A cuspy dark matter halo
Yong Shi, Zhi-Yu Zhang, Junzhi Wang, Jianhang Chen, Qiusheng Gu, Xiaoling Yu, Songlin Li
DDraft version January 6, 2021
Typeset using L A TEX twocolumn style in AASTeX63
A cuspy dark matter halo
Yong Shi,
1, 2
Zhi-Yu Zhang,
1, 2
Junzhi Wang, Jianhang Chen, Qiusheng Gu,
1, 2
Xiaoling Yu, and Songlin Li School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China. Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China. Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China
ABSTRACTThe cusp-core problem is one of the main challenges of the cold dark matter paradigm on smallscales: the density of a dark matter halo is predicted to rise rapidly toward the center as ρ ( r ) ∝ r α with α between -1 and -1.5, while such a cuspy profile has not been clearly observed. We have carriedout the spatially-resolved mapping of gas dynamics toward a nearby ultra-diffuse galaxy (UDG), AGC242019. The derived rotation curve of dark matter is well fitted by the cuspy profile as describedby the Navarro-Frenk-White model, while the cored profiles including both the pseudo-isothermaland Burkert models are excluded. The halo has α = -(0.90 ± M halo =(3.5 ± × M (cid:12) and a small concentration of 2.0 ± < × − eV or > × − eV , the cross section of self-interacting dark matter to be < /g, and the particlemass of warm dark matter to be > INTRODUCTIONThe cosmological model of cold dark matter and darkenergy, i.e., ΛCDM, has achieved tremendous success inunderstanding the cosmic structure across time on largescales, but this model is challenged by observations onsmall scales such as the cusp-core problem, the missingdwarf problem, the too-big-to-fail problem etc. (for areview, see Weinberg et al. 2015).In cosmological simulations of cold and collisionlessdark matter, a dark matter halo has a density profilethat rises toward the center with a power index of -1to -1.5 (Moore 1994; Burkert 1995; Navarro et al. 1997;Moore et al. 1998; Ghigna et al. 2000; Jing & Suto 2002;Wang et al. 2020), referred as a cuspy profile. However,over the past decades, much shallower or even flat core-like profiles toward centers have been found in most, ifnot all, observed data of nearby galaxies through map-ping dynamics of gas and stars. Early studies withHI interferometric data reveal shallow dark-matter cen-
Corresponding author: Yong [email protected] tral profiles in individual galaxies (Carignan & Beaulieu1989; Lake et al. 1990; Jobin & Carignan 1990). Stud-ies with higher spatial resolutions for a larger sample ofdwarfs and low-surface-brightness galaxies further con-firm the central flatness of the rotation curve, and de-rived a median dark-matter density slope of about -0.2toward centers (de Blok et al. 2001; Oh et al. 2011,2015). Optical observations of ionized gas such as H α can achieve higher spatial resolutions than the HI data,and confirmed the median density slope of about -0.2with long-slit spectra for a large sample of dwarf galax-ies (Spekkens et al. 2005). Although the rotation curvesfrom the long-slit spectroscopic data are sensitive to theassumed dynamical center, the position angle of thekinematic major axis etc, further studies with opticalintegral field unit have suggested insignificance of theabove effects and validated dark-matter profiles withcentral shallower slopes (Kuzio de Naray et al. 2008;Adams et al. 2014).Among few galaxies whose halos can be described bycuspy profiles, the one with the highest signal-to-noise-ratio is DDO 101 that has a central dark-matter slopeof -1.02 ± σ ) (Oh et al. 2015). However, thisobject can be fitted equally well with a cored profile; a r X i v : . [ a s t r o - ph . GA ] J a n Shi et al. the difference in the reduced χ is only 0.02 for a degreeof freedom (d.o.f) of six (see Oh et al. 2015). This isbecause of the limitation on the spatial resolution andspatial extent of the observed rotation curve, as well asdifferent models for cuspy and cored profiles (see § σ and 6.8- σ by Jardel et al. (2013)and Hayashi et al. (2020), respectively. However, it hasnot been demonstrated whether a cored model can fitthe data or not for this object, as done for DDO 101.And furthermore, systematic uncertainties are compli-cated for these studies in which the velocity dispersionamong individual stars as a function of the galactic ra-dius are used to measure the dark matter distribution.The orbital anisotropy, the method to model stellar or-bits, the dark-matter shape and the limited number ofthe member stars are all found to affect the conclusions(Evans et al. 2009). Especially, a recent study of simu-lated galaxies by Chang & Necib (2020) emphasized theimportance of a large amount of stars in order to unam-biguously measure the dark matter distribution. If thenumber of stars is less than 10000, an intrinsic cored pro-file cannot be differentiated reliably from a cuspy profile.For Draco, there are only ∼
468 member stars (Walkeret al. 2015). Fortunately, these systematic uncertaintiesare significantly eliminated for HI interferometric dataas seen in simulated galaxies (Kuzio de Naray et al. 2009;Kuzio de Naray & Kaufmann 2011). This is mainly be-cause gas is collisional and their dynamics can be rela-tively easily described by titled-ring models (Begeman1989; Di Teodoro & Fraternali 2015). In summary, theabove few claimed cuspy profiles of dark halos are notdefinitive.There are two solutions to the small scale controversiesof ΛCDM. One is to modify the nature of dark matterso that it is no longer cold but instead self-interacting,fuzzy or warm. These modifications can retain the prop-erties of the universe on large scales as predicted byΛCDM but resolve its small-scale challenges. Fuzzy cold dark matter, also known as ultralight scalar particles,has a mass around 10 − eV (e.g. Hu et al. 2000; Schiveet al. 2014). The uncertainty principle of its wave natureacts on kpc scales, smoothing the density fluctuationsand preventing the growth of small halos and formationof central cusps. The self-interacting dark matter trans-ports “heat” (higher velocity dispersion) from the outerregion to the “cooler” inner region of a halo. It leads toa constant density core with isothermal velocity disper-sion (e.g. Spergel & Steinhardt 2000; Rocha et al. 2013;Tulin & Yu 2018). If dark matter is warm, it decou-ples from the primordial plasma at relativistic velocity,thus free streaming out of small density peaks. As aresult, the structure formation at small scales are sup-pressed (e.g. Avila-Reese et al. 2001; Lovell et al. 2014).Warm dark matter scenario is found to have a “catch-22” problem, i.e., if the “cusp-core” problem is resolved,the requirement for the particle mass cannot form smallgalaxies at the first place, and vice versa (e.g. Macci`oet al. 2012).Another solution is to keep cold dark matter paradigmbut invoke efficient gravitational interaction betweendark matter and baryonic matter through stellar feed-back (Navarro et al. 1996; Governato et al. 2010). Over-all, as gas falls into the inner region of the halo, star for-mation takes place and the subsequent feedback throughsupernovae expels an appreciable amount of gas andstars to large radii. These baryonic matter pulls darkmatter particles to migrate outward through pure gravi-tational interaction, lowering the central density of darkmatter. The overall efficiency of this interaction andits effects are still under investigations (Di Cintio et al.2014; Read et al. 2016a; Tollet et al. 2016; Bose et al.2019; Ben´ıtez-Llambay et al. 2019; Read et al. 2019). Inaddition to the stellar feedback, other mechanisms havealso been proposed, such as dynamical friction betweengas clouds and dark matter particles (Nipoti & Binney2015).A definitive cuspy profile from observations will provethe validity of the cold dark matter paradigm on sub-galactic scales while challenging other types of dark mat-ter. As shown in Figure 1, the object AGC 242019 isan ultra-diffuse galaxy (UDG) identified by the AreciboLegacy Fast ALFA (ALFALFA) survey of H i galaxies(Leisman et al. 2017). This galaxy has a stellar massof (1.37 ± × M (cid:12) , a HI mass of (8.51 ± × M (cid:12) and a star formation rate of (8.2 ± × − M (cid:12) yr − as listed in Table 1. Its receding velocity is2237 ±
25 km s − after correcting for the Virgo, GreatAttractor and Shapley supercluster (Mould et al. 2000).In the cosmological frame of h =0.73, Ω =0.27 andΩ Λ =0.73, the corresponding distance is 30.8 Mpc and cuspy halo § Figure 1. False-color image of AGC 242019.
Thefalse-color image of the g , r and z bands. The white contoursof the H i intensity are at levels of 0.4, 0.6 and 0.8 mJy kms − . The yellow ellipse indicates the H i beam size.2. OBSERVATIONS AND DATA REDUCTION2.1.
Radio interferometric observation of H i gas withthe VLA Our L-band observations were performed with theKarl G. Jansky Very Large Array (VLA) through twoprojects corresponding to the D configuration (19B-072;PI: Y. Shi) and to the C and B configurations (20A-004;PI: Y. Shi). The D-configuration observations were per-formed on 21 st and 23 rd Sep. 2019, each with a 1.5-hrobserving time and a ∼
66 min on-source time. A to-tal of 27 antennas were employed in both executions,with the first under overcast weather conditions and thesecond under clear weather conditions. The projectedbaselines of the D-configuration array are in the rangeof 34–1050 m. The flux and bandpass calibrator was3C 295, and the gain calibrator was J 1419+0628. Weconfigured the spectrometers with a mixed setup. Thespectral line window that covers the H i ∼ − ), while the remaining windows covera frequency range between 963.0 MHz and 2017.0 MHzto optimize the sensitivity for the radio continuum.The C-configuration observations were performed on4 th , 9 th and 13 th Feb. 2020, each with a 2-hr observingtime and a ∼
90 min on-source time. Two observations h m s s s Right Ascension (J2000) D e c li n a t i o n ( J ) (a) m J y K m / s
200 100 0 100 200
V-V sys [km/s] f l u x d e n s i t y ( m J y ) (b) VLAArecibo0 2 4 6 8 10 12
Radius [kpc] g a s [ M p c ] (c) Figure 2. The HI data of AGC 242019. a,
The H i intensity. b, The integrated spectra of the VLA compared tothe Arecibo’s spectrum (Haynes et al. 2018). c, The radialprofile of the gas mass surface density corrected for inclina-tion and helium. were conducted with 25 antennas, while the rest wereobserved with 26 antennas, mostly under overcast orclear weather conditions. The projected baselines of theC configuration are in the range of 40–3200 m. In the C-configuration observations, the 21-cm emission line wasobserved with a spectral line widow that has a channelwidth of 5.682 kHz ( ∼ − ) and a bandwidthof 32 MHz. The remaining windows cover a frequencyrange between 963.0 MHz and 2017.0 MHz, for opti-mizing the radio continuum bandwidth. The flux andbandpass calibrator was 3C 286 and the gain calibratorwas J 1419+0628. Shi et al. h m s s R.A. D e c . v=1888.32 km/s + h m s s D e c . v=1881.23 km/s + h m s s D e c . v=1874.14 km/s + h m s s D e c . v=1867.06 km/s + h m s s R.A. D e c . v=1859.97 km/s + h m s s D e c . v=1852.88 km/s + h m s s D e c . v=1845.79 km/s + h m s s D e c . v=1838.71 km/s +14 h m s s R.A. D e c . v=1831.62 km/s + 14 h m s s R.A. D e c . v=1824.54 km/s + 14 h m s s R.A. D e c . v=1817.45 km/s + 14 h m s s R.A. D e c . v=1810.37 km/s + Figure 3. The channel map of the H i data . The contour levels are 1, 3, 4.5 and 6 mJy km s − . The white and redcontours indicate the observations and best-fitted models, respectively. The B-configuration observation was performed withfive executions during July and August of 2020, eachwith a 2-hr observing time and a ∼
90 min on-sourcetime. In most executions, 27 antennas were employedunder cloudy conditions. The projected baselines of theB-configuration are in the range of 230–11000 m. Thehardware setup and the calibrators were the same asthose used for the C-configuration observations.We reduced all data manually with the Common As-tronomy Software Applications (CASA) package (Mc-Mullin et al. 2007), v5.6.1 . Both D-configuration ob-servations were severely affected by radio frequency in-terferences (RFI). Therefore we calibrated and flaggedthe data manually. Approximately 20% of the data haveto be flagged, across different frequency ranges. The data obtained on 21 st Sep. 2019, including the cali-brators, has about three times higher noise than thatestimated from the VLA sensitivity estimator, due tounknown reasons. However, the fluxes, line profile, andspatial distributions are highly consistent with the dataobtained on 23 rd Sep. 2019. Whether combine it to thefinal data does not change the results. All gain calibra-tors were checked carefully to ensure a flat bandpass, apoint-source distribution and excellent calibration solu-tions. The 1.4 GHz flux densities of 3C 286, 3C 295 andJ 1419+0628 were 15.1 ± ± ± statwt task. Then,through the uvcontsub task, we fitted and subtracted cuspy halo h m s s s (a) J y Radius [kpc] s t a r [ M p c ] t (b) h m s s RA (J2000) D E C ( J ) (c) e r g / s / c m
1e 17 6555 6560 6565 6570 rest [ Å ] f [ e r g / s / c m / Å ]
1e 15 (d)
Figure 4. The infrared and optical data of AGC 242019 . a, The 3.6 µ m flux density with overlaid WiFeS IFU pointings. b, The radial profile of the stellar mass surface density corrected for the inclination. c, Individual H α clumps for which the lineof sight velocities are measured. Each clump has a circular radius of 2.0 (cid:48)(cid:48) . The small red circle indicates the dynamical center. d, The integrated spectrum of H α emissions. I n c li n a t i o n [ d e g ] (a) P . A . [ d e g ] (b) V r o t [ k m / s ] (c) H I [ k m / s ] (d) Figure 5. The two-stage operation of D Barolo . Thegray points are the first stage, where four parameters of eachring are set as free. The red points are the second stage,where the inclination and position angles are regularized byfitting a Bezier function to the results from the first stage. the radio continuum from the visibility data, with line-free channels on both sides of H i , using the first-orderlinear function. We resampled the C- and B-array datato match with the spectral resolution of the D-arraydata, using the mstransform task. The final spectrallymatched line-only data from the D, C, and B configura-tions were combined together with the concat task.In the end, we inverted the visibility data to the imageplane and cleaned the data cube with Briggs’ robust pa-rameter of 2.0 using the tclean task. The final D+C+Bdatacube has a velocity coverage of 500 km s − and achannel width of 32 kHz, corresponding to a velocityresolution of ∼ − . The r.m.s. noise level reaches0.26 mJy beam − per channel with a restoring synthesisbeam size of 9.85 (cid:48)(cid:48) × (cid:48)(cid:48) and a position angle of 17.56 ◦ .The H i intensity map is shown in Figure 2 (a). Theintegrated spectrum of the B+C+D configuration has aflux of 3.8 ± − , which is comparable to theflux of 3.4 Jy km s − obtained with the Arecibo (Leis-man et al. 2017), as shown in Figure 2 (b). The total Shi et al.
HI gas mass is (8.51 ± × M (cid:12) . A slightly higherflux as seen by VLA in the red wing may be caused by asmall offset of the Arecibo beam from the galaxy center,given a large size of the galaxy that is two arcmins inthe diameter. The radial profile of the gas mass surfacedensity is shown in Figure 2 (c). The channel map isshown in Figure 3.We also tested tilted-ring modeling with the combineddata using the B+C configurations and found essentiallyno difference, except for a higher noise level. Though thebetter velocity resolution leads to better estimates of thepressure support, which has a very minor contributionto the decomposition of the rotation curve.2.2. Broad-band images
The co-added images at 3.6 and 4.5 µ m were obtainedfrom the Wide-field Infrared Survey Explorer (WISE)archive (Wright et al. 2010) with spatial resolutions of6.0 (cid:48)(cid:48) and 6.8 (cid:48)(cid:48) , respectively. The target is well detectedat 3.6 µ m as presented in Figure 4 (a), but shows al-most no detection at 4.5 µ m. To derive the radial profileof the stellar mass surface density as presented in Fig-ure 4 (b), a few nearby bright stars in the field were sub-tracted using the stellar point spread functions . Thestellar mass based on the 3.6 µ m image is estimatedto be (1.37 ± × M (cid:12) for a Kroupa stellar ini-tial mass function and a mass-to-light ratio Υ . µm =0.6( M (cid:12) /L (cid:12) , . µm ) (see below).The optical g and r images were obtained fromthe Dark Energy Camera Legacy Surveys (Dey et al.2019). The far-ultraviolet image was retrieved from theGALEX data archive. The integrated flux was convertedto a star formation rate of (8.2 ± × − M (cid:12) yr − fora Kroupa stellar initial mass function (Leroy et al. 2008).2.3. Integral field unit observation of H α by ANU 2.3m Integral field unit observations of H α were carried outwith the Wide-Field Spectrograph (WiFeS) onboard theAustralian National University 2.3 m telescope on thenights of 21 st -22 nd Mar., 26 th May and 20 th -22 nd Jul.,2020. An R7000 grism (5290-7060˚A) at a resolution of7000 was adopted to cover the H α emission line. WiFeShas a field of view of 25 (cid:48)(cid:48) × (cid:48)(cid:48) . As shown in Figure 4 (a),exposures were taken at several positions to cover thewhole optical extent of the galaxy, with one pointingtoward a nearby bright star for astrometric calibration.Each exposure was 30 mins, and the total on-source inte-gration time varied from 2.0 hrs to 3.5 hrs at each pixel. http://wise2.ipac.caltech.edu/docs/release/allsky/expsup/sec4 4c.html For every 1–2 hrs, an off-target blank sky and a stan-dard star were observed. The seeing was between 1.5 (cid:48)(cid:48) and 2.0 (cid:48)(cid:48) . Each frame was first reduced following thestandard procedure by pyWiFeS (Childress et al. 2014),and then was subtracted by the median value of thesky frame at each wavelength. Individual frames werealigned with each other to produce the final mosaic im-age using the positions of the H α clumps, as the con-tinuum emission was too faint. The absolute astrom-etry was obtained through alignment with the bright-est star in the mosaic field of view. Since H α clumpsare not perfect point sources, we estimated the final as-trometric uncertainty to be about 1 (cid:48)(cid:48) . To correct thebarycentric velocity offset and any possible intrinsic in-strumental shift, the wavelength solution of each nightwas further cross-calibrated based on H α lines of thesame clumps observed during different nights. The in-tegrated H α flux map is presented in Figure 4 (c) withthe integrated spectrum shown in Figure 4 (d). DATA ANALYSIS3.1.
The derivation of the rotation curve of darkmatter
We derived the rotation curve of the dark matter byfirst obtaining the total rotation curve from the H i andH α data with a correction for the pressure support, andthen quadratically subtracting the gas and stellar grav-ity contributions as detailed below.3.1.1. The observed rotation curve from titled-ring fittingto the H i datacube We fitted the tilted-ring model to the H i D Barolo (Di Teodoro & Fraternali 2015) to ob-tain the rotation curve as listed in Table 2. The radialwidth of each ring is set to be 9 (cid:48)(cid:48) , which is roughly thebeam size so that each ring is independent. We adopteduniform weighting (WFUNC=0) and least-squared min-imization (FTYPE=1). All other optional parametersare set as default. The full list of the set-up is includedin Table A1.The angular resolution and the signal-to-noise ratiowere good enough to set the rotation velocity, the veloc-ity dispersion, the position angle and the inclination an-gle as free parameters for each ring. Before running thissetup, we first ran a model by also setting the dynamiccenter and the systematic velocity as free parameters,and then fixed them to the mean values of all rings aslisted in Table 1.The scale height was fixed to 100 pc, independent ofthe radius. If the height is varied by a factor of five(see below), the conclusion remains unchanged. We donotice that D Barolo is not able to remove effects fully cuspy halo DATA I N T E N S I TY MODEL I n t e n s i t y ( m J y * K M / S ) RESIDUAL I n t e n s i t y ( m J y * K M / S ) V E L O C I TY V L O S ( k m / s ) V L O S ( k m / s ) D I S P E R S I O N ( k m / s ) ( k m / s ) Figure 6. The moment maps and tilted-ring modeling achieved through D Barolo . First row: the moment-0 map,best-fitting model and residual for the H i intensity. Note that the run of D Barolo adopted the setting NORM=LOCAL whichnormalizes the model’s flux to the observed one pixel by pixel. As a result, the flux residual is zero.
Second row: the moment-1map, best-fitting model and residual for the velocity.
Third row: the moment-2 map, best-fitting model and residual for thevelocity dispersion. In all panels, the filled ellipse on the left lower corner indicates the beam, while the large red ellipse marksthe outer boundary of the last ring. due to the disk height for a thick disk (Iorio et al. 2017).When running D Barolo , we adopted the “twostage”fitting method, which allows a second fitting stage afterregularizing the first-stage parameter sets. As shownin Figure 5, the radial profiles of the inclination andposition angles from the first-stage fitting are regularizedby fitting a Bezier function, based on which the second-stage fitting is performed. The errors of the derivedrotation velocity and velocity dispersion in D Barolo (Di Teodoro & Fraternali 2015) are estimated through aMonte Carlo method. We also ran D Barolo with onlythe receding and approaching sides, respectively. Thevelocities from three runs are within 1- σ errors, whilethe velocity dispersions are also within errors except forthe innermost radius, which is shown in Figure B1 ofAppendix. As shown in Figure 5, the derived rotationvelocity and velocity dispersion vary well within the 1- σ error bars before and after regularization. As shown in Figure 6, within the outer boundary ofthe last ring, the residual of the H i intensity is domi-nated by the observed flux noise, and the residuals ofthe velocity and velocity dispersions show small ampli-tudes with medians of 1.9 km s − and 1.8 km s − , re-spectively. Figure 7 further shows the position-velocitydiagram along the major and minor axes. Overall, theobservation matches the best-fitted model well.As gas is collisional, pressure support is a driver of gasmotion in addition to gravity (Bureau & Carignan 2002;Oh et al. 2015; Iorio et al. 2017; Pineda et al. 2017). Agas disk in equilibrium satisfies: V = V + V , (1)where the circular velocity V circ = (cid:113) R ∂ Φ ∂R reflects solelythe effect of the gravitational potential, V rot is the ob-served rotation velocity and V P is the velocity driven by Shi et al.
Table 1.
The Properties of AGC 242019
Parameters Mean 1- σ error 3% HPD ¶
97% HPD ¶ M star (10 M (cid:12) ) 1.37 0.05 - - M HI (10 M (cid:12) ) 8.51 0.36 - -SFR (10 − M (cid:12) /yr) 8.2 0.4 - -Dynamical center ∗ (J2000) 14:33:53.38, 01:29:12.5 5.0 (cid:48)(cid:48) , 2.2 (cid:48)(cid:48) - - V ∗ sys (km s − ) 1840.4 1.9 - -Distance (Mpc) 30.8 5% - -log(Υ . µm ( M (cid:12) /L (cid:12) , . µm )) -0.22 0.1 - - R (NFW) (kpc) 65.0 7.4 54.6 74.3 R S (NFW) (kpc) 33.3 9.1 17.4 50.6concentration (NFW) 2.0 0.36 1.45 2.72 M halo (NFW) (10 M (cid:12) ) 3.5 1.2 1.5 5.8 V norm (ISO) (km s − ) 16.7 1.9 13.1 20.3 R C (ISO) (kpc) 2.5 0.5 1.6 3.3 Note — ∗ The dynamical center and systematic velocity are obtained by setting them as free parameters when running D Barolo . ¶ HPD refers to the Highest Posterior Density.
Table 2.
The results of the tilted-ring modeling
Rad V rotobs V circobs σ disp V circgas V circstar V circdark–matter P.A. Inclination(kpc) (km s − ) (km s − ) (km s − ) (km s − ) (km s − ) (km s − ) ( ◦ ) ( ◦ )H I ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± α ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± the pressure. V P is related to the gas density ( ρ ) andthe velocity dispersion ( σ v ) following V = − Rσ ∂ ln( ρσ ) ∂R . (2)Assuming that the scale height is independent of theradius and that the disk is thin, the above equation be-comes V = − Rσ ∂ ln( σ Σ obs cos i ) ∂R , (3)where Σ obs is the observed gas mass surface density, and i is the inclination angle. Following Equation 3, thepressure support correction is implemented in D Barolo (Iorio et al. 2017). For AGC 242019, the correction issmall, with | V circ − V rot | being smaller than ∼ − across the radius. In this case, only the error of V rot is propagated to V circ while ignoring the error of V A . The radial profile of V circ is referred to as the observedrotation curve.Note that our H i data have a spectral resolution of7 km s − which is comparable to or even lower thanthe derived velocity dispersion shown in Figure 5. Tocheck the effect of the limited spectral resolution, wealso performed tilted-ring fitting on the H i datacubefrom the B and C configurations that has a spectralresolution of 2.4 km s − . The difference in the velocitydispersion between the two data is within the 1- σ error,and the difference in the derived rotation curve is withinthe 1- σ error too.3.1.2. The observed rotation curve from the H α map The H α emission is clumpy and sporadic across thedisk. As a result, tilted-ring modeling cannot be per-formed for the H α data in the same way as for the H i data. Instead, we adopted the same ring parameters cuspy halo
75 50 25 0 25 50 75
Offset (arcsec) V L O S ( k m / s ) = 1
75 50 25 0 25 50 75
Offset (arcsec) V L O S ( k m / s ) = 91 V L O S ( k m / s ) V L O S ( k m / s ) Figure 7. The position-velocity diagram of thegalaxy along the major and minor axes of the H i gas disk. The blue and red contours are the observationand best-fitted model, respectively. The yellow symbol isthe derived line-of-sight velocity of each ring after correctingthe beam smearing. The contour levels are 0.73, 1.44, 2.88and 5.77 mJy km s − . from the H i data, including the center position, incli-nation angle, and position angle, to convert the line-of-sight velocities into the rotation curve. As shownin Figure 4 (c), we used a circular aperture with a ra-dius of 2 (cid:48)(cid:48) to extract the spectrum of the H α region andmeasured the line-of-sight velocity, whose error was es-timated through a Monte Carlo method, by randomlyinserting a Gaussian error at each data point and iter-ating one thousand times.The velocity was then converted into the rotationcurve with the following steps. 1) We first correct for theabsolute wavelength of the optical IFU data by compar-ing the H α velocity to the H i velocity at the same posi-tion. 2) The line-of-sight velocity is then converted intothe rotation velocity by V rot = V l . o . s /sin( i )/cos( θ ), where i is the inclination angle and θ is the azimuthal anglefrom the major axis in the plane of the un-projecteddisk. Here, the inclination angle and position angle arealso interpolated based on the result from the H i fitting.3) The dynamical center from the H i ring is adopted. V r o t [ k m / s ] (c) HII regions (reshifted side)HII regions (blueshifted side)Binned Data
Figure 8.
The rotation velocities from the line of sightvelocities based on the H i ’s ring parameters. These velocitiesare further rebinned with a radial bin of 1.0 kpc.
4) Individual velocities are then rebinned at a width of1.0 kpc to have enough points to obtain the mean value.Except for the outermost bin where only one datapointis available, the error of the mean is obtained throughthe Monte Carlo method, which fits a mean value afterit randomly inserts a Gaussian error at each data pointand iterates one thousand times. To obtain the pres-sure support correction for the H α , we assumed thatthe radial shape of the H α gas mass surface density andvelocity dispersions are similar to those of H i (Adamset al. 2014). The velocity dispersions of the H α clumpsafter correcting for the thermal broadening (10 km s − )are about the same as that of H i at similar locations. Asa result, we adopted the V P from the H i data to correctfor the H α rotation velocity.Since our H α emission is sporadic and the errors of therotation curve are only based on those detected regions,we used the H α rotation curve as a sanity check of theH i curve.3.1.3. The contribution to the rotation curve from thegravity of gas
The radial profile of the gas mass surface density de-rived from the tilted-ring modeling is shown in Fig-ure 2 (c). We used this profile to estimate the gas ro-tation curve using the
ROTMOD task (Begeman 1989) inthe GIPSY package. The gas mass has been correctedfor helium by multiplying it by a factor of 1.36. Thevertical distribution of the gas disk is assumed to be0
Shi et al. “SECH-SQUARED” with a scale height of 0.1 kpc. Byvarying the gas mass surface density with its error, weused the Monte-Carlo method to estimate the uncer-tainty of the rotation curve. We then varied the scaleheight from 0.02 kpc to 0.5 kpc to quantify the effect ofthe scale-height uncertainty on the rotation curve. Thetwo types of errors were summed quadratically to getthe final error.3.1.4.
The contribution to the rotation curve from thegravity of stars
To estimate the stellar mass distribution, we first de-rived the radial profile of the 3.6 µ m surface brightnessat a bin width of 3 (cid:48)(cid:48) , half of the resolution FWHM.The ellipse parameters derived by the above H i dynam-ical modeling were used, for which the inclination angleand the position angle were interpolated. We then con-verted this brightness into the stellar mass profile usingΥ . µm =0.6 M (cid:12) /L (cid:12) , . µm with a Kroupa stellar initialmass function (see below), as shown in Figure 4 (b). Toextend the profile to 10 kpc, an exponential disk wasfitted to the part outside a radius of 4 kpc.To measure the stellar contribution to the rotationcurve, we use the ROTMOD task (Begeman 1989) inGIPSY, where the vertical distribution was assumed tobe “SECH-SQUARED” with a scale height of 0.2 kpc.The errors of the stellar mass surface density are dom-inated by the subtraction of the sky background andits effects on the rotation curve are simulated with theMonte Carlo method. We also varied the scale heightfrom 0.01 kpc to 0.5 kpc to obtain the uncertainty dueto the scale height. The above two errors were quadrat-ically added to represent the final error of the stellarrotation curve.3.2.
The models of dark matter halo
The models of cuspy dark matter halo are moti-vated by numerical simulations of dark matter, includinga Navarro-Frenk-White (NFW) model (Navarro et al.1997) and an Einasto model (Einasto 1965). The modelsof cored dark matter halo are observatinally motivated,including the pseudo-isothermal (ISO) model (Begemanet al. 1991) and Burkert model (Burkert 1995).1) A NFW halo model (Navarro et al. 1997) has adensity profile of ρ NFW ( R ) = ρ c δ char ( R/R S )(1 + R/R S ) , (4)where ρ c =3 H /(8 π G) is the present critical density, δ char is dimensionless density contrast and R S is thescale radius. R S is related to R through the con-centration c = R / R S , where R is the radius within which the halo average density is 200 times the presentcritical density. δ char = c g with g = c ) − c/ (1+ c ) .The halo mass with R is M = H R G . The in-ner density profile of the NFW model shows a cusp with ρ ∝ R − . The corresponding rotation velocity of theNFW model is V NFW ( R ) = V (cid:115) ln(1 + cx ) − cx/ (1 + cx ) x [ln(1 + c ) − c/ (1 + c )] , (5)where V is the circular velocity at R with V = R h , and x = R / R .2) A pseudo-isothermal halo model (Begeman et al.1991) is observationally motivated to describe the pres-ence of a central core: ρ ISO ( R ) = ρ R/R C ) , (6)where ρ is the central density and R C is the core radiusof the halo. The rotation velocity is V ISO ( R ) = (cid:115) πGρ R C (cid:20) − R C R arctan( RR C ) (cid:21) = V norm (cid:115) ( R C R ) (cid:20) − R C R arctan( RR C ) (cid:21) , (7)where V norm = R √ πGρ and R =1 kpc.3) The Burkert density profile (Burkert 1995) is de-scribed by ρ Bk ( R ) = ρ R C ( R + R C )( R + R C ) , (8)where ρ and R C are the central core density and coreradius, respectively.The corresponding circular velocity is given by V Bk ( R ) = V C (cid:115) x ) + ln(1 + x ) − x ) x [3 ln(2) − , (9)where x = R/R c and V C is the circular velocity at thecore radius.4) With the rotation curve of a dark matter halo, thedensity profile of the halo can be derived through (deBlok et al. 2001): ρ ( R ) = 14 πG (cid:20) VR ∂V∂R + ( VR ) (cid:21) (10) cuspy halo V c i r c o b s [ k m / s ] (a) Circular Velocity (HI)Circular Velocity (H )Gas-GravityStellar-Gravity
Radius[arcsec] 0 2 4 6 8Radius[kpc]5101520253035 V c i r c d a r k m a tt e r [ k m / s ] (b) Dark-Matter (HI)Dark-Matter (H )NFW ModelISO ModelBurkert Model
Radius[arcsec]
Figure 9. The rotation curves of AGC 242019. a,
The observed rotation curves from the H i data (black circles) andthe H α data (red rectangles), along with the rotation curves due to the gas and stellar gravity contributions. b, The derivedrotation curve of dark matter. Different curves indicate the best-fitted NFW, ISO and Burkert models to the H I only rotationcurve. Radius[kpc] ( o b s - m o d e l ) / e rr o r NFW Model
Dark-Matter (HI)Dark-Matter (H )
Radius[kpc]
Dark-Matter (HI)Dark-Matter (H )
Radius[kpc]
Dark-Matter (HI)Dark-Matter (H )
Figure 10. The residuals of the fitting to the dark-matter rotation curve for three models . Table 3.
The priors for the dark-matter modeling
Parameters Bounded Normal ( µ , σ )NFW model R (75, 150) kpc R S (7.5, 15) kpcISO model V norm (25, 50) km s − R C (2, 4) kpcBurkert model V C (35, 70) km s − R C (2, 4) kpc Priors and set-ups of the rotation curve modeling
We fitted the above three dark-matter models to therotation curve of dark matter through Bayesian infer-ence with the Python code
PyMC3 (Salvatier et al. 2016).We adopted the No-U-Turn Sampler (NUTS) with 5000samples, 2000 tune, 4 chains and target accept=0.99.As listed in Table 3, the prior of each parameter ofa dark-matter model is described with a bounded nor-mal distribution, whose lower-limit bound is set to bezero. The mean of the distribution is set with someprior knowledge as detailed below, and the standard de-viation is set to be twice the mean.The prior R of the NFW model is set to have amean of 75 kpc, corresponding to a halo mass of 5 × Shi et al. log(Radius[kpc]) l o g ( d a r k m a tt e r [ M / p c ]) Dark-Matter (HI)Dark-Matter (H )NFW ModelISO Model
Figure 11. The density profile of dark matter ofAGC 242019.
Two curves represent the best-fitted NFWand ISO models to the H I rotation curve. M (cid:12) as expected by the stellar mass vs. halo mass rela-tionship given our stellar mass of (1.37 ± × M (cid:12) (Santos-Santos et al. 2016). At this halo mass, the haloconcentration is about 10 in simulations (Macci`o et al.2007), giving the mean prior R S of 7.5 kpc.Since the core radius of a cored dark-matter halo ison a kpc scale (Oh et al. 2015), the mean prior R C of the ISO model is set to be 2 kpc. The mean prior V norm of the ISO model is thus set to be 25 km s − , sothat the galaxy is on the baryonic Tully-Fisher relation-ship (Mancera Pi˜na et al. 2020) given its stellar mass of(1.37 ± × M (cid:12) and a HI mass of (8.51 ± × M (cid:12) . Similarly, the mean prior R C of the Burkert modelis set to be 2 kpc, and the mean prior V C of the modelis set to be 35 km s − .The fitting is convergent. By exploring the full pos-terior distribution, the best-fitted parameter is given asthe mean of the distribution, and the error is given bythe standard deviation, as well as the 3% and 97% High-est Posterior Density (HPD). The percentage of smallPareto shape diagnostic values ( k < RESULTS4.1.
The rotation curve of dark matter
Figure 9 (a) shows the observed rotation curve of AGC242019. The H i measurements cover a radial extent of 9kpc while the H α data cover the central 5 kpc in radius. m + 3 (a) m - 3 (b) Distance + 3 (c)
Distance - 3 (d)
Height x 5 (e)
Height/5 (f)
Radius[arcsec]Radius[kpc] V r o t d a r k m a tt e r [ k m / s ] Figure 12. The rotation curves of dark matter byvarying different parameters . a-b, The results with themass-to-light ratio in 3.6 µ m varying by ± σ . c-d, Theresults with the distance varying by ± σ . e-f, The resultswith the scale height of the H i gas disk varying by a factorof five. Two datasets are overall consistent with each other inthe overlap region. The rotation curve rises all the wayup to the last measurable radius.Figure 9 (b) shows the rotation curve of dark matterafter quadratically subtracting the gas and stellar grav-ity contributions. The sampling of the H i curve is atthe beam size, while the H α curve has a radial bin of1.0 kpc. As shown in the figure, the H α curve is overallwell consistent with the H i curve over the spatial extentwhere both dataset cover. As mentioned before, sincethe H α emission is sporadic and the errors of the H α ro-tation curve are only based on the detected regions, itsrotation curve is only used for the sanity check of the H i curve. We fitted the H i curve with two spherical halomodels, namely, the NFW model (Equation 5, Navarroet al. 1997) and the ISO model (Equation 7, Begemanet al. 1991), through the Python code PyMC3 (Salvatieret al. 2016) to represent the cuspy and cored profiles, re-spectively. The ISO model is too steep at the inner radii,whereas the NFW model matches the observed data inthe overall fitting, which is also illustrated by the fit- cuspy halo radius[kpc] c m ( r ) [ k m / s ] (a) c c radius[kpc] s m ( r ) [ k m / s ] (b) s s s radius[kpc] A m ( r ) [ k m / s ] (c) A A A radius[kpc] A t o t ( r ) [ k m / s ] (d) radius[kpc] A t o t ( r ) / V c i r c o b s ( r ) (e) radius[kpc] p o t s i n ( (f) Figure 13. The results of harmonic expansion of non-circular motions. a,
The individual c component up to theharmonic order of 3. b, The individual s component up to the harmonic order of 3. c, The amplitude of each harmonic order. d, The total harmonic amplitude. e , The ratio of the total harmonic amplitude to the circular velocity. f, The elongation ofthe potential as described by (cid:15) pot sin2 φ . ting residuals as shown in Figure 10. To quantitativelydiscriminate the two models, we performed a leave-one-out (LOO) predictive check (Vehtari et al. 2015). Wefound that the estimated effective number of parame-ters ( p loo =1.6) of the NFW model is smaller than thereal number of free parameters, i.e., two, while that ofthe ISO model has p loo =6.3, significantly larger thantwo (see Table 4). This quantitatively indicates thatthe NFW model is valid, while the ISO model is ruledout. The corresponding difference in the reduced χ is1.9 for d.o.f.=5, equivalent to a Gaussian significance of3.0- σ . As listed in Table 4, some Pareto diagnostic value k of the ISO model is larger than 0.7, further indicatingthat the model is mis-specified. We also run the fittingby including the H α , and obtained similar difference in p loo values and ∆( χ /d.o.f.)=2.8 for d.o.f.=9 betweenthe two models, equivalent to a Gaussian significance of5.0- σ .The Burkert model is another observationally-motivated formula to describe a cored dark matter halo(Equation 9, Burkert 1995). Its density profile is flattoward the center, which is like the ISO model, but hasa power law index of -3.0 toward infinity, which is likethe NFW model. As shown in Figure 9 (b), the Burk- ert model gives a bad fitting too, with p loo =4.4, muchlarger than (1.6) that of the NFW model.The best-fitted NFW model has a halo scale radiusof ∼
33 kpc. Such a large cusp is well spatially resolvedgiven that its size relative to the H i beam is 24. Thissuggests that the presence of the cusp is not an artifactcaused by a limited spatial resolution (de Blok et al.2001; Oh et al. 2015). The dark-matter halo has a massof (3.5 ± × M (cid:12) within R , the radius at whichthe average halo density is 200 times the average cosmicdensity. Due to the fact that the rotation curve doesnot reach the flat part, the constraints on the R (or M halo ) and R S do not reach a small error. But a smallhalo concentration of only 2.0 ± § α innermost of the dark matter halo, we derived the density profileof dark matter from the rotation curve (de Blok et al.2001) as shown in Figure 11. Following de Blok et al.(2001) and Oh et al. (2015), we measured α innermost bycarrying out the least-square fitting to the density pro-4 Shi et al. file within the break radius, and defined the error of α innermost as the difference between the result includingthe first data-point outside the break radius and the re-sult including only data-points within the break radius.Here the break radius is the radius where the densityprofile shows a maximum change in the slope. If adopt-ing the core radius of the best-fitted ISO model as thebreak radius, α innermost =-(0 . ± . α innermost =-(0.90 ± σ that thedark matter halo of AGC 242019 is cuspy down to a ra-dius of 0.67 kpc .4.2. The systematic uncertainties of the dark-matterrotation curve
The identification of a cuspy profile in AGC 242019is robust against systematic uncertainties from severalaspects. (1) Position and inclination angles:
Since thesetwo angles have been set as free parameters during thefitting, their uncertainties have already been includedin the derived rotation curve. For comparison, we fur-ther estimated the photometric position and inclinationangles based on the r -band image. The galaxy in the r -band is somewhat asymmetric with clumpy features,but outside the radius of 8 (cid:48)(cid:48) , the position angle and incli-nation angles converge to (2 ± ◦ and (67 ± ◦ , respec-tively. These results are consistent with the values de-rived from the dynamic fitting of the H i data, as shownin Figure 5 and listed in Table 2. (2) Mass-to-light ratio in 3.6 µ m: The rotationvelocity due to stellar gravity is proportional to thesquare root of the stellar mass-to-light ratio. The over-all stellar contribution to the observed rotation curve issmall, so our result is not sensitive to the mass-to-lightratio Υ . µm , as is detailed here. The mass-to-light ratioin a broad band is obtained by fitting a synthetic spec-trum to the observed spectrum or a broad-band spec-tral energy distribution. The uncertainties of the stellarpopulation synthesis model, the star formation history,the dust extinction, the metallicity and the initial massfunction all result in the variation of the derived mass-to-light ratio.With the measured stellar mass and star formationrate of AGC 242019, we assumed a low metallicitywith [Fe/H]=-1 and suppressed asymptotic giant branch(AGB) stars, as seen in some low surface brightnessdwarfs (Schombert et al. 2019), to derive the Υ . µm ofour target to be 0.6 (Schombert et al. 2019). The pre- ceding assumption about AGB stars causes the mass-to-light ratio to be ∼
30% larger than that in normalgalaxies. Υ . µm is also color dependent (Bell et al. 2003;Zibetti et al. 2009; Jarrett et al. 2013; Meidt et al. 2014;Shi et al. 2018; Schombert et al. 2019; Telford et al.2020). As the object is not detected in 4.5 µ m, we usedthe radial variation in g − r color with a median of 0.3and standard deviation of 0.03. Converting to B − V = 0.98*( g − r ) + 0.22 (Jester et al. 2005), the variationin Υ . µm is expected to have a standard radial devia-tion as small as 0.02 dex (Schombert et al. 2019), andthus, its effects on the rotation curve of dark matter arenegligible.The systematic uncertainties due to the difference instar formation histories, initial mass functions and stel-lar population models among different studies are muchlarger even at a fixed color (Bell et al. 2003; Zibetti et al.2009; Jarrett et al. 2013; Meidt et al. 2014; Schombertet al. 2019; Telford et al. 2020). As a result, we adopted0.1 dex as the 1- σ error to encompass the results indifferent studies. We then varied Υ . µm by ± σ to in-vestigate its effect on the rotation curve of dark matter.As listed in Table 4 and shown in Figure 12, the cusp-like NFW model is a more reasonable model than thecore-like ISO model for both Υ . µm values. (3) Distance: The velocities due to baryonic gravityvary with the square root of the distance, while the ob-served rotation curve is independent of the distance. Asthe object is an isolated galaxy with no close-by com-panion that is brighter than m r = 17.7 (or M r = -14.8)within 500 kpc and 1000 km s − in the Sloan DigitalSky Survey, the error in the distance is dominated bythe uncertainty of the Hubble constant and the peculiarvelocity. The heliocentric velocity is 2237 km s − af-ter correcting for the Virgo, Great Attractor and Shap-ley supercluster (Mould et al. 2000). We estimated theresidual error to be 100 km/s, which corresponds to aninfalling velocity toward an imaginary dark matter halowith 10 M (cid:12) at a separation of 50 kpc. By adoptinga local Hubble constant of 73 km s − with 2.5% uncer-tainty (Riess et al. 2016), the distance thus has a 1- σ uncertainty of 5%. We varied this distance by ± σ toinvestigate its effect on the dark-matter rotation curve.As listed in Table 4 and shown in Figure 12, the cusp-like NFW model is again advocated against the core-likeISO model for both distances. (4) The scale height of the gas disk: For a thickdisk, emissions from adjacent rings are projected to bein the same pixels, which causes a beam smearing andaffects the measurements of the inclination and positionangles. By increasing and decreasing the scale heightby a factor of five to 500 pc and 20 pc, respectively, the cuspy halo Table 4.
The fitting result of NFW and ISO models to the dark-matter rotation curve with
PyMC3
Rotation Curves NFW model ISO model R R s p LOO k ∗ ( < χ /d.o.f. ¶ V norm R C p LOO k ∗ ( < χ /d.o.f. ¶ (kpc) (kpc) (km s − ) (kpc)HI (fiducial) 65.0 ± ± ± ± α (fiducial) 64.6 ± ± ± ± . µm +3 σ ) 59.5 ± ± ± ± . µm -3 σ ) 67.2 ± ± ± ± σ ) 62.1 ± ± ± ± σ ) 73.8 ± ± ± ± × Height) 60.7 ± ± ± ± ± ± ± ± Note — ∗ The percentage of the Pareto diagnostic values that are “good” and “ok” ( k < ¶ The reduced χ is given for thebest-fitted result with PyMC3 . Table 5.
The fitting result of the Burkert model to theHI’s curve of dark matter V C [km/s] R C [kpc] p LOO k ∗ ( < χ /d.o.f. ¶ ± ± Note — ∗ The percentage of the Pareto diagnostic values thatare “good” and “ok” ( k < ¶ The reduced χ is given forthe best-fitted result with PyMC3 . NFW model is still much better than the ISO model, asshown in Figure 12 and Table 4. (5) The noncircular motion:
As shown in Figure 6,the median amplitude of the residual velocity obtainedby D Barolo fitting is small, i.e., ∼ − . Duringthe fitting, the line-of-sight velocities are assumed tobe entirely circular motions, while noncircular motionscause the real circular velocity to be underestimated. Toquantify the amplitude of noncircular motions, we car-ried out harmonic decomposition with the GIPSY task RESWRI (Begeman 1989). As detailed in previous stud-ies (Schoenmakers et al. 1997; Trachternach et al. 2008),the line-of-sight velocity of each ring can be decomposedinto v los ( r ) = v sys ( r ) + Σ Nm =1 ( c m ( r )cos mψ + s m ( r )sin mψ ) , (11)where r is the radial distance of each ring from thedynamical center, v sys ( r ) is the system velocity, ψ isthe azimuthal angle in the plane of the disk. v los ( r ) = v sys ( r )+ c ( r )cos ψ corresponds to a pure circular motionscenario. In this study, we expanded the velocity up to m =3, as has been adopted in other studies (Schoenmak-ers et al. 1997; Trachternach et al. 2008).To run RESWRI , we used the 2-D velocity field pro-duced by D Barolo as the input, fixed the dynami-cal center and system velocity to those determined by D Barolo , and set the rotation velocity, inclination an-gle and position angle of each ring as free parameters.With the derived c m and s m , the amplitude of each non-circular harmonic component with m > A m ( r ) = (cid:112) c m ( r ) + s m ( r ) (12)and for m =1 where c is the circular motion, A ( r ) = (cid:113) s ( r ) . (13)The total amplitude of noncircular motion is given by A tot ( r ) = (cid:113) s ( r ) + c ( r ) + s ( r ) + c ( r ) + s ( r ) . (14)The measured harmonic component can be used toquantify the elongation of the potential (cid:15) pot at each ra-dius as follows: (cid:15) pot sin2 φ = ( s − s ) 1 + 2 q + 5 q c (1 − q ) , (15)where φ is an unknown angle between the minor axis ofthe elongated ring and the observer in the plane of thering and q = cos i , with i being the inclination angle ofthe disk.Figure 13 shows the result of the harmonic decomposi-tion. As shown in Figure 13 (a), the radial c m fluctuatesaround 0 km/s with amplitudes (cid:46) s m shows similar behaviors with amplitude (cid:46) A m of each harmonic component is ingeneral (cid:46) A tot is only about 2 km/s. As shown in Fig-ure 13 (c), all the amplitudes are small fractions of thecircular velocities at the corresponding radii that are < Shi et al. indeed a galaxy without stronger noncircular motions.Compared to simulated galaxies where noncircular mo-tions result in noticeable underestimations of circularvelocities (Oman et al. 2019), the noncircular amplitudeof ∼ (cid:15) pot sin2 φ , as shown in Figure 13 (f), suggestsa spherical gravitational potential. (6) The triaxiality of a dark matter halo: In thisstudy a spherical dark matter halo is assumed, while ahalo has been found to be moderately triaxial in nu-merical simulations (Jing & Suto 2002; Bailin & Stein-metz 2005). However, a typical triaxial mass distribu-tion results in only a small deviation in the density fromthe spherical assumption. Within the scale radius ofthe halo, the difference is only 10-20%, which is muchsmaller than the required variation of a factor of 3 to de-crease the inner slope by 0.5 (Knebe & Wießner 2006).Numerical modeling of the rotation curve further sug-gests that the halo triaxiality cannot significantly changethe shape of the curve to make an intrinsic cusp to bea core (or vice versa) in the observed data (Kuzio deNaray et al. 2009; Kuzio de Naray & Kaufmann 2011). (7) Beam smearing: D Barolo takes the beamsmearing into account in the tilted-ring fitting (DiTeodoro & Fraternali 2015). D Barolo first builds agas disk in three spatial dimensions and three veloc-ity dimensions, and then convolves this artificial diskto the observed spatial resolution for comparison withthe observed 3-D datacube to derive the best-fitting pa-rameters. It has been shown that D Barolo is able torecover the rotation curve even at a low spatial reso-lution, i.e., two resolution elements over the semimajoraxis (Di Teodoro & Fraternali 2015). The semimajoraxis of AGC 242019 is resolved into ∼ i map. In addition, the H α map has a rebinned spa-tial resolution (4 (cid:48)(cid:48) in diameter) that is two times higherthan the H i map. Although our H α clumps are mainlydistributed along the major axis with a narrower spa-tial extent, the overall shape of the H α -derived curveis consistent with the H i curve, demonstrating that thebeam smearing effect on the H i ’s rotation curve has beenlargely removed by D Barolo (Di Teodoro & Fraternali2015). (8) The contributions from molecular gas andionized gas:
The very low surface density of AGC242019 results in a low mid-plane pressure with a P ext /k (cid:46) cm − K, which gives a molecular-to-atomic gas ra-tio of (cid:46)
5% (see their Figure 3 and Equation 12 in stud-ies of nearby galaxies (Blitz & Rosolowsky 2006; Leroy et al. 2008)). We also checked that the atomic gas aloneis sufficient to place the galaxy in the extended Schmidtlaw (Shi et al. 2011, 2018), consistent with a negligiblefraction of molecular gas.The ionized gas mass can be estimated from the H α luminosity by M HII = m p L Ha . × − N e (Osterbrock &Ferland 2006), where m p is the proton mass, L Ha is theH α luminosity in erg/s, and N e is the electron volumedensity in cm − . By assuming a low N e of 100 cm − ,we found that, even at the peak spaxel as shown in Fig-ure 4 (c), an ionized gas mass surface density of 0.025 M (cid:12) /pc is too small to affect the derived rotation curveof dark matter. DISCUSSIONSThrough measurements of the dynamics of atomic andionized gas, we demonstrate that the dark matter haloof AGC 242019 can be well fitted by the cuspy profile asdescribed the NFW model, while excluding cored mod-els including ISO and Burkert ones. We here discussits constraints on the alternatives of standard cold darkmatter, implications for the role of feedback and impli-cations for formation of UDGs.5.1.
Implications for the alternatives of standard colddark matter
Fuzzy cold dark matter
Through numerical simulations, a halo of fuzzy colddark matter is found to be composed of a soliton coresuperposed on an extended halo (Hu et al. 2000; Schiveet al. 2014). The latter can be represented by the NFWmodel, while the former can be approximately describedby: ρ c ( r ) ≈ . m ψ / − eV) − ( R c / kpc) − [1 + 9 . × − ( r/R c ) ] M (cid:12) pc − , (16)where m ψ is the particle mass and R c is the core radius.The soliton core is linked to the total halo through thecore-halo relationship (Schive et al. 2014), which gives R c = 1 . m ψ / − eV) − ( M h / M (cid:12) ) − / kpc . (17)As shown in Figure 9 (b), a NFW model fits the den-sity profile of AGC 242019 very well, which shows negli-gible residual density for the soliton core to account for.As a result, the possible contribution to the dark-matterdensity from the soliton core should not exceed the ob-served error at all radii, which can be used to constrainthe m ψ . For each radius, we estimated the density ofthe soliton core as a function of the particle mass m ψ with the best-fitted M h =(3.5 ± × M (cid:12) through cuspy halo m ψ . The 3- σ observed errors at two radiiconstrain the m ψ range to be < × − eV or > × − eV . Compared to the constraint from Ly α forest in which m ψ < × − eV or m ψ > × − eV, the dynamics of AGC 242019 gives a factor of about20 times smaller constraint on the upper-bound of thelower range. As shown in Figure 14 (a), if adopting the3% HPD halo mass of 1.5 × M (cid:12) , m ψ < × − eV or m ψ > × − eV whose upper-bound of thelower range is still 10 times lower than the constraint bythe Ly α forest. If somehow our error is underestimatedby a factor of 2, the upperbound of the lower range onlyincreases by a factor of ∼ ∼ α forest, is inconsistent with the typical m ψ of ∼ − eV that is required to explain the dynamics ofother galaxies with cored dark-matter halos (Hu et al.2000; Schive et al. 2014). It is thus found that thereis no m ψ value that can reconcile all the observationalfacts. 5.1.2. Self-interacting Dark Matter
Self-interacting dark matter transmit the kinetic en-ergy from the outer part inward to form a constant den-sity core. For the interaction to be efficient, the scat-tering rate per particle should be important, that is atleast once over the galaxy age (Spergel & Steinhardt2000; Rocha et al. 2013; Tulin & Yu 2018):Γ( r ) t age ≈ ρ ( r )( σ/m ) v rms ( r ) t age ∼ , (18)where is Γ( r ) is the scattering rate per particle, ρ ( r ) isdark matter density at a radius of r , σ/m is the crosssection per particle mass and v rms is the relative velocityof dark matter particles. The above equation can be re-written as: σ/m / g ≈ ( 9 . t age )( 0 .
01 M (cid:12) / pc ρ ( r ) )( 50 km / s v rms ) . (19)For AGC 242019, the NFW model fits the rotationcurve well down to the innermost radius. This sets theupperlimit to the radius of a possible density core andthe lowerlimit to the above ρ ( r ) if dark matter is self-interacting in AGC 242019. If the halo forms around z =2, we got t age = 10 Gyr. The v rms is set to be thevirial velocity at the virial radius, which is 47 km s − .we have σ /m < /g for AGC 242019.As shown in Figure 14 (b), existing studies prefersomewhat larger σ/m on galaxy scales (Elbert et al.2015; Kaplinghat et al. 2016; Zavala et al. 2013) and smaller values on cluster scales (Kahlhoefer et al. 2015;Randall et al. 2008; Harvey et al. 2015; Kaplinghat et al.2016). Such a velocity dependence of the cross sectionseems to reconcile results over different scales (Kapling-hat et al. 2016). However, the cuspy dark matter halo ofAGC 242019 may challenge this simple picture, whoseupperlimit to σ/m is somewhat in tension with thelowerbound of the σ/m range as required by cored halosof other dwarf galaxies.5.1.3. Warm Dark Matter
In warm dark matter, the density core has a size thatcan be approximately by Hogan & Dalcanton (2000);Macci`o et al. (2012) r = √ πGQ max < v > . , (20)where v rms is the velocity dispersion (i.e. the mass) ofthe halo. Q max is the maximum phase density as givenby Q max = 1 . × − ( ρ L ρ cr )( m keV ) M (cid:12) pc − (km s − ) , (21)where m is the mass of warm dark matter, ρ L ρ cr is thelocal density relative to the critical density. m keV = 0 .
37 1( ρ L /ρ cr ) . ∗ ( kpc r core ) . ( km / s < v > . ) . . (22)As shown in Figure 14 (c), the innermost radius ofthe cuspy halo of AGC 242019 leads to m > The modified Newtonian dynamics
Unlike massive disk galaxies whose baryonic disk sur-face density rises exponentially toward their galacticcenters, AGC 242019 shows a much flatter profile witha density deficit in the central region. Such a distinctspatial offset between the baryonic matter and the dy-namical mass leads to an increasing baryonic matter8
Shi et al. m /10 eV S o li t o n C o r e [ M p c ]
3- observed error at R=0.67kpc3- observed error at R=2.02kpc (a)
Fiducial (Hu+00) Armengaud+17Schive+14This Work V max [km/s] / m [ c m / g ] (b) This Work
Dwarf+LSB (Kaplinghat+16) Clusters (Kaplinghat+16)Local Spheroids (Zavala+13)Local Spheroids (Elbert+15) Bullet Cluster (Randall+08)Abell 3827 (Kahlhoefer+15)Clusters (Harvey+15) m[keV] r c o r e [ k p c ] This Work(c)
LITTLE THINGS 13.5 13.0 12.5 12.0 11.5 11.0 log(g baryon [m/s ]) l o g ( g o b s [ m / s ]) (d) d e e p - M O N D ( g o b s g . b a r y o n ) L a t e - T y p e G a l a x i e s : ( N o D a r k M a tt e r ) HIHLelli+17
Figure 14. Test of the alternatives of standard cold dark matter with AGC 242019 . a, The test of the fuzzy darkmatter. Two curves represent the soliton core mass density as a function of the dark matter particle mass at the innermost andsecond innermost rings, respectively. The dotted lines are the observed 3- σ errors at the corresponding two radii. Color barsare the constraint of the particle mass by different studies (see text). b, The test of the self-interacting dark matter. Brownsymbols are the constraints in the literature. The density of the innermost ring of AGC 242019 results in the upperlimit of thecross section of self-interacting dark matter. c, The test of warm dark matter. The radius of the innermost ring of AGC 242019marks the lowerbound of the particle mass of warm dark matter. d, The test of MOND through the relationship between theobserved radial acceleration and the baryonic radial acceleration. The black/red symbols are the results of AGC 242019, wherea larger symbol size corresponds to the ring with a larger radius. The blue solid line is the best linear fit to the observations.Lines labeled with “Late-Type Galaxies” and orange symbols are the best-fitted line plus its scatter and individual late-typegalaxies in Lelli et al. (2017). The dotted line labeled with “deep-MOND” is the MOND prediction in the low accelerationregime with a slope of 0.5. The dashed line labeled with “No Dark Matter” is the no dark matter line. relative to dark matter at larger radii, in contrast togalaxies in general. This can be quantified by the loga-rithmic relationship between the observed radial accel-eration and the baryonic radial acceleration as shownin Figure 14 (d). From the inner (smaller symbols) tolarger radii (larger symbols), the data are more closer tono dark matter line, demonstrating the increasing bary-onic matter relative to the dark matter at larger radii. The modified Newtonian dynamics (MOND)paradigm (Milgrom 1983) has been proposed as analternative to dark matter theory for interpreting dy-namical features. However, MOND cannot explain thedynamics of AGC 242019 as specified by the radialacceleration relationship shown in Figure 14 (d). Therelationship has a slope of 0.15 ± cuspy halo a . The slope of AGC 242019 is thus 3.2- σ below thethreshold in the MOND, although it lies on the theextrapolation of the relationship defined by late-typegalaxies (Lelli et al. 2017). log( M star / M halo ) s l o p e ( . < r / R v i r < . ) Di Cintio+14Tollet+16Benitez-Llambay+19 (low SF threshold)AGC 242019 (This Work)
Figure 15. Implications of AGC 242019 for the stel-lar feedback model.
Three lines represent different modelsabout the dark-matter central density slope as a function ofthe M star /M halo ratio. The red diamond symbol representsAGC 242019 in this study. Implications for the role of feedback on producingcored dark matter halo
It is found that the effect of baryonic feedback on thedensity profile of dark matter depends on the stellar-to-halo mass ratio ( M ∗ /M halo ) (e.g. Di Cintio et al.2014; Tollet et al. 2016), as shown in Figure 15. Inthe low M ∗ /M halo regime ( < M ∗ /M halo regime ( > M ∗ /M halo =0.6%.The above result is independent of model parameterssuch as star formation threshold, initial mass func-tion, supernovae energy etc. However, AGC 242019has a stellar mass of (1.37 ± × M (cid:12) and a halomass of (3.5 ± × M (cid:12) , leading to M ∗ /M halo of(0.39 ± M baryon [M ] j b a r y o n [ k p c k m s ] (a) J M / J M / AGC 242019dwarf irregular M halo [M ] j b a r y o n [ k p c k m s ] J M / J M / (b) AGC 242019dwarf irregular log( [(M pc ) ]) l o g ( S F R [ M / y r / k p c ]) R e g i o n s I n S p i r a l s (c) Outer Regions Of DwarfsNGC1614 Ring+NucleusIC4687 RegionsLocal U/LIRGs Nuc.High-z Integrated main-sequenceHigh-z Integrated starburstsGMCs in M33AGC 242019
Figure 16. The implication of AGC 242019 for theUDG formation. a,
The specific angular momentum ofthe baryons as a function of the baryonic mass, where redstar symbol is AGC 242019 as compared to dwarf irregu-lars (Butler et al. 2017). b, the same as a, but versus thehalo mass. c, The location of AGC 242019 in the extendedSchmidt law (Shi et al. 2018). Shi et al. thus inconsistent with the above M ∗ /M halo dependenceof the inner dark-matter profile.Some other studies emphasize the importance of starformation threshold on the effect of the stellar feed-back (e.g. Governato et al. 2010; Ben´ıtez-Llambay et al.2019). If the threshold for gas to form stars is high, alarge amount of gas can accumulate in the center of ahalo and dominate the potential before star formationtakes place. The subsequent feedback-driven massiveoutflows or repeated multiple outflows alter the orbitsof dark matter to produce a dark-matter core. On theother hand, for low star formation threshold, gas is ex-pelled by feedback before it contributes significantly tothe potential. A dark-matter cuspy profile thus pre-serves. This scenario at least partly explains the forma-tion of dark-matter cores in some simulations (Di Cintioet al. 2014; Fitts et al. 2017; Macci`o et al. 2017) while notin others (Bose et al. 2019). The simulation by Ben´ıtez-Llambay et al. (2019) predicts an intact cuspy dark-matter density profile independent of the M ∗ /M halo ifstar formation threshold is low. As shown in Figure 15,AGC 242019 is consistent with their prediction. As aUDG, AGC 242019 does have a low gas and stellar masssurface density with ongoing star formation as revealedby the GALEX far-UV image. This indicates that onsub-kpc scales star formation in AGC 242019 can pro-ceed at a very low gas mass surface density that is onthe order of 1 M (cid:12) /pc or 0.4 × (100pc/ h ) cm − , where h is the scale height of the gas disk. However, its star for-mation efficiency (SFR/gas=0.03 Gyr − ) is much lowerthan that in spiral galaxies ( ∼ − ) (Leroy et al.2008; Shi et al. 2011), inconsistent with that adopted inthe simulations. Therefore, low star formation thresh-old should not be simply the physical cause for a cuspyprofile in AGC 242019.Besides the star formation threshold, the duration ofstar formation may be also important as recognized insome simulations (Read et al. 2016a). As long as starformation proceeds long enough, e.g., ∼
10 Gyr for a halomass of 10 M (cid:12) and a longer timescale for a larger halo,a halo core always form. AGC 242019 has a halo mass of(3.5 ± × M (cid:12) , and its low concentration implieslate formation time (see next section), two of which mayexplain its cuspy profile given the above scenario.5.3. Implications for formation of UDGs
UDGs are low-stellar-mass dwarfs but with sizes typ-ical of spiral galaxies (Abraham & van Dokkum 2014;Leisman et al. 2017). They were found in all environ-ments including galaxy clusters, galaxy groups and field.Many mechanisms have been proposed to understandtheir origin: they may be normal dwarf galaxies but ex- perience star-formation feedback that re-distributes gasand stars to larger radii (Governato et al. 2010; Di Cintioet al. 2014; Jiang et al. 2019); they may live in a high-spin dark matter halo with extended gas distributionsand low efficiencies in converting gas into stars (Amor-isco & Loeb 2016); some environmental effects such asram-pressure stripping or tidal puffing may be also im-portant in formation of UDGs (Yozin & Bekki 2015;Jiang et al. 2019).The dark matter halo of AGC 242019 has a mass of(3.5 ± × M (cid:12) , which is typical of a halo hostinga dwarf. This suggests that AGC 242019 is not a failedmassive galaxy, unlike other UDGs found in clusters(van Dokkum et al. 2016; Beasley et al. 2016). AlthoughUDGs may have diverse origins, our measurements aremore reliable. In studies of van Dokkum et al. (2016)and Beasley et al. (2016), the halo mass was inferredfrom 1-2 velocity data-points by assuming a halo shapeespecially the concentration.The cuspy halo of AGC 242019 also suggests that thefeedback has not been strong over its history to expela large amount of baryonic matter to large distances.Otherwise, a cored halo should have already formed likein other dwarf galaxies as suggested by feedback mod-els (Navarro et al. 1996; Governato et al. 2010). AGC242019 thus has experienced weak feedback over its his-tory. This seems to be consistent with the deviation ofUDGs from the Tully-Fisher relationship: gas and starsare not expelled out of the disk so that a UDG containmore baryons at a given maximum circular velocity thatroughly represents the halo mass (Mancera Pi˜na et al.2020). AGC 242019 has a maximum circular velocityof 47 km s − and a baryonic mass of (9.88 ± × M (cid:12) , placing it slightly above the Tully-Fisher relation-ship too. With the accurate measurement of the halomass, AGC 242019 is found to be off both the M ∗ vs. M halo and M baryon vs. M halo relationships (Santos-Santos et al. 2016). The low SFR of AGC 242019 alsosuggests weak ongoing stellar feedback as implied by therelationship between the SFR and the ionized gas veloc-ity dispersion (e.g. Yu et al. 2019). Our regular velocityfield with small non-circular motion as shown in § ± cuspy halo
21A “young” halo thus suggests late formation of AGC242019, which seems to be consistent with the findingsof UDGs in cosmological simulations (Rong et al. 2017).The specific angular momentum can be derived fromthe rotation curve combined with the stellar and gasmass profiles. The mass profiles were extended to 15 kpcfollowing the same procedure that estimates the bary-onic contribution to the rotation curve (see § § § CONCLUSIONSWe have carried out the spatially-resolved mapping ofgas dynamics toward a nearby UDG, AGC 242019. It isfound that AGC 242019 has a cuspy dark matter haloat a high confidence, which demonstrates the validityof the cold dark matter paradigm on subgalactic scales.Our main conclusions are(1) AGC 242019 has an overall regular velocity field.After subtracting the baryonic contribution, the rota-tion curve of dark matter is well fitted by the cuspy pro-file as described by the Navarro-Frenk-White (NFW)model, while the cored profiles including both thepseudo-isothermal and Burkert models are excluded.The result is robust against various systematic uncer-tainties. (2) The central density slope of dark matter halo isfound to be -(0.90 ± < × − eV, or > × − eV , the cross section of self-interacting darkmatter to be < /g, and the particle mass ofwarm dark matter to be > ± σ below the threshold(0.5) of the modified Newtonian dynamics.(5) In the cold dark matter paradigm, the cuspy haloof AGC 242019 thus supports the feedback scenariothat transforms cuspy halos to cored halos as frequentlyseen in other galaxies. However, the detailed physicalprocess is unclear. The cuspy halo of AGC 242019 isinconsistent with the stellar-to-halo-mass-ratio depen-dent model, while consistent with the star-formation-threshold dependent model. But even for the later, theobserved star formation efficiency (SFR/gas) is muchlower than what is adopted in simulations. It may beconsistent with the scenario that the duration of starformation is the key driver.(6) As a UDG, AGC 242019 has a halo mass of(3.5 ± × M (cid:12) , implying its formation in a dwarf-size halo. The cuspy halo further suggests weak feedbackover the history. The small concentration of its halo isconsistent with late formation time. Its specific angularmomentum of baryons is consistent with the average ofdwarf irregulars at a given halo/baryonic mass. Its starformation efficiency (SFR/gas) is low, probably due tothe low stellar mass surface density.2 Shi et al.
Table A1.
The full parameter list for D Barolo
Parameters ValuesChecking for bad channels in the cube........[checkChannels] falseUsing Robust statistics?...................[flagRobustStats] trueWriting the mask to a fitsfile....................[MAKEMASK] falseSearching for sources in cube?......................[SEARCH] falseSmoothing the datacube?.............................[SMOOTH] falseHanning smoothing the datacube?....................[HANNING] falseWriting a 3D model?.................................[GALMOD] falseFitting a 3D model to the datacube?..................[3DFIT] trueNumber of radii..................................[NRADII] 7Separation between radii (arcsec)................[RADSEP] 9X center of the galaxy (pixel).....................[XPOS] 415.19Y center of the galaxy (pixel).....................[YPOS] 405.49Systemic velocity of the galaxy (km/s).............[VSYS] 1840.46Initial global rotation velocity (km/s)............[VROT] 30Initial global radial velocity (km/s)..............[VRAD] -1Initial global velocity dispersion (km/s).........[VDISP] 5Initial global inclination (degrees)................[INC] 69.90Initial global position angle (degrees)..............[PA] 2.0025Scale height of the disk (arcsec)....................[Z0] 0.7Global column density of gas (atoms/cm2)...........[DENS] -1Parameters to be minimized.........................[FREE] VROT,VDISP,PA,INCType of mask.......................................[MASK] SEARCHSide of the galaxy to be used......................[SIDE] BType of normalization..............................[NORM] LOCALLayer type along z direction......................[LTYPE] gaussianResiduals to minimize.............................[FTYPE] chi-squaredWeighting function................................[WFUNC] uniformWeight for blank pixels.........................[BWEIGHT] 1Minimization tolerance..............................[TOL] 0.001What side of the galaxy to be used.................[SIDE] BTwo stages minimization?.......................[TWOSTAGE] trueDegree of polynomial fitting angles?............[POLYN] bezierEstimating errors?...........................[FLAGERRORS] trueRedshift of the galaxy?........................[REDSHIFT] 0Computing asymmetric drift correction?...........[ADRIFT] trueOverlaying mask to output plots?...............[PLOTMASK] falseRMS noise to add to the model..................[NOISERMS] falseUsing cumulative rings during the fit?.......[CUMULATIVE] falseFull parameter space for a pair of parameters.....[SPACEPAR] falseGenerating a 3D datacube with a wind model?........[GALWIND] falseFitting velocity field with a ring model?............[2DFIT] falseDeriving radial intensity profile?.................[ELLPROF] false
APPENDIX A. THE FULL SETUP TO RUN D BAROLO
In Table A1, the full list of the parameters to run D Barolo is listed. cuspy halo B. DIFFERENT RUNS OF D BAROLO
In Figure B1, we compared the derived rotation velocities (before the pressure support correction) and velocitydispersions by running D Barolo with both sides of the gas disk, only the receding side and only the approaching side,respectively.
Radius [kpc] V r o t o b s [ k m / s ] Both SidesReceding SideApproaching Side
Radius [kpc] o b s [ k m / s ] Both SidesReceding SideApproaching Side
Figure B1.
The comparison in the derived rotation velocities and velocity dispersions by using both sides, only the recedingside and only the approaching side of HI data, respectively.
ACKNOWLEDGMENTSWe thanks a lot the referee for his/her very helpful and constructive comments that improve the paper signif-icantly. Y.S. acknowledges the support from the National Key R&D Program of China (No. 2018YFA0404502,No. 2017YFA0402704), the National Natural Science Foundation of China (NSFC grants 11825302, 11733002 and11773013), and the Tencent Foundation through the XPLORER PRIZE. The National Radio Astronomy Observatoryis a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint projectof the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,funded by the National Aeronautics and Space Administration.REFERENCES
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