---
title: Cosmic Star Formation Rate Density
url: https://www.emergentmind.com/topics/cosmic-star-formation-rate-density-sfrd
type: topic
---

# Cosmic Star Formation Rate Density

Cosmic star formation rate density (SFRD) is the comoving rate at which the Universe forms stars, usually expressed as a function of redshift or cosmic time. It is a central summary statistic of galaxy evolution because it links luminosity functions, stellar-mass build-up, metal production, dust-obscured activity, and, at the highest redshifts, reionization-era star formation. The canonical multiwavelength synthesis places the maximum of the SFRD near “cosmic noon,” but recent radio, submillimeter, generative-model, tomographic, and cosmological reconstructions show that the precise peak redshift, amplitude, and high-redshift decline remain tracer- and model-dependent [1403.0007] [2509.20430] [2604.17660].

## 1. Definition and mathematical representations

In catalog-based analyses, the SFRD is the sum of galaxy star-formation rates divided by comoving volume. For submillimeter galaxies (SMGs), one explicit estimator is
$$
\mathrm{SFRD}_{\mathrm{SMG}}(z)=\frac{1}{V}\sum_{i\in \mathrm{SMGs}} \mathrm{SFR}_i ,
$$
while in the generative-model analysis of COSMOS2020 galaxies the redshift-bin estimator is
$$
\Psi_b=\frac{N f_b}{V_{{\rm co},b}}\,\frac{1}{N_b}\sum_{n=1}^{N_b}\psi_n ,
$$
with the sum taken over accepted mock galaxies in bin $b$ [2501.19327] [2509.20430].

Luminosity-function methods instead write the SFRD as an integral over a tracer-dependent luminosity function,
$$
\mathrm{SFRD}(z)=\int \Phi(L,z)\,\mathrm{SFR}(L)\,d\log_{10}L ,
$$
after specifying an observational calibration from luminosity to star-formation rate. This is the standard construction in radio, UV, IR, and H$\alpha$ work [2311.08975] [2204.04167] [1610.03441].

Several analytic forms are used to summarize the redshift evolution. A widely adopted empirical fit is the Madau–Dickinson form,
$$
\psi(z)=0.015\,\frac{(1+z)^{2.7}}{1+\left[(1+z)/2.9\right]^{5.6}}
\quad \left[\mathrm{M}_\odot\,\mathrm{yr}^{-1}\,\mathrm{Mpc}^{-3}\right] ,
$$
which was designed to capture the rise, turnover, and late decline of the global history [1403.0007]. A more physically motivated cosmological fit uses the Hernquist–Springel prescription,
$$
\dot{\rho}_\star(z)=\dot{\rho}_{\star,0}\,
\frac{\chi^2}{1+\alpha(\chi-1)^3\exp(\beta\chi^{7/4})},
\qquad
\chi=\left(\frac{H(z)}{H_0}\right)^{2/3},
$$
thereby making the SFRD explicitly cosmology-dependent [2604.17660]. A separate phenomenological study argues that the observed CSFRD can be described by only two parameters and a function that has the form of a Gamma distribution,
$$
\mathrm{CSFRD}(t)=\frac{b^a}{\Gamma(a)}\,t^{a-1}e^{-bt},
$$
with the parameters connected to the star formation rate depletion time and cosmic baryonic gas density [2107.02733].

## 2. Measurement strategies across wavelengths and methodologies

The observational route to the SFRD depends on tracer physics. UV emission directly traces young, massive stars, but dust attenuation is a first-order systematic. IR emission is an excellent probe of obscured star formation, especially in massive dusty systems, while H$\alpha$ provides an emission-line census of lower-SFR galaxies where mild obscuration occurs. Radio wavelengths offer a dust-unbiased tracer of the total SFR, provided AGN contamination and the IR–radio calibration are handled consistently [1610.03441] [2311.08975] [1703.09724].

Classical SFRD compilations are built by integrating dust-corrected UV and IR luminosity functions over redshift [1403.0007]. Radio studies refine this strategy by fitting analytic radio luminosity functions directly to source catalogs with completeness corrections and AGN rejection or subtraction. In the VLA-COSMOS and GOODS-N analyses, the radio luminosity function is fit with modified Schechter or LADE-type models, and the SFRD follows from integrating the fitted function after applying a redshift-dependent IR–radio conversion [2311.08975] [2202.00019]. One recent radio study emphasizes that fitting the luminosity function directly to the data, rather than to binned $1/V_{\max}$ points, reduces bias in the inferred SFRD [2311.08975].

Other methods depart from luminosity-function fitting altogether. The pop-cosmos framework trains a score-based diffusion model on 26-band photometry of $\sim420{,}000$ COSMOS2020 galaxies, generates mock catalogs with 16 SPS parameters, and computes the SFRD by directly integrating individual galaxy SFRs. The stated advantage is that this approach avoids constructing and fitting a parametric luminosity function and evades extrapolation below observational limits, dust corrections, and SFR conversion factors associated with luminosity-function methods [2509.20430]. At low redshift, fossil-record analyses reconstruct star-formation histories of nearby galaxies with delayed-$\tau$ models and infer the cosmic SFRD, sSFR, and stellar-mass density from spatially resolved stellar populations [1802.10118].

Tomographic and intensity-mapping methods extend the observable. Cross-correlation of the cosmic infrared background with KiDS galaxy samples was detected at $43\sigma$ and modeled with a halo framework to recover the SFRD up to $z\sim1.5$, or to $z=4$ when external SFRD measurements are added [2204.01649]. CO intensity mapping with one-point $P(D)$ statistics was proposed as a high-redshift route to the SFRD at $z\simeq2$–$7$; for pessimistic model uncertainty the forecast error is of order $50\%$, while improved CO–SFR calibration yields roughly $5$–$10\%$ precision [1507.06304].

## 3. Global evolutionary history and the location of the peak

The canonical synthesis by Madau and Dickinson places the SFRD peak approximately $3.5$ Gyr after the Big Bang, at $z\sim1.9$, followed by an exponential decline with an e-folding timescale of $3.9$ Gyr. In that framework, half of the stellar mass observed today was formed before $z=1.3$, about $25\%$ formed before the SFRD peak, another $25\%$ formed after $z=0.7$, and less than $\sim1\%$ of today’s stars formed during the epoch of reionization [1403.0007].

Recent measurements do not yield a single unique peak. The pop-cosmos generative reconstruction finds that the SFRD peaks at $z=1.3\pm0.1$ with peak value $0.08\pm0.01~\mathrm{M}_\odot\,\mathrm{yr}^{-1}\,\mathrm{Mpc}^{-3}$, about $\Delta z\approx0.6$ later than Madau and Dickinson, with a broader, later maximum and a flatter high-redshift decline [2509.20430]. The tomographic CIB–galaxy analysis, when combined with external SFRD measurements, yields a peak SFRD of $0.09^{+0.003}_{-0.004}\,\mathrm{M}_\odot\,\mathrm{yr}^{-1}\,\mathrm{Mpc}^{-3}$ at $z=1.74^{+0.06}_{-0.02}$, corresponding to a lookback time of $10.05^{+0.12}_{-0.03}$ Gyr [2204.01649].

Radio reconstructions tend to place the maximum somewhat later than the canonical UV+IR fit or to broaden it into a plateau, but not uniformly. The COSMOS-XS analysis finds that the radio-based SFRD rises steeply out to $z\sim1.8$ and then declines more rapidly than previous radio-based estimates [2204.04167]. A separate radio luminosity-function study concludes that the SFRD peaks between $z=2$ and $z=2.5$ and falls more rapidly toward high redshift once density evolution is included [2311.08975]. By contrast, the GOODS-N radio-selected analysis reports a rise up to $z\sim2$ and then an almost flat plateau up to $z\sim3.5$ [2202.00019].

At the modeling end, cosmological reconstruction with SFRD data compiled over $z\in[0,15]$ gives a robust peak at
$$
z_{\rm peak}=2.600^{+0.114}_{-0.087}
$$
within $\Lambda$CDM, with similar values in $w$CDM [2604.17660]. Taken together, these results indicate that the existence of a broad “cosmic noon” is secure, whereas the precise peak location and width remain sensitive to tracer choice, dust treatment, luminosity-function parameterization, and the adopted mapping between astrophysical and cosmological parameters.

## 4. Dust-obscured star formation and high-redshift revisions

A major current issue is how much of the SFRD is missed by rest-UV selection. The ASPIRE JWST+ALMA program provides a spectroscopically complete census of dusty star-forming galaxies at $z=4$–$6$ over $\sim35$ arcmin$^2$ and measures
$$
\log\left[\rho_{\rm SFR,IR}/(\mathrm{M}_\odot\,\mathrm{yr}^{-1}\,\mathrm{Mpc}^{-3})\right]
= -1.52^{+0.14}_{-0.13}.
$$
It further concludes that the majority, $(66\pm7)\%$, of cosmic star formation at $z\sim5$ is still obscured by dust, and that the IR luminosity function flattens toward the faint end with slope $\alpha=0.59^{+0.39}_{-0.45}$ [2412.06894].

At even earlier times, the REBELS ALMA survey at $z\sim7$ infers a mass-dependent obscured fraction $f_{\rm obs}\approx0.3$–$0.6$ for galaxies with $\log_{10}(M_\star/M_\odot)=9.4$–$10.4$ and an obscured cosmic SFRD
$$
\log_{10}(\mathrm{SFRD}_{\rm IR})=-2.24^{+1.18}_{-0.61},
$$
with a lower limit of $-2.56\pm0.30$. The survey argues that dust-obscured star formation still contributes $\sim30\%$ at $z\sim7$ [2208.08243].

Radio and submillimeter surveys also imply a substantial obscured component at high redshift. The VLA-COSMOS 3 GHz analysis finds evidence that UV-based SFRD estimates at $z\gtrsim4$ underestimate the true SFRD by $15$–$20\%$ because of appreciable star formation in highly dust-obscured galaxies [1703.09724]. The COSMOS-XS study pushes this discrepancy further at matched luminosity limits, reporting that the radio-based SFRD exceeds the UV-based, dust-corrected SFRD by approximately $1$ dex for $z\gtrsim3$ [2204.04167].

The dust problem is not one-sided. A semi-analytic reassessment of UV-derived CSFRD argues that standard dust obscuration corrections and UV-to-SFR conversions can overestimate the CSFRD by $\sim0.1$–$0.3$ dex and $\sim0.1$–$0.2$ dex, respectively, compared with the model’s intrinsic values, and presents new redshift-dependent calibrations for both effects [1208.0489]. Taken together, these results indicate that discrepancies among high-redshift SFRD determinations arise from both genuinely obscured populations that UV surveys miss and from biases introduced when dust and SFR calibrations are assumed to be redshift-invariant.

## 5. Decomposition by galaxy mass, morphology, halo mass, and submillimeter population

The SFRD can be resolved by stellar mass, morphology, galactic radius, or halo mass. At $z\sim1$, the ROLES survey finds that the shape of the SFRD as a function of stellar mass does not evolve between $z\sim1$ and $z\sim0.1$, even though the normalization declines by a factor of $\sim2.6$ in the corrected [OII]-based estimate and by $\sim6$ in the UV-based estimate [1002.3170]. In the nearby Universe, MUSE and GAMA extend the SFRD–mass relation down to $10^{5.5}\,M_\odot$ and find a constant low-mass slope in log SFRD versus log stellar mass, with no turn-over in the galaxy stellar mass function [2410.08036].

Spatially resolved fossil-record work adds a structural dimension. In CALIFA, most star formation at $z<0.5$ takes place in the outer regions of late spiral galaxies, whereas at $z>2$ the inner regions of the progenitors of current E and S0 galaxies are the major contributors to the SFRD. The same analysis finds that the inner regions are the major contributor to stellar-mass density at $z>0.5$, consistent with inside-out growth [1802.10118].

Halo-based analyses assign the SFRD to characteristic environments. Tomographic CIB–galaxy cross-correlation yields a best-fit maximum star formation efficiency of $\eta_{\rm max}=0.41^{+0.09}_{-0.14}$ at $\log_{10}(M_{\rm peak}/M_\odot)=12.14\pm0.36$, shifting to $12.42^{+0.35}_{-0.19}$ when external SFRD measurements are included [2204.01649]. This places the most efficient star formation in halos near $10^{12}\,M_\odot$.

Submillimeter-selected systems provide a direct population-level contribution to the SFRD around the peak epoch. In cosmological simulations calibrated to reproduce observed SMG counts and redshift distributions, SMGs with $S_{850}>1\,\mathrm{mJy}$ contribute up to $27\%$ of the total cosmic SFRD at $z\sim2.6$ in FLAMINGO. The same study finds that the abundance of SMGs rises from $z=6$ to $2.5$ and then declines sharply, and that sources with $S_{850}>3\,\mathrm{mJy}$ are exclusively starburst galaxies. For the TolTEC Ultra Deep Survey over $0.8$ deg$^2$, the forecast is $\sim80{,}000$ detections at $1.1$ mm and about $50\%$ of the cosmic SFRD captured at $z=2.5$ [2501.19327].

## 6. Physical interpretations from simulations and semi-analytic models

Semi-analytic and hydrodynamic models generally interpret the SFRD as the convolution of galaxy star-formation histories with the growth of the halo population. A simple semi-analytic treatment writes the SFRD as the sum over galaxies of different masses and types, weighted by the evolving number density of dark matter halos $N(M,z)$. In that framework, the “time-delayed” star-formation history,
$$
\Psi(t)\propto \frac{t}{\tau}e^{-t/\tau},
$$
is essential for reproducing the broad asymmetric shape of the observed SFRD, while artificially fixing $N(M,z)$ destroys the agreement. Moderate, prolonged feedback and winds modulate the normalization, but the principal drivers are the evolving halo mass function and the delayed shape of galaxy star-formation histories [1711.03416].

Cosmological SPH simulations refine this picture by making the feedback prescription explicit. A comparison of UV, IR, and H$\alpha$ SFR functions with P-GADGET3(XXL) simulations at $z\sim1$–$4$ shows that AGN feedback decreases the simulated CSFRD at $z<3$ but is not sufficient to reproduce the observed evolution at higher redshift. The best overall match comes from variable galactic winds that are efficient at decreasing the SFRs of low-mass objects at high redshift and become less efficient with time [1610.03441].

Submillimeter galaxy modeling further reveals that not all large-volume simulations reproduce the same obscured contribution. When parametric radiative-transfer-based flux prescriptions are applied to EAGLE, IllustrisTNG, and FLAMINGO, only FLAMINGO reproduces the observed SMG number counts and redshift distributions without requiring a top-heavy IMF; EAGLE and IllustrisTNG show a deficit of bright SMGs and a lower SFRD at $z\sim2$–$3$ [2501.19327]. This emphasizes that reproducing the global SFRD is not equivalent to reproducing the distribution of dust-obscured star formation across galaxy populations.

## 7. Cosmological applications and persistent controversies

The SFRD is increasingly treated as a cosmological observable rather than only a galaxy-evolution summary. A recent compilation over $z\in[0,15]$ fits SFRD jointly with cosmological parameters in $\Lambda$CDM and $w$CDM. In that analysis, SFRD combined with BBN alone gives $H_0=65\pm11$ km\,s$^{-1}$\,Mpc$^{-1}$, while adding DESI-DR2 BAO yields $H_0=68.28\pm0.18$ km\,s$^{-1}$\,Mpc$^{-1}$; joint analyses reduce uncertainties in astrophysical parameters by $30$–$50\%$ and preserve a robust $z_{\rm peak}$ near $2.6$ [2604.17660].

At the same time, fundamental methodological disagreements remain. One study argues that UV-corrected and IR-derived star-formation-rate functions are described by different distributions—Schechter for UV$_{\rm corr}$ and double power law for IR—and compare differently with the stellar-mass density evolution, even though both indicate a plateau rather than a sharp peak at $z\sim1$–$4$ [2107.02733]. Radio luminosity-function modeling reaches a related conclusion: pure luminosity evolution cannot describe the high-redshift radio LF, whereas luminosity+density evolution is genuinely indispensable, and the adopted FIR–radio calibration can shift the SFRD by factors of $\sim2$ [2311.08975]. A stellar-mass-selected radio analysis likewise finds that beyond $z\sim1$ the inferred decline of the SFRD depends crucially on whether the IR–radio correlation is allowed to evolve [2012.09797].

The present state of the field is therefore one of broad agreement on the existence of a cosmic noon, but continuing disagreement on its exact timing, on the magnitude of the obscured component at high redshift, and on the appropriate functional form for luminosity-function evolution and luminosity-to-SFR conversion. These disagreements are not peripheral: they determine whether the SFRD is interpreted primarily as a tracer of dust-hidden star formation, as a convolution of halo growth and feedback-regulated star formation, or as a complementary probe of background cosmology.

Source: https://www.emergentmind.com/topics/cosmic-star-formation-rate-density-sfrd