But What About... Cosmic Rays, Magnetic Fields, Conduction, & Viscosity in Galaxy Formation
Philip F. Hopkins, T. K. Chan, Shea Garrison-Kimmel, Suoqing Ji, Kung-Yi Su, Cameron B. Hummels, Dusan Keres, Eliot Quataert, Claude-Andre Faucher-Giguere
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 11 February 2020 (MN L A TEX style file v2.2)
But What About... Cosmic Rays, Magnetic Fields, Conduction, &Viscosity in Galaxy Formation
Philip F. Hopkins (cid:63) , T. K. Chan , Shea Garrison-Kimmel , Suoqing Ji , Kung-Yi Su ,Cameron B. Hummels , Dušan Kereš , Eliot Quataert , Claude-André Faucher-Giguère TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA Department of Physics, Center for Astrophysics and Space Sciences, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093 Department of Astronomy and Theoretical Astrophysics Center, University of California Berkeley, Berkeley, CA 94720 Department of Physics and Astronomy and CIERA, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
Working Document
ABSTRACT
We present and study a large suite of high-resolution cosmological zoom-in simulations, using the FIRE-2 treatment ofmechanical and radiative feedback from massive stars, together with explicit treatment of magnetic fields, anisotropicconduction and viscosity (accounting for saturation and limitation by plasma instabilities at high- β ), and cosmic rays(CRs) injected in supernovae shocks (including anisotropic diffusion, streaming, adiabatic, hadronic and Coulomblosses). We survey systems from ultra-faint dwarf ( M ∗ ∼ M (cid:12) , M halo ∼ M (cid:12) ) through Milky Way/Local Group(MW/LG) masses, systematically vary uncertain CR parameters (e.g. the diffusion coefficient κ and streaming velocity),and study a broad ensemble of galaxy properties (masses, star formation [SF] histories, mass profiles, phase structure,morphologies, etc.). We confirm previous conclusions that magnetic fields, conduction, and viscosity on resolved ( (cid:38) M ∗ (cid:28) M (cid:12) , M halo (cid:46) M (cid:12) ), or at high redshifts ( z (cid:38) − any physically-reasonable parameters.However at higher masses ( M halo (cid:38) M (cid:12) ) and z (cid:46) −
2, CRs can suppress SF and stellar masses by factors ∼ − κ (cid:38) × cm s − . At lower κ , CRs take too long to escape dense star-forming gas and lose their energy to collisional hadronic losses, producingnegligible effects on galaxies and violating empirical constraints from spallation and γ -ray emission. At much higher κ CRs escape too efficiently to have appreciable effects even in the CGM. But around κ ∼ × cm s − , CRs escapethe galaxy and build up a CR-pressure-dominated halo which maintains approximate virial equilibrium and supportsrelatively dense, cool ( T (cid:28) K) gas that would otherwise rain onto the galaxy. CR “heating” (from collisional andstreaming losses) is never dominant.
Key words: galaxies: formation — galaxies: evolution — galaxies: active — stars: formation — cosmology: theory
Galaxy and star formation are intrinsically multi-physics processesthat involve a competition between gravity, collisionless dynam-ics, fluid dynamics and turbulence, radiation-matter coupling andchemistry, relativistic particles, magnetic fields, and more. Manyof these processes enter most dramatically via “feedback” frommassive stars, whereby the radiation, winds, and explosions frommassive stars dramatically alter subsequent generations of star andgalaxy formation.In recent years, there has been tremendous progress in mod-eling and understanding the effects of multi-phase gas in the inter-stellar and circum/inter-galactic medium (ISM, CGM, and IGM),radiative cooling, turbulence, and self-gravity, and how these pro-cesses couple to stellar feedback. For example, it is now well-established that without feedback, these processes lead to runawayfragmentation and gravitational collapse that turns dense gas intostars on a single gravitational free-fall time (on both giant molec-ular cloud [GMC] and galactic scales; Tasker 2011; Hopkins et al.2011; Dobbs et al. 2011; Harper-Clark & Murray 2011), and trans-forms most of the baryons in the Universe into stars (Katz et al.1996; Somerville & Primack 1999; Cole et al. 2000; Springel &Hernquist 2003; Kereš et al. 2009), both in stark contrast to obser- (cid:63)
E-mail:[email protected] vations. Moreover direct and indirect effects of feedback are ubiq-uitously observed in outflows and enrichment of the CGM and IGM(Martin 1999; Heckman et al. 2000; Pettini et al. 2003; Songaila2005; Martin et al. 2010; Sato et al. 2009; Weiner et al. 2009;Steidel et al. 2010). Of course, feedback (even restricting just tofeedback from stars) comes in a variety of forms, including radia-tive (ionization, photo-heating and radiation pressure), mechanical(thermal and kinetic energy associated with supernovae Types Ia &II, stellar mass-loss, and protostellar jets), the injection of magneticfields, and acceleration of cosmic rays. Numerical simulations havebegun to directly resolve the relevant scales of some of these pro-cesses and therefore have begun to explicitly treat some of thesefeedback channels and their interactions on ISM and galactic orinter-galactic scales (e.g. Tasker 2011; Hopkins et al. 2011, 2012;Wise et al. 2012; Kannan et al. 2014; Agertz et al. 2013; Roškaret al. 2014).One example is the suite of simulations studied in the “Feed-back In Realistic Environments” (FIRE) project (Hopkins et al.2014). These simulations have been used extensively in recent See the FIRE project website: http://fire.northwestern.edu
For additional movies and images of FIRE simulations, see: c (cid:13) a r X i v : . [ a s t r o - ph . GA ] F e b Hopkins et al.
Table 1.
Zoom-in simulation volumes run to z = Simulation M virhalo M MHD + ∗ M CR + ∗ m i , (cid:104) (cid:15) gas (cid:105) sf NotesName [ M (cid:12) ] [ M (cid:12) ] [ M (cid:12) ] [1000 M (cid:12) ] [pc] m09 m10v m10q m10y m10z m11a m11b m11i m11e m11c m11q m11v z ∼ m11h m11d m11f m11g m12z z ∼ m12r m12w m12i m12b m12c z ∼ m12m m12f ∼ ( − ) comoving, so there are actually several hundred resolved galaxies in total. (1) Simulation Name: Designation used throughout this paper. (2) M virhalo : Virial mass (following Bryan & Norman 1998) of the“target” halo at z = (3) M MHD + ∗ : Stellar mass of the central galaxy at z =
0, in our non-CR, but otherwise full-physics(“MHD+”) run. (4) M CR + ∗ : Stellar mass of the central galaxy at z =
0, in our “default” (observationally-favored) CR+ ( κ = e (5) m i , : Mass resolution: the baryonic (gas or star) particle/element mass, in units of 1000 M (cid:12) . The DM particle mass isalways larger by the universal ratio, a factor ≈ (6) (cid:104) (cid:15) gas (cid:105) sf : Spatial resolution: the gravitational force softening(Plummer-equivalent) at the mean density of star formation (gas softenings are adaptive and match the hydrodynamic resolution,so this varies), in the MHD+ run. Time resolution reaches ∼ − ∼ − cm − . (7) Additional notes.
Table 2.
Additional high-redshift, massive-halo simulations ( M virhalo (cid:38) M (cid:12) ). All units are physical. Simulation z z f M MHD + ∗ M CR + ∗ m i , Reference Notes m12z10
10 10 2e9 3e9 7 Ma et al. (2018b) clumpy, multiply-merging, no defined center m12z7 m12z5 m12z4 z ∼ − m12z3 z ∼ − m12z2 z ∼ − m12z1 z < m12q in Hopkins et al. (2014))Halo/stellar properties (as Table 1) of simulations which feature halos that reach ∼ M (cid:12) at various redshifts z (i.e. are more massive at z ∼ ∼ − M (cid:12) . Columns show: (1) Simulation name. (2) z : Redshift at which thehalo virial mass reaches ∼ M (cid:12) . (3) z f : Lowest redshift to which the simulation is evolved. (4) M MHD + ∗ ( z f ) : Stellar mass (in M (cid:12) ) of the primarygalaxy at z f , in the MHD+ run. (5) M CR + ∗ ( z f ) : Stellar mass (in M (cid:12) ) of the primary galaxy at z f , in the CR+( κ = e
29) run. (6) m i , : Mass resolutionin 1000 M (cid:12) . (7) Reference (paper in which this IC first appeared). (8)
Additional notes.
Table 3.
Default physics “suites” in this paper
Name NotesHydro+ “Default” FIRE-2 physics. Includes stellar feedback (SNe, radiation, mass-loss), cooling, gravity, but no MHD or CRsMHD+ Includes all “Default” FIRE-2 physics, plus ideal MHD, with anisotropic Spitzer-Braginskii conduction & viscosityCR+ Includes all “MHD+” physics, with CR injection in SNe, CR streaming, diffusion (with coefficient κ in cgs), hadronic and Coulomb losses- in this category we survey these properties, e.g. “CR+( κ = e κ = e κ as labeledDescription of the basic physics in the simulation sets described here. This notation is used throughout the paper. c (cid:13) , 000–000 osmic Rays on FIRE years to explore the interplay between a multi-phase ISM and CGMand both radiative (Hopkins et al. 2018a) and mechanical feed-back (Hopkins et al. 2018c) processes from stars. These processesalone, coupled to the physics of radiative cooling, gravity, and starformation, appear to explain a wide variety of observed phenom-ena in galaxies, including their abundances (Ma et al. 2016; Es-cala et al. 2018), star formation “main sequence” and fluctuations(Sparre et al. 2017), satellite mass functions (Wetzel et al. 2016;Garrison-Kimmel et al. 2018), color distributions (Feldmann et al.2016), stellar (Wheeler et al. 2015, 2017) and gas-phase (El-Badryet al. 2018b,a) kinematics, radial gradients and internal thick/thindisk structure (Ma et al. 2017a,b; Bonaca et al. 2017), stellar ha-los (Sanderson et al. 2017; El-Badry et al. 2018c), multi-phase fastoutflows (Muratov et al. 2015, 2017), dark matter profiles (Oñorbeet al. 2015; Chan et al. 2015), and more.However, essentially all of the conclusions above were basedon simulations that treated the gas in the hydrodynamic limit. Thedetailed plasma physics of the ISM and CGM/IGM is of coursequite complex and still a subject of active research. But it is well-established that magnetic fields are ubiquitous and important forlocal particle transport, that anisotropic transport processes (e.g.conduction and viscosity) can become significant in hot, tenuousgas, and that the ISM and IGM contain a spectrum of relativis-tic charged particles (cosmic rays [CRs]). In a very broad sense atleast in the local Solar-neighborhood ISM, magnetic field and CRenergy densities are order-of-magnitude comparable to thermal andturbulent energy densities (Ginzburg & Ptuskin 1985; Boulares &Cox 1990), suggesting they may not be negligible dynamically.Magnetic fields have been studied and discussed in the galac-tic and extra-galactic context for decades (for reviews, see Becket al. 1996; Beck 2009). They can, in principle, slow star formationin dense gas (Piontek & Ostriker 2005, 2007; Wang & Abel 2009;Beck et al. 2012; Pakmor & Springel 2013; Kim & Ostriker 2015),or alter fluid mixing instabilities (Jun et al. 1995; McCourt et al.2015; Armillotta et al. 2017) and the evolution of SNe remnants(Jun & Norman 1996b,a; Jun & Jones 1999; Thompson 2000; Kim& Ostriker 2015), and of course determine the actual dynamics ofanisotropic transport. Anisotropic thermal conduction and viscosityin hot gas have also been studied extensively in the past (albeit notquite as widely), and it has been widely suggested that both couldbe important for plasma heating and dynamics on galaxy clusterscales (Reynolds et al. 2005; Sijacki & Springel 2006; Markevitch& Vikhlinin 2007; Sharma et al. 2009, 2010; Parrish et al. 2012;Choi & Stone 2012; Armillotta et al. 2017) or, again, in SNe rem-nants (see references above), or for the mixing/survival of coolclouds in hot galactic outflows or the CGM (Brüggen & Scanna-pieco 2016; Armillotta et al. 2017). However, it is not clear if theseprocesses are particularly important for galaxy properties. In fact,most studies in the past have argued the effects of these physics indwarfs and Milky Way (MW)-mass ( ∼ L ∗ ) galaxies are relativelysmall – perhaps not surprising since the magnetic dynamo in super-sonic turbulence appears to saturate with magnetic fields alwayssub-dominant to turbulence (effectively, passively-amplified; seeFederrath et al. 2014; Su et al. 2018a; Squire & Hopkins 2017; Col-brook et al. 2017; Rieder & Teyssier 2017; Banda-Barragán et al.2018; Martin-Alvarez et al. 2018) and conduction/viscosity dependstrongly on gas temperature and are weaker in the cooler gas ofsub- L ∗ galaxy halos. In Su et al. (2017), we attempted to study the Throughout this manuscript, we use “halo” to refer to the extendedcircum-galactic gas, stars, and/or dark matter extending from outside the effect of magnetic fields, anisotropic conduction, and viscosity inaddition to the physics described above in FIRE simulations, andconcluded the effects were minimal. However, the simulations inthat paper were mostly non-cosmological (although it did includetwo cosmological cases), so might not capture all the important ef-fects in the CGM. Moreover, a swathe of recent work in plasmaphysics has argued that conductivity and viscosity of dilute plas-mas might be self-limiting under exactly the relevant conditions ofthe ISM and CGM (see Kunz et al. 2014; Riquelme et al. 2016;Roberg-Clark et al. 2018; Squire et al. 2017b; Komarov et al. 2018,and references therein), and these effects were not accounted forin previous studies (although they generally act to weaken the con-ductivity and viscosity).The situation with CRs is much less clear. In the MW (and, it iswidely believed, most dwarf and ∼ L ∗ or star-forming galaxies), theCR pressure and energy density (and correspondingly, effects onboth gas dynamics and heating/cooling rates of gas via hadronic orCoulomb collisions or excitation of Alfvén waves in various plasma“streaming instabilities”; Mannheim & Schlickeiser 1994; Enßlinet al. 2007; Guo & Oh 2008) are dominated by mildly-relativistic ∼ GeV protons accelerated primarily in supernova remnants (with ∼
10% of the SNe ejecta energy ultimately in CRs; Bell 2004). Inmore massive galaxies, which host supermassive black holes (BHs)and have little star formation, most of the CR production appearsto be associated with AGN jets and “bubbles.” CRs and their influ-ence on galaxy evolution have been a subject of interest in both an-alytic (Ipavich 1975; Breitschwerdt et al. 1991, 1993; Zirakashviliet al. 1996; Socrates et al. 2008; Everett et al. 2008; Dorfi & Bre-itschwerdt 2012; Mao & Ostriker 2018) and numerical simulation(Jubelgas et al. 2008; Uhlig et al. 2012; Booth et al. 2013; Salem& Bryan 2014; Ruszkowski et al. 2017; Farber et al. 2018a) stud-ies for decades – with an explosion of work in recent years. Thiswork has argued that CRs could, in principle, drive galactic out-flows (Simpson et al. 2016; Girichidis et al. 2016; Pakmor et al.2016; Wiener et al. 2017), suppress star formation in low (or high)mass galaxies (Hanasz et al. 2013; Chen et al. 2016; Jacob et al.2018), provide additional pressure to “thicken” galactic gas disks(Wiener et al. 2013b; Salem et al. 2014), alter the phase structureof the CGM (Salem et al. 2016; Girichidis et al. 2018; Butsky &Quinn 2018), “open up” magnetic field lines or otherwise alter thegalactic dynamo (Parker 1992; Hanasz et al. 2009; Kulpa-Dybełet al. 2011, 2015), and more.However, a number of major uncertainties and limitations re-main in this field. First, the actual CR transport processes, andtheir coupling to the gas, remain deeply uncertain (owing to theextremely complicated plasma processes involved) – the physicsthat gives rise to some “effective diffusivity” and/or CR stream-ing is still debated (see Strong et al. 2007; Zweibel 2013; Gre-nier et al. 2015), and there is no widely-accepted a priori modelwhich predicts the relevant transport coefficients in the way of, say,Spitzer-Braginskii conductivity and viscosity. There are some em-pirical constraints from e.g. γ -ray emission in nearby galaxies ormore detailed products (e.g. spallation) in the MW, but critically “luminous” galaxy to around the virial radius, i.e. from ∼ − ∼ − (cid:46) − ∼ (cid:13) , 000–000 Hopkins et al. any inferred constraint on the “effective diffusion coefficient” or“streaming speed” of CRs is strongly model-dependent (as it de-pends on the density distribution the CRs propagate through, themagnetic field configuration, etc.). Really, one must forward-modelthese constraints in any galaxy model, to test whether the adoptedCR transport assumptions are consistent with the observations. Sec-ond, almost all previous studies of CRs on galaxies either focusedon (a) idealized “patches” of the ISM or CGM, ignoring the globaldynamics of accretion, outflows, star formation, etc., or (b) galaxysimulations with (intentionally) highly-simplified models for theturbulent, multi-phase ISM, star formation, stellar feedback fromsupernovae, stellar winds, radiation, and more. But these details arecritical for determining the balance of CR heating and cooling, howCRs will be trapped or escape galaxies, whether CRs will influenceoutflows or gas in the CGM “lofted up” by other processes, andwhether CRs ultimately matter compared to the order-of-magnitudelarger energy input in mechanical (thermal+kinetic) form in SNe.To give an extreme example: almost anything will have a large ef-fect relative to a “baseline” model which includes weak or no stellarfeedback. It is much less clear whether the inclusion of CR physicswill “matter” once mechanical and radiative feedback from stars isalready accounted for.Working towards this goal, Chan et al. (2018) performed andpresented the first simulations combining the specific physics fromthe FIRE simulations, described above, with explicit CR injec-tion and transport, accounting for advection and fully-anisotropicstreaming and diffusion, as well as hadronic and Coulomb colli-sional and streaming (Alfvén) losses. They systematically variedthe transport coefficients and treatment, and compared with ob-servational constraints, to argue that – at least given this particu-lar physics set and treatment of CR transport – the observationsrequired diffusivities (cid:38) cm s − , and that within the allowedrange of diffusivities, the effects on galaxy star formation rates andgas density distributions were modest. However, these simulationswere restricted to non-cosmological, isolated galaxies, representa-tive of just a couple of z = >
150 fully-cosmological simulations spanning halo masses from ultra-faintdwarfs through MW-mass systems at a range of redshifts (reach-ing ∼ pc-scale resolution), and systematically exploring all of thephysics above. Specifically, we compare our standard physics as-suming hydrodynamics, to simulations with explicit MHD andanisotropic conduction and viscosity as in Su et al. (2017), and sim-ulations with all of the above plus explicit treatment of cosmic raysas in Chan et al. (2018). We moreover systematically survey thetreatment of CR transport physics and coefficients, and comparewith observations where possible to constrain the allowed rangeof assumptions. Our intention here is to identify which physicsmight have an influence on bulk galaxy properties (e.g. SFRs, stel-lar masses, morphologies), and where uncertain parameters exist(e.g. CR diffusivities), what range of those parameters is allowedand how the effects (if any) on galaxies depend on them within theallowed range. We also limit our study to dwarf and ∼ L ∗ galax-ies where it is widely believed that SNe dominate the CR injection.In companion papers (e.g. Su et al. 2018d, Su et al., in prep.) wewill study the complementary role of AGN injecting CRs in muchmore massive galaxies, and in other companion papers (e.g. Ji et al.2019 and Chan et al., in prep) we will study the (potentially much larger) effects of CRs on the CGM around galaxies and the originand properties of the weak, CR-driven outflows.In § 2 we review the numerical methods and describe the sim-ulation suite. Before analyzing the simulations, § 3 presents a sim-ple analytic model for the effects and equilibrium distribution ofCRs, given our assumptions in the simulations, which allows us topredict and estimate (with surprising accuracy) many of the scal-ings we will observe in the cosmological simulations. § 4 brieflypresents the key results from the simulations, which we discuss andanalyze in more detail – attempting to break down the effects of dif-ferent physics on different scales – in § 5. Finally, we summarizeand conclude in § 6. The simulations in this paper were run with the multi-physics code
GIZMO (Hopkins 2015), in its meshless finite-mass MFM mode.This is a mesh-free, finite-volume Lagrangian Godunov methodwhich provides adaptive spatial resolution together with conser-vation of mass, energy, momentum, and angular momentum, andthe ability to accurately capture shocks and fluid mixing insta-bilities (combining advantages of both grid-based and smoothed-particle hydrodynamics methods). We solve the equations of idealmagneto-hydrodynamics (MHD), as described and tested in de-tail in Hopkins & Raives (2016); Hopkins (2016), with fully-anisotropic Spitzer-Braginskii conduction and viscosity and otherdiffusion operators implemented as described in Hopkins (2017).Gravity is solved for gas and collisionless (stars and dark matter)species with fully-adaptive Lagrangian force softening (so hydro-dynamic and force resolutions are consistently matched).Our simulations are fully-cosmological “zoom-in” runs, witha high-resolution Lagrangian region identified surrounding a z = We note that the high-resolution vol-umes reach as large as ∼ (
10 Mpc ) in the largest runs, so there aremany galaxies present (our set has hundreds of galaxies with > ∼
30 zoom-in vol-umes, because we systematically vary the physics and CR parame-ters like the diffusion coefficient, our default simulation set includeswell over 100 full-physics high-resolution simulations. However, tosimplify our analysis and presentation, avoid ambiguities in galaxymatching and separating systematic differences between satelliteand field galaxies, and to focus on the best-resolved galaxies possi-ble, we focus only on the most massive “primary” galaxies in eachbox. We note though that a brief comparison indicates that our con-clusions appear to apply to all galaxies in the box. Unless otherwisespecified (e.g. Table 2), all are run to z = All our simulations here include the physics of cooling, star for-mation, and stellar feedback from the FIRE-2 version of the Feed-back in Realistic Environments (FIRE) project, described in detailin Hopkins et al. (2018b), but briefly summarized here. A public version of GIZMO is available at For the MUSIC (Hahn & Abel 2011) files necessary to generate all ICshere, see: c (cid:13) , 000–000 osmic Rays on FIRE [ Gyr ] − − S F R [ M (cid:30) y r − ] m09 a l o g ( M ∗ ) [ M (cid:30) ] ( κ = e ) CR+ ( κ = e ) a − − − (cid:28) [ Z ∗ / H ] (cid:27) -3.2-3.1-3.4-3.2 − ( r ) [ kpc ] l o g ( ρ [ r ] ) [ M (cid:30) k p c − ] r [ kpc ] V c [ k m s − ] m i ,1000 = [ Gyr ] − − − m10v a a − − − − ( r ) [ kpc ]
468 0.40.30.40.50 2 4 r [ kpc ] m i ,1000 = [ Gyr ] − − − m10q a
56 6.46.46.06.30.0 0.25 0.5 0.75 1.0Scale Factor a − − − ( r ) [ kpc ]
468 0.70.80.50.80 2 4 r [ kpc ] m i ,1000 = [ Gyr ] − − m11b a
678 7.97.97.87.90.0 0.25 0.5 0.75 1.0Scale Factor a − − − − ( r ) [ kpc ]
468 2.22.41.72.10 10 20 30 r [ kpc ] m i ,1000 = Figure 1.
Properties of our default physics “suites.” We compare different galaxies (columns, from Table 1), simulated with different physics “suites” (linesas labeled, from Table 3). Recall, all include the same cooling, stellar feedback, etc, but the MHD+ runs include MHD and fully-anisotropic conduction& viscosity, and the CR+ runs include MHD, conduction, viscosity, CR injection, losses, streaming, and diffusion (with fixed CR diffusion constant κ aslabeled). We compare: Top:
Star formation history (averaged in 100Myr intervals) of the primary ( z =
0) galaxy.
Second:
Total stellar mass in box (dominatedby primary) vs. scale factor ( a = / ( + z ) ). The logM ∗ / M (cid:12) value at z = Middle:
Stellar mass-weightedaverage metallicity vs. scale factor ( z = Fourth:
Baryonic ( thick ) and total ( thin ) mass density profiles (averaged in spherical shells) as afunction of radius around the primary galaxy at z =
0. Number is the stellar effective (1 / z = Bottom:
Rotation curves (circularvelocity V c versus radius) in the primary galaxy. Value m i , = m i / M (cid:12) of the mass resolution is shown. The galaxies here are dwarfs, from lowest-to-highest mass. Though there are some effects (e.g. m10v rises to higher stellar mass in its initial high- z burst in the CR runs, and m10q has a lower M ∗ in one CR run), they are well within the range of stochastic run-to-run variations. There does not appear to be a large systematic effect of CRs or MHD orconduction/viscosity (perhaps a small suppression of M ∗ with CRs and low diffusion coefficients, but see Fig. 15). This Figure is continued in Figs. 2-3. Gas cooling is followed from T = − K including free-free, Compton, metal-line, molecular, fine-structure and dust col-lisional, and we also follow gas heating from photo-electric andphoto-ionization by both local sources and a uniform meta-galacticbackground including the effect of self-shielding. We explicitlyfollow 11 different abundances, including explicit treatment ofturbulent diffusion of metals and passive scalars as in Colbrooket al. (2017); Escala et al. (2018). Gas is turned into stars usinga sink-particle prescription: gas which is locally self-gravitating As detailed in Hopkins et al. (2018b) Appendix B, in our runs that do not include explicit CR transport (“Hydro+” and “MHD+”), the cool-ing/ionization tables do assume a uniform MW-like CR background ( ∼ − ) for gas at densities > . − . This is generally negligiblefor heating, but is important for e.g. the small ionized fraction in GMCs.In our runs with explicit CR transport, these terms are replaced with theexplicitly-evolved CR background and collisional+streaming heating ratesdescribed below. at the resolution scale following Hopkins et al. 2013c, self-shielding/molecular following Krumholz & Gnedin 2011, Jeans un-stable, and denser than n crit > − is converted into star par-ticles on a free-fall time. Star particles are then treated as single-age stellar populations with all IMF-averaged feedback propertiescalculated from STARBURST99 (Leitherer et al. 1999) assuming aKroupa (2001) IMF. We then explicitly treat feedback from SNe(both Types Ia and II), stellar mass loss (O/B and AGB mass-loss), and radiation (photo-ionization and photo-electric heatingand UV/optical/IR radiation pressure), with implementations at theresolution-scale described in Hopkins et al. (2018b), Hopkins et al.(2018c), and Hopkins et al. (2018a).The simulations labeled “Hydro+” in this paper include all ofthe physics above, but do not include magnetic fields, physical con-duction or viscosity, or explicit treatment of cosmic rays. c (cid:13)000
Rotation curves (circularvelocity V c versus radius) in the primary galaxy. Value m i , = m i / M (cid:12) of the mass resolution is shown. The galaxies here are dwarfs, from lowest-to-highest mass. Though there are some effects (e.g. m10v rises to higher stellar mass in its initial high- z burst in the CR runs, and m10q has a lower M ∗ in one CR run), they are well within the range of stochastic run-to-run variations. There does not appear to be a large systematic effect of CRs or MHD orconduction/viscosity (perhaps a small suppression of M ∗ with CRs and low diffusion coefficients, but see Fig. 15). This Figure is continued in Figs. 2-3. Gas cooling is followed from T = − K including free-free, Compton, metal-line, molecular, fine-structure and dust col-lisional, and we also follow gas heating from photo-electric andphoto-ionization by both local sources and a uniform meta-galacticbackground including the effect of self-shielding. We explicitlyfollow 11 different abundances, including explicit treatment ofturbulent diffusion of metals and passive scalars as in Colbrooket al. (2017); Escala et al. (2018). Gas is turned into stars usinga sink-particle prescription: gas which is locally self-gravitating As detailed in Hopkins et al. (2018b) Appendix B, in our runs that do not include explicit CR transport (“Hydro+” and “MHD+”), the cool-ing/ionization tables do assume a uniform MW-like CR background ( ∼ − ) for gas at densities > . − . This is generally negligiblefor heating, but is important for e.g. the small ionized fraction in GMCs.In our runs with explicit CR transport, these terms are replaced with theexplicitly-evolved CR background and collisional+streaming heating ratesdescribed below. at the resolution scale following Hopkins et al. 2013c, self-shielding/molecular following Krumholz & Gnedin 2011, Jeans un-stable, and denser than n crit > − is converted into star par-ticles on a free-fall time. Star particles are then treated as single-age stellar populations with all IMF-averaged feedback propertiescalculated from STARBURST99 (Leitherer et al. 1999) assuming aKroupa (2001) IMF. We then explicitly treat feedback from SNe(both Types Ia and II), stellar mass loss (O/B and AGB mass-loss), and radiation (photo-ionization and photo-electric heatingand UV/optical/IR radiation pressure), with implementations at theresolution-scale described in Hopkins et al. (2018b), Hopkins et al.(2018c), and Hopkins et al. (2018a).The simulations labeled “Hydro+” in this paper include all ofthe physics above, but do not include magnetic fields, physical con-duction or viscosity, or explicit treatment of cosmic rays. c (cid:13)000 , 000–000 Hopkins et al. [ Gyr ] − − S F R [ M (cid:30) y r − ] m11q a l o g ( M ∗ ) [ M (cid:30) ] ( κ = e ) CR+ ( κ = e ) a − − (cid:28) [ Z ∗ / H ] (cid:27) -0.6-0.6-0.7-0.6 − ( r ) [ kpc ] l o g ( ρ [ r ] ) [ M (cid:30) k p c − ] r [ kpc ] V c [ k m s − ] m i ,1000 = [ Gyr ] − − m11d a
789 9.69.79.49.20.0 0.25 0.5 0.75 1.0Scale Factor a − − − ( r ) [ kpc ]
468 6.55.55.24.00 10 20 30 r [ kpc ] m i ,1000 = [ Gyr ] − − m11h a
789 9.69.79.49.50.0 0.25 0.5 0.75 1.0Scale Factor a − − − ( r ) [ kpc ]
468 4.03.43.93.50 10 20 30 r [ kpc ] m i ,1000 = [ Gyr ] − m11f a
810 10.410.510.510.10.0 0.25 0.5 0.75 1.0Scale Factor a − −
10 -0.00.10.0-0.2 − ( r ) [ kpc ] r [ kpc ] m i ,1000 = Figure 2.
Fig. 1, continued to higher stellar and halo masses. There is no systematic discernable effect of MHD/conduction/viscosity. CRs may have a smallsystematic effect at low diffusion coefficient. At high κ ∼ × cm s − , they have a larger effect, most pronounced at late times ( z <
1) in the SFRs (cid:29) . M (cid:12) yr − . The effect gets larger at larger stellar masses. Our simulations labeled “MHD+” in this paper include all of the“Hydro+” physics (e.g. radiative cooling, star formation, stellarfeedback), but add magnetic fields and physical, fully-anisotropicconduction and viscosity. These are described in Su et al. (2017)but we briefly summarize here.As noted above, for magnetic fields we solve the equationsof ideal MHD, as described in Hopkins & Raives (2016); Hopkins(2016). We can optionally include ambipolar diffusion, the Hall ef-fect, and Ohmic resistivity as in Hopkins 2017, but these are com-pletely negligible at all resolved scales in our simulations here. For conduction and viscosity we include the physical scalingsfor fully-anisotropic Spitzer-Braginskii transport, with the conduc-tive heat flux κ cond ˆ B (ˆ B · ∇ T ) , where κ cond ≡ . k B ( k B T ) / m / e e ln Λ c f i + ( . + β/ ) (cid:96) e /(cid:96) T (1) We have, in fact, run one m10q and one m12i run from z = . z = ∼ (Spitzer & Härm 1953; Braginskii 1965), where m e and e arethe electron mass and charge, k B the Boltzmann constant, f i isthe ionized fraction, ln Λ c ∼
37 is a Coulomb logarithm and (cid:96) e = / ( k B T ) / n e π / e ln Λ c is the electron deflection length(Sarazin 1988), (cid:96) T ≡ T / | ˆ B · ∇ T | is the parallel temperature gradi-ent scale-length, and the plasma β ≡ P thermal / P magnetic is the usualratio of thermal-to-magnetic pressure. Note that the (cid:96) e /(cid:96) T term en-sures proper behavior in the saturated limit (and a smooth transitionbetween un-saturated and saturated limits, e.g. Cowie & McKee1977), while the β term accounts for micro-scale plasma instabil-ities (e.g. the Whistler instability) limiting the heat flux in high- β plasmas (see Komarov et al. 2018). For viscosity we modify themomentum and energy equations with the addition of the viscousstress tensor Π ≡ − η visc (ˆ B ⊗ ˆ B − I / ) (ˆ B ⊗ ˆ B − I / ) : ( ∇ ⊗ v ) (where ⊗ is the outer product, I the identity matrix, and : thedouble-dot-product), with η visc ≡ . m / i ( k B T ) / ( Z i e ) ln Λ c f i + ( + β − / ) (cid:96) i /(cid:96) | v | (2) − P magnetic < η visc (ˆ B ⊗ ˆ B ) : ( ∇ ⊗ v ) < P magnetic (3)where like the above m i , (cid:96) i , and Z i e are the mean ion mass,deflection-length, and charge, and (cid:96) | v | ≡ | v | / | ˆ B ⊗ ˆ B : ∇ ⊗ v | isthe parallel velocity gradient scale-length. The upper and lower al-lowed values of the anisotropic stress are limited (capped) accord-ing to the latter expression, to account again for plasma instabilities c (cid:13) , 000–000 osmic Rays on FIRE [ Gyr ] − S F R [ M (cid:30) y r − ] m11g a l o g ( M ∗ ) [ M (cid:30) ] ( κ = e ) CR+ ( κ = e ) a − − (cid:28) [ Z ∗ / H ] (cid:27) − ( r ) [ kpc ] l o g ( ρ [ r ] ) [ M (cid:30) k p c − ] r [ kpc ] V c [ k m s − ] m i ,1000 =
12 0 5 10Cosmic Time [ Gyr ] m12i a
810 10.810.810.910.40.0 0.25 0.5 0.75 1.0Scale Factor a − −
10 0.10.10.3-0.1 − ( r ) [ kpc ] r [ kpc ] m i ,1000 = [ Gyr ] − m12f a
810 10.910.911.010.50.0 0.25 0.5 0.75 1.0Scale Factor a − −
10 0.10.10.2-0.2 − ( r ) [ kpc ]
510 3.94.42.73.40 20 40 60 r [ kpc ] m i ,1000 = [ Gyr ] − m12m a
810 11.111.111.010.50.0 0.25 0.5 0.75 1.0Scale Factor a − −
10 0.20.20.2-0.2 − ( r ) [ kpc ]
510 4.64.85.27.50 20 40 60 r [ kpc ] m i ,1000 = Figure 3.
Figs. 1-2, continued to Milky Way-mass ( M halo ∼ M (cid:12) ) halos. Again MHD/conduction/viscosity have no discernable effect. CRs with lowerdiffusion coefficients (cid:46) cm s − also produce no systematic effect. But CRs with higher diffusion coefficients produce substantial suppression of theSFRs and stellar masses above M ∗ (cid:38) M (cid:12) . This in turn strongly reduces the central "spike" in the rotation curves in these galaxies. The final ( z =
0) stellarmasses and SFRs are suppressed by factors ∼ (the mirror and firehose at positive and negative anisotropy, respec-tively) limiting the flux (see Squire et al. 2017c,a,b).These numerical methods have been extensively tested anddiscussed in previous papers, to which we refer for details (e.g.Hopkins & Raives 2016; Hopkins 2016, 2017; Hopkins & Conroy2017; Su et al. 2017, 2018a; Lee et al. 2017; Colbrook et al. 2017;Seligman et al. 2018). We note that, given the mass resolution andLagrangian nature of the code, the Field (1965) length λ F (approx-imately, the scale below which thermal conduction is faster thancooling) is resolved ( ∆ x < λ F ) in the simulations here in gas hotterthan T (cid:38) × K ( n / .
01 cm − ) . . Our simulations labeled “CR+” in this paper include all of the“MHD+” physics, and add our “full physics” treatment of CRs.The CR physics is described in detail in Chan et al. (2018) but weagain summarize here: we include injection in SNe shocks, fully-anisotropic CR transport with streaming and advection/diffusion,CR losses (hadronic and Coulomb, adiabatic, streaming), and self-consistent CR-gas coupling.CRs are treated as an ultra-relativistic fluid (adiabatic index The fluid limit adopted here is motivated by the fact that ∼ GeV CRgyro radii r L (cid:28) au are always much smaller than any resolved spatial γ cr = /
3) in the “single bin” approximation, which we can thinkof either as evolving only the CR energy density ( e cr ) at ∼ GeV en-ergies that dominate the CR pressure ( P cr ≡ ( γ cr − ) e cr ), or equiv-alently assuming a universal CR energy spectral shape. CR pres-sure contributes to the total pressure and effective sound speed inthe Riemann problem for the gas equations-of-motion accordingto the local strong-coupling approximation , i.e. P = P gas + P cr and c s , eff = ∂ P /∂ρ = c s + γ cr P cr /ρ . Integrating over the CR distributionfunction and energy spectrum, we evolve the CR energy density asMcKenzie & Voelk (1982): ∂ e cr ∂ t + ∇ · F cr = (cid:104) v cr (cid:105) · ∇ P cr + S cr − Γ cr (4)where S cr and Γ cr are source and sink terms; (cid:104) v cr (cid:105) ≡ v gas + v stream is the bulk CR advection velocity, de-composed into the gas ve-locity v gas and the “streaming velocity” v stream ; and F cr is the lab-frame or total CR energy flux which can be de-composed into scales, and CR “deflection lengths” or scattering lengths ∼ κ/ c ∼ pc arealso always smaller than e.g. the CR pressure gradient scale lengths. How-ever, we emphasize that fundamental questions remain about the validityof other assumptions (like the form of the CR distribution function) whenthe CR diffusivity is large (scattering rates are low), particularly when | κ (cid:107) ∇ (cid:107) P cr | (cid:29) v A ( e cr + P cr ) , which occurs in our favored models at galacto-centric radii (cid:28) (cid:13) , 000–000 Hopkins et al. F cr ≡ (cid:104) v cr (cid:105) ( e cr + P cr ) + F di ≡ v gas ( e cr + P cr ) + ˜ F cr , where ˜ F cr is theflux in the fluid frame (with F di the “diffusive” flux).For S cr , we assume CR injection in SNe shocks, with a fixedfraction (cid:15) cr of the initial ejecta kinetic energy ( ≈ erg) of ev-ery SNe going into CRs. In our default simulations, (cid:15) cr = . Γ cr , we fol-low Guo & Oh (2008) and account for both hadronic/catastropicand Coulomb losses with Γ cr = e cr n n ( + . x e ) / t n , where n n is the nucleon number density, t n ≡ . × s cm − , and x e is the number of free electrons per nucleon. A fraction ∼ / Q gas = e cr n n ( . + . x e ) / t n .As shown in Chan et al. (2018) the remaining terms in Eq. 4can be decomposed (in Lagrangian form) into simple advection(automatically handled in our Lagrangian formulation) and adia-batic “PdV work” terms (solved in an exactly-conservative man-ner with our usual MFM solver), the ˜ F cr term (see below), anda “streaming loss” term v A · ∇ P cr which is negative definite andrepresents energy loss via streaming instabilities that excite high-frequency Alfvén waves (frequency of order the CR gyro fre-quency, well below our resolution; Wentzel 1968; Kulsrud &Pearce 1969) that damp and thermalize almost instantaneously,so the energy lost via this term is added to the gas thermal en-ergy each timestep. The streaming velocity always points downthe CR pressure gradient, projected along the magnetic field, so v stream = − v stream ˆ B (ˆ B · ˆ ∇ P cr ) .It is widely argued that micro-scale instabilities regulate thestreaming speed to of order the Alfvén speed (Skilling 1971; Kul-srud 2005; Yan & Lazarian 2008; Enßlin et al. 2011), althoughsuper-Alfvénic streaming can easily emerge in self-confinementmodels for CR transport (Wentzel 1968; Holman et al. 1979;Achterberg 1981; Wiener et al. 2013a; Lazarian 2016); moreoverin partially-neutral gas (ionized fraction f ion <
1) there is an am-biguity about whether the appropriate Alfvén speed is the ideal-MHD Alfvén speed v A ≡ | B | / ( π ρ ) / or the (larger) ion Alfvénspeed v ion A ≡ | B | / ( π ρ ion ) / ∼ f − / v A (Skilling 1975; Zweibel2013; Farber et al. 2018b). So we simply adopt the ad-hoc v stream ≈ v A as our default, although we vary this widely below setting v stream = v A , disabling streaming entirely, or allowing highly super-sonic/Alfvénic streaming with v stream = ( v A + c s ) / (several timesthe fastest possible MHD wavespeed), and show (both here and inChan et al. 2018) it has little effect on our conclusions.Following Chan et al. (2018) we treat ˜ F cr using a two-momentscheme (similar to other recent implementations by e.g. Jiang & Oh2018 and Thomas & Pfrommer 2018), solving1˜ c (cid:20) ∂ ˜ F cr ∂ t + ∇ · (cid:0) v gas ⊗ ˜ F cr (cid:1)(cid:21) + ∇ (cid:107) P cr = − ( γ cr − ) κ ∗ ˜ F cr (5)where ∇ (cid:107) P cr = (ˆ B ⊗ ˆ B ) · ( ∇ P cr ) = ( γ cr − ) ˆ B (ˆ B ·∇ e cr ) is the paral-lel derivative of the CR pressure tensor, ˜ c is the maximum allowedCR “free streaming” speed, and κ ∗ ≡ κ (cid:107) + γ cr v st P cr / |∇ (cid:107) P cr | is As discussed at length in Chan et al. (2018) and studied in the Appendicestherein, we have considered simulations where the “streaming losses” scaleeither as v stream · ∇ P cr or v A · ∇ P cr . The latter (streaming losses limited tothe Alfvén speed, as they arise from damped Alfvén waves) is our defaultchoice, even if v stream > v A . However we do not find this choice significantlyalters any of our conclusions in this paper. Following Chan et al. (2018) we note that ˜ c is a nuisance parameter (the the effective parallel diffusivity (we are implicitly taking the per-pendicular κ ⊥ = F cr → − κ ∗ ∇ (cid:107) e cr = κ (cid:107) ∇ (cid:107) e cr + v stream ( e cr + P cr ) ), but unlike a pure-diffusion equation (where one forces ˜ F cr to always be exactly − κ ∗ ∇ (cid:107) e cr ) ) it correctly handlesthe transition between streaming and diffusion and prevents un-physical super-luminal CR transport. As discussed in Chan et al. (2018), if the “streaming loss” termis limited or “capped” to scale with the Alfvén speed ( ∼ v A ∇ P cr ;see above), and streaming is super-Alfvénic ( v stream (cid:29) v A ), thenonly the “effective” diffusivity κ ∗ (which can arise from a combina-tion of microphysical diffusion and/or streaming) – as compared to κ (cid:107) or v stream individually – enters the large-scale dynamics. This ef-fective κ ∗ , or equivalent CR transport speed v cr , eff ∼ κ ∗ |∇ P cr | / P cr ,is what we actually constrain in our study here.There are many approximations in this description, and theeffective “diffusion coefficient” κ ∗ for CRs on these (energy, spa-tial, and time) scales remains both theoretically and observationallyuncertain. Therefore, we treat κ as a constant but vary it systemati-cally in a parameter survey, with values motivated by the compari-son with observational constraints in Chan et al. (2018). In this section, we develop a simple theoretical “toy model” forCRs, which provides considerable insight into the phenomena wesee in the simulations.Assume a galaxy has a quasi-steady SFR ˙ M ∗ , so the as-sociated CR injection rate is ˙ E cr = (cid:15) cr (cid:15) SNe ˙ M ∗ , where (cid:15) SNe ∼ ( erg / M (cid:12) ) is the time-and-IMF-averaged energetic yield persolar mass of star formation from SNe (for the IMF here includingType-II and prompt Ia events). Also assume that (since we are pri-marily interested in large scales) the CR injection is concentratedon relatively small scales, so it can be approximated as point-like,and assume – for now – that diffusion with constant effective co-efficient ˜ κ ∼ (cid:104)| ˆ B · ˆ ∇ e cr | (cid:105) κ ∗ ∼ κ ∗ / e cr = ˙ E cr π ˜ κ r (6)This gives rise to the CR pressure gradient ∇ P cr = − e cr ˆ r / ( r ) .Now also assume, for simplicity, the (diffuse) gas and DMare in an isothermal sphere (the detailed profile shape is not im-portant) with gas fraction f gas and circular velocity V c ∼ V max ,with some characteristic halo scale radius R s ∼ R vir / c . The ra-tio of the CR pressure gradient to the gravitational force at R s is |∇ P cr | / | ρ ∇ Φ | ∼ ˙ E cr G R s / ( κ f gas V ) (this is identical up to an simulations evolve to identical solutions independent of ˜ c , so long as it isfaster than other bulk flow speeds in the problem), so rather than adopt themicrophysical ˜ c = c (speed of light), we adopt ˜ c = − by default(but show below that our solutions are independent of ˜ c over a wide range). Super-luminal CR transport would occur in the “pure-diffusion” approx-imation for ˜ F cr wherever the resolution scale ∆ x (cid:46) κ/ c ∼ ( κ/ × cm s − ) . The simulations in this paper routinely reach this or betterspatial resolution, so this distinction is important. Eq. 6 is still approximately valid (up to an order-unity constant) in mostcases for a non-constant ˜ κ or effective transport/streaming speed v cr (re-placing ˜ κ → v cr r ). This follows from the fact that the radial CR flux (ina spherical hydrostatic system with no losses) is F cr → − ¯ κ ( ∂ e cr /∂ r )ˆ r ∼− ¯ κ ( e cr / r )ˆ r ∼ v cr e cr ˆ r . c (cid:13) , 000–000 osmic Rays on FIRE Figure 4.
Top:
Predicted ratio of γ -ray luminosity from hadronic collisionsof CRs ( L γ ) to luminosity from star formation/massive stars ( L SF ), as afunction of the central gas surface density of the galaxy. Points show eachsnapshot (every ∼ z <
1, while dashed lines show the 1 σ el-lipsoid for each galaxy (labeled). We show the galaxies in Figs. 1-3 (ha-los (cid:46) M (cid:12) continue the trend but fall off the plot to smaller Σ central ).We compare the observed points from the MW, Local Group, and otherFermi detections (black squares with error bars) from Lacki et al. (2011).Horizontal dashed line is the steady-state, constant-SFR calorimetric limit.The κ ∼ × cm s − runs appear to agree well with observations;lower κ produces excessive γ -ray flux. Middle:
Same, for just three galax-ies (each at distinct Σ central , as labeled) varying κ . Increasing κ systemati-cally lowers L γ , on average. Bottom:
Same, comparing runs with lower κ but default (Alfvénic) streaming ( blue ) or super-sonic/Alfvénic streaming( v stream = ( v A + c s ) / ; red ). Faster streaming reduces L γ (as CRs escapefaster and thermalize energy in streaming losses which do not appear in γ -rays), but for the values here the effect is equivalent to a small (factor < κ . order-unity constant if we assume the gas is in e.g. an NFW profileor a Mestel disk). Now recall, ˙ E cr ∝ ˙ M ∗ . For star-forming (sub- L ∗ )galaxies, ˙ M ∗ = α M ∗ / t Hubble where t Hubble ( z ) is the Hubble time atredshift z and α ≈ − for halo R s , R vir ,and M halo = M vir , we can re-write this ratio as: |∇ P cr || ρ ∇ Φ | ∼ ˙ E cr G R s κ f gas V ∼ . α (cid:15) cr , . f gas , . ˜ κ ( + z ) / (cid:18) M ∗ f b M halo (cid:19) (7)where (cid:15) cr , . ≡ (cid:15) cr / . f gas , . ≡ f gas / .
1, ˜ κ ≡ ˜ κ/ cm s − ,and f b ≡ Ω b / Ω m is the universal baryon fraction, so M ∗ / f b M halo isthe stellar mass relative to what would be obtained converting allbaryons into stars.This leads immediately to the prediction that CRs can have alarge effect in relatively massive (intermediate, LMC-like throughMW-mass; M halo ∼ − M (cid:12) ) halos, at low-to-moderate red-shifts ( z (cid:46) − M ∗ / M halo is a strongly-increasing function ofmass (roughly, M ∗ ∝ M at low masses). So for a MW mass halo,CR pressure is approximately able to balance gravitational forcesfor ˜ κ ∼
1, while for a true dwarf halo like m10q , the CR pres-sure will be order-of-magnitude too small, because the SFR (hencethe CR injection rate, which is proportional to the CR energy den-sity/pressure/pressure gradient in steady state) is several orders ofmagnitude smaller in such a tiny dwarf. We also show below thatin dwarfs, CRs escape efficiently from both the galaxy and CGM,while SNe cool less efficiently, all of which make CRs relativelyless-dominant compared to mechanical stellar feedback.Note also that our derivation above, at face value, would implythat a lower diffusion coefficient would give stronger effects of CRpressure (because the CRs are more “bottled up” so the steady-statepressure is larger). However, we neglected collisional and stream-ing losses: we show below that if ˜ κ (cid:28)
1, these quickly dominateand prevent CRs from having any significant effects. κ & Observational Constraints We neglected collisional losses above. Assuming ionized gas (thedifference is small, this just determines the contribution fromCoulomb terms) these scale as ˙ e loss = e cr n / ( t n ) with t n ∼ . × s cm − (§ 2.3). If we take the same isothermal spheremodel above, and calculate the steady-state volume-integrated CRloss rate ˙ E loss , we obtain: ˙ E loss ˙ E cr ≈ ln ( r max / r min ) π ˜ κ m p t n f gas V G ∼ . κ (cid:18) Σ gas R gas .
01 g cm − kpc (cid:19) (8)where in the latter equality we use f gas V / G = M gas / R gas ≈ π Σ gas R gas (and ln ( r max / r min ) ∼ L ∗ , star-forming) galactic disks have approximatelyconstant effective Σ gas ∼ − M (cid:12) pc − and R gas ∼ R ∗ ∼ For Eq. 7, we approximate the system around the radii of interest asan isothermal sphere with V c ≈ V max , gas ρ = f gas V c / ( π Gr ) , |∇ Φ | = GM enc / r = V c / r , and V max ≈ ( GM vir / R vir ) / (a reasonable approx-imation for NFW-type halos with concentrations ∼ − ˙ E cr = (cid:15) cr u SNe ˙ M ∗ = (cid:15) cr u SNe ( α M ∗ / t Hubble [ z ] ) (with u SNe ≈ erg / M (cid:12) ).For simplicity we adopt redshift scalings for a matter-dominated Universe: M halo = M vir ≡ ( π/ )∆ c ρ c R with ∆ c ≈
180 (and ρ c = H ( + z ) / π G the critical density) and t Hubble [ z ] ≈ ( + z ) − / Gyr, and eval-uate everything at a halo scale radius r ∼ R s ∼ . R vir , but the choice ofcosmology or exact radius do not qualitatively alter our conclusions.c (cid:13) , 000–000 Hopkins et al.
Figure 5.
Gas pressure profiles for m10q ( top ), m11b ( middle ), m12i ( bottom ), runs “Hydro+”, “MHD+”, “CR+( κ = e κ = e z =
0. In each, solid lines show volume-averaged profiles (in spherical shells); shaded range shows the 5 −
95% inclusion interval of all resolutionelements (gas-mass-weighted sampling) at each radius around the galaxy. We compare thermal ( nk B T ), magnetic ( | B | / π ), CR ( ( γ cr − ) e cr ), “gravitational”( ≡ ρ V c / ≡ (cid:104) ρ | v | /
2) pressures. Magnetic pressure is sub-dominant to thermal ( β (cid:29)
1) especially at large radii, as expected. Both (andCRs) are well below gravity within the disk, where gas is primarily rotation-supported ( | v | ∼ V c ) – i.e. gas is primarily in a thin or turbulent structure inside < r (in massive galaxies), supporting denser and/or cooler gas which wouldotherwise accrete onto the galaxy. Dashed line shows the analytic prediction from § 3 for CR pressure with negligible collisional losses. This is an excellentapproximation at high- κ (the “turnover” at ∼ −
100 kpc is where streaming begins to dominate transport). At low- κ , the CR pressure is much lower,indicating that most CR energy has been lost. Although CRs still balance gravity, the lower energy means gas has already cooled onto the galaxy (formingstars) so the remaining “weight” (magnitude of gravitational pressure to be supported) is much lower.
10 kpc ( M ∗ / M (cid:12) ) / (e.g. Courteau et al. 2007), we can equiv-alently write: ˙ E loss ˙ E cr ∼ κ (cid:18) Σ gas M (cid:12) pc − (cid:19) (cid:18) M ∗ M (cid:12) (cid:19) / (9)Of course, ˙ E loss cannot exceed ˙ E cr in steady-state: this gives an up-per bound to the expression above at the“calometric limit” ( ˙ E loss = ˙ E cr ) at which point all input CR energy is lost to collisions.The ratio ˙ E loss / ˙ E cr is directly related to the observed ratio ofGeV γ -ray flux or luminosity ( F γ ∝ L gamma ∝ ˙ E loss ) to bolometricflux from massive stars ( F SF ∝ L SF ∝ ˙ M ∗ ∝ ˙ E cr ). Using the valuesgiven in § 4 below for conversion factors between collisional lossesand γ -ray luminosity or flux, Eq. 8 becomes: F γ F SF ∼ × − ˜ κ (cid:18) Σ gas R gas .
01 g cm − kpc (cid:19) (10)with the calorimetric limit “capping” this at F γ / F SF ∼ × − .Note that the geometry of the gas is not especially importanthere. If we assume instead the gas is in a thin, exponential diskwith scale height H / R (cid:28)
1, and effective radius R e , but the CRs diffuse approximately spherically (as a random walk), then we ob-tain, a result which differs only by an order-unity constant from theisothermal-sphere scaling above even as H / R → Σ , the three-dimensional density n in the disk, hencethe CR loss rate while within it, increases inversely with H / R ).Three immediate consequences follow from this. First, es-sentially all observed galaxies with SFRs below ∼
10 M (cid:12) yr − (including the MW, LMC, SMC, M31, and M33) have observed F γ / F SF ≤ − (Lacki et al. 2011), so we predict that a diffusioncoefficient ˜ κ (cid:38) required to match the observations. The same˜ κ is required to reproduce the canonical Milky Way constraints –most recent studies agree that for a MW with a diffuse gaseoushalo of scale length (cid:38)
10 kpc (appropriate for the simulations here,since the galaxies have extended halos and the diffusivity is, by as-sumption, constant) an effective, isotropically-averaged diffusivity˜ κ (cid:38) c (cid:13) , 000–000 osmic Rays on FIRE Figure 6.
Profiles of gas heating/cooling rate around the central galaxy, at z =
0, as Fig. 5. We show the total gas cooling rate, vs. the heating rate fromCRs via collisional (hadronic+Coulomb) and streaming losses. Hadroniclosses ( ∝ n gas ) dominate in dense gas (e.g. within the disks at r (cid:46) T (cid:46) K and so cools relatively rapidly). Streaming rates for theruns with super-Alfvénic streaming are larger but the difference is negligi-ble compared to gas cooling.
Cummings et al. 2016; Korsmeier & Cuoco 2016; Evoli et al. 2017;Amato & Blasi 2018). Second, we see why low diffusion coefficients cannot be in-voked to increase the CR pressure (as noted above) – if one lowers˜ κ (cid:28)
1, then not only will the model fail to reproduce the observa-tions, but the collisional losses will quickly dominate. If ˙ E loss (cid:38) ˙ E cr ,it means that all CR energy is rapidly lost in the ISM, so there isno steady-state, high-pressure CR halo (which requires they escapethe galaxy in the first place). In small dwarfs with low densities For the MW, we note that the observations (e.g. secondary-to-primary ra-tios and the like) do not really constrain the “diffusion coefficient” or “resi-dence time” (these are model-dependent inferences), but rather the effectivecolumn density or “grammage” X s ≡ (cid:82) CRpath n gas d (cid:96) CR = (cid:82) CRpath n gas cdt integrated over the path of individual CRs from their source locationsto the Earth, with X s ≈ × cm − measured. Repeating our calcu-lation above for either an isothermal sphere or thin-disk gas distribution,it is straightforward to show that the grammage (integrated to infinity, asopposed to the solar circle) is directly related to the hadronic losses as ˙ E loss / ˙ E cr = X ∞ s / ( t n c ) ∼ . ( X ∞ s / × cm − ) . Thus the con-straints from matching the direct MW observations are essentially equiva-lent to matching the observationally “inferred” F γ / F SF ∼ ( . − ) × − in the MW. (low Σ gas R gas ) one may be able to make ˜ κ slightly lower withoutlosing all of the CR energy, but we show below this still leads tolarge violations of the observational constraints.Third, this means we do not expect CRs to (at least locally)have strong effects in extreme starburst-type galaxies, at either lowor high redshift (where they are more common), owing to the veryhigh Σ gas observed in such systems, which should lead to efficientlosses (see Chan et al. 2018 for more discussion). Now consider the streaming terms explicitly. If the streaming ve-locity is ∼ v A ∼ β − / c s , and β is approximately constant (as onemight expect from e.g. a transsonic turbulent dynamo), then in anisothermal halo v A is also constant, so the radial streaming “flux” is ∼ v A e cr . The diffusive flux is ∼ ˜ κ ∇ e cr ∼ ˜ κ e cr / r , so the streamingdominates at r (cid:38) r stream ∼ ˜ κ/ v A (note we could have derived thisinstead in terms of the transport timescales to reach a given radius,and would reach the identical conclusion). This gives: r stream ∼
30 kpc ˜ κ v A , ∼
30 kpc β / ˜ κ M / , ( + z ) / (11)where v A , ≡ v A /
10 km s − , β ≡ β/ M vir , ≡ M halo / M (cid:12) .Beyond this radius, if we continue to neglect losses, the solu-tion would become that for a steady-state wind with constant ve-locity, i.e. e cr ( r > r stream ) ∼ ˙ E cr / ( π v A r ) , so it falls more steeplycompared to the diffusion-dominated case ( e cr ∝ r − ). This hasseveral important consequences. First (1) if trans-Alfvénic stream-ing dominates transport, then the streaming transport timescale ∼ r / v stream is always of the same order as the streaming loss timescale ∼ e cr / ˙ e streamloss ∼ e cr / | v stream · ∇ P cr | ∼ ( / ) r / v stream . So the CRs will,by definition, lose energy to Alfvénic and ultimately thermal en-ergy as they stream at the same rate they propagate out, at this ra-dius. This means the energy density must eventually decay, furtheraccelerating streaming losses. So (2) a non-negligible fraction ofthe CR energy is thermalized within a factor of a few of this radius.And (3) since the CR pressure drops more rapidly, the CRs even-tually provide small pressure support at r (cid:29) r stream , even if theydominate over gravitational pressure at r (cid:46) r stream according to ourarguments above.Thus, in general, the dominant role of CRs is predicted to beconfined to an “inner CGM” CR “halo” at (cid:46) −
100 kpc. For fixed β and ˜ κ , lower v A means that this radius extends further in smallerhalos, but (as we argued above), the CR injection and energy den-sity also declines rapidly in smaller halos, so this simply means thatthe (relatively small) CR energy density more efficiently escapes,rather than being thermalized, in smaller halos. In massive halos, r stream becomes comparable to the halo scale-radius.Streaming losses still occur in the inner parts of the halo,even if streaming does not dominate the transport. Integrating thestreaming loss rate = v stream · ∇ P cr within a maximum radius r , weobtain: ˙ E streamloss ˙ E cr ( r (cid:28) r stream ) = v A r κ = (cid:18) rr stream (cid:19) (12)so this is always a small (but not negligible) fraction of ˙ E cr within r < r stream . However, from the scaling of r stream , we see that stream-ing imposes yet another reason why CRs will not have a large effectat very low diffusion coefficients. If ˜ κ is too low, then even if thegas density is so low that collisional losses are negligible, the CRswill lose much of their energy to Alfvénic damping (“streaming c (cid:13) , 000–000 Hopkins et al. losses”) at very small r . Effectively, if r stream is smaller than the gasdisk effective radius, it means that the CR energy is thermalizedbefore it can escape the star-forming disk.Note that if we make v stream much larger and allow the “stream-ing loss” term to increase with ∼ v stream ∇ P cr (instead of limiting itat ∼ v A ∇ P cr ), then although streaming moves CRs faster , it alsomeans streaming losses occur much faster and closer to the galaxy,where they can be radiated away more efficiently. In Eq. 11, forexample, if we set v = v A + c s , then for β (cid:29) r stream ∼ κ M − / , ( + z ) − / . Thus, especially ifcombined with a lower diffusion coefficient, this particular super-Alfvénic streaming model means that streaming losses would ther-malize most of the CR energy within the galaxy and ISM, where itwould be efficiently radiated away.Note that because v A and r stream depend directly on | B | or β ,if magnetic fields are stronger (weaker) it shifts our predictions forCR transport accordingly. For CRs, this is effectively degeneratewith how we treat the scaling of v stream with v A : our default model( v st ≈ v A ) is akin to a model with v st ≈ v A but 10 times lower β .Our experiments with different v st therefore give some insight intohow our predictions depend on the ultimate strength of magneticfields. We have argued for the importance of the CR pressure/adiabaticterms above, and discussed CR losses. But can the CRs also beimportant as a thermal heating mechanism?If we assume 100% of the CR energy is thermalized (of coursean upper limit, since some CRs escape, some energy is lost doingadiabatic work, and for hadronic losses 5 / γ -rays which escape rather than thermalize), we cancompare this to the cooling luminosity L totcool = (cid:82) Λ n d x , where Λ ∼ Λ − erg s − cm is roughly constant over the temperaturerange of interest. Using this and the various scalings above (forfully-ionized gas), we obtain: ˙ E cr L totcool ∼ . α (cid:15) cr , . ( + z ) / Λ ( f gas / f b ) (cid:18) M ∗ f b M halo (cid:19) (cid:18) − cm − (cid:104) n (cid:105) cool (cid:19) (13)where (cid:104) n (cid:105) cool is the cooling-luminosity-weighted density (i.e. den-sity where most of the cooling occurs). We immediately see, again,that CR heating cannot have a large effect in dwarf galaxies, ow-ing to their very low M ∗ / M halo (and correspondingly low SFRs andCR energy production), even if the CRs couple at very low CGMdensities. In massive halos, at ISM densities ( (cid:104) n (cid:105) cool ∼ In the very lowest-density CGM near the virial radius (aroundthe mean gas density of the halo, (cid:104) n (cid:105) cool ∼ − ( + z ) cm − ),Eq. 13 suggests CR heating could become significant, but that as-sumes all the CR energy couples in the least-dense gas just inside R vir (and ignores gas outside/inside). More accurately, if we assumestreaming losses dominate (with the equilibrium e cr for streaming at CR heating can be non-negligible in cold gas with T (cid:28) K, where Λ is much smaller, and the gas is strongly self-shielded so photo-electricheating is negligible, provided the local CR energy density is high. How-ever this has a negligible effect on the dynamics of the cold gas (it is mostlyimportant for accurate ionization fraction calculations), or on the total cool-ing budget of the ISM which is dominated by warmer gas cooling down tothese low temperatures. large r > r stream , defined above), and that these are instantly ther-malized, and that the gas is in an isothermal sphere, we can thencalculate the ratio of the local thermal heating rate from CRs to thecooling rate: ˙ e streamheat ˙ e cool ∼ . α (cid:15) cr , . Λ ( f gas / f b ) ( + z ) / (cid:18) M ∗ f b M halo (cid:19) (cid:18) rR vir (cid:19) (14)So the CR thermal heating is unlikely to be relevant at any radius(at least for the halos of interest here). The model above immediately implies that CRs have very weak ef-fects within the star-forming galaxy disk at any mass scale. In orderto avoid losing all the energy to collisional losses, it is required thatthe CR diffusion time ( ∼ .
03 Myr ( L /
100 pc ) ˜ κ for diffusion onscale L ) is much faster than dynamical times in the disk. This es-sentially means CRs “diffuse out” of any locally dynamically inter-esting region of the disk (e.g. a GMC or strong shock) well beforethey can do interesting adiabatic work on that region. As a result,the CRs form (as assumed here) a quasi-spherical profile.For a MW-like galaxy, our Eq. 6 predicts aCR energy density at the solar circle of e cr ∼ − ( ˙ M ∗ / M (cid:12) yr − ) (cid:15) cr , . ( r / ) − ˜ κ − , more or lessexactly the canonical value, and in order-of-magnitude equiparti-tion with other disk-midplane energy densities. However, becauseof rapid diffusion, the CR pressure gradients are necessarilyweak in the disk. If we assume a vertically-exponential gasdisk balancing gravity in vertical hydrostatic equilibrium (diskmidplane pressure P mid ≈ π G Σ ), then the ratio of the verticalpressure gradients | ∂ P cr /∂ z | / | ∂ P othermid /∂ z | (where ∂ P othermid /∂ z isthe gradient of thermal/magnetic/turbulent pressure in verticalhydrostatic equilibrium) scales as ∼ ( P cr / P othermid ) ( H / R ) – i.e. theCR pressure forces (or gradients) are sub-dominant by a factor of ∼ ( R / H ) (cid:29)
100 in the MW midplane. The difference is evenmore dramatic if we consider still smaller-scale sub-structure (e.g.turbulent substructure in the ISM or clumps/cores in GMCs, wherethe relevant turbulent or magnetic/thermal support terms havestructure on sub-pc scales).
This toy model illustrates that, although for realistic (or observa-tionally allowed) parameters we do not expect CRs to be dynam-ically dominant in the evolution of small dwarf galaxies, thereis a potential “sweet spot” of galaxy parameter space in whichCRs (from SNe) might be quite important for intermediate (LMC)through massive (MW-mass) galaxies ( M halo ∼ − M (cid:12) ) atlow-to-intermediate redshifts ( z (cid:46) − cosmicray transport parameter space. If ˜ κ is too low, the CRs are trappedand collisional+streaming losses dissipate all their energy rapidly In Eq. 14, because we assume streaming dominates the transport ( r > r stream ), the steady-state CR energy e cr ∝ ˙ E cr / ( v stream r ) while the stream-ing losses scale as ∼ v A ∇ P cr ∝ ( v A / v stream ) ˙ E cr / r . For v stream ∼ v A (as-sumed in Eq. 14), the dependence on v A vanishes, while for super-Alfvénicstreaming, the heating rate from streaming losses is reduced by a factor ∼ v A / v stream . Likewise, for streaming inside of r < r stream where diffusiondominates transport, the heating rate is reduced by a factor ∼ v A r /κ ∼ r / r stream <
1. c (cid:13) , 000–000 osmic Rays on FIRE (in contradiction to all present observational constraints for LocalGroup galaxies). If ˜ κ is too high (which may be observationallyallowed), or is not constant but rises very rapidly outside of thegalaxy (certainly allowed observationally), CRs will simply “freestream” out of the CGM without building up a significant pressuregradient or thermalizing their energy – although this may requireextremely high ˜ κ . If the CR injection fraction (cid:15) cr (cid:28) .
1, there issimply not enough energy in CRs to have an effect at any massscale, while if it is too large ( (cid:15) cr ∼
1) it would violate direct obser-vational constraints.That is not to say this “sweet spot” requires very special fine-tuning. In fact, the characteristic parameters in the “sweet spot”:˜ κ ∼ − cm s − , v stream ∼ v A , (cid:15) cr ∼ . We now present the results of our simulations. Across all the pa-rameter surveys described below, we study >
150 high-resolutionfull-physics cosmological zoom-in simulations run to z =
0. Wediscuss the implications of these results, and use them to test oursimple model expectations, in § 5 below.
Figs. 1, 2, & 3, summarize a number of basic properties of the sim-ulated galaxies: the “archeological” star formation history (distri-bution of star formation times); galaxy stellar mass and metallic-ity versus redshift; dark matter and baryonic mass profiles, stel-lar effective radii, and circular velocity curves at z =
0. A numberof other properties can be directly inferred from this (or are triv-ially related to information here), including e.g.: the “burstiness” ofstar formation, the distribution of stellar ages, the evolution of themass-metallicity relation and stellar mass-halo mass relation, theexistence of “cusps” or “cores” in the dark matter profile, baryoncontent of the halo, baryonic mass concentration, etc. We explicitlynote the z = (1) our default FIRE-2 physics “Hy-dro+” run, (2) the FIRE-2 + MHD + anisotropic conduction andviscosity “MHD+” run, (3) the FIRE-2 + MHD + anisotropic con-duction and viscosity + full CR physics “CR+” run with a “lower”parallel diffusion coefficient κ ≡ × cm s − , and (4) a “CR+”run with higher parallel diffusion coefficient κ ≡ × cm s − .Simulations from Table 1 not shown explicitly in Figs. 1-3, aswell as the (many) less-massive galaxies in each box, are omittedfor brevity, but exhibit very similar behavior to those shown at sim-ilar masses. Likewise, Appendix A shows additional simulationswith varied resolution: the qualitative conclusions are similar to thesurvey in Figs. 1-3.Qualitatively, we see that “MHD+” and “Hydro+” runs are ( M halo ) [ M (cid:31) ] − − − − l o g ( M ∗ / f b a r y o n M h a l o ) Hydro+MHD+CR+( κ = κ = Brook et al. 2014Moster et al. 2013Garrison-Kimmel et al. 2017 ( M ∗ ) [ M (cid:30) ] − − − Hydro+MHD+CR+( κ = κ = S t e ll a r M e t a lli c i t y [ F e / H ] Kirby et al. 2013Gallazzi et al. 2005
Figure 7.
Top:
Stellar mass (normalized to the universal baryon fraction)vs. halo mass relation of the simulated galaxies (Table 1) at z =
0. Linescompare observational estimates of the median relation from abundance-matching to either isolated massive galaxies (Moster et al. 2013, extrapo-lated to low masses) or Local Group dwarfs (Brook et al. 2014; Garrison-Kimmel et al. 2017a).
Bottom:
Stellar mass vs. metallicity of the samegalaxies at z =
0. Lines show observational estimates for Local Group satel-lites (Kirby et al. 2013) and isolated massive galaxies (Gallazzi et al. 2005).Definitions of mass and metallicity match Figs. 1-3. Large symbols showeach “primary” galaxy. To illustrate that the results are robust across thelarger set of halos in our simulated volumes, as well as resolution, smallpoints show ∼
50 randomly-selected non-satellite halos (distance > κ = e m12i,f,m (resolution m i , =
7, much poorer than our “primary”small-dwarf galaxies). The “CR+” runs with higher- κ exhibit systemati-cally lower stellar masses at M halo (cid:38) . − M (cid:12) ” (other suites do notdiffer significantly). However, the galaxies move along , not off of, a tightmass-metallicity relation. very similar in all respects and “CR+( κ = e κ = e M halo (cid:38) M (cid:12) ) systems at low redshifts( z (cid:46) − γ -Ray Emission Fig. 4 compares the γ -ray emission predicted in our simulationsto observational constraints. With the exception of the MW, wheremore detailed constraints from spallation and other measurementsexist (see § 3), γ -ray emission represents one of the most direct ob-servational constraints available on the CR energy density in nearbygalaxies. This was studied in detail in Chan et al. (2018), in non- c (cid:13) , 000–000 Hopkins et al. cosmological simulations, so we extend that comparison here withthe addition of our cosmological runs. Many of the γ -ray obser-vations (and the equivalent constraint for the MW, essentially themeasured grammage or “residence time”) are collected in Lackiet al. (2011) (note that more recent studies, e.g. Tang et al. 2014;Griffin et al. 2016; Fu et al. 2017; Wojaczy´nski & Nied´zwiecki2017; Wang & Fields 2018; Lopez et al. 2018, find consistent re-sults). We compare directly to these constraints in Fig. 4, mim-icking the observations as best as possible (see Chan et al. 2018for additional details). Specifically, we compute the predicted lu-minosity L γ in ∼ GeV γ -rays produced by hadronic CR collisions.To do so, we follow Guo & Oh (2008); Chan et al. (2018) andtake the exact hadronic loss rate computed in-code, assume 5 / / π ,which decay to γ -rays with a spectrum that gives ∼
70% of theenergy at > ∼ M halo < × M (cid:12) ) or ∼
10 kpc aperture for more mas-sive galaxies (similar to the effective areas used for observations,although this has a relatively small effect). We also compute thecentral sightline-averaged gas surface density Σ central , and the lu-minosity from young stars L SF (computed with STARBURST99 con-volving all star particles <
100 Myr old with their appropriate agesand metallicities). This defines the ratio L γ / L SF , versus Σ central , asmeasured in Lacki et al. (2011).If all of the CR energy injected by SNe were lost collision-ally, in steady-state with a time-constant SFR, this would produce asteady-state L γ / L SF ∼ × − , which we label as the “calorimet-ric limit.” Of course, galaxies can violate this in transient events(or by a small systematic amount if e.g. star formation is non-constant in time). The ratio of L γ / L SF to calorimetric gives, approx-imately, the fraction of CR energy lost collisionally. We comparethis for our “default” suite from Figs. 1-3, as well as a detailed sur-veys varying κ (discussed below) and the streaming speed. For thestreaming-speed study, we specifically re-run simulations m10q , m11b , m11q , m11d , m11h , m11f , m11g , m12f , m12i , and m12m ,all with κ = × cm s − , either with our default streamingspeed, or with an arbitrarily much larger speed ( = ( c s + v A ) / ,chosen ad hoc to be faster by a factor of a few than the largestMHD wavespeed).We see that increasing κ leads to more-efficient CR escapefrom the dense galactic gas, lowering L γ (as expected). Increasedstreaming speed produces a similar but much weaker effect, forthe values we consider. The observations appear to strongly ruleout κ (cid:28) cm s − : reproducing them requires κ ∼ − × cm s − . In this regime, galaxies with dense nuclei (e.g. star-bursts and bulge centers) are approximate proton calorimeters, butless-extreme systems (essentially all dwarfs and much of the vol-ume of ∼ L ∗ galaxies) see the large majority ( > γ -rays. Fig. 5 shows the gas thermal ( = n k B T ), magnetic ( = | B | / π ), CR( = ( γ cr − ) e cr ), and kinetic ( = ρ | v | /
2) pressures (equivalent toenergy densities), as a function of radial distance r from the galaxy “Central” radius is defined here following Lacki et al. (2011) as a pro-jected radius ∼ ∼ L ∗ galaxies. However us-ing a constant ∼ Σ HI + H [M (cid:31) pc − ]10 − − − − − − Σ S F R [ M (cid:31) y r − k p c − ] Hydro+MHD+CR+( κ = e κ = e Figure 8.
Location of the suite of runs from Figs. 1-3 on the Schmidt-Kennicutt relation at z =
0. For each zoom-in region (different symbols), weplot the z = Σ HI + H ) andstar formation (averaged over the last < Σ SFR ),averaged over ∼
10 different random line-of-sight projections, within a cir-cular aperture containing 90% of the V -band luminosity. We compare (col-ors, as labeled) different physics sets as Figs. 1-3. For reference we show(shaded contour) the 5 −
95% inclusion interval at each Σ HI + H of observedgalaxies compiled from Kennicutt et al. (2007); Bigiel et al. (2008); Genzelet al. (2010). We note that the scatter in time for any individual simula-tion is large, comparable to the scatter observed (see Orr et al. (2018) fora detailed study), so the deviations between individual runs are all con-sistent within this scatter. The robust conclusion is that there is no sys-tematic trend towards lower/higher “star formation efficiency” (normaliza-tion of the relation here) with different physics studied: to the extent thatsome physics produce higher/lower SFRs, galaxies move along the relationrather than off of it (e.g. m12i , in pentagons, which shows lower SFR in the“CR+( κ = e center at z = −
95% in-clusion contour of these properties at each galacto-centric radius.Because of our Lagrangian numerical method, this is effectivelya gas-mass-weighted distribution of these values. To contrast, wetherefore also plot the volume-averaged values in spherical shells.Note that where most of the gas is in a thin disk (e.g. in m12i at r (cid:46)
10 kpc), this means the spherically-volume-weighted averagevalue of certain quantities (if they are concentrated in the thin disk)will be lower than its midplane value (closer to the mass-weightedaverage value) by a factor ∼ H / R (the ratio of the disk scale-height H to radius R ). We compare these to the “gravitational” or approx-imate local virial energy density, ≡ ρ V c / ≡ ρ G M enc ( < r ) / r .Using a different definition based on the total potential gives quitesimilar results for our purposes here.Note that the kinetic energy density here is defined as ρ | v | ,with v defined relative to the mean velocity of the whole galaxy, i.e.this includes rotation. This is done in part for simplicity because thedwarfs do not have strong rotation and separating rotation vs. dis-persion (even in simulations) is in general quite challenging (see El-Badry et al. 2018b). It also allows us to immediately see (whetherprimarily in rotation or dispersion) where the gas is primarily “heldup” by kinetic energy.We see (discussed below) that magnetic pressure is almostalways sub-dominant ( β ≡ P thermal / P magnetic (cid:29) c (cid:13) , 000–000 osmic Rays on FIRE − − − − log ( n ) [ cm − ] − − − − M a ss F r a c t i o n ( d m / d l o g n ) MHD+m10q − − − − log ( n ) [ cm − ] CR+( κ = e − − − log ( n ) [ cm − ] − − − M a ss F r a c t i o n ( d m / d l o g n ) MHD+m11b − − − log ( n ) [ cm − ] CR+( κ = e − − − log ( n ) [ cm − ] − − − M a ss F r a c t i o n ( d m / d l o g n ) MHD+m11f − − − log ( n ) [ cm − ] CR+( κ = e − − − log ( n ) [ cm − ] − − − M a ss F r a c t i o n ( d m / d l o g n ) MHD+m12f − − − log ( n ) [ cm − ] CR+( κ = e − − − log ( n ) [ cm − ] − − − M a ss F r a c t i o n ( d m / d l o g n ) MHD+m12i − − − log ( n ) [ cm − ] CR+( κ = e Figure 9.
Phase distribution of gas in the ISM (within < mass-weighted distribution of gas as a func-tion of density (per log ( n ) ), normalized so the integral over all curvesequals unity, in representative galaxies ( m10q , m11b , m11f , m12f , m12i ),comparing “MHD+” and “CR+( κ = e T < T > T < K), hot ionized medium (HIM; ionized and T > K). The qualitative features are similar in CR+ and Hydro+/MHD+runs; however the CR+ runs (including “CR+( κ = e los, especially in our high- κ runs, at radii outside the galaxy from ∼ −
200 kpc.Fig. 6 considers a similar comparison of the radial profilesof the gas cooling rate, computed in-code at z = ∼ − K, near the peak of the cooling curve).
We next survey a more detailed set of internal galaxy properties.Fig. 7 shows each of the “primary” galaxies in our suite (Ta-ble 1) on the stellar mass-halo mass and stellar mass-stellar metal-licity relations at z =
0. These re-affirm the trends suggested inFigs. 1-3: our “MHD+” and “CR+( κ = e κ = e M halo (cid:38) . − M (cid:12) ) by a factor ∼ −
3, but all runs move along a single(relatively tight) mass-metallicity relation. In addition to the “pri-mary” galaxies, we randomly select ∼
50 additional non-satellitegalaxies within the box surrounding our m12 halos, to comparewith the “primary” dwarfs. Although the dwarfs in the larger m12 boxes are simulated at lower resolution ( m i , =
7) compared tothe smaller m09 or m10 boxes around a single dwarf (reaching m i , = . ∼ −
95% scatter inferred around the me-dian M ∗ − M halo relation at all masses shown (Garrison-Kimmelet al. 2017a), although interestingly at the highest masses ( M halo (cid:38) . M (cid:12) ) the low- κ CR and Hydro/MHD runs appear to be sys-tematically high in M ∗ while the high- κ CR runs are systematicallylow (bracketing the observed median). In metallicity the simula-tions agree extremely well with observations at M ∗ (cid:38) M (cid:12) butappear to fall more steeply in very faint dwarfs compared to the Lo-cal Group satellites in Kirby et al. (2013): this is explored in detailin Wheeler et al. (2018).Fig. 8 shows the locations of the example galaxies fromFigs. 1-3 on the observed Schmidt-Kennicutt relation, at z =
0. Wespecifically follow Kennicutt (1998) and measure the SFR (definedas the average over the last <
10 Myr) and total neutral (HI+H )gas mass inside a projected circular aperture enclosing 90% of thestarlight (approximately equivalent to the RC2 sizes used therein,for MW-mass systems, but this allows us to include low-surface-brightness dwarfs), to define Σ SFR and Σ gas . We also show the These galaxies are chosen within the high-resolution “zoom-in” region( <
1% contamination by low-resolution dark matter particles inside theirvirial radii), but >
500 kpc away from the “primary” and outside the virialradius of any more massive halo. We select from the boxes m12i , m12f , m12m (each with resolution m i , =
7) because these have relatively largezoom-in regions.c (cid:13) , 000–000 Hopkins et al.
Figure 10.
Morphologies of dwarfs ( m10q , m11b , m11q ), at z =
0. Visualis a ugr composite, ray-tracing starlight (attenuated by dust in the simu-lation), with a log-stretch ( ∼ T (cid:29) K; red ), “warm/cool” ( T ∼ − K; green ),and “cold (neutral)” ( T (cid:28) K; magenta ) phases. We compare “Hydro+”( left ) and “CR+( κ = e right ); there is no large systematic difference. ∼
90% inclusion contour of observed galaxies compiled from Ken-nicutt et al. (2007); Bigiel et al. (2008); Genzel et al. (2010), al-though this is again intended only for reference, as we are not at-tempting a rigorous comparison with observations (such a compar-ison is presented in Orr et al. 2018, for those interested) but onlyto discern systematic effects between different runs. Briefly, we seeno apparent offset in the relation between any of our physics suites.Fig. 9 compares the distribution of gas phases (densities andtemperatures) in the interstellar medium. We plot the distribution ofmass as a function of density, in the different traditional ISM phases(molecular, cold neutral, warm neutral, warm ionized, hot ionized),where for simplicity we define “ISM” as gas within r <
10 kpc(the exact threshold makes little difference). For simplicity, weonly compare “MHD+” and “CR+( κ = e κ = e κ , the differences are very subtle (tens of percent shift in cold-to-warm-neutral medium mass). In a temperature-density diagram,this subtle shift is nearly undetectable. We see similar effects ex-amining the phase structure of outflows, specifically, but defer adetailed study of this to future work.Figs. 10, 11, 12, 13 shows the visual morphologies of thegalaxies at z = u / g / r composite ray-tracingimages determined using STARBURST99 to compute the age-andmetallicity-dependent spectrum of each star (the same assumptionsused in-code) and adopting a constant dust-to-metals ratio to usethe gas and metals distribution determined in-code to attenuate andextinct the light; the gas images are volume-renderings with iso-temperature contours centered on broad (log-normal) temperaturebands with dispersion ∼ . κ CR runs are later-type, consistentwith their lower mass, and we see more warm (as compared to hot)gas in the CGM (beyond the disk). The CGM properties will bestudied in greater detail in future work.We have also examined the morphology of the magnetic fields,specifically, but find they are highly tangled on all scales here, inboth “MHD+” and “CR+” runs, consistent with our more detailedstudies in Su et al. (2017) and Su et al. (2018a).Fig. 14 shows the “quantitative kinematic morphology” (an-gular momentum distribution) of stars (weighted by stellar massor visual luminosity) and neutral gas. In all of these, we againcompare our ensemble of galaxies from Table 1 across our “corephysics variations” from Table 3. Again, the variations with physicsare weak, and (where present) consistent with the morphologicalchanges described above.
We have widely-varied the CR transport coefficients κ and v stream ina subset of our simulations. The effect on γ -ray emission is shownin Fig. 4.Fig. 15 shows the effects on galaxy properties (using the samestyle as Fig. 1) of varying κ more widely and densely in our galax-ies m10q , m11b , m11q , m12i . This confirms that CRs have weakeffects for κ (cid:46) cm s − and maximal effect in massive halos at z (cid:46) − κ ∼ − × cm s − .Fig. 16 varies the fraction (cid:15) CR of SNe energy which is injectedinto CRs, focusing on a low-resolution version of run m12i with c (cid:13) , 000–000 osmic Rays on FIRE Figure 11.
Visual & gas morphologies of the intermediate-mass galaxy m11f ( M ∗ ∼ . − × M (cid:12) , M halo ∼ × M (cid:12) ), as Fig. 10. We compare“Hydro+”, “MHD+”, “CR+( κ = e κ = e left-to-right ). “Hydro+” and “MHD+” exhibit no differences. In stellar/visual and gas disk morphology, all runs are broadly similar. The “CR+( κ = e κ = e ∼ κ where it is warm/cool gas-dominated (this extends beyond the region shown, but we defer CGM studies tofuture work). Similar-mass runs (e.g. m11g , m11h , m11d ) show very similar systematic effects. Figure 12.
Visual & gas morphologies of the MW-mass galaxy m12f ( M ∗ ∼ − × M (cid:12) , M halo ∼ M (cid:12) ), as Fig. 10. Similar trends appear as Fig. 11.“Hydro+” and “MHD+” runs do not differ, while “CR+( κ = e κ = e κ run resembles the “Hydro+” or “MHD+” run froman earlier time when the masses were more similar. All CR runs exhibit substantially enhanced warm/cool gas in the inner CGM. Here this is more dramaticat high- κ , because the gas is so heavily depleted in the low- κ run that what remains is very tenuous.c (cid:13) , 000–000 Hopkins et al.
Figure 13.
Visual & gas morphologies of the MW-mass galaxy m12m , as Fig. 10. Similar trends appear as in m12f . Note that as this halo is more massive,even the “CR+” runs are more dominated by hot gas in the CGM, and the Hydro+/MHD+/CR+( κ = e
28) runs (with higher masses) are notably redder incolor (with older, more metal-rich stellar populations formed at z ∼ . −
1) and have prominent stellar bars (MHD+ is being perturbed by a minor mergerpassage whose timing is slightly different owing to the different mass). CR+( κ = e
29) is less massive and later-type. κ = × cm s − (since MW-mass halos with this diffusivityare where we see the most dramatic CR effects). As expected, in-creasing the CR energy input increases their effect, but the effect isquite weak (sub-linear). Fig. 17 compares m10q , m11q , m12i in a survey of basicCR physics: comparing our default “CR+” implementation (§ 2)to runs (1) without MHD (where lacking B directions we assumeCR diffusion/streaming, and conduction/viscosity, are isotropic,i.e. take the projection tensor ˆ B ⊗ ˆ B → I ); (2) without collisional(hadronic or Coulomb) CR losses; (3) without streaming (set-ting v stream → v stream → ( c s + v A ) / . These effects are dis-cussed in detail below but their effects are generally small com-pared to including/excluding CRs at all.Note that for Fig. 4 we noted we have actually run a largeensemble of simulations with lower κ = × cm s − and thelarger v stream → ( c s + v A ) / : we have also compared the resultinggalaxy properties but omit them for brevity as the difference owingto faster streaming is very small. We have also run a large suite with Throughout, we have assumed CRs are injected at the sites of SNe: be-cause the efficiency of CR acceleration scales with the shock velocity, mod-els (and SN remnant observations) generally assign most of the accelera-tion to the “fastest” SNe shocks with v (cid:38) − and “swept up” ISMmass ∼ M ejecta ∼ − M (cid:12) , which is always un-resolved in our simula-tions. However one might imagine that if CRs are injected primarily outsidethe galaxy from e.g. structure-formation or superbubble-CGM shocks, thenthis would allow one to match the observations with lower κ – but only ifwe dramatically lowered the injection from SNe themselves (otherwise thiswould simply increase L γ ). So this is unlikely to change our conclusionshere, but could be important for CRs from AGN (where jet terminationshocks reach into the CGM). κ = × cm s − varying v stream = ( − ) v A more modestly,and find (unsurprisingly) essentially no effect.Additional purely-numerical tests are given in Chan et al.(2018) and Appendix A. This includes modifications of the CRpressure tensor, form of the flux equation, and other detailed as-sumptions that stem from any two-moment expansion of the CRtransport equations. We do not see large effects from these on ourpredictions, but of course can only survey a limited range of possi-bilities. Figs. 18-19 repeats our earlier exercises comparing galaxy prop-erties and magnetic+CR energy densities (from Figs. 1 and 5) forthe suite of halos in Table 2, which reach large masses ( M halo > M (cid:12) ) at increasingly higher redshifts z ∼ −
10. These simu-lations are not run to z =
0, hence we analyze them separately.We discuss the results in detail below, but briefly, we see noappreciable effects of CRs or MHD above z (cid:38) − We now explore the implications of the results presented in § 4.
In Su et al. (2017), we used similar FIRE-2 simulations to studythe effects of magnetic fields and anisotropic Spitzer-Braginskiiconduction and viscosity on galaxy properties. The study there in-cluded much more detailed measurements of properties like the gasphase distributions of the ISM and CGM, outflow properties, turbu-lence and energy balance in the ISM, magnetic field amplification,and more. There, we concluded that there was no appreciable sys-tematic effect on any global galaxy properties from these physics.The “MHD+” simulations here improve on those studied in Suet al. (2017) in three significant ways. (1)
Our mass resolution is an c (cid:13) , 000–000 osmic Rays on FIRE Figure 14.
Kinematic morphology of our simulated galaxies. We plot the distribution of specific angular momentum ( j , specifically the component j z along thetotal angular momentum axis) versus the specific angular momentum of a test particle on a perfectly-aligned circular orbit ( j c [ (cid:15) ]) with the same specific energy( (cid:15) ) – so + − M ∗ ), V -band luminosity ( L V ), and neutral gas inside the halo scale radius, for different galaxies. For each, we compare “Hydro+” ( red ), “MHD+”( green ), “CR+( κ = e blue ), “CR+( κ = e black ). The results here largely mirror those from the visual morphologies in Figs. 10-13. Low-massdwarfs are primarily dispersion-supported in all cases, higher-mass galaxies and younger stars (more prominent in L V vs. M ∗ weighting) are systematicallymore disk-dominated. There are minor run-to-run variations (largely stochastic), and when the disk forms at late times, some of the “CR+( κ = e m11f ) have a smaller fraction of their stellar mass in the disk, as expected (although note the gasis still disky, and the L V -weighted distribution, since it is dominated by the younger stars, is also strongly disk-dominated). In no case do any physics studiedhere change spheroidal galaxies to disky or vice versa. order-of-magnitude better, allowing us to much better-resolve theField length and other small-scale effects. (2) The simulation suitehere is an order-of-magnitude larger, and all fully-cosmological, al-lowing us to assess and improve the statistics and avoid uncertain-ties owing to inevitable run-to-run stochastic variations in galaxyproperties (discussed therein or in Keller et al. 2018; Genel et al.2018). (3)
Our treatment of anisotropic conduction & viscosityis more accurate. Specifically, a large body of recent work in theplasma physics literature (both theoretical and experimental) hasshown that the parallel transport coefficients for heat and momen-tum (e.g. κ cond and η visc ) are strongly self-limited by micro-scaleplasma instabilities (e.g. the Whistler, firehose, and mirror insta-bilities) in high- β plasmas like the ISM and CGM (Kunz et al.2014; Komarov et al. 2016, 2014; Riquelme et al. 2016; Santos-Lima et al. 2016; Roberg-Clark et al. 2016, 2018; Tong et al. 2018;Squire et al. 2017c,a,b; Komarov et al. 2018). Our treatments in§ 2.2 include these effects, while our previous work did not.Despite this (or perhaps because of it), we confirm the con-clusions of Su et al. (2017): in every property we measure, the“MHD+” runs do not appear to systematically differ significantlyfrom the “Hydro+” runs. This is perhaps not surprising: the only physical change of (1) - (3) above is (3) , which has the effect of uni-formly decreasing the magnitude of conduction and viscosity athigh- β (e.g. the ISM and CGM). Because of this, we will not dis- cuss these variations further (for much more detailed discussion of why the results do not change, we refer to Su et al. 2017).Our goal here is to understand effects of magnetic fields ongalaxies, not to consider a detailed comparison with observations.However, Guszejnov et al. (2019) directly compared the values of B predicted in our MW-like m12i simulation in atomic/molecularclouds to Zeeman observations from Crutcher et al. (2010) andfound remarkably good agreement at all observed densities ( n (cid:38)
10 cm − ). In the ionized/warm ISM and thick disk/inner halo, ob-servations are much more uncertain and model-dependent (see Han2017), e.g. assuming gas density profiles and equipartition be-tween CR and magnetic energy to infer | B | from rotation measures(RMs) or synchrotron emission ( I s ). But preliminary estimates fromour simulations show the combination of large “clumping factors”(e.g. (cid:104)| B | (cid:105) / (cid:104)| B |(cid:105) ), violation of equipartition, and more extendedgaseous halos can produce similar RM and I s to observations,often with much lower median B compared to simpler models.More detailed forward-modeling is clearly needed. In the extendedCGM/halo, observations are lacking, but comparing our Fig. 5 toother cosmological simulations of MW and dwarf galaxies includ-ing stellar feedback shows that we predict similar, or somewhathigher, values of | B | as a function of either baryon density ( n / (cid:104) n (cid:105) )or galacto-centric distance r / r vir (Vazza et al. 2014; Marinacci et al.2015; Martin-Alvarez et al. 2018; Katz et al. 2019). Older simula-tions not including stellar feedback show substantially lower | B | c (cid:13) , 000–000 Hopkins et al. [ Gyr ] − − − S F R [ M y r − ] m10q a l o g ( M ∗ ) [ M ] ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) a − − [ Z ∗ / H ] -1.9-1.8-2.0-1.9-1.9-1.8 − ( r ) [ kpc ] l o g ( ρ [ r ] ) [ M k p c − ] r [ kpc ] V c [ k m s − ] m i ,1000 = [ Gyr ] − − m11b a
678 8.18.07.97.9 CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) a − − − − ( r ) [ kpc ]
468 2.82.32.12.10 10 20 30 r [ kpc ] m i ,1000 = [ Gyr ] − − m11q a
789 9.59.59.09.39.19.1 CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) a − − − ( r ) [ kpc ]
468 4.44.83.43.93.23.40 10 20 30 r [ kpc ] m i ,1000 = [ Gyr ] m12i a
810 10.910.410.4 CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) a − −
10 0.3-0.1-0.0 − ( r ) [ kpc ] r [ kpc ] m i ,1000 = Figure 15.
As Figs. 1-3, varying the CR diffusion coefficient κ more extensively, in a subset of our dwarf-through-MW mass runs. Very low κ (cid:46) cm s − produce essentially no systematic effect at any mass scale. Intermediate 10 (cid:46) κ (cid:46) × may produce a small suppression of SF (factor of ∼ . − . (cid:29) , produce factor ∼ − z (cid:46) − (Dubois & Teyssier 2008; Dolag et al. 2008; Donnert et al. 2018),consistent with recent studies of the role of turbulence and windsin field amplification (Su et al. 2018a; Martin-Alvarez et al. 2018). In essentially every galaxy property we examine, the effects of CRphysics on dwarf galaxies ( M ∗ (cid:28) M (cid:12) , M halo (cid:46) M (cid:12) ) arerelatively small. Above in § 3, we argued that this is generically ex-pected: in equilibrium for realistic star-forming galaxies on the ob-served “main sequence” of star formation, the ratio of CR pressureforces on gas in the halo, compared to gravitational forces, scales ∝ M ∗ / M halo . Of course, other forms of feedback scale this way aswell. But with other forms of stellar feedback in place (notablySNe), CR pressure is relatively inefficient at re-accelerating windsor stalling accretion on scales of order the galaxy and halo scaleradius. One key point is that on small (ISM) scales, in dwarfs, withlow metallicities and densities, SNe cool relatively inefficiently andcan convert a large fraction of their ejecta energy into work and/orthermal energy which is not immediately radiated (see Hopkinset al. 2018c). Since the CR energy is only ∼
10% of the mechanicalenergy, by construction, this means mechanical energy will alwaysdominate, if it is not efficiently radiated (as it would be in densegas, in more massive galaxies).A second key point follows directly from the observations. In dwarf galaxies, almost all of the CRs escape efficiently from theISM: but unlike massive galaxies where there is a large CGM “hothalo” extending to >
300 kpc which can “confine” the CRs and al-low them to build up a substantial pressure profile, in dwarf galaxiesthere is no such hot halo (it cannot be built up from virial shocks,and outflows are much less confined as well so simply escape) andthe size of the virial radius is much smaller ( (cid:28)
100 kpc). So CRsin dwarfs simply escape from their CGM as well (and even if theystay, they have relatively little mass to support and prevent fromaccreting).We do see some effects of CRs, but these are small comparedto the effects of different treatments of SNe and stellar radiation(see Hopkins et al. 2018b,c,a), and appear primarily only at lower κ than allowed observationally. Table 1 and Figs. 1-3 (also 7, 15)show that at ultra-faint masses (e.g. m09 and m10v or m10q at veryearly times, with M ∗ (cid:46) × M (cid:12) ), runs with CRs have slightly increased stellar mass by ∼ . − . κ within therange κ ≈ − cm s − . This effect is maximal for m10q , buteven there is only a factor of ∼ (cid:28)
1% of their baryons into stars). By slightly highermasses ( m11b , m11q at M ∗ ∼ . − . M (cid:12) ) the effect weakensto ∼ . M ∗ (cid:38) M (cid:12) in m11v , m11c , see c (cid:13) , 000–000 osmic Rays on FIRE [ Gyr ] SF R [ M (cid:31) y r − ] m12i a l og ( M ∗ ) [ M (cid:31) ] ε cr = . ε cr = . ε cr = . a − − (cid:28) [ Z ∗ / H ] (cid:27) − ( r ) [ kpc ] . . . l og ( ρ [ r ] ) [ M (cid:31) kp c − ] r [ kpc ] V c [ k m s − ] m i , = Figure 16.
As Figs. 1-3, comparing our low-resolution MW-mass m12i runwith “full CR physics” (CR+), and the high diffusion coefficient ( κ = e (cid:15) cr (fraction of SNe ejecta kineticenergy assumed to go into CRs). The default value is (cid:15) cr = .
1. Increas-ing/decreasing this produces systematically stronger/weaker suppression ofSF, as expected, but the effect is relatively small compared to changes in thediffusion coefficient in Fig. 15.
Table 1). Figs. 8, 9, 10 & 14, show that even where the effect onstellar masses is maximal it has no systematic qualitative effect onthe visual/stellar or gas morphology, phase structure, or kinematicseither within the galaxy itself or the inner CGM, nor on the starformation efficiency of the galaxy.Figs. 5-6 illustrate why this is the case, in the context ofthe arguments above. In all cases, CR “heating” is vastly sub-dominant to gas cooling (as expected – recall the virial tempera-tures of these halos are near the peak of the cooling curve). More- over, for higher- κ ( (cid:38) cm s − ), even with zero losses, the CRpressure in the galaxy and CGM is always sub-dominant to ther-mal pressure (SFRs are simply too low to support an energetically-dominant CR halo). By making κ lower and trapping CRs one canbuild up more CR pressure close to the galaxy (keeping warm gas“puffy” and non-star-forming), but for κ (cid:46) cm s − diffusionis too slow and much of the CR energy is lost collisionally. Evenat the “sweet spot” where the CR pressure is maximized (about κ ∼ × cm s − ), CR pressure only becomes comparable tothermal pressure (never strongly dominant) in the more massivedwarfs.Because these effects are somewhat fine-tuned, and CRs arenever strongly dominant, they are also sensitive to other details ofthe physical treatment: Fig. 17 shows that if we allow modestlysuper-Alfvénic streaming (increasing both the streaming speed and“streaming loss” term in these runs), the combination of enhancedlosses and faster escape from the galaxy (extending but loweringthe CR pressure) eliminates the (already small) effect of CRs withingalaxies. Likewise, isotropic transport (effectively allowing slightlyfaster diffusion) allows the CRs to escape and weakens the effects.These conclusions are discussed and demonstrated in more detailin Chan et al. (2018) (and also consistent across our more extensivestudies for e.g. Fig. 4).Finally, and perhaps most important, Fig. 4 shows that the rel-atively low κ required to produce an effect on dwarfs leads to afactor ∼
10 over-prediction of the observed γ -ray luminosity inlow-surface-density galaxies. Recall, the observed points includethe LMC, SMC, and M33, analogous to several of our simulateddwarfs. In order to match these, Fig. 4 shows κ > cm s − isrequired, at which point the effect of CRs on most internal dwarfgalaxy properties vanishes.Briefly, we note that our conclusions here appear to contradictsome claims in the literature that CRs can have a strong effect on SFin dwarf galaxies (e.g. Jubelgas et al. 2008; Booth et al. 2013; Chenet al. 2016). However, to our knowledge, all such claims either (a)adopted low diffusion coefficients, κ (cid:28) cm s − , without com-paring to the constraints from γ -ray fluxes which we argue prohibitsuch coefficients (in several such studies, collisional losses werenot included at all, which allows CRs to artificially “build up” atlow κ when their energy should be lost); or (b) used idealized (non-cosmological) simulations, or simulations with very weak (or non-existent) stellar feedback from other sources (e.g. mechanical SNeand/or radiative feedback), such that the SFR and ratio M ∗ / M halo was much higher than observed (which, according to the scalingsin § 3, would allow CRs to have a large effect, but directly violatesobservations of galaxy stellar masses and SFRs).We do stress that CRs could still have an effect in the CGM,or ICM further away from the galaxy, at least at intermediate massscales. This will be studied in detail in future work, but briefly, wenote that even in the outer CGM or IGM out to ∼ R vir , we seeno obvious systematic effect of CRs in very low-mass halos (e.g. M halo (cid:46) M (cid:12) , i.e. m10q and smaller galaxies). However we dosee some effects on the velocity field of gas even at surprisinglysmall halo mass scales (down to M halo ∼ × M (cid:12) , correspond-ing to m11b ). In these intermediate-mass systems, Fig. 5 shows theCR pressure is not completely negligible around ∼ R vir (although itis not dominant), so this is plausible. But since our primary focushere is galaxy properties, we defer a more detailed investigation tofuture work.In summary, for any observationally-allowed CR parametersexplored here, the effects on any galaxy property studied are small c (cid:13) , 000–000 Hopkins et al. [ Gyr ] − − − S F R [ M (cid:30) y r − ] m10q a l o g ( M ∗ ) [ M (cid:30) ] Isotropic/No MHDNo CR Cooling No StreamingStream (cid:28) v A a − − (cid:27) [ Z ∗ / H ] (cid:26) -2.0-1.7-1.7-1.7-2.1-1.9 − ( r ) [ kpc ] l o g ( ρ [ r ] ) [ M (cid:30) k p c − ] r [ kpc ] V c [ k m s − ] m i ,1000 = [ Gyr ] − − m11q a
789 9.09.19.29.29.09.20.0 0.25 0.5 0.75 1.0Scale Factor a − − − ( r ) [ kpc ]
468 3.43.03.63.63.54.10 10 20 30 r [ kpc ] m i ,1000 = [ Gyr ] m12i a
810 10.710.910.710.510.710.80.0 0.25 0.5 0.75 1.0Scale Factor a − −
10 0.10.20.30.00.20.2 − ( r ) [ kpc ] r [ kpc ] m i ,1000 = Figure 17.
As Fig. 15, varying the CR physics in “CR+” runs with otherwise identical physics, at a few different mass scales. Here we choose κ = e
28 for thedwarfs ( m10q and m11q ) and κ = e
29 for the MW-mass halo, because these values give some of the strongest effects seen at each mass (making differenceshere more obvious). We compare: (1) “CR Default”: the default physics shown in all CR+ runs elsewhere. (2) “No CRs (MHD+)”: the default MHD+ runs(without CR transport) for reference. (3) “Isotropic/No MHD”: CR runs without MHD, where (lacking a magnetic field) CR streaming and diffusion, as wellas Spitzer-Braginskii conduction and viscosity, are assumed to be isotropic. (4) “No CR Cooling”: Turning off hadronic & Coulomb losses from CRs. (5) “No Streaming”: Disabling CR streaming (setting the streaming velocity to zero). (6) “Stream (cid:29) v A ”: Allowing super-Alfvénic/sonic streaming (streamingvelocity = ( c s + v A ) / , a multiple of the fastest MHD wavespeed). In m10q the stellar mass varies by a factor ∼ z burst around z ∼
3, so it is difficult to interpret. These choices generally have small effects in m11q (LMC-mass). In m12i (MW-mass), artificially removing CR losses/cooling leads to significantly stronger suppression of SF, as expected, while allowing highly super-Alfvénicstreaming actually produces less suppression of SF (owing to enhanced CR streaming losses leaving a less-energetic CR halo), and allowing isotropic streaming(without MHD) produces the least effective suppression of SF, as the CRs too efficiently escape the galaxy and halo. (much smaller than effects of e.g. mechanical or radiative feed-back).
As we look at progressively more massive low-redshift galaxies,above M halo (cid:38) M (cid:12) (at z = κ (cid:46) cm s − ,the effects on galaxy properties (in e.g. Figs. 2-3, 7-9, 11-14, etc.)are very weak – typically ∼ . κ are even weaker with super-Alfvénic streaming, as seen with dwarfs (see also Chan et al. 2018).The reason is obvious in Fig. 4: for κ (cid:46) cm s − , the massivegalaxies become proton calorimeters, i.e. lose most of their CR en-ergy to collisions within the galaxy – also as predicted in § 3. Note c (cid:13) , 000–000 osmic Rays on FIRE that the massive galaxies have higher central densities compared tothe dwarfs (almost all have Σ central (cid:38) − g cm − ), so it requireslarger κ for CRs to escape without losing most of their energy. Thisis also obvious in Fig. 5 – for these lower κ values, the CR pres-sure/energy density outside the galaxy is order-of-magnitude belowthe predicted value if the CRs diffused without losses.Fig. 15 shows that, as a result, the “sweet spot” where CRshave maximal effect occurs at κ ∼ − × cm s − . Forthese κ , Figs. 2-3 show that as the the galaxies approach stellarmasses M ∗ ∼ M ∗ , or halo mass ∼ M (cid:12) (SFRs (cid:38) . − (cid:12) yr − ) the CRs begin to have an effect suppressing SF. Thesuppressed SFRs in the “CR+( κ = e κ = e z ∼ m11 runs, or z ∼ m12 runs) the difference in SFR can be as large asfactor ∼ −
10, although the integrated difference in stellar massby z ∼ ∼ − z (cid:46) − κ (discussed further in § 5.6).In Fig. 16 we systematically vary the fraction of SNe energy in-jected as CRs from (cid:15) cr ∼ . − .
2: as expected, more efficientCR production produces stronger SFR suppression, but the effectis highly sub-linear (a factor of ∼ (cid:15) cr produces a factor ∼ . sufficient CRs reachlarge radii to set up a halo that can pressure-support the cool gas,adding somewhat more produces little effect. Of course eventuallyif (cid:15) cr is too low, the CR halo cannot maintain pressure support, andthe effect of CRs should rapidly vanish.Fig. 4 shows that for these MW-mass systems, the same κ ∼ × cm s − which produces large effects on their SF historiesappears to agree well with the spallation constraints in the MW andobserved γ -ray luminosities in the MW and local galaxies, whilethe lower κ (cid:46) cm s − (which produces weak effects on thegalaxies) predicts excessive γ -ray flux.The effects of CRs within the galaxies, even at high- κ , appearquite weak, as expected (see § 3) – properties like the stellar/visualand/or gas morphology and kinematics within the galactic disks,abundances/metallicities, baryonic and dark matter mass profiles,disk sizes and rotation curves, outflow rates, and star formation ef-ficiencies (location on e.g. the Kennicutt-Schmidt relation) do notappear to be strongly altered at any κ or mass scale we study, exceptinsofar as the total galaxy mass and SFR shift up or down (chang-ing e.g. the total gas supply, or total mass in metals produced, ortotal mass in baryons contributing to V c ). This is not surprising: inthe disk midplane, we confirm the CR pressure gradients are sub- Note that although the low-resolution ( m i , = m12i shown inFig. 15 shows increasing effects of CRs going from κ ∼ × to κ ∼ × , the low-resolution version of this simulation is a relativelyhigh-density galaxy with Σ central ∼ − . gcm − in Fig. 4, so the higher κ improves escape. We have run both higher resolution ( m i , = m12i (where the galaxy is systematically less dense), and m11f with κ ∼ × , to limited redshift ( z ∼ Σ central ∼ − ( . − . ) gcm − ) systems that the CR effects become weaker at thisstill-higher κ , as predicted if they escape “too” efficiently. dominant to thermal and turbulent forces. But in the CGM aroundthe galaxies, the CRs appear to have a direct and dramatic effect.We discuss this further in § 5.5 below.However Figs. 5 and 11-13 demonstrate that the effects of CRson the CGM around these galaxies, at radii ∼ −
100 kpc, are dra-matic. In the cases where CRs suppress SF, they establish a high-pressure CR halo outside of the galactic disk (extending to or evenpast the virial radius), supporting a large reservoir of gas which ismuch cooler ( T (cid:28) K) than would be required to maintain ther-mal pressure equilibrium. In contrast in the “MHD+” or “Hydro+”cases denser and/or cooler halo gas cools rapidly, then falls onto thegalaxy, leaving a virialized halo of only the “leftover” gas which ismore tenuous. In the “CR+” runs around massive galaxies, Fig. 5shows the CR pressure is dominant over thermal and magnetic pres-sure outside the disk, all the way to the virial radius (Fig. 6 showsthe direct CR heating is negligible). For low- κ the CR pressure issuppressed owing to losses, but for the high- κ runs the CR pressureprofile agrees remarkably well with the analytic predictions in § 3,which also predict accurately the mass scale where this can supportenough gas mass to suppress gas inflows and (ultimately) SF in thegalaxies at a significant level. We discuss the dynamics of the CRsin the CGM in more detail below and in future work. L ∗ MassiveGalaxies)
The galaxies of interest in § 5.3, where CRs have appreciable ef-fects, have M halo ∼ − M (cid:12) at z ∼
0. It is natural to ask whethergalaxies in this mass range at higher redshifts also experience sim-ilar effects of CR transport.First, note that high-redshift dwarfs are represented in oursample already, in Figs. 1-3, etc. These are simply the progenitorsof the z = all of thedifferences owing to CRs (at any mass scale) manifest only at rela-tively low redshifts z (cid:46) −
2. In most cases, it is quite notable: e.g.for galaxies m11i , m11d , m11h , m11f , m11g , m12i , m12f , m12m ,the SFH, stellar mass, and metallicity (as well as all other proper-ties, such as morphology, outflow properties, etc.) in the CR+ runsclosely track the MHD+ runs until z (cid:46) −
2, where they begin todiverge significantly. For the lower-mass galaxies, this is less sur-prising: their progenitors at z (cid:38) − z = m12m and m12f ) reach stellar masses (cid:38) M (cid:12) and halo masses (cid:38) . M (cid:12) long before they begin todiverge – well above the threshold where we see effects appear at z =
0. This indicates that there is some additional redshift depen-dence here.To explore more massive halos at higher redshifts, i.e. thosewith M halo ∼ − M (cid:12) at z ∼ −
10, Figs. 18-19 consider theextended suite of high-redshift, high-mass galaxies from Table 2.These are halos which reach ∼ M (cid:12) (comparable to our mostmassive z = z = z ∼ − z ∼
0. In Fig. 18 we can see essentially no de-tectable systematic effects of MHD or CRs on these halos at anyredshift, except for the lowest-redshift example ( m12z1 ) which be-gins to show a modest suppression of its SFR and stellar mass onlyat z (cid:46) .
5. We have also examined the other properties in this paper(e.g. morphologies, gas phase distributions, outflow velocities) andsimilarly find no differences above z (cid:38) − c (cid:13) , 000–000 Hopkins et al.
Figure 18.
As Fig. 1, but comparing our high-redshift massive halos from Table 2. Each halo labeled m12zX exceeds a halo mass (cid:38) M (cid:12) at a redshift z = z ∼ X . We run to at least this redshift, and in some cases somewhat further. Consistent with the lower-mass halos in Figs. 1-Fig. 3 (where z (cid:46) z (cid:38) −
2. This is consistent with our analytic expectations (§ 3): at high- z the SFRs (and correspondingCR injection rates) are higher at a given mass (some reaching ∼ M (cid:12) yr − here), but the CGM densities/pressures are much higher so the CRs are notable to support the halo in virial equilibrium. Denser gas in galaxies also produces large CR losses during the peak starburst epochs (all the systems with ˙ M ∗ (cid:38) M (cid:12) yr − have γ -ray losses near calorimetric, and Σ central (cid:38) . − as in Fig. 4. As shown in Su et al. (2018b), lacking AGN (or some other)feedback, feedback from SNe alone cannot “quench” SF in massive halos and they over-cool, producing the extremely large central V c in several of the runs. high redshift, SFRs are higher at a given stellar mass (as obviousin Fig. 18), so the CR production/injection rate is also higher (scal-ing ∝ t − ∼ ( + z ) / ). However, the density of the CGM andIGM, and ram pressure of inflowing gas, is much higher (scaling ∼ ( + z ) ). So CRs are unable to maintain a super-virial pres-sure which can suppress inflow/accretion in this dense gas (as inFig. 5). We have directly confirmed this, in fact, comparing the CRand virial pressure in Fig. 19 – for the high-redshift massive ha-los, most of the inflow is under-pressurized relative to what wouldbe needed to maintain virial equilibrium, with or without CRs. Moreover, owing to their high inflow rates, most of the star for-mation in the high-redshift systems occurs in starbursts with veryhigh SFRs and, correspondingly, very high gas densities within thegalaxy (obeying the extension of the observed Schmidt-Kennicuttlaw in Fig. 8 to higher densities) – we see this directly in Fig. 18where all the massive high- z systems reach ˙ M ∗ (cid:38) M (cid:12) yr − andthe couple most extreme reach ˙ M ∗ (cid:38) M (cid:12) yr − . During thesephases, the gas surface densities reach (cid:29) . − (see Fig. 8 andthe more detailed studies in Hopkins et al. 2011; Orr et al. 2018),or (cid:29) M (cid:12) pc − . At these densities, we see in Fig. 4 that essen-tially all observed galaxies, and all of our simulations (even withextremely high ˜ κ > cm s − ) approach the proton-calorimetriclimit – in other words, a substantial fraction of the CR energy is lost In the high-redshift halos in Table 2, there is some low-density gas in theCGM for which the CR pressure exceeds or is comparable to that neededfor virial equilibrium. This may imply that CRs have an effect in the CGMeven where they have little effect on the bulk galaxy properties. However,this under-dense gas represents very little of the gas which accretes onto thegalaxy (with or without CRs). to collisions in the dense ISM. We confirm this directly in the sim-ulations in Fig. 18 in their “peak starburst” phases. Interestingly,there is some evidence that away from the starburst peaks, at thelowest SFRs, the CR+ runs exhibit slightly more-suppressed SFRs,but these phases of course contribute negligibly to the total SF andstellar mass (and therefore most other galaxy properties).Importantly, these simulations also highlight the critical needfor some additional feedback beyond the stellar feedback mecha-nisms (SNe types Ia and II, stellar mass loss from O/B and AGBstars, radiation, cosmic rays accelerated in SNe) and microphysicalprocesses (magnetic fields, conduction, viscosity) studied here, inorder to explain the suppression of SF (i.e. “quenching”) in mas-sive ( (cid:38) L ∗ ) galaxies. In Su et al. (2018b), we show this explicitly,in non-cosmological simulations of halos with M halo ∼ − M (cid:12) at z ∼ z ∼ −
10, when their halo masses were in the range M halo ∼ − M (cid:12) ). Not only do these massive halos sustain highSFRs ( ˙ M ∗ (cid:38) − M (cid:12) yr − ) as long as we run them (includingto z ∼
0) – i.e. clearly fail to quench – but they also form extremelydense central bulges in their starbursts with (in the most extremecases) central circular velocities approaching ∼ − (wellin excess of the most massive galaxies observed). But these are pre-cisely the systems ( M halo (cid:38) − M (cid:12) at z ∼
2, and (cid:38) M (cid:12) at z ∼
0, with bulge-dominated M ∗ (cid:38) × M (cid:12) ) expected to hostextremely massive super-massive black holes and quasars. And in-deed, Anglés-Alcázar et al. (2017b) presented preliminary studiesof the most extreme two halos here ( m12z4 and m12z3 ) including c (cid:13) , 000–000 osmic Rays on FIRE Figure 19.
Gas pressure profiles (as Fig. 5) divided into ther-mal/magnetic/CR/kinetic energy and “gravitational” pressure needed forhydrostatic balance, for three representative examples of our high-redshifthalos from Table 2 & Fig. 18. In m12z10 at z =
10 ( top ; m12z7 , m12z5 , and m12z4 at z = , ,
4, respectively, are similar), the CR pressure is slightlysub-dominant to thermal pressure, but both are much less (factors ∼ P cr ∼ P gravity , but this is a small fraction of the mass.In m12z4 at its latest time run, z = middle ; m12z3 at z = . ∼ − m12q at z = bottom ), CR pressure isclose to virial in the halo ( r (cid:38) models for black hole growth (but not AGN feedback), where theblack holes reached masses (cid:38) − M (cid:12) . So it will clearly be ofparticular interest in the future to explore the effects of AGN feed-back in these systems. We noted in § 5.3 above that CRs appear to have very weak effectson instantaneous properties within galaxies, even in the regime(e.g. large- κ and high- M ∗ ) where they have a large cosmologically-integrated effect on galaxy masses and SFHs. For example, in Figs. 1-3 & 7, in the systems where CRs suppressSF, the metallicity and central circular velocity are also lower, ac-cordingly, but these are essentially consistent with moving along ,not off of, the observed mass-metallicity relation and Tully-Fisherrelations, respectively (for more detailed studies of those, see Ma et al. 2016; El-Badry et al. 2018a). From these plots, we see thatthe galaxies with higher (lower) star formation rates have system-atically higher (lower) baryonic masses in their centers, i.e. they ap-pear to be shifting with the “supply” of gas. Fig. 8 shows this morerigorously, demonstrating that the different physics runs do not sys-tematically differ in the normalization of the Schmidt-type relations– i.e. they are not consuming gas faster/slower or more/less “effi-ciently.” Rather, the galaxies with suppressed star formation havemoved along the relation. This is quite different from what occursif we increase/decrease the number or mechanical energy of su-pernovae, which systematically moves the relation down/up (forthe same gas mass, fewer/more massive stars are required to regu-late against gravitational collapse; see Hopkins et al. 2011; Ostriker& Shetty 2011; Faucher-Giguère et al. 2013; Agertz & Kravtsov2015; Orr et al. 2018). Furthermore, although we defer a detailedstudy of the effect of CRs on galactic winds to future work, we findthat gas outflow rates and velocities in the immediate vicinity of thegalaxy are not strongly influenced by CRs – again unlike the caseif we were to change the energetics or rate of SNe (see Hopkinset al. 2012, 2013a,b, 2018c). It is possible, of course, that CRs con-tribute to outflows via “slow” or “gentle” acceleration of materialat sub-virial velocities, and this increases at larger and larger radii(discussed below) but they do not directly launch “fast” outflows( V launch (cid:38) V escape ).Similarly, Figs. 10-13 show that the galaxy-scale stellar/visualand gas morphologies are only weakly modified even in any ofour core suite. Where the stellar masses are strongly suppressedat high- κ and high- M ∗ , the galaxies do tend to have slightly later-type visual morphologies, but this is entirely consistent with theirlower masses – they simply resemble the earlier-time versions oftheir “Hydro+” counterparts (i.e. they have simply evolved lessalong the mass-morphology sequence; for more details of that seeGarrison-Kimmel et al. 2017b). Conversely, some of the low- κ runswhich produce slightly higher stellar masses exhibit earlier-typemorphologies. In the gas within the disks there are some slight dif-ferences in the “sharpness” of features in the cool gas in Figs. 10-13, but nothing qualitatively distinct. Fig. 9 shows some non-trivial(but still quantitative, rather than qualitative) effects on the ISMdistribution of phases: namely, CRs can support more warm neu-tral gas within the galaxy. This is not surprising (they both heatand pressure-stabilize this phase of gas, without ionizing it signifi-cantly), but it is quantitatively non-negligible for detailed compar-isons of ISM phase structure. Also, in Fig. 14 we see the CRs donot radically alter the kinematics of gas or stars (again, except inso-far as they suppress the amount of low-redshift star formation; seeEl-Badry et al. 2018b for a detailed study of how this varies as afunction of mass in our “Hydro+” runs).All of this is expected from § 3 and Figs. 5-6. Within thestar-forming galactic disk, CRs can have roughly equipartition-level energy densities, but their large diffusivity means that (as ob-served) the CR scale-height/length is much larger than the cold star-forming gas scale height (let alone the size of structures like GMCs,cores, etc). This means that CR pressure gradients – which actuallydetermine the forces – are usually order-of-magnitude smaller thanthe small-scale forces from gravity, magnetic fields, gas thermalpressure, and turbulent ram pressure, in the multi-phase ISM. On the other hand, we see strong CR effects in the CGM, even inmany simulations (e.g. with lower κ ) where the CRs do not have alarge effect on galaxy masses.This is especially evident in Figs. 10-13: for dwarfs with halo c (cid:13) , 000–000 Hopkins et al. masses M halo (cid:38) M (cid:12) , the “CR+” runs often feature much moreprominent cool gas in the CGM. Figs. 1-3 show that the total bary-onic mass density on CGM scales ( ∼ −
100 kpc) is not dras-tically modified by CRs, although there are subtle (order-unity)changes evident in many cases (e.g. some of the intermediate-mass m11 halos with lower- κ feature enhanced gas density on largescales). The runs where CRs dominate over thermal pressure in theCGM (see Fig. 5) typically feature modest (factor ∼ −
3) overallenhancements of gas density at some intermediate range of radii oforder the halo scale radius ( R s ∼ R vir / ∼ R vir .As noted above, Fig. 6 shows this enhancement is not drivenby CR “heating” via either collisional or streaming processes. Mostof the CGM mass in these runs is in “cool” or “warm” CGMphases, where the cooling times are relatively short and the tem-perature is maintained largely by photo-ionization equilibrium (Jiet al. 2019), hence their low (sub-virial) thermal pressure. Rather,Fig. 5 shows that the “maintenance” of this gas density profile owesto the CRs establishing a quasi-virial-equilibrium pressure profileon large scales. This is qualitatively similar to the findings of e.g.Salem et al. (2014, 2016); Chen et al. (2016) in their cosmologicalsimulations with CRs, despite their adopting different numericalmethods and treatment of the CR and star formation/feedback “mi-crophysics,” suggesting the conclusion is robust to these details.These, plus the weak effects of CRs on essentially all star for-mation and outflow and internal galaxy properties discussed above,demonstrate that the CRs primarily operate as a “preventive” feed-back mechanism, rather than an “ejective” or “responsive” feed-back mechanism. Rather than launching strong outflows, or remov-ing gas from the halo, or preventing gas which has already accretedinto the galaxy from efficiently forming stars, the primary effect ofCRs appears to be preventing gas in the halo from actually accret-ing rapidly onto the galaxy. These CR effects result in a more subtlere-arrangement of gas mass within the halo and between differentphases. Even when (cid:38) L ∗ halos become dominated by a hot, viri-alized halo gas, cooling of that hot atmosphere or re-accretion ofpreviously ejected gas can provide a substantial gas supply for latetime star formation (Kereš et al. 2005; Muratov et al. 2015; Anglés-Alcázar et al. 2017a). Preventing such accretion (or re-accretion)has important consequences for the late time star formation andgrowth of galaxies.Note that even a modest (factor ∼
2) effect on the gas densityprofile around the halo scale radius is significant – this is the ra-dius where most of the halo gas mass resides, and most of the totalbaryon supply is in the halo (especially in the lower-mass galaxies)rather than in stars, so this relatively modest effect can easily ac-count for differences of factors ∼ − within the galaxies. We now discuss the different detailed CR transport processes mod-eled here, their effects, and sensitivity to uncertain physics. Note that it is not possible here to completely dis-entangle the role ofCRs preventing “new” gas from accreting onto the galaxy at all, versus“lofting” or “gently/slowly accelerating” cool gas in galactic fountains (atvelocities (cid:28) V c ) and preventing it from re-accreting. In an instantaneoussense these are the same thing: suppressing cool gas from falling into thegalaxy. Future work following trajectories of individual gas elements willallow us to better disentangle these possibilities. In § 5.2 & 5.3, we noted the existence of a “sweet spot” in theCR diffusion coefficient κ , demonstrated throughout but especiallyin Fig. 15. As predicted analytically in § 3 and confirmed in thesimulations in Fig. 4, at too-low κ , CRs take too long to es-cape dense gas, and lose their energy rapidly to collisional pro-cesses. Advection alone (e.g. CRs being “carried” out of galax-ies in super-bubbles and cold outflows) is sub-dominant even atquite low κ . Even without collisional losses, it requires a rela-tively high κ for CRs to build up in the CGM before trans-Alfvénicstreaming (which is also “lossy”) takes over. On the other hand, atarbitrarily high- κ , CRs would escape “too efficiently” even fromthe CGM/extended halo and the steady-state CR pressure ( ∝ /κ )would become too small to support gas or do any interesting work.For dwarfs, this “special” value of κ ∼ × cm s − , buteven there the effects of CRs are weak, and moreover the obser-vations of γ -ray luminosities from dwarfs like the SMC, LMC,and M33 favor higher κ (cid:38) × cm s − (where they escapelow-density dwarfs more efficiently and do little work). But formassive (MW-mass) systems, interestingly, the “sweet spot” in κ ∼ − × cm s − appears to neatly coincide with theobservationally-favored values.It is also encouraging that the more detailed study of CR trans-port physics in isolated (non-cosmological) simulations in Chanet al. (2018) identified approximately the same critical κ needed toreproduce the γ -ray observations. In detail we favor slightly higher κ here, owing to the larger gaseous halos present in cosmologicalsimulations, which contribute to increasing the probability of CRsre-entering the disk (hence “residence time”) before they escape,but this is expected. We also note that the value we quote hereis the parallel diffusivity. If one assumed isotropic CR diffusion,the corresponding isotropically-averaged ˜ κ would be a factor ∼ κ spans ap-proximately an order-of-magnitude in range, and within this range,the predictions are not especially sensitive to κ (i.e. changing κ bya factor ∼ κ models here. If, in nature, the “ef-fective” κ simply varies systematically from galaxy-to-galaxy (e.g.with cosmological environment or halo mass) this is not so prob-lematic: it simply amounts to choosing a different “effective κ ” as“most appropriate” for our different simulations. If the effective κ varies on small spatial or time scales, this is also not such a con-cern, as the CRs diffuse sufficiently rapidly that these variations areaveraged out on the large CGM scales ( ∼ −
100 kpc) on whichthey act (and indeed, because the transport is anisotropic, local tur-bulent magnetic field-line variations already mean that the actuallocal diffusivity is constantly varying by factors of several on smallscales). c (cid:13) , 000–000 osmic Rays on FIRE The biggest cause for concern is if, in reality, there are largesystematic variations in effectively diffusivity with some propertythat varies between galaxies and CGM (e.g. spatial scale, density,magnetic field strength). It is plausible to imagine models where κ ∼ × cm s − in the ISM (producing the same γ -ray lumi-nosity), but CRs “de-couple” or “free-stream” or escape much morerapidly in the CGM, dramatically reducing their effects on galaxyformation (see § 6). We confirm the conclusion in Chan et al. (2018), that streaming attrans-Alfvénic speeds (as parameterized here) appears to be sub-dominant in transport. For observationally-favored (high) κ val-ues, transport is sufficiently fast that streaming at speeds ∼ v A can only take over as a dominant transport mechanism, and be-gin to dissipate a large fraction of the CR energy, outside a radius (cid:38) −
100 kpc (§ 3 and Fig. 6). Thermalized energy from stream-ing losses are never, in the simulations here, able to compete withgas cooling losses (Fig. 6). As a result, turning off “streaming” en-tirely (Fig. 17) produces only minor effects on galaxy properties,around the favored κ .Above, we argued that for CRs to have large effects, high “ef-fective” κ ∗ is needed in order for CRs to escape the dense star-forming regions of galaxies, and these high- κ values are also fa-vored by observations of dwarfs and MW-like galaxies (which ap-pear to be well below calorimetric). Because the actual physicswhich regulates streaming speeds remains uncertain (and streaminglosses will not appear in the γ -ray constraints), one might reason-ably wonder whether the same effects might be accomplished bymaintaining a relatively low diffusivity (e.g. κ (cid:107) = × cm s − )but increasing the streaming velocity by a modest factor. This is theexperiment performed in Figs. 4 and 17. If the “streaming losses”scale with v stream – as assumed in this test for the sake of illustration– then increasing v stream is not equivalent to increasing κ . In fact,increasing v stream and decreasing κ in this manner leads to weaker,not stronger, effects of CRs on galaxy properties. And while in-creasing v stream (with κ constant) does slightly reduce the predicted L γ / L SF (Fig. 4), the effect is minor for the values considered here.There are two fundamental reasons for this. First, if the “streamingloss” term scales with v stream , then more of the CR energy is dis-sipated close to the galaxy (where cooling is efficient). Second, ifstreaming dominates then the effective diffusivity is κ ∗ ∼ v stream (cid:96) cr (see § 2.3; where (cid:96) cr ≡ P cr / |∇ (cid:107) P cr | ), so for the large κ ∗ favored werequire v stream ∼ − ( κ ∗ / × cm s − ) ( (cid:96) cr / kpc ) − ),much larger than v A or c s .So, if we set v stream to be much larger ( (cid:38) − − ),and limit streaming losses to scale with the Alfvén speed ( ∼ v A ∇ P cr , as these come from the excited Alfvén waves which can-not propagate faster than this), then streaming will become essen-tially degenerate with high- κ diffusion. We see this indirectly viathe fact that the maximum “transport speed” of CRs has little effecton the results once it is sufficiently large (Fig. A3). How, micro-physically, these transport speeds arise remains uncertain. As shown in Fig. 6, thermalized CR collisional or streaming lossesare essentially never important as a source of heating the gas(they are always highly sub-dominant to cooling). However, theycan be important as loss mechanism for the CRs themselves. Al-though it is sometimes (incorrectly) assumed that CRs are “loss-less” (or “don’t cool”), the collisional loss timescale for CRs indense gas on length scale ∼ (cid:96) is shorter than the diffusion time if n (cid:38)
10 cm − ˜ κ ( (cid:96)/ kpc ) − . And indeed, essentially all observedstarburst galaxies, which have very high nuclear gas densities (seeFig. 4) are consistent with being proton-calorimeters (i.e. all ormost of the CR energy appears to be lost). For that reason, at eitherlow- κ ( (cid:28) cm s − ) or high gas surface densities, collisionallosses play an important role in limiting CR energetics and effectson galaxies. Otherwise, in “steady state” very low- κ would producemore trapping and allow one to artificially build up essentially arbi-trarily high CR energy densities. On the other hand, once κ is rela-tively high, in galaxies with central densities comparable to or lessthan the MW, then most of the CRs escape without decaying (e.g. L γ is well below calorimetric in Fig. 4). In these cases, which in-clude most of the cases of greatest interest above, collisional lossesnecessarily play a minor role, and so de-activating these losses inFig. 17 produces relatively small effects under these conditions.Streaming losses are similarly not dominant if κ ∗ is suffi-ciently large, at least out to radii r stream ∼ κ/ v stream where trans-Alfvénic streaming can begin to dominate the transport. Remov-ing streaming losses with similar v stream and κ simply removes thelosses at large r , making the equilibrium CR pressure profile fallsomewhat less rapidly. But because, in this regime, it is alreadyfalling more rapidly than in the halo center (see Fig. 5), this is asecond-order effect. Of course, as discussed above, if we substan-tially increase the streaming velocity beyond the Alfvén speed, andcorrespondingly increase the streaming loss rate (and shrink r stream so the losses happen closer to the galaxy), then it can become im-portant limiting the effect of CRs. Of course, as noted above, if weincrease v stream but remove or suppress the loss term so it does notscale accordingly, this is just equivalent to increasing κ . Fig. 17 considers some experiments where we remove magneticfields, and assume all transport processes are isotropic (instead ofprojecting them along field lines). There, and in the other galaxyand CGM properties studied here, we see no radical or qualitativechange in behavior. In the dwarfs, the results are very similar; in theMW-mass experiment ( m12i ), switching to isotropic transport andremoving magnetic fields (Fig. 17) leads to a slightly weaker effectfrom CRs here, largely because the CRs escape more efficiently(but the difference is small compared to removing CRs entirely).In general, as shown in Chan et al. (2018), the leading-order effectof anisotropic transport is to alter the effective (galaxy-averaged) κ , by a modest (order-unity) factor, with a net effect of somewhatlower diffusion coefficients required in isotropic diffusion case forsimilar effect.Some of the reason for this is that magnetic fields are highlydisordered on most scales, especially in the CGM. This is shownmore rigorously in previous studies (e.g. Su et al. 2018a), and willbe explored in subsequent work as well (Ji et al. 2019). This isnot surprising, as the CGM of ∼ L ∗ and dwarf galaxies is generi-cally trans-sonically turbulent and has plasma β (cid:29) c (cid:13) , 000–000 Hopkins et al.
We have noted in several places above various numerical tests inour simulations. For example, we have run our standard physicsset (“Hydro+”, “MHD+”, “CR+( κ = e κ = e m10q (with resolu-tion m i , = . , . , m11q ( m i , = . , , m12i , m12f , and m12m (each with m i , = , , m10q and m11q show very weak resolution-dependence in “Hy-dro+” runs (or “MHD+”, in Su et al. 2017). We find the same for“CR+” runs, reinforcing our conclusions. As studied extensively inHopkins et al. (2018b), our massive halos m12i , m12f , m12m do show systematic resolution dependence in all the runs. This is ofcourse important and the scalings and reasons for this are discussedin Hopkins et al. (2018b). What is important, for our purposes here,is that our qualitative conclusions about the systematic effects (orlack thereof) of MHD and CRs, are independent of resolution. Likethe dwarf runs, at every resolution we find the MHD runs and low- κ CR runs have little or no systematic effect on the massive ha-los, while the high- κ CR runs suppress SF significantly, via thesame physical mechanisms. The effect is slightly stronger in thehigher-resolution runs because the galaxy is overall lower-densityand lower-mass even in the “Hydro+” runs (this owes to better res-olution of galactic winds “venting” and mixing in the CGM), whichmeans (per Fig. 4) the collisional losses are somewhat less.Chan et al. (2018) also considered extensive tests of the im-posed maximum CR free-streaming speed ˜ c , and showed that aslong as this is faster than typical bulk velocities of gas in the simu-lated systems, it has no effect on the results: Appendix A shows thisholds in our simulations so long as ˜ c (cid:38)
500 km s − . More detailedresolution tests and idealized validation tests of our numerical CRmethods, and explicit comparison of the results of two-moment vs.one-moment approximations for the CR transport flux solver, areall presented in Chan et al. (2018): none of these presents obviousnumerical concerns here (see also Appendix A).We stress that extensive numerical tests of almost every otheraspect of these simulations (resolution, force softening, hydrody-namic solvers, radiation-hydrodynamics solvers, etc.) are presentedin Hopkins et al. (2018b), Hopkins et al. (2018c), and Hopkinset al. (2018a), for our “Hydro+” models. For more detailed reso-lution and physics tests of the “MHD+” models, we refer to Suet al. (2017, 2018a), and for extensive numerical validation tests ofthe MHD, conduction, viscosity, and anisotropic transport solverswe refer to Hopkins & Raives (2016); Hopkins (2016, 2017). We present and study a suite of >
150 new high-resolution ( ∼ − M (cid:12) , ∼ −
10 pc, 10 −
100 yr, ∼ − cm − )FIRE-2 cosmological zoom-in simulations, with explicit treat-ment of stellar feedback (SNe Types Ia & II, O/B & AGB mass-loss, photo-heating and radiation pressure), magnetic fields, fully-anisotropic conduction and viscosity (accounting for saturation andtheir limitation by plasma instabilities at high- β ), and cosmic rays.Our CR treatment accounts for injection in SNe shocks; advec-tion, fully-anisotropic streaming and diffusion; and losses from col-lisional (hadronic+Coulomb), streaming, and adiabatic processes.We systematically survey different aspects of the CR physics anduncertain transport coefficients. We examine the effects of these physics on a range of galaxy properties including: stellar masses,star formation rates and histories, metallicities and abundances,stellar sizes and baryonic mass profiles, dark matter mass pro-files, rotation curves, visual morphologies and kinematics of starsand gas, and gas phase distributions. We survey these propertiesacross a suite of simulations spanning all redshifts, and masses at z ∼ −
10 ranging from ultra-faint dwarf ( M ∗ ∼ M (cid:12) , M halo ∼ M (cid:12) ) through Milky Way mass. We summarize our conclusionsas follows: • We confirm the growing body of work showing that mag-netic fields, physical conduction, and viscosity on resolved scaleshave little effect on any galaxy property studied (Figs. 1-3 and7-8). These simulations reach higher resolution (sufficient to re-solve the Field length in warm and hot gas with T (cid:38) × K ( n / .
01 cm − ) . ), and include more detailed treatment of theeffect of plasma instabilities on transport coefficients, compared toour previous work (Su et al. 2017), but this only serves to rein-force that conclusion. It is of course possible there are importantun-resolved effects which could be important via “sub-grid” effects(e.g. altering the effective cooling rates or stellar initial mass func-tion). • Magnetic fields are highly-tangled, at all mass scales we sur-vey, with or without CR physics (see also Ji et al. 2019). PerFig. 5, the plasma β ≡ P thermal / P magnetic varies enormously in theISM ( β ∼ . − ∼ −
10 in the diffuse ISM in MW-mass galaxies, and larger ∼ −
30 or ∼ −
300 in lower-mass M halo ∼ M (cid:12) and ∼ M (cid:12) dwarfs, respectively), consistent with many recent stud-ies of amplification (see e.g. Martin-Alvarez et al. 2018, and refer-ences therein). It rises in the CGM with galacto-centric distance (toa median ∼ − (cid:38) R vir , with local variations reaching ∼ ). • CRs have relatively weak effects on the galaxy properties stud-ied, in dwarfs with M halo (cid:28) M (cid:12) ( M ∗ (cid:28) M (cid:12) ), for essen-tially any physical CR parameters considered, once realistic me-chanical and radiative feedback are already included (Figs. 1, 7,10). This is both predicted and easily understood from basic en-ergetic considerations (§ 3). Previous claims to the contrary haveeither failed to account for (a) the fact that realistic dwarfs havevery low M ∗ / M halo (and correspondingly low SFRs, hence SNeand CR energy injection rates), without much “hot halo” gas, (b)realistic supernova and radiative feedback which easily overwhelmthe effects of CRs in dwarfs, and/or (c) observational constraintsfrom γ -ray emission in dwarfs which place upper limits on the col-lisional loss rate and require that > −
99% of the CR energyescape without hadronic losses (prohibiting low transport speeds).It remains possible CRs modify the CGM in these dwarfs, or mod-ify processes like ram-pressure stripping in dwarf satellites aroundmassive galaxies (not studied in detail here). • CRs (from SNe) also have relatively weak effects on galaxyproperties, at any mass scale, at high redshifts z (cid:38) − z , the CGM density and ram pressure ofdense, cold inflows (carrying most of the mass) is much higher,and so CR pressure is insufficient to strongly suppress those in-flows (Fig. 19). Moreover the periods of strongest inflow are oftenaccompanied by dense starbursts within the galaxy where surfacedensities exceed (cid:29) . − ( (cid:29) M (cid:12) pc − ), during whichCRs experience strong hadronic losses. Consistent with observedlow-redshift starbursts at these densities, they become approximateproton calorimeters. c (cid:13) , 000–000 osmic Rays on FIRE • CRs can have significant effects in massive galaxies ( M halo (cid:38) M (cid:12) , M ∗ (cid:38) M (cid:12) ) at relatively low redshifts ( z (cid:46) − ∼ z = ∼ − κ ∗ ∼ × − × cm s − (which can arise from acombination of microphysical diffusion and/or streaming). • In these systems, the CRs primarily operate on the CGM, andhave relatively little direct effect within the galaxies (e.g. Figs. 8,9, 14). Essentially any model where CRs are sufficiently well-trapped in galaxies to, e.g. slow down SF directly in relative dense( > − ) disk gas, or launch strong outflows (velocities (cid:38) V c ) isruled out observationally and results in excessive CR losses (greatlylimiting their net effect). We see essentially all galaxy propertiesmove along , not off of, standard scaling relations. This is dramat-ically different from e.g. increasing the mechanical energy or rateof SNe. Instead, CR feedback is primarily “preventive.” In massivehalos at low redshifts, with the appropriate κ ∗ , the CRs establish aquasi-virial-equilibrium pressure profile, which dominates over thegas thermal and magnetic pressure from ∼ −
200 kpc, and sup-ports warm+cool gas ( T (cid:28) K) which would otherwise cool andrain onto the galaxy (Fig. 5 and Ji et al. 2019). • Given present uncertainties, the most important-yet-uncertainparameter determining the effects of CRs (especially in massivegalaxies) is the effective diffusivity κ ∗ . If too low, CRs are trappedand (a) lose energy to collisional processes in dense gas (negatingtheir ability to do work, and directly violating observational con-straints from spallation and γ -ray emission), and (b) cannot prop-agate to large-enough radii to slow accretion in the outer halo. If κ ∗ is too high, CRs should simply escape and their equilibriumpressure, even in the outer halo, would be too low to do interestingwork. The “sweet spot” is approximately an order-of-magnitude inwidth, and agrees well in the simulations and simple analytic mod-els. • We show that with κ ∗ ∼ − × cm s − , cosmologi-cal simulations reproduce observed γ -ray emissivities (and simi-lar constraints from spallation in the MW), at dwarf through LocalGroup through starburst-galaxy density/star formation rate scales(Fig. 4). This echoes the conclusion from recent non-cosmologicalsimulations (Chan et al. 2018), for the first time in cosmologicalsimulations. • Cosmic ray “heating” of the gas (via energy transfer throughcollisional or streaming losses) is orders-of-magnitude smaller thangas cooling rates and never important for the gas (Fig. 6). It is how-ever important as a loss mechanism for CRs, especially in densestarburst-type systems (which become proton calorimeters) or theouter halo (where trans-Alfvénic streaming can dominate). • Around the effective diffusivities of interest, anisotropy andstreaming at the Alfvén speed play second-order roles in CR trans-port (Fig. 17). Anisotropy does not have radical obvious effects, butdoes lead to somewhat suppressed diffusion, especially in densegas where losses can be large (necessitating larger- κ ∗ to enableescape). Trans-Alfvénic streaming is too slow to account for therequired κ ∗ in the ISM and inner CGM, although it can dominatetransport outside of some radius ∼ κ ∗ / v A (in the outer halo). • We present a simple analytic equilibrium model in § 3, whichis able to at least qualitatively predict essentially all of the relevantCR effects here. Particularly in the mass and κ ∗ range of great-est interest, this analytic model provides a remarkably accurate de- scription of when CRs become important, the favored values of κ ∗ ,and the resulting CR pressure profile and equilibrium gas densityprofile in the CGM of CR-dominated halos. This study – despite its length – is far from comprehensive. Infuture work, we plan to explore a number of properties of thesesimulations in greater detail, including: their more detailed outflowand gas phase structure, properties of GMCs in the ISM and ac-cretion onto the galactic disk, observable ions in the CGM, proper-ties of resolved satellites around massive galaxies in the modifiedCR-dominated CGM. There may be more detailed tracers whichcan distinguish between different models in greater detail, or areaswhere MHD/conduction/viscosity/CRs have large effects which wehave failed to identify here. Moreover, even assuming our simpleCR treatment is a reasonable representation, there are a number ofbasic physical processes meriting further investigation. For exam-ple it is unclear how the non-linear thermal instability operates in astratified halo supported primarily by a CR pressure gradient (noris it likely this can be fully resolved in the simulations here). It isnot obvious how much of the additional cool gas in the halo in thehigh- κ ∗ , high- M ∗ CR runs owes to “pure suppression” of new in-flowing material vs. CRs “gently” (and slowly) re-accelerating or“lofting” cool gas which would otherwise recycle in galactic foun-tains. And it is worth exploring how the virial shock and transition“out of” the CR dominated-regime at large radii occur. These andmany additional questions clearly merit exploration in idealized,high-resolution numerical experiments.Future and parallel work will also explore the role of CRs inmore massive ( > L ∗ ) galaxies. We predict and find in our simu-lations that the strength of the effects from CRs sourced via SNescales as ∝ ˙ M ∗ / M halo , which is maximized around ∼ L ∗ . In verymassive halos (in nature), this declines both because (a) M ∗ / M halo drops, and (b) star formation is “quenched.” As a result, our par-allel study in Su et al. (2018b) demonstrates that CRs sourced bySNe (including Ia’s) cannot possibly solve the “cooling flow prob-lem” and resist excessive cooling and star formation in very mas-sive ( M halo ∼ M (cid:12) ) halos. And here we show essentially thesame at higher redshifts when these halos initially form most oftheir stars and dense, compact bulges (when their progenitor halomasses are (cid:46) − M (cid:12) ).However, in those same massive galaxies, super-massive blackholes (SMBHs) and AGN appear to dominate CR production (seenin e.g. jets and “bubbles”) by orders-of-magnitude compared toSNe, and in this paper we only accounted for CRs produced inSNe. In a companion study (Su et al. 2018c) we show that injec-tion of ∼ erg s − in CRs in a ∼ M (cid:12) halo – modest for theSMBHs and AGN in those systems, but ∼ − ∼
10% of the SNe energy goesinto CRs – can have a dramatic effect on the halo gas and cool-ing flow in these massive galaxies. Obviously it will be importantto revisit these high-redshift halos with an explicit model for AGNfeedback.
We wish to strongly emphasize that our conclusion is only thatCRs could be important to massive galaxies at low redshifts, notthat they are necessarily. We have parameterized our ignorance byadopting a simple two-moment model with fixed parallel diffusiv-ity κ (cid:107) , but the reality is that deep physical uncertainties remain. It isnot clear what actual physics determines the “effective diffusivity” c (cid:13) , 000–000 Hopkins et al. or transport parameters of CRs in all the regimes here, let alone howthese parameters should scale as a function of local plasma proper-ties (e.g. magnetic field strength, density, mean-free-path, strengthof turbulence, etc.). Although constraints from detailed modelingof the Milky Way CR population and γ -ray observations of nearbygalaxies favor an effective diffusivity κ ∗ (cid:38) cm s − , this con-straint is (a) only measured in a few z = n cr n nucleon ,is maximized). It is completely plausible to imagine models where κ ∗ is ∼ × cm s − in this gas, but once CRs reach the CGMat r (cid:29)
10 kpc (where the gas has low-density, very weak magneticfields β ∼ − , and long mean-free paths ∼ −
100 pc), thetransport speed rises dramatically and CRs simply “escape” ratherthan forming a high-pressure halo. This is, essentially, the classic“leaky box” model. Even if the CRs are confined in the halo on ∼
100 kpc scales, it is not clear whether the local tight-coupling ap-proximation is valid in the tenuous CGM – i.e. can CR “pressure”actually be simply “added” to to the local gas stress tensor in theMHD equations? Or can CRs “slip” or couple with non-negligiblelag? Fundamentally, our treatment of CRs as a fluid depends onassuming something about their ability to reach a micro-scale gy-rotropic equilibrium distribution function, which may not be validwhen scattering rates are low, i.e. transport speeds are very large(see e.g. Holcomb & Spitkovsky 2019). These are critical ques-tions which could completely alter our conclusions (in the exam-ples above, they would make the effects of CRs much weaker), andmay fundamentally require PIC-type simulations that can followexplicit kinetic plasma processes to answer.Observationally, direct constraints on CRs in the regimes ofinterest are unfortunately very limited. Essentially the only suchmeasure outside the MW is the γ -ray luminosity discussed exten-sively here. Recall, most of the CR energy/pressure is in ∼ GeVprotons. So constraints on the CR electrons (e.g. synchrotron),which tend to dissipate their energy far more efficiently and closerto sources, are not particularly illuminating for this specific ques-tion, nor are constraints on the high-energy CR population (whichhas very different diffusivity and contains negligible CR pressure).However, given that the regime of greatest interest is preciselywhere we predict CRs should have a large effect on the tempera-ture and density distribution of CGM gas around ∼ L ∗ galaxies – atopic of tremendous observational progress at present – it is likelythat indirect observational constraints from the CGM will repre-sent, in the near future, the best path towards constraining the CRphysics of greatest interest here. ACKNOWLEDGMENTS
We thank the anonymous referee for a number of insightfulcomments and suggestions. Support for PFH and co-authorswas provided by an Alfred P. Sloan Research Fellowship, NSFCollaborative Research Grant
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APPENDIX A: ADDITIONAL NUMERICAL TESTS
In this Appendix we present some examples of additional numeri-cal tests, discussed in § 5.7 in the main text.Figs. A1-A2 show resolution tests, for MW-mass halos andvaried diffusivity κ . As noted in § 5.7 we have also extensively var-ied the resolution in our dwarf halos m10q and m11q , but since(a) those halos exhibit no large effect from either MHD or CRs(at any κ ) and (b) our extensive resolution studies in Hopkins et al.(2018b) (as well as Su et al. 2017 and Chan et al. 2018) have shownthe dwarf properties in “Hydro+” and “MHD+” runs are extremely [ Gyr ] S F R [ M (cid:31) y r − ] m12i a l o g ( M ∗ ) [ M (cid:31) ] ( κ = e ) CR+ ( κ = e ) a − − (cid:28) [ Z ∗ / H ] (cid:27) − ( r ) [ kpc ] l o g ( ρ [ r ] ) [ M (cid:31) k p c − ] r [ kpc ] V c [ k m s − ] m i ,1000 =
56 0 5 10Cosmic Time [ Gyr ] − m12f a
810 11.111.111.110.90.0 0.25 0.5 0.75 1.0Scale Factor a − −
10 0.20.20.20.1 − ( r ) [ kpc ]
510 3.33.92.93.30 20 40 60 r [ kpc ] m i ,1000 =
56 0 5 10Cosmic Time [ Gyr ] − m12m a
810 11.111.211.110.80.0 0.25 0.5 0.75 1.0Scale Factor a − −
10 0.20.20.20.1 − ( r ) [ kpc ]
510 5.35.64.24.80 20 40 60 r [ kpc ] m i ,1000 = Figure A1.
As Fig. 3, at order-of-magnitude lower resolution. At lower res-olution for our MW-mass systems, all the galaxies exhibit higher SFRs andstellar masses, as studied extensively in Hopkins et al. (2018b). However,the systematic effects (or lack thereof) of MHD and CRs are robust acrossresolution. Lower-resolution runs of our dwarfs ( m10v , m10q , m11q ) showno significant resolution dependence with or without CRs (as shown in Hop-kins et al. 2018b). insensitive to resolution, it is not surprising that we find there is nochange at any resolution level for these halos (this is also demon-strated in Fig. 7). Therefore we focus on MW mass where both CRphysics and resolution have much larger effects.As discussed in § 5.7, Figs. A1-A2 do show a systematic reso-lution dependence in our massive m12 halos (compare also Fig. A1to 3). In massive halos, lower-resolution runs produce larger stellarmasses and, correspondingly, higher central baryonic densities andenhanced circular velocities. This is all studied in extensive detailin Hopkins et al. (2018b). Critically, for our study here, we see thesame systematic effects (or lack thereof) of MHD and CRs at all resolution levels.In Fig. A3, we systematically compare halos m10q , m11q , m12i , chosen with the values of κ where effects on the galaxyare maximal, varying the numerically-imposed maximum CR free-streaming speed ˜ c . As expected, we see that this has little effect onour conclusions (variations are smaller than stochastic run-to-runvariations) for essentially all values ˜ c (cid:29)
200 km s − , fast enoughthat CRs can (when they should , according to the diffusivity κ and streaming speed v stream ) diffuse or stream faster than local gasrotation/outflow velocities. Otherwise, CR effects are somewhatweaker, as the CRs can spend more time artificially trapped indense gas (where collisions sap energy). This is consistent with ourmore detailed numerical study in Chan et al. (2018).Chan et al. (2018) also describe and test in detail the CR fluxequation (Eq. 5), i.e. the equation for the diffusive CR flux ˜ F cr .They showed that using alternative forms of this equation, or sim-ply omitting it entirely and solving directly a single 0th-moment(pure diffusion) CR energy equation (setting ˜ F cr ≡ κ ∗ ∇ (cid:107) e cr ) giveessentially identical results. We confirm this in our cosmologicalsimulations in Fig. A4. We re-run a low and high-mass halo whereCRs have a large effect ( m11i and m12i ), adopting the alternative c (cid:13) , 000–000 Hopkins et al. S F R [ M y r − ] l o g ( M ∗ ) [ M ][ Z ∗ / H ] l o g ( ρ [ r ] ) [ M k p c − ] V c [ k m s − ] [ Gyr ] m12i (LR) a
810 11.111.010.710.7 CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) a − −
10 0.30.30.10.1 − ( r ) [ kpc ] r [ kpc ] m i ,1000 =
56 0 5 10Cosmic Time [ Gyr ] m12i (HR) a
810 10.910.410.4 CR+ ( κ = e ) CR+ ( κ = e ) CR+ ( κ = e ) a − −
10 0.3-0.1-0.0 − ( r ) [ kpc ] r [ kpc ] m i ,1000 = Figure A2.
As Fig. A1, varying the CR diffusion coefficient κ more exten-sively, in both our default (high-resolution “HR”; right ) and low-resolution(“LR” from Fig. A1; left ) versions of m12i . Although the stellar massesand central densities do depend on resolution, the systematic effect of CRsis similar at all resolution levels, for all values of κ explored. [ Gyr ] − − − SF R [ M (cid:30) y r − ] m10q a l og ( M ∗ ) [ M (cid:30) ] ( ˜ c = ) CR+ ( ˜ c = ) CR+ ( ˜ c = ) CR+ ( ˜ c = ) a − . − . (cid:28) [ Z ∗ / H ] (cid:27) -1.7-1.9-1.7-1.7 − ( r ) [ kpc ] l og ( ρ [ r ] ) [ M (cid:30) kp c − ] r [ kpc ] V c [ k m s − ] m i , = .
25 0 5 10Cosmic Time [ Gyr ] − − m11q a
789 9.39.29.29.20.0 0.25 0.5 0.75 1.0Scale Factor a − − − ( r ) [ kpc ]
468 4.14.13.04.20 10 20 30 r [ kpc ] m i , = . [ Gyr ] m12i a
810 10.810.710.70.0 0.25 0.5 0.75 1.0Scale Factor a − −
10 0.20.10.1 − ( r ) [ kpc ] . . . r [ kpc ] m i , = Figure A3.
As Fig. A1, comparing our “CR+” runs ( κ = e
28 for thedwarfs m10q & m11q , and κ = e
29 for m12i ), varying the numerical“maximum CR free-streaming speed” ˜ c (values in kms − ). For values˜ c (cid:38) − (fast enough that CRs can “outpace” most bulk rotationand outflow motion), we see excellent agreement, so our results the maintext should be insensitive to this. [ Gyr ] − − S F R [ M (cid:30) y r − ] m11i a l o g ( M ∗ ) [ M (cid:30) ] v stream = v A CR+: v stream = a − − (cid:28) [ Z ∗ / H ] (cid:27) -0.6-1.0-1.2-1.0-1.1 − ( r ) [ kpc ] l o g ( ρ [ r ] ) [ M (cid:30) k p c − ] r [ kpc ] V c [ k m s − ] m i ,1000 = [ Gyr ] m12i a
810 10.910.710.710.710.70.0 0.25 0.5 0.75 1.0Scale Factor a − −
10 0.20.10.20.20.2 − ( r ) [ kpc ] r [ kpc ] m i ,1000 = Figure A4.
As Fig. A1, comparing our default “CR+( κ = e m11i and m12i , with (1) a different formfor the CR flux equation (Eq. A1), (2) streaming speed = v A (instead of ourdefault = v A ), and (3) streaming speed = ∼ v A ∇ P cr ). None of these variations have appreciable effects onour conclusions. formulation of the flux equation from Thomas & Pfrommer (2018): ˆ ˜ F cr ˜ c (cid:104) ∂ | ˜ F cr | ∂ t + ∇ · (cid:0) v gas | ˜ F cr | (cid:1) + ˜ F cr · { (ˆ ˜ F cr · ∇ ) v gas } (cid:105) (A1) + ∇ (cid:107) P cr = − ( γ cr − ) κ ∗ ˜ F cr with ˆ ˜ F cr ≡ ˆB by definition. As discussed in Chan et al. (2018),this is essentially the same as our formulation, up to whether ˆ ˜ F cr appears inside or outside of the derivative terms. The Thomas &Pfrommer (2018) formulation arises if we assume the perpendicu-lar fluxes that arise when magnetic fields rotate are instantaneouslyconverted into gyrotropic motion by micro-scale instabilities; thisand the assumption of frame in which the motion is gyrotropic pro-duce the small “pseudo-force” correction ˜ F cr · { (ˆ ˜ F cr · ∇ ) v gas } . Inpractice, the choice of Eq. A1 instead of our default Eq. 5 onlyproduces differences below the CR mean free path, and has no ap-preciable effect here. Unsurprisingly, the flux equation from Jiang& Oh (2018), which gives behavior “in-between” our Eq. 5 andEq. A1 above (as shown in Chan et al. 2018) is even more similarto our default.We have also experimented with a modified CR pressure termin the hydrodynamic equation motivated by Jiang & Oh (2018): re-placing ∇ P cr → ∇ ⊥ P cr − ( γ cr − ) ( F − v st [ e cr + P cr ] ) /κ ∗ ≈ ∇ P cr + ˜ c − [ ∂ ˜ F /∂ t + ∇· ( v gas ˜ F cr ) ], i.e. keeping the perpendicular CR pres-sure but only the parallel component contributing to CR scattering. c (cid:13) , 000–000 osmic Rays on FIRE Because this correction only enters in the ∼ / ˜ c terms, and be-cause field lines are tangled, this has only weak effects.Finally, Fig. A4 also considers two additional variants of thestreaming speed: taking v stream = v A (as compared to our default = v A ), or setting v stream = retaining the “streaming loss”term: in all of these the loss term scales as v A ∇ P cr . Given the muchlarger variations to the streaming speed considered in the main text(which produce small effects), it is expected that the effect of thesevariations is very small. c (cid:13)000