MNRAS 000,1{19 (2020) Preprint 4 September 2020 Compiled using MNRAS LATEX style file v3.0 A comparison of H2 formation models at high redshift Alexander Sch¨abe1;⋆, Emilio Romano-D´ıaz1, Cristiano Porciani1, Aaron D. Ludlow2 and Matteo Tomassetti3 1Argelander Institut fur¨ Astronomie , Auf dem Hugel¨ 71, D-53121 Bonn, Germany 2International Centre for Radio Astronomy Research, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia, 6009, Australia 3Marks and Spencer Group plc. Waterside House 35 North Wharf Road London W2 1NW Accepted 2020 July 27. Received 2020 July 3; in original form 2020 March 7 ABSTRACT Modelling the molecular gas that is routinely detected through CO observations of high-redshift galaxies constitutes a major challenge for ab initio simulations of galaxy formation. We carry out a suite of cosmological hydrodynamic simulations to compare three approximate methods that have been used in the literature to track the formation and evolution of the simplest and most abundant molecule, H2. Namely, we consider: i) a semi-empirical procedure that associates H2 to dark-matter haloes based on a series of scaling relations inferred from observations, ii) a model that assumes chemical equilibrium between the H2 formation and destruction rates, and iii) a model that fully solves the out-of-equilibrium rate equations and accounts for the unresolved structure of molecular clouds. We study the impact of finite spatial resolution and show that robust H2 masses at redshift z ≈ 4 can only be obtained for galaxies that are sufficiently metal enriched in which H2 formation is fast. This corresponds to H2 reservoirs with ≳ × 9 masses MH2 6 10 M⊙. In this range, equilibrium and non-equilibrium models predict similar molecular masses (but different galaxy morphologies) while the semi- empirical method produces less H2. The star formation rates as well as the stellar and H2 masses of the simulated galaxies are in line with those observed in actual galaxies at similar redshifts that are not massive starbursts. The H2 mass functions extracted from the simulations at z ≈ 4 agree well with recent observations that only sample the high-mass end. However, our results indicate that most molecular material at high z 9 < < 10 lies yet undetected in reservoirs with 10 MH2 10 M⊙. Key words: methods: numerical - ISM: molecules - galaxies: evolution - galaxies: formation 1 INTRODUCTION has enabled detections of molecular-gas reservoirs at red- shifts as high as z 4-7 (e.g. Riechers et al. 2014; Capak Observations of molecular gas at high redshift (see e.g. Car- ∼ et al. 2015; Maiolino et al. 2015; Decarli et al. 2016; Both- illi & Walter 2013, for a review) are shaping our knowledge well et al. 2017; Santini et al. 2019). of the early phases of galaxy formation. To fully appreci- arXiv:2003.04329v2 [astro-ph.GA] 3 Sep 2020 ate their implications, it is vital to develop a theoretical Opening a window on the molecular Universe also moti- framework within which the experimental findings can be vates new theoretical efforts to gain insight into how galax- interpreted. However, ab initio simulations of galaxy forma- ies grow their stellar component. This requires developing tion generally do not resolve the spatial scales and densities a coherent picture that links molecular gas in the turbu- (nor capture the physics) that characterize molecular clouds lent interstellar medium (ISM) to the various feedback pro- in the interstellar medium (ISM), and therefore fall short of cesses that regulate the supply of gas available to form stars. modelling the molecular content of galaxies. The need for Stellar nurseries in the Milky Way appear to be associated more sophisticated models is therefore becoming increas- with dusty and dense molecular clouds. Spatially resolved ingly important, particularly with the advent of the Ata- observations of nearby galaxies show that the surface den- cama Large Millimeter/submillimeter Array (ALMA) which sity of star formation (SF) better correlates with the sur- face density of molecular gas than with the total gas density (e.g. Wong & Blitz 2002; Kennicutt et al. 2007; Leroy et al. ⋆ E-mail: [email protected] 2008; Bigiel et al. 2008). A possible interpretation of these © 2020 The Authors 2 Sch¨abe et al. findings is that the presence of molecular material is nec- analytic framework (Fu et al. 2010; Lagos et al. 2011; Fu essary to trigger SF (Krumholz & McKee 2005; Elmegreen et al. 2012; Krumholz & Dekel 2012; Somerville et al. 2015) 2007; Krumholz et al. 2009b), although other viewpoints are and in numerical simulations of small-to-intermediate cos- also plausible. One possibility, advocated by Krumholz et al. mological volumes (Kuhlen et al. 2012; Jaacks et al. 2013; (2011) and Glover & Clark(2012), is that H 2 and SF are spa- Kuhlen et al. 2013; Hopkins et al. 2014; Thompson et al. tially correlated due to the ability of the gas to self shield 2014; Lagos et al. 2015; Dav´eet al. 2016). A more com- from interstellar ultraviolet (UV) radiation. That SF primar- plex equilibrium model in which the dust abundance and ily takes place in molecular clouds would, in that case, be grain-size distribution evolve with time has been recently coincidental rather than a consequence of some fundamental employed in simulations of an isolated disc galaxy (Chen underlying relation between H2 and SF. et al. 2018). In numerical simulations, the two scenarios generate The third option on the market is to model the out-of- different galaxies: H2-regulated SF is delayed in the low- equilibrium evolution of the H2 abundance. The main mo- metallicity progenitors of a galaxy where dust and central tivation for doing this is that the formation of H2 on dust- gas densities are too low to activate an efficient conversion grains can be a slow process and the chemical rate equations of Hi into H2 (Kuhlen et al. 2012; Jaacks et al. 2013; Kuhlen reach equilibrium only if the ISM presents favourable condi- et al. 2013; Thompson et al. 2014; Tomassetti et al. 2015). tions (e.g. high dust content and long dynamical timescales). The resulting galaxies are thus characterized by lower stellar As a result, the equilibrium models described above may masses, younger stellar populations, and a smaller number over-predict the abundance of H2 in certain scenarios. To of bright satellites (Tomassetti et al. 2015). In addition, the overcome this problem, one can directly integrate the system fact that the energy due to stellar feedback is injected at dif- of chemical rate equations without resorting to approximate ferent locations gives rise to different galaxy morphologies equilibrium solutions. This approach, however, requires ac- (Tomassetti et al. 2015; Pallottini et al. 2017). counting for the complex interplay between velocity and In the ISM, H2 primarily forms due to the catalytic density in a turbulent medium that ultimately determines action of dust grains and is destroyed by resonant absorp- the column density of the gas and dust. While such a line tion of photons in the Lyman and Werner (LW) bands. This of attack characterizes state-of-the-art simulations of small is why H2 is abundant in the densest and coldest regions ISM patches (see, e.g., Seifried et al. 2017, and references of the ISM where far-UV radiation is heavily attenuated therein), it cannot yet be fully implemented in cosmological (Draine 1978; Hollenbach & McKee 1979; van Dishoeck & simulations of galaxy formation as they do not yet resolve Black 1986; Black & van Dishoeck 1987; Draine & Bertoldi the relevant length, time and density scales. The simplest 1996; Sternberg 2005). The main difficulty in tracking molec- approach is to solve the chemical rate equations after coarse- ular gas within galaxy formation models is the huge dynamic graining them at the level of the single resolution elements range between the scales that tidally torque galaxies and and introduce a clumping factor in the H2 formation rate to those that regulate the turbulent ISM and on which SF and account for unresolved density fluctuations. The best pos- stellar feedback take place. sible spatial resolution is then achieved by focusing on ide- One way to overcome this limitation is to use empirical alized (Pelupessy et al. 2006; Robertson & Kravtsov 2008; laws inferred from observations in order to predict the abun- Pelupessy & Papadopoulos 2009; Hu et al. 2016; Richings dance of molecular gas within galaxies. For example, the & Schaye 2016; Lupi et al. 2018) or cosmological simula- ratio between the surface densities of molecular and atomic tions of individual galaxies (Gnedin et al. 2009; Feldmann hydrogen is found to scale quasi-linearly with the interstellar et al. 2011; Christensen et al. 2012; Katz et al. 2017; Pal- gas pressure in the mid-plane of disc galaxies (Wong & Blitz lottini et al. 2017; Nickerson et al. 2018; Lupi et al. 2019; 2002; Blitz & Rosolowsky 2004, 2006; Leroy et al. 2008), a Pallottini et al. 2019). Alternatively, physics at the unre- fact that has been exploited to develop semi-analytic models solved scales can be dealt with by introducing a sub grid (SAMs) of galaxy formation (Dutton & van den Bosch 2009; model that takes into account the probability distribution Obreschkow et al. 2009; Obreschkow & Rawlings 2009a; Fu of local densities and the temperature-density relation ob- et al. 2010; Lagos et al. 2011; Fu et al. 2012; Popping et al. tained in high-resolution simulations of the turbulent ISM. 2014, 2015; Somerville et al. 2015; Lacey et al. 2016; Stevens In this case, a 1D slab approximation is used to associate an et al. 2016; Lagos et al. 2018) and numerical simulations optical depth to each microscopic density (see Tomassetti (Murante et al. 2010, 2015; Diemer et al.