Galaxy-wide Initial Mass Function Research Articles

The Variation in the Galaxy-wide Initial Mass Function for Low-mass Stars: Modeling and Observational Insights

The stellar initial mass function (IMF) characterizes the mass distribution of newly formed stars in various cosmic environments, serving as a fundamental assumption in astrophysical research. Recent findings challenge the prevalent notion of a universal and static IMF, proposing instead that the IMF’s shape is contingent upon the star formation environment. In this study, we analyze the galaxy-wide variation in the IMF for low-mass stars in both dwarf and massive galaxies with diverse observational methods. Despite systematic discrepancies between different approaches, an IMF model with a metallicity-dependent slope for the low-mass stars aligns with the majority of observations, indicating a high degree of uniformity in the star formation processes across the Universe. We also emphasize the need for a more comprehensive understanding of the variation in the low-mass IMF, considering measurement biases and factors beyond metallicity.

Variation of the stellar initial mass function in semi-analytical models

Context. In our previous work, we derived the CR-IGIMF, which is a new scenario for a variable stellar initial mass function (IMF) that combines numerical results on the role played by cosmic rays (CR) in setting the thermal state of star-forming gas with the analytical approach of the integrated galaxy-wide IMF (IGIMF). Aims. In this work, we study the implications of this scenario for the properties of local early-type galaxies (ETG) as inferred from dynamical, photometric, and spectroscopic studies. Methods. We implemented a library of CR-IGIMF shapes in the framework of the galaxy evolution and assembly (GAEA) model. GAEA provides predictions for the physical and photometric properties of model galaxies and for their chemical composition. Our realization includes a self-consistent derivation of the synthetic spectral energy distribution for each model galaxy, which allows a direct derivation of the mass fraction in the mean IMF of low-mass stars (i.e., the dwarf-to-giant ratio, fdg) and a comparison with IMF-sensitive spectral features. Results. The predictions of the GAEA model implementing the CR-IGIMF confirm our previous findings: It correctly reproduces both the observed excess of z ∼ 0 dynamical mass (mass-to-light ratio) with respect to spectroscopic (photometric) estimates assuming a universal MW-like IMF, and the observed increase in [α/Fe] ratios with stellar mass in spheroidal galaxies. Moreover, this realization reproduces the increasing trends of fdg and IMF-sensitive line strengths with velocity dispersion, although the predicted relations are significantly shallower than the observed ones. Conclusions. Our results show that the CR-IGIMF is a promising scenario that reproduces at the same time dynamical, photometric, and spectroscopic indications of a varying IMF in local ETGs. The shallow relations found for spectral indices suggest that either a stronger variability as a function of galaxy properties or additional dependences (e.g., as a function of star forming gas metallicity) might be required to match the strength of the observed trends.

The most massive stars in very young star clusters with a limited mass: Evidence favours significant self-regulation in the star formation processes

The stellar initial mass function (IMF) is commonly interpreted to be a scale-invariant probability density distribution function (PDF) such that many small clusters yield the same IMF as one massive cluster of the same combined number of stars. Observations of the galaxy-wide IMF challenge this as dwarf galaxies do not form as many massive stars as expected. This indicates a highly self-regulated star formation process in which stellar masses are not stochastically sampled from the IMF and are instead related to the environment of star formation. Here, we study the nature of star formation using the relation between the most massive star born in a star cluster and its parental stellar cluster mass (the mmax − Mecl relation). This relation has been argued to be a statistical effect if stars are sampled randomly from the IMF. By comparing the tightness of the observed mmax − Mecl distribution with synthetic star clusters with stochastically sampled stellar masses, we find that the expected dispersion of the mock observations is much larger than the observed dispersion. Assuming that mmax and Mecl uncertainties from the literature are correct, our test rejects the hypothesis that the IMF is a PDF at a more than 4.5σ confidence level. Alternatively, we provide a deterministic stellar mass sampling tool that reproduces the observed mmax − Mecl distribution and compares well with the luminosities of star-forming molecular clumps. In addition, we find that there is a significant flattening of the mmax − Mecl relation near mmax = 13 M⊙. This may suggest strong feedback of stars more massive than about 13 M⊙, and/or that the ejections of the most massive stars from young clusters in the mass range 63 to 400 M⊙ are likely important physical processes in forming clusters.

Has JWST Already Falsified Dark-matter-driven Galaxy Formation?

The James Webb Space Telescope (JWST) discovered several luminous high-redshift galaxy candidates with stellar masses of M * ≳ 109 M ⊙ at photometric redshifts z phot ≳ 10, which allows to constrain galaxy and structure formation models. For example, Adams et al. identified the candidate ID 1514 with located at and Naidu et al. found even more distant candidates labeled as GL-z11 and GL-z13 with at and at , respectively. Assessing the computations of the IllustrisTNG (TNG50-1 and TNG100-1) and EAGLE projects, we investigate if the stellar mass buildup as predicted by the ΛCDM paradigm is consistent with these observations assuming that the early JWST calibration is correct and that the candidates are indeed located at z ≳ 10. Galaxies formed in the ΛCDM paradigm are by more than an order of magnitude less massive in stars than the observed galaxy candidates implying that the stellar mass buildup is more efficient in the early universe than predicted by the ΛCDM models. This in turn would suggest that structure formation is more enhanced at z ≳ 10 than predicted by the ΛCDM framework. We show that different star formation histories could reduce the stellar masses of the galaxy candidates alleviating the tension. Finally, we calculate the galaxy-wide initial mass function (gwIMF) of the galaxy candidates assuming the integrated galaxy IMF theory. The gwIMF becomes top-heavy for metal-poor star-forming galaxies decreasing therewith the stellar masses compared to an invariant canonical IMF.

Downsizing revised: Star formation timescales for elliptical galaxies with an environment-dependent IMF and a number of SNIa

Previous studies of the stellar mean metallicity and [Mg/Fe] values of massive elliptical (E) galaxies suggest that their stars were formed over a very short timescale that cannot be reconciled with estimates from stellar population synthesis (SPS) studies and with hierarchical assembly. Applying the previously developed chemical evolution code, GalIMF, which allows an environment-dependent stellar initial mass function (IMF) to be applied to the integrated galaxy initial mass function theory instead of an invariant canonical IMF, the star formation timescales (SFT) of E galaxies are re-evaluated. The code’s uniqueness lies in it allowing the galaxy-wide IMF and associated chemical enrichment to evolve as the physical conditions in the galaxy change. The calculated SFTs become consistent with the independent SPS results if the number of type Ia supernovae (SNIa) per unit stellar mass increases for more massive E galaxies. This is a natural outcome of galaxies with higher star formation rates producing more massive star clusters, spawning a larger number of SNIa progenitors per star. The calculations show E galaxies with a stellar mass ≈109.5 M⊙ to have had the longest mean SFTs of ≈2 Gyr. The bulk of more massive E galaxies were formed faster (SFT ≈ 1 Gyr) leading to domination by M dwarf stars and larger dynamical mass-to-light ratios as observed, while lower mass galaxies tend to lose their gas supply more easily due to their shallower potential and therefore also have similarly-short mean SFTs. This work achieves, for the first time, consistency of the SFTs for early-type galaxies between chemical-enrichment and SPS modelling. Equally, it leads to an improved understanding of how the star formation environment may affect the total number of SNIa per unit stellar mass formed.

Chemical evolution of elliptical galaxies with a variable IMF

Growing evidence in recent years suggests a systematic variation of the stellar initial mass function (IMF), being top-heavy for starburst galaxies and possibly bottom-heavy for massive ellipticals. Galaxy chemical evolution simulations adopting an invariant canonical IMF face difficulty in simultaneously reproducing the metallicity and α-enhancement of the massive elliptical galaxies. Applying a variable IMF that changes with time is a promising solution, however, it is non-trivial to couple a variable IMF theory with the existing galaxy evolution codes. Here we present the first open source simulation code which recalculates the galaxy-wide IMF at each time step according to the integrated galactic IMF (IGIMF) theory where the galaxy-wide IMF depends on the galactic star formation rate and metallicity. The resulting galaxy-wide IMF and metal abundance evolve with time. With this pilot work, we explore the effect of the IGIMF theory on galaxy chemical evolution in comparison with an invariant IMF.

Impact of metallicity and star formation rate on the time-dependent, galaxy-wide stellar initial mass function

The stellar initial mass function (IMF) is commonly assumed to be an invariant probability density distribution function of initial stellar masses. These initial stellar masses are generally represented by the canonical IMF, which is defined as the result of one star formation event in an embedded cluster. As a consequence, the galaxy-wide IMF (gwIMF) should also be invariant and of the same form as the canonical IMF; gwIMF is defined as the sum of the IMFs of all star-forming regions in which embedded clusters form and spawn the galactic field population of the galaxy. Recent observational and theoretical results challenge the hypothesis that the gwIMF is invariant. In order to study the possible reasons for this variation, it is useful to relate the observed IMF to the gwIMF. Starting with the IMF determined in resolved star clusters, we apply the IGIMF-theory to calculate a comprehensive grid of gwIMF models for metallicities, [Fe/H] ∈ (−3, 1), and galaxy-wide star formation rates (SFRs), SFR ∈ (10−5, 105) M⊙ yr−1. For a galaxy with metallicity [Fe/H] < 0 and SFR > 1 M⊙ yr−1, which is a common condition in the early Universe, we find that the gwIMF is both bottom light (relatively fewer low-mass stars) and top heavy (more massive stars), when compared to the canonical IMF. For a SFR < 1 M⊙ yr−1 the gwIMF becomes top light regardless of the metallicity. For metallicities [Fe/H] > 0 the gwIMF can become bottom heavy regardless of the SFR. The IGIMF models predict that massive elliptical galaxies should have formed with a gwIMF that is top heavy within the first few hundred Myr of the life of the galaxy and that it evolves into a bottom heavy gwIMF in the metal-enriched galactic centre. Using the gwIMF grids, we study the SFR−Hα relation and its dependency on metallicity and the SFR. We also study the correction factors to the Kennicutt SFRK − Hα relation and provide new fitting functions. Late-type dwarf galaxies show significantly higher SFRs with respect to Kennicutt SFRs, while star-forming massive galaxies have significantly lower SFRs than hitherto thought. This has implications for gas-consumption timescales and for the main sequence of galaxies. We explicitly discuss Leo P and ultra-faint dwarf galaxies.

On the shape and evolution of a cosmic-ray-regulated galaxy-wide stellar initial mass function

In this paper, we present a new derivation of the shape and evolution of the integrated galaxy-wide initial mass function (IGIMF), incorporating explicitly the effects of cosmic rays (CRs) as regulators of the chemical and thermal state of the gas in the dense cores of molecular clouds. We predict the shape of the IGIMF as a function of star formation rate and CR density and show that it can be significantly different with respect to local estimates. In particular, we focus on the physical conditions corresponding to IGIMF shapes that are simultaneously shallower at high-mass end and steeper at the low-mass end than those of a Kroupa IMF. These solutions can explain both the levels of α-enrichment and the excess of low-mass stars as a function of stellar mass, observed for local spheroidal galaxies. As a preliminary test of our scenario, we use idealized star formation histories to estimate the mean IMF shape for galaxies of different |$z$| = 0 stellar mass. We show that the fraction of low-mass stars as a function of galaxy stellar mass predicted by these mean IMFs agrees with the values derived from high-resolution spectroscopic surveys.

Star formation in the outskirts of DDO 154: a top-light IMF in a nearly dormant disc

We present optical photometry of Hubble Space Telescope (HST) ACS/WFC data of the resolved stellar populations in the outer disc of the dwarf irregular galaxy DDO 154. The photometry reveals that young main sequence stars are almost absent from the outermost HI disc. Instead, most are clustered near the main stellar component of the galaxy. We constrain the stellar initial mass function (IMF) by comparing the luminosity function of the main sequence stars to simulated stellar populations assuming a constant star formation rate over the dynamical timescale. The best-fitting IMF is deficient in high mass stars compared to a canonical Kroupa IMF, with a best-fit slope $\alpha = -2.45$ and upper mass limit $M_U = 16\ M_{\odot}$. This top-light IMF is consistent with predictions of the Integrated Galaxy-wide IMF theory. Combining the HST images with HI data from The HI Nearby Galaxy Survey Treasury (THINGS) we determine the star formation law (SFL) in the outer disc. The fit has a power law exponent $N = 2.92 \pm0.22$ and zero point $A=4.47 \pm 0.65 \times 10^{-7} \ M_{\odot} \ \text{yr}^{-1} \ \text{kpc}^{-2}$. This is depressed compared to the Kennicutt-Schmidt Star Formation Law, but consistent with weak star formation observed in diffuse HI environments. Extrapolating the SFL over the outer disc implies that there could be significant star formation occurring that is not detectable in H$\alpha$. Last, we determine the Toomre stability parameter $Q$ of the outer disc of DDO 154 using the THINGS HI rotation curve and velocity dispersion map. 72% of the HI in our field has $Q\leq 4$ and this incorporates 96% of the observed MS stars. Hence 28% of the HI in the field is largely dormant.

Variations of the stellar initial mass function in semi-analytical models II: the impact of Cosmic Ray regulation.

Recent studies proposed that cosmic rays (CR) are a key ingredient in setting the conditions for star formation, thanks to their ability to alter the thermal and chemical state of dense gas in the UV-shielded cores of molecular clouds. In this paper, we explore their role as regulators of the stellar initial mass function (IMF) variations, using the semi-analytic model for GAlaxy Evolution and Assembly (GAEA). The new model confirms our previous results obtained using the integrated galaxy-wide IMF (IGIMF) theory: both variable IMF models reproduce the observed increase of $\alpha$-enhancement as a function of stellar mass and the measured $z=0$ excess of dynamical mass-to-light ratios with respect to photometric estimates assuming a universal IMF. We focus here on the mismatch between the photometrically-derived ($M^{\rm app}_{\star}$) and intrinsic ($M_{\star}$) stellar masses, by analysing in detail the evolution of model galaxies with different values of $M_{\star}/M^{\rm app}_{\star}$. We find that galaxies with small deviations (i.e. formally consistent with a universal IMF hypothesis) are characterized by more extended star formation histories and live in less massive haloes with respect to the bulk of the galaxy population. While the IGIMF theory does not change significantly the mean evolution of model galaxies with respect to the reference model, a CR-regulated IMF implies shorter star formation histories and higher peaks of star formation for objects more massive than $10^{10.5} M_\odot$. However, we also show that it is difficult to unveil this behaviour from observations, as the key physical quantities are typically derived assuming a universal IMF.

The optimally sampled galaxy-wide stellar initial mass function

Here we present a full description of the integrated galaxy-wide initial mass function (IGIMF) theory in terms of the optimal sampling and compare it with available observations. Optimal sampling is the method we use to discretize the IMF into stellar masses deterministically. Evidence has been indicating that nature may be closer to deterministic sampling as observations suggest a smaller scatter of various relevant observables than random sampling would give, which may result from a high level of self-regulation during the star formation process. The variation of the IGIMFs under various assumptions are documented. The results of the IGIMF theory are consistent with the empirical relation between the total mass of a star cluster and the mass of its most massive star, and the empirical relation between a galaxy's star formation rate (SFR) and the mass of its most massive cluster. Particularly, we note a natural agreement with the empirical relation between the IMF's power-law index and a galaxy's SFR. The IGIMF also results in a relation between the galaxy's SFR and the mass of its most massive star such that, if there were no binaries, galaxies with SFR $<10^{-4}$ M$_\odot$/yr should host no Type II supernova events. In addition, a specific list of initial stellar masses can be useful in numerical simulations of stellar systems. For the first time, we show optimally-sampled galaxy-wide IMFs (OSGIMF) which mimics the IGIMF with an additional serrated feature. Finally, A Python module, GalIMF, is provided allowing the calculation of the IGIMF and OSGIMF in dependence on the galaxy-wide SFR and metallicity.

The chemical evolution of galaxies with a variable integrated galactic initial mass function

Standard analytical chemical evolution modelling of galaxies has been assuming the stellar initial mass function (IMF) to be invariant and fully sampled allowing fractions of massive stars to contribute even in dwarf galaxies with very low star formation rates (SFRs). Recent observations show the integrated galactic initial mass function (IGIMF) of stars, i.e. the galaxy-wide IMF, to become systematically top-heavy with increasing SFR. This has been predicted by the IGIMF theory, which is here used to develop the analytical theory of the chemical evolution of galaxies. This theory is non-linear and requires the iterative solution of implicit integral equations due to the dependence of the IGIMF on the metallicity and on the SFR. It is shown that the mass-metallicity relation of galaxies emerges naturally, although at low masses the theoretical predictions overestimate the observations by 0.3--0.4 dex. A good agreement with the observation can be obtained only if gas flows are taken into account. In particular, we are able to reproduce the mass--metallicity relation observed by Lee et al. (2006) with modest amounts of infall and with an outflow rate which decreases as a function of the galactic mass. The outflow rates required to fit the data are considerably smaller than required in models with invariant IMFs.

Galaxies as simple dynamical systems: observational data disfavor dark matter and stochastic star formation

According to modern theory, galactic evolution is driven by the dynamics of dark matter and stochastic star formation, but galaxies are observed to be simple systems. The existence of dark matter particles is a key hypothesis in present-day cosmology and galactic dynamics. Given the large body of high-quality work within the standard model of cosmology (SMoC), the validity of this hypothesis is challenged significantly by two independent arguments: (i) The dual dwarf galaxy theorem must be true in any realistic cosmological model. But the data now available appear to falsify this postulate when the dark-matter-based model is compared with the observational data. A consistency test for this conclusion comes from the significantly anisotropic distributions of satellite galaxies (baryonic mass &lt;108M⊙) that orbit in the same direction around their hosting galaxies in disk-like structures, which cannot be derived from dark matter models. (ii) The action of dynamical friction due to expansive and massive dark matter halos must be evident in the galaxy population. The evidence for dynamical friction is poor or even absent. Indendently of this, the long history of failures of the SMoC have reduced the likelihood that it describes the observed Universe to less than 10−4%. The implication for fundamental physics is that exotic dark matter particles do not exist and that consequently effective gravitational physics on the scales of galaxies and beyond ought to be non-Newtonian and (or) non-Einsteinian. An analysis of the kinematic data in galaxies shows them to be described elegantly in the weak-gravitational regime by scale-invariant dynamics, as discovered by Milgrom. The full classical regime of gravitation is effectively described by Milgromian dynamics. This leads to a natural emergence of the simple laws that galaxies are indeed observed to obey. Such success has not been forthcoming in dark-matter-based models. Observations of stellar populations in galaxies suggest that the galaxy-wide initial mass function varies with star formation rate and that stochastic descriptions of star formation are inconsistent with the data. This requires a reinterpretation of the stellar mass assembly in galaxies and thus of the accretion rates onto galaxies. A consequence of this understanding of galactic astrophysics is that most dwarf satellite galaxies are formed as tidal dwarf galaxies in galaxy–galaxy encounters, that they follow the mass–metallicity relation, that galactic mergers are rare, that galaxies immersed in external potentials are physically larger than isolated galaxies, and that star-forming galaxies form a main sequence. Eight predictions are offered that will allow the concepts raised here to be tested. A very conservative, cold- and warm-dark-matter-free cosmological model may be emerging from these considerations.

The galaxy-wide initial mass function of dwarf late-type to massive early-type galaxies

Observational studies are showing that the galaxy-wide stellar initial mass function are top-heavy in galaxies with high star-formation rates (SFRs). Calculating the integrated galactic stellar initial mass function (IGIMF) as a function of the SFR of a galaxy, it follows that galaxies which have or which formed with SFRs > 10 Msol yr^-1 would have a top-heavy IGIMF in excellent consistency with the observations. Consequently and in agreement with observations, elliptical galaxies would have higher M/L ratios as a result of the overabundance of stellar remnants compared to a stellar population that formed with an invariant canonical stellar initial mass function (IMF). For the Milky Way, the IGIMF yields very good agreement with the disk- and the bulge-IMF determinations. Our conclusions are that purely stochastic descriptions of star formation on the scales of a pc and above are falsified. Instead, star formation follows the laws, stated here as axioms, which define the IGIMF theory. We also find evidence that the power-law index beta of the embedded cluster mass function decreases with increasing SFR. We propose further tests of the IGIMF theory through counting massive stars in dwarf galaxies.

The (galaxy-wide) IMF in giant elliptical galaxies: from top to bottom

Recent evidence based independently on spectral line strengths and dynamical modelling point towards a non-universal stellar Initial Mass Function (IMF), probably implying an excess of low-mass stars in elliptical galaxies with a high velocity dispersion. Here we show that a time-independent bottom-heavy IMF is compatible neither with the observed metal-rich populations found in giant ellipticals nor with the number of stellar remnants observed within these systems. We suggest a two-stage formation scenario involving a time-dependent IMF to reconcile these observational constraints. In this model, an early strong star-bursting stage with a top-heavy IMF is followed by a more prolonged stage with a bottom-heavy IMF. Such model is physically motivated by the fact that a sustained high star formation will bring the interstellar medium to a state of pressure, temperature and turbulence that can drastically alter the fragmentation of the gaseous component into small clumps, promoting the formation of low-mass stars. This toy model is in good agreement with the different observational constrains on massive elliptical galaxies, such as age, metallicity, alpha-enhancement, M/L, or the mass fraction of the stellar component in low-mass stars.

Top-heavy integrated galactic stellar initial mass functions in starbursts

Star formation rates (SFR) larger than 1000 Msun/ yr are observed in extreme star bursts. This leads to the formation of star clusters with masses > 10^6 Msun in which crowding of the pre-stellar cores may lead to a change of the stellar initial mass function (IMF). Indeed, the large mass-to-light ratios of ultra-compact dwarf galaxies and recent results on globular clusters suggest the IMF to become top-heavy with increasing star-forming density. We explore the implications of top-heavy IMFs in these very massive and compact systems for the integrated galactic initial mass function (IGIMF), which is the galaxy-wide IMF, in dependence of the star-formation rate of galaxies. The resulting IGIMFs can have slopes, alpha_3, for stars more massive than about 1 Msun between 1.5 and the Salpeter slope of 2.3 for an embedded cluster mass function (ECMF) slope (beta) of 2.0, but only if the ECMF has no low-mass clusters in galaxies with major starbursts. Alternatively, beta would have to decrease with increasing SFR >10 Msun/ yr such that galaxies with major starbursts have a top-heavy ECMF. The resulting IGIMFs are within the range of observationally deduced IMF variations with redshift.

THE FUNDAMENTAL GAS DEPLETION AND STELLAR-MASS BUILDUP TIMES OF STAR-FORMING GALAXIES

Stars do not form continuously distributed over star-forming galaxies. They form in star clusters of different masses. This nature of clustered star formation is taken into account in the theory of the integrated galactic stellar initial mass function (IGIMF) in which the galaxy-wide initial mass function (IMF) on galaxy-wide scales is calculated by adding all IMFs of young star clusters. For massive stars, the IGIMF is steeper than the universal IMF in star clusters and steepens with decreasing star formation rate (SFR) which is called the IGIMF effect. The current SFR and the total Hα luminosity of galaxies therefore scale nonlinearly in the IGIMF theory compared to the classical case in which the galaxy-wide IMF is assumed to be constant and identical to the IMF in star clusters. Here we apply for the first time the revised SFR–LHα relation on a sample of local volume star-forming galaxies with measured Hα luminosities. The fundamental results are: (1) the SFRs of galaxies scale linearly with the total galaxy neutral gas mass, (2) the gas depletion timescales of dwarf irregular and large disk galaxies are about 3 Gyr, implying that dwarf galaxies do not have lower star formation efficiencies than large disk galaxies, and (3) the stellar-mass buildup times of dwarf and large galaxies are only in agreement with downsizing in the IGIMF context, but contradict downsizing within the traditional framework that assumes a constant galaxy-wide IMF.

Diverging UV and Hα fluxes of star-forming galaxies predicted by the IGIMF theory

Although the stellar initial mass function (IMF) has only been directly determined in star clusters, it has been manifoldly applied on galaxy-wide scales. But taking the clustered nature of star formation into account the galaxy-wide IMF is constructed by adding all IMFs of all young star clusters leading to an integrated galactic initial mass function (IGIMF). The IGIMF is top-light compared to the canonical IMF in star clusters and steepens with decreasing total star formation rate (SFR). This discrepancy is marginal for large disc galaxies but becomes significant for Small Magellanic Cloud type galaxies and less massive ones. We here construct IGIMF-based relations between the total far- and near-ultraviolet luminosities of galaxies and the underlying SFR. We make the prediction that the Hα luminosity of star-forming dwarf galaxies decreases faster with decreasing SFR than the ultraviolet (UV) luminosity. This