Since the seminal results in the 1990s by Lilly et al. (1996) and Madau et al. (1996) the luminosity density and associated SFH of the Universe has been measured in progressively greater detail. This has been complemented by growing numbers of measurements of the stellar mass density (SMD) of the Universe from large-scale galaxy surveys, following Cole et al. (2001). These two major cosmic census methods, summarised in the review by Madau & Dickinson (2014), provide fundamental boundary conditions on an IMF for the galaxy population as a whole. Other census-style probes sensitive to the IMF include the extragalactic background light (another luminosity density metric), and the core-collapse supernova rate. Using constraints such as these, that effectively sample the entire galaxy population at a given epoch or series of epochs, provides a direct test of whether the IMF can be “universal”. A “universal” IMF must be able to reconcile the measurement of all such census metrics. With the increasing fidelity of and focus on the SFH and SMD, it is perhaps not surprising that this combination has been used to explore implications for the IMF. Such cosmic census approaches have an advantage over the analysis of individual galaxies in that the assumption of a relatively smooth SFH is more likely to be reasonable for the ensemble of a large galaxy population, and less of a source of systematic uncertainty.
The different sensitivity to an underlying IMF present in the cosmic SFH and the SMD in the universe allows these properties to be used in combination to infer a constraint on the IMF, averaged, in a sense, over the ensemble of galaxies sampled. The constraint arises because the SFH measurements are based on luminosity densities (such as the UV or far-infrared) sensitive to high mass stars (m > 5−10 M⊙), while the mass density measurements are based on luminosity densities (such as the optical and near-infrared) sensitive to the low mass stellar population. Different IMF shapes will affect these stellar populations, and their associated luminosity densities, differently. Even as early as Cole et al. (2001) there was a recognised tension between the SFH and SMD, with that study noting that the two could only be reconciled for a “universal” IMF with the assumption of surprisingly little dust obscuration affecting the overall SFH.
The approach was explored explicitly by Baldry & Glazebrook (2003) who inferred constraints on the high mass (m >0.5 M⊙) slope of an assumed “universal” IMF. They find a slope of αh = −2.15, slightly more positive than, but consistent with, that of the Salpeter slope, using the joint constraint of the cosmic SFH and the z≈0.1 luminosity density. Combining the SFH and the electron antineutrino upper limit arising from the core-collapse supernova background, Hopkins & Beacom (2006) place bounds on the high mass (m > 0.5 M⊙) slope of a “universal” IMF (−2.35 < αh < −2.15), and note (Hopkins & Beacom, 2008) that for consistency with the SMD an IMF slope of αh = −2.15 (from Baldry & Glazebrook, 2003) is favoured over the Salpeter slope of αh = −2.35. Fardal et al. (2007) also noted a need for an excess of high mass stars, proposing a “paunchy” IMF, with an excess of stars in the mass range 1.5 < m < 4 M⊙, (α = −1 for 0.1 < m / M⊙ < 0.5; α = −1.7 for 0.5 < m / M⊙ < 0.4; and α = −2.6 for 4 < m / M⊙ < 100) to reconcile joint measurements of the extragalactic background radiation density and the stellar mass density (or K-band luminosity density).
Building on these results, Wilkins et al. (2008a) showed that the SFH and SMD are inconsistent with a universal, unevolving IMF. Wilkins et al. (2008b) quantified a requirement for an IMF with a high mass slope of αh = −2.15 at low redshift that evolves to a high mass slope with a more positive index still (αh > −2.15) at z ≳ 1. This result has subsequently been questioned (Reddy & Steidel, 2009), with the key issues being the extent of obscured star formation at high redshift (z ≳ 2) and systematics in the estimates of the SMD.
The core collapse supernova rate density can also be used as a tracer of the cosmic SFH (e.g., Dahlen et al., 2012). Recent results (Strolger et al., 2015) suggest that the observed rates are consistent with the SFR densities derived from dust-corrected UV emission, and inconsistent with the higher SFH that has been used to infer IMF evolution. These results, which rely on 1.6 µm imaging for samples out to z = 2.5, may still suffer from incompleteness, however, due to extreme obscuration in high star formation regions. This has been demonstrated through the 2.15 µm detection of heavily obscured supernovae in the nuclei of nearby (z < 0.027) luminous infrared galaxies (Kool et al., 2018).
The extensive review of the cosmic SFH by Madau & Dickinson (2014) argues that the discrepancy between the SFH and the SMD is not significant enough to require an evolving IMF. They use a Salpeter IMF over the full mass range (α = −2.35 for 0.1 < m / M⊙ < 100) and a selective compilation of observations focusing on far-infrared and UV measurements, and argue that the discrepancy between the SFH and SMD is not as large as previously assserted. They note that their observed 0.2 dex (60%) overestimate between the SMD implied from the SFH and direct SMD measurements can be reduced to 0.1 dex with a Chabrier (2003a) or Kroupa (2001) IMF (as subsequently demonstrated, for example, by Davidzon et al., 2017), and argue that this residual discrepancy is not sufficient evidence for variations in the IMF. The more recent analysis of 570 000 galaxies by Driver et al. (2018) reaches a similar conclusion. In both cases, though, the highest redshifts are probed through galaxies that are rest-frame UV selected, and are not sensitive to heavily obscured systems.
The analysis of Madau & Dickinson (2014) omits high redshift (z ≈ 2.3) Hα measurements (e.g., Sobral et al., 2013), which are somewhat higher than those inferred from the compilation of UV measurements, perhaps by as much as ≈ 0.1 dex at z ≈ 2.3. Combined with the observation that the fitted functional form of Madau & Dickinson (2014) tracks closer to the lower envelope of their data compilation for 1 ≲ z ≲ 3 than the median, another offset of about 0.1 dex, there appears to remain scope for discussion of the consistency between the SFH and SMD. Subsequent updates include new high redshift (z > 4) SMD measurements (Grazian et al., 2015) that also renew the tension between the SMH and SMD. This was explored in more detail by Yu & Wang (2016) who highlight in particular a significant mismatch between the observed SFH and that inferred from the SMD in the range 0.5 ≲ z ≲ 6. It seems that there are still degrees of inconsistency between the SFH and SMD that remain to be resolved. This includes the steeper IMF slope (αh = −2.45−0.03+0.06) found in M31 by Weisz et al. (2015), which would be inconsistent with the required αh = −2.35 implied by Madau & Dickinson (2014) (but see Oh & Kroupa, 2016). There is also the evidence from Milky Way CEMP stars (Tumlinson, 2007) that seems to require an evolution in the IMF toward a larger proportion of high mass stars at higher redshift (increasing mc), which would be absent in the “universal” IMF scenario of Madau & Dickinson (2014).
Before the SFH and SMD can be used as an IMF constraint, their robustness must be established. At the lower redshift end, z ≲ 2−3, the SFH and SMD are well constrained to the level of 30−50%. At higher redshifts, especially z ≳ 4, there has been a growing tension over the past decade in the form of the SFH evolution. The differences arise depending on whether the SFH is measured using photometric dropout samples (e.g., Bouwens et al., 2015a), probes that may be sensitive to low mass galaxies (such as gamma ray bursts, GRBs, e.g., Kistler et al., 2013) or heavily obscured systems, (using far-infrared data, e.g., Gruppioni et al., 2013, Rowan-Robinson et al., 2016). Recent work has highlighted issues with these latter measurements, with Koprowski et al. (2017) arguing that they are overestimated because the inferred luminosity functions overpredict the observed 850 µm source counts. Koprowski et al. (2017) show that results from SCUBA-2 and ALMA are consistent with those inferred from the UV-selected photometric dropout samples. In contrast, recent results using deep radio observations (Novak et al., 2017) find SFR densities at z > 2 consistent with those of Gruppioni et al. (2013) and Rowan-Robinson et al. (2016). These are higher than inferred by Behroozi et al. (2013), who updated the SFH compilation of Hopkins & Beacom (2006) based on the UV selected samples at such redshifts that had appeared in the meantime, inferring a lower SFH fit beyond z > 3. Novak et al. (2017) conclude that there is substantial dust-obscured star formation at these high redshifts, finding marginal consistency with the dust-corrected SFH of Bouwens et al. (2015a).
It is clear from the comparison between the UV and radio luminosity functions by Novak et al. (2017) at ⟨ z ⟩ = 3.7 and ⟨ z ⟩ = 4.8 that there is a significant difference at the high luminosity (SFR) end, with the deep radio data picking up high SFR systems not seen in the UV luminosity functions of Bouwens et al. (2015a). This may be a consequence of the much larger survey area probed by the radio surveys (∼ 2 deg2) than the UV surveys (∼ 0.3 deg2). It is telling that in the comparison by Bouwens et al. (2015a) with their earlier work in much smaller (∼ 50 arcmin2) survey regions (their Figure 10), they find that the larger survey area (∼ 1000 arcmin2) reveals uniformly higher bright ends for the UV luminosity functions at z > 5, implying larger numbers of higher luminosity systems. It is perhaps not unreasonable to expect that trend to continue when much larger regions are sampled. An alternative is significant obscuration, optically thick at UV wavelengths, preventing the high luminosity systems from being detected at all, and unable to be accounted for when making obscuration corrections to the observed high redshift UV detected population. Of course, both effects may play a role here.
Behroozi & Silk (2015) used the updated SFH normalisation from Behroozi et al. (2013) to scale down the GRB inferred SFH of Kistler et al. (2013) at z > 4, making them more consistent with their inferred SFH fit. In a recent review, though, Chary et al. (2016) show that metallicity constraints at z > 2 from damped Lyα systems are consistent with the rather more elevated SFH inferred by Kistler et al. (2013), and consistent with that of Gruppioni et al. (2013) and Rowan-Robinson et al. (2016) than the lower SFH of Behroozi & Silk (2015). It is noteworthy in this discussion that the radio luminosity function results of Novak et al. (2017) are also consistent with the GRB inferred SFR densities (Chary et al., 2007, Yüksel et al., 2008, Kistler et al., 2009, Kistler et al., 2013) at these high redshifts, reinforcing the expectation of a steepening low luminosity tail to the high redshift galaxy luminosity function (Kistler et al., 2013, Bouwens et al., 2015a). Such a steep tail, implying the existence of a low mass population of star forming galaxies at z > 4, was argued for by Wyithe et al. (2014), who show a need for a 10% duty cycle for star formation based on observed sSFR at such high redshifts.
To return to the IMF constraints imposed by cosmic census approaches, the metal mass density of the universe is another worth considering (e.g., Dunne et al., 2003, Hopkins et al., 2005, Hopkins & Beacom, 2006), as well as average metallicities of galaxy populations (e.g., Driver et al., 2013, Chary et al., 2016). The limited use to date of such constraints reflects in part the challenge in observationally constructing large samples of such measurements at high redshift. There would seem to be significant power achievable through a joint cosmic census constraint combining the local luminosity density (Baldry & Glazebrook, 2003), the SFH/SMD (e.g., Hopkins & Beacom, 2006), the extragalactic background light (Fardal et al., 2007), and the metal mass density.
Taken together, these results suggest that there is further scope for refining our understanding of the high redshift end of the SFH and SMD, and the joint constraint they impose on the underlying IMF. In that light, I strongly endorse objectivity in the selection of observational datasets for future comparisons. There has been a clear tendency in the community to favour one form of observational constraint over another when comparing new work against old, or models against data, to present new results in the best light. The wealth of published measurements makes it easy to overlook or omit data that is inconsistent or introduces tension with the new results, rather than objectively comparing against the full range of observations, with a critical consideration of their limitations. With the now significant numbers of published measurements for the SFH and SMD, there is scope for a critical and thorough review to assess the reliability of each, in order that all published measurements are not simply each given equal weight in future compilations, and that future work does not have the scope to be selective in the published measurements against which they compare. Old results that have been superseded should be discarded, and careful consideration given to the origins of any tension in apparently conflicting results, rather than choosing to favour one over another. There is a valuable opportunity now to establish a new “gold standard” of SFH and SMD results for comprehensive future use.
It is worth considering that the observational constraints summarised by Madau & Dickinson (2014) set, in a sense, an absolute bound for a “universal” IMF. Taking the most robust measurements possible, they still find (and dismiss) a mild tension between the SFH and SMD. The observations not considered by Madau & Dickinson (2014), though, are those which imply higher values for the SFH, and hence exacerbate the SFH/SMD tension. If any weight is given at all to these other observations, the high redshift SFH tends to move upward and the tension with the SMD is increased. In that sense, either the IMF is universal, similar to Kroupa (2001) and Chabrier (2003a), and any higher SFH estimates must be overestimated, or there is evidence for an IMF that varies, with αh increasing as redshift increases.
The SFH/SMD constraint is, however, inconsistent with the very precise high mass IMF slope of M31 from Weisz et al. (2015), with αh = −2.45−0.03+0.06 (m > 1 M⊙), and the steeper slopes found for the LMC, SMC and other dwarf galaxies (e.g., Parker et al., 1998, Úbeda et al., 2007, Lamb et al., 2013, Bruzzese et al., 2015). This steep a high mass slope, if it were “universal,” would exacerbate the tension between the SFH and the SMD significantly, as the inferred SFH would need to be at least 30-50% higher. It seems reasonable to conclude on this point alone, then, that the IMF is not universal. It also bears reiterating here that most of the SFH/SMD tension is in the mid-range of redshifts, 1 ≲ z ≲ 4 (e.g., Yu & Wang, 2016), since there is too little time at the highest redshifts (z > 6) for appreciable stellar mass to form, compared to that assembled subsequently (e.g., Driver et al., 2013).
Observations at such high redshifts also begin to probe the epoch of reionisation (z ≳ 6). The reionisation of the universe now seems able to be well-explained by star formation in z > 6 galaxies (e.g., McLeod et al., 2015, Bouwens et al., 2015b). The contributions from Population III stars and the implications for their IMF are now also being explored (e.g., Salvador-Solé et al., 2017). The IMF in such high redshift galaxies is of critical interest. In an earlier analysis using the UV and V-band luminosity densities at z ≈ 6 and a constraint from the epoch of reionisation, Chary (2008) rule out a Salpeter-like IMF (α = −2.3 for 0.1 < m / M⊙ < 200) for z > 6 as not producing enough ionising photons per baryon. Depending on the details of the reionisation history, Chary (2008) argues that the high redshift (z>6) IMF must have a flatter slope, favouring α = −1.65 over 0.1 < m / M⊙ < 200. It is tantalising that such a conclusion is in the same sense as would be required from an evolving IMF from the SFH/SMD constraint, and there is clearly scope for a unified approach to link these observational constraints on the IMF.
I digress now to take step back and consider some logical inconsistencies in the argument for a “universal” IMF:
Since a universal IMF cannot have a high mass slope that is steeper, flatter or equal to the Salpeter value, the logical conclusion, then, is that the IMF is not “universal.” The limitations in this argument will be clear, and it is obviously not a formal proof, but the conclusion that a growing wealth of evidence points against a “universal” IMF is inescapable.
If the IMF is not universal, then authors must be wary of inconsistent usage of assumed IMFs. As a naive example, galaxy SFRs may be calculated assuming a nominal IMF, but then compared against SPS outputs assuming a variety of input IMFs in order to establish which (erroneously) better matches the data. Such analyses must be careful to ensure self-consistency of IMF assumptions throughout. This is true of cosmic census analyses as well.
If the IMF is not universal, then there are clearly many observational implications, that can be tested to further explore the extent of any IMF variation. For example, Ferreras et al. (2015) show that no single IMF with a fixed high mass (m > 0.5 M⊙) slope (αh = −2.3) and a low mass slope ranging from −2.8 ≤ αl ≤ −1.8 can reproduce the observational constraints from the stellar populations of massive early type galaxies, together with their observed metallicities. They conclude that an evolving IMF (Weidner et al., 2013) is required to explain the joint constraint. Some implications of a varying IMF were explored by Clauwens et al. (2016), who show the impact of assuming the metallicity-dependent IMF found by Martín-Navarro et al. (2015d) on the SFR of galaxies, the stellar mass function, mass-metallicity relation and reionisation. The results range from significant to minimal, depending on how the dwarf-to-giant ratio of the IMF is implemented, but define a clear set of observational constraints that can be used to begin ruling out particular IMF forms. The substantial variations in physical distributions seen for some of these comparisons, many already inconsistent with observation, highlight the significant existing scope to begin a focused program of quantifying any potential variation in the IMF.
As a thought experiment, consider whether the metallicity-dependent or σ-dependent “bottom heavy” IMF for spheroids (αl ≲ −2.35, m < 1 M⊙) and the SFR-dependent “top heavy” IMF for disk galaxies (αh ≳ −2.35, m > 0.5 M⊙) might both be consistent with the sense of a putative evolving IMF from the SFH/SMD constraint. We can use the two-phase model for the evolution of galaxies proposed by Driver et al. (2013). In this model, systems that will become spheroids dominate the SFH earlier (with a peak around 3 ≲ z ≲ 5) than those that become disks (with a peak around z ≈ 1). If the spheroids and disks of Driver et al. (2013) respectively have the “bottom heavy” and “top heavy” IMFs seen locally (as defined above), then in very qualitative terms, it would appear that the IMF evolution should be increasingly dominated by “bottom heavy” systems at higher redshift, inconsistent with the allowed evolution from the SFH/SMD constraint. Such a coarse analysis clearly neglects many effects that need to be investigated in more detail, but this illustration hopefully indicates the scope of opportunities for continuing to explore and refine our understanding of the IMF.
It might be argued, adopting the traditional approach, that all of the work above may be considered “consistent with a (poorly specified) universal IMF (with large uncertainties)”, given the variety of conflicting results, counter-claims, and limitations. I hope by this point that the specious nature of this conclusion is clear. There appears to be clear and growing evidence, albeit with a variety of associated limitations, for some degree of variation in the IMF, and it is appropriate for the conversation to move on to constraining such variations rather than dismissing them.
On that note, I briefly explore simulation work in § 7 below, aiming to highlight the need for modelers to focus not on reproducing a particular IMF behaviour, but on identifying which physical conditions lead to what kind of IMF behaviour, and under what assumptions. Only by reframing the question to one that asks how the IMF varies and how do different assumptions or physical conditions impact such variation can we begin to make self-consistent progress in understanding the IMF itself.