My aim with this review is to unify the various observational and simulation approaches for investigating the stellar initial mass function (IMF), the mass distribution of stars arising from a star formation event. I do this by summarising work over the past few decades focusing primarily on observational constraints, and presenting a self-consistent framework to support future research. I address issues of terminology, definition, and scope of results in a way not previously attempted, with the goal of minimising ambiguity and assessing the degree of consistency or otherwise in published results regarding the “universality” of the IMF.
The significance of understanding the IMF was highlighted by Kennicutt (1998) who wrote: “Accurate knowledge of the form and mass limits of the stellar initial mass function, and its variation in different star formation environments, is critical to virtually every aspect of star formation, stellar populations, and galaxy evolution.” And: “Testing the universality of this initial mass function remains as our primary challenge for the coming decade.” Despite this goal being set two decades ago, the question of the universality of the IMF is still unresolved with a variety of results over the past decade providing evidence in favour of some kind of variation (e.g., van Dokkum & Conroy, 2010, Treu et al., 2010, Gunawardhana et al., 2011). Kennicutt (1998) concluded that, while there was no clear physical reason to expect the IMF to be universal, there was also “no compelling evidence for large systematic IMF variations in galaxies.” A contrary view was expressed by Larson (1998) who summarised a broad range of circumstantial evidence in favour of a stellar IMF with proportionally more high-mass stars at high redshift compared to the low redshift IMF.
The challenge posed in understanding the IMF is highlighted through the range and frequency of review articles dedicated to it since the 1980s (Scalo, 1986, Scalo, 1998, Kennicutt, 1998, Larson, 1998, Kroupa, 2002, Chabrier, 2003a) with a growing number in recent years (McKee & Ostriker, 2007, Elmegreen, 2009, Bastian et al., 2010, Jeffries, 2012, Kroupa et al., 2013, Offner et al., 2014, Krumholz, 2014), each touching on different but crucial aspects of the problem. Major conferences, too, have focussed on the IMF, with a celebration of the 50th anniversary of the IMF concept in 2005, “The Initial Mass Function 50 Years Later” (Corbelli et al., 2005), updating work presented in 1998 at the “The Stellar Initial Mass Function (38th Herstmonceux Conference)” (Gilmore & Howell, 1998). This was followed in 2010 with “UP2010: Have Observations Revealed a Variable Upper End of the Initial Mass Function?” exploring evidence for the possibility of IMF variations (Treyer et al., 2011), and in 2016 with a Lorentz Centre workshop “The Universal Problem of the Non-Universal IMF” 1 to share updates on the status of the work on IMF variations. Such levels of activity provide further evidence for the significance of the IMF and the complexity involved in understanding its details.
The field of IMF studies is vast. A search using the SAO/NASA Astrophysics Data System for papers having abstracts containing “initial mass function” or “IMF” yields more than 15000 publications. No single reviewer could ever hope to comprehensively summarise such a prodigious volume of work. Fortunately, existing reviews cover a broad range of different aspects of the field, and provide a solid basis on which to build.
By way of illustration, Elmegreen (2009) summarises and compares the shape of the IMF (its slope and characteristic mass) as probed through an extensive range of measurements within and external to the Milky Way, and gives a high level review of the primary physical processes responsible for star formation and the IMF. Bastian et al. (2010) provides a comprehensive review into the question of the universality of the IMF, thoroughly summarising work in the Galaxy and Local Group along with much of the work that was developing at the time to explore novel extragalactic approaches. Subsequently these fields have evolved quickly, with a lot of attention on the IMF shape in early type galaxies in particular. Kroupa et al. (2013) present an extensive and detailed review ranging from defining the IMF through to the various approaches to measuring the IMF in both stellar and extragalactic regimes, and discuss the implications in the context of the “integrated galaxy IMF” (IGIMF) formalism of Kroupa & Weidner (2003). Offner et al. (2014) present a detailed summary of work measuring the IMF in Milky Way star clusters and nearby galaxies, along with an overview of extragalactic work, before providing a highly comprehensive analysis of analytical and numerical theories behind the form and origins of the IMF. Krumholz (2014) reviews in detail the physical processes and phenomenology of star formation, and the status of the theoretical framework used in addressing the problem.
This review is intended to complement these and other reviews, referring to the detailed summaries they provide as needed, without attempting to duplicate the scope of their work. The aim here is not to deliver a comprehensive review of a vast body of work, but rather to synthesise the key elements from the work to date in order to develop a self-consistent framework and set of terminology on which to base future work. It is inevitable that there will be incompleteness in the references covered below, but the hope is that the main elements are addressed, and that at least representative results are presented.
This review builds on earlier work by summarising traditional approaches and the growing range of more recent techniques used to measure or infer the IMF with the aim of establishing their strengths and limitations, and identifying the different regimes in which they are applicable. I explore issues around the nature of the problem itself, in particular the degree to which the IMF is even a well-posed concept and whether there is an alternative formalism that might lend itself better to observational measurement.
The strengths and limitations of different methods are highlighted, and comparisons made between the typical samples to which they are applied, and the corresponding range of physical conditions probed. Some examples include the approaches typically used in stellar investigations within the Milky Way and Local Group galaxies, contrasted against those now becoming routine in extragalactic analyses. The latter include metrics relying on stellar population synthesis (SPS) tools (e.g., Hoversten & Glazebrook, 2008, van Dokkum & Conroy, 2010, Gunawardhana et al., 2011), the comparison of stellar and dynamical mass-to-light ratios (e.g., Treu et al., 2010), kinematics of stellar populations to infer mass-to-light ratios (e.g., Cappellari et al., 2012), and galaxy census approaches such as the cosmic star formation history (SFH) and the cosmic stellar mass density evolution (e.g., Wilkins et al., 2008a, Wilkins et al., 2008b).
I investigate the potential for linking the results established from this broad range of approaches, highlighting areas of actual inconsistency and carefully defining areas where apparent inconsistencies are potentially a result of different physical conditions accessible to different methodologies. I then identify opportunities for development of the field through new approaches to measurement of the IMF to provide a self-consistent and uniform foundation for subsequent work.
The review is structured as follows. In § 2 I briefly summarise the history of the IMF, explore issues of nomenclature and propose some conventions to minimise ambiguities in future work. §§ 3 - 7 present an overview of the wide variety of measurement approaches taken to date to constrain the IMF. I present a updated approach to the IMF in § 8, followed by a discussion in § 9 of the constraints and implications from the numerous measurements to date, before concluding in § 10. I assume H0 = 70 km s−1 Mpc−1, ΩM = 0.3 and ΩΛ = 0.7 where necessary for converting between redshift and lookback time.
1 https://www.lorentzcenter.nl/lc/web/2016/841/info.php3?wsid=841 Back.