Observations of star-forming galaxies at high-zݑ§zitalic_z have suggested discrepancies in the inferred star formation rates (SFRs) either between data and models, or between complementary measures of the SFR. These putative discrepancies could all be alleviated if the stellar initial mass function (IMF) is systematically weighted toward more high-mass star formation in rapidly star-forming galaxies. Here, we explore how the IMF might vary under the central assumption that the turnover mass in the IMF, M^csubscript^ݑ€c\hatM_\rm cover^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT, scales with the Jeans mass in giant molecular clouds (GMCs), MJ^^subscriptݑ€J\hatM_\rm Jover^ start_ARG italic_M start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT end_ARG. We employ hydrodynamic simulations of galaxies coupled with radiative transfer models to predict how the typical GMC Jeans mass, and hence the IMF, varies with galaxy properties. We then study the impact of such an IMF on the star formation law, the SFR-M*subscriptݑ€M_{*}italic_M start_POSTSUBSCRIPT * end_POSTSUBSCRIPT relation, sub-millimetre galaxies (SMGs), and the cosmic star formation rate density. Our main results are: The H22{}_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT mass-weighted Jeans mass in a galaxy scales well with the SFR when the SFR is greater a few M⊙subscriptݑ€direct-productM_\odotitalic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT yr-11{}^-1start_FLOATSUPERSCRIPT - 1 end_FLOATSUPERSCRIPT. Stellar population synthesis modeling shows that this results in a nonlinear relation between SFR and Lbolbol{}_\rm bolstart_FLOATSUBSCRIPT roman_bol end_FLOATSUBSCRIPT, such that SFR ∝Lbol0.88proportional-toabsentsuperscriptLbol0.88\propto\mboxL${}_\rm bol$^0.88∝ L start_FLOATSUBSCRIPT roman_bol end_FLOATSUBSCRIPT start_POSTSUPERSCRIPT 0.88 end_POSTSUPERSCRIPT. Using this model relation, the inferred SFR of local ultraluminous infrared galaxies decreases by a factor ∼×2fragmentssimilar-to2\sim\times 2∼ × 2, and that of high-zݑ§zitalic_z SMGs decreases by ∼×3-5fragmentssimilar-to35\sim\times 3-5∼ × 3 - 5. At z∼similar-toݑ§absentz\simitalic_z ∼ 2, this results in a lowered normalisation of the SFR-M*subscriptݑ€M_{*}italic_M start_POSTSUBSCRIPT * end_POSTSUBSCRIPT relation in better agreement with models, a reduced discrepancy between the observed cosmic SFR density and stellar mass density evolution, and SMG SFRs that are easier to accommodate in current hierarchical structure formation models. It further results in a Kennicutt-Schmidt (KS) star formation law with slope of ∼1.6similar-toabsent1.6\sim 1.6∼ 1.6 when utilising a physically motivated form for the CO-H22{}_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT conversion factor that varies with galaxy physical property. While each of the discrepancies considered here could be alleviated without appealing to a varying IMF, the modest variation implied by assuming M^csubscript^ݑ€c\hatM_\rm cover^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT∝proportional-to\propto∝MJ^^subscriptݑ€J\hatM_\rm Jover^ start_ARG italic_M start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT end_ARG is a plausible solution that simultaneously addresses numerous thorny issues regarding the SFRs of high-zݑ§zitalic_z galaxies.

keywords:
stars:luminosity function, mass function - stars: formation - galaxies: formation -galaxies: high-redshift - galaxies: ISM - galaxies: starburst - cosmology:theory
††pubyear: 2010

The buildup of stellar mass over cosmic time is a central issue in understanding the formation and evolution of galaxies. A common approach to quantifying stellar growth is to measure the evolution of the star formation rates (SFRs) of galaxies. This is done using a wide variety of tracers from the ultraviolet (UV) to the radio. Generally, all such measures trace the formation rate of higher-mass (typically O and B) stars, while the bulk of the stellar mass forming in lower-mass stars is not directly detected. Hence measuring the true rate of stellar growth requires assuming a conversion between the particular tracer flux and the total stellar mass being generated (e.g. Kennicutt, 1998a; ken12). This requires assuming some stellar initial mass function (IMF), page_seo_titlely the number of stars being formed as a function of mass.

On global cosmological scales, multi-wavelength observations of galaxies are converging on a broad scenario for the cosmic star formation rate evolution (Madau et al., 1996). Galaxies at high-redshift appear to be more gas-rich and forming stars more rapidly at a given stellar mass than present day galaxies (see the recent review by Shapley, 2011). The cosmic star formation rate density rises slowly from early epochs to peak between redshifts z≈ݑ§absentz\approxitalic_z ≈1-3, and then declines toward zݑ§zitalic_z=0 (e.g. Hopkins & Beacom, 2006). This global evolution is also reflected in measurements of the star formation rates of individual galaxies, which grow significantly at a given stellar mass from today out to z∼2similar-toݑ§2z\sim 2italic_z ∼ 2, prior to which they show a much slower evolution (e.g. Daddi et al., 2005; Davé, 2008; González et al., 2010; Hopkins et al., 2010).

Meanwhile, recent advances in near-infrared capabilities have enabled measurements of the buildup of stellar mass out to high redshifts, as traced by more long-lived stars (typically red giants). Nominally, the integral of the cosmic star formation rate, when corrected for stellar evolution processes, should yield the present-day cosmic stellar mass. Analogously, the time differential of the stellar mass evolution should be the same as measured star formation rate. Thus in principle there is now a cross-check on the rate of high-mass stars forming relative to lower-mass stars.

Preliminary comparisons along these lines have yielded general agreement out to z∼1similar-toݑ§1z\sim 1italic_z ∼ 1 (e.g. Bell et al., 2007). However, moving to higher redshifts into the peak epoch of cosmic star formation, there are growing hints of discrepancies: the integrated cosmic star formation rate density seems to exceed the observed stellar mass function (accounting for stellar mass loss; Hopkins & Beacom, 2006; Elsner et al., 2008; Wilkins et al., 2008; Pérez-González et al., 2008). These discrepancies are relatively mild, at the factor of two to three level, so could perhaps be resolved by a more careful consideration of systematic uncertainties in SFR and M*subscriptݑ€M_{*}italic_M start_POSTSUBSCRIPT * end_POSTSUBSCRIPT measures. Indeed, some analyses fail to show strong discrepancies (e.g. Sobral et al., 2012). Nevertheless, it is interesting that when discrepancies are seen, they unanimously favor the idea that the observed stellar mass growth appears slower than expected from the observed SFR.

Theoretical models have made a number of advances toward understanding the observed properties of star-forming galaxies at high redshifts. Simulations advocate a picture in which continual gas accretion from the intergalactic medium (IGM) feeds galaxies with fresh fuel (e.g. Mo et al., 1998; Robertson et al., 2004; Governato et al., 2009; Ceverino et al., 2010a; Agertz et al., 2011) and ultimately drives star formation (Kereš et al., 2005; Dekel et al., 2009a, b; Kereš et al., 2009). In such models, stars typically form at rates proportional to the global gas accretion rate from the IGM, regulated by feedback processes (Springel & Hernquist, 2003; Davé et al., 2011). Such a scenario predicts a fairly tight and nearly linear relation between star formation rate and stellar mass for star-forming galaxies (e.g. Davé et al., 2000; Finlator et al., 2006). Observations have now identified and quantified such a relation out to z∼similar-toݑ§absentz\simitalic_z ∼ 6-8 (Noeske et al., 2007a, b; Daddi et al., 2007; Stark et al., 2009; McLure et al., 2011), which has come to be called the “main sequence” of galaxy formation. The bulk of cosmic star formation appears to occur in galaxies along this main sequence, while merger-driven starbursts contribute ≲20%less-than-or-similar-toabsentpercent20\la 20\%≲ 20 % globally (e.g. Rodighiero et al., 2011; Wuyts et al., 2011).

Although this scenario broadly agrees with observations, a small but persistent discrepancy has been recently highlighted between the evolution of the main sequence in simulations and observations, particularly during the peak epoch of cosmic star formation (1≲z≲3less-than-or-similar-to1ݑ§less-than-or-similar-to31\la z\la 31 ≲ italic_z ≲ 3). In it, the rate of stellar mass growth at these redshifts in models is typically smaller by ∼2-3×fragmentssimilar-to23\sim 2-3\times∼ 2 - 3 × than the observed star formation rates. This discrepancy exists for both cosmological hydrodynamic simulations (Davé, 2008), as well as semi-analytic models (SAMs; Daddi et al., 2007; Elbaz et al., 2007), and is to first order independent of model assumptions about feedback (Davé, 2008). This is further seen in both the global cosmic star formation rate (Wilkins et al., 2008) as well as when comparing individual galaxies at a given M*subscriptݑ€M_{*}italic_M start_POSTSUBSCRIPT * end_POSTSUBSCRIPT (Davé, 2008). Similarly, galaxy formation simulations that utilise a variety of methods all show a paucity of galaxies that form stars as rapidly as galaxies with the highest star formation rates at z∼similar-toݑ§absentz\simitalic_z ∼ 2, the sub-millimetre galaxies (SMGs) (Baugh et al., 2005; Davé et al., 2010; Hayward et al., 2011). In all cases, the models tend to favor lower true SFRs than implied by using available tracers and using conversion factors based on a canonical IMF.

One possible but speculative solution to all these discrepancies is that the stellar IMF in galaxies at z∼2similar-toݑ§2z\sim 2italic_z ∼ 2 is different than what is measured directly in the Galaxy (e.g. kan10). The discrepancies described above, between the various observations as well as between models and data, would all be mitigated by an IMF that forms somewhat more high-mass stars than low-mass ones at those epochs compared to the present-day IMF111This could be described as a “top-heavy” IMF, which we specify as an IMF whose high-mass slope is different than local, or a “bottom-light” IMF, which we define as retaining the same high-mass slope but forming fewer low-mass stars. This paper focuses on bottom-light IMFs.. Nevertheless, it is important to point out that at present, no firm evidence that the IMF varies strongly from the locally observed one (see the review by Bastian et al., 2010). Locally, some observations suggest that a top-heavy/bottom-light IMF may apply to the Galactic Centre (Nayakshin & Sunyaev, 2005; Stolte et al., 2005). Similarly, Rieke et al. (1993) and Förster Schreiber et al. (2003) suggest a turnover mass a factor of ∼2-6similar-toabsent26\sim 2-6∼ 2 - 6 larger than in a traditional Kroupa (2002) IMF in the nearby starburst galaxy M82. Simultaneous fits to the observed cosmic star formation rate density, integrated stellar mass measurements, and cosmic background radiation favour a “paunchy” IMF that produces more stars at intermediate masses (Fardal et al., 2007). ≈0.8) based on an analysis of the evolution of the colours. Mass-to-light ratios of early-type galaxies.8) based on an analysis of the evolution of the colours and mass-to-light ratios of early-type galaxies. However, these observations can all be interpreted without the need for IMF variations (Bastian et al., 2010). Beyond this, some observations find evidence for a bottom-heavy IMF in zݑ§zitalic_z=0 early-type galaxies(van Dokkum & Conroy, 2011; Cappellari et al., 2012; con12). It is therefore interesting to examine whether an IMF-based solution is viable and consistent with a broad suite of observations, both locally and in the distant Universe.

In this paper, we explore the cosmological consequences of a physically-based model for IMF variations. Past work has generally focused on empirically determining the amount of IMF variation needed in order to solve one (or more) of the above problems (e.g. Fardal et al., 2007; van Dokkum, 2008; Davé, 2008; Wilkins et al., 2008). Here, instead, we make a single critical assumption, first forwarded by Jeans, and later expanded upon by Larson (2005) and tum07: The IMF critical mass (M^csubscriptnormal-^ݑ€normal-c\hatM_\rm cover^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT) scales with the Jeans mass in a giant molecular cloud (GMC). For reference, we call this the Jeans mass conjecture. We employ hydrodynamic simulations of isolated galaxies and mergers including a fully radiative model for the interstellar medium (ISM) to predict the typical Jeans mass of GMCs in galaxies with different physical conditions, corresponding to quiescent and starbursting systems both today and at z=2ݑ§2z=2italic_z = 2. Applying the Jeans mass conjecture, we then make a prediction for how the IMF varies with global galaxy properties, and explore the implications for such variations on the discrepancies noted above.

We emphasise that the main purpose of this paper is to utilise numerical models of the molecular ISM in galaxies to investigate the consequences of an IMF in which M^csubscript^ݑ€c\hatM_\rm cover^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT scales with MJsubscriptݑ€JM_\rm Jitalic_M start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT. We do not directly argue for such a scaling relationship; this is taken as an assumption, and we only seek to study its implications in a cosmological context. For the reader’s edification, we present arguments both for and against the Jeans mass conjecture in § 5.

This paper is outlined as follows. In § 2, we detail our numerical models. In § 3, we discuss variations of the IMF versus galaxy star formation rates. In § 4, we investigate the effect of this model on derived star formation rates, paying particular attention to the star formation law (§ 4.1 and § 4.2), the SFR-M*subscriptݑ€M_{*}italic_M start_POSTSUBSCRIPT * end_POSTSUBSCRIPT relation at high-redshift (§ 4.3), and the evolution of the cosmic star formation rate density (§ 4.5). In § 5, we present a discussion, and in § 6, we summarise.

2 Numerical Methods

Our main goal is to simulate the global physical properties of GMCs in galaxies, and understand the effect of the varying Jeans mass on observed star formation rates. These methods and the corresponding equations are described in significant detail in Narayanan et al. (2011b, c); for the sake of brevity, we summarise the relevant aspects of this model here, and refer the interested reader to those papers for further detail.

We first simulate the evolution of galaxies hydrodynamically using the publicly available code gadget-3 (Springel, 2005; Springel et al., 2005). In order to investigate a variety of physical environments, we consider the evolution of disc galaxies in isolation, major (1:1 and 1:3) mergers, and minor (1:10) mergers at both low (zݑ§zitalic_z=0) and high (zݑ§zitalic_z=2) redshifts. These simulations are summarised in Table A1 of Narayanan et al. (2011c). The discs are initialised according to the Mo et al. (1998) model, and embedded in a live dark matter halo with a Hernquist (1990) density profile. Galaxy mergers are simply mergers of these discs. The halo concentration and virial radius for a halo of a given mass is motivated by cosmological NݑNitalic_N-body simulations, and scaled to match the expected redshift-evolution following Bullock et al. (2001) and Robertson et al. (2006).

For the purposes of the hydrodynamic calculations, the ISM is modeled as multi-phase, with cold clouds embedded in a hotter phase (McKee & Ostriker, 1977). The phases exchange mass via radiative cooling of the hot phase, and supernova heating of cold gas. Stars form within the cold gas according to a volumetric Kennicutt-Schmidt star formation relation, SFR∼ρcold1.5similar-toSFRsuperscriptsubscriptݜŒcold1.5\rm SFR\sim\rho_\rm cold^1.5roman_SFR ∼ italic_ρ start_POSTSUBSCRIPT roman_cold end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1.5 end_POSTSUPERSCRIPT, with normalisation set to match the locally observed relation (Kennicutt, 1998a; Springel, 2000; Cox et al., 2006).

In order to model the physical properties of GMCs within these model galaxies, we perform additional calculations on the SPH models in post-processing. We first project the physical properties of the galaxies onto an adaptive mesh with a 53superscript535^35 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT base, spanning 200 kpc. The cells recursively refine in an oct-subdivision based on the criteria that the relative density variations of metals should be less than 0.1, and the Vݑ‰Vitalic_V-band optical depth across the cell τV<1subscriptݜV1\tau_\rm V<1italic_τ start_POSTSUBSCRIPT roman_V end_POSTSUBSCRIPT <1. The smallest cells in this grid are of order ∼70similar-toabsent70\sim 70∼ 70 pc across, just resolving massive GMCs.

The GMCs are modeled as spherical and isothermal. The H22{}_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT-HI balance in these cells is calculated by balancing the photodissociation rates of H22{}_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT against the growth rate on dust grains following the methodology of Krumholz et al. (2008, 2009a) and Krumholz et al. (2009b). This assumes equilibrium chemistry for the H22{}_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT. The GMCs within cells are assumed to be of constant density, and have a minimum surface density of 100 M☉ pc-2M☉ superscriptpc2\mboxM${}_\sun$ \rm pc^-2M start_FLOATSUBSCRIPT ☉ end_FLOATSUBSCRIPT roman_pc start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT. This value is motivated by the typical surface density of local group GMCs (McKee & Ostriker, 2007; Bolatto et al., 2008; Fukui & Kawamura, 2010), and exists to prevent unphysical conditions in large cells toward the outer regions of the adaptive mesh. In practice, the bulk of the GMCs in galaxies of interest for this study (i.e., galaxies where the galactic environment has a significant impact on the Jeans mass) have surface densities above this fiducial threshold value. With the surface density of the GMC known, the radius (and consequently mean density) is known as well. Following Narayanan et al. (2011b), to account for the turbulent compression of gas, we scale the volumetric densities of the GMCs by a factor eσρ2/2superscriptݑ’superscriptsubscriptݜŽݜŒ22e^\sigma_\rho^2/2italic_e start_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 end_POSTSUPERSCRIPT, where σρ≈ln(1+M1D2/4)subscriptݜŽݜŒln1superscriptsubscriptݑ€1ݐ·24\sigma_\rho\approx\rm ln(1+M_1D^2/4)italic_σ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ≈ roman_ln ( 1 + italic_M start_POSTSUBSCRIPT 1 italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 4 ), and M1Dsubscriptݑ€1DM_\rm 1Ditalic_M start_POSTSUBSCRIPT 1 roman_D end_POSTSUBSCRIPT is the 1 dimensional Mach number of turbulence (Ostriker et al., 2001; Padoan & Nordlund, 2002).

Because this study centres around understanding the Jeans scale in GMCs, the thermal state of the molecular ISM must be known. Following Krumholz et al. (2011b); Narayanan et al. (2011b) and Narayanan et al. (2011c), we consider the temperature of the H22{}_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT gas as a balance between heating by cosmic rays, grain photoelectric effect, cooling by CO or CII line emission, and energy exchange with dust. The cosmic ray flux is assumed to be that of the mean Galactic value222Tests described by Narayanan et al. (2011c) show that even under the assumption of a scaling where cosmic ray flux scales with the star formation rate of a galaxy, the typical thermal profile of H22{}_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT gas in a given galaxy is unchanged. This is because, in these environments, energy exchange with dust tends to dominate the temperature.. Formally, if we denote heating processes by ΓΓ\Gammaroman_Γ, cooling by ΛΛ\Lambdaroman_Λ, and energy exchange with ΨΨ\Psiroman_Ψ, we solve the following equations:

Γpe+ΓCR-Λline+Ψgd=0subscriptΓpesubscriptΓCRsubscriptΛlinesubscriptΨgd0\displaystyle\Gamma_\rm pe+\Gamma_\rm CR-\Lambda_\rm line+\Psi_\rm gd=0roman_Γ start_POSTSUBSCRIPT roman_pe end_POSTSUBSCRIPT + roman_Γ start_POSTSUBSCRIPT roman_CR end_POSTSUBSCRIPT - roman_Λ start_POSTSUBSCRIPT roman_line end_POSTSUBSCRIPT + roman_Ψ start_POSTSUBSCRIPT roman_gd end_POSTSUBSCRIPT = 0 (1)

Γdust-Λdust-Ψgd=0subscriptΓdustsubscriptΛdustsubscriptΨgd0\displaystyle\Gamma_\rm dust-\Lambda_\rm dust-\Psi_\rm gd=0roman_Γ start_POSTSUBSCRIPT roman_dust end_POSTSUBSCRIPT - roman_Λ start_POSTSUBSCRIPT roman_dust end_POSTSUBSCRIPT - roman_Ψ start_POSTSUBSCRIPT roman_gd end_POSTSUBSCRIPT = 0 (2)
We refer the reader to Krumholz et al. (2011b) and Narayanan et al. (2011b) for the equations regarding the photoelectric effect and cosmic ray heating terms.

The dust temperature is calculated via the publicly available dust radiative transfer code sunrise (Jonsson, 2006; Jonsson et al., 2010; Jonsson & Primack, 2010). We consider the the transfer of both stellar light from clusters of stars as well as radiation from a central active galactic nucleus (though this plays relatively little role in these simulations). The stars emit a starburst99 spectrum (Leitherer et al., 1999; Vázquez & Leitherer, 2005), where the metallicity and ages of the stellar clusters are taken from the hydrodynamic simulations. The radiation then traverses the galaxy, being absorbed, scattered and reemitted as it escapes. The evolving dust mass is set by assuming a constant dust to metals ratio comparable to the mean Milky Way value (Dwek, 1998; Vladilo, 1998; Calura et al., 2008), and takes the form of the R=3.15ݑ…3.15R=3.15italic_R = 3.15 Weingartner & Draine (2001) grain model as updated by Draine & Li (2007). The dust and radiation field are assumed to be in equilibrium, and the dust temperatures are calculated iteratively. Energy exchange between gas and dust becomes important typically around n∼104cm-3similar-toݑ›superscript104cm-3n\sim 10^4\mboxcm${}^-3$italic_n ∼ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT cm start_FLOATSUPERSCRIPT - 3 end_FLOATSUPERSCRIPT. Henceforth, we shall refer to this as the density where grain-gas coupling becomes important.

The gas cools either via CII or CO line emission. The fraction of H22{}_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT where most of the carbon is in the form of CO (versus atomic form) is determined following the semi-analytic model of Wolfire et al. (2010), which is a metallicity dependent model (such that at lower metallicities, most of the carbon is in atomic form, and at higher metallicities, it is mostly in the form of CO):

fCO=fH2×e-4(0.53-0.045lnG0′nH/cm-3-0.097lnZ′)/Avsubscriptݑ“COsubscriptݑ“H2superscriptݑ’40.530.045lnsuperscriptsubscriptݐº0′subscriptݑ›Hsuperscriptcm30.097lnsuperscriptݑ′subscriptݐ´vf_\rm CO=f_\rm H2\times e^-4(0.53-0.045\rm ln\fracG_0^\primen_% \rm H/\rm cm^-3-0.097\rm lnZ^\prime)/A_\rm vitalic_f start_POSTSUBSCRIPT roman_CO end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT H2 end_POSTSUBSCRIPT × italic_e start_POSTSUPERSCRIPT - 4 ( 0.53 - 0.045 roman_ln divide start_ARG italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_n start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT / roman_cm start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT end_ARG - 0.097 roman_ln italic_Z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / italic_A start_POSTSUBSCRIPT roman_v end_POSTSUBSCRIPT end_POSTSUPERSCRIPT (3)
where G0′superscriptsubscriptݐº0′G_0^\primeitalic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is the UV radiation field with respect to that of the solar neighbourhood, and AVsubscriptݐ´VA_\rm Vitalic_A start_POSTSUBSCRIPT roman_V end_POSTSUBSCRIPT is the visual extinction, AV=NH1.87×1021cm-2Z′subscriptݐ´VsubscriptݑH1.87superscript1021superscriptcm2superscriptݑ′A_\rm V=\fracN_\rm H1.87\times 10^21\rm cm^-2Z^\primeitalic_A start_POSTSUBSCRIPT roman_V end_POSTSUBSCRIPT = divide start_ARG italic_N start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT end_ARG start_ARG 1.87 × 10 start_POSTSUPERSCRIPT 21 end_POSTSUPERSCRIPT roman_cm start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT end_ARG italic_Z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT (Watson, 2011). If fCO>0.5subscriptݑ“CO0.5f_\rm CO>0.5italic_f start_POSTSUBSCRIPT roman_CO end_POSTSUBSCRIPT >0.5, we assume that the gas cools via CO line emission; else, CII. The line emission is calculated via the publicly available escape probability code as detailed in Krumholz & Thompson (2007).

While the temperature model is indeed somewhat complicated, the typical temperature of a cloud can be thought of in terms of the dominant heating effects at different densities. At low densities (n∼10-100cm-3similar-toݑ›10100cm-3n\sim 10-100\ \mboxcm${}^-3$italic_n ∼ 10 - 100 cm start_FLOATSUPERSCRIPT - 3 end_FLOATSUPERSCRIPT), the gas cools via line cooling to ∼8similar-toabsent8\sim 8∼ 8 K, the characteristic temperature imposed by cosmic ray heating. At high densities (n≳104cm-3greater-than-or-similar-toݑ›superscript104cm-3n\ga 10^4\ \mboxcm${}^-3$italic_n ≳ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT cm start_FLOATSUPERSCRIPT - 3 end_FLOATSUPERSCRIPT), grain-gas coupling becomes efficient, and the gas rises to the dust temperature. At intermediate densities, the temperature is typically in between 8888 K and the dust temperature.

3 The Relation Between Jeans Mass and Star Formation Rate

We begin by examining the conditions under which the Jeans mass in GMCs in a galaxy may vary. We define the characteristic Jeans mass, MJ^^subscriptݑ€J\hatM_\rm Jover^ start_ARG italic_M start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT end_ARG, as the H22{}_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT mass-weighted Jeans mass across all GMCs in a galaxy, and consider the deviations of this quantity. The density term in the Jeans mass calculation is the mean density of the GMC. In Figure 1, the top panel shows the SFR as a function of time for an unperturbed zݑ§zitalic_z=0 disc galaxy, a zݑ§zitalic_z=0 major 1:1 galaxy merger, and a zݑ§zitalic_z=2 disc. The second panel shows the evolution of MJ^/250M☉^subscriptݑ€J250M☉\mbox$\hatM_\rm J$/250{}\mboxM${}_\sun$over^ start_ARG italic_M start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT end_ARG / 250 M start_FLOATSUBSCRIPT ☉ end_FLOATSUBSCRIPT. 250 M☉☉{}_\sunstart_FLOATSUBSCRIPT ☉ end_FLOATSUBSCRIPT is the Jeans mass for physical conditions as found in local discs like the Milky Way (MW), page_seo_titlely n=1-2×102cm-3ݑ›12superscript102cm-3n=1-2\times 10^2\mboxcm${}^-3$italic_n = 1 - 2 × 10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT cm start_FLOATSUPERSCRIPT - 3 end_FLOATSUPERSCRIPT and T=10ݑ‡10T=10italic_T = 10 K. Our disc galaxy has a baryonic mass of ∼5×1010similar-toabsent5superscript1010\sim 5\times 10^10∼ 5 × 10 start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT M☉☉{}_\sunstart_FLOATSUBSCRIPT ☉ end_FLOATSUBSCRIPT, and is the fiducial MW model in Narayanan et al. (2011b). We note that the fiducial zݑ§zitalic_z=0 MW model naturally produces densities, temperatures, and MJ^^subscriptݑ€J\hatM_\rm Jover^ start_ARG italic_M start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT end_ARG as observed in the MW, thereby providing an important check for our ISM model. The major merger is a merger of two Mbar=1.5×1011subscriptݑ€bar1.5superscript1011M_\rm bar=1.5\times 10^11italic_M start_POSTSUBSCRIPT roman_bar end_POSTSUBSCRIPT = 1.5 × 10 start_POSTSUPERSCRIPT 11 end_POSTSUPERSCRIPT M☉☉{}_\sunstart_FLOATSUBSCRIPT ☉ end_FLOATSUBSCRIPT galaxies. The high-zݑ§zitalic_z disc model is an unperturbed Mbar=1011M☉ subscriptݑ€barsuperscript1011M☉ M_\rm bar=10^11\mboxM${}_\sun$ italic_M start_POSTSUBSCRIPT roman_bar end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT 11 end_POSTSUPERSCRIPT M start_FLOATSUBSCRIPT ☉ end_FLOATSUBSCRIPT disc. The remaining panels show the evolution of the characteristic temperatures and the characteristic GMC densities333Note that because MJ^^subscriptݑ€J\hatM_\rm Jover^ start_ARG italic_M start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT end_ARG is calculated as the H22{}_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT mass-weighted Jeans mass, one cannot convert from the characteristic temperatures and densities shown in Figure 1 to MJ^^subscriptݑ€J\hatM_\rm Jover^ start_ARG italic_M start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT end_ARG..

For the bulk of the zݑ§zitalic_z=0 disc’s life, and during non-interacting stages of the galaxy merger, the GMCs are relatively quiescent, retaining surface densities near 100 M☉ pc-2M☉ superscriptpc2\mboxM${}_\sun$ \rm pc^-2M start_FLOATSUBSCRIPT ☉ end_FLOATSUBSCRIPT roman_pc start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT, and temperatures near the floor established by cosmic ray heating of ∼10similar-toabsent10\sim 10∼ 10 K. During the merger-induced starburst, however, the story is notably different. Gas compressions caused by the nuclear inflow of gas during the merger cause the average density to rise above 104cm-3superscript104cm-310^4\mboxcm${}^-3$10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT cm start_FLOATSUPERSCRIPT - 3 end_FLOATSUPERSCRIPT in GMCs. At this density, grain-gas energy exchange becomes quite efficient. At the same time, dust heating by the massive stars formed in the starburst causes the mass-weighted dust temperature to rise from ∼30similar-toabsent30\sim 30∼ 30 K to ∼80similar-toabsent80\sim 80∼ 80 K. Consequently, the kinetic temperature of the gas reaches similar values. While the mean gas density also rises during the merger, the Jeans mass goes as T3/2/n1/2superscriptݑ‡32superscriptݑ›12T^3/2/n^1/2italic_T start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT / italic_n start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT. As a result, MJ^^subscriptݑ€J\hatM_\rm Jover^ start_ARG italic_M start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT end_ARG rises by a factor ∼5similar-toabsent5\sim 5∼ 5 during the merger-induced starburst. The zݑ§zitalic_z=2 disc represents an intermediate case: high gas fractions and densities drive large SFRs, and thus warmer conditions in the molecular ISM.

The characteristic Jeans mass in a galaxy is well-parameterised by the SFR. In Figure 2, we plot the star formation rates of all of the galaxies in our simulation sample against their MJ^^subscriptݑ€J\hatM_\rm Jover^ start_ARG italic_M start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT end_ARG. The points are individual time snapshots from the different galaxy evolution simulations. At high SFRs, dust is warmed by increased radiation field, and thermally couples with the gas in dense regions. The increased temperature drives an increase in the Jeans mass, resulting in roughly a power-law increase of MJ^^subscriptݑ€J\hatM_\rm Jover^ start_ARG italic_M start_POSTSUBSCRIPT roman_J end_POSTSUBSCRIPT end_ARG with SFR.

This relation does not extend to SFR≲3-5M☉yr-1less-than-or-similar-toSFR35M☉yr-1\rm SFR\la 3-5\ \mboxM${}_\sun$yr${}^-1$roman_SFR ≲ 3 - 5 M start_FLOATSUBSCRIPT ☉ end_FLOATSUBSCRIPT yr start_FLOATSUPERSCRIPT - 1 end_FLOATSUPERSCRIPT. Here, the mean densities become low enough (n<104cm-3ݑ›superscript104cm-3n<10^4\mboxcm${}^-3$italic_n <10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT cm start_FLOATSUPERSCRIPT - 3 end_FLOATSUPERSCRIPT) that energy exchange with dust no longer keeps the gas warm. Cosmic rays dominate the heating in this regime, and the gas cools to ∼8-10similar-toabsent810\sim 8-10∼ 8 - 10 K, the typical temperature that results from the balance of CO line cooling and cosmic ray heating. Because the temperatures are relatively constant, and the characteristic densities of order ∼10-100cm-3similar-toabsent10100cm-3\sim 10-100\ \mboxcm${}^-3$∼ 10 - 100 cm start_FLOATSUPERSCRIPT - 3 end_FLOATSUPERSCRIPT,

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