Cosmic SFR density histories. The observational trend from Wilkins et al. (2008) is compared to model predictions assuming a constant SMF (filled circles assuming the Bell et al. 2003 SMF for a color-selected sample, filled triangles assuming the AMF), or using for each redshift the SMF observed at this redshift by Ilbert et al. (2010) (open circles) or Arnouts et al. (2007) (open squares). In the vertically shaded area (z > 1.1), the results from the models are mere extrapolations showing the limits of our approach beyond this redshift.