Stellar Initial Mass Function

• Stellar Initial Mass Function (IMF) is a statistical distribution detailing the masses of stars at formation, modeled by broken power-law and lognormal functions.
• IMF studies employ advanced observational techniques and corrections for multiplicity, sampling biases, and metallicity to derive accurate star formation histories.
• Variations in the IMF across environments impact feedback, chemical enrichment, and galaxy evolution, making it a cornerstone of astrophysical research.

The stellar initial mass function (IMF) is a fundamental statistical distribution specifying the number of stars formed in a single star formation event as a function of their initial mass. Formally, it is expressed as $dN = \xi(m) dm$, where $\xi(m)$ is the IMF and $m$ denotes the stellar mass. The IMF directly influences the feedback, chemical enrichment, and baryonic cycle in galaxies, shaping virtually all subsequent astrophysical processes from the birth of stars to the evolution of galaxies and the Universe. Its functional form, physical origin, dependence on environment, and observational determination have been central topics of debate and research in astrophysics for decades.

1. Concept and Mathematical Formulation

The definition of the IMF encompasses the mass distribution of stars at the time they are formed—immediately after cessation of accretion, before significant mass loss. The canonical form in the field population of the solar neighborhood is the Salpeter power law: $\xi(m) \propto m^{-\alpha}$, with $\alpha \simeq 2.35$ for $m\gtrsim1\,M_\odot$ (Kroupa et al., 2019). More sophisticated descriptions introduce multiple power-law breaks or a lognormal component at low masses. For example, the contemporary IMF is often parametrized as:

$$\xi(m) = \begin{cases} C_1 m^{-\alpha_1}, & m < m_{\text{break}} \ C_2 m^{-\alpha_2}, & m \geq m_{\text{break}} \end{cases}$$

where empirical constraints for the solar neighborhood include $\alpha_1 \sim 0.8$, $\alpha_2 \sim 2.1$, and $m_{\text{break}}\sim0.4\,M_\odot$ (Wang et al., 15 Jun 2025). Alternatively, the lognormal form is used for the low-mass regime:

$$\xi(\ln m) \propto \exp\left[-\frac{(\ln m - \ln m_c)^2}{2\sigma^2}\right]$$

with transition to a power law at high mass (Paladini et al., 2019, Gabrielli et al., 13 Sep 2024). The IMF is a "probability density distribution function" in some physical interpretations, but recent work also suggests an "optimal sampling" scenario under regulation by the physical conditions of the forming region (Kroupa et al., 9 Oct 2024).

2. Physical Origin and Theoretical Frameworks

The IMF’s shape is determined by the interplay of gravitational collapse, supersonic turbulence, thermal and magnetic pressure, and additional microphysical processes. Gravo-turbulent fragmentation is a key mechanism where supersonic turbulence in molecular clouds generates a lognormal density distribution, resulting in clumps whose mass reservoir for star formation is set by the local Jeans mass:

$M_J \propto c_s^3 / \sqrt{G^3 \rho}$

with $c_s$ the sound speed and $\rho$ the density (Hennebelle et al., 10 Apr 2024). The high-mass end is determined primarily by turbulent fragmentation: the density PDF’s power-law tail dominates where gravity overwhelms turbulent and magnetic support, producing the classical power-law IMF. The low-mass peak is regulated by dust opacity and molecular physics: as densities increase, dust opacity quenches cooling, leading to the formation of the first hydrostatic core and a well-defined minimum fragmentation mass ($M_{\text{peak}}\sim10\,M_{\text{FHSC}}$) that depends only weakly on environment (Hennebelle et al., 10 Apr 2024). Theoretical and numerical studies confirm that the interplay of these processes naturally produces the observed IMF shape with high-mass power-law slope and a quasi-universal turnover (Veltchev et al., 2010, Offner et al., 2013, Hennebelle et al., 10 Apr 2024).

3. Observational Determination and Challenges

The extraction of the IMF from observed populations requires careful accounting for observational and astrophysical biases. Key corrections include:

• Unresolved multiplicity: A large fraction of stars form in binary or multiple systems. If not accounted for, unresolved binaries inflate stellar counts at higher luminosity and bias the inferred IMF, especially at the low-mass end. Modern analyses model the binary formation process and dynamical evolution to infer the true IMF from the "system IMF" (Wang et al., 15 Jun 2025, Kroupa et al., 2018).
• Sampling and dynamical evolution: Early gas expulsion, mass segregation, and dynamical ejections can alter the present-day mass function of clusters relative to the original IMF. Radial gradients in clusters must be corrected for (Jerabkova et al., 8 Sep 2025).
• Metallicity and star formation history: The conversion from luminosity function to mass function depends on metallicity, age, and extinction. Bayesian or forward-modelling approaches combine stellar evolution models, metallicity distributions, and detailed color-magnitude diagram (CMD) fitting to derive robust IMFs (Wang et al., 15 Jun 2025, Sollima, 2019).

Summing over the above, the current best estimate for the solar neighborhood IMF, carefully accounting for multiplicity and biases, is a broken power law with $\alpha_1 = 0.81^{+0.06}_{-0.05}$, $\alpha_2 = 2.12^{+0.04}_{-0.04}$, and $m_{\text{break}} = 0.41^{+0.01}_{-0.01}\,M_\odot$ with binary fraction $\sim25$\% (Wang et al., 15 Jun 2025).

4. Environmental Dependencies and Variation

The universality of the IMF remains debated. Multiple studies across the Milky Way and external galaxies provide evidence for both invariance and systematic variation. Key findings include:

• Star Formation Rate (SFR) and Surface Density: Galaxies with higher SFR or SFR surface density show flatter IMF slopes (i.e., "top-heavy"—more massive stars formed) (Gunawardhana et al., 2011). Quantitatively, $$\alpha \approx 0.36\,\log\langle\mathrm{SFR}\rangle - 2.6$$.
• Metallicity: The low-mass slope increases (becomes more bottom-heavy) with increasing metallicity, particularly for dynamically "cold" disc stars in the Milky Way (Li et al., 2023). At low metallicities ([Fe/H]$\lesssim-2$), a top-heavy IMF is favored in early halo populations (Suda et al., 2013).
• Galaxy Type and Compactness: Bottom-heavy IMFs—an excess of low-mass stars—are reported in the central regions of massive early-type galaxies, with the IMF mismatch parameter (ratio of observed to Milky Way M/L) positively correlated with velocity dispersion ($\sigma$), [Mg/Fe], [Z/H], and stellar surface density (Gu et al., 2021, Conroy et al., 2012).
• Cosmic Evolution: The cosmic mean high-mass IMF slope shows no significant redshift evolution out to $z\approx 2$; constraints from supernova and LGRB rates yield $\alpha\sim 2.5$ (Gabrielli et al., 13 Sep 2024, Aoyama et al., 2021).
• Star-forming Environment: In extreme environments (e.g., Galactic Center, starbursts, ultra-faint dwarfs), both top-heavy and bottom-heavy IMFs are observed, interpreted as a function of density, turbulence, metallicity, and feedback (Lu et al., 2013, Jr. et al., 2019, Kroupa et al., 2018).

5. The IMF–Core Mass Function Connection and Galaxy-Wide IMF

The functional similarity between the stellar IMF and the prestellar core mass function (CMF) found in molecular clouds has led to the view that the IMF is closely inherited from initial fragmentation (Paladini et al., 2019, Offner et al., 2013, Jerabkova et al., 8 Sep 2025). Empirically, the CMF mirrors the IMF but is offset to higher masses by a factor $\sim3$, reflecting a core-to-star efficiency of $\sim30\%$. The mapping is complicated by multiplicity—each core may spawn multiple stars—so the core-to-star correspondence is not strictly one-to-one.

When integrating star formation over a galaxy, the "galaxy-wide IMF" (gwIMF) or "integrated galactic IMF" (IGIMF) emerges. This is calculated as an average over all stellar IMFs (sIMFs) from individual molecular cloud clumps or embedded clusters (Kroupa et al., 9 Oct 2024, Jerabkova et al., 8 Sep 2025):

$$\xi_{\mathrm{IGIMF}}(m; t) = \int_{M_{\mathrm{ecl,min}}}^{M_{\mathrm{ecl,max}}(\psi(t))} \xi(m \leq m_{\max}(M_{\mathrm{ecl}}):Z)\, \xi_{\mathrm{ecl}}(M_{\mathrm{ecl}})\, dM_{\mathrm{ecl}}$$

where $m_{\max}(M_{\mathrm{ecl}})$ sets the upper stellar mass that can form in a cluster of mass $M_{\mathrm{ecl}}$. This formalism predicts that the gwIMF can be steeper or "top-light" in low-SFR galaxies or more top-heavy in starbursts, reflecting the mass distribution of the parent clusters (Kroupa et al., 9 Oct 2024, Jerabkova et al., 8 Sep 2025).

6. Impact of Multiplicity, Feedback, and Dynamical Processes

Stellar multiplicity (binaries and higher-order systems) has a profound effect on the observed IMF. Neglecting unresolved companions leads to systematic underestimation of low-mass stars and distortion of the derived IMF slope and break position (Wang et al., 15 Jun 2025, Kroupa et al., 2018). At birth, most stars reside in binaries; dynamical evolution (disruption and hardening) modifies the binary population over time.

Protostellar outflows and feedback mechanisms (e.g., jets, winds) contribute additional regulation, both limiting the star formation efficiency and shifting the characteristic mass by unbinding and redistributing gas within clumps (Hennebelle et al., 10 Apr 2024, Veltchev et al., 2010).

Cluster dynamical evolution—gas expulsion, mass segregation, two-body relaxation, and ejections—all reshape the observable stellar mass function (SMF) relative to the IMF. Mass loss mechanisms disproportionately affect low-mass stars, leading to observed differences between young clusters, field populations, and globular clusters (Jerabkova et al., 8 Sep 2025, Kroupa et al., 2018).

7. Future Directions and Unresolved Issues

Progress is anticipated from large, high-precision datasets (e.g., Gaia DR3, spectroscopic surveys), high-resolution near-IR integral-field spectroscopy, and the next generation of 30-meter class telescopes. Improved mapping of the IMF across a greater volume, and in varied environments, will allow direct investigation of its environmental dependencies (Jr. et al., 2019, Jerabkova et al., 8 Sep 2025).

Critical outstanding questions include:

• How universal is the IMF across cosmic time and different environments?
• What physical parameters (e.g., Mach number, metallicity, density, magnetic field) tightly regulate the IMF’s shape?
• Can the relation between the CMF and IMF be generalized across all star-forming regions, including those forming massive or metal-poor stars?
• What is the precise functional form and physical motivation for the break and turnover masses?

The IMF sets the foundation for models of galaxy evolution, star and planet formation efficiencies, and the mass-to-light ratios required for dynamical mass determinations. Future theoretical and observational work targeting IMF variation, the connection to cloud fragmentation and feedback, and robust correction for observational biases will continue to refine our understanding of this fundamental astrophysical function.