7.1. Simulating star formation
The physics of star formation is an enormous field, and I do not pretend to provide a thorough review here. The purpose of the current summary is to highlight the complexity of the field, and the challenge in directly linking fundamental astrophysical processes to the form of an IMF. For details of work in this area, interested readers are referred to reviews by Krumholz (2014), and Offner et al. (2014), work by Hopkins (2013a), Bate et al. (2014), Guszejnov & Hopkins (2015), Bate & Keto (2015), Klishin & Chilingarian (2016), Guszejnov et al. (2016), Guszejnov et al. (2017), and references therein.
For ease of readability I refer below to “top heavy” or “bottom heavy” IMFs in reference to work that uses those terms. These correspond respectively to αh ≳ −2.35 (usually for m > 0.5 M⊙, sometimes α ≳ −2.35 for the full mass range) and αl ≲ −2.35 (often for m < 1 M⊙, but about as often also α ≲ −2.35 for the full mass range). Since this summary is largely qualitative, this usage should not be too ambiguous.
The recent work by Krumholz et al. (2016) provides a concise introduction to the key elements considered by most star formation simulations. In brief, the thermal Jeans mass, turbulence, magnetic fields, radiative feedback and mechanical feedback are all considered by various authors to play more or less significant roles. That analysis extends work by Krumholz (2011), who quantifies how radiative feedback can set the stellar mass scale, in turn building on earlier work by Bate (2009) and Krumholz (2006). He argues that radiative processes are the dominant mechanism in determining the gas temperature and ultimately the origin of the peak in the IMF.
Early work proposed an IMF characteristic mass determined by the thermal Jeans mass (e.g., Larson, 1998, Larson, 2005). Being temperature dependent, this would lead naturally to higher mc in extreme environments such as super star clusters or galactic nuclei, or high redshift galaxies. Other potential drivers, such as the role of metallicity, have subsequently been explored. There are arguments that, while metallicity plays an important role in cooling for the formation of Population III stars, it is unlikely to have a direct effect on the IMF for later stellar generations (Bate, 2014, Bate, 2005), apart from increasing the lower mass limit for lower metallicity systems. While Bate (2005) notes that metallicity may have an indirect impact because of its role in setting the Jeans mass during cloud fragmentation, to the degree that the Jeans mass of the cloud may affect the characteristic mass of the IMF, Bate (2014) find stellar mass functions that are consistent for metallicities ranging from 1/100 to 3 times solar. Similarly, using two numerical simulations corresponding to Jeans masses different by a factor of three, Bate & Bonnell (2005) argue that any potential IMF variation appears through a change in the characteristic mass of the system rather than a change in slope at the high-mass end.
It is clear that there is enormous complexity and interplay of the astrophysical processes involved in star formation. Given this complexity, it is easy to understand that a “universal” IMF is an attractive end-state to aim at achieving with models and simulations, as a form of validation. Introducing IMF variations removes this touchstone, making the work of the theorists more challenging, but as the observational constraints become more complex, so too do the models in their efforts at addressing them. This has led in more recent work to the goal of testing how particular models fare in reproducing the range of popular published IMF variations.
Hopkins (2013b) uses the excursion set formalism to calculate mass functions from the density field in a supersonically turbulent interstellar medium. This analysis predicts that IMF variations are most likely to appear at the low mass end, with remarkably uniform slope for high masses, for reasonable choices of temperature, velocity dispersion and gas surface density. A different approach, based on a Press-Schechter formalism, by Hennebelle & Chabrier (2013), extends their earlier work by including time dependence and the impact of magnetic field, and reaches similar conclusions. Chabrier et al. (2014) shows how the turbulent Jeans mass leads to the peak of the IMF shifting toward lower masses, to reproduce “bottom heavy” IMF shapes. Subsequently Bertelli Motta et al. (2016) identified conditions in two suites of hydrodynamic simulations that lead to IMF variations at the high mass end. Recently, though, Liptai et al. (2017) have argued against supersonic turbulence being the primary driver in the shape of the IMF, based on two sets of simulations with different turbulent modes, finding statistically indistiguishable differences in the resulting IMFs.
It seems that despite the growing sophistication of our theoretical understanding of star formation, there is still scope for refinement in identifying the various dominant physical mechanisms in different astrophysical environments. Elmegreen (2007) notes that “most detailed theoretical models reproduce the IMF, but because they use different assumptions and conditions, there is no real convergence of explanations yet.” In the subsequent decade, although the models have become more sophisticated, subtle and complex, so have the observational constraints, and the outcome remains largely the same.
7.2. Simulating galaxy evolution
Moving from the complexity of astrophysical processes in individual star formation to the larger scale of galaxies requires a different form of modeling and simulations. As above, this is a vast field in its own right, and only briefly and incompletely summarised here with the aim of identifying some of the developments and challenges.
Galaxy populations are typically modeled through semi-analytic recipes embedded in large cosmological simulations, and individual or small numbers of galaxies through detailed hydrodynamical simulations with better physical resolution than the cosmological models. In the absence of confirmed physical drivers underlying the shape of the IMF, such simulations tend to invoke a range of empirical or phenomenological relations that have some physical motivation. The outcome is that most observational evidence for IMF variations is able to be reproduced by a suitable choice of physical dependencies for the IMF, although not all results are consistent with each other, or necessarily self-consistent.
Baugh et al. (2005) modeled the abundance of Lyman break galaxies and submillimetre galaxies, successfully reproducing luminosity functions and the optical and infrared properties of local galaxy populations, but found a need for a “top-heavy” IMF to reproduce the observed 850 µm galaxy number counts. Without such a change to the IMF in the model, the constraint from the global SFR density led to the predicted number counts being too low. Allowing the IMF to be “top-heavy” increases the 850 µm flux for a given (lower) SFR because of the relative increase in the number of high mass stars, allowing the model to consistently reproduce both the number counts and the SFR density. Narayanan & Davé (2012) explore the impact of allowing mc to scale with the Jeans mass in giant molecular clouds, showing that this simple assumption leads to a reduction in the SMH/SMD discrepancy, as well as reducing the tensions in several other observational constraints. Narayanan & Davé (2013) extend this work to show that such an assumption leads to galaxies experiencing both “top heavy” and “bottom heavy” IMFs at different stages of their evolution, with the bulk of stars forming in a “top heavy” phase. Marks et al. (2012) use a model of rapid gas expulsion to produce more “top heavy” IMFs in systems with increasing density and decreasing metallicity. Bekki (2013a) show that such density and metallicity dependencies for the IMF can lead, among other effects, to lower SFRs than with a fixed IMF, and that [Mg/Fe] is higher for a given metallicity. Similarly, Bekki & Meurer (2013) are able to reproduce the “top heavy” IMF results of Gunawardhana et al. (2011) by allowing the IMF to depend on local densities and metallicities of the interstellar medium. In contrast, Bekki (2013b) show that “bottom heavy” IMFs can also be reproduced with suitable choices of metallicity and gas density in the star forming gas clouds.
Taking a different approach, Fontanot (2014) uses a semi-analytic model to test the impact of different IMF prescriptions, broadly falling into two classes of SFR-dependent “top heavy” models and stellar mass-dependent or σ-dependent “bottom heavy” models. He finds that the “bottom heavy” models lead to variations in stellar mass and SFR functions similar to the uncertainty in the determination of those quantities, while the “top heavy” models lead to an underestimate in the high mass end of the galaxy stellar mass function, compared to a fixed Kroupa (2001) IMF.
Blancato et al. (2017) also explore the impact of observed IMF variations on models. They implement various IMF dependencies by tagging stellar particles in their simulation with individual IMFs using observationally derived dependencies on velocity dispersion, metallicity or star formation rate. They then find that the IMFs recovered in the simulated z=0 galaxies no longer reproduce the imposed relations. This leads them to conclude that even more extreme physical IMF relations for some stellar populations are required to reproduce the observed level of variation. Sonnenfeld et al. (2017) explore the evolution of αmm (defined here as the ratio of the true stellar mass to that inferred assuming a Salpeter IMF) using cosmological N-body simulations. They find that dry mergers do not strongly impact the relation between αmm and σ. They note, though, that the underlying dependence of the IMF on stellar mass or σ is mixed through the dry merger process, making it observationally challenging to infer which quantity was originally coupled with the IMF. Schaye et al. (2010) tested the impact of a “top-heavy” IMF at high gas pressures, finding that it reduced the need to invoke self-regulated feedback from accreting black holes to reproduce the observed decline in the cosmic SFR density at z < 2. Gargiulo et al. (2015) argue that a “top-heavy” IGIMF best reproduces the [α/Fe]-stellar mass relation for elliptical galaxies when there is an SFR-dependence for the IGIMF slope. Fontanot et al. (2017) also explore the implications of the IGIMF method in their semi-analytic model, finding that it leads to a more realistic [α/Fe]-stellar mass relation than with a fixed IMF.
It is clear that numerical simulations and semi-analytic models can provide valuable insights into the way we understand the IMF. In particular, they can be used to test how different physical prescriptions for star formation manifest, and the properties of the observational constraints on IMFs that they produce, as well as what accessible observational tracers give the most discrimination in measuring the IMF. It is important that models are used to make predictions for how different IMF prescriptions should present observationally, defining observational tests to refine or rule out particular forms of physical dependencies or underlying variation. There is perhaps more value in using the models in this way than merely through tweaking some underlying dependencies to reproduce a select subset of observational constraints. Because of the fundamental nature of the IMF it is important that models and simulations are used to test as broad a suite as possible of observational implications, rather than merely focusing on one or two in particular. This is to ensure that some observational constraints are not violated in the models while attempting to assess the impact on others.
Large cosmological hydrodynamical simulations are now available, such as Eagle (Schaye et al., 2015), Illustris (Vogelsberger et al., 2014) and Magneticum (Dolag, 2015) within which detailed galaxy simulations can be created, for example. By selecting sub-volumes sampling a broad range of galaxy environments and re-simulating those subregions at high resolution, it should be possible to identify the impact of different simulated IMFs on the physical properties of the resulting galaxies. Simulation outputs should be produced that are directly comparable to observables (e.g., luminosities as well as stellar masses or SFRs) to avoid the need to reconstruct such derived properties from observational datasets, and potentially introducing inconsistent assumptions regarding the IMF in doing so. By incorporating population synthesis approaches that link directly to the observational metrics being used in inferring IMF properties, there may be the opportunity to directly assess how underlying IMF dependencies are subsequently quantified observationally.
In summary, there is an opportunity to begin linking the numerical, semi-analytic and population synthesis model approaches to self-consistently assess whether observational approaches for inferring IMFs are actually providing the quantitative conclusions expected, or whether other underlying effects may dominate.