The observed cosmic star formation rate density has an evolution which resembles a {\Gamma}(a, bt) distribution and can be described successfully by only 2 parameters

A debate is emerging regarding the recent inconsistent results of different studies for the Cosmic Star Formation Rate Density (CSFRD) at high-z. We employ UV and IR datasets to investigate the star formation rate function (SFRF) at ${\rm z \sim 0-9}$. We find that the SFRFs derived from the dust corrected ${\rm UV}$ (${\rm UV_{corr}}$) data contradict those from IR on some key issues since they are described by different distributions (Schechter vs double-power law), imply different physics for galaxy formation (${\rm UV_{corr}}$ data suggest a SFR limit/strong mechanism that diminish the number density of high star forming systems with respect IR) and compare differently with the stellar mass density evolution obtained from SED fitting (${\rm UV_{corr}}$ is in agreement, while IR in tension up to 0.5 dex). However, both tracers agree on a constant CSFRD evolution at ${\rm z \sim 1-4}$ and point to a plateau instead of a peak. In addition, using both indicators we demonstrate that the evolution of the {\it observed} CSFRD can be described by only {\bf 2} parameters and a function that has the form of a Gamma distribution (${\bf \Gamma(a,bt)}$). In contrast to previous parameterizations used in the literature our framework connects the parameters to physical properties like the star formation rate depletion time and cosmic baryonic gas density. The build up of stellar mass occurs in $\Gamma(a,bt)$ distributed steps and is the result of gas consumption up to the limit that there is no eligible gas for SF at t = ${\rm \infty}$, resulting to a final cosmic stellar mass density of $\sim 0.5 \times 10^9 \, {\rm \frac{M_{\odot}}{Mpc^3}}$.