models [31]. detailed description of the model and additional com- The rest of the paper is organized as follows. In parisons between model predictions and cosmological Sec. II, we present the SIDM fits to 135 galaxies from simulations. the SPARC sample, which exemplify the full range of We adopt two independent but complementary the diversity. In Sec. III, we show the radial acceleration approaches to perform the analysis. In the controlled relation and the distribution of the stellar mass-to-light sampling (CS) approach, we demand that the host halos ratios from our SIDM fits, compared to the MOND fits. follow the concentration-mass relation within a 2σ range In Sec. IV, we discuss the host halo properties and the predicted in cosmological simulations [37]. We model the origin of the acceleration scale. In Sec. V,weshowthe stellar distribution as an axisymmetric thin disk as in predicted stellar-halo mass relation and the baryonic Ref. [29], which directly enters into the calculation of Tully-Fisher relation (BTFR). We comment on future the density profile of SIDM through the gravitational directions and conclude in Sec. VI. In the Appendix, we potential ΦðR; zÞ. In the CS fits, we start with the outer provide detailed information about the model and the NFW halo and find the SIDM density profile that matches fitting procedure. In the Supplemental Material [32],we its mass and density at r1. In the second approach, we use present SIDM and MOND fits to 135 individual galaxies the Markov chain Monte Carlo (MCMC) sampling (MS) to from the SPARC sample and additional results that explore the full likelihood. To save computational time, we support the main text, including model fits to simulated assume spherical symmetry by spreading the mass within halos. the disk at radius R into a sphere of the same radius [25,28]. The rotation curves generated from two approaches II. THE DIVERSITY OF GALACTIC agree well, and the differences in the fits are small (see ROTATION CURVES Supplemental Material [32]). For our main results, we show inferences from both of the approaches. We select 135 out of 175 galaxies in the full SPARC In Fig. 1, we show the SIDM fits to the diverse rotation sample based on the criteria that they must have a curves from the controlled sampling with σ=m ¼ 3 cm2=g. recorded value for the flat part of the rotation curve, Vf . In each panel, galaxies are selected to have similar flat In our sample, 87, 42, and 6 galaxies have quality rotation velocities at their outermost data points. The rise up flags 1, 2, and 3, respectively. It spans a wide range to Vf within their central regions displays a wide variety of of galaxy masses and inner shapes of rotation curves slopes, and the SIDM halo model provides equally good 20 300 with Vf ranging from km=sto km=s. In fitting fits to the shallow and steeply rising rotation curves. The to the data, we utilize the analytical SIDM halo fits for the other galaxies in the sample are as good as those model [26,29], where we assume the dark matter in Fig. 1 (see Supplemental Material [32]). distribution in the inner halo follows the isothermal The success of the SIDM halo model stems from a density profile, combination of the following effects. First, SIDM thermal- ization ties the baryon and dark matter distributions together. ρ ð Þ¼ρ (½Φ ð0 0Þ − Φ ð Þ σ2 ) ð Þ iso R; z 0 exp tot ; tot R; z = v0 ; 1 For low-surface-brightness galaxies, thermalization leads to a shallow density core and a circular velocity profile that ρ σ – where 0 is the central dark matter density, v0 is the rises mildly with radius [38 44]. While, for high-surface- one-dimensional dark matter velocity dispersion, brightness galaxies, the core shrinks in response to the Φ ð Þ tot R; z is the total gravitational potential, and R, z deeper baryonic potential, and the central SIDM density are cylindrical coordinates aligned with the stellar disk. increases accordingly [25,28,30,36]. The galaxies in our

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100 140 120 80 100

60 80 (km/s) (km/s)

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V 40 V 40 20 20 Vf 79–91 km/s Vf 91–136 km /s 0 0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 18 20 22 Radius (kpc) Radius (kpc)

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250 150 /s) 200 km ( (km/s) 100 150 circ circ V V 100 50 50 Vf 139–172 km/s Vf 239–315 km/s 0 0 0 5 10 15 20 25 30 35 40 45 0 10 20 30 40 50 60 70 Radius (kpc) Radius (kpc)

FIG. 1. SIDM fits (solid) to the diverse rotation curves across a range of spiral galaxy masses, where we take σ=m ¼ 3 cm2=g. The data points with error bars are from the SPARC data set [18]. Each panel contains 14 galactic rotation curves that are selected to have similar flat rotation velocities at their furthest radial data points, and the corresponding Vf bins are 79–91, 91–126, 139–172, and 239–315 km=s, spanning the mass range of the galaxies considered in this work. The galaxies are colored according to their relative surface brightness in each panel from low (red) to high (violet). sample have a variety of central surface brightnesses, minor change in the stellar mass-to-light ratio, and many of resulting in diverse central dark matter densities. Second, these systems are actually compatible with a NFW profile. scatter in the cosmological halo concentration-mass relation The cross section may have a mild velocity dependence over leads to scatter in the characteristic SIDM core density and the sample, as implied by the constraint from galaxy clusters radius, which is reflected in the rotation curves [26]. [26,47,48], but it is impossible to extract it from the SPARC Reference [29] fitted 30 galaxies and illustrated the impor- data set given the reasons discussed above. In this work, we tance of these effects in explaining the diverse rotation present the results for fixed σ=m ¼ 3 cm2=g, and they curves. In this work, we fit a larger sample of galaxies and remain the same qualitatively for other values larger than demonstrate that the observed galaxies are fully consistent about 1 cm2=g on galaxy scales. with the SIDM predictions. An important consequence of the large cross section is We have assumed a constant cross section to fit the SPARC sample because it is hard to pin down the cross that the SIDM profile is driven quickly to be isothermal in section for individual galaxies. For low-surface-brightness the inner regions. This implies that the resultant SIDM fits galaxies with a large core, a large cross section, such as will not depend sensitively on the formation history of σ=m ¼ 3 cm2=g, is preferred [29]. However, since the individual galaxies [29] but on the final stellar and gas central SIDM density varies mildly with the cross section distributions [25]. This has been explicitly confirmed in in the range 1–10 cm2=g [45,46], a feature that is well recent hydrodynamical SIDM simulations [35] and those captured in our analytical model [26], an even larger cross with idealized disk growth [28]. Furthermore, in our fits r1 section may work as well. For high-surface-brightness is close to rs, which is well outside the stellar disk or budge galaxies, to which most of the galaxies with high Vf belong, in the galaxies. It is unlikely that a viable baryonic feedback the fits are insensitive to the cross section because of the process could change the halo mass profile significantly at degeneracy between σ=m and ϒ⋆ [29]. The effect in the that far distance. Thus, our analytical model takes into SIDM fits induced by varying σ=m can be compensated by a account the realistic baryon distribution for individual

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50 100 –8 SIDM SIDM 3012 points MOND 40 –9 10 2 s m/ –10 2 30 ) (–10 ) g†: 1.38 10 m/s SD: 0.10 dex

mod tot 1 g 400 d.o.f. (MOND)

20 / log( –11 300 2

200 Number of galaxies 0.1 100 Number of Points 10 0 –12 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 Residuals (dex) –12 –11 –10 –9 –8 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.1 1 10 100 mod 2 2 log(gbar )m/( s ) M/L (M /L ) /d.o.f. (SIDM)

mod mod σ ¼ 3 2 FIG. 2. Left panel: The radial acceleration relation from the SIDM fits, where gtot and gbar are inferred from the =m cm =g fits. The black solid line is the best fit to Eq. (2); the two red dashed curves correspond to the 1σ deviation from this fit. The black dotted line is the one-to-one reference line. Insets: Corresponding histograms of residuals after subtracting the fit function with the best-fitting scale −10 2 parameter g† ¼ 1.38 × 10 m=s , together with the Gaussian fits to the residuals, which have 1σ widths of 0.10 dex. Middle panel: 2 Inferred ϒ⋆;disk distributions for the SIDM and MOND fits. Right panel: Distribution of χ =d:o:f: values for individual galaxies from the SIDM and MOND fits.

−10 2 galaxies and encodes this effect on the SIDM halo profile we obtain g† ¼ 1.19×10 m=s and dispersion 0.12 dex, through the matching procedure. both in agreement with previous work [17]. For a more detailed comparison, we also fit the sample of 135 SPARC galaxies using the MOND relation in Eq. (2), III. THE RADIAL ACCELERATION ¼ 1 2 10−10 2 ϒ where we fix g† . × m=s but vary ⋆;disk and RELATION IN SIDM ϒ 0 1 ϒ ð Þ 10 ⋆;bulge in the range of . < ⋆= M⊙=L⊙ < (same as In the RAR described in Ref. [17], the gravitational MCMC SIDM fits) using MCMC sampling (see also acceleration gtot at radius r is found to be related to the Ref. [49]). The results look similar if we set g† to 1.0 × −10 −10 2 acceleration gbar at the same radius. This relation can be fit 10 or 1.4 × 10 m=s . The middle panel of Fig. 2 to a functional form with a single parameter g†: ϒ shows the ⋆;disk distribution from the MOND fits (dotted pffiffiffiffiffiffiffiffiffiffiffiffiffiffi line), which closely tracks the one from the SIDM fits. The −1 2 − g ðrÞ=g† right panel shows the distribution of minimum χ =d:o:f. g ðrÞ¼g ðrÞð1 − e bar Þ : ð2Þ tot bar values for individual galaxies from the SIDM and MOND fits. The SIDM model provides a better fit than MOND for −10 2 Their best-fit value of g† ¼ 1.2 × 10 m=s is the oft- most of the galaxies (∼77%) while maintaining a tight quoted MOND acceleration scale. RAR. In fact, 72% (45%) of them have χ2=d:o:f: ≤ 3ð1Þ in In the left panel of Fig. 2, we show the inferred total and the controlled SIDM fits, and those with a large χ2=d:o:f: baryonic acceleration values from the controlled sampling, value have either tiny errors or wiggles in the observed mod mod where gtot and gbar are calculated from the SIDM fits, rotation curves that cannot be reproduced by MOND either. using the halo parameters and the best-fit ϒ⋆ values for We have also compared our fits with the MOND fits in each galaxy. The intensity of color in Fig. 2 (left panel) Li et al. [19]. The major difference is that they marginalized reflects the density of points. After fitting the data with the over both the distance and inclination uncertainties, while empirical relation given in Eq. (2), we find that the best-fit we did not. The MOND fits in Ref. [19] are slightly better −10 2 value of g† is 1.38 × 10 cm =g, and the resulting than our MOND fits due to the two additional variables dispersion in the residuals is 0.10 dex. Figure 2 (middle (distance and inclination), but still only about 20% of the ϒ 2 panel) shows ⋆;disk distribution from the SIDM fits (solid galaxies are fit with χ =d:o:f: ≤ 1. For comparison, we ϒ ¼ 0 5 line). It is peaked toward ⋆;disk . M⊙=L⊙, in good have checked that over 60% of the MCMC SIDM fits agreement with predictions from stellar population synthe- have χ2=d:o:f: ≤ 1. sis models [31]. This result is remarkable because no We emphasize that the diversity in the inner rotation − priors based on the stellar population synthesis models curves is also reflected in the gtot gbar plane, as explicitly were used. We have also reproduced the analysis in demonstrated in the Supplemental Material [32], where we ϒ ϒ 0 5 Ref. [17], with ⋆;disk and ⋆;bulge fixed to . M⊙=L⊙ show the gtot vs gbar plot but now split the sample into two 0 7 ϒ 2 and . M⊙=L⊙, respectively. For this fixed ⋆ case, sets: radii outside and inside Rd, with Rd being the scale

031020-4 RECONCILING THE DIVERSITY AND UNIFORMITY OF … PHYS. REV. X 9, 031020 (2019) ) () () (

( ) (() )

− FIG. 3. Left panel: rmax Vmax distributions of the host halos in the SIDM fits with controlled (circles) and MCMC (squares) samplings. We also show the mean relation (black solid line) and 2σ scatter (gray shaded area) predicted in cosmological CDM simulations [37]. Middle panel: Halo virial mass vs galaxy stellar mass from the SIDM fits. The black solid line corresponds to the abundance matching inference from Ref. [58]. Right panel: Total baryonic mass vs flat circular velocity for the 135 galaxies, where Mbar ϒ ϒ is inferred from our SIDM fits (circles and squares). For comparison, we also show the case (triangles) when ⋆;disk and ⋆;bulge are fixed to 0.5 M⊙=L⊙ [54]. The black solid line is the mean baryonic Tully-Fisher relation from Ref. [54], derived from 118 SPARC galaxies with ϒ⋆;disk ¼ ϒ⋆;bulge ¼ 0.5 M⊙=L⊙, at which the scatter is minimized. radius of the stellar disk. The scatter is relatively large SIDM halo using the concentration and mass or, equiv- 2 for radii less than Rd, and this is due to the different alently, the maximal circular velocity (Vmax) and the shapes in the inner rotation curves and not just the associated radius (rmax) of its CDM counterpart. Ideally, result of random errors (see also Ref. [50]). On the other one would measure these halo parameters directly from the hand, there is a clear ordered behavior of gtot vs gbar curves kinematics data and compare them with simulations. 2 for radii > Rdisk, which is a reflection of the BTFR: the Unfortunately, most rotation curves do not have the radial tight correlation between the flat circular velocity Vf extent needed to sufficiently constrain them. In this work, and the total baryonic masspffiffiffiffiffiffiffiffiffiffiffiMbar for spiral galaxies [51]. we impose the cosmological concentration-mass relation ≈ ≈ 2 In this regime, gtot g†gbar, where gtot Vf =r and [37] as a prior similar to Ref. [57] and examine the 2 4 g ≈ GM =r ; hence, we have V =ðGM Þ ≈ g†. This consistency between its consequences and observations. bar bar f bar − ∝ 4 In Fig. 3 (left panel), we show the rmax Vmax distri- is the success of MOND; i.e., if one assumes Mbar Vf , then the normalization of the BTFR also predicts the butions from our controlled (circles) and MCMC (squares) rotation curve, which in many cases is a good fit to the samplings. For the former, we intend to seek the best ∝ s 3 4 SIDM fits to the rotation curves following the mean relation observed one. Many studies find Mbar Vf with

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CDM halo with its gravitational acceleration at r ¼ 0 completely in line with N-body simulations of structure ð0Þ¼ 2j ≈2π ρ ≈2π 2 ð1 26 Þ as gNFW GM=rpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir→0 G srs Vmax= . rmax , formation using the cosmological parameters from the ≈ 0 58 ρ 2 ≈ 2 16 Planck experiment [61,62]. In addition, the stellar mass- where Vmax . G srmax and rmax . rs. Taking − ¼ to-light ratios are consistent with the results from stellar the mean cosmological Vmax rmax relation, rmax 27 ð 100 Þ1.4 ð0Þ ≈ 1 0 population models [31]. kpc Vmax= km=s ,wehavegNFW . × 10−10 m=s2 ðV =240 km=sÞ0.6, which is close to the These results lead to a natural question: What is the max predicted halo mass for a given stellar mass in the SIDM MOND acceleration parameter g†. This result is the under- lying reason why the empirical MOND relation captures model? Since we assume the primordial matter power the overall stellar kinematics of spiral galaxies well. In the spectrum is unchanged from the CDM one for the scales presence of dark matter self-interactions and baryons, we are interested in, there should be a relation consistent ð0Þ with the abundance matching results in the literature. In the the actual central acceleration deviates from gNFW , but the general argument still holds. For example, we middle panel of Fig. 3, we show the stellar mass vs halo can characterize a halo with the acceleration at the mass relation derived using the mass-to-light ratios from scale radius r , where the impact of dark matter self- controlled (circles) and MCMC (squares) samplings. The s 1σ interactions and the influence of baryons tend to be small, error bars on the MCMC points denote the widths from ð Þ ≈ 0 39 ð0Þ the posteriors (16th and 84th percentiles). Our results are gNFW rs . gNFW , slightly smaller than gNFW at the center. The characteristic halo acceleration has a mild consistent with the overall trend in the relation from 300 abundance matching (solid line) [58] (see Ref. [50] for dependence on Vmax, ranging from 20 to km=sin the sample, and it also varies with the scatter in the the CDM case). We also note that there is a tendency for our cosmological relation. This variation is important, as shown data points to lie a bit to the left of the abundance matching in Fig. 3 (left panel). Since MOND does not have such line. The halo masses inferred from our fits seem to be flexibility, its overall fits are worse than the SIDM ones, as systematically lower than those inferred in Ref. [58]. This difference could be due to different assumptions on the illustrated in Fig. 2 (right panel). We emphasize that g† ¼ −10 2 cosmological parameters in deriving the halo concentra- 1.38 × 10 m=s inferred from our SIDM fits in Sec. III tion-mass relation, or other differences in the analyses. A is an average quantity over the sample after fitting to Eq. (2), not a universal value for all the galaxies as in systematic investigation of this subject would be a fruitful MOND (see also Ref. [59]). avenue for future research. In addition, there are a few The calculation of the acceleration due to dark matter outliers on the left side of the black line in the low-mass toward the center is more subtle. Inside a constant density regions, and many of them have low-quality observational ð Þ ∝ data, as we will discuss later. core, gSIDM r r, and we need to specify the radius where the acceleration is being computed. The half-light radius We have already alluded to the importance of the BTFR in our discussion of the RAR. Lelli et al. [54] selected 118 (r1=2), which encloses half the luminosity, is typically used − SPARC galaxies and found that their Vf Mbar inferences to characterize the size of the stellar distribution, so it is a ð Þ¼ ð Þþ can be fitted with a simple relation: log Mbar s log Vf natural radius to compute the acceleration at r1 2.On = logðAÞ, where s ¼ 3.71 0.08 and logðAÞ¼2.27 0.18 average, the stellar half-light radius is empirically observed ϒ ¼ ϒ ¼ 0 5 ≈ 0 015 for ⋆;disk ⋆;bulge . M⊙=L⊙. The right panel of to track the virial radius as r1=2 . rvir [60], and we − ϒ ≈ 1 7 Fig. 3 shows the Vf Mbar inferences with the ⋆;disk have r1=2 . Rd for an exponential disk model. Without a and ϒ⋆ values from the controlled (circles) and significant contribution from baryons to the gravitational ;bulge −11 2 MCMC (squares) fits. The error bars in Mbar on the potential, SIDM predicts that g ðr1 2Þ¼10 m=s SIDM = MCMC points denote the 1σ widths in the stellar mass- ð 100 Þ0.2 Vmax= km=s for the median halo concentration, and to-light ratios from the posteriors, and the errors in Vf are its dependence on the halo mass is extremely mild. When taken directly from the SPARC data set [18]. We also show baryons contribute, gtot does not increase linearly with gbar the fit from Ref. [54] as the solid line of Fig. 3 (right panel). since both the central SIDM density and the core radius Note that this fit used 118 galaxies, and a few outliers at the depend on the gravitational potential contributed by the low Vf end were not included. For comparison, we plot the baryons. The net result is a strong correlation between gtot 135 galaxies in our sample as triangles by fixing and g , which is clearly evident in Fig. 2. The model bar ϒ⋆ ¼ ϒ⋆ ¼ 0.5 M⊙=L⊙. We see that their distri- predictions have a definite width in the g vs g plane, ;disk ;bulge tot bar bution in the V − M plane is almost identical to the one and we have shown clearly that this scatter is required to f bar from our SIDM fits. This result is not surprising, as the fully explain the diversity in the rotation curve data. ϒ ⋆;disk values inferred from the SIDM fits are peaked toward 0.5 M⊙=L⊙ as shown in Fig. 2 (middle panel). Thus, we V. CORRELATIONS BETWEEN THE TOTAL conclude that the SIDM fits also lead to a tight BTFR LUMINOUS AND DARK MATTER MASSES relation. For our fits, we find s ≈ 3.46 (CS), 3.27 (MS), and We have seen that the SIDM fits to the rotation curves 3.58 (0.5 M⊙=L⊙), excluding six obvious outliers on the require values for the halo concentration parameter that are left side of the black line. Note that five of them, F 561-1,

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PGC 51017, UGC 04305, UGC06628, and UGC 09992, keeps the inner halo isothermal, and this makes the have either low-quality rotation curves or small inclination predictions for the central halo profile at later times angles, and they may not be well suited for dynamical insensitive to the star formation history, as confirmed in analysis. We have also checked that these galaxies are the recent hydrodynamical N-body simulations [35,69]. This − outliers in the low-mass regions of the Mstar Mhalo relation, result implies that a large variety of feedback models (e.g., shown in Fig. 3 (middle panel). Refs. [70–75]) can be compatible with the SIDM model we We note that there is no evidence in the data for s ¼ 4 have discussed here. The predictions are quantitatively the ∝ 4 σ ≳ 1 2 exactly, i.e., Mbar Vf , which is the motivation for same for =m cm =g. This makes our results robust, MOND, in either the constant ϒ⋆ fits or in the SIDM fits. but it is hard to precisely determine the cross section from We note that Vf may not be a good proxy for the asymptotic kinematic data sets on galaxy scales [29]. velocity of every galaxy in the sample, and systematic There are a number of promising directions that can effects could lead to a shallower BTFR slope [63]. Many further test SIDM and explore galaxy formation and of the recent CDM simulations with efficient baryonic evolution in this framework. Here, we highlight a few of feedback seem to obtain something akin to the BTFR with them. SIDM simulations predict a correlation between the s ≈ 3.6–3.8 [64–67], but it is fair to say that this is still not half-light radius of the stars and the dark matter core size in well understood theoretically; in particular, the smallness of dwarf and low-surface-brightness galaxies [27], which the scatter in the BTFR is equivalent to the one seen in the should be further explored and may provide an observa- RAR [68]. We expect that there will be interplay between tional test of SIDM. Similarly, the ultradiffuse galaxies in dark matter self-interactions and baryonic feedback in the clusters could be a test laboratory [76]. A related issue is changing the halo potential. Understanding how the the origin of the large spread in the surface brightness of BTFR emerges in SIDM is fertile territory for research galaxies, which remains poorly understood. Interestingly, in galaxy formation. hydrodynamical simulations of galaxy clusters show that the stellar density profiles in SIDM are more diverse than in VI. DISCUSSION AND CONCLUSIONS their CDM counterparts [35]. Is this a more general feature in SIDM due to the dynamical interplay between core In this work, we have investigated SIDM as a solution to formation and feedback? How does this interplay impact two puzzles that are present in galactic rotation curves: the emergence of the BTFR? Finally, at the lowest mass (1) the diversity of inner rotation curves in galaxies that end, the dwarf spheroidal galaxies, including the so-called have similar baryon content and similar flat circular ultrafaint dwarfs, in the local group could provide a key velocities, and (2) the small scatter in the radial acceleration test of SIDM (see Refs. [77,78]). Dedicated SIDM simu- relation between the total gravitational acceleration and lations with the baryons will be required to explore these the one inferred from the baryonic mass content, i.e., exciting topics. uniformity. The predictive power of the SIDM model, the clear We have fitted our SIDM halo model to the rotation connection to cosmology, and its rich implications for other curves of 135 SPARC galaxies and found that it reproduces astrophysical observations and particle physics phenom- the observed diversity in the inner regions. The distribution enology [79], all taken together, make a clear case that it of the resulting 3.6-μm stellar disk mass-to-light ratios for should be treated on the same footing as the CDM model. the sample peaks at ϒ⋆ ≈ 0.5 M⊙=L⊙, in good agree- ;disk The economical explanation, with the addition of just one ment with the stellar population models. Our fits lead to a parameter, for the diverse rotation curves across the entire radial acceleration relation described by the characteristic range of observed galaxies argues in favor of the idea that 10−10 2 acceleration scale of about m=s , with tight scatter of the dark matter particles have a large affinity for the self- 0.10 dex. The host halos are fully consistent with the interactions. Planck cosmology. The inferred stellar mass-halo mass relation agrees with the result from the abundance matching ACKNOWLEDGMENTS method, and the fits also predict a tight BTFR. These results provide compelling arguments in favor of the idea that the We thank the authors of Refs. [30,40,45] for the inner halos of galaxies are kinematically thermalized due to simulation data, which are used for comparison with the dark matter self-interactions. analytical model, shown in the Supplemental Material [32]. The SIDM model automatically inherits all of the This work was supported by the National Science successes of the CDM model on large scales, as the Foundation Grant No. PHY-1620638 (M. K.), the U.S. predictions are indistinguishable at distances larger than Department of Energy under Grant No. DE-SC0008541 about 10% of the virial radius of galactic halos. The (H. B. Y.), the Hellman Fellows Fund (H. B. Y.), UCR required cross section is similar to the proton-neutron Academic Senate Regents Faculty Development Award elastic scattering cross section, and this similarity may (H. B. Y.), and the National Science Foundation under be a strong hint that the dark matter sector replicates some Grant No. NSF PHY-1748958 as part of the KITP “The elements of the standard model. The large cross section Small-Scale Structure of Cold (?) Dark Matter” and “High

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” Σ Energy Physics at the Sensitivity Frontier workshops where 0 is the central surface density and Rd is the scale Σ (M. K. and H. B. Y.). H. B. Y. also acknowledges Tsung- radius. For each galaxy, we reconstruct the 0 and Rd values Dao Lee Institute, Shanghai, for its hospitality during the by fitting the profile to the disk contribution of the rotation completion of this work. curve as in Ref. [29]. We neglect the baryonic influence on the SIDM halo from the gas and bulge potentials, but we include all the mass components in modeling the total APPENDIX: METHODS circular velocity. This method is a reasonable approximation We provide a detailed description of the analytical model for the following reasons: (1) The gas is less centrally developed previously [26,29] and the fitting procedure. We concentrated, so its impact on the SIDM density profile is divide the halo into an inner and an outer region [40], with smaller; (2) the bulge (when presents) mainly affects the the aim that the outer halo is not significantly changed innermost region, while the disk contributes in this region as by the self-scattering process. In the inner region, well as at farther radii. Reference [29] solved Eq. (A1) with dark matter self-interactions thermalize the halo in the the thin-disk approximation and created numerical tem- presence of the baryonic potential, and we model the dark plates for the isothermal density profile on the grid of a ≡ 8π ρ 2 ð2σ2 Þ ≡ 8π Σ ð2σ2 Þ matter distribution using the isothermal density profile, G 0Rd= v0 and b G 0Rd= v0 . When the ρ ∝ ( − Φ ð Þ σ2 ) ’ iso exp tot R; z = v0 . Poisson s equation relates stellar profile is known, the parameters a and b give the Φ tot to the dark matter and baryon profiles as central density and dispersion of the isothermal dark matter halo, which completely specify the inner density profile. We ∇2Φ ð Þ¼4π ½ρ ð Þþρ ð Þ ð Þ tot R; z G iso R; z b R; z : A1 interpolate the templates to generate rotation curves for any ðρ σ Σ Þ set of 0; v0; 0;Rd . The fixed value of the cross section For the outer halo, where the self-scattering effect becomes allows us to match this density profile to the outer spheri- ∼ ≫ negligible, we model the dark matter distribution with a cally symmetric NFW density profile. Since r1 rs Rd in ρ ð Þ¼ρ 3 ð þ Þ2 NFW profile NFW r srs=r r rs . To construct the our fits, the influence of the baryons on the SIDM halo shape full SIDM halo profile, we define a radius r1, where dark becomes negligible at r1, and the SIDM halo recovers matter particles have one interaction on average over the spherical symmetry accordingly; see the Supplemental age of the galaxy. We join the spherically averaged Material [32]. ρ ρ ¼ isothermal ( iso) and spherical NFW ( NFW) profiles at r In fitting to the SPARC sample with the templates, we r1 such that the mass and density are continuous at r1. take a controlled sampling approach. For a given galaxy, ρ σ − Thus, the isothermal parameters ( 0, v0) directly map onto we start with the mean rmax Vmax relation from cosmo- ρ the NFW parameters (rs, s)or(rmax, Vmax). logical ΛCDM simulations [37] and a NFW profile that The value of r1 is determined by the following condition, matches the flat part of the rotation curve. Then, we choose ϒ ϒ an appropriate ⋆;disk ( ⋆;bulge) value to reproduce the inner hσ iρ ð Þ ¼ ð Þ 2 vrel NFW r1 tage=m Nsc; A2 rotation curve. We calculate a χ =d:o:f: value for each fit and iterate this process manually by adjusting the param- where σ is the self-scattering cross section, m is the dark eters until a good fit is achieved. For most galaxies, the very matter particle mass, vrel is the dark matter relative velocity first step provides decent fits, showcasing the simplicity of in the halo, h…i denotes averaging over the Maxwellian the model and its ability to fit the observed data simulta- velocity distribution, t is the age of the galaxy, and N is age sc neously. For each galaxy, we demand the (rmax, Vmax) a factor of order unity, to be determined by calibrating to values to be within a band of about 2σ. In this way, we have ¼ 10 simulations. In this work, we have set tage Gyr and good control over the halo parameters in the fits. The goal is ¼ 1 Nsc , which reproduce simulation results well; see the to see to what degree the galaxy halos of the SPARC sample Supplemental Material [32]. In principle, we should use are consistent with predictions of the hierarchical structure different ages for each galaxy, say, between 10 Gyr and formation scenario and to what degree the extracted ϒ⋆ 13 Gyr. However, our model can only constrain the values are consistent with stellar population synthesis combination of the cross section and the age. More impor- model results [31]. tantly, we have set σ=m to a large enough value that the In our second approach, we perform a MCMC sampling SIDM density profile is insensitive to small changes in the of the SIDM model parameter space. Since it is computa- cross section. We assume that this cross sectionpffiffiffi is a constant tionally expensive to use the templates, we use a spherical hσ i¼σð4 πÞσ over the SPARC sample, so vrel = v0. In this approximation to model the baryon distribution [25,28]. work, we present our results for σ=m ¼ 3 cm2=g on galaxy We create a spherical baryonic mass profile from the stellar scales, which can be naturally realized in particle physics and gas masses, such that the baryonic mass within a sphere – ð Þ¼ð 2 þ 2 þ 2 Þ models of SIDM [80 98]. of radius r is Mb r Vdisk Vbulge Vgas r=G, where We take two independent but complementary approaches. Vdisk is the contribution to the rotation curve from the disk In the first one, we assume a thin-disk profile for the stellar at radius r and similarly for the bulge and gas. Below the ρ ð Þ¼Σ ð− Þδð Þ disk in solving Eq. (A1), b R; z 0 exp R=Rd z , smallest radii at which the baryonic contribution is

031020-8 RECONCILING THE DIVERSITY AND UNIFORMITY OF … PHYS. REV. X 9, 031020 (2019) tabulated in the SPARC database, we assume that the inference of cores or cusps. We do not include this density in baryons is constant. Outside the last tabulated regularization error when quoting χ2 values. radii, we assume that the mass is constant. We have tested (ii) Wepffiffiffi impose a uniformpffiffiffi regulation prior on Vmax: some galaxies where we change the constant central density 1 2 2 =

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