Top panel: redshift evolution of the total IR luminosity density ρ_IR out to z ∼ 6. The results of integrating the best-fit modified Schechter function for our observed total IR LF only (based on the de-blended catalogue in COSMOS) are shown as red stars. The results of integrating the best-fit function for both our observed total IR LF and the Gruppioni et al. (2013) IR LF are shown as blue stars. Error bars on the stars represent the 1σ uncertainty. The measurements from Magnelli et al. (2013) are shown as green squares. The measurements from Gruppioni et al. (2013) are shown as black squares. The black solid line shows the predicted ρ_IR as a function of redshift from GALFORM. The black dotted line shows the predicted ρ_IR of the starburst galaxy population from GALFORM. The black dashed line shows the predicted ρ_IR of the quiescent galaxy population from GALFORM. Bottom panel: co-moving SFRD ρ_SFR as a function of redshift out to z ∼ 6. Estimates of the dust-obscured SFRD based on the best-fit function for our total IR LF only are shown as red stars. Estimates of based on the joint constraints from our total IR LF and the Gruppioni et al. (2013) IR LF are shown as blue stars. We also compare with other Herschel-based studies such as Rowan-Robinson et al. (2016) and Liu et al. (2018) and the ALMA-based study of Dunlop et al. (2017). The black dotted line shows the best-fit function of the evolution of from Koprowski et al. (2017). The black dashed line corresponds to the UV-based unobscured SFRD estimates from Parsa et al. (2016). The black solid line shows the parametric description of the evolution of the total SFRD ρ_SFR provided by Madau & Dickinson (2014).