The morphology of star-forming gas and its alignment with galaxies and dark matter haloes in the EAGLE simulations
Alexander D. Hill, Robert A. Crain, Juliana Kwan, Ian G. McCarthy
MMNRAS , 1–23 (2021) Preprint 1 March 2021 Compiled using MNRAS L A TEX style file v3.0
The morphology of star-forming gas and its alignment with galaxies anddark matter haloes in the EAGLE simulations
Alexander D. Hill, ★ Robert A. Crain, Juliana Kwan and Ian G. McCarthy. Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, United Kingdom
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present measurements of the morphology of star-forming gas in galaxies from the EAGLE simulations, and its alignmentrelative to stars and dark matter (DM). Imaging of such gas in the radio continuum enables weak lensing experiments that comple-ment traditional optical approaches. Star-forming gas is typically more flattened than its associated stars and DM, particularly forpresent-day subhaloes of total mass ∼ − . M (cid:12) , which preferentially host star-forming galaxies with rotationally-supportedstellar discs. Such systems have oblate, spheroidal star-forming gas distributions, but in both less- and more-massive subhaloesthe distributions tend to be prolate, and its morphology correlates positively and significantly with that of its host galaxy’s stars,both in terms of sphericity and triaxiality. The minor axis of star-forming gas most commonly aligns with the minor axis of its hostsubhalo’s DM, but often aligns more closely with one of the other two principal axes of the DM distribution in prolate subhaloes.Star-forming gas aligns with DM less strongly than is the case for stars, but its morphological minor axis aligns closely with itskinematic axis, affording a route to observational identification of the unsheared morphological axis. The projected ellipticitiesof star-forming gas in EAGLE are consistent with shapes inferred from high-fidelity radio continuum images, and they exhibitgreater shape noise than is the case for images of the stars, owing to the greater characteristic flattening of star-forming gas withrespect to stars. Key words: galaxies: ISM – cosmology: large-scale structure of Universe – methods: numerical – gravitational lensing: weak– radio observations
The currently preferred Λ -cold dark matter ( Λ CDM) cosmogonyposits that the large-scale cosmic matter distribution (spatial scales (cid:38) ★ E-mail: [email protected] (ADH) hydrodynamical simulations of large cosmic volumes (e.g. Crain et al.2017; McCarthy et al. 2017; Springel et al. 2018), can be considereda remarkable corroboration of the Λ CDM paradigm. However, beingsubject to the rich array of dissipative physical processes that governtheir growth, galaxies inevitably represent imperfect tracers of theirlocal environment (e.g. Kaiser 1984; White et al. 1987), such thattheir baryonic components do not necessarily trace the shape andorientation of their DM haloes in a simple fashion.Besides their potential use as a means to place constraints on theill-understood microphysics of galaxy formation, and to reveal thenature of the environment of galaxies (e.g. Zhang et al. 2015; Codiset al. 2015), differences in the shape and orientation of baryoniccomponents of galaxies with respect to those of their DM haloes areof particular interest because they represent sources of uncertaintyin observational inferences of the morphology of DM haloes, and oftheir orientation with respect to the large-scale matter distribution(e.g. Troxel & Ishak 2015). This is of consequence for efforts to con-strain cosmological parameters via the shape correlation function ofgalaxies, a key aim of ongoing optical/near-infrared weak lensingsurveys such as the Canada-France-Hawaii Telescope Lensing Sur-vey (CFHTLens, Erben et al. 2013), Kilo-Degree Survey (KiDS, deJong et al. 2015), the Hyper Suprime Cam Subaru Strategic Program(HSC, Aihara et al. 2018) and the Dark Energy Survey (DES, TheDark Energy Survey Collaboration 2005), and ambitious forthcomingsurveys with the Vera Rubin Observatory (LSST Science Collabo-ration et al. 2009), the
Euclid spacecraft (Laureijs et al. 2012), and © a r X i v : . [ a s t r o - ph . GA ] F e b A. D. Hill et al. the
Nancy Grace Roman Space Telescope (e.g. Spergel et al. 2015).Moreover, the severity of differences between the shape and align-ment of haloes and those of the observable structures used to inferthem, has a strong bearing on the accuracy of weak gravitationallensing predictions derived from dark matter-only simulations. Atpresent, such simulations are the only means of modelling the evolu-tion of cosmic volumes comparable to those mapped out by lensingsurveys.Simplified techniques such as halo occupation distribution (HOD)modelling, subhalo abundance matching (SHAM) and semi-analyticmodels have, in order of increasing sophistication, proven valuablemeans of understanding the connection between galaxies and thematter distribution (see e.g. Schneider & Bridle 2010; Joachimi et al.2013). However, such methods have been shown to exhibit significantsystematic differences with respect to the predictions of cosmologicalhydrodynamical simulations on small-to-intermediate spatial scales(e.g. Chaves-Montero et al. 2016; Springel et al. 2018), in large partbecause they (by design) do not self-consistently capture the back-reaction of baryon evolution on the structure of DM haloes (Bettet al. 2010; Guo et al. 2016). A comprehensive understanding of theinfluence of systematic uncertainties stemming from the differencesin the shape and orientation of galaxies and their host haloes there-fore requires self-consistent and realistic physical models of galaxyformation in a fully cosmological framework.There is a rich history of the use of numerical simulations toestablish the correspondence between the morphology, angular mo-mentum and orientation of galaxies, their satellite systems and theirhost DM haloes, with particular emphases on the roles played by gasaccretion (e.g. Chen et al. 2003; Sharma & Steinmetz 2005; Saleset al. 2012), mergers (e.g. Dubinski 1998; Boylan-Kolchin et al. 2006;Naab et al. 2006) and environment (e.g. Croft et al. 2009; Hahn et al.2010; Shao et al. 2016). However, prior studies have tended to sufferfrom one or more significant shortcomings, namely relatively poorspatial and mass resolution, relatively small sample sizes, and a poorcorrespondence between the properties of simulated galaxies withobserved counterparts. These shortcomings are significantly amelio-rated by the current generation of state-of-the-art hydrodynamicalsimulations, such as EAGLE (Schaye et al. 2015; Crain et al. 2015),HorizonAGN (Dubois et al. 2014), Illustris/IllustrisTNG (Vogels-berger et al. 2014; Pillepich et al. 2018, e.g.) and MassiveBlack-II(Khandai et al. 2015). Each of these simulations broadly reproduceskey observed properties of the present-day galaxy population, thusengendering confidence that they capture (albeit with varying de-grees of accuracy) the complexity of the interaction between thebaryonic components of galaxies and their DM haloes. The sim-ulations each follow a cosmological volume sufficient to a yieldrepresentative galaxy population ( ∼ cMpc ), and do so with amass resolution ( ∼ M (cid:12) ) and spatial resolution ( ∼ 𝐿 ★ galaxies. Moreover, they capture important second-order effects suchas the back-reaction of baryons on the structure and clustering of DMhaloes.The emergence of optical weak lensing surveys as a promisingmeans of constraining the nature of DM and dark energy has intensi-fied the need to assess the severity of systematic uncertainties afflict-ing cosmic shear measurements (specifically, the galaxy shape cor-relation function). Cosmological hydrodynamical simulations haveproven a valuable tool for this purpose, highlighting that galaxiescan be significantly misaligned with respect to their DM haloes (e.g.Bett et al. 2010; Hahn et al. 2010; Bett 2012; Tenneti et al. 2014;Velliscig et al. 2015a; Shao et al. 2016; Chisari et al. 2017) and thatthe shapes and alignments of galaxies and their haloes are corre- lated over large distances via tidal forces (Tenneti et al. 2014; Chisariet al. 2015; Codis et al. 2015; Velliscig et al. 2015b). The simu-lations have also been exploited to examine the morphological andkinematic alignment of galaxies with the cosmic large-scale structure(see e.g. Cuesta et al. 2008; Codis et al. 2018). The current generationof state-of-the-art simulations remains reliant on the use of subgridtreatments of many of the key physical processes governing galaxyevolution and, as noted by Joachimi et al. (2015) the details of theirparticular implementation can in principle influence the alignmentof cosmic structures (see also Velliscig et al. 2015a). However, inkey respects the simulations appear to be quantitatively compatiblewith extant observational constraints, e.g. the 𝑤 g + correlation func-tion of luminous red galaxies in SDSS and their analogues in theMassiveBlack-II simulation (Tenneti et al. 2015, their Fig. 21).A complementary approach to optical/near-IR weak lensing sur-veys is to measure shear at radio frequencies. The concept has beendemonstrated both by exploiting very large area, low source densityradio data (Chang et al. 2004), and deep, pointed observations withgreater source density (Patel et al. 2010). Ambitious future radiocontinuum surveys such as those envisaged for the Square KilometreArray (SKA) may therefore prove to be competitive with the largestoptical surveys. Brown et al. (2015) argue that, in the most optimisticcase, a full Phase-2 SKA survey over 3 𝜋 steradians would yield twicethe areal coverage of the Euclid ‘wide survey’, with a similar sourcedensity of (cid:39)
30 galaxies arcmin − . The characteristic redshift of sen-sitive radio continuum surveys may also prove to be significantlygreater than that of optical counterparts. By bridging the gap be-tween traditional shear measurements and those derived from mapsof the cosmic microwave background (CMB) radiation, radio weaklensing surveys offer the promise of tomographic mapping of cosmicstructure evolution in both the quasi-linear and strongly non-linearregimes. Shear mapping in the radio regime offers advantageouscomplementarity with optical surveys, in particular to suppress keysystematic uncertainties. For example, the use of kinematic and/orpolarisation information may enable improved characterisation of theintrinsic (unsheared) ellipticity, and suppress the influence of intrin-sic alignment, the deviation from random of the observed ellipticityof a sample (Blain 2002; Morales 2006; de Burgh-Day et al. 2015;Whittaker et al. 2015).Shear measurements in the radio regime are derived from imagesof the extended radio continuum emission from galaxies, which effec-tively traces the star-forming component of the interstellar medium(ISM). The morphology and kinematics of this component, and theirrelationship with those of the underlying DM distribution, can inprinciple differ markedly from the analogous quantities traced by thestellar component imaged by conventional lensing surveys. However,by design, leading models of the radio continuum sky (e.g. Wilmanet al. 2008; Bonaldi et al. 2019) do not account for such differences.This therefore motivates an extension of prior examinations of therelationship between galaxies and the overall matter distribution,and correlation of shapes and alignments of galaxies separated overcosmic distances, focusing on the use of the star-forming ISM to char-acterise the morphology and orientation of galaxies. The current gen-eration of state-of-the-art cosmological hydrodynamical simulationsare well suited to this application since, as for the stellar component,they self-consistently model the evolution of star-forming gas withingalaxies, including cosmological accretion from the intergalactic andcircumgalactic media (IGM and CGM, respectively), expulsion byfeedback processes, and its interaction with a dynamically ‘live’ DMhalo.In this study, we use the cosmological hydrodynamical simula-tions of the EAGLE project (Schaye et al. 2015; Crain et al. 2015) MNRAS , 1–23 (2021) orphology and alignment of star-forming gas to examine the correspondence between the morphology and orien-tation of the star-forming ISM of galaxies and those of their parentDM haloes. EAGLE is well suited to this application: although thesimulations do not explicitly model the balance between molecular,atomic and ionised hydrogen, the use of empirical or theoretical mod-els to partition gas into these phases indicates that the simulationsbroadly reproduce key properties of the atomic and molecular reser-voirs of galaxies (see e.g. Lagos et al. 2015; Bahé et al. 2016; Crainet al. 2017; Davé et al. 2020) including, crucially, the ‘fundamentalplane of star formation’ that relates their stellar mass, star formationrate and neutral hydrogen fraction (Lagos et al. 2016). This studycomplements prior examinations of the morphology of stars, hot gasand DM in the EAGLE simulations (e.g. Velliscig et al. 2015a,b;Shao et al. 2016). The morphology of the star-forming ISM of galax-ies in the IllustrisTNG-50 simulation (hereafter TNG50) was alsoexamined by Pillepich et al. (2019); whilst the motivation for thatstudy was quite different to that of ours, their findings are of directrelevance and offer an opportunity to assess the degree of concensusbetween different simulations.This paper is structured as follows. We discuss our numericalmethods in Section 2, as well as summarising briefly details of theEAGLE simulation and galaxy finding algorithms, and our sampleselection criteria. In Section 3 we examine the morphology of star-forming gas and its dependence on subhalo mass and redshift. InSection 4 we examine the internal alignment of star-forming gas withDM and stars, and its mutual alignment with its kinematic axis, againas a function of subhalo mass and redshift. In Section 5 we investigatethe shapes and alignments of the various matter components in 2D.In Section 6 we discuss and summarise our findings. In a series ofappendices, we examine the influence of a series of numerical andmodelling factors on our findings. In this section we briefly introduce the EAGLE simulation (Section2.1) and key numerical techniques for identifying haloes and sub-haloes (Section 2.2), and for characterising their morphology withshape parameters (Section 2.3). Our sample selection criteria arediscussed in Section 2.4. Detailed descriptions of the simulationsare provided by many other studies using them, so we present only aconcise summary of the most relevant aspects and refer the interestedreader to the project’s reference articles (Schaye et al. 2015; Crainet al. 2015).
The EAGLE project (the Evolution and Assembly of GaLaxies andtheir Environments) comprises a suite of hydrodynamical simula-tions that model the formation and evolution of galaxies and the cos-mic large-scale structure in a Λ CDM cosmogony (Schaye et al. 2015;Crain et al. 2015). Particle data, and derived data products, from thesimulations have been released to the community as detailed byMcAlpine et al. (2016). The simulations were evolved evolved with amodified version of the Tree-Particle-Mesh (TreePM) smoothed par-ticle hydrodynamics (SPH) solver Gadget-3 last described by (lastdescribed by Springel 2005). The main modifications include theimplementation of the pressure-entropy formulation of SPH intro-duced by Hopkins (2013), a time-step limiter as proposed by Durier& Dalla Vecchia (2012), switches for artificial viscosity and artificialconduction, as per Cullen & Dehnen (2010) and Price (2008), re-spectively, and the use of the Wendland (1995) 𝐶 smoothing kernel. Identifier
𝐿 𝑁 𝑚 g 𝜖 com 𝜖 phys [cMpc] [ M (cid:12) ] [ckpc] [pkpc]L025N0376 25 376 . × . × . × Table 1.
The box sizes and resolution details of the EAGLE simulations usedin this study. The columns are: comoving box side length, 𝐿 ; number of DMpartices (there is initially an equal number of baryon particles); the initialbaryon particle mass; the Plummer-equivalent gravitational softening lengthin comoving units; the maximum proper softening length. The influence of these developments on the properties of the galaxypopulation yielded by the simulations is explored by Schaller et al.(2015).EAGLE includes subgrid treatments of several physical processesthat are unresolved by the simulations. These include element-by-element radiative heating and cooling of 11 species (Wiersma et al.2009a) in the presence of a spatially-uniform, temporally-evolvingUV/X-ray background radiation field (Haardt & Madau 2001) and thecosmic microwave background (CMB); a model for the treatment ofthe multiphase ISM as a single-phase fluid with a polytropic pressurefloor (Schaye & Dalla Vecchia 2008); a metallicity-dependent den-sity threshold above which gas becomes eligible for star formation(Schaye 2004), with a probability of conversion dependent on the gaspressure (Schaye & Dalla Vecchia 2008); stellar evolution and massloss (Wiersma et al. 2009b); the seeding of BHs and their growth viagas accretion and mergers (Springel et al. 2005a; Booth & Schaye2009; Rosas-Guevara et al. 2015); and feedback associated with theformation of stars (Dalla Vecchia & Schaye 2012) and the growth ofBHs (Booth & Schaye 2009; Schaye et al. 2015). The simulationsadopt the stellar initial mass function (IMF) of Chabrier (2003). Theefficiency of stellar feedback was calibrated to reproduce the stellarmass function of the low-redshift galaxy population and, broadly,the sizes of local disc galaxies. The efficiency of AGN feedback wascalibrated to reproduce the present-day scaling relation between thestellar mass and central black hole mass of galaxies. The gaseousproperties of galaxies and their haloes were not considered duringthe calibration.EAGLE adopts values of the cosmological parameters derivedfrom the initial Planck data release (Planck Collaboration et al. 2014),namely Ω = . Ω b = . Ω Λ = . 𝜎 = . 𝑛 s = . ℎ = . 𝑌 = . 𝐿 =
100 cMpc, realised with 𝑁 = collisionless DM parti-cles of mass 𝑚 DM = . × M (cid:12) , and an initially equal num-ber of baryonic particles of mass 𝑚 g = . × M (cid:12) . ThePlummer-equivalent gravitational softening length is 1/25 of themean interparticle separation ( 𝜖 com = .
66 ckpc), limited to a max-imum proper length of 𝜖 com = . 𝐿 =
25 cMpc realised with 𝑁 = particles of each species,with masses 𝑚 DM = . × M (cid:12) and 𝑚 g = . × M (cid:12) .For these simulations the Plummer-equivalent gravitational softeninglength is 𝜖 com = .
33 ckpc, limited to a maximum proper length of 𝜖 com = .
35 pkpc. A summary of the simulations used in this paperis given in Table 1.The standard-resolution simulations marginally resolve the Jeans
MNRAS , 1–23 (2021)
A. D. Hill et al. scales at the density threshold for star formation in the warm anddiffuse photoionised ISM. They hence lack the resolution to modelthe cold, dense phase of the ISM explicitly, and so impose a tempera-ture floor to inhibit the unphysical fragmentation of star-forming gas.This floor takes the form 𝑇 eos ( 𝜌 ) , corresponding to the equation ofstate 𝑃 eos ∝ 𝜌 / normalised to 𝑇 eos = × K at 𝑛 H = − cm − .The temperature of star-forming gas thus reflects the effective pres-sure of the ISM, rather than its actual temperature. A drawback ofthe use of this floor is the suppression of the formation gas discswith scale heights much less than Jeans length of the gas on thetemperature floor ( ∼ 𝑚 dm / 𝑚 b ≡ ( Ω − Ω b )/ Ω b (cid:39) .
4. Thevertical support of the disc may also have physical causes, such asturbulence stemming from gas accretion and energy injection fromfeedback (Benítez-Llambay et al. 2018), although it is likely that thethese influences are artificially strong in the simulations. Thereforewe caution that both the gas and stellar discs of galaxies in EAGLEare generally thicker than their counterparts in nature (see also Tray-ford et al. 2017). We note however that these effects are unlikely toinfluence significantly the mutual alignment of the stellar and gaseousdiscs, nor their alignment with their parent DM halo.
We define galaxies as the cold baryonic component of gravitationallyself-bound structures, identified by the application of the subfind al-gorithm (Springel et al. 2001; Dolag et al. 2009) to DM haloes firstidentified with the friends-of-friends (FoF) algorithm (with a link-ing length of 0.2 times the mean interparticle separation). Subhaloesare identified as overdense regions in the FoF halo bounded by sad-dle points in the density distribution. Within a given FoF halo, thesubhalo comprising the particle (of any type) with the lowest gravi-tational potential energy is defined as the central subhalo, others arethen satellites.The position of galaxies is defined as the location of the particlein their subhalo with the lowest gravitational potential energy. Theposition of the central galaxy is used as a centre about which to com-pute the spherical overdensity mass (see Lacey & Cole 1993), 𝑀 ,for the adopted enclosed density contrast of 200 times the criticaldensity, 𝜌 c . In general, the properties of galaxies are computed byaggregating the properties of the appropriate particles located within30 pkpc of the galaxy centre, as this yields stellar masses compa-rable to those recovered within a projected circular aperture of thePetrosian radius (see Schaye et al. 2015). Following Thob et al. (2019), we obtain quantitative descriptionsof the morphology of galaxies and their subhaloes by modellingthe spatial distribution of their constituent particles as ellipsoids,characterised by their sphericity , 𝑆 = 𝑐 / 𝑎 , and triaxiality, 𝑇 = ( 𝑎 − 𝑏 )/( 𝑎 − 𝑐 ) , parameters, where 𝑎 , 𝑏 and 𝑐 are, respectively, the Thob et al. (2019) used the flattening, 𝜖 = − 𝑆 , rather than the sphericitybut, as is clear from their definitions, the two are interchangeable. moduli of the major, intermediate and minor axes of the ellipsoid .Therefore 𝑆 = 𝑆 = 𝑇 correspond, respectively, to oblate and prolate ellipsoids.Axis lengths are given by the square root of the eigenvalues ofa matrix describing the three-dimensional mass distribution of theparticles in question. The simplest choice is the mass distributiontensor (e.g. Davis et al. 1985; Cole & Lacey 1996), defined as: 𝑀 𝑖 𝑗 = (cid:205) 𝑝 𝑚 𝑝 𝑟 𝑝,𝑖 𝑟 𝑝, 𝑗 (cid:205) 𝑝 𝑚 𝑝 , (1)where the sum runs over all particles, 𝑝 , comprising the structure, 𝑟 𝑝,𝑖 denotes the 𝑖 th component ( 𝑖, 𝑗 = , ,
2) of each particle’s coordinatevector with respect to the galaxy centre, and 𝑚 p is the particle’s mass.As has been widely noted elsewhere, the mass distribution tensor isoften referred to as the moment of inertia tensor, as the two sharecommon eigenvectors.There are several well-motivated alternative choices to the massdistribution tensor and, as per Thob et al. (2019), we elect here to usean iterative form of the reduced inertia tensor (see also Dubinski &Carlberg 1991; Bett 2012; Schneider et al. 2012). The reduced formis advantageous because its suppresses a potentially strong influenceon the tensor of structural features in the outskirts of galaxies, bydown-weighting the contribution of particles at a large (ellipsoidal)radius. The use of an iterative scheme is further advantageous becauseit enables the scheme to adapt to particle distributions that deviatesignificantly from the initial particle selection. Since the latter isusually (quasi-)spherical, this is particularly relevant for stronglyflattened or triaxial systems. This form of the tensor is thus: 𝑀 𝑟𝑖 𝑗 = (cid:205) 𝑝 𝑚 𝑝 ˜ 𝑟 𝑝 𝑟 𝑝,𝑖 𝑟 𝑝, 𝑗 (cid:205) 𝑝 𝑚 𝑝 ˜ 𝑟 𝑝 , (2)where ˜ 𝑟 𝑝 is the ellipsoidal radius. In the first iteration, all particlesof the relevant species within a spherical aperture of a prescribed ra-dius are considered. This yields a initial estimate of the axis lengths( 𝑎, 𝑏, 𝑐 ). In the next iteration, particles satisfying the following con-dition relating to the ellipsoidal distance are considered: (cid:101) 𝑟 𝑝 ≡ 𝑟 𝑝,𝑎 (cid:101) 𝑎 + 𝑟 𝑝,𝑏 (cid:101) 𝑏 + 𝑟 𝑝,𝑐 (cid:101) 𝑐 ≤ , (3)where (cid:101) 𝑎, (cid:101) 𝑏 and (cid:101) 𝑐 are the re-scaled axis lengths calculated as (cid:101) 𝑎 = 𝑎 × 𝑟 𝑝 /( 𝑎𝑏𝑐 ) / . This ensures the ellipsoid maintains a constantvolume; in this respect, we differ from the scheme used by Thobet al. (2019), who maintained a constant major axis length betweeniterations. We opt for this scheme to avoid artificial suppressionof the major axis in cases of highly-flattened geometry, which ismore common when examining star-forming gas than is the case forstellar distributions. Iterations continue until the fractional change inthe axis ratios 𝑐 / 𝑎 and 𝑏 / 𝑎 falls below 1 percent. If this criterionis not satisfied after 100 iterations, or if the number of particlesenclosed by the ellipsoid falls below 10, the algorithm is deemedto have failed and the object’s morphology is declared unclassified.We find a failure to converge only in cases of low particle number(e.g. subhaloes with very few gas or star particles) and, crucially,our selection criteria (Section 2.4) ensure that no subhaloes withunclassified morphologies are included in our sample. Thob et al. (2019) present publicly-available Python routines for this pro-cedure at https://github.com/athob/morphokinematics.MNRAS000
2) of each particle’s coordinatevector with respect to the galaxy centre, and 𝑚 p is the particle’s mass.As has been widely noted elsewhere, the mass distribution tensor isoften referred to as the moment of inertia tensor, as the two sharecommon eigenvectors.There are several well-motivated alternative choices to the massdistribution tensor and, as per Thob et al. (2019), we elect here to usean iterative form of the reduced inertia tensor (see also Dubinski &Carlberg 1991; Bett 2012; Schneider et al. 2012). The reduced formis advantageous because its suppresses a potentially strong influenceon the tensor of structural features in the outskirts of galaxies, bydown-weighting the contribution of particles at a large (ellipsoidal)radius. The use of an iterative scheme is further advantageous becauseit enables the scheme to adapt to particle distributions that deviatesignificantly from the initial particle selection. Since the latter isusually (quasi-)spherical, this is particularly relevant for stronglyflattened or triaxial systems. This form of the tensor is thus: 𝑀 𝑟𝑖 𝑗 = (cid:205) 𝑝 𝑚 𝑝 ˜ 𝑟 𝑝 𝑟 𝑝,𝑖 𝑟 𝑝, 𝑗 (cid:205) 𝑝 𝑚 𝑝 ˜ 𝑟 𝑝 , (2)where ˜ 𝑟 𝑝 is the ellipsoidal radius. In the first iteration, all particlesof the relevant species within a spherical aperture of a prescribed ra-dius are considered. This yields a initial estimate of the axis lengths( 𝑎, 𝑏, 𝑐 ). In the next iteration, particles satisfying the following con-dition relating to the ellipsoidal distance are considered: (cid:101) 𝑟 𝑝 ≡ 𝑟 𝑝,𝑎 (cid:101) 𝑎 + 𝑟 𝑝,𝑏 (cid:101) 𝑏 + 𝑟 𝑝,𝑐 (cid:101) 𝑐 ≤ , (3)where (cid:101) 𝑎, (cid:101) 𝑏 and (cid:101) 𝑐 are the re-scaled axis lengths calculated as (cid:101) 𝑎 = 𝑎 × 𝑟 𝑝 /( 𝑎𝑏𝑐 ) / . This ensures the ellipsoid maintains a constantvolume; in this respect, we differ from the scheme used by Thobet al. (2019), who maintained a constant major axis length betweeniterations. We opt for this scheme to avoid artificial suppressionof the major axis in cases of highly-flattened geometry, which ismore common when examining star-forming gas than is the case forstellar distributions. Iterations continue until the fractional change inthe axis ratios 𝑐 / 𝑎 and 𝑏 / 𝑎 falls below 1 percent. If this criterionis not satisfied after 100 iterations, or if the number of particlesenclosed by the ellipsoid falls below 10, the algorithm is deemedto have failed and the object’s morphology is declared unclassified.We find a failure to converge only in cases of low particle number(e.g. subhaloes with very few gas or star particles) and, crucially,our selection criteria (Section 2.4) ensure that no subhaloes withunclassified morphologies are included in our sample. Thob et al. (2019) present publicly-available Python routines for this pro-cedure at https://github.com/athob/morphokinematics.MNRAS000 , 1–23 (2021) orphology and alignment of star-forming gas For consistency with the aperture generally used when computinggalaxy properties by aggregating particle properties (see e.g. Sec5.1.1. of Schaye et al. 2015), we adopt a radius of 𝑟 =
30 pkpc forthe initial spherical aperture. We use this aperture for all three mattertypes, star-forming gas, stars and DM, and note that for the latter, thisfocusses our morphology measurements towards halo centres, sincehaloes are in general much more extended than their cold baryons (seeSection 2.5). We retain the use of this aperture for the DM componentin order to focus on the DM structure local to star-forming gas discs,and note that the global morphology of DM haloes in EAGLE waspresented by Velliscig et al. (2015a). In Section 5 we examine the 2-dimensional projected morphology and alignment of galaxies. Whenperforming these measurements for star-forming gas and stars, weuse an initial circular aperture of 𝑟 = max (
30 pkpc , 𝑟 / , SF ) , where 𝑟 / , SF is the half-mass radius of star-forming gas bound to thesubhalo. This ensures a robust morphological characterisation of theimage projected by the most extended gas discs when viewed closeto a face-on orientation.Equation 2 can be generalised to be weighted by any particlevariable, rather than its mass. To crudely mimic the morphology ofcontinuum-luminous regions, when computing the tensor for star-forming gas, we weight by their star formation rate (SFR) rather thantheir mass, since it is well-established that the relationship betweenSFR and radio continuum luminosity is broadly linear (see e.g. Con-don 1992; Schober et al. 2017). We do not consider radio continuumemission due to AGN, since this is not extended. Pillepich et al.(2019) recently employed a similar approach to assess the morphol-ogy and alignment of H 𝛼 -luminous regions of star-forming galaxiesin the TNG50 simulation, via the use of the SFR as a proxy for theH 𝛼 luminosity. The recovered shape parameters and orientation arelittle changed with respect to the use of particle mass as the weight-ing variable, or indeed a uniform weighting, largely because the SFRof particles scales as (cid:164) 𝑚 ★ ∝ 𝑃 / for a Kennicutt-Schmidt law withindex 𝑛 s = . 𝑧 =
0, the 10 th and 90 th percentiles of the pressure of star-formingparticles in the Ref-L100N1504 volume spans less than two decadesin dynamic range.We define the orientation of galaxies and subhaloes as the unitvector parallel to the minor axis of the best-fit ellipsoid, and hencemeasure the relative alignment of structures as the angle betweenthese unit vectors. We note that it is more typical in the literature touse the unit vector parallel to the major axis; this is arguably the best-motivated choice for describing the alignment of systems that are ingeneral prolate (e.g. DM haloes), since in such systems the majoraxis is the most ‘distinct’. In contrast, it is the minor axis that is themost distinct in systems that are preferentially oblate, as is the casefor a flattened disc. In Section 4.2 we examine the correspondencebetween the morphological and kinematic axes of the star-forminggas distribution; we define the latter as the unit vector parallel to theangular momentum vector of all star-forming gas particles locatedwithin 30 pkpc of the galaxy centre. We identify subhaloes comprising a minimum of 100 each of star-forming gas particles, stellar particles and DM particles. This nu-merical threshold is motivated by tests, presented in Appendix B,that assess the fractional error on shape parameters induced whenperforming the measurement on sub-samples, randomly-selected andof decreasing size, of the particles comprising exemplar subhaloes.These tests indicate that a minimum of 100 particles are needed to − . − . − . − . − . . r/ < r > . . . . . . M ( < r ) / M ( < r ) SF - GasStarsDM10 M (cid:12) M (cid:12) M (cid:12) Figure 1.
Mean, spherically-averaged cumulative radial mass distributionprofiles of the star-forming gas (blue curve), stars (red) and DM (green) ofcentral subhaloes within our sample that have a halo mass 𝑀 ∼ M (cid:12) (solid curve), 10 M (cid:12) (dashed) and 10 M (cid:12) (dotted). The distributions arenormalised relative to the total mass of each component within 𝑟 . Star-forming gas is much more centrally concentrated than dark matter at allmasses. recover a measurement error of the flattening of star-forming gasdiscs of less than 10 percent, when using the iterative reduced inertiatensor. As noted by Thob et al. (2019), the sphericity and triaxi-ality shape parameters are poor descriptors of systems that deviatestrongly from axisymmetry, so we excise subhaloes with stronglynon-axisymmetric star-forming gas distributions. We quantify thischaracteristic by adapting the method of Trayford et al. (2019), bin-ning the mass of star-forming gas into pixels of solid angle about thegalaxy centre using Healpix (Górski et al. 2005). The asymmetry ofthe star-forming gas distribution, 𝐴 , is then computed by summingthe (absolute) mass difference between diametrically-opposed pixelsand normalising by the total star-forming gas mass. As per Trayfordet al. (2019), we use coarse maps of 12 pixels, and exclude systemswith 𝐴 SFG3D > .
6. This criterion excises 534 subhaloes, mostly oflow mass, and leaves us with a sample of 6,764 subhaloes at 𝑧 = ∼ M (cid:12) and a min-imum SFR of (cid:39) × − M (cid:12) yr − , where the latter assumes thestar-forming particles have a density of 0 . − and pressure cor-responding to a temperature of 8000 K. These criteria result in theselection of approximately (0.1, 10, 80) percent of all subhaloes ofmass log ( 𝑀 sub / M (cid:12) ) ∼ ( , , ) , respectively. Prior studies have demonstrated that the shape and orientation ofstars and DM in haloes can vary significantly as a function of radius(see e.g. Velliscig et al. 2015a). Fig. 1 shows the mean, spherically-averaged, cumulative radial mass distribution profiles of the star-
MNRAS , 1–23 (2021)
A. D. Hill et al. forming gas (blue curve), stars (red) and DM (green) comprisingpresent day central subhaloes with halo mass in ranges 𝑀 ∼ M (cid:12) (solid curves), 10 M (cid:12) (dashed) and 10 M (cid:12) (dotted). Asmight be naïvely expected, the baryonic components are much morecentrally concentrated than the DM, in each of the subhalo massbins: the median half-mass radius of star-forming gas is ( , . , . ) percent of 𝑟 for the low, middle and high mass bins respectively,compared with ( , , ) percent of 𝑟 for the DM . Owing tothis central concentration of the star-forming gas, we do not considerhere how the shape parameters of the star-forming gas distributionchange in response to the use of an initial aperture that envelops anever-greater fraction of the virial radius. We begin with an examination of the morphology of star-forminggas associated with subhaloes. To illustrate visually how the methoddescribed in Section 2.3 yields shape and orientation diagnosticsfor the simulated galaxies, we show in Fig. 2 the star formation ratesurface density, Σ SFR , of star-forming gas (upper row), in face-on andedge-on views, and the mass surface density of stars ( Σ ★ , bottom left)and DM ( Σ DM , bottom right) of a present-day star-forming galaxyfrom Recal-L025N0752. The galaxy is taken from the high-resolutionRecal-L025N0752 run, and its stellar mass is 𝑀 ★ = . × M (cid:12) ,with a subhalo mass of 𝑀 sub = . M (cid:12) . The galaxy’s sSFR is (cid:164) 𝑀 ★ = − . M (cid:12) yr − , and it exhibits reasonably strong rotationalsupport: stars residing within 30 pkpc of its centre of potential havea significant fraction of their kinetic energy invested in corotation( 𝜅 ★ co = . 𝜅 ★ co > . 𝜅 SFco = . ( Σ ★ ) =
6, 7, 8 M (cid:12) kpc − andlog ( Σ DM ) = . , . , .
75 M (cid:12) kpc − , respectively.As expected for a galaxy whose gas disc has strong rotational sup-port, the star-forming gas distribution is much more flattened thanthe corresponding distributions of stars and DM. In this example,the distributions of the three matter components are well aligned:the minor axis of the star-forming gas is misaligned with respect tothat of the stars by (cid:39) (cid:39) The figures for the low subhalo mass bin are significantly influenced byour sample selection criteria: removal of the minimum particle number cri-terion results in the inclusion of systems with less-extended star-forming gasdistributions, and further reduces the characteristic half-mass radius of thestar-forming gas. the minor axis in this example, as naïvely expected for an extended,rotationally-supported disc.
Fig. 3 shows probability distribution functions (PDFs) of the shapeparameters of the star-forming gas (blue curves), stellar (red) andDM (green) distributions of the subhaloes comprising our samplefrom Ref-L100N1504 at 𝑧 =
0. We reiterate that measurements ofthe stellar and DM distributions are included here, despite being pre-viously presented for EAGLE subhaloes by Velliscig et al. (2015a),because we use an alternative form of the mass distribution tensor.Thick and thin lines represent the sphericity and triaxiality param-eters respectively. Each panel shows subhaloes split by total massin bins of 0.5 dex, spanning 𝑀 sub = − M (cid:12) . For clarity,the PDFs of triaxiality have been artificially elevated in the verticalaxis by an increment of 0.4. Down arrows denote the median valueof each distribution. The bottom-right panel shows the volumetricsubhalo mass function, split into central and satellite subhalo popu-lations, highlighting that the sample is dominated by central galaxiesat all subhalo masses except for the lowest-mass bin. For clarity, wealso show the median values of the shape parameters for star-forminggas as a function of subhalo mass in Fig. 4. The solid and dashedcurves of that plot correspond to the samples, identified as discussedin Section 2.4, at 𝑧 = 𝑧 =
1, respectively. The lower panelof the figure shows the subhalo volumetric mass function at the twoepochs.These figures show that the distribution of sphericities of star-forming gas distributions is peaked at relatively low values for allsubhalo masses, but with a long tail towards high 𝑆 (i.e. quasi-spherical systems). The median value of the distributions, which isqualitatively similar to the peak value of the distribution, declinesfrom ˜ 𝑆 (cid:39) .
25 for subhaloes of 𝑀 sub ∼ M (cid:12) , to a minimum of˜ 𝑆 (cid:39) . 𝑀 sub ∼ . M (cid:12) . The sphericity of the star-forming gasis therefore systematically lower than is the case for that of the stars,and much more so than is the case for the DM, consistent with thenaïve expectation that this dissipational component is found primar-ily in flattened discs. Broadly, the peaks of the sphericity PDFs ofstars and DM are found at 𝑆 (cid:39) . − . 𝑆 (cid:39) . − .
75, respec-tively, irrespective of subhalo mass. Thob et al. (2019) noted thatpresent-day galaxies whose stellar component exhibit a sphericity of 𝑆 (cid:46) . 𝜅 ★ co > . 𝑀 sub (cid:38) M (cid:12) ). Similarly,the distribution of sphericities of star-forming gas discs are consis-tent with those recovered by Pillepich et al. (2019) when applyingthe standard mass distribution tensor to galaxies in the TNG50 sim-ulation. We remark that we have also computed the morphology ofstar-forming gas structures using an iterative form of the simple masstensor (equation 1), and do not find a significant systematic change.The sphericity of star-forming gas is most uniform in subhaloesof intermediate mass, 𝑀 sub ∼ . − M (cid:12) . In such structures, thedistribution of 𝑆 is strongly peaked at low values corresponding toflattened discs, albeit with a long tail to more spherical configura-tions. Owing to this asymmetry, which is most prominent for the MNRAS , 1–23 (2021) orphology and alignment of star-forming gas Star-Forming Gas: Face-on r = 22pkpc Edge-on
SF-gasKin: 2.14DM: 5.67Stars: 1.81
Stars: Edge-on r = 6pkpc Dark Matter: Edge-on r = 144pkpc r = 268pkpc [M yr kpc ] 10 [M yr kpc ]10 [M kpc ] 10 [M kpc ] Figure 2.
The star-forming gas, stars and dark matter (DM) comprising a star-forming central galaxy drawn from Recal-L025N0752, with stellar mass 𝑀 ★ (cid:39) × M (cid:12) . Each panel is 200 pkpc on a side. The galaxy’s subhalo mass is 𝑀 sub = . M (cid:12) , and its sSFR is (cid:164) 𝑀 ★ = − . M (cid:12) yr − . The upperpanels show the star formation rate surface density, a simple proxy for the radio continuum surface brightness, viewed face-on and edge-on. The green circlein the upper-left panel denotes the spherical half-mass radius of star-forming gas within 30 pkpc. Ellipsoids in the upper-right panel show projections of thebest-fit ellipsoids of the three matter components recovered by the iterative reduced inertia tensor. Overlaid solid lines show the minor axis of the stars (red) andDM (green), whilst the dotted white line corresponds to the rotation axis. The SF-gas is much flatter ( 𝑆 = .
06) than the stars ( 𝑆 = .
35) and DM ( 𝑆 = . ( Σ ★ ) =
6, 7, 8 M (cid:12) kpc − andlog ( Σ DM ) = . , . , .
75 M (cid:12) kpc − for the stars and DM respectively. The spherical half-mass radii for the stars and dark matter are 6 pkpc and 144 pkpcrespectively. These images have been made using the publicly available code Py-SPHViewer (Benitez-Llambay 2015) MNRAS , 1–23 (2021) A. D. Hill et al.
DMStarsSF - GasSphericityTriaxiality .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . log M sub [M (cid:12) ] = 10 . − . N sub = 157 .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . . − . N sub = 710 .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . P D F . − . N sub = 2452 .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . . − . N sub = 2305 .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . . − . N sub = 753 .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . . − . N sub = 223 .
00 0 .
25 0 .
50 0 .
75 1 . Shape Parameter (S , T) . . . . . . − . N sub = 117
10 11 12 13 14 log ( M sub [M (cid:12) ]) − . − . − . − . − . − . − . l og ( d n / d l og M s ub [ M p c − ] ) AllSatCen
Figure 3.
Probability distribution functions (PDFs) for the sphericity and triaxiality of the star-forming gas (blue), stars (red) and DM (green) comprisingsampled subhaloes in Ref-L100N1504 at 𝑧 =
0. Subhaloes are binned by their total mass. The triaxiality PDFs have been raised artificially by 0 . ∼ 𝐿 ★ galaxies. Star-forming gas does not exhibit a characteristic triaxiality, spanning a wide range of 𝑇 at all subhalo masses. star-forming gas, we quantify the diversity of the shape parame-ter distributions via the interquartile range (IQR) rather than theirvariance (see Table 2). The IQR of the star-forming gas sphericitydecreases from 0 .
13 for subhaloes of log ( 𝑀 sub / M (cid:12) ) = − . .
06 for subhaloes of log ( 𝑀 sub / M (cid:12) ) = − .
14 for the most massive haloes in oursample. The greater diversity in low-mass subhaloes is driven largelyby stochasticity in the structure of star-forming gas, with star forma-tion in many low-mass galaxies being confined to a small number ofgas clumps rather than being distributed throughout a well-defineddisc. In massive subhaloes, cold gas discs are readily disturbed byoutflows driven by efficient AGN feedback (see e.g. Bower et al.2017; Oppenheimer et al. 2020), and are less readily replenished with high-angular momentum gas from coherent circumgalactic in-flows (see e.g. Davies et al. 2020b,a).A potentially surprising finding highlighted by Figs. 3 and 4 is thatthe characteristic morphology of present-day star-forming gas discscan deviate significantly from that of a disc. The characteristic triaxi-ality of star-forming gas in subhaloes of mass 𝑀 sub ∼ − . M (cid:12) is 𝑇 < .
5, consistent with a flattened, oblate spheroid. Subhaloes ofall masses exhibit a broad distribution of 𝑇 , in marked contrast withthat of 𝑆 , and for subhaloes in the lower and higher mass bins, themedian value is 𝑇 > .
5, signifying that the characteristic morphol-ogy is prolate, such that even though the structures are flattened, theirisodensity contours when viewed face-on deviate significantly fromcircular. A similar finding from the TNG50 simulation was recently
MNRAS000
MNRAS000 , 1–23 (2021) orphology and alignment of star-forming gas . . . . . . Sh a p e P a r a m e t e r s ( S , T ) z = 0 z = 1Triax . Sph . log ( M sub [M (cid:12) ]) − − − l og ( d n / d l og M s ub [ M p c − ] ) Figure 4.
The median sphericity and triaxiality of the star-forming gas distri-bution of subhaloes, as a function of subhalo mass, at 𝑧 = 𝑧 = 𝑧 = 𝑧 =
1, respectively. The lower panel showsthe volumetric mass function of the sampled subhaloes.log 𝑀 sub Sphericity, 𝑆 Triaxiality, 𝑇 [ M (cid:12) ] SF-gas Stars DM SF-gas Stars DM10.0-10.5 0.13 0.21 0.16 0.28 0.27 0.2610.5-11.0 0.11 0.18 0.13 0.31 0.24 0.3111.0-11.5 0.12 0.19 0.14 0.31 0.23 0.3611.5-12.0 0.12 0.18 0.13 0.29 0.2 0.4112.0-12.5 0.06 0.15 0.12 0.29 0.23 0.412.5-13.0 0.06 0.14 0.11 0.26 0.48 0.4813.0-14.0 0.14 0.13 0.13 0.37 0.51 0.38
Table 2.
Interquartile ranges of the distributions of the sphericity ( 𝑆 ) andtriaxiality ( 𝑇 ) shape parameters of the star-forming gas (SF-gas), stars anddark matter (DM) comprising subhaloes in our sample, as a function ofsubhalo mass. reported by Pillepich et al. (2019). Inspection of face-on projectionsof the star-forming gas surface density highlights that this behaviouragain stems primarily from the stochasticity of star-forming gas struc-ture in low-mass subhaloes. In more massive subhaloes, stochasticityis also relevant, owing to the efficient disruption of well-sampled coldgas discs by AGN feedback. However we note that the stellar and DMcomponents tend towards more prolate configurations in more mas-sive subhaloes (as has been widely reported elsewhere, e.g. Vellisciget al. 2015a; Tenneti et al. 2014), suggesting that the morphologyof the gravitational potential may influence that of the cold gas. Weexamine this further in Section 3.3.As previously noted by Velliscig et al. (2015a), the triaxiality ofthe stars and DM in EAGLE subhaloes increases as a function ofthe subhalo mass, such that these components in the most-massive structures are strongly prolate. We note that our quantitative measuresare however slightly lower than those reported by Velliscig et al.(2015a), owing to our use of an initial 30 pkpc aperture and thereduced inertia tensor, which ascribes less weight to morphologyof these structures at large (elliptical) radius. It is well establishedfrom prior studies that the condensation of baryons in halo centresdrives the morphology towards a more spherical configuration thanis realised in dark matter-only simulations (see e.g. Kazantzidis et al.2004; Katz et al. 1994; Dubinski 1994; Springel et al. 2004; Zempet al. 2012). 𝑧 > 𝑧 >
0, for which wetake two approaches. First, we identify subhaloes at 𝑧 = . 𝑧 =
0. Clearly,these approaches require the examination of increasingly dissimilarsubhalo samples as one advances to higher redshift.The evolution of the characteristic values of the star-forminggas shape parameters (sphericity in blue, triaxiality in orange) foridentically-selected samples at 𝑧 = 𝑧 =
1, respectively, canbe assessed from comparison of the solid and dashed curves of Fig.4. These curves denote the median values of the shape parameters(sphericity in orange, triaxiality in blue) as a function of subhalomass, whilst the shaded regions correspond to the interquartile range.The darker (lighter) shaded areas for each parameter correspond to 𝑧 = 𝑧 = ( 𝑀 sub / M (cid:12) ) (cid:38)
10, are slightly more spherical (i.e. less flat-tened) at 𝑧 = < . 𝑆 at either epoch, which varies between 0 .
03 and0 .
22 at 𝑧 = .
11 and 0 .
19 at 𝑧 =
1, over the subhalo mass rangefrom log ( 𝑀 sub / M (cid:12) ) = −
14. Similarly, the star-forming gas insubhaloes of the same mass tends to be less oblate / more prolate at 𝑧 = 𝑧 = ( , , , , , ) , of present-daycentral subhaloes with mass log ( 𝑀 sub / M (cid:12) ) = . − .
5. Suchsubhaloes broadly correspond to those that host present-day ∼ 𝐿 ★ galaxies. The progenitors are identified using the D-Trees algorithm(Jiang et al. 2014); a full description of its application to the EAGLEsimulations is provided by Qu et al. (2017). The standard 30 pkpcaperture is used at all redshifts . Progenitor subhaloes are includedin the 𝑧 > We have assessed the impact of using an adaptive aperture of initial spher-ical radius 𝑟 = . 𝑟 ( 𝑧 ) , to account for the decreasing physical size ofprogenitors at early times, and do not recover significant differences.MNRAS , 1–23 (2021) A. D. Hill et al. . . . . . . . . . . P D F SF - Gas z = 5 , N = 198 z = 4 , N = 390 z = 3 , N = 575 z = 2 , N = 709 z = 1 , N = 725 z = 0 , N = 753 . . . M sub = 10 . − . M (cid:12) Stars . . . DM Figure 5.
The sphericities of the three matter components (star-forming gas, stars and DM, from left to right, respectively) comprising subhaloes of present-daymass log ( 𝑀 sub / M (cid:12) ) = − .
5, and their main progenitor subhaloes at 𝑧 = ( , , , , ) . The latter are included only whilst still satisfying the 𝑧 = 𝑁 , shown in the legend of the left-hand panel, is a monotonically-declining function of redshift. The arrows represent thelocation of the median for each distribution. The characterisation of the matter distributions with the iterative reduced inertia tensor indicates that DM becomesmore spherical at late cosmic epochs, the stellar distribution evolves mildly towards a more flattened configuration, and the star-forming gas typically evolvesstrongly towards a very flattened configuration by the present day. sample size, 𝑁 , is a monotonically declining function of redshift, asdenoted in the legend of the left-hand panel of the figure.We saw from Fig. 3 that present-day galaxies hosted by subhaloesin this mass range typically exhibit strongly-flattened ( 𝑆 (cid:39) .
1) star-forming gas discs. The left-hand panel of Fig. 5 highlights that,although star-forming gas discs are predominantly flattened evenat early epochs, the median sphericity at 𝑧 = 𝑆 SF − gas (cid:39) . 𝑆 (cid:46) .
2) is generally limited to 𝑧 <
2: the mediansphericity evolves from ˜ 𝑆 SF − gas (cid:39) .
33 at 𝑧 = 𝑆 SF − gas (cid:39) . 𝑧 =
0. The strong evolution of the star-forming gas sphericity of theseprogenitors is broadly coincident with the growth of the gas disc’smedian scale length, which grows only from (cid:39) (cid:39) 𝑧 = 𝑧 =
2, but by 𝑧 = (cid:39) ∼ 𝐿 ★ galaxy sample is largely insensitiveto redshift. The majority of the galaxies comprising our sample re-main actively star-forming at 𝑧 =
0, and are characterised by flattened For context, we reiterate that, as noted in Section 3.1, present-day galaxieswith a stellar component sphericity of 𝑆 (cid:46) . discs ( ˜ 𝑆 ★ (cid:39) . − .
45 at all redshifts examined). Such galaxies willtherefore have assembled primarily via in-situ star formation (Quet al. 2017) and will not have experienced the strong morphologicalevolution that typically follows internal quenching (see e.g. Davieset al. 2020b,a). Furlong et al. (2017) showed that the half-mass radiusof the stellar component of present-day star-forming ∼ 𝐿 ★ galaxiesgrows only from (cid:39) . pkpc to (cid:39) . pkpc between 𝑧 = 𝑧 = ∼ 𝐿 ★ galaxies, even when focussing primarily on the halo centre bydefining the shape parameters via the use of the iterative reducedmass tensor. The resolved progenitors exhibit a median sphericity of˜ 𝑆 DM (cid:39) . 𝑧 =
5, and this median increases monotonically to˜ 𝑆 DM (cid:39) . 𝑧 = .
18 at 𝑧 = = .
06 at 𝑧 = 𝑧 = 𝑧 =
0, withvalues of 0.13 to 0.14 for the stars and 0.14 to 0.12 for the DM.
We noted in the Section 3.1 that the star-forming gas configurationin massive subhaloes is often well described by a flattened prolatespheroid, similar to the characteristic morphology of the stars andDM in such structures. Thob et al. (2019) previously demonstratedthat EAGLE galaxies with flattened stellar distributions are preferen-tially hosted by flattened DM haloes, motivating a closer examination
MNRAS000
MNRAS000 , 1–23 (2021) orphology and alignment of star-forming gas . . . . . . S S F G a s
10 11 12 13 14log ( M sub [M ]) ⇢ . . . . . . . S ? . . . . . . T S F G a s
10 11 12 13 14log ( M sub [M ]) ⇢ . . . . . T ? Figure 6.
The correlation between star-forming gas and stellar sphericity (left) and triaxiality (right), as a function of subhalo mass. Black curves denote therunning medians of the star-forming gas shape parameters, computed via the locally-weighted scatterplot smoothing method (LOWESS). Colours represent themedian sphericity/triaxiality for the stellar mass distributions of the subhaloes residing within each hexagonal bin. The lower panels display the running Spearmanrank correlation coefficient, 𝜌 , between the residual shape parameters about 𝑀 sub for the two matter distributions, i.e. Δ 𝑆 SF − gas − Δ 𝑆 ★ and Δ 𝑇 SF − gas − Δ 𝑇 ★ ,where for instance Δ 𝑆 ★,𝑖 = 𝑆 ★,𝑖 − ˜ 𝑆 ★ ( 𝑀 sub ,𝑖 ) , with ˜ 𝑆 ★ ( 𝑀 sub ,𝑖 ) computed via LOWESS. Grey shaded regions indicate mass ranges for which the correlationis recovered at low significance ( 𝑝 > . here of the degree to which the morphology of the star-forming gascorrelates with that of the other matter components. Since the densityof stars typically dominates over the density of dark matter withinthe region traced by the star-forming gas, we focus on the corre-spondence between the morphology of the star-forming gas and thestars.Fig. 6 shows, as a function of subhalo mass, the correlation betweenthe sphericity (left) and triaxiality (right) shape parameters of thestar-forming gas and stars in the subhaloes comprising our sample.Black curves denote the running median of the shape parameters,˜ 𝑆 ( 𝑀 sub ) and ˜ 𝑇 ( 𝑀 sub ) , computed via the locally weighted scatterplotsmoothing method (LOWESS, e.g. Cleveland 1979). The LOWESScurves are plotted within the interval for which there are at least10 measurements at both lower and higher 𝑀 sub . The colour of thebackground hexagonal pixels denotes the median value of the stellarsphericity (left) or triaxiality (right) of galaxies residing in each pixel.Subhaloes in bins denoted by red (blue) colours therefore typicallyhave a stellar component with a high (low) value of the correspondingshape parameter.Clearly, the shape parameters of the star-forming gas and stellardistributions are strongly and positively correlated at effectively allsubhalo masses: flattened star-forming gas distributions are gener-ally found in subhaloes with flattened stellar components, and moreprolate star-forming gas distributions are found in subhaloes withmore prolate stellar components. We quantify the strength and sig-nificance of these correlations by computing a ‘running’ Spearmanrank correlation coefficient, 𝜌 ( 𝑀 sub ) , for the Δ 𝑆 SF − gas − Δ 𝑆 ★ and Δ 𝑇 SF − gas − Δ 𝑇 ★ relations, where Δ 𝑋 m represents the residual of shapeparameter 𝑋 for matter distribution 𝑚 about the LOWESS median. Hence, in the case of sphericity, Δ 𝑆 ★,𝑖 = 𝑆 ★,𝑖 − ˜ 𝑆 ★ ( 𝑀 sub ,𝑖 ) for the 𝑖 th subhalo. The running Spearman rank correlation coefficient is com-puted in subhalo mass-ordered sub-samples: for bins with a mediansubhalo mass 𝑀 sub < . M (cid:12) , we use samples of 200 subhaloeswith starting ranks separated by 50 subhaloes (e.g. subhaloes 1-200,51-250 and 101-300). For bins with median 𝑀 sub > . M (cid:12) , weuse samples of 50 subhaloes with starting ranks separated by 25subhaloes, to ameliorate the effect of the relative paucity of massivesubhaloes. This running 𝜌 ( 𝑀 sub ) is plotted in the lower sub-panel.Regions shaded in grey denote a Spearman rank 𝑝 -value is > . 𝜌 (cid:38) .
4) for thesphericity over a wide range in subhalo mass indicates that the degreeof flattening of the two components is indeed strongly and positivelycorrelated. The correlation is weaker ( 𝜌 (cid:38) .
3) for the triaxialityparameter, but remains positive and significant over a wide range ofsubhalo masses. We have also examined the correlation of the shapeparameters of star-forming gas with those of their host subhalo’sDM, and we find that the correlation is not formally significant atany subhalo mass.There is not currently a consensus amongst observational studies,which are necessarily limited to comparisons of projected elliptici-ties, in regard to correlations between the morphologies of the radiocontinuum and optical components of galaxies. Battye & Browne(2009) report a strong, positive correlation of the two in late-typegalaxies, and a weak negative correlation for early-type galaxies,whilst complementary studies using a smaller sample (Patel et al.2010), or a sample of fainter, more-distant galaxies (Tunbridge et al.
MNRAS , 1–23 (2021) A. D. Hill et al.
In this section we examine the orientations of the 3-dimensionaldistribution of star-forming gas in galaxies with respect to the stellarand DM components of their host subhaloes. We begin in Section4.1 with an examination of the morphological alignment of subhalocomponents as a function of subhalo mass, triaxiality and cosmicepoch. In Section 4.2 we consider the alignment of the morphologicalminor axis of the star-forming gas with its kinematic axis.
We quantify the morphological alignment of the various componentsvia the angle, 𝜃 , between the minor axes of the ellipsoids describingeach matter distribution, such that 𝜃 = ◦ indicates perfect align-ment and 𝜃 = ◦ indicates orthogonality. As noted in Section 2.3,we consider the minor axis to be the natural choice when focussingon discs, as the minor axis is the most distinct axis for oblate discs(though we reiterate the finding from Section 3.1 that many flattenedstar-forming structures are mildly prolate). Moreover, as seen in Sec-tion 3.1, the central regions of the stellar and DM distributions (towhich the iterative reduced mass distribution tensor is more stronglyweighted) also tend to be mildly oblate.Fig. 7 shows the alignment between the star-forming gas distri-bution and that of the DM. In the left-hand panel the alignmentis shown as a function of subhalo mass ( 𝑀 sub ) and in the right-hand panel it is shown as a function of the triaxiality of the DM.The thick orange curve and associated shading denotes the medianalignment angle, and the 10 th − th percentiles of the distribution,when considering the minor axes of the two components. In general,the alignment is strong, with the median alignment angle typically (cid:39) ◦ for 𝑀 sub = M (cid:12) , declining to (cid:39) ◦ for 𝑀 sub = M (cid:12) and (cid:39) ◦ for 𝑀 sub = − . M (cid:12) . In more massive subhaloes,the alignment is typically (marginally) poorer, rising to (cid:39) ◦ for 𝑀 sub (cid:38) M (cid:12) .Examination of the right-hand panel shows that the characteristicalignment of the minor axes of the star-forming gas and the DM ofits host subhalo is a strong function of the latter’s triaxiality, withoblate subhaloes exhibiting close alignment of the two components( ˜ 𝜃 < ◦ for 𝑇 DM (cid:46) .
4) but prolate subhaloes exhibiting muchpoorer alignment ( ˜ 𝜃 > ◦ for 𝑇 DM (cid:38) . All 𝑇 DM > . SF-gas
Min. Int. Maj. Min. Int. Maj. DM Min.
Int.
Maj.
Stars
Min.
Int.
Maj.
Table 3.
The frequency with which each star-forming gas axis is best alignedwith minor, intermediate or major axis of the inner regions of its DM halo.‘All’ relates to our full sample, while ‘ 𝑇 DM > . set of columns display the fractions computed for those subhaloesin our sample with prolate DM distributions ( 𝑇 DM > . 𝜃 (cid:48) . This exercise shows that in prolate subhaloes forwhich the alignment quantified by the standard measure is poor, thereoften remains reasonable alignment with one of the other principlemorphological axes (typically 𝜃 (cid:48) (cid:46) ◦ ).One might reasonably advance the argument that, in cases wherethe minor axis of the star-forming gas closely aligns with the inter-mediate or major axis of the DM, the terminology ‘misalignment’ isunhelpful. If the shape parameters of the two components are dis-similar, as is the case for the common configuration of an oblatedisc within a prolate subhalo, alignment of the minor axes mightnot be the most likely scenario, since in such cases the minor andintermediate DM axes are not distinct. Indeed, the axes that shouldbe ‘expected’ to align are likely to be those most closely aligned withthe angular momenta of the respective components (as we discuss inSection 4.2). However, for simplicity, hereafter we retain the originaldefinition of component alignment as the angle between the minoraxes of both of the two components.Fig. 8 shows the cumulative distribution function of the alignmentangle 𝜃 for the three pairs of matter components, namely star-forminggas and DM (pink), star-forming gas and stars (blue), and stars andDM (green). We plot the distribution as a function of log ( + 𝜃 ) because the bulk of the misalignments (for all component pairs)are small, but there are long tails to severe misalignments. Thicklines denote our fiducial measurement, whilst the thin lines showthe alignments inferred when the initial characterisation of the massdistribution considers all particles of the relevant matter componentbound to the subhalo, rather than only those within 30 pkpc of thesubhalo’s centre. We show the latter in order to highlight the influ-ence of the initial aperture, since an influence is to be expected: forexample, Velliscig et al. (2015a) showed that the alignment of thestellar and DM components is stronger closer to the subhalo centre,i.e. that galaxies are best aligned with the local, rather than global,distribution of matter in the subhalo. For reference, the dotted blackline shows the distribution function of alignment angles betweenrandomly-oriented vectors.For our fiducial measurements, half of the sampled subhaloes have MNRAS000
The frequency with which each star-forming gas axis is best alignedwith minor, intermediate or major axis of the inner regions of its DM halo.‘All’ relates to our full sample, while ‘ 𝑇 DM > . set of columns display the fractions computed for those subhaloesin our sample with prolate DM distributions ( 𝑇 DM > . 𝜃 (cid:48) . This exercise shows that in prolate subhaloes forwhich the alignment quantified by the standard measure is poor, thereoften remains reasonable alignment with one of the other principlemorphological axes (typically 𝜃 (cid:48) (cid:46) ◦ ).One might reasonably advance the argument that, in cases wherethe minor axis of the star-forming gas closely aligns with the inter-mediate or major axis of the DM, the terminology ‘misalignment’ isunhelpful. If the shape parameters of the two components are dis-similar, as is the case for the common configuration of an oblatedisc within a prolate subhalo, alignment of the minor axes mightnot be the most likely scenario, since in such cases the minor andintermediate DM axes are not distinct. Indeed, the axes that shouldbe ‘expected’ to align are likely to be those most closely aligned withthe angular momenta of the respective components (as we discuss inSection 4.2). However, for simplicity, hereafter we retain the originaldefinition of component alignment as the angle between the minoraxes of both of the two components.Fig. 8 shows the cumulative distribution function of the alignmentangle 𝜃 for the three pairs of matter components, namely star-forminggas and DM (pink), star-forming gas and stars (blue), and stars andDM (green). We plot the distribution as a function of log ( + 𝜃 ) because the bulk of the misalignments (for all component pairs)are small, but there are long tails to severe misalignments. Thicklines denote our fiducial measurement, whilst the thin lines showthe alignments inferred when the initial characterisation of the massdistribution considers all particles of the relevant matter componentbound to the subhalo, rather than only those within 30 pkpc of thesubhalo’s centre. We show the latter in order to highlight the influ-ence of the initial aperture, since an influence is to be expected: forexample, Velliscig et al. (2015a) showed that the alignment of thestellar and DM components is stronger closer to the subhalo centre,i.e. that galaxies are best aligned with the local, rather than global,distribution of matter in the subhalo. For reference, the dotted blackline shows the distribution function of alignment angles betweenrandomly-oriented vectors.For our fiducial measurements, half of the sampled subhaloes have MNRAS000 , 1–23 (2021) orphology and alignment of star-forming gas θ T DM (r < DM min DM int DM maj DM best M sub [M (cid:12) ])0123 l og N s ub s . . . . . . T DM ( r < Figure 7.
The misalignment angle, 𝜃 , of the minor axis of star-forming gas of our sample of subhaloes with respect to each of the principal axes of their DMdistribution. Solid lines indicate the binned median values of 𝜃 , whilst the shading denotes the 10th - 90th percentiles. The values are shown as a function ofsubhalo mass ( 𝑀 sub , left) and DM triaxiality ( 𝑇 DM , right). The orange, cyan and magenta curves correspond to the alignment of the star-forming gas minoraxis with respect to the minor, intermediate and major axis of the DM, respectively, whilst the blue curve is with respect to the principle DM axis with whichthe star-forming gas minor axis is most closely aligned. The dotted lines indicate where the sampling drops below 30 subhaloes per bin. Sub-panels show thenumber of subhaloes per bin. In general the minor axis of star-forming gas is well aligned with the minor axis of its corresponding DM, but in strongly triaxialsubhaloes the former often aligns more closely with the intermediate or major axis of the DM distribution. star-forming gas distributions misaligned with their stellar compo-nents by more than 5 ◦ , and half have star-forming gas distributionsmisaligned with their DM component by more than 9 . ◦ . Half of thesubhaloes have stellar components misaligned with their DM compo-nent by more than 6 ◦ . Assessing the alignments recovered when con-sidering all the particles of a given type associated with subhaloes, wefind that half of the subhaloes have stellar-DM misalignments greaterthan 17 ◦ . The poorer star-forming gas - DM alignment with respect tothe stars - DM alignment might be expected; since the stars and DMare collissionless components, their relevant evolutionary timescaleis the gravitational dynamical time, 𝑡 dyn = / √︁ 𝐺 𝜌 ∼ yr, suchthat their morphologies and orientation effectively ‘encode’ their for-mation and assembly history over an appreciable fraction of a Hubbletime. In contrast, the phase-space structure of the collissional, dissi-pative gas is not preserved as it accretes onto galaxies and condensesinto star-forming clouds. Its morphology and orientation therefore re-flects a more instantaneous snapshot of the evolution of the subhalothan is the case for the collissionless components.We note that the stellar - DM alignment shown in Fig. 8 (thickgreen curve) is significantly better than that inferred by Vellisciget al. (2015a), who found that half of all the subhaloes they exam-ined had misalignments worse than the 40 ◦ . This follows primarilyfrom our use of an initial particle selection within a 30 pkpc sphereand the iterative reduced inertia tensor (which weights more stronglytowards the halo centre), and also in part due to their measurementof the misalignment angle relative to the major axes of the massdistribution, and the slightly different sample selections. The influ- ence of the initial particle selection can be assessed by comparisonof the thick and thin solid curves: as expected, when one considersall matter bound to the subhalo (as opposed to only that within a30 pkpc sphere) when initialising the iterative characterisation of themass distribution, the misalignments with respect to the DM becomesignificantly more pronounced. As is clear from the thinner curves ofFig. 8, in this case half of the sampled subhaloes have star-forminggas distributions misaligned with their DM components by more than (cid:39) ◦ , and half have stellar components misaligned with their DMcomponent by more than 15 ◦ . The misalignment of star-forming gasand the stars is however largely unaffected, since the bulk of bothcomponents is typically found within the central 30 pkpc.Having noted that misalignments are typically most severe in mas-sive, prolate subhaloes, which tend to host quenched elliptical galax-ies (see e.g. Thob et al. 2019), it is reasonable to hypothesise that sub-haloes hosting star-forming disc galaxies (i.e. those with 𝜅 ★ co > . (cid:39) ◦ .Fig. 9 shows the temporal evolution of the misalignment angle, 𝜃 , of the minor axes of star-forming gas and DM mass distributions(left) and the star-forming gas and stars (right). Here, as was thecase for Fig. 5, we consider at all epochs subhaloes that satisfy MNRAS , 1–23 (2021) A. D. Hill et al. . . . . . (1 + θ )0 . . . . . . C D F ( N o r m a li s e d ) SF - Gas − DMSF - Gas − StarStar − DM κ ? co > . ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ θ Figure 8.
The misalignment of the star-forming gas, DM and stellar distri-butions within the subhaloes of our sample. The figure shows the cumulativedistribution function of the misalignment angle, 𝜃 , between the minor axes ofthe matter distribution pairs in the legend. Thick curves correspond to fidu-cial measurements, thin curves denote alignments inferred when the initialcharacterisation of the mass distribution considers all particles of the rele-vant type bound to the subhalo, rather than only those within 30 pkpc of thesubhalo centre. Thick dashed lines correspond to the subset of galaxies with 𝜅 ★ co , which broadly identifies star-forming disc galaxies. The dotted black lineindicates the distribution of angles between randomly orientated vectors in3D. Star-forming gas is a poorer tracer of the orientation of the subhalo DMdistribution than are the stars. the selection criteria specified in Section 2.4, however we do nothere focus solely on main branch progenitors of 𝐿 ★ subhaloes. It isimmediately apparent that the orientation of the star-forming gas is amuch poorer tracer of the orientation of both the DM and the stars atearly cosmic epochs than at the present day (though the characteristicalignment is always much better than random). As noted above, at 𝑧 = ◦ , but at 𝑧 = ◦ and at 𝑧 = ◦ .Similarly, at 𝑧 = ◦ , but at 𝑧 = ◦ and at 𝑧 = ◦ . The deterioration of thealignment of the star-forming gas distribution with both the DM andthe stars at earlier times is to be expected, since all three componentstend to be more spherical (less flattened) at higher redshift. Althoughin principle even highly spherical distributions can exhibit perfectalignment, as 𝑆 → 𝜃 , of the star-forminggas distribution with those of DM and stars, enabling subhaloes indark matter-only simulations to be populated with galaxies whosestar-forming gas has a realistic misalignment distribution. A novel aspect of radio continuum lensing surveys is that comple-mentary observations of the 21cm hyperfine transition emission linefrom atomic hydrogen can, in principle, be obtained simultaneouslywith little or no extra observing time. The Doppler shift of the 21cmline is widely used to infer the kinematics of the atomic phase ofthe ISM (e.g. Bosma 1978; Swaters 1999) and hence affords an in-dependent means of assessing galaxy orientation. As noted by Blain(2002), Morales (2006) and de Burgh-Day et al. (2015), the kine-matic axis can be used as a proxy for the unsheared morphologicalaxis, and hence affords a means to suppress the influence of galaxyshape noise and intrinsic alignments.Clearly, the naïve application of this method assumes perfectalignment of the kinematic and minor morphological axes. To as-sess the accuracy of this assumption, we define the morphokine-matic misalignment angle, 𝛽 , as the angle between the minor axis ofthe star-forming gas distribution, and the unit vector of its angularmomentum. Fig. 10 shows the cumulative distribution function oflog ( + 𝛽 ) , with solid curves denoting present-day measurementsand dashed lines denoting measurements at 𝑧 =
1. The blue curvescorrespond to the fiducial sample, whilst red curves correspond to thesubset of galaxies with 𝜅 ★ co > .
4. For reference, the dotted black lineagain shows the distribution function of alignment angles betweenrandomly-oriented vectors.As naïvely expected, the star-forming gas minor axis and angularmomentum vector of star-forming gas are well aligned for present-daysubhaloes: 80 percent of systems exhibit morphokinematic misalign-ments of less than 10 ◦ . However, similarly to the internal componentalignments, the distribution function exhibits a long tail to severe,but rare, misalignments. The morphokinematic alignment improvesif one restricts the analysis to the 𝜅 ★ co > . ◦ , andthe tail to severe misalignments is strongly diminished. At might beexpected when considering the reduced prevalence of strongly flat-tened star-forming discs at 𝑧 =
1, the morphokinmatic alignment ispoorer at this earlier epoch, with 80 percent of subhaloes aligned tobetter than 30 ◦ , and 12 ◦ when restricting to the 𝜅 ★ co > . 𝛽 , and quote inTable 4 the median values of key characteristics of subhaloes in eachquartile, namely the star-forming gas sphericity, subhalo mass, starformation rate, stellar mass, the star-forming gas co-rotation param-eter and the half-mass radius of the star-forming gas. This exerciseillustrates that poor alignment of the minor axis of the star-forminggas with its angular momentum vector is more typical in subhaloeshosting a spheroidal central galaxy, with a low star-formation rateand a less flattened and less extended star-forming gas distribution.In principle, such systems can be readily identified from either opticalor radio continuum imaging. In this section we examine the morphologies, alignments and ori-entations of star-forming gas and DM when projected ‘on the sky’in 2-dimensions, affording a direct connection with observationaltests. In Section 5.1 we consider the ellipticity of the matter compo-nents, i.e. their projected morphology. In Section 5.2 we consider theprojected alignments of galaxies.
MNRAS000
MNRAS000 , 1–23 (2021) orphology and alignment of star-forming gas . . . . . (1 + θ )0 . . . . . . C D F ( N o r m a li s e d ) SF - Gas − DM z = 5 , N = 3784 z = 4 , N = 7712 z = 3 , N = 13258 z = 2 , N = 19765 z = 1 , N = 19500 z = 0 , N = 6764 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ θ . . . . . (1 + θ )0 . . . . . . C D F ( N o r m a li s e d ) SF - Gas − Stars ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ θ Figure 9.
The temporal evolution of the misalignment of the minor axes of the star-forming gas and DM distributions (left) and of the star-forming gas andstellar distributions (right) comprising the subhaloes of our sample, as computed while adopting a 30 pkpc aperture. The figure shows the cumulative distributionfunction of the misalignment angle, 𝜃 . Colour indicates the redshift, and the thin black dotted line shows the distribution of angles between randomly-orientatedthree-dimensional vectors. . . . . . (1 + β )0 . . . . . . C D F ( N o r m a li s e d ) Full Sample κ ? co > . z = 0 z = 1Random ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Figure 10.
A histogram of the alignment between the morphological andkinematic axes of the star-forming gas within our sample. Alignment anglesare given in terms of log ( + 𝛽 ) , as the majority of the subhaloes havesmall 𝛽 . Solid (dashed) lines correspond to subhaloes at redshift 𝑧 = ( ) .The black dotted line shows the distribution of angles between randomly-orientated three-dimensional vectors. Red lines correspond to the subset ofgalaxies with 𝜅 ★ co , which broadly identifies star-forming disc galaxies, whileblue lines correspond to the full sample. It is via measurement of the morphology of galaxies in projection, i.e.their ellipticity, that the weak gravitational shear is estimated. Sincegalaxies are intrinsically ellipsoidal (i.e. non-circular), the observed
Quartile 1 st nd rd th 𝑆 SF − gas ± ± ± ± 𝑀 sub ± ± ± ± ± ± ± ± 𝑀 ★ ± ± ± ± 𝜅 SFco ± ± ± ± 𝑟 SF − gas ± ± ± ± Table 4.
The median and standard deviation in various subhalo propertiesfor the systems binned into quartiles based on the alignment angle betweenthe star-forming gas kinematic and morphological minor axes. In degrees, the0 the , 25 th , 50 th , 75 th and 100 th percentile values are 0.0 ◦ , 2.14 ◦ , 4.36 ◦ , 9.15 ◦ and 89.75 ◦ respectively. The values from top to bottom are: the star-forminggas sphericity; total subhalo mass (as log 𝑀 sub [ M (cid:12) ] ); star-formation ratewithin 30 pkpc (in M (cid:12) yr − ); stellar mass within 30 pkpc (as log 𝑀 ∗ [ M (cid:12) ] );the fraction of kinetic energy in the star-forming gas invested in co-rotation;and the star-forming gas half-mass radius (in pkpc). ellipticity is due to both the intrinsic ellipticity of the galaxy, andthe lensing shear. The former can therefore be considered as a noiseterm when measuring the shear, and is often referred to as ‘shapenoise’. Since the variance of the observed ellipticity, 𝜖 obs is the sumof the variances of the intrinsic ellipticity and the (reduced) shear,i.e. 𝜎 ( 𝜖 obs ) = 𝜎 ( 𝜖 int ) + 𝜎 ( 𝜖 sh ) , the signal-to-noise ratio of shearmeasurements is markedly sensitive to the diversity of the intrinsicellipticity of the galaxy population being surveyed.To measure the intrinsic ellipticity of matter distributions, we adaptthe iterative reduced inertia tensor algorithm presented in Section 2.3to consider only two spatial coordinates and so recover the best-fittingellipse. The intrinsic ellipticity is then | 𝜖 int | = ( 𝑎 − 𝑏 )/( 𝑎 + 𝑏 ) , where 𝑎, 𝑏 are the major and minor axis lengths of this ellipse, respectively,such that low ellipticity corresponds to near-circular morphology,and high ellipticity corresponds to a strongly flattened configuration.Hereafter we omit the subscript for brevity, such that 𝜖 ≡ 𝜖 int . Asnoted in Sec 2.3, the first iteration of the algorithm considers all MNRAS , 1–23 (2021) A. D. Hill et al. . . . . . . | (cid:15) | = ( b − a ) / ( b + a )0 . . . . . . . P D F Circular Flattened
Source:SF - GasStarsTunbridge et al . - onEdge - on Figure 11.
Probability distribution functions of the projected 2-dimensionalellipticities of the present-day mass distributions of star-forming gas (bluecurves) and stars (red curves) bound to the subhaloes of our sample. Solidcurves denote the aggregated ellipticities recovered from projection of the3-dimensional mass distributions along 100 random axes of projection. Forreference, the dashed and dotted curves show the distributions recoveredwhen the galaxies are oriented face-on and edge-on, respectively, to the axisof projection. An observational comparison (green curve with crosses) issourced from Tunbridge et al. (2016), who provide a best-fitting model to thedistribution of observed ellipticities of galaxies in the radio VLA COSMOSdata set. particles of the relevant type within a circular aperture of radius 𝑟 ap = max (
30 pkpc , 𝑟 sfg ) , where 𝑟 sfg is the 2-dimensional half-mass radius of star-forming gas within a circular aperture of 30 pkpc.The use of this additional criterion ensures a robust morphologicalcharacterisation of the image projected by the most extended gasdiscs when viewed close to a face-on orientation. At each iteration,the elliptical aperture adapts to maintain a constant area.Fig. 11 shows the probability distribution function of the projectedellipticity of star-forming gas (blue curves) and the stars (red curves)associated with the subhaloes of our sample. Solid curves denotethe distribution of aggregated ellipticities recovered from projectionof the 3-dimensional mass distributions along the line-of-sight of100 ‘observers’ randomly positioned on a unit sphere, thus crudelymimicking a real lightcone (albeit without noise or degradation frominstrumental limitations). The dashed and dotted curves show the el-lipticity distributions recovered when the subhaloes are first orientedsuch that the projection axis is parallel to, respectively, the minorand major principal axes of the respective 3-dimensional mass dis-tribution, in order to show the ellipticities when viewed face-on andedge-on.The distribution of ellipticities when projected along the simula-tion axes is significantly broader for the star-forming gas than is thecase for the stars: the IQRs of two distributions are 0 .
30 and 0 . 𝜖 sfgedge = .
70 and ˜ 𝜖 starsedge = . minimise these differences, since galaxies with significant star-forming gas reservoirs preferentially exhibit flattened stellar discs,i.e. elliptical and spheroidal galaxies are under-represented by oursample.The solid green curve of Fig. 11 denotes the best-fitting functionalform of the galaxy ellipticity distribution recovered from the appli-cation of the im3shape algorithm (Zuntz et al. 2013) to Very LargeArray (VLA) 𝐿 -band observations of galaxies in the COSMOS field(Tunbridge et al. 2016, see their equation 8). The iterative algorithmfinds the best-fitting two-component Sèrsic (disc and bulge) model,yielding two-component ellipticities 𝜖 = ( 𝑒 , 𝑒 ) , and is similar inconcept, if not in detail, to the approach used here to characterise thesimulated galaxies. There is a remarkable correspondence betweenthe observed ellipticity distribution and that recovered from EAGLE.The qualitative similarity is a reassuring indication that the elliptic-ity distribution of star-forming gas yielded by EAGLE is realistic,however we caution that the degree of agreement is likely to be,in part, coincidental: besides the differences in shape measurementalgorithms and the absence of noise or smearing by a point spreadfunction in the simulated shape measurements, the observed samplealso spans a wide range of redshifts.Tunbridge et al. (2016) noted that the dissimilar diversity of theprojected ellipticities of the star-forming gas and stellar mass distri-butions is of practical relevance, because it governs the shape noise.This difference is analogous to the difference in shape noise in theoptical regime expected for samples of early- and late-type galaxies:Joachimi et al. (2013) estimate that the former exhibit up to a factorof two less shape noise than the latter at fixed number. We assess themagnitude of this effect in EAGLE, by defining the shape noise of asample of galaxies, 𝜎 𝑒 , as: 𝜎 𝑒 = 𝑁 ∑︁ 𝑖 | 𝑒 𝑖 | , (4)where 𝑁 is the total number subhaloes in the sample. The quantityin the summation is often referred to as the polarisation (see e.g.Blandford et al. 1991) and is defined as | 𝑒 | = ( 𝑎 − 𝑏 )/( 𝑎 + 𝑏 ) . Itis thus related to the ellipticity via 𝑒 = 𝜖 /( + | 𝜖 | ) .We compute 𝜎 𝑒 for the star-forming gas and stellar distributionsof subhaloes as a function of subhalo mass. These measurements areshown in Fig. 12. Solid curves denote measurements for the star-forming gas (blue) and stars (red) considering all subhaloes com-prising our sample. To place the difference in shape noise betweenthe two matter types into context, we also show the shape noise ofthe stellar component when splitting the main sample into two sub-samples separated about 𝜅 ★ co = .
4, thus broadly separating the mainsample into late- and early-type galaxies. The shape noise of thestar-forming gas associated with subhaloes of all masses probed byour sample is systematically greater than is the case for their stars, by Δ 𝜎 𝑒 (cid:39) . − .
25, an offset comparable to the difference betweenthe shape noise (at fixed subhalo mass) of the stellar component ofsubhaloes comprising our crudely defined early- and late-type sub-samples. Tunbridge et al. (2016) report a qualitatively similar offset
MNRAS000
MNRAS000 , 1–23 (2021) orphology and alignment of star-forming gas . . . . . . . . . σ e SF - GasStarsStars − κ ? co < . − κ ? co > . ( M sub [M (cid:12) ]) - − − l og ( d n / d l og M s ub [ M p c − ] ) Figure 12.
The scatter in the projected ellipticities of stars and star-forminggas, calculated as the standard deviation of the minor-major axis ratios withina given mass bin. Blue lines relate to the star-forming gas, and red lines tothe stars. Solid lines correspond to the shape error in the projected galaxyshapes and the dotted line corresponds to the edge-on star-forming gas particlecoordinate projection for the star-forming gas. The red dashed and dot-dashedlines correspond to the standard deviations for 𝜅 ★ co < . 𝜅 ★ co > . 𝜅 ★ co is here the fraction of kinetic energy invested in co-rotation for the stars , as opposed to the star-forming gas, within 30pkpc, asoutlined by Thob et al. (2019). The lower panel displays the volume densityof subhaloes in a given mass bin, given as log ( dn / dlog M sub [ Mpc − ]) .At all masses, the shape noise is systematically greater for galaxy populationsthat are intrinsically flatter. of the shape noise of radio continuum sources relative to their opticalimages (see their Table 3). In practice, it is only the misalignment angle of the various mat-ter types in projection that can be measured observationally. Wetherefore extend the exploration of 3-dimensional misalignments pre-sented in Section 4, to examine misalignments in projection. Fig. 13shows the cumulative distribution function of 𝜃 , the alignmentangle of the three pairs of matter components when viewed in pro-jection. As with Fig. 8, we plot the distribution as a function oflog ( + 𝜃 ) since the bulk of the misalignments are small, butshow long tails to severe misalignments. Thick lines denote our fidu-cial measurement, whilst thin lines show the alignments inferredwhen the initial characterisation of the projected mass distributionconsiders all particles of the relevant matter component bound to thesubhalo. Thick dashed lines repeat the fiducial measurement for thesub-sample of subhaloes hosting late-type galaxies, i.e. those with 𝜅 ★ co > .
4. For reference, the dotted black line shows the distributionfunction of alignment angles between randomly-oriented vectors.The plot reveals that the projected alignments are qualitativelysimilar to those recovered in 3-dimensions, insofar that the star- . . . . . (1 + θ )0 . . . . . . C D F ( N o r m a li s e d ) SF - Gas − DMSF - Gas − StarStar − DM κ ? co > . ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ θ Figure 13.
The projected 2D internal alignment between the stars, DM andstar-forming gas within the subhaloes of our sample. The figure displays anormalised cumulative distribution function of the angle 𝜃 between theminor axes of various matter distributions within subhaloes. The line colourindicates the two matter types assessed, thick dashed and thin solid linescorrespond to the aperture used in the computation of the iterative reducedinertia tensor. The black dotted line indicates the distribution of angles be-tween randomly orientated vectors in 2D. Star-forming gas is a poorer tracerof the underlying DM distribution than the stars in terms of orientation. forming gas and DM are most weakly aligned (half of all subhaloesare aligned to better than 16 . ◦ ), whilst the star-forming gas - starsand stars - DM alignments are aligned significantly more closely (halfof all subhaloes aligned to better than 10 . ◦ and 8 . ◦ , respectively).Discarding the initial aperture weakens the alignment between themore centrally-concentrated baryons and the DM but, in a similarfashion to the 3-dimensional case, has little impact on the alignmentbetween star-forming gas and stars. Restricting the sample to late-type galaxies improves the alignment of all component pairs, withhalf of all subhaloes being aligned to better than 12 . ◦ , 7 . ◦ and 7 . ◦ for, respectively, the star-forming gas - DM, star-forming gas - stars,and stars - DM pairs.We note that, in contrast to Tenneti et al. (2014, their Fig 10.)and Velliscig et al. (2015a, their Fig. 13), we find that the projectedalignments are in general weaker in projection than in 3-dimensions.For example, the median alignment angle of star-forming gas andDM using our fiducial aperture choices are 9 . ◦ in 3-dimensionsand 16 . ◦ in projection. This is a consequence of our choice, moti-vated in Section 2.3, to measure misalignments relative to the minoraxis rather than the major axis; whilst the projected misalignmentis insensitive to this choice, the choice has a significant bearing onthe alignments in 3-dimensions. We have explicitly confirmed thatswitching from the use of the minor axis to the major axis to definethe misalignment angle results in smaller misalignments when pro-jecting from 3-dimensions, consistent with the findings of Tennetiet al. (2014) and Velliscig et al. (2015a). Although not shown in thefigure, we have further examined the misalignment angles of all mat-ter component pairs at 𝑧 =
1, and find more severe misalignments atthe earlier cosmic epoch. This result is largely insensitive to the useof the axisymmetry criterion.
MNRAS , 1–23 (2021) A. D. Hill et al.
We have investigated the morphology of, and mutual alignments be-tween, the star-forming gas, stars and dark matter bound to subhaloesthat form in the EAGLE suite of simulations (Schaye et al. 2015;Crain et al. 2015; McAlpine et al. 2016). Our study is motivated bythe complementarity of weak lensing experiments conducted usingradio continuum surveys with traditional optical surveys. In simu-lations like EAGLE, gas that has a non-zero star formation rate isa good proxy for gas that is bright in the radio continuum. EAGLErepresents a judicious test-bed for an assessment of this kind, as thesimulations were calibrated to ensure a good reproduction of thegalaxy stellar mass function and the size-mass relation of late-typegalaxies. We focus primarily on present-day subhaloes, but also ex-amine the simulations at earlier times to explore evolutionary trends.A summary of our results is as follows:(i) The star-forming gas distribution of present-day subhaloes istypically flattened (i.e. low sphericity) along its minor axis. Flatteningis most pronounced in subhaloes of 𝑀 sub ∼ . M (cid:12) , for whichthe median sphericity is ˜ 𝑆 SF − gas = .
1. The distribution of star-forming gas sphericities is significantly narrower than that of starsand dark matter at all subhalo masses, but particularly for those of 𝑀 sub = − . M (cid:12) , for which the interquartile ranges of star-forming gas, stars and DM are 0 .
06, 0 .
15 and 0 .
12, respectively (Fig.3).(ii) Star-forming gas exhibits a diverse range of triaxiality param-eters. Subhaloes of mass 𝑀 sub ∼ − . M (cid:12) typically host oblatedistributions consistent with classical gas discs, but in both low andhigh mass subhaloes, the distributions are more often prolate (Fig.3).(iii) Star-forming gas is less flattened at earlier epochs, for allsubhalo masses examined, irrespective of whether one considers asample selected in a similar fashion to the present-day sample, orconsiders the progenitors of the latter. Strongly flattened star-forminggas structures ( 𝑆 (cid:46) .
2) emerge only at 𝑧 (cid:46)
2, broadly coincidentwith the growth of the disc’s scale length (Figs. 4 and 5).(iv) The shape parameters describing the morphology of star-forming gas are strongly and positively correlated with those describ-ing the stellar morphology of the host galaxy, such that e.g. flattenedgas structures are associated with flattened stellar structures (Fig. 6).(v) The minor axis of the star-forming distribution preferentiallyaligns most closely with the minor axis of the (inner) DM halo.However, in prolate subhaloes 𝑇 DM ( 𝑟 < ) (cid:38) .
7, a significantfraction of galaxies have star-forming gas distributions whose minoraxis most closely aligns with one of the other principal axes of theDM (Fig. 7).(vi) Characterised by the angle between the minor axes of therespective components of subhaloes, star-forming gas tends to alignwith the DM (i.e. the alignment is stronger than random), but thealignment is weaker than is the case for stars and the DM. This isthe case for both the 3-dimensional matter distributions (Fig. 8) andtheir projections on the sky (Fig. 13). The alignments are strongestwhen considering the inner DM halo, and in general the alignmentsare stronger for late-type galaxies.(vii) The alignment of the star-forming gas distribution with thoseof both the stars and the DM bound to its parent subhalo is typicallyweaker at early cosmic epochs (Fig. 9).(viii) The kinematic axis of star-forming gas aligns closely withits minor morphological axis, with most galaxies being aligned tobetter than 10 ◦ at the present-day, and better than 6 ◦ if only late-typegalaxies are considered. The alignment is poorer at 𝑧 =
1, with thesecharacteristic misalignment angles doubling (Fig. 10). (ix) The more pronounced flattening of star-forming gas struc-tures leads to them exhibiting a broader distribution of projectedellipticities than is the case for stellar structures, analogous to thediffering ellipticity distributions of optical images of late-type andearly-type galaxies. The ellipticity distribution of star-forming gasin EAGLE corresponds closely to that recovered from high-fidelityVLA radio continuum images of galaxies in the COSMOS field (Fig.11). For a fixed subhalo sample, the ‘shape noise’ of its star-forminggas is therefore systematically greater than that of its stars 12).Our analyses reveal that the morphology of star-forming gas distri-butions, and their orientation with respect to the DM of their parentsubhalo, are more complex than might be naïvely assumed. Thiscomplexity is particularly relevant in the context of using extendedstar-forming gas distributions, which can be imaged in the radiocontinuum, to conduct weak lensing experiments.Forecasts for the outcomes of the next generation of the ‘megasur-veys’ require that very large cosmic volumes are modelled. The asso-ciated expense of including the baryonic component forces the use ofempirical, analytic or semi-analytic models grafted onto treatmentsof the evolving cosmic dark matter distribution. By construction,such techniques do not capture the full complexity of the evolutionof the baryonic component resulting from the diverse range of phys-ical processes that influence galaxies, nor do they capture the ‘backreaction’ of the baryons on the DM, and so can mask the importanceof key systematic uncertainties.In the specific case of modelling the radio continuum sky, the mostpopular approach has been to couple observed source populationswith either a Press-Schechter or 𝑁 -body treatment of the evolvingcosmic DM distribution (see e.g. Wilman et al. 2008; Bonaldi et al.2019). By construction, such models invoke no explicit connectionbetween the properties of star-forming gas structures and their parentDM haloes, and often relate (or equate) the properties of the formerto those of the host galaxy’s stellar component. Our analyses high-light shortcomings of these approximations: the characteristic mor-phology of star-forming gas is a strong function of the mass of itshost subhalo and, although the simulations indicate that it correlatesstrongly with the morphology of its associated stellar component, wefind that the respective morphologies can differ significantly.We also find that star-forming gas structures are imperfectlyaligned with both the stellar and DM components of their host sub-halo. Although the misalignment angle is generally small (partic-ularly with respect to the stellar component), there is a long-tail tosevere misalignments, and we find that the misalignment is most pro-nounced in early-type galaxies. We also find that the misalignmentof the star-forming gas with the DM of its host subhalo becomesmore pronounced if the outer halo is considered (for instance, if dis-abling the use of the 30 pkpc spherical aperture). Therefore, whenconstructing semi-empirical radio sky models based on 𝑁 -body sim-ulations, we caution against naïvely orienting star-forming discs withthe principle axes of the DM distribution.Our analyses also highlight that the shape noise of images of afixed sample of galaxies seen in the radio continuum should be sig-nificantly greater than when seen in the optical. This follows naturallyfrom the lower characteristic sphericity (or, alternatively, the greaterflattening) of star-forming gas structures than their stellar counter-parts. A systematic offset in shape noise was previously reportedby Tunbridge et al. (2016) following the examination of a relativelysmall sample of galaxies with high-fidelity radio and optical imaging.The corollary of this finding is that radio continuum weak lensingexperiments will require a greater source density in order to obtaina signal-to-noise ratio equal to optical experiments. However, our MNRAS , 1–23 (2021) orphology and alignment of star-forming gas analyses also corroborate the hypothesis that the use of the kinematicaxis (revealed by ancillary 21 cm observations) affords an effectivemeans of estimating the unsheared orientation of the minor axis,and thus mitigating the systematic uncertainty in radio weak lensingexperiments.An interesting consequence of the poorer alignment of star-forming gas structures with the DM of their host subhaloes than is thecase for the stars - DM alignment, is that it implies that the intrinsicalignment signal may be less severe in radio weak lensing surveysthan is the case for optical counterparts. In a follow-up paper, Hillet al. (in prep), we examine the two key ‘intrinsic alignment’ signalsrecoverable from radio continuum imaging, namely the orientationof star-forming gas distributions with respect to the directions to, andorientations of, the star-forming gas structures of its neighbouringgalaxies. ACKNOWLEDGEMENTS
DATA AVAILABILITY
Particle data, and derived data products from the simulations havebeen released to the community as detailed by McAlpine et al. (2016).Further derived data used in this article will be shared on reasonablerequest to the corresponding author.
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APPENDIX A: ANALYTIC FITS TO THE MISALIGNMENTANGLE DISTRIBUTIONS
We provide fitting functions to the distribution of internal misalign-ment angles between star-forming gas and DM for present-day sub-haloes in three mass bins from the EAGLE Ref-L100N1504 simula-tion, in both 2- and 3-dimensions. The fits enable users of 𝑁 -bodysimulations to populate subhaloes with galaxies oriented with respectto the minor axis of the subhalo in a realistic fashion. We fit to 𝑃 ( 𝜃 ) using the following functional form: M( 𝜃 ) = 𝐶 exp (cid:16) − 𝜃 𝜎 (cid:17) + 𝐷 exp (cid:16) − 𝜃 𝜎 (cid:17) + 𝐸, (A1)where 𝐶, 𝐷, 𝜎 , 𝜎 , 𝐸 are the free parameters, and 𝜃 is the mis-alignment angle. The same form was used by Velliscig et al.(2015a) to fit to the misalignment angle of stars and DM in pro-jection. We calcuate the best fit parameters with the Python packagescipy.optimize.curve_fit, using 1 𝜎 Poisson errors.The best fit parameters are quoted in Tables A1. Parameters arerecovered for the misalignment angles in both the cases of i) applyingour fiducial aperture to the initial step of the iterative algorithm, andii) applying no initial aperture, i.e. considering all particles bound tothe subhalo. In addition to presenting best fit parameters for all sub-haloes in our sample (‘All’), we provide fits to subsamples ‘M1’ and‘M2’, which are subject to the additional criteria log 𝑀 sub [ M (cid:12) ] ≤ . . < log 𝑀 sub [ M (cid:12) ] ≤
13 (M2). This is motivatedby two factors. Firstly, below 𝑀 sub = . M (cid:12) our selection crite-ria result in significant incompleteness. Secondly, the misalignmentof the minor axes of the star-forming gas and the DM components be-comes large for 𝑀 sub > M (cid:12) (see discussion in Sec 4.1), severelydegrading the value of the fits. The best fits for the 3-dimensionalfiducial aperture case are shown in Fig. A1.We find that the fitting is able to recover the profile of the inputdistribution fairly successfully. As an example we find the percent-age difference in the retrieved median as compared with the inputdistribution to be ( . , . , . ) percent for the three cases dis-played in the panels of Fig A1, while for the standard deviation thisbecomes ( . , . , . ) percent. For the no aperture version of MNRAS000
13 (M2). This is motivatedby two factors. Firstly, below 𝑀 sub = . M (cid:12) our selection crite-ria result in significant incompleteness. Secondly, the misalignmentof the minor axes of the star-forming gas and the DM components be-comes large for 𝑀 sub > M (cid:12) (see discussion in Sec 4.1), severelydegrading the value of the fits. The best fits for the 3-dimensionalfiducial aperture case are shown in Fig. A1.We find that the fitting is able to recover the profile of the inputdistribution fairly successfully. As an example we find the percent-age difference in the retrieved median as compared with the inputdistribution to be ( . , . , . ) percent for the three cases dis-played in the panels of Fig A1, while for the standard deviation thisbecomes ( . , . , . ) percent. For the no aperture version of MNRAS000 , 1–23 (2021) orphology and alignment of star-forming gas . . . . P ( θ D ) AllDataFit 0 30 60 90 θ M1 0 30 60 90M2
Figure A1.
Probability distribution functions 𝑃 ( 𝜃 ) , where 𝜃 is the angle between the morphological minor axes of stars and DM within the sample ofsubhaloes. A fiducial aperture of 30 pkpc is imposed. The faded step functions show the raw histograms, while the smooth lines are their respective analyticfits described by equation A1. Panels correspond to different mass bins: the full sample (left), 𝑀 sub ≤ . M (cid:12) (middle) and 10 . M (cid:12) ≤ 𝑀 sub < M (cid:12) (right). During fitting, errors in the 𝑦 -axis were taken to be the 1 𝜎 Poisson errors. these cases we find errors of ( . , . , . ) percent for the medianand ( . , . , . ) percent for the standard deviation. When no aper-ture is applied, we find that the errors in the 2-dimensional fittingsare comparable to the 3-dimensional case. However with the fiducialaperture the errors are noticeably larger for the 2-dimensional case,the largest being ∼
10 percent for the median and standard deviationof the M2 bin. Twelve figures comprising all variations displayed inTable A1 (two dimensions × two apertures × three mass bins) in thestyle of Fig A1 may be found at the author’s website . APPENDIX B: THE INFLUENCE OF PARTICLESAMPLING ON SHAPE CHARACTERISATION
The morphological characterisation of structures defined by particledistributions is unavoidably influenced by sampling error. It is there-fore crucial to establish the reliability of such characterisations as afunction of particle number. A common methodology is to realise amass distribution of a known analytic form with a particle distribu-tion, and assess the deviation of the recovered shape from the inputshape as the distribution is progressively sub-sampled (see e.g. Ap-pendix A2 of Velliscig et al. 2015a). We adopt a similar approach herebut, since star-forming gas distributions are not readily characterisedby a simple analytic form, we instead draw 20 central subhaloes fromthe sample described in Section 2.4, with dynamical mass compara-ble to that of the Milky Way ( 𝑀 sub (cid:39) . − . M (cid:12) ). We computetheir ‘true’ shape parameters by applying the algorithm defined inSection 2.3 using all star-forming gas particles ( 𝑁 part (cid:39) 𝑁 part ,generating 10 realisations at each value of 𝑁 part , and recompute theshape parameters.Fig. B1 shows the median of the relative error on the sphericityof the star-forming gas distribution recovered from the 10 sub-samplings of the particle distribution as a function of 𝑁 part . Thecurves are coloured by the ‘true’ value of the triaxiality parameterof the subhalo’s star-forming gas. The dashed black curve shows the‘grand median’ recovered by aggregating the measurements from all
20 subhaloes. Down arrows show the 10 th , 50 th and 90 th percentilevalues of the number of star-forming gas particles within the subhalosample. Velliscig et al. (2015a) noted that poor particle samplingleads to a systematic underestimate of the sphericity parameter ofthe DM; we find this is also the case for the star-forming gas. A shapeerror of less than 10 percent typically requires at least 𝑁 part = 𝑇 > .
5) distributions. As shown in Fig. 4, the star-forming gas oflow-mass subhaloes is preferentially prolate, hence a minimum of 𝑁 part =
100 can be considered a conservative choice. For complete-ness, the sub-panel shows the ‘running’ value of the Spearman rankcoefficient recovered from 𝑁 part -ordered sub-samples, of the correla-tion between the absolute shape error and the true shape parameters, 𝑇 (solid curve) and 𝑆 (dashed curve). The solid curve highlights thata negative correlation between the shape error on sphericity and thetrue triaxiality persists to over 1000 particles. The dashed curve in-dicates that there is a very mild positive correlation of the relativeshape error on sphericity with the true input sphericity. APPENDIX C: INFLUENCE OF SUBGRID ISMTREATMENTS
In this section we examine the sensitivity of star-forming gas mor-phologies to aspects of EAGLE’s subgrid models that in principleinfluence the structure of interstellar gas directly, namely the form ofthe temperature floor equation of state and the star formation law. Toacheive this, we compare the Ref-L025N0376 simulation with twopairs of complementary L025N0376 simulations. The first pair, intro-duced by Crain et al. (2015), varies the slope of the equations of statefrom the reference value of 𝛾 eos = / 𝛾 eos =
1) and adiabatic ( 𝛾 eos = /
3) equations of state.Schaye & Dalla Vecchia (2008) used simulations of idealised discsto show that a stiffer equation of state generally leads to smootherstar-forming gas distributions with a larger scale height. Crain et al.
MNRAS , 1–23 (2021) A. D. Hill et al.
2D 3D
Aperture & Mass-Bin
𝐶 𝐷 𝐸 𝜎 𝜎 𝐶 𝐷 𝐸 𝜎 𝜎 Fiducial
All M1 -27.0 9.21 0.0205 0.042 0.00688 -13.3 3050.0 0.0714 9.06 -9.05 M2 No Aperture
All -17.4 17.4 -87.3 87.3 0.0122 29.3 -29.3 -157.0 157.0 0.0125 M1 -17.7 17.7 -71.7 71.7 0.0124 31.2 -31.2 -181.0 181.0 0.013 M2 -17.3 17.3 -79.8 79.8 0.0124 26.9 -26.9 -114.0 114.0 0.0121 Table A1.
Best fitting parameters for equation A1, used to fit the probability distribution functions of the intrinsic 3-dimensional and projected 2-dimensionalmisalignment angle between star-forming gas and DM within present-day subhaloes of three mass bins (denoted by italics). Parameters are provided for theangles recovered using our fiducial initial aperture for the iterative reduced inertia tensor, and for no aperture (denoted by font weight). Number of Particles − − − − Sh a p e E rr o r( % ) Grand MedianIndividual MedianAperture = 30 pkpcSF - Gas Number of Particles − ρ Tri Sph . . . . . . . Subh a l o T r i a x a li t y Figure B1.
The sphericity shape error recovered as a function of the degree ofparticle sub-sampling for star-forming gas in 20 present-day subhaloes withdynamical mass similar to that of the Milky Way. Down arrows correspondto the 10 th , 50 th and 90 th percentile values of the number of star-forminggas particles within the subhalo sample. Curves show the median shapeerror recovered from 10 random sub-samplings of the true star-forminggas particle distribution, and are coloured by the latter’s true triaxiality. Thedashed curve shows the median recovered by aggregating measurements fromall 20 subhaloes. The sub-panel shows the running Spearman rank correlationcoefficients relating the shape error to the true value of the triaxiality (solidcurve) and sphericity (dashed curve) of the star-forming gas. (2015) showed that in EAGLE, a stiffer equation of state also sup-presses accretion onto the central BH in massive galaxies. The secondpair, introduced by Crain et al. (2017), varies the normalisation ofthe Kennicutt-Schmidt law (the variable 𝐴 in equation 1 of Schayeet al. 2015) from its fiducial value of 1 . × − M (cid:12) yr − kpc − by ± . . . . . . . . . . . . . . . P D F RefEOS1p000EOS1p666KSNormHiKSNormLo
Figure C1.
Probability distribution function of the sphericity parameter of thestar-forming gas of present-day subhaloes drawn from the Ref-L025N0376simulation (black curve) and two pairs of simulations that incorporate vari-ations of the reference model: pair with alternative equation of state slopes(EOS1p00, solid blue; and EOS1p666, dotted blue) and with normalisa-tions of the star formation law adjusted by ± . Fig. C1 shows probability distribution function of the sphericitystar-forming gas for the reference model (solid black curve) and thesimulations with differing equations of state ( 𝛾 eos =
1, solid blue; 𝛾 eos = /
3, dotted blue), and with higher (solid red) and lower (dot-ted red) normalisations of the star formation law with respect to thereference model. The subhaloes shown are selected according to thestandard sampling criteria outlined in the Section 2.4. Down arrowsdenote the median sphericity of the distribution of each simulation.Inspection reveals that the distributions are not strongly influencedby changes to the subgrid modelling of the ISM. The median valueof the sphericity of the star-forming gas in the Ref-L025N0376 sim-ulation is 0.15. As can be clearly seen from the figure, the mediansphericity in the three variation simulations shifts by < .
05 withrespect to the reference simulation, a value that is much smaller thanthe interquartile range of the reference case. Although not shown
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MNRAS000 , 1–23 (2021) orphology and alignment of star-forming gas here, we recover similar behaviour when focussing on the triaxialityparameter. APPENDIX D: NUMERICAL CONVERGENCE
In this section we examine the influence of the numerical resolutionof the EAGLE simulations on the recovered sphericity of the star-forming gas, stars and DM comprising subhaloes. We follow Schayeet al. (2015) and adopt the terms ‘strong convergence’ and ‘weak con-vergence’, where the former denotes a comparison at different reso-lutions of a fixed physical model, and the latter denotes a comparisonat different resolutions of two models calibrated to recover the sameobservables. We use three 𝐿 =
25 cMpc simulations introduced bySchaye et al. (2015): Ref-L025N0376, which is identical to the flag-ship Ref-L100N1504 simulation with the exception of the boxsize;Ref-L025N0752, which adopts the same Reference physical modelbut has a factor of 8 more particles each of both baryons and DM; andRecal-L025N0752 which also adopts values for subgrid parametersgoverning stellar and AGN feedback that have been recalibrated toimprove the match to the galaxy stellar mass function and galaxysizes at this higher resolution. Comparison of Ref-L025N0376 withRef-L025N0752 and Recal-L025N0752 thus affords simple tests of,respectively, the strong and weak convergence behaviour.Fig. D1 shows the probability distribution functions of the spheric-ities of the star-forming gas (left), stars (centre) and DM (right) foreach of the three 𝐿 =
25 cMpc simulations. The subhaloes shownare selected according to the standard sampling criteria outlinedin the Section 2.4, irrespective of the resolution of the simulation.Down arrows denote the median sphericity of the distribution of eachsimulation. Inspection shows that the distributions are not stronglyinfluenced by the change in resolution. The median values of thesphericity of the three matter components in the Ref-L025N0376simulation are 0.15, 0.50 and 0.69 for the star-forming gas, starsand DM, respectively. As can be clearly seen from the figure, whenmoving to the high resolution simulations, the shift in median val-ues is much smaller than the associated interquartile ranges of theRef-L025N0376 simulation (IQR = 0 .
14, 0 .
15, 0 .
14 for the threecomponents, respectively). Although not shown here, we recoversimilar behaviour when focussing on the triaxiality parameter.
MNRAS , 1–23 (2021) A. D. Hill et al. . . . . . . . . . . . P D F DMStarsSF - Gas 0 . . . . . . Ref − L025N0376Ref − L025N0752Recal − L025N0752 . . . . . . Figure D1.
Probability distribution function of the sphericity parameter of the star-forming gas (left panel), stars (centre) and DM (right) of present-day subhaloesdrawn from the Ref-L025N0376 (solid dark-coloured curve), Ref-L025N0752 (dashed medium) and Recal-L025N0752 (dotted light) simulations. Down arrowsdenote the median sphericity of the distribution of each simulation. Comparison of Ref-L025N0376 with Ref-L025N0752 and Recal-L025N0752 affords simpletests of, respectively, the strong and weak convergence behaviour of the star-forming gas sphericity.MNRAS000