Extreme star formation events in quasar hosts over ${\bf0.5<\textit{z}<4}$

We explore the relationship between active galactic nuclei and star formation in a sample of 513 optically luminous type 1 quasars up to redshifts of $\sim$4 hosting extremely high star formation rates (SFRs). The quasars are selected to be individually detected by the \textit{Herschel} SPIRE instrument at $> $3$\sigma$ at 250 $\mu$m, leading to typical SFRs of order of 1000 M$_{\odot}$yr$^{-1}$. We find the average SFRs to increase by almost a factor 10 from $z\sim0.5$ to $z\sim3$, mirroring the rise in the comoving SFR density over the same epoch. However, we find that the SFRs remain approximately constant with increasing accretion luminosity for accretion luminosities above 10$^{12}$ L$_{\odot}$. We also find that the SFRs do not correlate with black hole mass. Both of these results are most plausibly explained by the existence of a self-regulation process by the starburst at high SFRs, which controls SFRs on time-scales comparable to or shorter than the AGN or starburst duty cycles. We additionally find that SFRs do not depend on Eddington ratio at any redshift, consistent with no relation between SFR and black hole growth rate per unit black hole mass. Finally, we find that high-ionisation broad absorption line (HiBAL) quasars have indistinguishable far-infrared properties to those of classical quasars, consistent with HiBAL quasars being normal quasars observed along a particular line of sight, with the outflows in HiBAL quasars not having any measurable effect on the star formation in their hosts.