Stellar Initial Mass Function: Origins and Implications

• The sIMF is the mass distribution of individual stars formed in a cluster, defined over finite mass ranges and typically modeled using broken power laws or a lognormal function.
• It plays a critical role in connecting star formation physics, such as core fragmentation and efficiency, to observable galactic properties like chemical enrichment and stellar demographics.
• Environmental conditions, multiplicity, and dynamical evolution influence the sIMF, affecting both local star clusters and the integrated galaxy-wide initial mass function.

The stellar initial mass function (sIMF) is the mass distribution of stars formed in a single star-formation event within a gravitationally bound molecular cloud clump, often identified with an embedded cluster. In differential form, it is written as $\xi(m)=dN/dm$, so that $dN=\xi(m)\,dm$, and its normalization over a star-formation event fixes both the number of stars and the total stellar mass formed. Because it enters the interpretation of unresolved stellar populations, galactic chemical enrichment, habitable-zone demographics, and black-hole growth, the sIMF is a fundamental link between star-birth physics and galaxy evolution (Jerabkova et al., 8 Sep 2025).

1. Definition, notation, and standard forms

The sIMF is defined over a finite mass interval. Typical lower limits are $m_{\min}\approx 0.01$–$0.08\,M_\odot$, spanning the substellar/stellar boundary, while typical upper limits are $m_{\max}\approx 100$–$150\,M_\odot$. Observations suggest a physical upper stellar mass limit $m_{\max *}\approx 150\,M_\odot$; objects apparently above this are plausibly merger products rather than direct IMF samples (Jerabkova et al., 8 Sep 2025). For a given star-formation event,

$$\int_{m_{\min}}^{m_{\max}} \xi(m)\,dm = N, \qquad \int_{m_{\min}}^{m_{\max}} m\,\xi(m)\,dm = M,$$

where $N$ is the number of stars and $M$ the stellar mass formed (Jerabkova et al., 8 Sep 2025).

Several parameterizations are standard. The Salpeter form for the high-mass regime is $dN=\xi(m)\,dm$0 with $dN=\xi(m)\,dm$1 for $dN=\xi(m)\,dm$2–$dN=\xi(m)\,dm$3, equivalent to $dN=\xi(m)\,dm$4; in this notation, $dN=\xi(m)\,dm$5 when the negative sign is written explicitly in $dN=\xi(m)\,dm$6 (Jerabkova et al., 8 Sep 2025). The canonical Kroupa form is a broken power law with $dN=\xi(m)\,dm$7 for $dN=\xi(m)\,dm$8 and $dN=\xi(m)\,dm$9 for $m_{\min}\approx 0.01$0 (Jerabkova et al., 8 Sep 2025). A Chabrier-like description uses a lognormal form below $m_{\min}\approx 0.01$1 matched to a power-law tail at higher mass (Jerabkova et al., 8 Sep 2025).

A central terminological distinction is that the sIMF refers to individual stars, not unresolved systems. The “system IMF” inferred from unresolved photometry differs from the individual-star IMF, particularly at low mass, and the difference has no direct meaning for birth conditions unless multiplicity is explicitly modeled (Kroupa et al., 2018). The same caution applies to brown dwarfs: the white paper argues that a separate brown-dwarf mass function is required, with a substellar slope $m_{\min}\approx 0.01$2, because the stellar and brown-dwarf IMFs overlap in mass but form via different channels (Jerabkova et al., 8 Sep 2025).

The parameterized forms are empirical compressions of a more complex formation problem. This suggests that the widely used canonical sIMF should be read as a regulated birth distribution rather than as a complete physical theory of star formation.

2. From filaments and cores to stellar masses

The mapping from gas structure to the sIMF is usually discussed through the prestellar core mass function (CMF), filament statistics, and the efficiency of converting cores into stars. In nearby metal-rich clouds, the prestellar CMF broadly resembles the IMF in shape and mass scale, often with a lognormal peak and a Salpeter-like high-mass tail (Jerabkova et al., 8 Sep 2025). By contrast, CO cloud and clump mass functions are shallower than Salpeter, whereas filament mass and line-mass functions are steeper and resemble the IMF high-mass tail. The white paper gives $m_{\min}\approx 0.01$3 for supercritical filaments with $m_{\min}\approx 0.01$4, and $m_{\min}\approx 0.01$5 above $m_{\min}\approx 0.01$6 (Jerabkova et al., 8 Sep 2025).

A commonly used first-order mapping is

$m_{\min}\approx 0.01$7

with a core-to-star efficiency $m_{\min}\approx 0.01$8–$m_{\min}\approx 0.01$9, consistent with loss to protostellar outflows (Jerabkova et al., 8 Sep 2025). In this picture, the IMF is a convolution of the CMFs of individual filaments with the filament line-mass distribution; higher-$0.08\,M_\odot$0 filaments form higher-mass cores, broadening the CMF and shifting its peak (Jerabkova et al., 8 Sep 2025). A statistical formulation calibrated on simulations found that sink masses are predominantly drawn from their parent bound-core reservoir with a characteristic dispersion of order one-third of the core mass, which preserves a close CMF–sIMF resemblance while broadening the low-mass tail (Chabrier et al., 2010).

The CMF-to-sIMF mapping is not universal across environments. In massive protoclusters, including ALMA-IMF targets, CMFs are reported to be top-heavy and to evolve with age; fragmentation below $0.08\,M_\odot$1 AU, variable core lifetimes, and multiplicity weaken any one-to-one mapping (Jerabkova et al., 8 Sep 2025). The near absence of subfragmentation in many Herschel cores suggests that CMFs retain predictive power in nearby low-mass regions, but the same inference is less secure in dense, massive systems (Jerabkova et al., 8 Sep 2025).

A complementary argument addresses the origin of the Salpeter slope itself. A hierarchical fragmentation model proposes an intrinsic clump-scale IMF with linear slope $0.08\,M_\odot$2, while the aggregate cluster IMF steepens toward the Salpeter value because the smallest star-forming clumps cannot form the highest-mass stars. In that model, Salpeter-like behavior arises when the lower stellar and clump mass limits overlap, $0.08\,M_\odot$3 (Oey, 2011). This suggests that the observed upper-mass slope may encode both local fragmentation physics and the mass hierarchy of the parent structure.

3. Multiplicity, dynamical evolution, and measurement biases

The sIMF is a birth function, whereas most observations sample a dynamically processed present-day mass function (PDMF). This distinction is methodological rather than semantic. Early gas expulsion from mass-segregated clusters preferentially unbinds low-mass stars, mass segregation modifies the census of massive stars through concentration and ejection, and two-body relaxation drives evaporation and radial mass-function gradients (Jerabkova et al., 8 Sep 2025). In clusters with top-heavy sIMFs, subsequent mass loss and expansion can be so large that survival depends on the fraction of mass in stars above $0.08\,M_\odot$4 (Jerabkova et al., 8 Sep 2025).

Multiplicity is a first-order correction. Binary and multiple fractions vary with mass and environment and therefore alter the core-to-star mapping and the inference of $0.08\,M_\odot$5 from star counts. The white paper summarizes birth-population binary fractions exceeding $0.08\,M_\odot$6 at $0.08\,M_\odot$7 Myr in low-density embedded clusters, about $0.08\,M_\odot$8 at $0.08\,M_\odot$9 Myr in open clusters born at densities $m_{\max}\approx 100$0, and about $m_{\max}\approx 100$1 by $m_{\max}\approx 100$2 Myr in globular clusters born at densities $m_{\max}\approx 100$3 (Jerabkova et al., 8 Sep 2025). Kroupa and Jerabkova likewise emphasize that the birth binary fraction is approximately unity across stellar masses and that cluster dynamical evolution then imprints population-dependent unresolved-binary biases on observed mass functions (Kroupa et al., 2018).

Unresolved companions flatten or steepen inferred slopes depending on the mass range and adopted mass–luminosity relation. The white paper notes that theoretical mass–luminosity relations misrepresent the sharp derivative near the radiative/convective transition at $m_{\max}\approx 100$4, generating spurious IMF features and biased slopes; empirically gauged relations are therefore required (Jerabkova et al., 8 Sep 2025). This point has become sharper in Gaia-based work, where the local IMF can be recovered only through full forward modeling of unresolved binaries, metallicity distributions, star-formation history, and selection effects (Sollima, 2019).

These corrections matter for the universality debate. Some apparent IMF variations can be produced by unmodeled multiplicity, gas-expulsion history, or stellar dynamics. A plausible implication is that the strongest claims of sIMF variation should come from mono-age populations with tailored multiplicity and dynamical corrections, rather than from raw luminosity functions.

4. Environmental variation and the universality problem

The white paper’s synthesis is explicit: low metallicity and high birth density favor top-heavy sIMFs, while low metallicity also tends to produce bottom-light behavior below $m_{\max}\approx 100$5 and high metallicity tends to produce bottom-heavy low-mass IMFs (Jerabkova et al., 8 Sep 2025). For the low-mass slopes, it cites the trend

$m_{\max}\approx 100$6

and for the high-mass end it argues that three independent lines of evidence from ultra-compact dwarfs and globular clusters converge on $m_{\max}\approx 100$7 (Jerabkova et al., 8 Sep 2025).

The theoretical link is usually expressed through fragmentation scales and thermodynamics. In the isothermal approximation,

$m_{\max}\approx 100$8

so higher temperatures or lower densities imply larger characteristic masses (Jerabkova et al., 8 Sep 2025). The white paper also emphasizes that in supercritical filaments of common width $m_{\max}\approx 100$9 pc, the effective Bonnor–Ebert mass increases with line mass, while protostellar radiative heating, opacity limits to fragmentation, cosmic rays, and magnetic fields regulate the low-mass scale (Jerabkova et al., 8 Sep 2025).

Observed systems do not all point in the same direction, but they do define a structured pattern. In 30 Dor/R136, once massive-star ejections are accounted for, the high-mass slope is reported as $150\,M_\odot$0, i.e. top-heavier than canonical (Jerabkova et al., 8 Sep 2025). In the young Galactic Center cluster, Bayesian inference from the Kp luminosity function gives $150\,M_\odot$1, flatter than Salpeter and consistent with a factor of $150\,M_\odot$2 fewer X-ray emitting pre-main-sequence stars than expected for a Salpeter IMF (Lu et al., 2013). In the Small Magellanic Cloud outskirts, over $150\,M_\odot$3–$150\,M_\odot$4, the IMF is well fit by a single power law with slope $150\,M_\odot$5 in the paper’s sign convention and shows no turnover in that interval (Kalirai et al., 2012).

At the low-mass end in metal-poor dwarf systems, the picture is mixed rather than null. Deep HST analyses of Reticulum II, Ursa Major II, Triangulum II, and Segue 1 reject many IMF choices but still permit Milky Way-like low-mass IMFs in all four systems; Ursa Major II appears more bottom heavy, although contamination from two known background galaxy clusters complicates that inference (Filion et al., 2024). By contrast, in the Solar neighbourhood, a star-counting study of $150\,M_\odot$6 M dwarfs reports that present-day populations become increasingly bottom-heavy with metallicity in the range $150\,M_\odot$7, while early-time populations contain fewer low-mass stars and show little metallicity trend over the same interval (Li et al., 2023).

Massive early-type galaxies furnish an indirect but influential line of evidence. A sample of $150\,M_\odot$8 ETGs from SPIDER shows a monotonic increase of IMF-sensitive Na I 8190 Å, TiO1, and TiO2 with central velocity dispersion, with low-$150\,M_\odot$9 systems better fit by Kroupa/Chabrier-like IMFs and high-$m_{\max *}\approx 150\,M_\odot$0 systems requiring bottom-heavy IMFs that exceed Salpeter in a unimodal parameterization above $m_{\max *}\approx 150\,M_\odot$1 (Ferreras et al., 2012). Against these environmental trends stands an important control sample: in 27 old Galactic globular clusters, cluster-to-cluster PDMF differences over $m_{\max *}\approx 150\,M_\odot$2–$m_{\max *}\approx 150\,M_\odot$3 are reproduced by a universal Kroupa-like IMF plus two-body relaxation, without requiring distinct birth IMFs (Leigh et al., 2012).

Taken together, these results do not support a strictly universal sIMF. They support a weaker statement: near-canonical behavior is common in Milky Way-like conditions, but low-$m_{\max *}\approx 150\,M_\odot$4, high-density, starburst, nuclear, and some high-$m_{\max *}\approx 150\,M_\odot$5 environments depart from it in systematic directions.

5. From the sIMF to the galaxy-wide IMF

The sIMF and the galaxy-wide IMF are not identical objects. In IGIMF theory, the galaxy-wide IMF is

$m_{\max *}\approx 150\,M_\odot$6

where $m_{\max *}\approx 150\,M_\odot$7 is the embedded-cluster mass function, integrated over a star-formation epoch $m_{\max *}\approx 150\,M_\odot$8 Myr (Jerabkova et al., 8 Sep 2025). The ECMF is taken as a power law truncated at $m_{\max *}\approx 150\,M_\odot$9, with $$\int_{m_{\min}}^{m_{\max}} \xi(m)\,dm = N, \qquad \int_{m_{\min}}^{m_{\max}} m\,\xi(m)\,dm = M,$$0, and the framework imposes a deterministic $$\int_{m_{\min}}^{m_{\max}} \xi(m)\,dm = N, \qquad \int_{m_{\min}}^{m_{\max}} m\,\xi(m)\,dm = M,$$1–$$\int_{m_{\min}}^{m_{\max}} \xi(m)\,dm = N, \qquad \int_{m_{\min}}^{m_{\max}} m\,\xi(m)\,dm = M,$$2 relation under optimal sampling, $$\int_{m_{\min}}^{m_{\max}} \xi(m)\,dm = N, \qquad \int_{m_{\min}}^{m_{\max}} m\,\xi(m)\,dm = M,$$3 (Jerabkova et al., 8 Sep 2025).

This construction yields immediate consequences. Low-SFR galaxies form only low-mass clusters and therefore produce top-light galaxy-wide IMFs with few or no O stars, a regime described as H$$\int_{m_{\min}}^{m_{\max}} \xi(m)\,dm = N, \qquad \int_{m_{\min}}^{m_{\max}} m\,\xi(m)\,dm = M,$$4-dark star formation (Jerabkova et al., 8 Sep 2025). Modern IGIMF formulations further allow each cluster to have its own $$\int_{m_{\min}}^{m_{\max}} \xi(m)\,dm = N, \qquad \int_{m_{\min}}^{m_{\max}} m\,\xi(m)\,dm = M,$$5 before convolution with the ECMF, which the white paper argues helps reproduce H$$\int_{m_{\min}}^{m_{\max}} \xi(m)\,dm = N, \qquad \int_{m_{\min}}^{m_{\max}} m\,\xi(m)\,dm = M,$$6 versus UV discrepancies, mass–metallicity relations, and chemical constraints in ellipticals and bulges (Jerabkova et al., 8 Sep 2025). A simulation study of cosmic IGIMF evolution likewise finds that the high-mass IGIMF slope becomes steeper for $$\int_{m_{\min}}^{m_{\max}} \xi(m)\,dm = N, \qquad \int_{m_{\min}}^{m_{\max}} m\,\xi(m)\,dm = M,$$7–$$\int_{m_{\min}}^{m_{\max}} \xi(m)\,dm = N, \qquad \int_{m_{\min}}^{m_{\max}} m\,\xi(m)\,dm = M,$$8, flatter for $$\int_{m_{\min}}^{m_{\max}} \xi(m)\,dm = N, \qquad \int_{m_{\min}}^{m_{\max}} m\,\xi(m)\,dm = M,$$9–$N$0, and steeper again beyond $N$1, with sensitivity to the ECMF slope $N$2, $N$3, and SFR (Chattopadhyay et al., 2014).

Observational probes of galaxy-wide IMF variability include gravitational lensing, stellar and gas kinematics, and spectral diagnostics sensitive to dwarf-to-giant ratios, notably Na I 8190 Å, the Ca II triplet, the FeH Wing–Ford band, and TiO indices (Jerabkova et al., 8 Sep 2025). These methods generally indicate bottom-heavy central IMFs in massive early-type galaxies and top-heavy behavior in starbursts and low-metallicity systems (Jerabkova et al., 8 Sep 2025). On still larger scales, a multi-messenger constraint combining cosmic core-collapse supernova rates with FUV/IR luminosity densities finds that the cosmic-average high-mass IMF slope at $N$4 lies in the range $N$5–$N$6 at $N$7 confidence, consistent with Salpeter and with no significant redshift evolution detected within current uncertainties (Aoyama et al., 2021).

The interpretation of this hierarchy depends on sampling. The white paper adopts a shared framework in which the sIMF is a physically regulated, optimally sampled distribution rather than a purely stochastic probability density (Jerabkova et al., 8 Sep 2025). A recent variational argument derives the power-law sIMF from a maximum-entropy principle with a fragmentation constraint and uses the tight $N$8–$N$9 relation as evidence against large Poisson scatter from stochastic sampling (Gjergo et al., 28 Jan 2026). This suggests that gwIMF variability is not merely a bookkeeping effect but an expected consequence of clustered star formation plus environment-dependent clump physics.

6. Consequences, observational strategy, and open questions

A variable sIMF has direct consequences for chemical enrichment, mass-to-light ratios, remnant production, and black-hole growth. The white paper notes that bottom-heavy low-mass IMFs increase $M$0 through dwarf stars, while top-heavy IMFs can also increase $M$1 through remnants; interpreting dynamical or lensing $M$2 therefore requires explicit accounting of remnants and possible dark matter (Jerabkova et al., 8 Sep 2025). It also argues that top-heavy phases in dense, high-redshift environments increase the production of massive-star remnants and can facilitate rapid supermassive black-hole growth (Jerabkova et al., 8 Sep 2025).

The same framework reshapes stellar archaeology. Chemical IMF indicators involving $M$3, CNO isotopes, Zn, and Mn are powerful, but the white paper cautions that they are entangled with uncertain stellar yields, explosion physics, mixing, rotation, and inhomogeneous enrichment (Jerabkova et al., 8 Sep 2025). Hence IMF inference from abundance ratios should be combined with photometric and spectroscopic diagnostics rather than treated as standalone proof (Jerabkova et al., 8 Sep 2025).

The observational program proposed for the next decade is correspondingly multi-scale. The white paper recommends combining IMF-sensitive indices such as Na I, FeH, TiO, and Ca II with dynamical and lensing constraints; using Gaia-quality astrometry with tailored Lutz–Kelker and Malmquist corrections; calibrating empirically gauged mass–luminosity relations across metallicity and age; and using ALMA and JWST to measure CMFs, filament mass functions, core lifetimes, and fragmentation scales (Jerabkova et al., 8 Sep 2025). A complementary white paper emphasizes that JWST, Roman, and thirty-meter telescopes should enable direct star counts to $M$4 in young massive clusters across the Milky Way and Local Group, breaking degeneracies between lognormal and broken-power-law low-mass forms and directly measuring both the IMF peak and high-mass slope in extreme environments (Jr. et al., 2019).

Several questions remain explicitly open. The CMF-to-sIMF mapping in massive protoclusters is time-dependent and environment-dependent; the physical upper mass $M$5 and its possible variation at extremely low metallicity are unsettled; chemical IMF indicators remain degenerate; and mass–luminosity systematics near $M$6 remain a critical limitation (Jerabkova et al., 8 Sep 2025). The controversy is therefore no longer well framed as a binary choice between “universal” and “non-universal.” The more precise issue is which parts of the sIMF are stable under Milky Way-like conditions, which respond to metallicity and density, and how those clump-scale responses propagate into the galaxy-wide IMF.

In current usage, the sIMF is best understood as a clump-scale birth distribution whose observed form is shaped by fragmentation physics, thermodynamics, multiplicity, and cluster dynamics, and whose galaxy-scale consequences emerge only after convolution over the embedded-cluster population (Jerabkova et al., 8 Sep 2025).