On the galaxy-halo connection in the EAGLE simulation
Harry Desmond, Yao-Yuan Mao, Risa Wechsler, Robert Crain, Joop Schaye
MMNRAS , 1–5 (2017) Preprint 3 July 2017 Compiled using MNRAS L A TEX style file v3.0
On the galaxy–halo connection in the EAGLE simulation
Harry Desmond , (cid:63) , Yao-Yuan Mao , , Risa H. Wechsler , , Robert A. Crain , andJoop Schaye Kavli Institute for Particle Astrophysics and Cosmology, Physics Department, Stanford University, Stanford, CA 94305, USA SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA Pittsburgh Particle Physics, Astrophysics, and Cosmology Center (PITT PACC), Pittsburgh, PA 15260, USA Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool, L3 5RF, UK Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, the Netherlands
ABSTRACT
Empirical models of galaxy formation require assumptions about the correlations betweengalaxy and halo properties. These may be calibrated against observations or inferred fromphysical models such as hydrodynamical simulations. In this
Letter , we use the EAGLE sim-ulation to investigate the correlation of galaxy size with halo properties. We motivate this anal-ysis by noting that the common assumption of angular momentum partition between baryonsand dark matter in rotationally supported galaxies overpredicts both the spread in the stellarmass–size relation and the anticorrelation of size and velocity residuals, indicating a prob-lem with the galaxy–halo connection it implies. We find the EAGLE galaxy population toperform significantly better on both statistics, and trace this success to the weakness of thecorrelations of galaxy size with halo mass, concentration and spin at fixed stellar mass. Usingthese correlations in empirical models will enable fine-grained aspects of galaxy scalings tobe matched.
Key words: galaxies: formation – galaxies: fundamental parameters – galaxies: haloes –galaxies: kinematics and dynamics – galaxies: statistics – dark matter
Accurate semi-analytic and empirical modelling of galaxy forma-tion is challenging, in part because the correlations of key galaxyand halo variables remain unknown. Observational manifestationsof these correlations include galaxy scaling relations, and throughdetailed investigations of these relations we may hope to buildknowledge of the galaxy–halo connection.Over the past decades, two models which have proven use-ful for capturing aspects of the galaxy–halo connection are subhaloabundance matching (SHAM; Kravtsov et al. 2004; Behroozi et al.2010), and the angular momentum model of Mo, Mao & White(1998, hereafter MMW). SHAM asserts a nearly monotonic rela-tionship between stellar mass and a halo proxy, establishing the de-pendence of galaxy mass on halo mass and concentration requiredto reproduce galaxy clustering (e.g. Conroy, Wechsler & Kravtsov2006; Reddick et al. 2013). The MMW model sets galaxy and halospecific angular momentum proportional and assumes galaxies’ ve-locities to be entirely rotational, making galaxy size a function ofgalaxy mass and halo mass, concentration and spin. This agreeswell with observed average galaxy sizes over a wide range of mass(Kravtsov 2013; Desmond & Wechsler 2015, hereafter DW15). (cid:63)
E-mail: [email protected]
Despite these successes, however, the conjunction of thesemodels (hereafter “SHAM + MMW”) is known to make incorrectpredictions for two properties of the galaxy population. The first isthe scatter s MSR in the stellar mass–size relation (MSR; de Jong &Lacey 2000; Gnedin et al. 2007). SHAM + MMW sets galaxy sizeproportional to halo spin, λ , and hence requires the scatter in size atfixed mass to be at least as large as that in λ . In fact, these scattersare ∼ . ∼ .
25 dex in observed galaxies and simulationsrespectively (DW15). The second is the correlation of residuals ofthe mass–size and mass–velocity relations ( ρ ∆ R − ∆ V ), which is negli-gible in observations but predicted to be negative (McGaugh 2005;Dutton et al. 2007; DW15). These discrepancies indicate that thegalaxy–halo correlations on which s MSR and ρ ∆ R − ∆ V depend are in-adequately captured by the model.This issue is relevant also for semi-analytic models. Manysuch models set galaxy size proportional to halo virial radius(e.g. Somerville et al. 2008; Croton et al. 2006; Lu et al. 2011),sometimes with a single value for all halo spins. Others that useadditional physical assumptions find important correlations of sizewith variables beyond halo mass and spin, but neglect the scatter insizes (e.g. Lu, Mo & Wechsler 2015). The empirical identificationof the aspects of the galaxy–halo connection responsible for real-istic size distributions – and correlations with velocity – will be ofuse in constraining such models and guiding the choice of inputs. c (cid:13) a r X i v : . [ a s t r o - ph . GA ] J un H. Desmond, Y.-Y. Mao, R. H. Wechsler, R. A. Crain and J. Schaye
The failure of SHAM + MMW may be due either to inaccura-cies in the properties of the halo populations on which the modelswere based (e.g. their neglect of baryonic physics), or incorrect pre-diction of the models themselves for the galaxy–halo connection.To resolve this dilemma, we turn in this
Letter to hydrodynamicalsimulations, which enable the prediction of galaxy properties with-out prior assumptions on galaxy–halo correlations. In particular,we investigate s MSR and ρ ∆ R − ∆ V in the EAGLE simulation (Schayeet al. 2015; Crain et al. 2015), which has previously been shownto match the galaxy size distribution as well as many other aspectsof galaxy phenomenology (Furlong et al. 2015). Sales et al. (2009)and Stevens et al. (2016) showed that the MMW model fails tomatch the output of the EAGLE simulation and its ancestor OWLS.Zavala et al. (2016) found the angular momentum of stars to corre-late with that of the inner halo in EAGLE, and Sales et al. (2012)reported weak correlation of galaxy properties with halo spin in therelated GIMIC simulation. This is in contrast with other simula-tions in which halo spin correlates more strongly with galaxy spinand morphology, especially at low mass (e.g. Teklu et al. 2015;Rodriguez-Gomez et al. 2017). Finally, Ferrero et al. (2016) stud-ied the EAGLE Tully–Fisher and mass–size relations.The structure of this paper is as follows. In Section 2 we de-scribe the EAGLE simulation and our methods to measure and ex-plore the origin of s MSR and ρ ∆ R − ∆ V . In Section 3.1 we show thatboth s MSR and ρ ∆ R − ∆ V are significantly nearer their observed val-ues in EAGLE than in the SHAM + MMW model, and close tothe predictions of SHAM alone. We show the success of EAGLEover SHAM + MMW to be due not to di ff erences in underlying haloproperties caused by baryons (Section 3.2), but rather to the corre-lations of halo variables with galaxy size (Section 3.3). In EAGLE,the sizes of low-redshift galaxies are only weakly correlated at fixedstellar mass with the mass, concentration and spin of their haloes,violating the assumption of angular momentum partition. Section 4discusses the broader implications of these results, and summarises. EAGLE is a recently completed set of cosmological hydrodynami-cal simulations, run with a modified version of G adget -3 (Springel2005) and including hydrodynamics, radiative cooling, star forma-tion, stellar feedback and black hole dynamics. The subgrid mod-els were calibrated against the present-day stellar mass functionand the normalisation of the mass–size relation. The simulationsused a flat Λ CDM cosmology with Ω m = . Ω b = . h = . σ = . n s = . z = darkmatter and gas particles from z =
127 to the present day in a boxwith comoving side length 100 Mpc, in addition to the correspond-ing dark matter only (DMO) run in which baryonic e ff ects wereswitched o ff . We refer the reader to Schaye et al. (2015) and Crainet al. (2015) for further information about the simulation. To enable direct comparison with the results of DW15, we per-form halo finding on both the DMO and hydrodynamical (hereafter“hydro”) runs of the EAGLE simulation using R ockstar (Behroozi http://eagle.strw.leidenuniv.nl et al. 2013). We define spin as λ ≡ J | E | / G − M − / , where J is a halo’s angular momentum and E its total energy (Peebles1969), and calculate concentration ( c ) using r s , klypin (derived from V max / V vir ; Klypin et al. 2001) rather than fitting an NFW profile.We include only dark matter when calculating c and λ . We multi-ply the DMO haloes’ virial masses by 1 − Ω b / Ω m to compare to thehydro haloes, where again we include dark matter only ( M DM ).Next, we match the DMO R ockstar catalogue to both the hy-dro R ockstar catalogue and the S ubfind catalogue (Springel et al.2001; Dolag et al. 2009) made by the EAGLE pipeline, as follows.Both halo finders produce a list of particles associated with eachhalo that they identify. Since the two runs share the same dark mat-ter particle IDs, we can match the haloes by finding common par-ticles. In practice, given a halo in the DMO run (halo A), we firstfind the halo (halo B) in the hydro run that contains the most parti-cles of halo A. If halo A also contains the most particles of halo B,we identify a “match” between them. Since the S ubfind catalogueof the hydro run also provides the connection between the haloesand galaxies, this method establishes a link between the haloes inthe DMO and hydro R ockstar catalogues, and the galaxies in theS ubfind catalogue. The fraction of haloes in the hydro run hostinggalaxies with M ∗ > M (cid:12) that are matched by our procedure is 91per cent; these haloes are not significantly biased in M DM , c or λ . We compare our models with the observations of Pizagno et al.(2007, hereafter P07) for compatibility with DW15. Althoughlarger samples with well-measured sizes now exist (e.g. Huang etal. 2017, Somerville et al. 2017), they produce similar MSRs. P07require an apparent axis ratio b / a ≤ . α rotationcurve, which they find not to significantly bias the admitted galaxypopulation in colour or concentration. We therefore do not makea morphology cut on the EAGLE galaxies in our fiducial analysis,although we have checked that our results change at no more thanthe ∼ σ level – and our qualitative conclusions remain unchanged– when only including galaxies with a substantial fraction of theirkinetic energy in ordered corotation ( κ co ≥ .
4; Correa et al. 2017).We compare the EAGLE results with two semi-empirical models,denoted “SHAM” and “SHAM + MMW” as in Section 1.For both data and models, we take s MSR to be the Gaussianscatter in radius of the best-fitting power-law relation between stel-lar mass ( M ∗ ) and half-mass radius ( R e ff ; measured for stars in a30 kpc aperture), over the range 9 < log( M ∗ / M (cid:12) ) < .
5. We haveverified that restricting to log( M ∗ / M (cid:12) ) >
10 does not a ff ect ourconclusions. We measure ρ ∆ R − ∆ V as the Spearman rank correlationcoe ffi cient of the ∆ R e ff − ∆ V max relation, where ∆ x denotes the resid-ual of quantity log( x ) after subtracting the value expected at that M ∗ given a power-law fit to the log( M ∗ ) − log( x ) relation, f x ( M ∗ ): ∆ x ≡ log( x ) − f x ( M ∗ ) . (1)We quantify the dependence of R e ff on halo variables with theSpearman correlation coe ffi cients ρ ∆ R − ∆ X , where X ∈ { M DM , c , λ } .We record in Table 1 the median and 1 σ spread of the statistics over100 Monte Carlo mock data sets of galaxies with M ∗ values within0.01 dex of those of the observational sample (see Section 3.1). Note that the use of a power law in this definition means that s MSR isincreased by curvature in the MSR; thus s MSR for the EAGLE relation (seeFig. 1a) may be considered an upper bound on the “true” intrinsic scatter.MNRAS , 1–5 (2017) n the galaxy–halo connection in the EAGLE simulation P07 EAGLE SHAM SHAM + MMW s MSR . ± . . ± . ρ ∆ R − ∆ V − . − . ± . − . ± . − . ± . ρ ∆ R − ∆ M DM – 0 . ± . . ± . ρ ∆ R − ∆ c – − . ± . − . ± . ρ ∆ R − ∆ λ – 0 . ± . . ± . Table 1.
Comparison of statistics of the galaxy–halo connection in ob-servations (P07), the EAGLE simulation, an abundance matching modelwith sizes chosen to match the stellar mass–size relation by construction(“SHAM”), and an analogous model with sizes set by angular momentumpartition (“SHAM + MMW”; Desmond & Wechsler 2015). s MSR is the scat-ter in size of the stellar mass–size relation, ρ denotes Spearman rank cor-relation coe ffi cient, and ∆ is defined in Eq. 1. Entries in italics are by con-struction. The SHAM + MMW model overpredicts both s MSR and | ρ ∆ R − ∆ V | due to the strong correlations it implies between R e ff and M DM , c and λ atfixed M ∗ . The EAGLE galaxy–halo connection, in which these variables areonly weakly correlated, performs significantly better on both statistics. ∆ R − ∆ V relations Figure 1a shows the MSR of the EAGLE galaxies, and Figure 1bthe correlation of their size and velocity residuals. That both EA-GLE relations are in approximate agreement with the P07 obser-vations is verified quantitatively in the first two rows of Table 1,which list the s MSR and ρ ∆ R − ∆ V values.The 3 rd and 4 th columns of Table 1 show analogous results fortwo alternative models. In “SHAM,” M ∗ is set by SHAM using the V peak proxy and 0.2 dex scatter (Reddick et al. 2013; varying theSHAM parameters within the bounds set by clustering has a negli-gible e ff ect on our results), and galaxy sizes are chosen randomlyfrom a normal distribution at given M ∗ to match the P07 MSRby construction. In “SHAM + MMW,” sizes are set by the MMWmodel after SHAM has been performed, using the procedure andbest-fitting parameter values of DW15.As mentioned in Section 1 (and discussed in detail inDW15), the SHAM + MMW model compares poorly with obser-vations in both s MSR and ρ ∆ R − ∆ V . This appears to be in conflictwith Somerville et al. (2017), who claim the model generates an s MSR in agreement with that of a compilation of GAMA and CAN-DELS data. However, they include only the contribution to s MSR from scatter in λ ( ∼ .
25 dex) and neglect the contributions fromscatter in M DM and c at fixed M ∗ . ρ ∆ R − ∆ V has contributions bothfrom baryonic mass (higher surface density means larger rotationvelocity), and from the dark matter, since in the MMW model moreconcentrated haloes, which generate larger rotation velocities, hostsmaller galaxies at fixed angular momentum. The SHAM model,which includes only the first contribution, predicts a ∆ R − ∆ V an-ticorrelation that is weaker but still stronger than the data’s. It isimportant to note, however, that these models assume negligiblevelocity dispersion σ . An decrease of σ/ V rot with λ – as producedin some hydrodynamical simulations (e.g. Rodriguez-Gomez et al.2017) – could reduce the predicted | ρ ∆ R − ∆ V | and s MSR . Only with theassumption that σ/ V rot does not vary systematically with λ does theMMW model follow uniquely from proportionality of galaxy andhalo specific angular momentum.We now investigate the origin of the di ff erence between theEAGLE and SHAM + MMW results.
A possible reason for the apparent failure of the SHAM + MMWmodel is its application in DW15 to haloes from an N-body simula-tion in which baryonic e ff ects were neglected. In Figure 2 we showthe fractional di ff erences in M DM , c and λ of all matched haloes inthe EAGLE DMO and hydro runs, and compare in the insets theiroverall distributions. We find the haloes to be a few per cent lessmassive on average in the hydro run, and their c and λ values tobe similar. (Schaller et al. 2015 reported larger di ff erences in halomass because they included baryons as well as dark matter in themass definition.) The spin distribution is slightly wider in the hy-dro run, which goes in the wrong direction to account for the lower s MSR in EAGLE than in the SHAM + MMW model.In Figure 3 we compare the correlations of λ with M DM and c in the two runs, finding them to be very similar. If spin was morepositively correlated with M DM or c with baryonic e ff ects included,then the corresponding increase in rotational velocity caused bydark matter for larger galaxies would compensate for the reductionin the rotation velocity caused by baryons, which could allow theSHAM + MMW model to agree with the measured ρ ∆ R − ∆ V . That wedo not find such an increased correlation leads us to conclude thatthe di ff erences between the EAGLE and SHAM + MMW models intheir predictions for s MSR and ρ ∆ R − ∆ V arise not from underlying darkmatter halo structure, but rather from di ff erences in the correlationsof galaxy and halo variables. It is to these that we now turn. Rows 3-5 of Table 1 record the Spearman rank coe ffi cients of thecorrelations between size residual ( ∆ R e ff ) and M DM , c and λ resid-ual in the EAGLE, SHAM, and SHAM + MMW models. As haloproperties cannot be observed, there are no corresponding entries inthe first column. By construction, the SHAM model does not cor-relate galaxy size with any halo property at fixed stellar mass. Asdescribed in Section 1, however, the SHAM + MMW model impliesa strong correlation of ∆ R e ff with ∆ λ and a strong anticorrelationwith ∆ c , and the latter in particular is responsible for the stronglynegative value of ρ ∆ R − ∆ V . In the EAGLE simulation, galaxy sizecorrelates only weakly with each halo variable, with the result thatthe predicted ρ ∆ R − ∆ V is similar to the SHAM case. In addition, thisprevents s MSR from receiving the full contributions from the scatterin halo variables at fixed M ∗ , allowing it to remain below the P07value. These correlations are shown explicitly in Figure 4. While the principle component of the galaxy–halo connection – therelation between galaxy mass and halo mass and concentration – isbecoming well constrained by abundance matching studies, sec-ondary components, such as the dependence of galaxy size on haloproperties, remain uncertain. A leading model for galaxy size (Moet al. 1998; MMW) assumes σ = s MSR ) and the strength of the correlation of sizeand velocity residuals ( ρ ∆ R − ∆ V ). This indicates a problem with thegalaxy–halo connection it implies. MNRAS , 1–5 (2017)
H. Desmond, Y.-Y. Mao, R. H. Wechsler, R. A. Crain and J. Schaye (a) M ∗ − R e ff (b) ∆ R e ff − ∆ V max Figure 1.
The M ∗ − R e ff relation and correlation of R e ff and V max residuals in the EAGLE simulation, compared to the observations of Pizagno et al. (2007). Asin DW15, stellar masses for the latter were taken from the NASA Sloan Atlas. The red lines in the left panel show the best-fitting power-law to the data, and itsscatter. In this plot and those that follow, points indicate medians and error bars 16 th and 84 th percentiles. We stack the 100 mock data sets (Section 2.3) to makecontour plots, and the levels enclose 90, 80, 60, 40 and 20 per cent of galaxies. The spread in the sizes of EAGLE galaxies is as low as is observed, and theycorrectly exhibit no significant ∆ R e ff − ∆ V max correlation. In Fig. 1b, the observations have ρ ∆ R − ∆ V = − .
07, and the EAGLE galaxies have ρ ∆ R − ∆ V = − . M DM (b) c (c) λ Figure 2.
The di ff erences in M DM , c and λ between all M ∗ > M (cid:12) haloes in the hydro runs of the EAGLE simulation and their counterparts in the DMOrun, as a function of the DMO variable. The insets compare the overall distributions (hydro in red and DMO in blue). With baryonic e ff ects included, M DM isreduced by a few per cent on average (the catalogue is incomplete for M DM (cid:46) M (cid:12) ), and λ increased slightly at low values. c is largely una ff ected. Figure 3. λ correlates in the same way with both M DM and c in the DMOand hydro runs of EAGLE. Together with Fig. 2, this shows that the di ff er-ence between the EAGLE and SHAM + MMW models in their predictionsfor s MSR and ρ ∆ R − ∆ V are not due to changes to the haloes caused by baryons.They must therefore be due to di ff erent galaxy–halo correlations. In this
Letter , we investigated this discrepancy in the contextof the EAGLE hydrodynamical simulation. We found the galaxypopulation in this simulation to exhibit near-agreement with mea-surements of both s MSR and ρ ∆ R − ∆ V . We showed that this di ff erencewith the SHAM + MMW prediction is due not to modifications tothe haloes themselves by baryons, but rather to the weakness of thecorrelations of galaxy size with M DM , c and λ . While the MMWmodel strongly correlates R e ff with M DM ( ρ = . c ( ρ = − . λ ( ρ = .
80) at fixed M ∗ , the Spearman rank coe ffi cients for thecorresponding EAGLE correlations are only 0 . − .
19 and 0 . σ .Our results have implications for both galaxy formation the-ory and semi-analytic and empirical modelling. On one hand, thebreakdown of the MMW model requires explanation. Galaxy prop-erties may become weakly correlated with halo spin due to stochas-tic transfer of angular momentum between baryons and dark matter,or a significant loss or redistribution through feedback or coolingprocesses (Brook et al. 2011; Zjupa & Springel 2016). On the otherhand, the EAGLE galaxy–halo correlations may be used to inform MNRAS , 1–5 (2017) n the galaxy–halo connection in the EAGLE simulation (a) ∆ R e ff − ∆ M DM (b) ∆ R e ff − ∆ c (c) ∆ R e ff − ∆ λ Figure 4.
The correlation of residuals of the stellar mass–size relation with dark matter mass, concentration and spin residuals in the hydro run of the EAGLEsimulation. Haloes were randomly selected from the catalogue to reproduce the stellar mass distribution of the P07 sample (see Section 2.3). In contrast to theSHAM + MMW model, EAGLE predicts these correlations to be weak, which accounts for the better agreement of the predicted s MSR and ρ ∆ R − ∆ V values withthe observations. The Spearman rank correlation coe ffi cients of these relations are 0 . − .
19 and 0 .
17, respectively (see Table 1). empirical models where galaxy sizes are added by hand. To match s MSR and ρ ∆ R − ∆ V , at least in a SHAM framework, R e ff should corre-late at most weakly with M DM , c and λ at fixed M ∗ . This is tacitlyassumed by several existing models (e.g. Dutton et al. 2011, 2013;Desmond & Wechsler 2016), and implied also by aspects of themass discrepancy–acceleration relation (Desmond 2017). We sug-gest such correlations be used by default from now on. Finally, ourresults facilitate the testing of galaxy formation theories: if a the-ory’s e ff ective galaxy–halo connection exhibits correlations com-patible with those of EAGLE, its success in matching the fine-grained statistics that we investigate here is assured. ACKNOWLEDGEMENTS / K00042X /
1, STFC capitalgrant ST / K00087X /
1, DiRAC Operations grant ST / K003267 / / / ERC Grant agreement 278594-GasAroundGalaxies. RACis a Royal Society University Research Fellow.
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