Dust Mass Function: Evolution & Implications

• Dust Mass Function is a statistical measure quantifying the space density of galaxies as a function of their dust mass, derived from far-infrared/submillimeter observations.
• It links dust content with galaxy properties such as star formation rate, stellar mass, and ISM characteristics, revealing strong evolutionary trends over cosmic time.
• Methodologies involve SED fitting, modified Schechter functions, and stacking analyses to constrain dust production, destruction, and grain growth in evolving galaxies.

The dust mass function (DMF) is a fundamental statistical descriptor quantifying the space density of galaxies as a function of their dust mass. It serves as a key benchmark for models of galaxy evolution, chemical enrichment, and interstellar medium (ISM) processing, analogous to the role of the stellar mass function for the buildup of stars. Empirical determination of the DMF relies heavily on far-infrared (FIR) and submillimeter observations that trace the optically thin thermal dust emission reprocessed from starlight, most recently enabled by surveys such as Herschel-ATLAS. The DMF encodes the outcome of the integrated history of dust production (in stellar ejecta and ISM grain growth), destruction (supernova shocks, outflows), and the dynamics of gas reservoirs across cosmic time, and reveals critical constraints and challenges for theoretical dust and galaxy evolution models.

1. Definition and Quantification of the Dust Mass Function

The DMF, $\Phi_d(M)$, specifies the comoving number density of galaxies per unit (typically logarithmic) dust mass interval,

$$\Phi_d(M)\,dM = \text{number density of galaxies with dust mass in } [M,\,M+dM].$$

In practice, dust masses ($M_d$) are inferred by fitting the far-infrared/submillimeter spectral energy distribution (SED) of each galaxy with one or more modified blackbody (“grey-body”) components: $$M_d = \frac{S_{\nu,\text{obs}}\, D_L^2 (1+z) K}{\kappa_\nu\, B_\nu(T)}$$ where $S_{\nu,\text{obs}}$ is the observed flux at frequency $\nu$ (rest), $D_L$ is the luminosity distance, $K$ is the SED-derived $K$-correction, $\kappa_\nu$ is the dust mass absorption coefficient, $B_\nu(T)$ is the Planck function at the dust temperature $T$, and $z$ is the redshift. For cases lacking full SEDs, empirical conversions such as

$\log M_d \approx \log L_{250} - 16.47$

are employed, with $L_{250}$ the monochromatic luminosity at 250 μm (Dunne et al., 2010). The DMF is commonly parameterized by a Schechter function: $$\Phi_d(M) dM = \phi_d^*\,\Big(\frac{M}{M_d^*}\Big)^{\alpha_d}\,\exp\Big(-\frac{M}{M_d^*}\Big)\frac{dM}{M_d^*},$$ where $M_d^*$ is the “knee” or characteristic mass, $\alpha_d$ the faint-end slope, and $\phi_d^*$ the normalization (Clemens et al., 2013, Beeston et al., 2017, Pozzi et al., 2019).

2. Observed Evolution and Scaling Relations

Analysis of Herschel-ATLAS data for $z<0.5$ demonstrates rapid evolution in the high-mass end of the DMF over the past $\sim5$ Gyr: the most massive galaxies at $z\simeq0.4-0.5$ have dust masses $\sim 4$–$5\times$ higher than their present-day analogues (Dunne et al., 2010). Evolution is most pronounced for galaxies in the upper decile of the DMF (by $M_d$), with the integrated dust mass density, $\rho_d$, scaling as $\rho_d \propto (1+z)^{2.6\pm0.6}$ at $z<0.5$ (Beeston et al., 7 Nov 2024).

Derived dust-to-stellar mass ratios ($M_d/M_*$) are also strongly redshift-dependent, peaking at $z\sim0.3-0.4$ with values up to $\sim7\times10^{-3}$. These ratios decline toward $z=0$, tracing the well-established decrease in gas fractions and fueling capacity for star formation in massive galaxies.

The DMF is tightly linked to galaxy global properties:

• The dust-to-gas ratio scales with metallicity (for star-forming galaxies, $$\log(\mathrm{DGR}) \sim 2.45\,\log(Z/Z_\odot) -2.03$$) (Li et al., 2019).
• The SFR–dust mass relation follows $M_d \propto \mathrm{SFR}^{1.11\pm0.01}$, rooted in the Schmidt–Kennicutt exponent characterizing star formation efficiency; at high SFR, this relation bends, reflecting a maximum attainable dust mass set by chemical enrichment timescales and ISM processing (Hjorth et al., 2014).
• Robust correlations of $M_d$ with H I mass and with stellar mass are observed, with the dust-to-stellar mass ratio anti-correlated with $M_*$ (Clemens et al., 2013, Beeston et al., 2017).

3. Methodological Frameworks for DMF Measurement

Empirical DMFs are constructed using volume-limited or flux-limited, multi-band FIR-selected samples. Corrections for incompleteness, flux limits, and $K$-corrections are critical. Typical approaches include:

• Modified $1/V_\text{max}$ or Page–Carrera-type estimators, accounting for individual SED $K$-corrections and completeness in submillimeter, optical ID, and spectroscopy (Dunne et al., 2010, Beeston et al., 2017, Pozzi et al., 2019).
• Bivariate or multivariate techniques linking dust mass bins to FIR luminosity bins (using known luminosity functions), or copula-based methods exploiting joint distributions with stellar proxies (Andreani et al., 2018).
• Stacking analyses to recover average SEDs and dust properties below direct detection thresholds (Beeston et al., 7 Nov 2024).
• SED fitting codes (MAGPHYS, CIGALE, Stardust) to derive $M_d$, using energy balance and flexible treatment of cold and warm dust temperatures (Dunne et al., 2010, Traina et al., 12 Jul 2024).

Derived DMFs are frequently fitted with a Schechter function, characterized by $M_d^*$ (location of the “knee”), $\alpha$ (low-mass slope, steep for many DMFs: $\alpha\sim1.3$–$1.5$ at $z\sim0$ (Clemens et al., 2013, Beeston et al., 2017, Pozzi et al., 2019)), and $\phi^*$ (normalization). Current studies integrate down to $M_d\sim10^{4-6}\,M_\odot$ depending on sample depth, though uncertainty increases at the faint end.

4. Physical Implications and Constraints on Models

The observed rapid evolution of the DMF, especially at the high-mass end, presents significant challenges for chemical and dust evolution models. Standard closed-box models, utilizing Milky Way-like yields, ISM grain growth, and dust destruction efficiencies, systematically underpredict dust masses and dust-to-stellar mass ratios at $z\sim0.4-0.5$ (Dunne et al., 2010). Matching the observed DMF amplitude and evolution requires introducing:

• More top-heavy IMFs (increasing the yield $p$ of metals/dust per unit stellar mass).
• Enhanced mantle growth efficiencies in dense ISM regions or rapid dust grain accretion.
• Reduced dust destruction via supernovae or stronger ISM shielding.
• Moderately strong but not excessive dust/metal outflows, as indicated by observed halo dust content.

Moreover, the DMF directly traces the ISM gas mass evolution: high $M_d$ implies abundant molecular gas, which is the immediate reservoir for star formation. The observed simultaneous decline in SFR density, cold gas fraction, and DMF at $z<0.5$ supports a picture where exhaustion of the ISM through star formation and feedback governs both dust and stellar mass assembly (Dunne et al., 2010, Beeston et al., 2017).

5. Relation to Galaxy Type and the Broader Context

Splitting the DMF by morphological type shows that late-type (star-forming) galaxies overwhelmingly dominate the cosmic dust mass budget; the dust mass density in late-types exceeds that of early-types by nearly an order of magnitude (Beeston et al., 2017). The scaling of the late-type DMF with the galaxy stellar mass function (GSMF), offset by a factor $\sim8\times10^{-4}$, underscores the tight coupling between dust and stellar/gas assembly in disks.

Comparisons between local DMFs from different selection methods (FIR-selected vs. optically-selected) reveal shape differences, but the integrated dust density is broadly consistent after volume and selection corrections (Beeston et al., 7 Nov 2024). At higher redshift, direct comparisons to model predictions reveal substantial discrepancies: current cosmological hydrodynamical simulations and semi-analytic models respectively under- and overproduce the space density of high-$M_d$ galaxies, indicating that grain growth timescales and stellar condensation efficiencies remain poorly constrained (Beeston et al., 2017, Millard et al., 2020).

6. Performance Metrics, Systematics, and Future Directions

Determining the DMF with high accuracy requires careful attention to modeling systematics:

• Assumptions on $\kappa_\nu$ (the dust mass opacity coefficient) can shift $M_d$ and $\rho_d$ estimates by factors of two (Clemens et al., 2013).
• Uncertainties in the adopted dust temperature, particularly at high redshift where only the Rayleigh–Jeans tail is observed, affect the derived dust masses.
• Survey incompleteness, confusion at the faint end, and uncertainties in SED modeling strongly influence constraints on the low-mass slope $\alpha$ (Pozzi et al., 2019).

Forthcoming deep surveys (e.g., PRIMA, enhanced ALMA and submillimeter facilities) promise to constrain the DMF at both the faint end and high redshift ($z>2$), while multi-component SED modeling may improve decomposition of cold and warm dust contributions (Traina et al., 1 Sep 2025).

A major theoretical challenge persists: standard dust chemistry and ISM models cannot currently reproduce the combined normalization, slope, and redshift evolution of the observed DMF without invoking non-standard IMFs, rapid grain growth, altered dust destruction rates, or environment-dependent feedback.

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In summary, the dust mass function provides an essential census of the dust content of galaxies and its evolution, encoding key aspects of ISM physics, star formation fueling, and metal enrichment across cosmic time. Its rapid and strong evolution, particularly at the high-mass end, constitutes a stringent test for models of galaxy evolution and constrains the interplay between dust formation, destruction, and the cosmic decline in star formation (Dunne et al., 2010, Beeston et al., 2017, Clemens et al., 2013, Beeston et al., 7 Nov 2024). The DMF thus remains a critical observable for understanding both the buildup of baryonic structures and the lifecycle of metals and dust in the universe.