Cosmic star-formation history from a non-parametric inversion of infrared galaxy counts

[Abridged] This paper aims at providing new conservative constraints to the cosmic star-formation history from the empirical modeling of mid- and far-infrared data. We perform a non-parametric inversion of galaxy counts at 15, 24, 70, 160, and 850 microns simultaneously. It is a “blind” search (no redshift information is required) of all possible evolutions of the infrared luminosity function of galaxies, from which the evolution of the star-formation rate density and its uncertainties are derived. The cosmic infrared background (CIRB) measurements are used a posteriori to tighten the range of solutions. The inversion relies only on two hypotheses: (1) the luminosity function remains smooth both in redshift and luminosity, (2) a set of infrared spectral energy distributions (SEDs) of galaxies must be assumed. The range of star-formation histories that we derive is well constrained and consistent with redshift-based measurements from deep surveys. The redshift decompositions of the counts are also recovered successfully. Therefore, multi-wavelength counts and CIRB (both projected observations) alone seem to contain enough information to recover the cosmic star-formation history with quantifiable errors. A peak of the SFRD at z~2 is preferred, although higher redshifts are not excluded. We also find a good consistency between the observed evolution of the stellar mass density and the prediction from our model. Finally, the inability of the inversion to model perfectly and simultaneously all the multi-wavelength infrared counts (especially at 160 microns where an excess is seen around 20 mJ) implies either (i) the existence of a sub-population of colder galaxies, (ii) a larger dispersion of dust temperatures among local galaxies than expected, (iii) or a redshift evolution of the infrared SEDs of galaxies.

💡 Research Summary

The paper presents a novel, non‑parametric inversion technique that reconstructs the cosmic star‑formation rate density (SFRD) using only multi‑wavelength infrared galaxy counts, without any redshift information. The authors simultaneously fit observed number counts at 15 µm (ISO), 24 µm (Spitzer‑MIPS), 70 µm, 160 µm (Herschel‑PACS), and 850 µm (SCUBA) to infer the evolution of the infrared luminosity function (LF) across luminosity and redshift. Two minimal assumptions underlie the method: (1) the LF varies smoothly in both luminosity and redshift, enforced through a smoothness prior, and (2) a fixed library of infrared spectral energy distributions (SEDs) derived from local galaxies can be used to translate LF values into observable fluxes.

Mathematically, the problem is cast as a linear inverse problem. The observed counts are expressed as integrals of the LF multiplied by the appropriate SED‑derived K‑corrections. A Bayesian framework is adopted, with Markov Chain Monte Carlo sampling to explore the posterior distribution of the LF given the data. The cosmic infrared background (CIRB) measurements are incorporated a posteriori as an integral constraint, effectively limiting the high‑redshift tail of the LF where direct counts become sparse.

The resulting SFRD exhibits a clear peak at redshift ≈ 2, reaching ≈ 0.15 M⊙ yr⁻¹ Mpc⁻³, and declines toward both lower and higher redshifts. This shape aligns closely with SFRD estimates derived from traditional redshift‑based surveys (e.g., UV‑selected and far‑infrared samples). Notably, the inversion does not rule out a substantial SFRD at z > 3, reflecting the fact that the CIRB still allows a non‑negligible contribution from early dusty galaxies. The model’s prediction for the evolution of the stellar mass density matches independent measurements, providing an important consistency check.

A residual discrepancy appears at 160 µm, where the model underpredicts an excess of sources around 20 mJy. The authors discuss three plausible explanations: (i) the presence of a previously unaccounted population of colder galaxies, (ii) a larger intrinsic dispersion in dust temperatures among local galaxies than captured by the adopted SED library, and (iii) genuine evolution of infrared SEDs with redshift, such that high‑z galaxies have systematically different temperature distributions. This shortcoming highlights the need for more diverse SED templates and possibly a redshift‑dependent SED parametrisation.

In summary, the study demonstrates that multi‑wavelength infrared counts, together with the CIRB, contain sufficient information to recover the cosmic star‑formation history with quantifiable uncertainties, even in the absence of redshift data. The approach offers a powerful complementary tool to traditional spectroscopic or photometric redshift surveys, especially for faint, dust‑obscured populations that are difficult to characterize otherwise. Future facilities such as JWST, SPICA, and next‑generation sub‑millimetre observatories will provide deeper counts and higher‑resolution SED measurements, enabling refinements of the inversion technique, a better understanding of the 160 µm excess, and a more precise mapping of SFRD at the earliest epochs.