2.3. Are the cosmic star formation rates and histories consistent?

It is interesting to test for consistency between the cosmic SFR and cosmic SFH - a lack of consistency between these two may give important insight into the form of the stellar IMF between ~ 1 M (the mass range which optical/NIR light is most sensitive to) and 5 M (the stellar mass range probed by SFR indicators), or the calibrations of or systematic errors in stellar mass and/or SFR determinations.

In order to integrate the cosmic SFR, I choose to roughly parameterize the form of the cosmic SFR following [Cole et al. (2001)]. The cosmic SFR = (0.006 + 0.072 z^1.35) / [1 + (z/2)^2.4] M yr^-1 Mpc^-3 provides a reasonable fit to the cosmic SFR (Fig. 1). The data are insufficient to constrain whether or not the cosmic SFR declines towards high redshifts from a maximum at z ~ 1.5; I choose to impose a mild decrease towards high redshift, primarily because that matches the evolution in integrated stellar mass slightly better than a flat evolution. This cosmic SFR is then integrated using the PEGASE stellar population model assuming a [Kroupa (2001)] IMF and an initial formation redshift z_f = 5. The exact integration in PEGASE accounts explicitly for the recycling of some of the initial stellar bass back into the interstellar medium; for the [Kroupa (2001)] IMF this fraction is ~ 1/2, i.e., stellar mass in long-lived stars is 1/2 of the stellar mass initially formed (for a [Kennicutt (1983)] IMF the fraction is similar). I show the expected cosmic SFH as the solid line in Fig. 2.

It is clear that the form of the cosmic SFR required in Fig. 1 reproduces rather well the cosmic SFH as presented in Fig. 2. There are some slight discrepancies; a cosmic SFR as flat as that shown in Fig. 1 appears to overpredict the amount of stellar mass that one sees at z ~ 3. This might indicate that a drop-off in cosmic SFR towards higher redshift is required, or may indicate that an IMF richer in high-mass stars is favored for high-redshift starbursts. Yet, it is important to remember that estimates of cosmic SFR and SFH are almost impossible to nail down with better than 30% accuracy at any redshift. At z 1 the constraints are substantially weaker still, owing to large uncertainties from large scale structure, uncertainties in the faint-end slope of the stellar mass or SFR functions used to extrapolate to total SFRs or masses, and the difficulty in measuring SFRs and masses of intensely star-forming, dusty galaxies. Therefore, I would tend to downweight this disagreement at z 1.5 until better and substantially deeper data are available, focusing instead on the rather pleasing overall agreement between these two independent probes of the cosmic SFH.