Star Formation Rate Density (SFRD)

• Star Formation Rate Density (SFRD) is defined as the stellar mass formed per unit comoving volume per unit time, measured in M⊙ yr⁻¹ Mpc⁻³.
• It is determined using observational tracers like [OII] emission, UV continuum, and IR data, combined with techniques such as nod-and-shuffle spectroscopy.
• The evolution of SFRD reveals a mass-dependent turnover and downsizing effects, providing essential constraints for galaxy evolution and feedback models.

Star Formation Rate Density (SFRD) quantifies the amount of stellar mass formed per unit comoving volume per unit time in the Universe, typically expressed in units of $M_\odot\,\mathrm{yr}^{-1}\,\mathrm{Mpc}^{-3}$. It is a fundamental metric for reconstructing the cosmic history of galaxy formation and evolution, integrally tied to the processes that regulate star formation across different mass regimes, epochs, and environments.

1. Observational Strategies for Measuring SFRD

Various tracers and survey methodologies are employed to estimate SFRD, each with specific selection effects, calibration requirements, and systematics. In the context of redshift $z\sim1$, efficient spectroscopic acquisition for faint, low-mass star-forming galaxies necessitates deep, targeted multi-object spectroscopy coupled to robust photometric pre-selection. The LDSS-3 instrument on the 6.5-m Magellan telescope, as implemented in the K-band selected sample of dwarf galaxies (22.5 < $K_\mathrm{AB}$ < 24.0, $8.4 < \log(M_*/M_\odot)<10$), leverages a custom band-limiting filter (7040–8010 Å) such that the [OII] $\lambda3727$ emission line is observable for $0.889<z\leq1.149$ sources (0902.3997). Nod-and-shuffle techniques are adopted for optimal sky subtraction at very faint flux levels, with emission-line detection achieved statistically (via image convolution with emission-line kernels and a ≥4.5σ threshold). Final line identification cross-validates spectroscopic and photometric redshifts. The overall calibration is verified by comparing measured [OII] fluxes against external samples (agreement within ∼30%).

For SFR estimation, the prescription directly relates [OII] luminosity (with correction for light lost outside extraction apertures and for dust extinction) to SFR using: $$\text{SFR} = 7.9 \times 10^{-42} \frac{L(\mathrm{[OII]})}{0.5} \quad [M_\odot\,\mathrm{yr}^{-1}]$$ where $L(\mathrm{[OII]})$ is in erg s$^{-1}$. The above assumes an [OII]$/\mathrm{H}\alpha=0.5$ ratio, extinction $A_\mathrm{H\alpha}=1$ mag, and a Baldry & Glazebrook (2003) IMF.

The cosmic SFRD is then determined using the $1/V_\mathrm{max}$ method: $$\rho_{\mathrm{SFR}} = \sum_{i} \frac{\mathrm{SFR}_i}{V_{\mathrm{max},i}}$$ with $V_\mathrm{max} \simeq 3.2 \times 10^4$ Mpc$^3$ reflecting the survey’s effective volume for each galaxy.

2. Mass-Dependent Behavior and Evidence for Turnover

Analysis of the SFRD as a function of stellar mass at $z\sim1$—covering $8.4 < \log(M_*/M_\odot) < 10$ using the ROLES and GDDS datasets—reveals a clear turnover in [OII] luminosity (hence SFRD) near $M_* \simeq 10^{10}\,M_\odot$. For $8.4 < \log(M_*/M_\odot) < 9.8$, the integrated SFRD is measured as: $$\rho_\mathrm{SFR} = (4.8 \pm 1.7) \times 10^{-3}\,M_\odot\,\mathrm{yr}^{-1}\,\mathrm{Mpc}^{-3}$$ This corresponds to a marked decline in the SFRD contribution from systems below $10^{10}\,M_\odot$ at this epoch; above this mass, massive galaxies dominate the global SFRD (0902.3997). The turnover signature persists using higher-mass, complementary spectroscopic samples (e.g., GDDS).

The [OII] luminosity function faint-end slope ($\alpha_\mathrm{faint} \sim -1.5$) at $z\sim1$ matches local SDSS values, validating consistency in the low-mass slope across cosmic time (Gilbank et al., 2010).

3. Evolutionary Trends: Downsizing and SFRD Temporal Evolution

Comparisons to local [OII] SFRD (SDSS legacy data) and broadband indicators (Spitzer 24μm, UV continuum, and SED-fitting) highlight the differential evolution of SFRD with mass. The SFRD in the lowest-mass galaxies evolves negligibly over cosmic time, while that in massive systems ($M_*\gtrsim10^{10}\,M_\odot$) declines sharply from $z\sim1$ to $z\sim0$ (0902.3997). This trend is emblematic of “downsizing,” wherein the locus of star formation shifts from high-mass galaxies at early epochs to lower-mass galaxies at late times.

Quantitative changes in the normalization of the SFRD-mass relation between $z\sim1$ and $z\sim0.1$ depend on the tracer: [OII]-derived SFRDs decrease by a factor ≈2.6, while UV-based SFRDs decrease by a factor ≈6, but the shape of the SFRD-mass distribution remains invariant, implying a uniform scaling down in SFR with time regardless of mass (Gilbank et al., 2010).

This uniformity in shape but evolution in normalization suggests that the simplest formulation of downsizing—interpreted as a moving peak in SFRD toward lower mass—does not fully describe the dataset. Rather, massive and less-massive galaxies alike reduce their SFR by comparable factors since $z\sim1$.

4. Cross-Indicator Comparisons and Systematics

Strong correlations but systematic offsets are observed between SFRD estimates based on nebular emission lines ([OII], Hα), UV continuum (SED fits), and mid-IR luminosities. Discrepancies arise from extinction corrections, sensitivity to dust-obscured star formation (especially for the most massive galaxies), metallicity, aperture effects, and IMF variations.

For instance, complete SFR census in high-mass galaxies requires IR tracers, as [OII] can underestimate the SFR by missing highly embedded regions (0902.3997). At low mass, these systematics are less severe as galaxies tend to be less dust-obscured, lending confidence to [OII]-based SFR estimators in the dwarf regime.

In the ROLES framework, SFRD uncertainties are further characterized by adopting both line-based and SED-based methods for the same sample and by performing empirical mass-dependent corrections (Gilbank et al., 2010). The overall factor-of-two divergence between SFRD estimators aligns with systematic differences seen in other contemporary $z\sim1$ studies.

5. Implications for Galaxy Evolution Models and Feedback Regimes

The measured turnover in SFRD at $M_* \sim 10^{10}\,M_\odot$ at $z\sim1$ indicates that the star formation regulatory mechanisms (e.g., supernova/AGN feedback, gas accretion) differ across mass scales. Below this threshold, the minor evolutionary change in SFRD with redshift implies that feedback and gas depletion do not catastrophically quench star formation in dwarfs—possibly because of ongoing accretion in low-density environments or lower efficiency of outflows.

These empirical constraints necessitate mass-dependent feedback implementations in theoretical models of galaxy formation and star formation histories, as well as explicit treatment of environmental and accretion processes.

6. Formulaic Framework for [OII]-Based SFRD Calculation

The translation from observed [OII] flux to an SFRD measurement is encapsulated by: $$\text{SFR (OII)} = 1.58 \times 10^{-41} L(\mathrm{[OII]})\quad [M_\odot\,\mathrm{yr}^{-1}]$$

$$\rho_{\mathrm{SFR}} = \sum_{i} \frac{\mathrm{SFR}_i}{V_{\mathrm{max},i}}$$

where [OII] fluxes are corrected for aperture losses and extinction. Volume limits per source are dictated by redshift and detection-filter constraints, ensuring completeness to an unobscured SFR limit.

7. Synthesis and Astrophysical Significance

Deep, K-band-selected spectroscopy at $z\sim1$ robustly establishes that the cosmic SFRD is mass-dependent and peaked at $M_* \sim 10^{10}\,M_\odot$ at this epoch; the contribution from less-massive (dwarf) galaxies decreases toward lower mass and does not show strong evolution relative to massive counterparts over $0

Multiwavelength, spectroscopically complete surveys employing matched selection and consistent methodology are essential for further constraining the evolution of low-mass star formation, quantifying dust-obscured activity, and informing hierarchical galaxy assembly frameworks. These measurements yield crucial boundary conditions for semi-analytic and hydrodynamical models of galaxy evolution, particularly with respect to the efficiency and universality of feedback at low masses and the mechanisms controlling “downsizing.”