The radial acceleration relation in a ΛCDM universe
MMNRAS , 1–19 (0000) Preprint 1 March 2021 Compiled using MNRAS L A TEX style file v3.0
The radial acceleration relation in a Λ CDM universe
Aseem Paranjape (cid:63) & Ravi K. Sheth , † , Inter-University Centre for Astronomy & Astrophysics, Ganeshkhind, Post Bag 4, Pune 411007, India Center for Particle Cosmology, University of Pennsylvania, 209 S. 33rd St., Philadelphia, PA 19104, USA The Abdus Salam International Center for Theoretical Physics, Strada Costiera, 11, Trieste 34151, Italy draft
ABSTRACT
We study the radial acceleration relation (RAR) between the total ( a tot ) and baryonic( a bary ) centripetal acceleration profiles of central galaxies in the cold dark matter(CDM) paradigm. We analytically show that the RAR is intimately connected with thephysics of the quasi-adiabatic relaxation of dark matter in the presence of baryons indeep potential wells. This cleanly demonstrates how a near-universal mean RAR andits scatter emerges in the low-acceleration regime (10 − m s − (cid:46) a bary (cid:46) − m s − )from an interplay between baryonic feedback processes and the distribution of CDM indark halos. Our framework allows us to go further and study both higher and loweraccelerations in detail, using analytical approximations and a realistic mock catalogof ∼ ,
000 low-redshift central galaxies with M r ≤ −
19. We show that, while theRAR in the baryon-dominated, high-acceleration regime ( a bary (cid:38) − m s − ) is verysensitive to details of the relaxation physics, a simple ‘baryonification’ prescriptionmatching the relaxation results of hydrodynamical CDM simulations is remarkablysuccessful in reproducing the observed RAR without any tuning . And in the (currentlyunobserved) ultra-low-acceleration regime ( a bary (cid:46) − m s − ), the RAR is sensitiveto the abundance of diffuse gas in the halo outskirts, with our default model predictinga distinctive break from a simple power-law-like relation for Hi -deficient, diffuse gas-richcentrals. Our mocks also show that the RAR provides more robust, testable predictionsof the ΛCDM paradigm at galactic scales, with implications for alternative gravitytheories, than the baryonic Tully-Fisher relation. Key words: galaxies: formation - cosmology: theory, dark matter - methods: analytical,numerical
Gravitational interactions at galactic scales offer a fertile test-ing ground for competing theories of gravitation. The highlysuccessful Lambda-cold dark matter (ΛCDM) paradigm at-tributes all gravitational interactions at these scales to theNewtonian limit of general relativity, but postulates the ex-istence of a collisionless (or dark) matter component thatpervades the cosmos. In stark contrast, alternative proposalssuch as Modified Newtonian Dynamics (MOND, Milgrom1983) attempt to explain extra-Galactic observations, partic-ularly galactic rotation curves, using Standard Model physicsalone (i.e., without a dark component), but alter the natureof gravity at these scales. MOND, in particular, postulatesa new, fundamental acceleration scale a ∼ − m s − tosegregate the high-acceleration regime of Newtonian dynam-ics from the low-acceleration regime where the nature ofgravity is modified. MOND is just one of a growing numberof modified gravity models (for a recent review, see Bertone& Tait 2018). (cid:63) E-mail: [email protected] † E-mail: [email protected]
Observationally, such competing ideas are potentiallyamenable to testing using empirical correlations betweenthe dynamical, gravitating mass of a system and the lightwe observe from it. Among the several such mass-to-lightscalings that are known to exist for galaxies of different types(Faber & Jackson 1976; Tully & Fisher 1977; McGaugh et al.2000), the ‘radial acceleration relation’ (RAR, McGaugh et al.2016) has recently emerged as an intriguing new potentialtest of gravity.The RAR is usually expressed as the relation between thecentripetal acceleration profile a tot ( r ) due to all gravitatingcomponents (in ΛCDM, these would be baryonic and darkmatter), and the Newtonian contribution a bary ( r ) to thisprofile from the baryonic components alone. In terms of thegalactic rotation curve v rot ( r ) and its baryonic contribution v bary ( r ) (these will be defined below), we have a tot ( r ) = v ( r ) /r , (1)and a bary ( r ) = v ( r ) /r . (2)The RAR and its close cousin, the baryonic Tully-Fisher re-lation (BTFR, McGaugh et al. 2000), have been extensively © a r X i v : . [ a s t r o - ph . GA ] F e b Paranjape & Sheth discussed in the literature, especially in the context of MONDversus ΛCDM (see, e.g., the review by McGaugh 2015, seealso below) In the ΛCDM framework, unlike MOND, thereis no fundamental acceleration scale. Correlations such asthe RAR and BTFR, to the extent that they are predictedby ΛCDM, are necessarily emergent phenomena that resultfrom a complex combination of many underlying correlations.The fact that the observed
BTFR and especially the RARhave low scatter, makes it very interesting to ask how theemergence of these relations in ΛCDM fares against observa-tions (see, e.g., Courteau et al. 2007, for a discussion of theconstraints on physical models of the Tully-Fisher relation).Several studies have followed this line of reasoning and usedhydrodynamical CDM simulations of, both, small samplesof objects as well as cosmological volumes, to quantify theBTFR and RAR expected in ΛCDM (e.g., Sorce & Guo 2016;Sales et al. 2017; Keller & Wadsley 2017; Ludlow et al. 2017;Tenneti et al. 2018; Garaldi et al. 2018).Focusing on the RAR (we discuss the BTFR separatelylater), a general trend is that most hydrodymanical CDMsimulations that broadly reproduce observed galaxy prop-erties do, in fact, also naturally produce a tight RAR (e.g.,Keller & Wadsley 2017, although see Milgrom 2016). How-ever, the details of the median trend and the scatter aroundit do not always agree with the observed ones (e.g., Ludlowet al. 2017; Tenneti et al. 2018), and it is usually difficult toassess whether the differences are fundamental (e.g., due tospecifics of baryonic feedback physics) or caused by widelydifferent sample definitions and other technical choices inmeasuring rotation curves. For example, the EAGLE simula-tions produce an RAR similar to the observed one but withan inferred acceleration scale a higher by about a factor 2(Ludlow et al. 2017), while the RAR in the MassiveBlack-IIsimulation is closer to a power law with no intrinsic accelera-tion scale (Tenneti et al. 2018).Several authors have attempted to build an analyticalunderstanding of the RAR in a ΛCDM universe. Wheeleret al. (2019) have argued that the RAR is a simple algebraicoutcome of the BTFR, although they do not address theemergence of the BTFR itself. Grudi´c et al. (2020) haveattempted to explain the emergence of a characteristic accel-eration scale from the physics of stellar feedback, expressing a using fundamental constants. The emergence of the RARand related scalings in ΛCDM is, in general, easier to appreci-ate using empirical models to connect dark matter to baryons,along with (semi-)analytical modelling for producing rotationcurves. This approach has been adopted by several authorsrecently using the subhalo abundance matching (SHAM)technique (e.g., Desmond & Wechsler 2015; Desmond 2017;Navarro et al. 2017). A common thread in these studies isthat the ΛCDM RAR is a complicated but natural outcomeof a combination of the SHAM association of stellar mass todark halos, the requirement that galaxy disk sizes obey theobservationally constrained scaling with halo properties, andthe magnitude of the ‘backreaction’ of the baryonic materialon the dark matter profile in the inner halo.In this work, we present new analytical insights into thestructure of the RAR, and the underlying physics that deter-mines this structure, in the ΛCDM paradigm. Specifically,we show that the physics of quasi-adiabatic relaxation of thedark matter profile in the presence of baryons, particularlyin the inner, baryon-dominated regions of the halo, plays akey role in establishing both the median and scatter of theRAR for any galaxy sample. Although previous work (e.g., Desmond 2017) has noticed the relevance of this relaxationphysics to the RAR, its full impact on the RAR has not beenappreciated to date (e.g., Navarro et al. 2017, discuss theRAR in the absence of any baryonic effect on the dark mat-ter). We believe this is largely due to the common practiceof expressing the RAR as the functional dependence of a tot on a bary (e.g., McGaugh et al. 2016; Lelli et al. 2017; Keller& Wadsley 2017; Ludlow et al. 2017; Navarro et al. 2017;Desmond 2017; Tenneti et al. 2018; Di Paolo et al. 2019;Tian et al. 2020), which can easily mask small but significantdifferences between alternative physical models in the pre-dicted approach of a tot → a bary at large a bary . As argued byChae et al. (2019), the baryon-dominated, high-accelerationregime ( a bary (cid:38) − m s − ) of the RAR is better probedby expressing the quantity∆ a ≡ a tot /a bary − , (3)as a function of a bary . In the language of McGaugh (1999),∆ a can be thought of as a ‘residual mass discrepancy’. Weexclusively use this formulation of the RAR in the presentwork.We augment our analytical calculations with measure-ments of the RAR in a mock galaxy catalog containing acosmologically representative sample of central galaxies withrealistic baryonic properties, including stellar mass and coldas well as hot gas, along with their spatial distributions. Thismock is based on the algorithm recently presented by Paran-jape et al. (2021) and is described below. The use of mockgalaxies with numerically sampled rotation curves allows usto extensively explore the sensitivity of the RAR to changesnot only in the underlying physics and baryon-dark matterscalings, but also to effects of sample selection and othertechnical aspects of rotation curve estimation. Our primarygoal is to emphasize and disentangle conceptual issues, ratherthan perform a detailed comparison with observations. Wetherefore ignore observational errors and focus on the intrin-sic predictions that follow from our analytical argumentsand mock catalogs. As such, we deal only with ‘perfectlymeasured’ rotation curves in this work (see Desmond 2017,for more careful comparisons with observed data sets).The paper is organized as follows. In section 2, we brieflydescribe the numerical algorithm and N -body simulation boxunderlying the mock galaxy catalog we use in this work. Insection 3, we present analytical calculations that show howany prescription for quasi-adiabatic relaxation and the associ-ated baryon-dark matter scalings (section 3.1) leads directlyto a prediction for the RAR of each individual galaxy, andhence of any population of galaxies (section 3.2). Appendix Abuilds on these analytical results to construct an approxi-mate but fully analytic RAR which allows us to predict theshape and tightness of the RAR in various limits. In sec-tion 4, we explore the RAR of our mock galaxies for variouschoices of relaxation physics prescription, sample selection,baryon-dark matter scaling, and technical details such asrotation curve sampling. This exercise allows us to put allour analytical arguments to the test. In section 5, we discussin detail the predictions of our mocks for the BTFR, high-lighting the pitfalls of over-interpreting BTFR measurementswhich, unlike the RAR, are inherently unstable to variationsin technical details of the analysis. We conclude in section 6.Throughout, m vir and R vir refer to the total halo massand virial radius. In keeping with the literature on quasi-adiabatic relaxation, on which we rely heavily, we define R vir ≡ R , the radius at which the enclosed halo-centricdensity becomes 200 times the critical density ρ crit of the MNRAS , 1–19 (0000)
AR in LCDM Universe, so that m vir = (4 π/ R × ρ crit . All our re-sults assume a spatially flat ΛCDM background cosmol-ogy, with parameters { Ω m , Ω b , h, n s , σ } given by { } , compatible with the 7-year resultsof the Wilkinson Microwave Anisotropy Probe experiment(WMAP7, Komatsu et al. 2011). We will denote the base-10(natural) logarithm as log (ln).
Our results are based on a mock galaxy catalog constructedusing the algorithm described in detail by Paranjape et al.(2021, hereafter, PCS21). Below, we briefly summarise thisalgorithm and the N -body simulation that is populatedwith mock galaxies, followed by a discussion of the baryoniccomponents and associated rotation curve of each mockcentral galaxy. We use one realisation of the L300 N1024 simulation configu-rations discussed by PCS21. This is a gravity-only simulationwith 1024 particles in a (300 h − Mpc) cubic box, performedusing the code gadget-2 (Springel 2005) with halos identi-fied using the code rockstar (Behroozi et al. 2013). Furtherdetails of the simulation can be found in Paranjape & Alam(2020).The PCS21 algorithm, which is based on the halo occu-pation distribution (HOD) models calibrated by Paul et al.(2018) and Paul et al. (2019), populates host halos in thisbox with mock central and satellite galaxies, producing aluminosity-complete sample of galaxies with an r -band ab-solute magnitude threshold M r ≤ −
19. In addition to the r -band magnitude, each mock galaxy is assigned realistic val-ues of g − r and u − r colours and stellar mass m ∗ . A fractionof these galaxies is also assigned non-zero values of neutralhydrogen ( Hi ) mass m Hi . The HOD models underlying thisalgorithm are constrained by the observed abundances andclustering of optically selected galaxies in the Sloan DigitalSky Survey (SDSS, York et al. 2000), and of Hi -selectedgalaxies in the ALFALFA survey (Giovanelli et al. 2005).PCS21 presented extensive tests of the algorithm, along witha detailed discussion of cross-correlation statistics betweenoptical and Hi -selected samples that are predicted by thealgorithm.In this work, we focus only on central galaxies, whosehost halos are ‘baryonified’ by the PCS21 algorithm as dis-cussed below. The L300 N1024 box described above containsapproximately 342 ,
000 central galaxies with M r ≤ − The PCS21 algorithm uses a modified version of the baryoni-fication prescription of Schneider & Teyssier (2015, hereafter,ST15) to model the spatial distributions of a number ofbaryonic components in each central galaxy and its host halo.These include: https://bitbucket.org/gfcstanford/rockstar • A spherical distribution of stars in the central galaxy(‘cgal’) with half-light radius R hl . The corresponding massfraction is f cgal = m ∗ /m vir . In principle, we could also modelthe stellar distribution as a combination of a disk and abulge, which we leave for future work. • A 2-dimensional axisymmetric Hi disk (‘ Hi ’) with scalelength h Hi , for centrals with m Hi >
0. The correspondingmass fraction is f Hi = 1 . m Hi /m vir , with the prefactoraccounting for Helium correction. • A spherical distribution of bound hot gas (‘bgas’) inhydrostatic equilibrium. The corresponding mass fraction f bgas is constrained by X-ray cluster observations at m vir (cid:38) h − M (cid:12) and extrapolated to lower masses where needed. • Expelled gas (‘egas’) or the circum-galactic medium(CGM). As discussed by PCS21, for rotation curve modellingthis is essentially a uniform density distribution inside R vir ,with a mass fraction f egas constrained by baryonic massconservation by demanding f bary ≡ f cgal + f Hi + f bgas + f egas = Ω b / Ω m (cid:39) . . (4)Further details of the numerical implementation, as well asall the underlying scalings of baryonic mass fractions andgalaxy sizes with halo properties, can be found in section 3.2of PCS21. Baryonification schemes of this type have beenshown to successfully reproduce the small-scale matter powerspectrum and bispectrum of cosmological hydrodynamicalsimulations (e.g., Chisari et al. 2018; Aric`o et al. 2020).The rotation curve v rot ( r ) for each mock galaxy is calcu-lated using equation (11) of PCS21, which can be rewrittenas v ( r ) = v Hi ( r ) + (cid:88) α Gm α ( < r ) r + Gm rdm ( < r ) r ≡ v ( r ) + Gm rdm ( < r ) r , (5)where, in the first line, v Hi ( r ) is the Hi disk contribu-tion (equation 10 of PCS21), the sum runs over α ∈{ bgas , cgal , egas } , m α ( < r ) is the mass of component α en-closed in radius r and m rdm ( < r ) is the corresponding massof the ‘relaxed’ dark matter component which we discuss indetail in the next section, and the second line defines thebaryonic contribution v ( r ).Below, we will also use the total (sphericalised) massprofile contained in radius r , which can be split into contribu-tions from baryons and the relaxed dark matter component, m tot ( < r ) = m bary ( < r ) + m rdm ( < r )= (cid:88) χ m χ ( < r ) + m rdm ( < r ) , (6)where the sum in the second line runs over χ ∈{ bgas, cgal, egas, Hi } .For later use, we also calculate an integrated baryonicmass M bary (e.g., Lelli et al. 2017; Sales et al. 2017) foreach central as the sum of the masses of stars and cold gascontained inside the radius r = 2 R h , bary , where R h , bary is This is violated by a small fraction ( ∼ M r ≤−
19 for which the sum f cgal + f Hi + f bgas exceeds Ω b / Ω m (which inturn are dominated by objects having f cgal + f Hi > Ω b / Ω m ). Forsuch objects, we follow PCS21 and set f egas = 0 without changingany of the other baryonic mass fractions, so that f bary > Ω b / Ω m .Overall mass conservation then implies that the correspondingdark matter fraction f rdm = 1 − f bary (see section 3) is smallerthan 1 − Ω b / Ω m for these objects.MNRAS000
19 for which the sum f cgal + f Hi + f bgas exceeds Ω b / Ω m (which inturn are dominated by objects having f cgal + f Hi > Ω b / Ω m ). Forsuch objects, we follow PCS21 and set f egas = 0 without changingany of the other baryonic mass fractions, so that f bary > Ω b / Ω m .Overall mass conservation then implies that the correspondingdark matter fraction f rdm = 1 − f bary (see section 3) is smallerthan 1 − Ω b / Ω m for these objects.MNRAS000 , 1–19 (0000) Paranjape & Sheth the radius which encloses half the mass of stars and coldgas : M bary = m cgal ( < R h , bary ) + m Hi ( < R h , bary ) , (7)and where m Hi ( < r ) includes the Helium correction men-tioned above, so that m Hi ( < R vir ) = 1 . m Hi . Our use ofa 3-dimensional half-mass radius to define M bary can, inprinciple, lead to systematic effects when comparing withobservations which typically use projected sizes for measur-ing M bary . For such analyses below, we have checked thatreplacing M bary with the total m ∗ + 1 . m Hi for each galaxyleads to identical conclusions, i.e., our results are expectedto be insensitive to the exact definition of M bary .As discussed in the Introduction, the radial accelerationrelation is then the dependence of ∆ a = a tot /a bary − a bary , with a tot and a bary given by equations (1) and (2),respectively. Notice that v bary , and hence a bary , containscontributions from both spherical as well as axisymmetriccomponents. This is consistent with observational analysesof the RAR (see, e.g., McGaugh et al. 2016). Thus far, we have not commented on the shape of the relaxeddark matter profile m rdm ( < r ). As we discuss in this section,this is a key component in determining the shape of the meanRAR. In the default PCS21 model, m rdm ( < r ) is calculated assum-ing complete spherical symmetry for all components, andassuming that the dark matter quasi-adiabatically relaxes(approximately conserving angular momentum) in responseto the baryonic components. The details of the procedure canbe found in ST15 or Appendix A of PCS21 and are brieflysummarised below. This relaxation can be described using afunction ξ ( r ) defined as ξ ≡ r/r in , (8)where r in is the initial radius of a spherical dark matterelement which eventually relaxes to a final radius r . Theequation governing the form of ξ can be written in generalas ξ = 1 + X (cid:18) m udm ( < r in ) m tot ( < r ) (cid:19) , (9)where m udm ( < r in ) is the unrelaxed dark matter profile. Weapproximate this using the Navarro, Frenk & White (1996,NFW) form in this work (although see below). The function X ( y ) in the ST15 model, which was adopted by PCS21, isgiven by X ( y ) = q rdm ( y − . (10)Here q rdm is a parameter controlling the level of angularmomentum conservation, with q rdm = 1 for perfect conserva-tion and q rdm = 0 for no baryonic backreaction. The defaultmodel from PCS21 follows the ST15 prescription and sets In practice, we determine 2 R h , bary by sampling the rota-tion curve using 200 logarithmically spaced points in the range(0 . , × R vir for each central galaxy. q rdm = 0 .
68. Equation (9) is then solved iteratively to obtain ξ ( r ), using which the relaxed dark matter profile satisfies(see Appendix A of PCS21) m rdm ( < r ) = f rdm m udm ( < r/ξ ) , (11)where f rdm is the mass fraction of dark matter inside thehost halo’s virial radius; due to equation (4), this is set to f rdm = 1 − Ω b / Ω m in this work for all but the small fractionof objects discussed in footnote 4.Figure 1 shows the numerically computed relaxationratio ξ in the ST15 model (right panel) for three examplesof baryonified halos whose mass profiles are shown in the left panel . The right panel shows that there is a lower limitto ξ because y ≥ y cannot be negative at any r ). Moreover, while theST15 model leads to a contraction of the dark matter profilethroughout the least massive halo, it predicts an expansionin the outskirts of more massive halos. A comparison withthe left panel shows that this happens in regions where thefraction of bound and/or expelled gas is higher than thatof stars and the Hi disk (compare the thin solid lines whichshow all baryons with the dash-dotted lines showing only thestellar and Hi component). The dashed red curve and bandin the right panel respectively show the median and central95% range of ξ for the entire mock catalog used below.Strictly speaking, the assumption of perfect sphericalsymmetry is not valid due to the presence of the axisymmet-ric baryonic disk, as well as the fact that dark matter halosin gravity-only simulations are triaxial in general. Includingthese non-spherical effects analytically and calculating a tri-axial (cid:126)ξ ( (cid:126)r ) is quite difficult. Interestingly, though, the resultsof hydrodynamical simulations show that baryonic backre-action actually tends to make the dark matter distributionafter relaxation more spherical (Dubinski 1994; Kazantzidiset al. 2004; Abadi et al. 2010; Cataldi et al. 2021). We there-fore expect that, in practice, our spherical assumption willlead to an accurate average description of quasi-adiabaticrelaxation. We intend to explore the effects of asphericity inthe relaxation process, along with detailed comparisons tohydrodynamical simulations, in future work. Equation (9) allows us to appreciate an intimate connectionbetween the level of angular momentum conservation andthe shape of the RAR. For the spherically symmetric caseassumed above, equations (8) and (11) give the identity m udm ( < r in ) m tot ( < r ) = 1 f rdm a tot ( r ) − a bary ( r ) a tot ( r ) . (12)Using this, equation (9) can be formally inverted and, aftersome straightforward algebra, brought to the form∆ a = Λ1 − Λ , (13)where ∆ a was defined in equation (3), andΛ ≡ f rdm X − ( ξ − , (14)with X − ( z ) = y being the inverse function of X ( y ) = z .Equation (13) is remarkable because it shows that, as afunction of the relaxation ratio ξ , the RAR has zero scatter in the spherical baryonification model, regardless of the exactfunctional form of X ( y ) which sets the mean relation. Forour default choice of f rdm = 1 − Ω b / Ω m , it is clear fromequation (14) that the scatter in the RAR as defined in MNRAS , 1–19 (0000)
AR in LCDM Figure 1. Relaxation physics . (Left panel:) Mass profiles for three examples of baryonified halos hosting an NGC99-like galaxy (curveswith different colours), with different combinations of components shown using the linestyles indicated in the legend. Halo concentrations,stellar masses and Hi disk sizes were set using scaling relations from the literature (see Paranjape et al. 2021, PCS21), while the Hi mass was fixed to m Hi = 10 . h − M (cid:12) in each case. Other baryonic fractions were set as described in the text. The two lower masshalos are the same as shown in figure 4 of PCS21. (Right panel:) Relaxation ratio ξ = r/r in for the dark matter profile computed usingequations (9) and (10) with q rdm = 0 .
68. The thick solid curves show ξ for the three halos from the left panel. The dashed red curve andband respectively show the median and central 95% range of ξ for the entire luminosity-complete mock catalog used in the text. Thelower horizontal line shows the theoretical lower bound of 1 − q rdm (see text). The upper horizontal line indicates unity, the solution whenbaryons do not affect the dark matter profile. Values of ξ less (greater) than unity correspond to contraction (expansion) of the darkmatter profile due to the presence of baryons. the literature arises solely from the scatter between ξ and a bary . If we think of these functions as ξ = ξ ( r | bary , dm) and a bary = a bary ( r | bary), then, at fixed r and for a given bary-onic configuration, this scatter is caused predominantly bythe object-to-object variation in halo mass and concentrationfor different central galaxies with this baryonic configura-tion. There could be some additional scatter at fixed halomass and concentration if, for example, multiple baryonicconfigurations happen to lead to the same value of ξ butdifferent a bary , or vice-versa. This is, of course, very differentfrom MOND which predicts that the RAR should have no intrinsic scatter.For the specific choice of X in equation (10) adopted inthis work, equation (13) simplifies to∆ a = ( ξ −
1) + q rdm q rdm /f rdm − ( ξ − − q rdm . (15)To glean some analytical insights into the implications ofequation (13) or equation (15), it is useful to analyse theresult perturbatively in the case 0 < q rdm (cid:28)
1, i.e., in thelimit of small baryonic backreaction. This is the same limitas studied by Navarro et al. (2017), who ignored baryonicbackreaction and focused on explaining the origin of theRAR in the low-acceleration regime using various baryon-dark matter scalings. At lowest order in q rdm , this leadsto ξ − (cid:39) q rdm (cid:18) f bary m udm ( < r ) − m bary ( < r ) f rdm m udm ( < r ) + m bary ( < r ) (cid:19) . (16)Plugging this into equation (15) gives, after some simplifica-tion, ∆ a (cid:39) f rdm m udm ( < r ) m bary ( < r ) , (17)which is an eminently sensible result. This also shows that the RAR in the limit of no baryonic backreaction can beexpected to have a large scatter as a function of a bary , sincethe dark matter profile m udm ( < r ) in the numerator of equa-tion (17) is decoupled from a bary ( r ) ∼ m bary ( < r ) /r , apartfrom the baryon-dark matter scalings that relate halo massand concentration to baryonic mass fractions and sizes. Ap-pendix A shows that, if the initial mass distribution is similarto an NFW profile, then the RAR is amenable to analytictreatment, even when backreaction is large. In particular,one can analytically estimate the RAR of individual galaxiessuch as the ones depicted by the thick solid lines in figure 1.The resulting dependence of the median and scatter of theRAR on various halo and galaxy properties then provides ananalytic understanding of the trends we discuss below usingnumerically sampled mock galaxies. With these analytical arguments in hand, we now explorethe RAR in the mock catalogs described in section 2 byvarying the underlying baryonification choices of the PCS21algorithm, as well as selecting galaxy samples using variouscriteria.
The coloured histogram in the top panel of figure 2 shows theRAR of the full sample of central galaxies with M r ≤ −
19 inone mock ( ∼ ,
000 objects) for our default baryonificationmodel. We calculated a tot ( r ) and a bary ( r ) on 20 logarith-mically spaced values of r in the range (0 . , × R vir foreach central galaxy (we explore the effects of changing thissampling choice below). Our results therefore explore not MNRAS000
000 objects) for our default baryonificationmodel. We calculated a tot ( r ) and a bary ( r ) on 20 logarith-mically spaced values of r in the range (0 . , × R vir foreach central galaxy (we explore the effects of changing thissampling choice below). Our results therefore explore not MNRAS000 , 1–19 (0000)
Paranjape & Sheth
Figure 2. Default RAR. (Top panel:)
The RAR (∆ a fromequation 3 as a function of a bary ) defined by central galaxies inone mock with our default baryonification model having q rdm =0 .
68 (see sections 2.2 and 3.1). The coloured histogram countsmeasurements from all centrals, with rotation curves sampled at20 logarithmically spaced points in the range (0 . , × R vir foreach object. Yellow solid and dashed lines show the median andcentral 68% region of the distribution in bins of a bary . Solid anddashed purple curves show F ( a bary /a ) − a bary . The break in the median relationat ultra-low accelerations ( a bary (cid:46) − m s − ) is discussed indetail in the text. (Bottom panel:) Residuals relative to the solidpurple curve, computed as log[∆ a / ( F ( a bary /a ) − F ( x )from equation (18), for each point ( a bary , a tot ) having a bary > − m s − , shown as a function of M bary which is assignedfor each galaxy using equation (7) (so each galaxy contributes avertical streak). Yellow curves now show the median and central68% of the distribution of the residuals in bins of M bary . only the inner, baryon-dominated parts of each halo, but alsothe halo outskirts corresponding to the ultra-low-accelerationregime ( a bary (cid:46) − m s − ) which is as yet observationallyunconstrained.The horizontal axis in the top panel of figure 2 showslog[ a bary ( m s − )] and the vertical axis shows log[∆ a ]. Thesolid purple curve shows log[ F ( a bary /a ) −
1] with F ( x )being the MOND-inspired calibration for a tot /a bary fromequation (4) of Chae et al. (2019), i.e., F ( x ) = (cid:34)
12 + (cid:114)
14 + 1 x ν (cid:35) /ν . (18)Similarly, the dashed purple curve shows equation (4) ofMcGaugh et al. (2016), namely, F ( x ) = 11 − e −√ x , (19)which is another MOND-inspired relation. For both of these,we used a = 1 . × − m s − (McGaugh et al. 2016 denotethis as g † ), and in equation (18) we set ν = 0 .
8, which Chaeet al. (2019) argue fits the observed RAR well, particularly at a bary ≥ a . For x (cid:28) F → / √ x ,and both tend to unity when x (cid:29) a = F − → x − ν /ν whereas∆ a → e −√ x for equation (19).The solid yellow curve shows the median (cid:104) ∆ a, mock (cid:105) of the distribution in bins of a bary , while the dashed yellowcurves show the corresponding 16th and 84th percentiles. Wesee that the median RAR of our default mock is in remarkablygood agreement with the solid purple curve for a bary (cid:38) − m s − , i.e., throughout the low- and high-accelerationregimes. (Quantitatively, | (cid:104) ∆ a, mock (cid:105) / ∆ a, eqn 18 − | (cid:46) this is a non-trivial success of our default model, withno additional tuning beyond what was already discussedby PCS21 to match other observations. The scatter aroundthe median relation is typically σ log[ a tot ] ∼ .
075 dex for a bary > − m s − . (Note that the scatter in log[∆ a ] seenin the figure is considerably larger; we report the scatter inlog[ a tot ] in the text for ease of comparison with the literature.)The RAR in the (as yet unobserved) ultra-low accelerationregime of a bary (cid:46) − m s − sharply breaks away fromthe extrapolation of equations (18) and (19). Below, wewill explore the nature of the galaxies which lead to thisdeparture. For now, we simply note that our results constitutepredictions for the ultra-low-acceleration regime (see alsoOman et al. 2020). The bottom panel of figure 2 shows the residuals ofthe RAR ratio data in the top panel with the solid purplecurve, as a function of baryonic mass M bary (equation 7).Specifically, on the vertical axis we plot log[( a tot /a bary − / ( F ( a bary /a ) − F ( x ) from equation (18) with a = 1 . × − m s − and ν = 0 .
8. This is conceptuallysimilar to figure 5 of Lelli et al. (2017), who define theresiduals using log[( a tot /a bary ) / ( F ( a bary /a ))], i.e., withoutsubtracting unity in the numerator and denominator in-side the logarithm. This difference is important because theresiduals calculated by Lelli et al. (2017) will be artificiallysuppressed in the high-acceleration regime where the numer-ator and denominator both approach unity. By subtractingthis leading behaviour, our definition of the residuals offersa sharper characterisation of the scatter around the medianrelation. We see from the bottom panel of figure 2 that thisscatter is nevertheless small, with a typical value of ∼ . a ] in the top panel .We have checked that using the Lelli et al. (2017) definitionof residuals instead, the typical scatter in the bottom panel is even smaller, closer to ∼ . To calculate the yellow curves, we use 17 linearly spaced binsin log[ a bary / ( m s − )] in the range ( − . , − . a bary of each bin. Recently, Lelli et al. (2017) and Di Paolo et al. (2019) havepresented RAR observations of ultra-faint dwarf spheroidal galax-ies which probe values a bary (cid:46) − m s − (see Garaldi et al.2018, for the corresponding predictions from ΛCDM simulations).This, however, is different from our predictions which hold for theoutskirts of much more massive systems and are hence relevanton very different length scales. MNRAS , 1–19 (0000) AR in LCDM Figure 3. RAR and relaxation physics.
Same as top panelof figure 2, but assuming q rdm = 0 (no baryonic backreactionon the dark matter profile; top panel ) or q rdm = 1 (perfectangular momentum conservation; bottom panel ). The medianand scatter are both different compared to our fiducial choice, q rdm = 0 .
68, from figure 2, especially in the high-accelerationregime ( a bary (cid:38) − m s − ). These trends can be understoodanalytically (Appendix A). phenomenon rather than a universal law (Keller & Wadsley2017; Desmond 2017; Navarro et al. 2017; Ludlow et al. 2017;Tenneti et al. 2018). That discussion also shows that galaxiespopulating halos of different masses and concentrations mightbe expected to define different RARs, in general. The RAR isadditionally expected to be sensitive to the physics of quasi-adiabatic relaxation of dark matter in the presence of baryons.In the following subsections, we explore the sensitivity ofthe RAR to differences in the physical content of galaxies,observational selection criteria and, importantly, differencesin the physical modelling of baryonification. Unless otherwisementioned, the plots below are formatted identically to the top panel of figure 2, with the solid and dashed purple curvesbeing repeated from that figure. Figure 3 shows the effect of changing the details of the quasi-adiabatic relaxation scheme (see the discussion in section 3.2).The top panel shows the RAR obtained if the baryonic matterhad no backreaction on the dark matter profile (e.g., Navarroet al. 2017), i.e., setting q rdm → ξ → high-acceleration regime, on the other hand, we see adramatic effect: the median RAR is substantially lower, andthe scatter is substantially higher, than in the default case.The bottom panel shows the RAR in the opposite limitwhere baryonic backreaction perfectly conserves angular Figure 4. RAR and cold gas content.
Same as top panelof figure 2, but for ‘bulge-dominated’ galaxies having m Hi = 0 (top panel) or gas-rich ‘spirals’ with m gas /m ∗ ≥ . (bottompanels) . Here, m gas = 1 . m Hi , and the threshold value cor-responds to the 95 th percentile of m gas /m ∗ in the mass range9 . ≤ log[ m ∗ / ( h − M (cid:12) )] ≤ .
9. The break in the median relationat ultra-low accelerations ( a bary (cid:46) − m s − ) is restricted tobulge-dominated objects. momentum, which we model by setting q rdm = 1 in equa-tions (15) and (9). As expected, the ultra-low-accelerationregime is unaffected. In the high-acceleration regime, theRAR is now substantially higher than in the default case,with a substantially smaller scatter. Appendix A providesanalytic understanding of the strong dependence on q rdm .Considering the sharp sensitivity of the high-accelerationRAR to the physics of quasi-adiabatic relaxation, the goodagreement between the default case and the observed relationis truly remarkable. We emphasize again that our defaultmodel (used by PCS21) simply reproduces the prescriptionof ST15 which focused on describing relaxed dark matterprofiles in hydrodynamical ΛCDM simulations and made noreference to the RAR. We next investigate the sensitivity of the RAR to the baryoniccontent of galaxies. We focus here on the presence/absence ofan Hi disk, and on the relative contribution of the expelledgas (‘egas’) component, which is a proxy for the circum-galactic medium. Observations suggest that galaxies withdifferent morphologies obey the same RAR in the low- andhigh-acceleration regimes (Lelli et al. 2017; Chae et al. 2019),which makes it interesting to ask how our mock galaxiesfare in comparison. On the other hand, galaxies with differ-ent ‘egas’ fractions may be expected to behave quite differ-ently in the halo outskirts and consequently in the ultra-low-acceleration regime of the RAR.The top panel of figure 4 shows the RAR of mock ‘bulge-dominated’ galaxies without Hi disks, while the bottom panel shows the RAR of gas-rich, disk-dominated ‘spiral’ galaxies.We see that bulge-dominated galaxies span a wider range MNRAS000
9. The break in the median relationat ultra-low accelerations ( a bary (cid:46) − m s − ) is restricted tobulge-dominated objects. momentum, which we model by setting q rdm = 1 in equa-tions (15) and (9). As expected, the ultra-low-accelerationregime is unaffected. In the high-acceleration regime, theRAR is now substantially higher than in the default case,with a substantially smaller scatter. Appendix A providesanalytic understanding of the strong dependence on q rdm .Considering the sharp sensitivity of the high-accelerationRAR to the physics of quasi-adiabatic relaxation, the goodagreement between the default case and the observed relationis truly remarkable. We emphasize again that our defaultmodel (used by PCS21) simply reproduces the prescriptionof ST15 which focused on describing relaxed dark matterprofiles in hydrodynamical ΛCDM simulations and made noreference to the RAR. We next investigate the sensitivity of the RAR to the baryoniccontent of galaxies. We focus here on the presence/absence ofan Hi disk, and on the relative contribution of the expelledgas (‘egas’) component, which is a proxy for the circum-galactic medium. Observations suggest that galaxies withdifferent morphologies obey the same RAR in the low- andhigh-acceleration regimes (Lelli et al. 2017; Chae et al. 2019),which makes it interesting to ask how our mock galaxiesfare in comparison. On the other hand, galaxies with differ-ent ‘egas’ fractions may be expected to behave quite differ-ently in the halo outskirts and consequently in the ultra-low-acceleration regime of the RAR.The top panel of figure 4 shows the RAR of mock ‘bulge-dominated’ galaxies without Hi disks, while the bottom panel shows the RAR of gas-rich, disk-dominated ‘spiral’ galaxies.We see that bulge-dominated galaxies span a wider range MNRAS000 , 1–19 (0000)
Paranjape & Sheth of a bary values than the spirals do. In the low-accelerationregime of overlap between the two samples, the median RARfor the two samples is indeed very similar, consistent with theobservations quoted above. The scatter around the medianis somewhat smaller for spirals ( σ log[ a tot ] ∼ .
06 dex) thanfor bulge-dominated galaxies ( ∼ .
075 dex). Interestingly,the break from the MOND-inspired relations at ultra-lowaccelerations is restricted to the bulge-dominated systems.We discuss this further below.We have also checked that splitting our default galaxysample by the value of M bary (with the split defined at themedian value M bary ∼ × M (cid:12) ) leads to results quali-tatively very similar to figure 4, with the massive samplebehaving like the spirals and the low-mass sample behavinglike the bulge-dominated galaxies. The stark differences be-tween such samples at ultra-low accelerations motivate us tostudy the effect of the one baryonic component that reachesthe halo outskirts, namely the expelled gas (‘egas’) which wediscuss next.Figure 5 shows the RAR for galaxies with large (toppanel) and small (bottom panel) values of the expelled gasmass fraction f egas , i.e., galaxies rich and poor, respectively,in diffuse gas content. We see that diffuse gas-rich galax-ies tend to populate low and ultra-low accelerations, whilediffuse gas-poor galaxies populate high and low accelera-tions. As compared to the split between bulge-dominatedand spiral galaxies in figure 4, in this case we see distinctdifferences between the two samples already at low acceler-ations 10 − m s − (cid:46) a bary (cid:46) − m s − , with the medianrelation of gas-poor galaxies being lower than that of gas-rich galaxies. This indicates that diffuse gas content is moreimportant than morphology in determining the typical RARat low accelerations.We also see that the break from the MOND-inspiredrelations at ultra-low accelerations is restricted to diffuse gas-rich galaxies. It is not surprising that galaxies with a largeamount of diffuse gas dominate the RAR arising from the halooutskirts, since the ‘egas’ component of our default model isessentially a uniform density sphere at scales r < R vir (seefigure 4 of PCS21). In fact, this also explains the results offigure 4, since bulge-dominated galaxies with m Hi = 0 arelikely to have higher values of f egas due to the baryonic massconservation constraint.The specific form of the sharp break in the RAR fromMOND-like predictions at ultra-low accelerations for thehigh- f egas sample is a consequence of the choice of spatialdistribution of the ‘egas’ component, which is the same asmotivated by ST15 in modelling the matter power spec-trum of hydrodynamical ΛCDM simulations. This result,together with the difference between the median RARof galaxies rich and poor in diffuse gas at accelerations10 − m s − (cid:46) a bary (cid:46) − m s − , are testable predictionsof the ΛCDM+baryons framework. The results in the preceding two subsections focused onthe dependence of the median RAR at high and ultra-lowaccelerations on variations in the underlying baryon-darkmatter response physics and the baryonic content of galaxies.In this subsection, we aim to understand the scatter aroundthe median relation.As we saw in section 3.2, RAR as a function of therelaxation ratio ξ has zero scatter in the ΛCDM+baryons Figure 5. RAR and diffuse gas content.
Same as top panelof figure 2, but for galaxies with f egas ≥ . (top panel) or f egas ≤ . (bottom panel) . The threshold values for rich and poor systemsrespectively correspond to the 90 th and 10 th percentile of f egas inthe mass range 9 . ≤ log[ m ∗ / ( h − M (cid:12) )] ≤ .
9. The break in themedian relation at ultra-low accelerations ( a bary (cid:46) − m s − )is restricted to diffuse gas-rich objects. Section 4.3 argues that f egas is the primary variable controlling the form of the RAR inthis regime. framework. The scatter in the RAR as a function of a bary is therefore entirely due to the scatter between ξ and a bary ,which in turn is expected to be driven almost entirely by thevariation in halo mass m vir and concentration c vir for galaxieswith similar baryonic content. This means, if we focus ongalaxies in narrow ranges of ( m vir , c vir ), the resulting RARshould have very little scatter but a mean trend that dependson the values of m vir and c vir , in general.We test this idea in figure 6. The left (right) panels show the RAR for our default model, with galaxies selectedto be in a narrow range of low (high) m vir . Within eachsuch range, the top (bottom) panels further split the galaxiesinto narrow ranges of low (high) values of c vir . It is visuallyobvious that the RAR in each of these ( m vir , c vir ) bins hasvery low scatter (quantitatively, σ log[ a tot ] ∼ . . m vir and c vir . The median RAR tends to increase in amplitude as m vir increases and, at fixed m vir , as c vir increases. That isto say, galaxies in massive, high-concentration halos have amedian RAR normalisation that is slightly but significantlyhigher than that of galaxies in low-mass, low-concentrationhalos. We have checked that the results of using differentvalues of m cvir and c vir lead to smooth extrapolations ofthese trends (see also figure 8). Appendix A provides analyticunderstanding of these trends. The preceding subsections, together with Appendix A, giveus an essentially complete picture of how the median RARand its scatter emerges from the interplay between haloproperties, their scalings with baryonic content and the direct
MNRAS , 1–19 (0000)
AR in LCDM Figure 6. RAR and halo properties.
Same as top panel of figure 2, but for centrals with host halo mass m vir and concentration c vir restricted to narrow ranges. The left (right) panels show results for m vir values in the 20-25 (90-95) percentile range, i.e. for low (high)halo mass. In each bin of m vir , the top (bottom) panel shows results for the corresponding 10-20 (80-90) percentile range of c vir , i.e. forlow (high) concentrations. The scatter is very low in each bin of ( m vir , c vir ) but the median relation varies systematically with both m vir and c vir , consistent with analytical arguments that host halo mass and concentration are the primary variables responsible for the scatterin the RAR of any galaxy sample (see section 3.2 and Appendix A). cross-talk between baryons and dark matter through quasi-adiabatic relaxation. In this subsection, we explore a fewmore aspects of the RAR, including its sensitivity to theshape of the ‘un-baryonified’ dark matter profile, some ofthe scaling relations underlying our baryonification schemeand technical choices in sampling the rotation curve data.We also show how the RAR responds to systematic changesin optical sample selection for an SDSS-like galaxy sample. Our default model uses the NFW form to model the ini-tial, ‘un-baryonified’ dark matter profile. We have checkedthat using an appropriately matched Einasto profile instead(Einasto 1965; Cardone et al. 2005; Retana-Montenegro et al.2012; Dutton & Macci`o 2014; Klypin et al. 2016) leads toessentially no change in the median RAR or its scatter. Inother words, while the RAR is sensitive to the overall massand concentration of halos (section 4.4), it is relatively insen-sitive to changes in the inner and outer slope of the initialdark matter profile.
The mass fraction f bgas in hot, bound gas in our defaultmodel from PCS21 is the same as used by ST15 and is givenby f bgas = (Ω b / Ω m ) × (cid:104) M c /m vir ) β (cid:105) − , (20)with M c = 1 . × h − M (cid:12) and β = 0 .
6. ST15 showed thatthere is considerable room for variation in the values of M c and especially β when considering the effects of baryonifi-cation on the matter power spectrum alone. Moreover, as Figure 7. RAR and bound gas fraction.
Same as top panelof figure 2, but with the baryonification parameter β defining thebound gas fraction in equation (20) increased (top panel) anddecreased (bottom panel) by ± . . a bary (cid:46) − m s − ) respondsto low values of f bgas . The text relates this to the relative massfractions and hence spatial distributions of bound and diffuse gasin the halo outskirts (section 4.5.2). discussed by PCS21, the relation above has been extrapo-lated to halos with m vir ∼ h − M (cid:12) in our mocks, wellbelow the scale m vir (cid:38) h − M (cid:12) at which ST15 calibratedtheir results. It is therefore interesting to ask how the RAR is MNRAS000
Same as top panelof figure 2, but with the baryonification parameter β defining thebound gas fraction in equation (20) increased (top panel) anddecreased (bottom panel) by ± . . a bary (cid:46) − m s − ) respondsto low values of f bgas . The text relates this to the relative massfractions and hence spatial distributions of bound and diffuse gasin the halo outskirts (section 4.5.2). discussed by PCS21, the relation above has been extrapo-lated to halos with m vir ∼ h − M (cid:12) in our mocks, wellbelow the scale m vir (cid:38) h − M (cid:12) at which ST15 calibratedtheir results. It is therefore interesting to ask how the RAR is MNRAS000 , 1–19 (0000) Paranjape & Sheth
Figure 8. RAR and stellar profile.
Similar to figure 6, showing results when the baryonification parameter R hl /R vir (whose defaultvalue is 0 . (left panels) or divided (right panels) by ( c vir / (cid:104) c vir | m vir (cid:105) ) . , with (cid:104) c vir | m vir (cid:105) being the median concentrationat fixed halo mass, which leads to a scatter of ∼ . R hl /R vir around a median value of 0 .
015 in each case. For this figure, we showonly the median and central 68% scatter of the RAR for centrals selected by m vir and c vir : top (bottom) panels correspond to halos in the20-25 (90-95) percentile ranges of m vir , and the coloured lines in each panel further split the samples into the indicated percentiles of c vir (c.f. figure 6). While galaxies at fixed ( m vir , c vir ) trace out the same RAR in each case, the range of a bary explored depends sensitively onwhether the correlation between R hl and c vir is positive or negative. See section 4.5.3 for a discussion. affected by variations in these model parameters. We explorethis in figure 7, focusing on β since the pivot scale M c isreasonably well constrained by the X-ray cluster observationscited by ST15. We see that variations in β primarily affectthe ultra-low-acceleration regime, changing the slope of themedian RAR. This is sensible, because an increase in f bgas at the mass scales of our interest (by decreasing β ) will corre-spondingly decrease f egas due to baryonic mass conservationand hence change the relative spatial behaviour of the ‘bgas’and ‘egas’ components in the halo outskirts (see figure 4 ofPCS21). Thus, the ultra-low-acceleration regime of the RARis, in principle, sensitive to the physics of both hot and coldgas in the outer halo. Our default model treats the stellar profile as a bulge withhalf-light radius R hl = 0 . R vir , which is approximately theresult obtained by Kravtsov (2013) using a power-law fit to ∼
180 galaxies. We have not included the scatter of ∼ . R hl /R vir and c vir ).Halo concentrations in our mocks have a Lognormaldistribution at fixed mass, with a median (cid:104) c vir | m vir (cid:105) anda scatter 0 .
16 dex taken from Diemer & Kravtsov (2015).The left (right) panels of figure 8 show the RAR after mul-tiplying (dividing) the default R hl /R vir for each galaxy by( c vir / (cid:104) c vir | m vir (cid:105) ) µ , thus leading to a positive (negative) cor-relation between stellar bulge size and initial halo concen-tration. In this toy model, the entire variation in R hl /R vir is explained by halo concentration; more realistic models would allow room for other variables (such as halo angularmomentum, or some unspecified source of stochasticity) toalso play a role. By construction, the modified set of R hl /R vir values obey a Lognormal distribution at fixed halo mass, withmedian 0 .
015 and a scatter of 0 . × µ dex. We thereforeset µ = 1 .
3, which gives a scatter of (cid:39) . R hl /R vir for both choices of the correlation, consistent with Kravtsov(2013).The top (bottom) panels of figure 8 use the same rangesof m vir shown in the left and right panels, respectively, offigure 6. Similarly to that figure, we further split these fixed- m vir samples into narrow ranges of c vir . For this figure alone,so as to highlight differences between the subsamples, weonly show the median and central 68% of each RAR using thedifferently coloured lines. As expected from the discussionin sections 3.2 and 4.4, galaxies at fixed m vir and c vir tracethe same RAR regardless of the sign of the bulge size-haloconcentration correlation. We do see a very interesting traceof this signature however, in that the range of values of a bary explored by any sample responds sensitively to whether thecorrelation is positive or negative. In the former case, galaxiesin high-concentration halos explore lower values of a bary than low-concentration ones, and vice-versa for a negativecorrelation. This trend can be understood as follows. Considera specific galaxy with stellar mass m ∗ in an ( m vir , c vir ) host.For a positive correlation, a large c vir implies a larger R hl forthis galaxy than in the absence of the correlation. Since the(now flatter) stellar density profile must enclose the same m ∗ inside the same R vir , its inner parts are forced to be lower,thus contributing less to m cgal ( < r ) and hence a bary in theinner region, than in the absence of the positive correlation.A negative correlation between R hl and c vir has exactly theopposite effect.While this shows that there is clearly no new physicsexplored by such a correlation beyond the dependence of theRAR on m vir and c vir through the relaxation ratio ξ , it doeslead to a curious degeneracy. It is clear from the left hand MNRAS , 1–19 (0000)
AR in LCDM Figure 9. RAR and luminosity.
Same as top panel of figure 2, for centrals selected by r -band absolute magnitude as indicated, withthe samples increasing in luminosity from top left → bottom left → top right → bottom right . There is a clear luminosity dependence,with the RAR shifting vertically upwards for brighter samples. This is a natural consequence of the halo mass dependence seen in figure 6(see section 4.5.4). panels of figure 8 that the RAR obtained from averaging overall c vir values will tend to curve downwards at large a bary inthe case of a positive R hl - c vir correlation. Further integrationover m vir will not change this curvature, so the resultingRAR will be qualitatively similar to that in which there is no R hl - c vir correlation but q rdm is smaller (compare top panelof figure 3). Conversely, the right hand panels of figure 8show that an R hl - c vir anti-correlation will result in an RARthat would imply a larger q rdm if one assumed there wasno R hl - c vir correlation (e.g. bottom panel of figure 3). Thatcurvature in the RAR may arise from the R hl - c vir relationrather than q rdm must be kept in mind during any analysiswhich aims to probe the physics of quasi-adiabatic relaxationusing the RAR. Using a more physically motivated ‘size-mass’correlation, such as the one between galaxy size and haloangular momentum alluded to above, can potentially add anew dimension to such degeneracies (see also the discussion inDesmond 2017). It will be interesting to study such effects inhydrodynamical simulations of cosmological volumes, whichwe leave to future work. Observational analyses of the RAR are typically limited bythe quality of rotation curve (e.g., Lelli et al. 2016a) or veloc-ity dispersion (e.g., Chae et al. 2019) measurements, whichcan introduce inhomogeneities in the statistical properties ofthe associated galaxy sample. Since our mock catalogs have‘perfect’ rotation curve measurements, we can use them toask how the RAR responds to systematic variations in, say,optical sample selection for SDSS-like galaxies.Figure 9 shows the RAR using our default model formock centrals chosen to lie in bins of luminosity (one bin ineach panel), represented by r -band absolute magnitudes M r (see PCS21 for a detailed definition). The samples increase inluminosity going from top left → bottom left → top right → bottom right , with the faintest bin corresponding to sub- L ∗ centrals (median m vir (cid:39) . h − M (cid:12) ) and the brightestto BCGs of massive clusters (median m vir (cid:39) . h − M (cid:12) ).We see a clear indication that the normalisation of the me-dian RAR increases with increasing luminosity, which issensible given the results of figure 6 and the fact that cen-tral luminosity correlates positively with halo mass in ourmocks. Interestingly, recent results suggest that the RARof observed galaxy clusters also has an elevated normalisa-tion relative to that of galaxy samples (Tian et al. 2020;see also Pradyumna et al. 2021), in qualitative agreementwith our results. We also find that the typical scatter ofthe RAR varies non-monotonically with luminosity, being σ log[ a tot ] ∼ . , . , . , .
066 in successively brighterbins.It is also interesting to split the galaxy sample at fixedluminosity by colour. We use a luminosity-dependent thresh-old on the g − r colour index of each of our mock centralgalaxies, given by (Zehavi et al. 2011)( g − r ) cut ( M r ) ≡ . − . M r . (21)We classify galaxies having g − r ≥ ( g − r ) cut ( M r ) as ‘red’and the rest as ‘blue’. Figure 10 shows the resulting RAR fortwo of the luminosity bins shown in figure 9. In the absenceof ‘beyond halo mass’ effects such as galactic conformity,galaxy colours in our default mocks correlate only with galaxyluminosity, not with halo mass or concentration. Naively,therefore, we should not expect any difference in the RAR ofred and blue galaxies at fixed luminosity. This is indeed thecase for the brighter luminosity bin shown in the right panels of figure 10. There is, however, a secondary correlation onemust account for in a luminosity-complete sample. This isthe fact that, due to a colour-dependent mass-to-light ratio,blue centrals of a given luminosity will have lower stellarmasses m ∗ than red centrals with similar luminosity (see,e.g., figure 3 of PCS21 and figure 4 of Paranjape et al. 2015).The decrease in m ∗ from red to blue objects is accompaniedby an increase in f egas due to baryonic mass conservation. MNRAS000
066 in successively brighterbins.It is also interesting to split the galaxy sample at fixedluminosity by colour. We use a luminosity-dependent thresh-old on the g − r colour index of each of our mock centralgalaxies, given by (Zehavi et al. 2011)( g − r ) cut ( M r ) ≡ . − . M r . (21)We classify galaxies having g − r ≥ ( g − r ) cut ( M r ) as ‘red’and the rest as ‘blue’. Figure 10 shows the resulting RAR fortwo of the luminosity bins shown in figure 9. In the absenceof ‘beyond halo mass’ effects such as galactic conformity,galaxy colours in our default mocks correlate only with galaxyluminosity, not with halo mass or concentration. Naively,therefore, we should not expect any difference in the RAR ofred and blue galaxies at fixed luminosity. This is indeed thecase for the brighter luminosity bin shown in the right panels of figure 10. There is, however, a secondary correlation onemust account for in a luminosity-complete sample. This isthe fact that, due to a colour-dependent mass-to-light ratio,blue centrals of a given luminosity will have lower stellarmasses m ∗ than red centrals with similar luminosity (see,e.g., figure 3 of PCS21 and figure 4 of Paranjape et al. 2015).The decrease in m ∗ from red to blue objects is accompaniedby an increase in f egas due to baryonic mass conservation. MNRAS000 , 1–19 (0000) Paranjape & Sheth
Figure 10. RAR and optical colour.
Same as top panels of figure 9, with the top (bottom) panels now focusing on red (blue) galaxies.The separation between red and blue uses the luminosity-dependent threshold on g − r colour given by equation (21). There is a sharpdifference between red and blue centrals in the fainter bin at ultra-low accelerations ( a bary (cid:46) − m s − , c.f. figure 5), with relativelylittle difference between the two samples in the brighter bin. These trends can be traced back to differences, or lack thereof, in the massfraction f egas of diffuse gas in these samples (see section 4.5.4). We saw already, in figure 5, that samples with higher f egas tend to break away from the smooth, MOND-inspired RARfunctional forms at ultra-low accelerations. Figure 6 alsoshowed that this break is prominent only for galaxies withlow-mass hosts, whose median RAR can reach the ultra-low-acceleration regime. Not surprisingly, then, we see in the leftpanels of figure 10 that faint blue centrals trace out exactlythe same break, which is correspondingly absent for faintred objects. At higher luminosity, the corresponding halomasses are higher, so that the median RAR does not reachthe ultra-low acceleration regime, leading to identical RARsfor red and blue galaxies as discussed above. All our results above have been based on an arbitrarilychosen sampling of the rotation curve of each galaxy, using 20logarithmically spaced values of r in the range (0 . , × R vir (section 4.1). Since observed rotation curves are typicallyinhomogeneous in the available sampling (e.g., Lelli et al.2017), it is important to check what role sampling plays inestablishing the median RAR and its scatter. We test thisin figure 11 by comparing our default results with thoseobtained using a different sampling choice, now using 40 linearly spaced points in the same range (0 . , × R vir for each galaxy. Visually, the resulting histogram is verydifferent from the default case, being over-sampled at ultra-low accelerations and under-sampled at high accelerations (asexpected from the fact that the linear sampling decreases thenumber of available points in the inner halo). Encouragingly,though, the median RAR as well as the scatter are relativelyunaffected across the entire range of a bary (nearly 6 orders ofmagnitude) probed in the plot. We conclude that samplingchoices are not expected to be a major source of systematicuncertainty in the median and scatter of the RAR. Figure 11. RAR and rotation curve sampling. (Top panel:)
Identical to top panel of figure 2, i.e. using rotation curves sampledwith 20 logarithmically spaced points between (0 . , × R vir foreach galaxy. (Bottom panel:) Same as top panel, but sampling eachrotation curve using 40 linearly spaced points between (0 . , × R vir . The density of points in different parts of the RAR dependsstrongly on the sampling of the rotation curves, but the medianrelation and its scatter are relatively insensitive to this choice. The RAR is closely linked with the so-called baryonic Tully-Fisher relation (BTFR, McGaugh et al. 2000) M bary ∝ V α c ,which generalises the classical Tully-Fisher relation m ∗ ∝ V α c (Tully & Fisher 1977) to include the mass in cold gas in MNRAS , 1–19 (0000)
AR in LCDM Figure 12. Baryonic Tully-Fisher relation (BTFR) using the default baryonification model. Coloured histograms show thedistribution of M bary , the ‘baryonic’ (i.e., stellar + cold gas) mass from equation (7) against circular velocity V c defined in different ways. Left panels use V c measured at r = 2 R h , bary (where R h , bary is the baryonic half-mass radius), while right panels use V c calculated as themean circular velocity in the flat part of the rotation curve using the algorithm of Lelli et al. (2016b). Note the difference in the range ofthe colour bars in the left and right panels. Top panels show ‘bulge-dominated’ galaxies with m Hi = 0, while bottom panels show gas-rich‘spiral’ galaxies with m gas /m ∗ ≥ .
72 (here, m gas = 1 . m Hi ). The threshold value is chosen as described in the caption of figure 4. Solidyellow curve in each panel shows the median V c in bins of baryonic mass. Dashed yellow curves show the corresponding 16 th and 84 th percentiles (i.e., the horizontal scatter). For comparison, the purple lines (repeated in each panel) show the observed relations using V c measured in the flat part of the rotation curve from McGaugh et al. (2000, dotted: M bary ∝ V . ) and Lelli et al. (2019, dashed: M bary ∝ V . ) and using V c measured at twice the observed half-light radius from Lelli et al. (2019, solid: M bary ∝ V . ). We see thatspirals in the mock catalog have steeper BTFR slopes as well as higher normalisations than bulge-dominated galaxies, with the effectbeing more pronounced when V c is measured in the flat part of the rotation curve. addition to stellar mass. The BTFR with a slope α = 4 anda small scatter ∼ . V c in the BTFR is an estimate of thecircular velocity in the outer parts of the galaxy and is meantto be a proxy for the total matter content of each system.The precise definition of V c has been the subject of somediscussion, and the inferred slope α of the BTFR is rathersensitive to the assumed definition of V c (Bradford et al.2016). Defining V c at some fixed multiple of the disk scalelength typically leads to α (cid:39)
3, while defining V c in the‘flat part’ of the rotation curve (see below), typically yieldssteeper slopes α (cid:39) V c tied tothe disk scale length. It has been further argued (Wheeleret al. 2019), that the RAR in the low-acceleration regime10 − m s − (cid:46) a bary (cid:46) − m s − is a simple algebraicconsequence of a BTFR with slope α = 4, so that modelswhich satisfy the BTFR with this slope are guaranteed tofollow the observed RAR at low accelerations.In this section, we use our mock galaxies to place theabove results in the context of the analytical and numericalarguments concerning the RAR from the preceding sections.Figure 12 shows the BTFR for our mock centrals using thedefault baryonification scheme and the same definition of M bary (equation 7) used in figure 4. The top panels of the figure focus on pure bulge-likegalaxies while the bottom panels show results for gas-rich,disk-dominated systems, with the split being identical to theone used in figure 4. The left panels show results when V c is defined as V c = v rot ( r = 2 R h , bary ), where R h , bary is thebaryonic half-mass radius, i.e. at the same location as usedto calculate M bary . In the right panels , we follow Lelli et al.(2016b) and discard galaxies for which the ‘outermost’ partof the rotation curve is either rising or falling too steeply.We pick r = 0 . R vir (cid:39) R hl as the outermost measuredradius and define the threshold steepness by requiring thatsuccessive residuals between v rot ( r ) at smaller radii andthe mean v rot in the outermost region be smaller than 2%.In other words, we implement the iterative algorithm ofLelli et al. (2016b) with a threshold of 2% instead of the5% those authors used. We only use galaxies with at least3 usable values of r , which was also done by Lelli et al.(2016b). Another detail is that we perform this exercise on alinearly sampled grid of r values containing 6 points between(0 . , . × R vir . The resulting mean value of v rot is then anestimate of V c (flat), the circular velocity in the ‘flat part’ ofthe rotation curve. The sample of galaxies selected by thisanalysis is an order of magnitude smaller than the one usedin the left panels .We will shortly discuss the dependence of our resultson the (admittedly arbitrary) technical choices in measur-ing V c (flat). We first note, however, that each of our mock MNRAS000
3, while defining V c in the‘flat part’ of the rotation curve (see below), typically yieldssteeper slopes α (cid:39) V c tied tothe disk scale length. It has been further argued (Wheeleret al. 2019), that the RAR in the low-acceleration regime10 − m s − (cid:46) a bary (cid:46) − m s − is a simple algebraicconsequence of a BTFR with slope α = 4, so that modelswhich satisfy the BTFR with this slope are guaranteed tofollow the observed RAR at low accelerations.In this section, we use our mock galaxies to place theabove results in the context of the analytical and numericalarguments concerning the RAR from the preceding sections.Figure 12 shows the BTFR for our mock centrals using thedefault baryonification scheme and the same definition of M bary (equation 7) used in figure 4. The top panels of the figure focus on pure bulge-likegalaxies while the bottom panels show results for gas-rich,disk-dominated systems, with the split being identical to theone used in figure 4. The left panels show results when V c is defined as V c = v rot ( r = 2 R h , bary ), where R h , bary is thebaryonic half-mass radius, i.e. at the same location as usedto calculate M bary . In the right panels , we follow Lelli et al.(2016b) and discard galaxies for which the ‘outermost’ partof the rotation curve is either rising or falling too steeply.We pick r = 0 . R vir (cid:39) R hl as the outermost measuredradius and define the threshold steepness by requiring thatsuccessive residuals between v rot ( r ) at smaller radii andthe mean v rot in the outermost region be smaller than 2%.In other words, we implement the iterative algorithm ofLelli et al. (2016b) with a threshold of 2% instead of the5% those authors used. We only use galaxies with at least3 usable values of r , which was also done by Lelli et al.(2016b). Another detail is that we perform this exercise on alinearly sampled grid of r values containing 6 points between(0 . , . × R vir . The resulting mean value of v rot is then anestimate of V c (flat), the circular velocity in the ‘flat part’ ofthe rotation curve. The sample of galaxies selected by thisanalysis is an order of magnitude smaller than the one usedin the left panels .We will shortly discuss the dependence of our resultson the (admittedly arbitrary) technical choices in measur-ing V c (flat). We first note, however, that each of our mock MNRAS000 , 1–19 (0000) Paranjape & Sheth
Figure 13. BTFR and rotation curve sampling.
Same asright panels of figure 12, except that the rotation curves weresampled with twice the number of points. We see that the BTFR,especially of massive spirals, is shallower than in figure 12, andmany more galaxies are included in the relation. The BTFR inthe flat part of the rotation curve is hence sensitive to samplingchoices. samples defines a reasonably tight BTFR in figure 12. Acloser comparison with results from the literature (purplelines) shows that (a) the BTFR of pure bulges has a slopeclose to α (cid:39)
3, decidedly shallower than that of gas-richspirals which are closer to α (cid:39)
4. Focusing on the latter(i.e., the lower panels), we also see some hint at the highestmasses that using V c (flat) leads to a slightly steeper slopethan V c (2 R h , bary ). The horizontal scatter around the medianrelation is ∼ .
055 dex.These trends are easily understood. For all objects, boththe stellar mass and V c (flat) are tightly correlated with halomass. For bulges, m bary on the relevant scales is dominatedby the stellar component, so the curvature in the top panel,which results in a shallower effective slope, is a consequence ofthe curvature in the m ∗ - m halo relation. Spirals in our mockshave the same m ∗ - m halo relation, but because our mockshave Hi gas fractions decreasing with mass, spirals do notprobe the higher halo masses where the curvature matters.This is why their m ∗ − V c relation appears to be steeper.Adding the Hi mass to m ∗ , so as to obtain m bary , lifts the Due to their luminosity-complete nature, our mocks are completein stellar and Hi mass only for thresholds m ∗ (cid:38) . h − M (cid:12) and m Hi (cid:38) . h − M (cid:12) (see PCS21 for details), which leads toa somewhat complicated completeness threshold as a functionof M bary . In order to avoid the resulting Malmquist bias effectsin characterising the BTFR, throughout this section we reportresults in bins of M bary rather than V c . See Bradford et al. (2016);Lelli et al. (2019) for a discussion of the complications in fittingBTFR slopes to observed data which, in addition to selectioneffects, also have errors on both variables. relation for spirals above that for the bulges, bringing themcloser to the observed BTFR.Thus far, our default BTFR results are in reasonableagreement with observations. This is already interesting, be-cause a comparison with figure 4 shows that, although gas-rich spirals with α (cid:39) so do pure pulges with α (cid:39)
3. Inother words, while being on the BTFR may guarantee beingon the RAR (Wheeler et al. 2019), the RAR is obeyed bya much wider class of galaxies. This implies, firstly, thatstatements such as ‘the RAR is a natural consequence of theBTFR’, which suggest that the BTFR is more fundamen-tal than the RAR, must be treated with caution. Secondly,the converse is also not true in our mocks: galaxies whichcontribute to the low-acceleration RAR need not obey theBTFR with α = 4 (compare the upper panels of figures 4and 12), in contrast with some claims in the literature (see,e.g., the discussion in section 7.1 of Lelli et al. 2017).Things become even more interesting when one startsto question the various technical choices used in defining‘good’ rotation curves. Figure 13 shows the results of anexercise identical to the one described above in estimating V c (flat), with the only difference being that we now used alinearly spaced array of r values with 12 instead of 6 pointsin the range (0 . , . × R vir , without changing the flatnessthreshold of 2%. It is obvious, upon some thought, thatthis change will relax the flatness restriction and allow moregalaxies to be used in the sample containing valid V c (flat)values. We see that the resulting BTFR for both bulges aswell as spirals are now very different from those in the rightpanels of figure 12. In particular, the BTFR of spirals is nowconsistent with α (cid:39)
3. We have checked that similar resultsare obtained upon relaxing the flatness threshold to 5% forour default sampling, as well as when modifying the definitionof the ‘flat part’ to use analytical slopes d ln v rot / d ln r inconstraining the degree of flatness. Considering the lack ofhomogeneity of rotation curve sampling in (otherwise veryhigh quality) data-bases such as SPARC (Lelli et al. 2016a)which has been used in many recent BTFR analyses, ourmock results call for a great deal of caution in interpretinga BTFR analysis in the context of competing gravitationaltheories.In contrast, the differences seen in the median RAR andits scatter in figure 11 due to (rather dramatic) changes insampling the rotation curves are relatively minor in com-parison. The RAR is therefore a much more observationallyrobust probe of the nature of gravity at galactic scales thanis the BTFR. We have presented new analytical insights into the structureand origin of the radial acceleration relation (RAR) betweenthe total ( a tot ) and baryonic ( a bary ) centripetal accelerationprofiles of galaxies in the ΛCDM framework.Our key result follows from the realisation (section 3)that the residual mass discrepancy ∆ a (equation 3) is com-pletely determined, with essentially no scatter , by the ratio ξ (equation 9) governing the quasi-adiabatic relaxation ofdark matter in the presence of baryons in any galactic halopotential, through equation (13). Since the physics of thisrelaxation can be approximated using simple fitting func-tions from the literature (equation 10), our framework allowsus to analytically estimate both the median and scatter of MNRAS , 1–19 (0000)
AR in LCDM the RAR (∆ a as a function of a bary ) in quantitative detail over a wide dynamic range in galaxy and halo properties(Appendix A).We augmented our analytical calculations with measure-ments of the RAR in a realistic mock catalog of ∼ , a bary : (i) high-acceleration ( a bary (cid:38) − m s − ), (ii) low-acceleration (10 − m s − (cid:46) a bary (cid:46) − m s − ) and (iii)ultra-low-acceleration ( a bary (cid:46) − m s − ). Our main re-sults can be summarized as follows. • The median RAR resulting from applying the relaxationprescription of Schneider & Teyssier (2015, ST15) – i.e.,setting the relaxation parameter q rdm = 0 .
68 in equation (10)– to our mock galaxies is within ∼ of the observedrelation at low and high accelerations a bary (cid:38) − m s − (figure 2). Since the ST15 prescription and value of q rdm were only tuned to reproduce the relaxation seen in halosin hydrodynamical CDM simulations, with no reference tothe RAR, this quantitative agreement over more than fourorders of magnitude in a bary represents a non-trivial successof the galaxy-dark matter association in ΛCDM. • This agreement is particularly remarkable in the high-acceleration regime, where we showed that the median andscatter of the RAR are both very sensitive to the value of q rdm ,and there is no a priori reason why the value q rdm = 0 . • The median RAR in the ultra-low-acceleration regime isvery sensitive to the expelled (or diffuse) gas fraction f egas ,and our default model predicts a distinctive break fromsmooth, MOND-inspired relations at a bary (cid:46) − m s − fordiffuse gas-rich systems (figure 5). This regime, correspondingto the outskirts of halos hosting sub- L ∗ galaxies, is currentlyunobserved, although future observations of the CGM couldbe promising in this regard (e.g., Cantalupo et al. 2014;Werk et al. 2014; Zahedy et al. 2019). Our results at ultra-low-accelerations constitute robust and testable predictions ofthe Λ CDM framework. • While the median RAR is set by a combination of baryon-dark matter scalings and relaxation physics (sections 4.2and 4.3, figures 4 and 5), we identified the primary sourceof scatter in the RAR to be host halo mass and concentra-tion, with a magnitude that depends on the value of therelaxation parameter q rdm (section 4.4 and Appendix A1, fig-ures 3 and 6). Specifically, the scatter in the high-acceleration,baryon-dominated regime is small when q rdm → q rdm → Why is q rdm closer to 1 than to 0, with small scatter, over the relevantmass range? • We used our mock galaxies to explore the sensitivityof the RAR to a number of details such as sample selec-tion, rotation curve measurement technicalities, as well asvariations in baryon-dark matter scalings and halo profileshape (section 4.5). Our framework, for example, providesa natural explanation for the observed offset (Tian et al.2020) between the RAR of cluster BCGs and fainter centrals(figure 9 and section 4.5.4), while predicting that the RAR isrelatively stable against variations in the chosen form of the‘un-baryonified’ dark matter profile (NFW versus Einasto; section 4.5.1) or technicalities of rotation curve sampling(section 4.5.5). • In contrast, we argued in section 5 that the baryonicTully-Fisher relation (BTFR) is substantially more suscep-tible to such technical details. As such, the RAR is a muchmore robust probe of galactic-scale gravitational physics thanis the BTFR.The intrinsic scatter of the RAR as inferred from obser-vations, after accounting for all sources of measurement error,is a matter of considerable interest and discussion. If thisscatter is indeed negligible, as reported by Lelli et al. (2017),it would pose a major challenge to the galaxy-dark matterassociation assumed in the ΛCDM paradigm. The robust-ness of this claim of a zero scatter RAR, however, remainsdebated. E.g., Stone & Courteau (2019) estimate an intrinsicscatter in the a tot - a bary relation (they focus on the stellarcontribution to a bary ) of 0 . ± .
02 dex, fully consistent withour results above as well as those from the earlier ΛCDMliterature (e.g., Keller & Wadsley 2017). Uncertainties inobserved rotation curves might also be sensitive to technicaldetails of extracting velocity profiles (e.g., Sellwood et al.2021). Finally, observational RAR (and BTFR) analysesoften focus on ‘good’ samples of inhomogeneously selectedrotation curves (e.g., Lelli et al. 2016b, 2017), making a directcomparison between predicted and observed scatter difficult.Our results above therefore suggest that the medianRAR (i.e., ∆ a as a function of a bary ), especially in theregimes of high and ultra-low accelerations, is likely to bethe most powerful discriminator between alternative gravi-tational models, as well as serving to constrain the physicsof baryon-dark matter interactions in ΛCDM. For this tobe successful, it will be important to perform observationalanalyses with well-defined, representative galaxy samples. ACKNOWLEDGEMENTS
AP thanks R. Srianand and Sowgat Muzahid for useful discus-sions. The research of AP is supported by the AssociateshipScheme of ICTP, Trieste and the Ramanujan Fellowshipawarded by the Department of Science and Technology, Gov-ernment of India. This work made extensive use of the opensource computing packages NumPy (Van Der Walt et al.2011), SciPy (Virtanen et al. 2020), Matplotlib (Hunter2007) and Jupyter Notebook. DATA AVAILABILITY
The mock catalogs underlying this work will be made avail-able upon reasonable request to the authors.
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APPENDIX A: ANALYTIC RELAXATION
Solving the relaxation problem boils down to describinghow the final radius r is related to the initial radius r i .Equation (10) of the main text considers a model in which rr i − q (cid:18) M i ( r i ) M ( r ) − (cid:19) (A1)where M i ( r i ) and M ( r ) are the initial and final enclosed massprofiles. Note that M ( r ) = M bary ( r ) + f d M i ( r i ) where f d isthe dark matter fraction (typically one sets f d = 1 − Ω b / Ω m ).In the main text, this problem was treated numerically. Themain purpose of this Appendix is to show that, for judi-cious (but realistic) choices of the profile shapes M i ( r i ) and M bary ( r ), much of the analysis can be done analytically. Weassume spherical symmetry in what follows.Previous analytic work (e.g. Keeton 2001) exploits thefact that, for simple choices of M bary ( r ), the relaxation equa-tion can be solved analytically for any M i ( r i ). This is at-tractive since, in practice, M i ( r i ) is unknown, so this is thequantity which one hopes to determine from detailed ob-servations of the baryons. In effect, such approaches solvefor r as a function of M i ( r i ), and hence of r i . However, aswe show below, for appreciating what sets the shape of theRAR, it is more illuminating to determine the inverse of thisrelation: r i as a function of M bary ( r ) and hence of r . Below,we exploit the fact that, for simple choices of M i ( r i ) thepost-relaxation profile can be written analytically for any M bary ( r ). If, in addition, the a bary ( r ) ≡ GM bary ( r ) /r vs r relation can be inverted analytically, the result will be a fullyanalytic expression for the RAR from baryonic relaxation. MNRAS , 1–19 (0000)
AR in LCDM A1 The RAR for an initially Hernquist profile
The main text used the NFW functional form to describe theinitial profile, but also showed that an appropriately scaledEinasto profile gave very similar results. Since the preciseparametrization does not matter, it is natural to ask if thereis a parametrization which simplifies the analysis. For scales r i < r vir , the NFW model is very well approximated by aHernquist profile, for which m i ( r i ) ≡ M i ( r i ) M vir = (cid:18) r i r vir r vir + r r i + r (cid:19) , (A2)provided that one sets c ≡ r vir /r = c . / √ m i ( r i ), the re-laxation equation reads qξ = (cid:20) − − qξ (cid:21) (cid:34) m bary ( r )(1 + 1 /c ) (cid:18) ξ + r vir rc (cid:19) + f d ξ (cid:35) , (A4)where ξ ≡ r/r i as in the main text. This is a cubic equationfor ξ which can be solved analytically. Since all the coefficientsare real, there is at least one real root. This root is given by1 ξ = S − QS − a , (A5)where S = (cid:104) P + (cid:112) P + Q (cid:105) / ,P = 3 a a − a − a and Q = a − a , with a = 2 µ (1 − q ) / ( r/r ) − ( µ + f d )3 a ,a = µr/r (1 − q ) / ( r/r ) − a ,a = − µ ( r/r ) a and a = q + (1 − q )( µ + f d ) , and µ ≡ m bary ( r ) / (1 + 1 /c ) . This follows from rearrangingthe cubic equation to a + 3 a x + 3 a x + x = 0 andassumes a (cid:54) = 0 which is guaranteed if q ≤
1. Inserting this ξ in equation (15) yields the RAR for any input m bary ( r ).Figure A1 shows the result if we use the m bary ( r ) profiles(i.e. the sum of the stellar, Hi , bound and expelled gasprofiles) returned by the baryonification procedure of PCS21for the two representative halo masses shown in figure 1.Note how the RAR changes as the quasi-adiabatic relaxationparameter varies from q = 0 (no conservation) to q = 1(exact conservation). The RAR associated with q = 0 .
68 israther well-described by equation (18) with ν = 0 .
8, as it wasfor the numerical (NFW-based) solutions presented in themain text. For the larger halo mass, the RAR turns over inthe outer regions (small a bary ). This is because a dm /a bary = m dm /m bary and figure 1 shows that m bary increases moresteeply than m dm in the outer regions of massive halos. Notealso that, at the smaller halo mass, the RAR flattens out,so that it lies below the scaling associated with M bary ∝ V c for which a tot ∝ a / . These results are in good quantitative agreement with those shown in the main text.Before moving on, it is worth noting that the structureof the cubic (equation A4) makes it easy to understandhow the RAR depends on q , f d , c and M vir in the baryon-dominated limit. For the PCS21 models with q >
0, this
Figure A1.
Dependence of the RAR relations, determined fromsolving the cubic equation (A4) with m bary ( r ) from PCS21, onthe relaxation parameter q = 1 , .
68 and 0, for two choices of halomass and associated halo concentration ( top and bottom panels ;the corresponding mass profiles were shown in the left panel offigure 1). Dashed curves show the predicted scaling in the baryondominated regime. Magenta and blue show scales smaller andlarger than that of the scale radius of the HI gas. Solid black curveshows equation (18) with parameters from the main text: dottedcurves show the associated scalings at large and small a bary . is where r → r (cid:28) r i . In this ξ (cid:28) ξ − ≈ − a . (Since a tot /a bary ≥ ξ ≥ − q ,we treat the q → q = 1, since then a = 1 and ξ = (1 + c ) / [ a vir /a bary ( r )] / , where a vir = GM vir /r .Moreover, we expect f d ≈
1, so equation (13) says that theRAR becomes∆ a ≈ f d (1 + c ) / [ a bary ( r ) /a vir ] / when q = 1 , (A6)where ∆ a was defined in equation (3). This scaling is shownas the dashed line in figure A1; clearly, it describes theapproach to baryon domination well.We now consider q → r, r i → ξ →
1. For this,it is useful to rewrite equation (A4) as q = ( ξ − q ) (cid:20) m bary ( r )(1 + c ) (cid:16) c + ξ r vir r (cid:17) + f d (cid:21) , (A7) MNRAS000
1. For this,it is useful to rewrite equation (A4) as q = ( ξ − q ) (cid:20) m bary ( r )(1 + c ) (cid:16) c + ξ r vir r (cid:17) + f d (cid:21) , (A7) MNRAS000 , 1–19 (0000) Paranjape & Sheth making∆ a = f d (cid:20) m bary ( r )(1 + c ) (cid:16) c + ξ r vir r (cid:17) (cid:21) − = f d m i ( r i ) m bary ( r ) (A8) → f d (cid:20) a bary ( r ) ξ a vir (1 + c ) (cid:21) − → f d (1 + c ) a bary ( r ) /a vir when q → . The second equality on the first line connects with equa-tion (17) of the main text, and the final expression is fromthe r i → ξ → a bary than when q = 1, but the dashed curve in figure A1(which is almost indistinguishable from the magenta part ofthe q = 0 curve) shows that it describes this limit well.Notice that the quantity a vir (1 + c ) plays a key role.Since all halos have density 200 ρ crit , a vir nm s − = 0 . r vir h − kpc = 0 . (cid:18) m vir h − M (cid:12) (cid:19) / . (A9)For m vir = 10 h − M (cid:12) and c vir = 10, typical of halos host-ing galaxies, a vir (1 + c ) ≈ .
13 nm s − . Since f d ≈ a = 0 .
12 nm s − scale that is usuallyassociated with the RAR from galactic dynamics. Of course,equation (A9) shows that this scale will be larger for clusters.Thus, our analysis shows that the RAR scale depends onhalo mass, concentration and dark matter fraction, but the shape of the RAR in the high-acceleration regime dependson the adiabatic parameter q .The analysis above is also useful for understanding whythe RAR has small scatter. At fixed f d (and q ) the scattercomes from M / (1+ c ) . However, in ΛCDM, more massivehalos are less concentrated, so averaging over a factor of ∼ M vir does not lead to large scatter in the RAR. (Of course,variations of order 10 in M vir will be more significant, whichis why the RAR of clusters is offset from that of galaxies – butby much less than a factor of 10 .) Moreover, equations (A6)and (A8) depend on different powers of this combination ofmass and concentration; this explains why the scatter aroundthe q = 0 relation is larger than around q = 1 (c.f. figure 3).This leaves variations in f d and q as possible additionalsources of scatter in the RAR. However, f d is expected tohave small scatter – and in our mocks it has no scatter (byassumption). So, in ΛCDM, the real puzzle posed by thetightness of the RAR is: Why is q closer to 1 than to 0, withsmall scatter, over the mass range relevant to the RAR ofgalaxies? A2 Fully analytic relaxation and the RAR
Although we used m bary ( r ) from PCS21 to make figure A1,the analysis of the previous subsection applies to any m bary .This means that, for judicious parameterizations of m bary ,it may be possible to provide fully analytic expressions forthe RAR. We now show that this is indeed possible over asubstantial fraction of the halo.Start with profiles of the form ρ β ( r ) ∝ ( r/r β ) − β (1 + r/r β ) − β , (A10)which scale as r − β on scales smaller than r β , and as r − onlarger scales. Simulations have shown that it is reasonableto approximate the stellar distribution with β = 2, the Hi gas with β = 1 and the bound and expelled gas profiles with β = 0 (but different scale radii r β ). Let M β ≡ π (cid:90) ∞ dx x ρ β ( x ) (A11)denote the total mass associated with this profile. Then,provided β <
3, the mass within r is given by M β ( < r ) = M β (cid:18) r/r β r/r β (cid:19) − β . (A12)If we ignore the two β = 0 components – in practice weassign their mass to the β = 1 component and modify r tomatch the profile of their sum – then m bary ( r ) ≡ M bary ( r ) M vir = F rr + r + F r ( r + r ) , (A13)where F β ≡ M β /M tot . It is conventional to work not with F β but with the mass fractions within the virial radius: f β ≡ M β ( < r vir ) M vir = F β (cid:18) r vir /r β r vir /r β (cid:19) − β . (A14)Then F β = f β (1 + r β /r vir ) − β and f d ≡ − (cid:88) β f β . (A15)Equation (A13) is substantially more realistic, and not muchmore complicated, than the model discussed in AppendixA of Teyssier et al. (2011) in which all the baryons areclubbed into a single component with ρ ∝ r − . Figure A2illustrates. Diamond symbols and curves compare the actualand approximated m bar profiles for the two halos shown infigure A1. To produce the curves, we set r = h Hi (the Hi disk scale length) for the lower mass halo and r = r s (theNFW scale radius) for the higher mass halo. For the lowermass halo, the agreement is good over almost the entirehalo, whereas, for the more massive halo, the agreement isgood only in the inner regions which are dominated by stars,and out to about the scale radius of the gas. Squares andassociated curves show that equation (A2) provides a gooddescription of the initial NFW profile for m i .If the baryonic profile is well described by equation (A13)then a bary ( r ) a vir = F r r ( r + r ) + F r ( r + r ) . (A16)Equation (A16) shows that, to invert the a bary - r relation onemust solve a quartic equation. Therefore, r can be writtenas a complicated but analytic function of a bary , which, wheninserted for r in equation (A5) yields ξ as a function of a bary . This ξ ( a bary ), when inserted in equation (15), yields afully analytic expression for the RAR. The accuracy of thisexpression depends on how well equation (A13) approximatesthe true m bary ( r ). Figure A2 shows that we expect this towork well out to approximately the scale where the gasdominates the baryonic component. As this fully analyticRAR is essentially indistinguishable from that shown by themagenta parts of the curves in figure A1, we have not shownit again.The procedure just described yields an analytic RARby first finding r i as a function of r , and then writing r asa function of a bary . For m bary ( r ) given by equation (A13),it is also possible to do the opposite. I.e., equations (9)and (10) yield a quartic equation for r , which can be solvedanalytically to yield r/r i for any m i ( r i ). This solution for r MNRAS , 1–19 (0000)
AR in LCDM Figure A2.
Comparison of the actual profiles (symbols) withthe analytic approximations (curves) described in the text for thetwo halos considered in figure A1. For both halos, equation (A2)describes the initial profiles (squares) well. Equation (A13) de-scribes m bary (diamonds) of the less massive halo well out to asubstantial fraction of the virial radius, but the gas in the moremassive halo is less centrally concentrated, so the agreement isonly good out to the scale radius of the gas. can be inserted in equation (A16) to yield a bary and then∆ a ≡ a tot a bary − f d m i ( r i ) m ( r ) + m ( r ) (A17)= f d m i ( r i ) (1 + r /r )(1 + r /r ) F (1 + r /r ) + F (1 + r /r )yields the corresponding fully analytic (but messy!) RAR.Over the range where equation (A13) provides a good descrip-tion of m bary ( r ), the r i ( r ) and r ( r i ) approaches are almostindistinguishable. However, the r i ( r ) approach, in which m i is given by equation (A2) and m bary is arbitrary, is moreefficient (solve a cubic rather than quartic).We conclude that we have analytic understanding of allthe RAR scalings presented in the main text. MNRAS000