Cosmological implications of a stellar initial mass function that varies with the Jeans mass in galaxies

Mon. Not. R. Astron. Soc. 423, 3601–3615 (2012) doi:10.1111/j.1365-2966.2012.21159.x Cosmological implications of a stellar initial mass function that varies with the Jeans mass in galaxies Desika Narayanan † and Romeel Davé Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ 85721, USA Accepted 2012 April 21. Received 2012 April 19; in original form 2012 February 29 Observations of star-forming galaxies at high 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 towards 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, M̂c , scales with the Jeans mass in giant molecular clouds (GMCs), M̂J . 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 ∗ relation, sub-millimetre galaxies (SMGs) and the cosmic SFR density. Our main results are: the H2 mass-weighted Jeans mass in a galaxy scales well with the SFR when the SFR is greater than a few M yr−1 . Stellar population synthesis modelling shows that this results in a non-linear relation between SFR and Lbol , such that SFR ∝ Lbol 0.88 . Using this model relation, the inferred SFR of local ultraluminous infrared galaxies decreases by a factor of ∼2, and that of high-z SMGs decreases by a factor of ∼3–5. At z ∼ 2, this results in a lowered normalization of the SFR–M ∗ 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 star formation law with a slope of ∼1.6 when utilizing a physically motivated form for the CO–H2 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 M̂c ∝ M̂J is a plausible solution that simultaneously addresses numerous thorny issues regarding the SFRs of high-z galaxies. Key words: stars: formation – stars: luminosity function, mass function – galaxies: formation – galaxies: high-redshift – galaxies: ISM – cosmology: theory. 1 I N T RO D U C T I O N 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 E-mail: †Bart J Bok Fellow.  C 2012 The Authors C 2012 RAS Monthly Notices of the Royal Astronomical Society  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; Kennicutt & Evans 2012). This requires assuming some stellar initial mass function (IMF), namely 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 SFR 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 SFR density rises slowly from early epochs to peak between redshifts z ≈ 1 and 3, and then declines towards z = 0 (e.g. ABSTRACT 3602 D. Narayanan and R. Davé highest SFRs at 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 favour 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 ∼ 2 is different from what is measured directly in the Galaxy (e.g. Kang et al. 2010). 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 IMF.1 Nevertheless, it is important to point out that at present there is no firm evidence that the IMF varies strongly from the locally observed one (see the review by Bastian, Covey & Meyer 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 ∼2–6 larger than in a traditional (Kroupa 2002) IMF in the nearby starburst galaxy M82. Simultaneous fits to the observed cosmic SFR density, integrated stellar mass measurements and cosmic background radiation favour a ‘paunchy’ IMF that produces more stars at intermediate masses (Fardal et al. 2007). van Dokkum (2008) suggested that the IMF may be more top-heavy at high redshift (z ≈ 0.8) 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 z = 0 early-type galaxies (van Dokkum & Conroy 2011; Cappellari et al. 2012; Conroy & van Dokkum 2012). 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; Davé 2008; van Dokkum 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 Tumlinson (2007): the IMF critical mass (M̂c ) 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 t (...truncated)