THE STAR FORMATION HISTORY OF MASS-SELECTED GALAXIES IN THE COSMOS FIELD

Radio observations of the full (2 deg²) COSMOS field were carried out with the Very Large Array (VLA) at 1.4 GHz (20 cm) in several campaigns between 2004 and 2006. The entire field was observed in A and C configurations (Schinnerer et al. 2007) where the 23 individual pointings were arranged in a hexagonal pattern. Additional observations of the central seven pointings in the more compact A configuration (Schinnerer et al. 2010) were obtained in order to achieve a higher 1.4 GHz sensitivity in the area overlapping with the COSMOS MAMBO millimeter observations (Bertoldi et al. 2007). In both cases, the data reduction was done using standard procedures from the Astronomical Imaging Processing System (AIPS; see Schinnerer et al. 2007, for details). At a resolution of 15 × 14 the final map has a mean rms of ∼8 μJy beam⁻¹ in the central 30' × 30' and ∼12 μJy beam⁻¹ over the full area, respectively. Using the SAD algorithm within AIPS, a total of 2865 sources were identified at more than 5σ significance in the final VLA–COSMOS mosaic (Schinnerer et al. 2010). As the outermost parts of the map are not covered by multiple pointings the noise increases rapidly toward the edges. In this study, we therefore exclude these peripheral regions resulting in a final useable area of 1.72 deg².

Deep Spitzer IRAC data mapping the entire COSMOS field in all four channels have been obtained during the S-COSMOS observations (Sanders et al. 2007). The data reduction yielding images and associated uncertainty maps for all the four channels is described in Ilbert et al. (2010, I10 hereafter). For the 3.6 μm channel a source catalog has been obtained by O. Ilbert & M. Salvato (2009, private communication) using the SExtractor package (Bertin & Arnouts 1996). Given the point-spread function of 17, a Mexican hat filtering of the 3.6 μm image within SExtractor was used in order to assure careful deblending of the sources.

The resulting sample of 3.6 μm sources down to a limiting magnitude of m_AB(3.6 μm) = 23.9 in the 2.3 deg² field, not considering the masked areas around bright sources (K_s < 12), areas of poor image quality and the field boundaries, consists of 306,000 sources.²⁵

As detailed in I10 photometric redshifts (hereafter photo-z's) were assigned to all 3.6 μm detected sources. The vast majority of sources is also detected at optical wavelengths and therefore contained in the COSMOS photo-z catalog²⁶ (Ilbert et al. 2009) so that in general photometric information from 31 narrow-, intermediate-, and broadband FUV-to-mid-IR filterbands was available.²⁷ Within the remaining 4% (i.e., a total of 8507) of the 3.6 μm sources 2714 are also contained in the COSMOS K-band selected galaxy sample (McCracken et al. 2010) and are also regarded as real sources. I10 assigned photo-z's to these extremely faint objects using the available NIR-to-IRAC photometry.

The quality of the photo-z's was estimated (for details see I10) by using spectroscopic redshifts for a total of 4148 sources at m_AB(i⁺) < 22.5 from the zCOSMOS survey (Lilly et al. 2009). At a rate of <1% of outliers the accuracy was found to be $\sigma _{(z_{\rm phot} - z_{\rm spec})/(1+z_{\rm spec})} = 0.0075$ down to the magnitude limit of the spectroscopic sample. For all objects within the 3.6 μm selected catalog—regardless of i-band magnitude the accuracy was derived by using the 1σ uncertainty on the photo-z's from the probability distribution function which yields a conservative estimate of the photo-z uncertainty as detailed in Ilbert et al. (2009). At 1.25 < z < 2 the relative photo-z uncertainty is 0.08 and thus higher by a factor of four compared to the median value for the full ($m_{\rm {AB}}(3.6\, \mu \rm {m}) \ge 23.9$) sample.²⁸ We account for this when binning the data in redshift by choosing increasing bin widths with increasing redshift.²⁹ It is worth noting that the photo-z accuracy is degraded at magnitudes fainter than m_AB(i⁺) = 25.5 (see Figure 12 in Ilbert et al. 2009). Our choice of lower stellar mass limits (see Section 2.6) and our stellar mass binning scheme (see Section 2.6) automatically ensures a low fraction (<15%) of these optically very faint objects within the lowest mass-bin above our mass limit at any redshift. The fraction of such faint objects effectively vanishes toward higher masses as also pointed out by I10.³⁰

Stellar masses for all objects within the 3.6 μm selected parent sample have been computed by I10. Here, we briefly summarize the method and the important findings. For the estimation of stellar masses based on a Chabrier IMF stellar population synthesis models generated with the package provided by Bruzual & Charlot (2003, hereafter BC03) have been used. Furthermore, an exponentially declining SFH and a Calzetti et al. (2000) dust extinction law have been assumed. Spitzer MIPS 24 μm flux densities (from LeFloc'h et al. 2009) have been included in the SED template fitting as an additional constraint on the stellar mass. Systematic uncertainties on the stellar masses, caused by the use of photo-z's, the choice of the dust extinction law and library of stellar population synthesis models, have been investigated. No systematic effect due to the use of photo-z's is apparent. Stellar masses derived from the BC03 templates are systematically higher by 0.13–0.15 dex compared to the newer Charlot & Bruzual (2007) versions (Bruzual 2007) that have an improved treatment of thermally pulsing asymptotical giant branch (TP-AGB) stars. As BC03 models are commonly used in the literature, both studies, I10 and this work, are based on BC03 mass estimates.

A number of studies suggest the existence of a bimodality in the SSFR–M_* plane (e.g., Salim et al. 2007; Elbaz et al. 2007; Santini et al. 2009; Rodighiero et al. 2010a) leading to a tight SSFR sequence to be in place only for SF galaxies. Therefore, a deselection of quiescent, i.e., non-SF, objects is needed.

Following I10 we classify galaxies with a best-fit BC03 template that has an intrinsic (i.e., dust unextincted) rest-frame color redder than $({\rm NUV}\hbox{--}r^+)_{\rm {temp}} = 3.5$ as quiescent. Several authors (e.g., Wyder et al. 2007; Martin et al. 2007b; Arnouts et al. 2007) suggest this color to be an excellent indicator for the recent over past average SFR as it directly traces the ratio of young (light-weighted average age of ∼10⁸ yr) and old (⩾10⁹ yr) stellar populations. Seeking for a color bimodality that discriminates galaxies with currently high from those with low star formation activity the NUV–r color appears therefore to be superior to purely optical rest-frame colors such as U − V (e.g., Bell et al. 2004).

Using a dust uncorrected ${\rm {NUV}}\hbox{--}r^+$ versus r⁺ − J rest-frame color–color diagram³¹ I10 showed that in the range 0 ⩽ z ⩽ 2 for $({\rm {NUV}}\hbox{--}r^+)_{\rm {temp}} > 3.5$ quiescent galaxies are well separated from the parent sample without severe contamination by dust-obscured SF galaxies. This quiescent population is therefore comparable to the one classified by Williams et al. (2009) based on a U − V versus V − J rest-frame color–color diagram.

Furthermore, our quiescent population shows a clear separation from the parent sample with respect to galaxy morphology. I10 visually classified a subset of 1500 isolated and bright galaxies from the 3.6 μm parent sample using HST/ACS images and found the quiescent population among those to be clearly dominated by elliptical (E/S0) systems. A further cut ($({\rm {NUV}}\hbox{--}r^+)_{\rm {temp}} < 1.2$) was shown to efficiently separate late-type spiral and irregular galaxies from early-type spirals as well as the remaining tail of elliptical systems. As any such color cut effectively is a cut in star formation activity we discuss the spectral pre-classification of SF systems in more detail in Appendix C.

A major concern arising in the context of using radio emission to trace star formation is contaminating flux from active galactic nuclei (AGNs). For some galaxies the total radio signal might even be dominated by an AGN. For our study, ideally, we should therefore remove all galaxies hosting an AGN from our sample.

Cross-matching the most recent XMM-COSMOS photo-z catalog (Salvato et al. 2009; Brusa et al. 2010) with the 3.6 μm selected parent sample delivered a total of 1711 (i.e., ∼1%) X-ray detected objects. Most of these sources exhibit best-fit composite AGN/galaxy SEDs³² while a minor fraction is fitted well by an SED showing no AGN contribution. However, here all X-ray detections are treated as potential AGN contaminants and thus removed from our sample.³³

Studies of the radio luminosity function (LF; e.g., Sadler et al. 2002; Condon et al. 2002) agree that radio-AGNs contribute half of the radio light in the local universe at radio luminosities slightly below $L_{1.4 \,\rm {GHz}} \sim 10^{23}$ W Hz⁻¹ and outnumber SF galaxies above ∼2 × 10²³ W Hz⁻¹. Detailed multi-wavelength studies (Hickox et al. 2009; Griffith & Stern 2010) yield that radio-AGNs are hosted by red galaxies. The evolution out to z ≈ 1.3 of the radio-AGN fraction for luminous (i.e., $L_{1.4 \,\rm {GHz}} > 4 \times 10^{23}$ W Hz⁻¹) radio-AGNs as a function of stellar mass has been presented by Smolčić et al. (2009b) who selected a parent sample of red galaxies with rest-frame U − B colors in a range close to our quiescent galaxy fraction. The derived AGN fractions at a given stellar mass within the red galaxy population are therefore applicable to our sample.

According to Smolčić et al. (2009b, see their Figure 11) the luminous radio-AGN fraction at 0.7 < z < 1.3 is well below 25% at all log(M_*[M_☉]) < 11.5 where it drops quickly to ∼1% at log(M_*[M_☉]) = 11 and continuously to lower levels as stellar mass decreases. At masses lower than log(M_*[M_☉]) = 11 the radio-AGN fractions are subject to non-negligible evolution between 0 < z < 1 while the fractions at higher masses increase only mildly. However, given that the radio-AGN fractions are well below 1% in the former (i.e., low) mass range out to z ∼ 1.3 it is unlikely that they rise above 10% at z ≫ 1. The evolution of the radio-AGN fraction at the high-mass end is much slower but the fractions are high already in the local universe. We therefore set an arbitrary but reasonable threshold and exclude all quiescent objects above log(M_*[M_☉]) = 11.6, where the expected radio-AGN fraction exceeds 50%, from our stacking analysis. As the radio-AGN fraction sharply drops below this limit the remainder of our full galaxy sample should be generally free from radio-AGN contamination. Within the highest mass bin probed here (M_*>10¹¹ M_☉; see Figure 2), however, the average fraction of radio AGNs among the quiescent galaxies could still be ∼25% at z>1. This fraction appears high but among the entire galaxy population (quiescent and SF sources) the percentage drops to at most 10% within our highest mass bin at z ∼ 1. As shown by I10 globally, but in particular at M_*>10¹¹ M_☉ the fraction of quiescent galaxies among the entire sample decreases strongly toward higher redshifts (see also Taylor et al. 2009).³⁴ An upper bound of 10% to the potential fraction of radio AGN within our highest mass bin hence is a well-justified number at z>1.

Due to prominent spectral features we regard the SED fits for quiescent objects as most trustworthy such that also the SED-derived SFRs are expected to be accurate for individual objects. These SFRs therefore serve as a prior for revealing potential radio AGN among the radio detections in our sample. Hence, we correlated our sample with the latest version of the VLA-COSMOS catalog (Schinnerer et al. 2010) and excluded those objects showing radio-derived SFRs more than twice as large as the SED-derived values. We find that the overall number of objects excluded in each sample to be stacked is negligible. The same holds for very luminous ($L_{1.4 \,\rm {GHz}} > 10^{25}$ W Hz⁻¹) radio sources among the radio detections that are most likely high-power radio AGNs. We therefore excluded also these objects relying on individual photo-z's in order to estimate the radio luminosity. The total fraction of galaxies among all objects in a given bin that we exclude by these two criteria amounts—on average—to less than 0.3% such that only a fraction of radio detections is rejected. We stress the smallness of this percentage as the advantage of our radio approach is its insensitivity to dust obscuration which might be challenged by relying on individual optical best-fit SEDs as we partially do when removing some of the radio-detected objects. It should be noted that the high-power radio-AGN candidates are exclusively hosted by red galaxies within our sample. Hence, X-ray detected sources are the only objects that have been removed from our SF samples.

As the radio-based SFR-results presented in this paper (see Section 4) are based on a median stacking approach (see Section 3) a minor fraction of contaminating outliers such as an AGN is even tolerable. We conclude that contamination of the stacked radio flux densities caused by AGN emission at radio frequencies is not a significant source of uncertainty in the context of this study and that our conclusions would not change if we included the radio-AGN candidates in our analysis.

In the following, we will discuss the completeness of our (sub-)samples. It is important to distinguish between two kinds of effects. While the full 3.6 μm selected source catalog (1) is subject to a flux density-dependent level of detection incompleteness we are interested in and (2) how representative for the underlying population a given subset of galaxies is at a given mass. Our lower mass limits hence need to be chosen such that the objects at hand remain sufficiently representative.

I10 evaluated the efficiency of the source extraction procedure (and hence the detection completeness) with Monte Carlo simulations of mock point sources inserted into the 3.6 μm mosaic. At the flux density cut of $1 \,\mu \rm {Jy}$ ($m_{\rm {AB}}(3.6\, \mu \rm {m}) = 23.9$) the catalog was found to be 55% complete; 90% completeness is reached at $F_{3.6\, \mu \rm {m}} \approx 5\, \mu$Jy ($m_{\rm {AB}}(3.6\, \mu \rm {m})=22.15$). This rather shallow decline in detection completeness toward the magnitude limit is due to source confusion.

Figure 1 shows the distribution of 3.6 μm flux density with stellar mass in narrow redshift slices for our source catalog, color coded by the spectral type of the galaxies (see Section 2.4). The Monte Carlo detection completeness levels of the catalog are indicated by horizontal dashed black lines starting from the flux density limit at the bottom to the 95% completeness limit at the top in each panel. Each sub-population shows a clear correlation between 3.6 μm flux density and stellar mass, and the quiescent population residing at the high-mass end at all flux densities. While SF sources (the union of all blue and green data points) span the entire range of 3.6 μm flux densities at all redshifts, hardly any quiescent objects with low flux densities are observed at intermediate and high redshifts. We consequently find fewer and fewer low-mass quiescent objects as redshift increases. This is certainly the combined effect of a general absence of such sources at higher redshifts plus the loss of these objects at low flux densities due to the global detection incompleteness of our catalog.

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Detection incompleteness affects all sources at a given 3.6 μm flux density, regardless of their spectral type. However, the different distribution of quiescent and SF sources with respect to 3.6 μm flux density necessitates that a different lower mass limit ("representativeness limit" hereafter) be adopted, depending on whether we consider the redshift evolution of SF galaxies or that of the entire galaxy population. We now discuss how the limiting mass is set for these two samples.

• 1.  
  In the case of the entire galaxy population, it is important to be working with a sample in which the fractional contribution of quiescent and SF sources reflects the true population fractions as closely as possible. The probability that this is the case becomes larger, the better the underlying population is sampled; i.e., it rises with increasing detection completeness. We therefore require an intrinsic catalog completeness of 90% (corresponding $m_{\rm {AB}}(3.6\, \mu \rm {m})=22.15$) at all masses considered. This is an arbitrary but reasonable threshold as the intrinsic catalog completeness rises rapidly toward higher flux densities.In order to evaluate the actual mass representativeness limit we need to define yet another type of completeness level, which we shall refer to as statistical completeness. By applying the analytical scheme described in detail in Appendix A we ensure that the statistical completeness of our sample always reaches at least 95%. This value sets the actual level of representativeness of a given sub-sample. In the following, we will also present results for sub-samples below the evaluated mass limits which will be indicated separately. Those results represent strict upper limits in (S)SFR.
• 2.  
  For studying the SF population we need not be as conservative because we are dealing with a single sub-population that is subject to less internal variation of SF activity as a (bimodal) sample including both quiescent and SF systems. We thus consider sources down to the limiting flux density of the 3.6 μm catalog when we compute the mass limits at a given redshift. Since this implies that at low stellar masses the flux distribution is sharply cut due to the magnitude limit of our catalog, we still need to use the scheme presented in Appendix A to identify stellar mass limits that provide a representative flux density distribution for SF galaxies. As visible in all panels of Figure 1, the lowest mass bin always contains objects over the full range of detection completeness, from 55% to 100%. One might expect—and the SED fits confirm this—that among galaxies of a given mass, those with the fainter fluxes have lower SSFRs. Failure to include them (due to detection incompleteness) would thus yield average radio-derived SSFRs that are biased toward higher values. We wish to emphasize, however, that our choice of the statistical completeness level ensures that this bias is small above our mass limit and that our samples hence are "representative" in the sense that they can be expected to render a meaningful measurement of, e.g., the average SSFR of the underlying population.

The stellar mass representativeness limits for the whole sample and the SF systems are marked in Figure 1 as vertical lines for each redshift bin in the range $0.2 < z_{\rm {phot}} < 3$ and listed in Table 1. Note that they increase with redshift. As a consequence, our results will be based on fewer mass bins at high redshift and the aforementioned bias in the lowest mass bin may therefore have a larger impact on fitting trends. Very conservatively speaking, our results for SF objects presented in the following should generally be regarded as most robust at z ≲ 1.5 while evolutionary trends inferred at the high mass end are robust out to our redshift limit of z = 3. We will also show results for SF galaxies obtained at masses lower than the individual mass limits and treat them as not entirely representative. Such measurements will be indicated with different symbols in our plots and we will discuss any further implications in Section 4.4.

The final sample of galaxies with $m_{\rm {AB}}(3.6\, \mu \rm {m}) = 23.9$ and $z_{\rm {phot}} < 3$ consists of 165,213 sources over an effective area of ∼1.5 deg². Figure 3 in I10 shows the redshift distribution with a median of $z_{\rm {phot}} \sim 1.1$. After adopting a lower redshift limit of $z_{\rm {phot}}=0.2$ in order to account for the small local volume sampled by our effective area and our binning scheme 113,610 sources³⁵ (90,957 SF galaxies) enter our analysis. This is by far the largest galaxy sample used for studying the dependence between SFR and stellar mass throughout cosmic time. Figure 2 shows the adopted binning scheme and the number of galaxies contained in each stellar mass and photo-z bin.

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Our stacking algorithm uses cutouts with sizes of 40'' × 40'', centered on the position of the optical counterpart. Since the COSMOS astrometric reference system was provided by the VLA-COSMOS observations the positional accuracy between radio and optical sources should be well within the errors of both data sets. As detailed in Schinnerer et al. (2007) the relative and absolute astrometry of the VLA data are 130 and <55 mas, respectively. In other words, the average distribution of radio flux follows the one at optical wavelengths and the central pixel in any stacked image was always the brightest one. Averaging over pixels located at the same position in each stamp hence is an astrometrically well-defined problem.

It can be approached by computing either the mean or the median of the mentioned set of pixels. The resulting stamp then shows the spatial distribution of the average radio emission for the sample studied. For an input sample of N galaxies its background noise level should correspond to ${\sim }1/\sqrt{N}$ of the noise measured in a single radio stamp.³⁷ Any sample of galaxies in a given bin of redshift and stellar mass likely contains also a fraction of sources with radio detections. Even if this fraction is small, the mean is sensitive to the large excess in 1.4 GHz flux density compared to the average radio emission of the individual non-detections. On the other hand, setting a threshold and excluding radio detections from the stack artificially changes the sample and the results, hence, depend on the threshold applied.

In addition, foreground objects and other extended radio bright features (e.g., lobes from radio galaxies) need to be handled with care and might be a source of contamination affecting the noise in the final stamp but also potentially the signal itself. It is therefore beneficial to exclude stamps showing these features from a mean stack. Typically, significantly less than 1% of objects in a sample are rejected and the effect of this artificial cut on the sample thus is negligible. However, by resorting to the median, the stacking technique becomes more robust against outliers allowing the use of the entire input sample.³⁸ Non-uniform noise properties within the radio map can also be addressed by applying a weighted scheme to compute the median (see Appendix B). While it is often argued that there is no straightforward way of interpreting the sample median compared to the sample mean, White et al. (2007) showed that the median is a well-defined estimator of the mean of the underlying population in the presence of a dominant noise background.

Although, strictly speaking, these arguments only apply to the case of pure point sources, the condition of a dominant noise background is given in our study. One has to be aware of the fact that there is in principle no possibility of accessing the intrinsic distribution of radio peak fluxes of the underlying population as a whole. The observed distribution merely is the intrinsic one as smeared out by the Gaussian noise background. However, it still contains information that needs to be used in order to find proper confidence limits for any statistic applied. Based on the above arguments, we expect the broadened distribution to be not only shifted but also skewed toward positive flux density values. As a result, the uncertainty for the obtained peak flux density is poorly estimated by the background noise in the final stamp. Using a bootstrapping technique (see Appendix B.2) allows us to obtain more realistic, asymmetric error bars for our measured peak flux densities.

So far we considered only the average peak flux density which, to the first order, would not require to stack individual cutouts but only their central pixel. However, the typical galaxy of a given sample might exhibit extended radio emission. In that case the peak flux density is no longer equivalent to the total source flux but underestimates the typical radio flux density and hence all other quantities derived from it.

The effect of bandwidth smearing (BWS), chromatic aberration caused by the finite bandwidth used during the VLA-COSMOS observations leads to a spatial broadening of a source even if it is intrinsically point-like. Within a single pointing the BWS increases with increasing radial distance from the pointing center and the effect is analytically well determined (e.g., Bondi et al. 2008). For a mosaic like the VLA-COSMOS map that consists of many overlapping pointings the effect becomes analytically unpredictable due to the varying uncertainties introduced by the calibration and observing conditions.

For all our samples we constructed median co-added cutout images (Figure 3) and determined accurate rms-noise estimates (hereafter $\sigma _{\rm {Stack}}$) for the image stacks as described in Section 3.1. These (115 × 115) pixel² dirty maps were processed within AIPS.³⁹

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We used the task PADIM to make the stacked images equal in size to a (512 × 512) pixel² image of the VLA-COSMOS synthesized (dirty) beam by filling the outer image frame with additional pixels of constant value. The task APCLN with a circular CLEAN box of radius of seven pixels (i.e., 245) around the central component was then used to CLEAN each dirty map down to a flux density threshold of $2.5 \times \sigma _{\rm {Stack}}$.⁴⁰

Integrated flux densities, as well as source dimensions and position angles after deconvolution with the CLEAN beam were obtained by fitting a single-component Gaussian elliptical model to the CLEAN image within a quadratic box of (15 × 15) pixel² around the central pixel using the task JMFIT. Errors on the integrated flux densities have been estimated according to Hopkins et al. (2003) and rely on the combined information on the best-fit source model and the bootstrapping results from the image stacking:

$$\begin{equation} \frac{\sigma _{\rm {Total}}}{\langle F_{\rm {Total}} \rangle }=\sqrt{\left(\frac{\sigma _{\rm {data}}}{\langle F_{\rm {Total}} \rangle } \right)^2 + \left(\frac{\sigma _{\rm {fit}}}{\langle F_{\rm {Total}}\rangle } \right)^2 }, \end{equation} \tag{ 1 }$$

where (Windhorst et al. 1984; Condon 1997; and also the explanations in Schinnerer et al. 2004, 2010)

$$\begin{eqnarray} \frac{\sigma _{\rm {data}}}{\langle F_{\rm {Total}} \rangle } &=& \sqrt{\left(\frac{S}{N}\right)^{-2} + \left(\frac{1}{100} \right)^2} \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} \frac{\sigma _{\rm {fit}}}{\langle F_{\rm {Total}} \rangle } &=& \sqrt{\frac{2}{\rho _S} + \left(\frac{\theta _{\rm {B}} \theta _{\rm {b}}}{\theta _{\rm {M}} \, \theta _{\rm {m}}} \right) \left(\frac{2}{\rho ^2_{\psi }} + \frac{2}{\rho ^2_{\phi }} \right)}. \end{eqnarray} \tag{ 3 }$$

$\theta _{\rm {M}}=1\farcs 5$ is the major axis and $\theta _{\rm {m}}=1\farcs 4$ the minor axis of the beam while $\theta _{\rm {M}}$ and $\theta _{\rm {m}}$ are the major and minor axes of the measured (hence convolved) flux density distribution. In order to include the bootstrapping error estimates we set ${\rm S}/{\rm N}=\langle F_{\rm {peak}} \rangle /\sigma _{\rm {bs}}$, i.e., the ratio of the peak flux density in the stacked dirty map and the 68% confidence interval resulting from the bootstrapping. The same applies to the parameter-dependent estimators of the fit entering Equation (3) that are given by

$$\begin{equation} \rho ^2_X = \frac{\theta _{\rm {M}} \, \theta _{\rm {m}}}{4 \, \theta _{\rm {B}} \theta _{\rm {b}}} \left(1 + \frac{\theta _{\rm {B}}}{\theta _{\rm {M}}} \right)^{a} \left(1 + \frac{\theta _{\rm {b}}}{\theta _{\rm {m}}} \right)^{b} \left(\frac{S}{N} \right)^2 \end{equation} \tag{ 4 }$$

and a = b = 1.5 for ρ_F, a = 2.5 and b = 0.5 for $\rho _{\rm {M}}$ as well as a = 0.5 and b = 2.5 for $\rho _{\rm {m}}$.

For a given sub-sample centered at a given median redshift $\langle z_{\rm {phot}} \rangle$ the average (median-stacking-based) integrated flux density $\langle F_{\rm {Total}} \rangle$ observed at 1.4 GHz can be directly converted into a rest-frame 1.4 GHz luminosity using a K-correction that depends on the radio spectral index $\alpha _{\rm {rc}}$ (here $\alpha _{\rm {rc}}=-0.8$, e.g., Condon 1992):

$$\begin{eqnarray} \nonumber \langle L_{1.4 \,{\rm GHz}} \rangle \,({\rm W\, Hz^{-1}}) &=& 9.52 \times 10^{12} \, \langle F_{\rm {Total}} \rangle \,(\mu {\rm Jy}) \\ &&\times (D_L \,({\rm Mpc}))^2 \, 4 \pi (1+\langle z_{\rm {phot}} \rangle)^{1+\alpha _{\rm {rc}}}\nonumber\\ \end{eqnarray} \tag{ 5 }$$

with D_L being the luminosity distance at this median photo-z of all objects inside the bin.

It was pointed out by Dunne et al. (2009) that the median redshift might not be appropriate for estimating the radio luminosity if the peak of the radio flux density distribution does not coincide with the median of the photo-z distribution. They overcame this problem by deriving (and subsequently effectively stacking) luminosities according to Equation (5) for all objects relying both on the individual photo-z's and peak flux density measurements at the pixel corresponding to the position in the input catalog. At z>0.2 they afterward applied a common (i.e., redshift independent) factor to the median of all obtained luminosities to correct for the difference between peak and total flux density as well as for the effect of BWS. A similar approach was recently also used by Bourne et al. (2011). The method by Dunne et al. (2009) is justified given their data as they find for z>0.2 that the ratio of total to peak flux density does not change significantly, in particular not as a function of K-band magnitude. However, our data do not yield such a uniform behavior with respect to mass in the correction factor as Figure 4 shows. Indeed, if we were to state an average peak to total flux density conversion it would be a function of mass. An explanation for the discrepancy of our findings compared to Dunne et al. (2009) can be found in the use of higher resolved A-array data in our case compared to the B-array data constituting their radio continuum imaging. Hence, both results are correct given the respective data used and show that higher resolved radio data needs to be treated differently. The spread in conversion factors within our sub-samples is large and lower redshift objects show a significantly larger $\langle F_{\rm {Total}}\rangle /\langle F_{\rm {Peak}}\rangle$ ratio⁴¹ (see Figure 4). Moreover, further variations might arise depending on the galaxy population studied. Hence, if high-resolution data are used, results are more robust when first total flux densities are individually derived for any radio-stacking experiment before computing radio luminosities. As it is apparent from the Dunne et al. (2009) results their method should be considered, however, if stacking is used to infer the average radio luminosity of an entire galaxy population with a broad redshift distribution (Δz ≳ 1). As our broadest bins in redshift have Δz = 0.5—and this only at z ≫ 1 where they span a much smaller range in time—it is indeed more accurate to rely on our approach given our radio imaging.

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In order to convert the derived average 1.4 GHz luminosities into average SFRs we use the calibration of the radio–FIR correlation by Bell (2003) scaled to a Chabrier IMF⁴²:

$$\begin{equation} \langle \rm {SFR} \rangle \,({\it M}_{\odot }\, \rm yr^{-1}) = \left\lbrace \begin{array}{@{}cl}3.18 \times 10^{-22} \, L, &\; L > L_c \\ \frac{3.18 \times 10^{-22} \, L}{0.1+0.9 \, (L/L_c)^{0.3}}, & \; L \le L_c\end{array}, \right. \end{equation} \tag{ 6 }$$

where $L=\langle L_{1.4 \,\rm {GHz}} \rangle$ is the average radio luminosity derived from the median stack according to Equation (5) and L_c = 6.4 × 10²¹ W Hz⁻¹ is the radio luminosity of an L_*-like galaxy. As Bell (2003) empirically argues the low-luminosity population needs to be treated separately from higher values of radio luminosities since non-thermal radio emission might be significantly suppressed in these galaxies. Even though our work exploits the radio-faint regime our derived average 1.4 GHz luminosities lie generally above this threshold. Only at the lowest masses and z ≲ 0.8 we find $\langle L_{1.4 \,\rm {GHz}} \rangle < L_c$ (see Tables 2 and 3). Any study relying on the calibration by Yun et al. (2001) is, consequently, directly comparable to our results as Yun et al. (2001) used a uniform normalization very similar to the case L>L_c in Equation (6).⁴³

Table 2. Radio Stacking Results for the Entire Mass-selected Sample

Δlog(M*)	〈log(M*)〉	$\Delta z_{\rm {phot}}$	$\langle z_{\rm {phot}} \rangle$	$\langle F_{\rm {Peak}}\rangle$	$\langle F_{\rm {Total}}\rangle$	rms	$\langle L_{1.4 \,\rm {GHz}}\rangle / L_c$	〈SFR〉
M*(M☉)	M*(M☉)			(μJy beam−1)	(μJy)	(μJy beam−1)	Lc = 6.4 × 1021 W Hz−1	(M☉ yr−1)
9.3–9.6	9.45a	0.2–0.4	0.27	1.7+0.3−0.4	4.9+1.4−2.2	0.399+0.015−0.002	0.2+0.0−0.1	0.6+0.1−0.2
	9.44a	0.4–0.6	0.49	2.0+0.4−0.3	2.5+1.1−1.0	0.388+0.006−0.006	0.3+0.1−0.1	0.9+0.3−0.3
	9.44a	0.6–0.8	0.69	1.4+0.2−0.3	2.2+1.2−1.9	0.276+0.004−0.006	0.6+0.4−0.6	1.5+0.6−1.0
	9.45a	0.8–1.0	0.89	1.3+0.2−0.3	1.9+0.5−0.9	0.246+0.007−0.013	1.0+0.3−0.5	2.1+0.5−1.0
	9.44a	1.0–1.2	1.11	1.1+0.3−0.2	1.9+1.6−1.2	0.275+0.005−0.010	1.7+1.4−1.1	3.5+2.8−2.3
	9.45a	1.2–1.6	1.40	0.7+0.2−0.1	0.9+0.5−0.4	0.179+0.001−0.003	1.4+0.9−0.6	2.8+1.8−1.2
	9.47a	1.6–2.0	1.78	0.5+0.2−0.3	0.9+0.8−1.0	0.243+0.004−0.006	2.6+2.3−2.8	5.2+4.6−5.7
9.6–9.9	9.74a	0.2–0.4	0.27	3.9+0.6−0.2	9.6+2.5−1.0	0.469+0.017−0.033	0.3+0.1−0.0	0.9+0.2−0.1
	9.74a	0.4–0.6	0.49	3.6+0.3−0.4	7.4+1.3−1.9	0.455+0.010−0.004	1.0+0.2−0.2	2.0+0.3−0.4
	9.74a	0.6–0.8	0.69	2.4+0.2−0.3	5.4+0.9−1.3	0.341+0.028−0.002	1.6+0.3−0.4	3.2+0.6−0.8
	9.74a	0.8–1.0	0.89	2.1+0.3−0.2	4.5+1.4−0.9	0.265+0.002−0.010	2.4+0.8−0.5	4.9+1.6−0.9
	9.75a	1.0–1.2	1.10	2.7+0.3−0.3	4.0+1.4−1.3	0.320+0.022−0.005	3.5+1.3−1.2	7.2+2.6−2.4
	9.73a	1.2–1.6	1.41	1.9+0.2−0.3	2.3+1.1−1.4	0.216+0.006−0.007	3.7+1.8−2.3	7.6+3.6−4.7
	9.75a	1.6–2.0	1.82	1.3+0.2−0.2	2.1+1.0−1.0	0.232+0.003−0.015	6.2+3.0−3.1	12.7+6.0−6.2
	9.75a	2.0–2.5	2.21	1.3+0.2−0.3	2.0+0.7−0.8	0.271+0.015−0.008	8.9+3.0−3.8	18.2+6.1−7.6
	9.76a	2.5–3.0	2.65	0.9+0.2−0.3	1.0+0.5−0.6	0.293+0.005−0.005	6.9+3.3−4.2	14.1+6.6−8.6
9.9–10.2	10.04b	0.2–0.4	0.27	5.1+0.6−0.5	13.8+2.9−2.4	0.504+0.014−0.008	0.5+0.1−0.1	1.2+0.2−0.2
	10.05b	0.4–0.6	0.49	5.4+0.5−0.5	11.8+2.1−2.2	0.472+0.016−0.022	1.5+0.3−0.3	3.1+0.6−0.6
	10.04a	0.6–0.8	0.68	5.3+0.2−0.5	8.5+0.7−1.7	0.360+0.011−0.013	2.4+0.2−0.5	5.0+0.4−1.0
	10.05a	0.8–1.0	0.89	3.9+0.2−0.4	5.7+0.8−1.5	0.288+0.005−0.009	3.1+0.4−0.8	6.3+0.9−1.7
	10.05a	1.0–1.2	1.09	4.0+0.3−0.4	7.0+1.4−1.6	0.364+0.002−0.004	6.1+1.2−1.4	12.5+2.5−2.9
	10.04a	1.2–1.6	1.39	3.3+0.2−0.2	4.7+1.0−1.0	0.260+0.003−0.006	7.3+1.6−1.5	14.8+3.2−3.0
	10.04a	1.6–2.0	1.87	2.9+0.2−0.2	5.5+0.8−0.8	0.274+0.005−0.003	17.0+2.6−2.4	34.6+5.2−4.9
	10.04a	2.0–2.5	2.28	2.2+0.2−0.3	3.2+0.9−1.1	0.277+0.006−0.003	15.9+4.3−5.5	32.4+8.8−11.2
	10.04a	2.5–3.0	2.73	1.7+0.3−0.3	2.5+1.3−1.2	0.308+0.004−0.007	18.8+9.3−8.8	38.2+18.9−17.9
10.2–10.5	10.35	0.2–0.4	0.29	7.5+0.5−0.4	18.1+2.4−1.7	0.534+0.016−0.005	0.7+0.1−0.1	1.6+0.2−0.1
	10.34	0.4–0.6	0.49	7.9+0.4−0.4	16.4+1.9−1.8	0.495+0.010−0.018	2.1+0.2−0.2	4.3+0.5−0.5
	10.35b	0.6–0.8	0.68	4.8+0.5−0.5	9.8+2.2−2.1	0.386+0.025−0.009	2.8+0.6−0.6	5.7+1.3−1.2
	10.35b	0.8–1.0	0.89	5.1+0.4−0.4	9.5+1.7−1.6	0.311+0.005−0.006	5.1+0.9−0.9	10.5+1.9−1.8
	10.35b	1.0–1.2	1.09	5.4+0.3−0.3	9.2+1.3−1.2	0.375+0.020−0.009	8.1+1.1−1.0	16.6+2.3−2.1
	10.34a	1.2–1.6	1.38	4.6+0.4−0.3	8.3+1.5−1.4	0.279+0.001−0.012	12.5+2.3−2.1	25.4+4.7−4.2
	10.33a	1.6–2.0	1.81	4.2+0.4−0.2	9.3+1.7−1.0	0.325+0.008−0.009	26.8+4.8−2.9	54.5+9.7−5.9
	10.33a	2.0–2.5	2.36	3.7+0.4−0.4	6.2+1.5−1.5	0.330+0.025−0.006	33.0+8.1−7.8	67.1+16.4−15.9
	10.34a	2.5–3.0	2.81	3.3+0.3−0.6	5.8+1.2−2.3	0.343+0.009−0.005	45.5+9.4−18.5	92.6+19.2−37.6
10.5–10.8	10.63	0.2–0.4	0.28	9.8+0.7−0.7	23.9+3.1−2.9	0.594+0.027−0.019	0.9+0.1−0.1	1.8+0.2−0.2
	10.64	0.4–0.6	0.48	8.1+0.8−0.4	18.6+3.4−2.0	0.555+0.010−0.026	2.4+0.4−0.2	4.8+0.9−0.5
	10.63	0.6–0.8	0.69	6.4+0.4−0.6	13.3+1.6−2.7	0.420+0.006−0.020	3.9+0.5−0.8	7.9+0.9−1.6
	10.64	0.8–1.0	0.89	6.3+0.4−0.5	11.1+1.5−2.0	0.331+0.002−0.010	6.0+0.8−1.1	12.2+1.6−2.2
	10.64	1.0–1.2	1.09	5.6+0.4−0.4	11.3+1.7−1.7	0.391+0.013−0.016	9.8+1.5−1.5	20.0+3.0−3.0
	10.64b	1.2–1.6	1.37	6.2+0.5−0.4	11.4+2.0−1.9	0.308+0.004−0.004	17.1+3.0−2.8	34.7+6.1−5.7
	10.63b	1.6–2.0	1.78	6.9+0.4−0.3	11.8+1.6−1.1	0.378+0.017−0.009	32.6+4.4−3.0	66.3+9.0−6.1
	10.64a	2.0–2.5	2.27	5.2+0.3−0.3	10.0+1.2−1.4	0.400+0.008−0.008	48.3+6.0−6.6	98.3+12.3−13.4
	10.62a	2.5–3.0	2.76	4.9+0.5−0.6	11.6+2.6−2.6	0.437+0.003−0.012	88.4+19.6−20.0	179.7+39.8−40.6
10.8–11.1	10.95	0.2–0.4	0.27	9.6+1.0−1.0	32.7+5.4−5.1	0.882+0.009−0.007	1.1+0.2−0.2	2.2+0.4−0.4
	10.92	0.4–0.6	0.48	9.2+0.6−0.4	22.5+2.6−2.0	0.733+0.030−0.009	2.8+0.3−0.2	5.7+0.7−0.5
	10.92	0.6–0.8	0.69	6.7+0.7−0.7	15.5+3.2−3.0	0.556+0.034−0.013	4.5+0.9−0.9	9.2+1.9−1.8
	10.91	0.8–1.0	0.90	7.1+0.5−0.3	12.8+1.8−1.1	0.412+0.025−0.014	7.0+1.0−0.6	14.3+2.0−1.3
	10.92	1.0–1.2	1.10	7.6+0.5−0.4	13.6+1.8−1.6	0.509+0.008−0.011	12.1+1.6−1.5	24.6+3.2−3.0
	10.91	1.2–1.6	1.36	6.3+0.6−0.6	13.7+2.5−2.5	0.417+0.010−0.008	20.2+3.7−3.7	41.0+7.4−7.5
	10.92	1.6–2.0	1.79	7.8+0.3−0.3	15.9+1.4−1.2	0.479+0.003−0.005	44.5+3.9−3.4	90.5+7.8−6.8
	10.93b	2.0–2.5	2.21	6.9+0.5−0.4	13.8+2.0−1.6	0.523+0.012−0.009	63.1+8.9−7.4	128.4+18.1−15.1
	10.93a	2.5–3.0	2.72	6.8+0.5−0.7	14.0+2.0−3.0	0.583+0.013−0.036	102.3+14.6−21.7	207.9+29.6−44.1
>11.1	11.20	0.2–0.4	0.27	9.2+0.9−1.9	36.0+5.5−10.8	1.250+0.047−0.038	1.2+0.2−0.4	2.5+0.4−0.8
	11.23	0.4–0.6	0.48	10.3+2.0−3.2	22.6+8.4−13.0	1.227+0.008−0.025	2.8+1.0−1.6	5.7+2.1−3.3
	11.20	0.6–0.8	0.69	7.3+1.2−1.2	24.4+6.1−6.5	0.897+0.010−0.007	7.1+1.8−1.9	14.4+3.6−3.8
	11.20	0.8–1.0	0.90	8.6+0.9−1.0	16.6+3.6−4.0	0.681+0.013−0.009	9.2+2.0−2.2	18.7+4.0−4.5
	11.20	1.0–1.2	1.10	10.1+0.7−0.8	21.6+2.8−3.1	0.881+0.018−0.013	19.3+2.5−2.8	39.2+5.1−5.7
	11.20	1.2–1.6	1.35	11.1+0.9−1.1	19.6+3.5−4.3	0.762+0.012−0.022	28.2+5.1−6.2	57.4+10.3−12.5
	11.20	1.6–2.0	1.78	14.3+0.7−1.0	28.7+3.0−4.0	0.835+0.022−0.009	79.7+8.4−11.0	162.0+17.1−22.4
	11.22	2.0–2.5	2.22	13.7+1.1−0.9	25.0+4.1−3.6	0.737+0.023−0.025	115.3+18.9−16.5	234.4+38.5−33.6
	11.23b	2.5–3.0	2.71	11.3+0.9−0.8	23.2+3.8−3.5	0.767+0.021−0.030	169.4+27.8−25.7	344.4+56.5−52.3

Notes. Median-stacking-based average 1.4 GHz radio flux densities and derived average quantities for all our bins in mass and redshift for the entire mass-selected sample. A Chabrier (2003) IMF is assumed. Radio luminosities are stated in units of L_c, the threshold luminosity below which Bell (2003) empirically found the non-thermal radio emission to be suppressed (see Equation (6)). Resulting SFRs from bins with lower radio luminosity are hence boosted compared to, e.g., the calibration of the radio–IR relation by Yun et al. (2001). The median stellar mass and median z for any given bin are also stated. ^aMass bin contains data below the limit of mass representativeness and yields an upper limit to the average SFR (see Section 2.6 for further details.) ^bFirst mass bin above the limit of representativeness (see Section 2.6) which contains a low fraction (<15%) of optically faint objects with m_AB(i⁺) ⩾ 25.5 for which the photo-z accuracy is degraded (see Section 2.2 for further details).

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Table 3. Radio Stacking Results for Star-forming Systems

Δlog(M*)	〈log(M*)〉	$\Delta z_{\rm {phot}}$	$\langle z_{\rm {phot}} \rangle$	$\langle F_{\rm {Peak}}\rangle$	$\langle F_{\rm {Total}}\rangle$	rms	$\langle L_{1.4 \,\rm {GHz}}\rangle / L_c$	〈SFR〉
M*(M☉)	M*(M☉)			(μJy beam−1)	(μJy)	(μJy beam−1)	Lc = 6.4 × 1021 W Hz−1	(M☉ yr−1)
9.4–9.8	9.58	0.2–0.4	0.28	3.1+0.5−0.5	8.7+2.3−2.2	0.408+0.001−0.002	0.3+0.1−0.1	0.9+0.2−0.2
	9.58	0.4–0.6	0.49	2.4+0.2−0.4	5.1+1.4−2.0	0.372+0.016−0.033	0.7+0.2−0.3	1.5+0.3−0.5
	9.58	0.6–0.8	0.69	2.1+0.3−0.2	3.6+1.2−0.7	0.258+0.009−0.002	1.0+0.3−0.2	2.1+0.7−0.4
	9.58	0.8–1.0	0.89	1.7+0.2−0.2	3.2+0.7−1.0	0.219+0.004−0.006	1.7+0.4−0.5	3.6+0.8−1.1
	9.58a	1.0–1.2	1.10	2.1+0.3−0.2	3.2+1.0−0.7	0.246+0.006−0.009	2.9+0.9−0.6	5.9+1.9−1.2
	9.58b	1.2–1.6	1.40	1.1+0.2−0.2	1.4+0.5−0.5	0.169+0.001−0.004	2.2+0.8−0.8	4.5+1.7−1.7
	9.60b	1.6–2.0	1.79	0.8+0.2−0.2	1.5+0.9−1.3	0.210+0.004−0.004	4.2+2.6−3.7	8.5+5.4−7.4
	9.62b	2.0–2.5	2.17	1.1+0.2−0.2	1.4+0.6−0.7	0.237+0.001−0.000	6.3+2.5−2.9	12.7+5.1−5.9
9.8–10.2	9.99	0.2–0.4	0.28	6.9+0.4−0.8	17.7+2.0−3.5	0.523+0.019−0.011	0.6+0.1−0.1	1.5+0.1−0.2
	9.99	0.4–0.6	0.49	6.2+0.5−0.5	12.6+2.0−1.9	0.441+0.011−0.016	1.7+0.3−0.3	3.4+0.5−0.5
	9.98	0.6–0.8	0.68	5.4+0.3−0.6	9.3+1.2−2.2	0.336+0.022−0.029	2.7+0.3−0.6	5.4+0.7−1.3
	9.99	0.8–1.0	0.89	3.9+0.2−0.4	6.2+0.9−1.5	0.264+0.002−0.005	3.4+0.5−0.8	6.8+1.0−1.6
	9.99	1.0–1.2	1.09	3.7+0.3−0.3	6.6+1.4−1.3	0.305+0.010−0.006	5.8+1.2−1.1	11.9+2.4−2.2
	9.97a	1.2–1.6	1.39	3.2+0.2−0.2	4.4+1.0−1.0	0.225+0.014−0.001	6.8+1.5−1.5	13.9+3.1−3.1
	9.98a	1.6–2.0	1.85	2.5+0.2−0.2	4.6+1.0−0.8	0.228+0.006−0.004	14.0+3.0−2.6	28.5+6.2−5.2
	9.98b	2.0–2.5	2.25	2.0+0.2−0.1	2.8+1.0−0.5	0.248+0.012−0.002	13.6+4.8−2.6	27.6+9.8−5.2
	9.99b	2.5–3.0	2.71	1.6+0.2−0.2	2.0+1.4−1.5	0.256+0.001−0.001	14.5+10.1−11.2	29.5+20.6−22.7
10.2–10.6	10.37	0.2–0.4	0.29	12.2+0.8−0.3	29.0+3.5−1.4	0.592+0.014−0.014	1.2+0.1−0.1	2.4+0.3−0.1
	10.37	0.4–0.6	0.49	11.5+1.1−0.6	23.3+4.4−2.6	0.542+0.007−0.011	3.0+0.6−0.3	6.2+1.2−0.7
	10.39	0.6–0.8	0.68	8.2+0.3−0.3	16.0+1.4−1.3	0.415+0.026−0.009	4.6+0.4−0.4	9.4+0.8−0.7
	10.38	0.8–1.0	0.89	7.8+0.3−0.4	14.5+1.4−1.5	0.324+0.006−0.002	7.9+0.8−0.8	16.0+1.5−1.6
	10.40	1.0–1.2	1.10	6.5+0.3−0.3	12.0+1.3−1.1	0.358+0.013−0.004	10.6+1.1−1.0	21.5+2.3−2.0
	10.38	1.2–1.6	1.38	5.3+0.2−0.2	9.7+0.9−0.8	0.267+0.005−0.004	14.7+1.3−1.2	29.9+2.7−2.5
	10.37	1.6–2.0	1.81	4.9+0.3−0.3	9.8+1.1−1.3	0.298+0.007−0.000	28.2+3.2−3.7	57.3+6.5−7.5
	10.37a	2.0–2.5	2.32	4.0+0.2−0.3	7.0+0.7−1.1	0.302+0.010−0.001	35.6+3.5−5.8	72.5+7.0−11.7
	10.38a	2.5–3.0	2.78	3.5+0.3−0.3	6.6+1.2−1.3	0.311+0.010−0.009	50.9+8.9−10.2	103.5+18.1−20.8
10.6–11.0	10.74	0.2–0.4	0.28	18.8+1.3−1.7	53.8+6.4−8.3	0.851+0.017−0.014	2.0+0.2−0.3	4.0+0.5−0.6
	10.75	0.4–0.6	0.48	13.8+1.2−1.3	30.2+5.0−5.4	0.707+0.012−0.015	3.9+0.6−0.7	7.9+1.3−1.4
	10.75	0.6–0.8	0.69	13.3+1.1−0.6	27.7+4.4−2.5	0.573+0.035−0.023	8.1+1.3−0.7	16.4+2.6−1.5
	10.75	0.8–1.0	0.89	10.5+0.6−0.5	19.5+2.6−2.2	0.433+0.009−0.001	10.5+1.4−1.2	21.3+2.8−2.4
	10.75	1.0–1.2	1.10	8.1+0.3−0.4	15.8+1.2−1.7	0.454+0.005−0.018	14.1+1.0−1.5	28.7+2.1−3.1
	10.75	1.2–1.6	1.37	8.6+0.4−0.4	16.8+1.6−1.8	0.333+0.002−0.003	25.0+2.3−2.7	50.9+4.7−5.4
	10.77	1.6–2.0	1.79	7.9+0.4−0.3	14.8+1.7−1.1	0.387+0.011−0.009	41.3+4.9−3.0	83.9+9.9−6.1
	10.77	2.0–2.5	2.22	6.3+0.7−0.5	11.9+2.6−1.9	0.397+0.002−0.004	55.2+11.9−8.7	112.3+24.2−17.7
	10.76	2.5–3.0	2.72	5.6+0.3−0.4	13.3+1.2−1.6	0.428+0.005−0.020	97.6+9.0−11.9	198.4+18.2−24.2
>11.0	11.10	0.2–0.4	0.29	19.8+3.7−4.1	75.4+20.8−23.1	1.640+0.084−0.100	2.9+0.8−0.9	5.9+1.6−1.8
	11.10	0.4–0.6	0.48	18.1+1.2−1.8	56.3+6.2−9.3	1.364+0.071−0.048	6.9+0.8−1.2	14.1+1.6−2.3
	11.10	0.6–0.8	0.69	12.7+1.1−1.4	32.6+5.1−6.4	1.086+0.040−0.011	9.6+1.5−1.9	19.5+3.1−3.8
	11.10	0.8–1.0	0.89	13.8+1.4−1.7	32.0+5.6−6.7	0.907+0.039−0.007	17.1+3.0−3.6	34.9+6.1−7.3
	11.13	1.0–1.2	1.10	13.4+1.5−1.3	26.9+5.6−4.9	0.881+0.013−0.009	24.2+5.0−4.4	49.2+10.2−9.0
	11.11	1.2–1.6	1.36	13.4+1.0−0.8	27.7+4.1−3.1	0.734+0.023−0.027	40.5+6.0−4.5	82.3+12.2−9.2
	11.11	1.6–2.0	1.80	15.6+1.1−1.6	30.3+4.3−6.3	0.701+0.013−0.014	85.9+12.1−17.9	174.6+24.6−36.4
	11.15	2.0–2.5	2.22	11.9+0.6−0.5	21.8+2.3−1.8	0.594+0.023−0.044	100.1+10.6−8.5	203.5+21.6−17.2
	11.17	2.5–3.0	2.71	11.1+0.7−1.1	22.5+2.9−4.6	0.645+0.007−0.004	164.5+20.8−33.7	334.4+42.4−68.5

Notes. Median-stacking-based average 1.4 GHz radio flux densities and derived average quantities for all our bins in mass and redshift for star-forming systems within our mass-selected sample. For details see the caption of Table 2. ^aFirst mass bin above the limit of representativeness (see Section 2.6) which contains a low fraction (<15%) of optically faint objects with m_AB(i⁺) ⩾ 25.5 for which the photo-z accuracy is degraded (see Section 2.2 for further details). The average SFR measured in this bin might be slightly overestimated toward higher values (see Section 2.6). ^bMass bin contains data below the limit of mass representativeness and yields an upper limit to the average SFR (see Section 2.6 for further details).

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Finally, for a given sample, specific SFRs are computed as the ratio of the SFR and the median stellar mass. Based on the same arguments as before we neither take into account an uncertainty in the median mass for the error estimates of our derived SSFRs. As we exclusively deal with average quantities in this work we omit the 〈〉-notation in the following.

We first consider the whole sample including all galaxies and show the redshift-dependent radio-based SSFRs that are distributed in the logarithmic SSFR–M_* plane as seen in the left panel of Figure 5. It is clear that the SSFR for a given stellar mass increases with redshift and that it generally decreases with increasing stellar mass.

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The data at the high-mass end (above log(M_*) ≈ 10.5) within all considered redshift slices suggest power-law relations between SSFR and stellar mass of the form

$$\begin{equation} \rm {SSFR}(\it M_*,\it z) =\it c(z) \times {\it M_*}^{\beta (z)}. \end{equation} \tag{ 7 }$$

In the following, we will refer to the index β also as a slope since the relation is commonly shown in log  space. The dashed lines in Figure 5 depict the best fit to the data in the mass-representative regime (see Section 2.6) and indicate that only the normalization evolves while the power-law index β_ALL of the individually fitted relations shows minor fluctuations but no clear evolutionary trend. Only at z ≳ 1.5 there is tentative evidence for a somewhat shallower slope. However, at the highest redshifts probed too few mass-representative data points exist to perform the linear fit. Our evidence is solely supported by the offset between the SSFR of the most massive galaxies and those of intermediate mass remaining the same as at z ∼ 1.8. Based on our data it therefore is justified to consider the index β_ALL in Equation (7) a constant at least for all z < 1.5 and log(M_*) ≳ 10.5.

At log(M_*) < 10–10.5 we see at practically all epochs that the measured SSFRs significantly deviate from the relation fitted to the high-mass end. The extrapolation toward lower masses overpredicts the measurement. In Section 2.6, we argued that all these data points—lying below the mass representativeness limits—likely represent upper limits. We hence believe this is a genuine deviation that is reminiscent of the bimodality (whereby quiescent galaxies preferentially populate the high-mass end) in the SSFR–M_* plane confirmed at various redshifts for galaxy samples with individually measured SFRs (e.g., Brinchmann et al. 2004; Salim et al. 2007; Elbaz et al. 2007; Santini et al. 2009; Rodighiero et al. 2010a).

Using our spectral classification scheme we separately study the SF galaxy population in order to break the afore mentioned bimodality. The right panel in Figure 5 shows that a power-law relation according to Equation (7) holds over the entire mass range probed, once quiescent galaxies are excluded. Linear fits exclusively to the mass-representative regime show that, at z ≲ 1.5: (1) SSFR declines toward higher mass, and that (2) the slope β_SFG is constant, as it was the case for the entire galaxy population. Compared to the entire sample, the slope is significantly shallower.⁴⁵ All these conclusions also hold at all other epochs probed but are supported by fewer data points significantly above the mass representativeness limits that enter the fits. Hence we regard our conclusions as most robust at z < 1.5.

Above z ∼ 1.4 and below log(M_*) ≈ 9.5–10, we again find that measurements in the regime not regarded as mass representative lie significantly below the linear fits. Since quiescent galaxies are even less frequent at these redshifts,⁴⁶ the bimodality argument is obviously insufficient to explain this observed trend. A possible explanation is that the magnitude limit of our catalog leads to a loss of dust-dominated systems with low masses but high star formation activity. If this were the case our previous statement that SSFRs in the underrepresented mass regime are upper limits would not necessarily hold. However, we do not expect a sufficiently high number density of low-mass dusty starbursts to make this scenario plausible. Another explanation could lie in the dynamical considerations presented in Section 4.2.

The fact that the aforementioned deviations from the linear fits at low masses steadily grow with redshift hints at a solid upper limit to the average SSFR. Local spiral galaxies have on average a dynamical timescale—i.e., the rotation timescale at the outer radius of a disk galaxy—of $\tau _{\rm {dyn}} \sim 0.37$ Gyr (Kennicutt 1998). Daddi et al. (2010b) show that this still holds at z ∼ 1.5. The inverse of this dynamical timescale, $1/\tau _{\rm {dyn}} \sim 2.7 \,\rm {Gyr}^{-1}$, is similar to the threshold that seems to prevent our average SSFRs from rising continuously with decreasing mass. Note also that this dynamical timescale approximately equals the free-fall time (Genzel et al. 2010) which is commonly used to relate SFR volume density with gas volume density (e.g., Schmidt 1959; Kennicutt 1998; Krumholz & McKee 2005; Krumholz et al. 2009; Leroy et al. 2008).

As indicated in Figure 5 the population of z>1.5 galaxies reaches average levels of star formation that enable these normal SF systems to double their mass within a dynamical timescale. Generally, star formation is thought to be limited by the rate at which cold gas is accreted onto the galaxy (e.g., Dutton et al. 2010; Bouché et al. 2010; and also, e.g., Kereš et al. 2005; Macciò et al. 2006, where simulations actually show the cold gas inflow), while the efficiency of star formation does not appear to change out to the highest redshifts accessible to molecular gas studies in normal disk galaxies to date (Daddi et al. 2010a; Tacconi et al. 2010). Consequently, even the highly elevated gas fractions—i.e., the amount of gas available for star formation over the sum of gas and stellar mass—compared to local disk systems (e.g., Daddi et al. 2010a, who find up to 60% at z = 1.5) might not suffice to sustain a star formation activity that proceeds faster than gravity permits. As average galaxies reach inverse SSFRs comparable to their inverse dynamical—and, most importantly, free fall—time it is hence likely that an effective gas accretion threshold is reached. Hence, the SSFR should stop its growth with redshift at some point. Lower mass galaxies reach this threshold at lower redshifts than the more massive systems leading to the flattening of the relation we observe at the lower mass end. We will henceforth refer to the transition from an inclined to a flat SSFR sequence as "crossing mass."

It is clear that carbon monoxide ALMA studies at z>1.5 of typical SF systems with M_* ⩽ 10¹⁰ M_☉ are required to understand their molecular gas properties and to test the star formation law of this population.

The redshift evolution of our data is shown in Figure 6 for all galaxies and for the SF population. Both panels suggest a co-evolution of the considered mass bins at least out to z ∼ 1.5; while all measured SSFRs increase with redshift, the high-mass end does not evolve faster compared to lower masses and it always has the lowest SSFRs. An offset between the typical SSFRs of different mass bins is also evident for SF galaxies but it is smaller than for the entire galaxy population which shows a wider spread of SSFRs at fixed mass. Clearly, all these aspects are the direct result of our previous findings.

• 1.  
  A constant slope β of the SSFR–M_* relation is observed for all galaxies (at the high-mass end) as well as for SF galaxies alone at least out to z ∼ 1.5.
• 2.  
  The slope β_SFG is shallower for SF systems.

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In Figure 6, we also plot the mass doubling line⁴⁷ (long dashed line) and the inverse age of the universe at any given redshift (dash-dotted line). Our measurement clearly shows that virtually all SF galaxies display a higher star formation activity than is required if their entire mass had been build up at a constant rate over the whole age of the universe.⁴⁸ All galaxies, generally, cross the dash-dotted line sooner or later depending on their stellar mass. The most massive systems enter the stage of subaverage star formation activity already at z ∼ 0.8.

At the high-mass end (log(M_*) ≳ 11) the SSFR for SF galaxies increases by almost a factor of 50 and is about twice as much for all galaxies within 0.2 ⩽ z ⩽ 3. For a given mass bin the redshift evolution is well described by a power law g(z) ∝ (1 + z)ⁿ as depicted by the dashed lines in Figure 6.⁴⁹ For the most massive SF galaxies this relation holds out to the highest redshifts probed, thus no flattening is observed. However, toward lower masses and z ≳ 2 significant deviation of the data from the best-fit relation with lower SSFRs toward lower masses is apparent. Again the argument of an upper SSFR limit due to dynamical reasons might explain such a deviation. For reference we also show the recent SED-based measurement by Magdis et al. (2010) for a Lyman Break Galaxy (LBG) sample at z ≈ 3. Their study probes M_* ≈ 10¹⁰ M_☉ and we see that their SSFR measurement is significantly below the extrapolation given by our evolutionary fit in the same mass regime even for SF galaxies. Basically, their data point is extending our measured data at the low-mass end if evolution were to stop at about z ≈ 1.5 (a scenario suggested by the data of Stark et al. 2009 and González et al. 2010).

Summarizing our findings a separable function of the form

$$\begin{equation} {\rm SSFR}(\it M_*,z) \propto \it f(\it M_*) \times \it g(z) = \it M_*^{\beta } \times (1+\it z)^n \end{equation} \tag{ 8 }$$

describes well the mass-dependent evolution of the SSFR given our data within the restrictions discussed. For SF galaxies we find β_SFG ≈ −0.4 and $n_{\rm {SFG}} \approx 3.5$. We emphasize again that the dynamical arguments discussed in Section 4.2 would give rise to a value of $\beta _{\rm {SFG}} = 0$ below the crossing mass of the average SSFR and the upper limiting SSFR. The results of the individual fits to our data yielding the parameters β and n for all and SF galaxies are presented in Tables 4 and 5.

Table 4. Two Parameter Fits to the Mass Dependence of the SSFR

	All Galaxies			SF Systems
Δz	log(cALL [1/Gyr])	βALL	χ2/dof	log(cSFG [1/Gyr])	βSFG	χ2/dof
0.2–0.4	−1.63 ± 0.04	−0.73 ± 0.03	0.11	−1.27 ± 0.03	−0.44 ± 0.03	0.03
0.4–0.6	−1.22 ± 0.04	−0.75 ± 0.04	0.24	−0.90 ± 0.03	−0.42 ± 0.03	0.78
0.6–0.8	−0.96 ± 0.08	−0.57 ± 0.08	0.17	−0.67 ± 0.03	−0.40 ± 0.03	1.37
0.8–1.0	−0.81 ± 0.07	−0.73 ± 0.06	0.07	−0.48 ± 0.03	−0.38 ± 0.03	1.42
1.0–1.2	−0.53 ± 0.05	−0.58 ± 0.05	0.64	−0.38 ± 0.03	−0.46 ± 0.03	0.61
1.2–1.6	−0.33 ± 0.13	−0.61 ± 0.12	0.12	−0.12 ± 0.03	−0.30 ± 0.03	1.08
1.6–2.0	0.04 ± 0.08	−0.33 ± 0.07	1.47	0.10 ± 0.07	−0.41 ± 0.07	2.21
2.0–2.5				0.22 ± 0.06	−0.42 ± 0.05	0.19
2.5–3.0				0.43 ± 0.16	−0.44 ± 0.15	0.81
	$\langle \beta _{\rm {ALL}} \rangle =$	−0.67 ± 0.02	+0.34−0.08	$\langle \beta _{\rm {SFG}} \rangle =$	−0.40 ± 0.01	+0.10−0.06

Notes. A power-law fit of the form c × (M_*/10¹¹ M_☉)^β (Equation (7)) was applied to the radio-stacking-based SSFRs as a function of mass within any redshift slice. Fits have only been applied if more than two data points remained above the mass limit where the individual sample is regarded mass representative. The results for all galaxies are shown in the left-half of the table while those for star-forming systems (see Section 2.4) are given in the right-half. The weighted average power-law index (over all accessible redshifts) found for each population is stated at the bottom along with the formal standard error and the scatter range yielding a more realistic uncertainty estimate.

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Table 5. Two Parameter Fits to the Redshift Evolution of the SSFR

	All Galaxies				SF Systems
Δlog(M*(M☉))	log(CALL [1/Gyr])	nALL	$\chi ^2/\rm {dof}$	Δlog(M*(M☉))	log(CSFG [1/Gyr])	nSFG	$\chi ^2/\rm {dof}$
				9.4–9.8	−0.92 ± 0.45	3.02 ± 0.15	0.65
10.2–10.5	−1.53 ± 0.56	4.18 ± 0.05	3.05	9.8–10.2	−0.92 ± 0.33	3.42 ± 0.07	1.63
10.5–10.8	−1.76 ± 0.93	4.28 ± 0.05	1.48	10.2–10.6	−1.11 ± 0.28	3.62 ± 0.04	5.24
10.8–11.1	−1.92 ± 1.48	4.27 ± 0.05	1.75	10.6–11.0	−1.28 ± 0.47	3.48 ± 0.04	1.73
>11.1	−2.12 ± 3.93	4.53 ± 0.07	2.37	>11.0	−1.41 ± 0.82	3.40 ± 0.06	1.52
	$\langle n_{\rm {ALL}} \rangle =$	4.29 ± 0.03	+0.24−0.11		$\langle n_{\rm {SFG}} \rangle =$	3.50 ± 0.02	+0.12−0.48

Notes. A power-law fit of the form C × (1 + z)ⁿ was applied to the radio-stacking-based SSFRs as a function of redshift within any mass bin. Fits have only been performed if more than two data points remained above the mass limit where the individual sample is regarded mass representative. The results for all galaxies are shown in the left-half of the table while those for star-forming systems (see Section 2.4) are given in the right-half. The weighted average power-law index (over all accessible masses) found for each population is stated at the bottom along with the formal standard error and the scatter range yielding a more realistic uncertainty estimate.

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It is worth noting that at z>1—where angular diameter distance is approximately constant—the evolutionary trend we find is very close to the redshift dependence of the radio luminosity of about (1 + z)^3.8 (see Section 3.2). As the SSFR is proportional to the radio luminosity this is yet another argument to support that our inferences are not challenged by systematic errors to the median redshift even in our broader bins at z>1 (see also the corresponding discussion in Section 3.2).

In order to alternatively probe our inferences in the redshift range below z = 1.5 where our data yield the most robust results we also stacked the same bins in redshift and mass into the Spitzer 24 and 70 μm COSMOS maps. We inferred SFRs from the total (8–1000 μm) IR luminosity predicted by the best-fitting IR SED (Chary & Elbaz 2001) given the joint flux density information. The results do not deviate significantly from those derived from the radio emission so that all our conclusions remain robust also when derived from the IR data. All these and further results will be presented and discussed in detail in a separate publication (M. T. Sargent et al. 2011, in preparation).

In this section, we compare our findings with results in the literature, with a particular focus on those least dependent on extinction corrections because they use either radio stacking or stacking of IR imaging by Spitzer and, most recently, Herschel. Literature data we show in the figures belonging to this section are based on a Salpeter IMF and have been converted to the Chabrier scale.

The evolutionary power law we derived for all mass-selected galaxies is in excellent agreement with the results presented by Damen et al. (2009a), both in terms of the evolutionary exponent and the normalization of the trend. We hence concur, in particular, with those findings of Damen et al. (2009a) resulting from a detailed comparison of their results with predictions from the SAM of Guo & White (2008). The study of Damen et al. (2009a), which is based on SFRs from 24 μm and UV detections in conjunction with deep K-band observation in the Chandra Deep Field South, is also in broad agreement with the 24 μm stacking analysis of Zheng et al. (2007a) at z < 1 in the same field. Consistent findings have also been presented recently in the Spitzer/MIPS stacking analysis at 70 and 160 μm by Oliver et al. (2010) whose data cover the largest on-sky area of all aforementioned surveys, albeit at a reduced depth of $F_{3.6\, \mu \rm {m}} = 10\, \mu$Jy (an order of magnitude shallower compared to our sample) which prevents them from reliably constraining the evolution beyond z ∼ 1. Based on a deep rest-frame NIR bolometric flux density selected galaxy sample in the northern Great Observatories Origins Deep Survey (GOODS; Giavalisco et al. 2004) field Cowie & Barger (2008) measure extinction corrected UV-based SSFRs for all individual objects out to z = 1.5. Their average trends with mass agree well with our results for all galaxies in the comparable redshift ranges and also on absolute scales both studies are consistent at all masses.

Radio-based measurements of the SSFR–M_* relation have been presented by Dunne et al. (2009). In terms of the evolution of the SSFR sequence both their and our study show a good agreement. The findings by Dunne et al. (2009) differ from ours (see Figure 7) as well as from most other studies not restricted to SF galaxies only in that they report an almost non-existent slope β_ALL at all reliably probed epochs. Their analysis and ours share some methodological similarities (e.g., the use of a mass-(in their case: K-) selected sample⁵⁰ and a radio-stacking approach) and should therefore be directly comparable. Despite the technical differences in the exact implementation of the image stacking as already discussed (see Section 3.2) it seems unlikely that an explanation for the different trends can be found in the radio data used. It appears more likely that the derivation of individual stellar masses causes the differences as Dunne et al. (2009) use a direct conversion from the rest-frame K-band magnitudes as measured from the best-fitting SED templates to stellar mass which exclusively depends on redshift. Such a conversion should not only be different for SF and quiescent sources (Arnouts et al. 2007) but even ceases to be applicable toward lower mass SF sources as discussed in Appendix D of I10. It is hence likely that low-mass SF sources with higher SSFRs have migrated to higher masses producing artificially elevated SSFRs at the high-mass end. This explanation is consistent with the generally higher deviations from our results at higher masses (see Figure 7). Since neither we nor Dunne et al. (2009) find a significant evolution of the slope β_ALL in the SSFR–M_* plane the pure evolutionary behavior reported in both studies is largely consistent.

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The SSFR–M_* relation for sBzK—and hence SF K-band selected—galaxies in the COSMOS field was derived by Pannella et al. (2009) based on radio stacks from the same VLA image. Our results for SF galaxies are (necessarily) in good agreement with their findings at z ≈ 2 (the left panel of Figure 7) where they do not probe the highest mass range presented here. A main conclusion of Pannella et al. (2009) is, however, a mass-independent SSFR at z>1.5 which is mainly inferred from a measurement on their entire SF sample (not further divided by redshift) and a measurement at z ≈ 1.6 both covering the same mass range as considered here. A similar tendency at z ∼ 2 has previously also been reported by Daddi et al. (2007) in a study carried out in the GOODS fields. Their work is based on 24 μm detected galaxies down to log(M_*) ≈ 9.5 but also based on radio stacks of their K-band selected sample. As galaxies at 1.3 < z < 1.5 substantially contribute to the photometric redshift distribution of the Pannella et al. (2009) sample, it is likely that the sBzK criterion no longer selects all SF objects at these low redshifts. In this context, we also refer to Appendix C where the upper left panel in Figure 14 shows that the sBzK criterion by construction fails to select all SF sources at z < 1.5. As we already pointed out, our SSFR–M_* relation for our SF sample tends to flatten toward lower M_*. When considering only low to intermediate masses, all measurements based on stacking into the VLA-COSMOS 1.4 GHz map are thus in good agreement. The steeper slope β_SFG of the SSFR–M_* relation for SF galaxies found in this study is thus a consequence of the fact that we span a larger mass range at z ≈ 2.

The left panel of Figure 7 also shows the results at z ≈ 1 presented by Elbaz et al. (2007) based on 24 μm detection resulting from deep Spitzer/MIPS observations of the GOODS fields and UV-corrected SFRs. Although this study too infers a nearly constant relation between SSFR and mass, the figure shows that the radio-derived results agree with the mid-IR measurements remarkably well. This illustrates that measurements of the slope β_SFG of the SSFR–M_* for SF galaxies are quite sensitive to deviations at the edges of the mass range even if measurements at individual masses do not significantly differ between different studies. Finally, it is also worth noting that, toward lower redshifts, our slope β_SFG agrees well with the measurements by Noeske et al. (2007b) which are based on SFRs from emission lines, UV as well as 24 μm imaging for a K-band selected sample of the DEEP2 spectroscopic survey. Also in the local universe GALEX/UV-based values around β = −0.35, consistent with our study, have been reported for galaxies taken from the Sloan Digital Sky Survey (SDSS; Salim et al. 2007; Schiminovich et al. 2007). It should be mentioned that, based on the SDSS emission lines study of Brinchmann et al. (2004), Elbaz et al. (2007) found a slightly shallower slope $\beta _{\rm {SFG}}=-0.23$ for SF galaxies in the local universe.

Image stacking results using Herschel/PACS data at 160 μm have been presented by Rodighiero et al. (2010a). Their GOODS-North Spitzer/IRAC data are only slightly shallower compared to our COSMOS imaging. As the right panel of Figure 7 shows, individual measurements by Rodighiero et al. (2010a) at z < 1 are in good agreement with our findings. (The one exception being their lowest redshift which extends to z = 0, explaining the overall slightly lower SSFRs.) At z>1.5 the Rodighiero et al. (2010a) results suggest that SSFRs cease to grow further at the high-mass end. While in this redshift range our radio-derived SSFRs agree with the far-IR based ones at the lowest masses probed, the radio measurement yields about 0.4 dex (i.e., significantly) higher SSFRs at the high-mass end as they do not show a different redshift trend than at lower z. As our highest redshift bin is centered at a slightly higher z compared to the corresponding one of Rodighiero et al. (2010a) the difference might be slightly lower if the bins were perfectly matched and given the high-mass SSFR evolution for SF galaxies is continuing at z>1.5. We therefore see no clear evidence for strong discrepancies of radio- and far-IR stacking derived SSFRs at high z as speculated by Rodighiero et al. (2010a) when comparing their results to those of Pannella et al. (2009) and especially those of Dunne et al. (2009). We emphasize, however, that future far-IR studies could test potential mass-dependent changes in the radio–IR correlation at z>1.5 responsible for the slight differences reported here.⁵¹ Based on power-law fits to their data, Rodighiero et al. (2010a) infer a steepening of β_SFG toward higher z, an effect they consequently term "upsizing." Note, however, that our measurements of β_SFG agree with those of Rodighiero et al. (2010a) within the uncertainties. Tentative evidence for upsizing is also reported by Oliver et al. (2010) who use Spitzer/MIPS stacking of late-type galaxies at 70 and 160 μm (see the right panel of Figure 7). In Appendix C, we show that we can mimic an upsizing trend, as well as the somewhat flatter evolution of the SSFR out to z ≈ 2 reported by, e.g., Rodighiero et al. (2010a) if we restrict our SF sample to those sources with the most active star formation.

At a given redshift and mass, the SFRD is computed as the product of (1) the comoving number density as inferred directly from the number of galaxies in the relevant mass bin and (2) their average SFR as measured in our stacking analysis.

As already pointed out, SFRs likely represent an upper limit at the smallest masses where the sampling of the underlying population is not representative. Consequently, SFRDs at low masses are also upper bounds, as we can correct the number counts in a given low-mass bin for the lost objects. This is done by computing the expected number from the observed mass functions derived for the same sample of SF galaxies that is used for this study (see I10 for further details). We account for the slightly smaller portion of the COSMOS field accessible to the radio stack compared to the area used for the derivation of the mass functions. The correction for the expected number counts is always small so that corrected and uncorrected values of SFRD(M_*, z) agree within the errors. Since it is a systematic correction it still needs to be taken into account.

All number-count corrected and uncorrected data points for the SFRD(M_*, z) are shown in Figure 8. There appears to be a characteristic mass of M_* = 10^10.6±0.4 M_☉ that contributes most to the total SFRD at a given redshift. Up to z ∼ 1.8 our data points sample below this characteristic mass and the peak is well constrained. At higher redshifts this is no longer the case.

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We want to motivate now that the underlying functional form for the distribution of data points in the SFRD–M_* plane is actually known because of two facts.

• 1.  
  There is a (possibly broken) power-law relation between (S)SFR and stellar mass for SF galaxies at all z < 3 as measured in this study.
• 2.  
  The functional form of the mass function for SF objects in the same redshift range is well determined.

Regarding the second point, the mass function of SF galaxies is commonly (e.g., Lilly et al. 1995; Bell et al. 2003, 2007; Zucca et al. 2006; Arnouts et al. 2007; Pozzetti et al. 2010; Ilbert et al. 2005, 2010) found to be parameterized well by a power law with an exponential cutoff at a characteristic mass M* as introduced by Schechter (1976) of the form

$$\begin{eqnarray} \Phi _{\rm {SFG}}(M_*) dM_* &=& \Phi ^*_{\rm {SFG}} \left(M_*/M^*_{\rm {SFG}}\right)^{\alpha _{\rm {SFG}}} \nonumber\\ &&\times \exp (-M_*/M^*_{\rm {SFG}})d(M_*/M^*_{\rm {SFG}}). \end{eqnarray} \tag{ 9 }$$

This Schechter function has recently been qualitatively as well as quantitatively been modeled to be the natural consequence of essentially two types of cessation of star formation (Peng et al. 2010).⁵²

Multiplying $\Phi _{\rm {SFG}}(M_*)$ by the SFR sequence, i.e., another power law in mass, again produces a Schechter function. Hence, we can write

$$\begin{equation} \rm {SFRD}\it (M_*,z) dM_*=\Phi _{\rm {SFRD}}(\Phi ^*_{\rm {SFRD}},\tilde{\alpha },\it M^*)\it dM_*, \end{equation} \tag{ 10 }$$

i.e., a distribution (SFRD function hereafter) of the same functional form as Equation (9) with the three parameters $\Phi ^*_{\rm {SFRD}}$, $\tilde{\alpha }$, and M*. While the exponential cutoff mass M* is the same as the one in the mass function (defined above as M*_SFG), $\tilde{\alpha }=\alpha _{\rm {SFG}} + \tilde{\beta }_{\rm SFG}$ is the sum of the low-mass slope of the mass function of SF galaxies and the slope⁵³ of the SFR sequence (see also Santini et al. 2009 for a similar parameterization). The parameter $\Phi ^*_{\rm {SFRD}}$ acts as a normalization and its role in the global evolutionary picture will be discussed in Section 5.2.

The index α_SFG and also the cutoff mass M* for SF galaxies are constant in the redshift regime considered (e.g., Bell et al. 2003, 2007; Arnouts et al. 2007; Pérez-González et al. 2008; Pozzetti et al. 2010; Ilbert et al. 2010). We assume here that α_SFG and M*_SFG stay constant also at z>2. These assumptions are tentatively supported by the few observational constraints reported for these high redshifts as detailed in Section 5.2. As detailed in Section 4.1, the power-law index β_SFG—that enters the parameter $\tilde{\alpha } = \alpha _{\rm {SFG}} + \beta _{\rm {SFG}} + 1$ in Equation (10)—is also found to be a constant. However, we explained in Section 4.2 that at masses lower than the crossing mass between the SSFR sequence and a possible SSFR-threshold $\beta _{\rm {SFG}}=0$ should be assumed. In the following, we will hence consider two possible scenarios below the suggested crossing mass at a given redshift:

$$\begin{eqnarray*} \rm {{Case \; A}:} &\quad & \tilde{\alpha } = \alpha _{\rm {SFG}} + \beta _{\rm {SFG}} + 1\\ \rm {{Case \; B}:} &\quad & \tilde{\alpha } = \alpha _{\rm {SFG}} + 1. \end{eqnarray*}$$

Figure 8 shows that at z < 1.5 the parameterization of the SFRD function in Equation (10) can reproduce our data at all masses sampled and irrespective of the exact value of $\tilde{\alpha }$. For this redshift range, Figure 8 also includes results of two other studies that rely on different SFR tracers. At z ∼ 1 the dependence of the SFRD on stellar mass has recently been measured using the [O ii]λ3727 line to trace star formation (Gilbank et al. 2010b). We overplot these data points in the corresponding redshift bins in Figure 8 and find that our SFRD function accurately fits these measurements as well.⁵⁴ The same holds for the UV-derived results based on a Salpeter IMF by Cowie & Barger (2008) in the GOODS-North field at 0.9 < z < 1.5. Given our results, the global evolution of the SFRD-function between 0.9 < z < 1.4 is mild such that we can plot these data in Figure 8 in the bin 1.2 < z < 1.6. It is worth noting that the Cowie & Barger (2008) measurements at z < 0.9 equally support our finding that the peak of the SFRD does not shift with redshift to higher values.

Below the limiting stellar mass (dashed red lines in Figure 8) our data points are lower than the prediction of Equation (10) if we assume case A for $\tilde{\alpha }$ (even though we have applied a number density correction). Moreover, we remind the reader that—in keeping with our previous discussion—these data points are likely upper limits. Given the comparatively large uncertainties of our SFRD functions at high z these deviations are not highly significant but the trend is systematic and suggest a steepening of the low-mass slope of the SFRD function. It is directly related to the fact that the corresponding data points deviate from the best-fit (S)SFR–M_* relation at lower mass. In Section 4.2, we proposed an upper limit to the average SSFR due to dynamical reasons as a possible explanation for the trends. Taking into account this limit of $\rm {SSFR} = 1/\tau _{\rm {dyn}} \sim 2.7$ Gyr⁻¹ yields an index $\beta _{\rm {SFG}} = 0$ below the mass at which our fitted high-mass SSFR–M_* relation crosses the supposed SSFR limit at a given epoch. We plot the SFRD function for $\tilde{\alpha }=\alpha _{\rm {SFG}} + 1$ as dashed blue lines in Figure 8. As the crossing mass increases with redshift and lies below the mass representativeness threshold at z < 1 it has little impact on the mass-integrated SFRD. The reason is that the mass-dependent SFRD has declined already by at least an order of magnitude from the peak value before reaching the crossing mass. The changes are largest at z>1.8 where the dashed SFRD function (case B) drops quickly toward lower masses right after the peak and, in doing so, traces our data points better than before for $\tilde{\alpha }=\alpha _{\rm {SFG}} + \beta _{\rm SFG}$ (case A). However, we emphasize that our data cannot clearly favor case A or B proposed given the large uncertainties of the Schechter parameters and the lack of representativeness of our data at such low masses at high z. Any scenario suggested, hence, awaits confirmation based on deeper data in the selection band once they are available. We robustly conclude that a single Schechter function is a good model for the SFRD function over all masses at least out to z ∼ 1 and that the distribution of SFRDs peaks in the same mass range at all z probed in both cases A and B.

Figure 9 shows the time evolution of our above constructed SFRD function. This non-logarithmic plot clearly illustrates the existence of a characteristic mass of star formation at all epochs although it should be kept in mind that our results are most robust at z < 1.5. Our findings exclude an evolution of this characteristic mass toward lower values with cosmic time. At z < 1.5 this important result is supported by independent observations at different wavelengths (Cowie & Barger 2008; Gilbank et al. 2010a, 2010b) and, toward z ≈ 2, also by empirical arguments (Peng et al. 2010) while the recent model of Boissier et al. (2010) predicts a mild evolution of this characteristic mass by about 0.3 dex. In Figure 9 again dashed lines of corresponding color denote the low-mass trends of the SFRD functions once we assume the SSFR-limit discussed. As pointed out above, within the important mass range above 10¹⁰ M_☉—at which the SFRD function peaks at any epoch—a mass-independent proportional increase is apparent even out to z>2. In contrast to that, the mass-dependent SFRD appears to evolve mildly below 10⁹ M_☉. As a result galaxies in the stellar mass range between 10¹⁰ and 10¹¹ M_☉ contribute most to the global—i.e., mass-integrated—SFRD at any epoch but low-mass systems gain more relative importance toward low redshifts. Evidently, low-mass systems show the same evolutionary trends as those of higher mass if case A is assumed.

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Earlier observational findings appear to be at odds with the existence of a characteristic stellar mass for star formation we measure. While they are all based on shallower survey data compared to our COSMOS imaging, the limiting magnitude of these samples is not necessarily the reason for the different conclusions. Juneau et al. (2005)—using a mass-independent extinction correction and hence a linear conversion from [O ii] luminosity to SFR—report that out to z ∼ 2 the contribution to the increase of the global SFRD is more rapid, the more massive the galaxies are. While this is a similar trend we find if we assume an upper limit in SSFR (case B), their lowest stellar mass bin—centered at 10^9.6 M_☉—always appears to contribute most to the SFRD integrated over all masses. Hence, there is no clear evidence for a peak in the SFRD mass-distribution function and certainly not in the higher mass range our data supports. However, based on their findings in the local universe (Gilbank et al. 2010a), Gilbank et al. (2010b) empirically showed that a mass-dependent calibration between [O ii] luminosity and SFR is more appropriate. If true, this would modify the low-mass dominance in the derived SFRDs of Juneau et al. (2005) to correspond more closely to our result.⁵⁵

Independently, also the radio-stacking-based study by Pannella et al. (2009) suggests a strong dependence of the dust-attenuation correction on stellar mass at higher (z>1.5) redshifts. Similarly to Juneau et al. (2005), Bauer et al. (2005) derive their [O ii]-based (S)SFRs from a mass-independent calibration albeit neglecting extinction corrections.⁵⁶ Finally, Bundy et al. (2006), who favor a shift of the characteristic mass toward lower values with cosmic time, entirely base their conclusion on the stellar mass functions they derive for SF galaxies (selected by rest-frame (U − B) color and [O ii] equivalent line width) within the DEEP2 survey sample. At z ≲ 1.4 these mass functions do not show an evolution of the faint-end slope, but the Schechter parameter M*_SFG appears to decrease with cosmic time, in contrast to the already discussed broad agreement in the more recent literature according to which neither α_SFG nor M*_SFG change in this redshift range.

To conclude, we summarize our findings as follows.

• 1.  
  Up to normalization, the mass distribution function of the SFRD is a universal Schechter function at z < 1 with possible deviations only below 10⁸ M_☉.
• 2.  
  We explain this surprising constancy in shape of this SFRD function by the non-evolving slope of the SFR sequence and the constant shape of the mass function for SF galaxies.
• 3.  
  Our data at z < 1.5 clearly disfavor a strong "downsizing" scenario in which the characteristic mass of SF galaxies that contribute most to the overall SFRD—integrated over all masses at a given time—shifts toward lower values over cosmic time. The situation does not appear to change even at z>1.5 where, however, deeper data are needed to confirm our results. The characteristic mass is M_* ∼ 10^10.6 M_☉.
• 4.  
  If we assume an upper limit to the average SSFR of the order of the inverse dynamical scale ($\tau _{\rm {dyn}} \sim 0.37$ Gyr) the SFRD of galaxies less massive than 10⁹ M_☉ evolves less rapidly than the one of the dominant higher mass range above 10¹⁰ M_☉.

As all introduced parameters show this remarkable constancy throughout the redshift range probed (at least out to z ≈ 1) the redshift dependence of Equation (10) is entirely contained in the normalization $\Phi ^*_{\rm {SFRD}}$. Equation (10) becomes a separable function so that we rewrite

$$\begin{equation} \rm {SFRD}\it (M_*,z) dM_*=\Phi ^*_{\rm {SFRD}}(z)\,\varphi (\tilde{\alpha },\it M^*)\it dM_*, \end{equation} \tag{ 11 }$$

where mass dependence consequently is solely contained in the universal SFRD function φ. The global SFRD at a given redshift, integrated over all masses, is simply given by

$$\begin{equation} \rm {SFRD}(\it z) =\Phi ^*_{\rm {SFRD}}(\it z)\,\int \varphi (\tilde{\alpha },\it M^*)\it dM_*. \end{equation} \tag{ 12 }$$

In the following, we will motivate that the evolution of the integrated SFRD follows a simple power law of the form

$$\begin{equation} \Phi ^*_{\rm {SFRD}}(\it z)\,\left[M_{\odot }\, \rm {yr^{-1}}\, \rm {Mpc}^{-3}\right] \propto (1+z)^{n_{\Phi ^*_{\rm {SFG}}} + n_{\rm {SSFR}}}, \end{equation} \tag{ 13 }$$

where the two power-law indices result from the change in stellar mass density contained in SF galaxies and the increase of the (S)SFR-sequence with redshift.

As detailed in I10 and previously also found by other studies (e.g., Bell et al. 2003, 2007; Arnouts et al. 2007; Pozzetti et al. 2010) the stellar mass density of SF galaxies grows after the big bang only until z ∼ 1. At lower redshifts it stays constant. Consequently, as the shape of the mass functions does not evolve, also the Schechter parameter Φ*_SFG in the mass function is constant in this redshift regime, apart from fluctuations due to large-scale density fluctuations. As shown in Figure 10, these fluctuations are consistent with cosmic variance as estimated by Scoville et al. (2007b) and detailed in I10.⁵⁷ It is clear that cosmic variance effects are strongest at low redshifts as the effective volume sampled in a redshift bin with Δz = 0.2 increases with redshift. In the interest of simplicity and to avoid systematic errors caused by cosmic sampling variance we adopt a constant Φ*_SFG at z < 1.

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If we therefore set $n_{\Phi ^*_{\rm {SFG}}}(z<1)=0$ in Equation (13) the evolution of the integrated SFRD in the range 0 < z < 1 is entirely described by the global, i.e., mass-uniform, decline of the average SFR of SF galaxies with cosmic time. It is important to emphasize once again that this strong decline is definitely not caused by a decreasing number of SF galaxies, in particular not at all at the high-mass end. As the efficiency of star formation within SF systems appears not to change over cosmic time (Daddi et al. 2010a; Tacconi et al. 2010), we conclude that below z ∼ 1 the strongly evolving integrated SFRD must be exclusively caused by a strongly declining mass density of cold molecular gas available for star formation toward the local universe. Recent theoretical model predictions (Obreschkow & Rawlings 2009; Dutton et al. 2010) indeed support such an evolutionary behavior of the cosmic mass density of molecular hydrogen. As we already explained in Section 4.3, systematic redshift uncertainties will not influence our conclusions either as they would only be propagated along the redshift trend inferred.

At earlier epochs (i.e., z>1) we monitor the redshift dependence of the parameter Φ*_SFG as measured by I10 (Table 6). Figure 10 shows that the above choice of a power law at z>1 is a reasonable assumption and we find $n_{\Phi ^*_{\rm {SFG}}}(z>1) = -2.38 \pm 0.02$ from a fit to our COSMOS data points at 1 < z < 3. It should be mentioned that the two highest redshift points in Figure 10 (open circles) were obtained by assuming that α_SFG and M*_SFG stay constant also at z>2 where I10 did not directly measure the mass function. The normalization $\Phi ^*_{\rm {SFG}}(z>2)$ was therefore obtained by matching the number densities to those observed in the mass-complete regime of our data when fixing the parameters α_SFG and M*_SFG to their average values at z < 2. This extrapolation is supported by the total mass function at z>2 measured by Marchesini et al. (2009). At these high redshifts we assume the total mass function to be representative for the SF population as quiescent galaxies are not expected to significantly contribute to the number density. The study by Marchesini et al. (2009) was carried out based on data taken in various survey fields for which deep NIR imaging is available allowing them to estimate Schechter parameters for the total mass function also in two high-z bins in the ranges 2 < z < 3 and 3 < z < 4. Within the errors all our extrapolated Schechter parameters at 2 < z < 3 are in good agreement with their results. In particular, the exponential cutoff mass at 2 < z < 3, $M^*_{\rm {ALL}} \equiv M^*_{\rm {SFG}}=10.96$, they find⁵⁸ agrees remarkably well with our assumption (see Table 6). Their normalization $\Phi ^*_{\rm {ALL}} \equiv \Phi ^*_{\rm SFG}$ does not deviate significantly from our prediction and the evolutionary trend we suggest is also supported by their data (see Figure 10). It is clear, however, that future measurements of the stellar mass function at z>1.5—based on deeper NIR or mid-IR data—are critical to confirm the validity of our assumptions. Based on the currently available data we conclude that the stellar mass build-up in the SF galaxy population inevitably leads to a shallower decline of the integrated SFRD between 3>z>1 compared to the steep decline between 1>z>0.

Table 6. Schechter Parameters for the Stellar Mass Function of Star-forming Galaxies

$\Delta z_{\rm {phot}}$	αSFG	log(M*SFG)	Φ*SFG
		(M☉)	(10−3 Mpc−3 dex−1)
0.2–0.4	−1.32+0.01−0.01	11.00+0.03−0.03	1.15+0.08−0.07
0.4–0.6	−1.32+0.01−0.01	11.04+0.03−0.03	0.70+0.05−0.04
0.6–0.8	−1.32+0.01−0.01	10.95+0.02−0.02	0.86+0.05−0.05
0.8–1.0	−1.16+0.01−0.01	10.86+0.02−0.02	1.38+0.06−0.06
1.0–1.2	−1.19+0.02−0.02	10.92+0.02−0.02	0.94+0.05−0.05
1.2–1.5	−1.28+0.02−0.02	10.91+0.02−0.02	0.68+0.03−0.03
1.5–2.0	−1.29+0.02−0.02	10.96+0.02−0.02	0.46+0.02−0.02
2.0–2.5	−1.29+0.03−0.03	10.95+0.03−0.03	0.32+0.06−0.06
2.5–3.0	−1.29+0.03−0.03	10.95+0.03−0.03	0.27+0.05−0.05

Notes. At z < 2 all parameters have been derived by I10. At z>2 we assume a non-evolving shape so that α_SFG and M*_SFG are taken to be the average of the respective lower z values. Φ*_SFG was then derived by matching the number densities to those observed in the mass-representative regime of our data.

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In order to validate our parameterization of the evolution of the integrated SFRD we proceed as follows. First, we simply add up all SFRDs measured in different mass bins in a given redshift bin (all data points shown in the corresponding panels in Figure 8) obtaining a lower limit to the integrated SFRD at that epoch. Second, in order to account for the contribution of the low-mass (log(M_*) < 9.5) SF population that we cannot directly measure, we integrate Equation (10) from our mass limit down to 10⁵ M_☉. As we discuss in Section 5.1 a single value of the index $\tilde{\alpha }$ in Equation (10) might not be valid over the entire low-mass range as the SSFR–M_* relation might flatten as soon as an upper limiting SSFR is reached. The upper left panel in Figure 11 hence shows the two alternative extrapolations and all obtained data are given in Table 7. The contribution of the integral generally lifts the SFRD at a given redshift by a linear factor of ∼1.4 if we assume an SSFR limit (red filled circles) and ∼1.7 if a single low-mass end slope $\tilde{\alpha }$ is used (red open circles), suggesting the stacking analysis missed ∼30% or ∼40%, respectively, of the integrated SFRD. The differences between both extrapolations are largest at z>1 below which either method yields practically the same results. As pointed out in Section 5.1 our data cannot clearly rule out any of the two alternative low-mass Schechter functions proposed. Consequently, our extrapolations overlap within their individual uncertainty ranges at all redshifts. In the following, we favor, however, the extrapolation that includes the SSFR limit as it assures a more conservative approach compared to the generally larger values the alternative extrapolation yields.

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Table 7. Total SFR Density as a Function of Redshift (Cosmic Star Formation History)

z	SFRD$_{\rm {obs}}(z)$ (M☉ yr−1 Mpc−3)	SFRD$_{\rm {int}}(z)$ (M☉ yr−1 Mpc−3)
0.30	0.011 (0.011)+0.001−0.001	$0.018^{+0.002}_{-0.006}\qquad\quad \left(0.019^{+0.004}_{-0.007}\right)$
0.50	0.015 (0.015)+0.001−0.001	$0.023^{+0.002}_{-0.006}\qquad\quad \left(0.025^{+0.004}_{-0.008}\right)$
0.70	0.025 (0.025)+0.002−0.002	$0.039^{+0.003}_{-0.009}\qquad\quad \left(0.043^{+0.007}_{-0.015}\right)$
0.90	0.043 (0.043)+0.003−0.003	$0.055^{+0.002}_{-0.008}\qquad\quad \left(0.058^{+0.005}_{-0.010}\right)$
1.10	0.048 (0.047)+0.004−0.003	$0.061^{+0.003}_{-0.005}\qquad\quad \left(0.073^{+0.010}_{-0.016}\right)$
1.40	0.048 (0.048)+0.004−0.004	$0.063^{+0.004}_{-0.007}\qquad\quad \left(0.070^{+0.008}_{-0.018}\right)$
1.80	0.070 (0.069)+0.007−0.008	$0.095^{+0.014}_{-0.014}\qquad\quad \left(0.119^{+0.040}_{-0.048}\right)$
2.25	0.066 (0.062)+0.007−0.006	$0.098^{+0.011}_{-0.009}\qquad\quad \left(0.115^{+0.043}_{-0.046}\right)$
2.75	0.077 (0.077)+0.009−0.011	$0.121^{+0.017}_{-0.017}\qquad\quad \left(0.175^{+0.236}_{-0.099}\right)$

Notes. The central column states the number density corrected (raw) sum over all mass bins at a given redshift of the product of the average SFR and the total number of galaxies contained in the corresponding bin down to the redshift-dependent limiting masses of this study. It is hence the sum of the data points within each panel of Figure 8 and—at least out to z = 1.5 a robust direct measurement of the total dust unbiased SFRD for galaxies more massive than ∼3.2 × 10⁹ M_☉. The values in the right column additionally take into account the not directly measured low-mass end where we integrate over the SFRD function at a given redshift as introduced in Section 5.1 while we assume a potential upper SSFR limit (see Section 4.1 for details). The values in brackets result from deriving the low-mass end contribution by integrating the single Schechter models of the SFRD functions and hence assuming no upper limit in SSFR. Like all other results presented in this work all values are based on a Chabrier (2003) IMF.

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It is evident that the number density corrections at our low-mass end discussed in Section 5.1 have almost no impact on our direct measurements (depicted as black open circles in the upper left panel of Figure 11) of the SFRDs. Even more, at z < 1.5 practically no corrections were necessary which highlights again that our inferences are most robust at these redshifts. It is therefore justified to regard the corrected values (depicted as black filled circles in the upper left and lower panels of Figure 11) as a direct and dust unaffected average measurement of the SFRD for SF galaxies to a lower mass limit of M_* ∼ 3.2 × 10⁹ M_☉. The evolutionary power law, scaled to our data points and with the indices in the two redshift regimes, matches well the observed CSFH with respect to our measured data points presenting lower limits as well as to the integrated ones. This may seem surprising as our evolutionary model does not take into account differential effects with mass as introduced by assuming an SSFR limit (case B in Section 5.1). However, since the bulk of the mass-integrated SFRD is contained in our direct-stacking-based measurements even at high z—which show a mass-independent evolution—the model represents a good approximation.

In Figure 11, we also compare⁵⁹ our data to the CSFH derived from confirmed SF radio sources within the COSMOS field in conjunction with extrapolations based on two distinct evolved local radio LFs (see Smolčić et al. 2009a for details). This comparison shows how good a deep radio survey constrains the CSFH and leads us to slightly favor the extrapolations based on the Condon (1989) radio LF in the Smolčić et al. (2009a) study as already our mass-limited, direct, stacking-based measurements of the SFRD on their own already reach the values which they inferred using the Sadler et al. (2002) radio LF at any z < 1.

The upper right panel of Figure 11 shows our data along with other radio-based measurements (Haarsma et al. 2000; Machalski & Godlowski 2000; Condon et al. 2002; Sadler et al. 2002; Serjeant et al. 2002; Smolčić et al. 2009a) and the radio-stacking derived CSFH by Dunne et al. (2009). Since the referenced measurements based on radio detections have been extensively discussed in Smolčić et al. (2009a), we will focus our comparison on the study of Dunne et al. (2009) as it is methodologically closest to our study. Here, the extrapolations toward faint sources are based on the evolving K-band LF with the fixed faint-end slope presented by Cirasuolo et al. (2010). As we previously pointed out, the evolutionary trends Dunne et al. (2009) find are in good agreement with our results. Also on absolute scales the results of both studies are basically indistinguishable with significant⁶⁰ deviations only at the highest redshifts probed in our study. However, at z>1.5 the trends observed tend to suggest different conclusions as Dunne et al. (2009) find a clear peak of the CSFH around z = 1.5–2 followed by a strong decline of the SFRD with redshift. Indeed, at the highest redshifts probed in our study the SFRD extrapolated by Dunne et al. (2009) does not exceed our direct measurement (without extrapolations, see the top left panel in Figure 11). Hence, one would have to assume that galaxies below ∼10¹⁰ M_☉ do not contribute at all to the mass-integrated SFRD at z>2 in order to support this peak based on our data.

Especially when compared to dust extinction corrected UV-based studies the existence and location of the CSFH peak as measured by a dust-unbiased star formation tracer is important. Recent UV-based measurements (Reddy & Steidel 2009; Bouwens et al. 2010) suggest a peak of the CSFH at around z = 2–2.5 and assume that the dust obscuration for the bulk of SF galaxies does not dramatically change out to z ∼ 4 (as shown by e.g., Bouwens et al. 2009; Finkelstein et al. 2010; McLure et al. 2010; Wilkins et al. 2010). Our rising SFRD at z>2 suggests that SF galaxies at these redshifts have a somewhat higher dust content or obey a different reddening law than the corresponding sources at lower redshifts. Whether dust-obscured sources at z>2 are lost in optical/UV based measurements of the CSFH cannot be definitely answered given the mentioned large error bars our data show. Future Herschel studies of the total IR luminosity evolution at z>2 should reveal potentially larger dust reservoirs in these systems. The even more rapid decline of the radio-based SFRD as derived by Dunne et al. (2009) appears to support the picture drawn by, e.g., Bouwens et al. (2010) that obscured star formation does not significantly contribute to the global SFRD at z ≫ 4. Such conclusions should, however, be treated with caution as it is highly unclear at such early cosmic times what extrapolation to the directly measured radio derived SFRDs are needed given the high stellar mass limits (see also Gallerani et al. 2010).

Our integrated CSFH is further supported by recent studies carried out at mid-IR (Rodighiero et al. 2010b) and far-IR (Gruppioni et al. 2010) wavelengths. The latter study is based on ∼210–240 Herschel/PACS detections at 100 and 160 μm in 150 arcmin² within the GOODS-N field and provides us with a deep view of the dust-unbiased CSFH. The lower panel in Figure 11 shows that the agreement of the Herschel-based results presented by Gruppioni et al. (2010) with our radio-stacking derived mass-integrated CSFH is striking. Out to z ∼ 1 we also find a broad agreement with the measurements of Rodighiero et al. (2010b). While Gruppioni et al. (2010) show only lower limits at z ≳ 1 below this redshift both studies measure an evolution of (1 + z)ⁿ with n = 3.8 ± 0.3 (0.4). A recent 24 μm based study by Rujopakarn et al. (2010) confirms this result measuring n = 3.4 ± 0.2. All these values are in remarkable agreement with our average measured evolution of the SSFR sequence of $\langle n_{\rm {SFG}} \rangle = 3.5 \pm 0.02$ (see Section 4 and Table 5⁶¹) and hence are a strong support for both our measurement and our parameterization given in Equation (13), especially out to z = 1. It is, however, worth mentioning that our work, compared to the other studies mentioned, draws on a far larger sample.

Finally, we want to stress the fact that any shallower high redshift trend in the evolution of the SSFR sequence also at the high-mass end would indeed lead to a decline in the evolution of the SFRD as Equation (13) suggests. This scenario cannot be ruled out given our data as the SSFR at the low-mass end of our sample tends to flatten and the high-mass end might follow at slightly higher redshifts based on the dynamical time arguments presented in Section 4.2 that result in a global upper limit to the average SSFR. Hence, again, a deeper mid-IR selected sample of SF galaxies is needed to accurately probe the regime above z = 1.5 where the CSFH is supposed to peak.

Due to the non-uniform noise distribution in the VLA-COSMOS map, the input samples used for stacking are ill-defined to some extent. Solely discarding the high-noise edge regions does not remedy the fact, that there is significant variation of the rms background noise in the cutout postage stamps originating from a broad spatial distribution across the field. Our aim is to find the best estimator for the representative value of the underlying population. Therefore, the sample should consist of a random and independent set of sources drawn from this population under equal circumstances. To approach the last condition, that is not achievable in observational reality, we have to compare the outcome of the stacked sample to that of a weighted sample, in which those stamps gain more influence, that lie in low noise regions.

Regarding the mean-stacking technique it is statistically well known, that appropriate weights are found in the reciprocal variance of each particular stamp's noise pixel sample where the variance of the ith stamp is defined as ${\rm Var}_i \equiv {\rm {Var}}\left(X^{{\rm bg},i}_{N^{\rm {bg}}}\right)=\sigma _{\rm {bg}}^2\left(X^{{\rm bg},i}_{N^{\rm {bg}}}\right).$ As explained above $X^{\rm {bg},i}_{N^{\rm {bg}}} \equiv \lbrace x_{i_1}^{\rm {bg}},\ldots,x_{i_N^{\rm {bg}}}^{\rm {bg}}\rbrace$. The noise-weighted mean of the sample X_N of peak fluxes can thus be considered the mean of the weighted sample $\widetilde{X}_{N}$, where the constituents $\widetilde{x}_i$ of $\widetilde{X}_{N}$ are defined as

$$\begin{equation} \widetilde{x}_i = \widetilde{w}_i x_i \equiv N\frac{ {\rm Var}_i^{-1}}{\sum _i {\rm Var}_i^{-1}} x_i, \qquad {\rm where} \quad x_i \in X_{N}. \end{equation} \tag{ B1 }$$

With the definitions W_i = Var⁻¹_i and w_i = W_i/∑_iW_i it is easily shown, that the mean of the $\widetilde{x}_i$ defined in Equation (B1) indeed equals the noise-weighted mean of the x_i:

$$\begin{equation} \langle \widetilde{X} \rangle =\frac{1}{N} \sum _i \widetilde{x}_i = \frac{1}{N} \sum _i \widetilde{w}_i x_i = \sum _i w_i x_i = \frac{\sum _i W_i x_i}{\sum _i W_i}. \end{equation} \tag{ B2 }$$

The above discussion leads to the suggestion, that in the presence of varying rms noise in a given sample, the sample $\widetilde{X}_{N}$ is the appropriate one to consider not only with respect to the mean value of the sample. It seems reasonable, that also its median is the best estimator for the median of the underlying population, because both computed quantities, the median and the mean, are then referring to the same sample. We will refer to this choice of an estimator in the following as a noise-weighted median.

In the above discussion, we justified that neither the observed nor the intrinsic distribution of peak fluxes is expected to be Gaussian. This needs to be taken into account no matter if we are looking for an appropriate uncertainty range to the median or mean estimator for a given sample. In order to obtain a 68% confidence interval not relying on normality of the underlying parent distribution we therefore chose a bootstrapping technique for the statistical parameter of choice.

In each case, we obtain the limits of the confidence interval by a bootstrapped Student's t-distribution. This technique is called studentizing. A (1 − α) confidence interval for a parameter $\overline{X}$ in traditional statistics is given by

$$\begin{equation} {\rm CI}_{\alpha /2} = \overline{X} \pm t_{\alpha /2} \frac{s_{\overline{X}}}{\sqrt{N}}\,, \qquad s_{\overline{X}}: \; \mbox{standard deviation of }\overline{X}. \end{equation} \tag{ B3 }$$

Here t_α/2 denotes the α/2 percentile of the classical t-distribution which is equal to the (1 − α/2) percentile due to the symmetry of Student's distribution. Bootstrapping a t-distribution means to circumvent the assumption of a normally distributed population by deriving the quantity

$$\begin{equation} t_i^{\ast } = \frac{\overline{X}^{\ast }_i - \overline{X}}{s^{\ast }_{\overline{X}^{\ast }_i}/\sqrt{N}} \end{equation} \tag{ B4 }$$

for $i=1,\ldots,N_{\rm {bootstrap}}$ samples drawn from the original sample of peak fluxes with replacement. In case of $\overline{X} \equiv \langle X \rangle$ being the sample mean one thus has to compute the sample mean 〈X〉 as well as all means of the bootstrapped samples 〈X*〉_i including standard deviations $s^{\ast }_{\langle X^{\ast } \rangle _i}$. The resulting distribution of $N_{\rm {bootstrap}}$ t*-values is then used to compute the upper and lower confidence limits by taking its α/2 and (1 − α/2) percentiles:

$$\begin{eqnarray} CI_{\rm {up}}^{1-\alpha } &=& \overline{X} + t^{\ast }_{\alpha /2} \frac{s_{\overline{X}}}{\sqrt{N}} \end{eqnarray} \tag{ B5 }$$

$$\begin{eqnarray} CI_{\rm {low}}^{1-\alpha } &=& \overline{X} - t^{\ast }_{1-\alpha /2} \frac{s_{\overline{X}}}{\sqrt{N}}, \end{eqnarray} \tag{ B6 }$$

where in general we chose 1 − α = 0.68 obtaining thus a 68% confidence interval. Here $s_{\overline{X}} \equiv s_{\langle X\rangle }$ still is just the standard deviation of the original sample. In case of $\overline{X} \equiv \rm {Med}(X)$ denoting the sample median as the parameter of choice we have to face the problem that the denominator of Equation (B4) does not provide us with an estimator of the standard error of the median. In order to estimate this latter quantity we need to access the empirical standard deviation of a sample of medians being representative for the median of the current bootstrapped sample. Starting from this sample we thus generate a number of new bootstrapped samples hence performing a bootstrapping within the bootstrapping procedure.⁶² The standard deviation $s^{\ast }_{\rm Med(X^{\ast \ast }_i)}$ of these sub-samples' medians is then used as an estimator for the standard error of the single outer bootstrapped median as given by the denominator of Equation (B4). In order to use Equations (B5) and (B6) we furthermore need the standard error of the original sample's median. This is estimated by computing the median of each outer bootstrapped sample and taking the standard deviation $s_{\rm Med(X^{\ast })}$ of this sample of medians as the standard error.

I10 have shown that the color selection criterion (${\rm NUV}\hbox{--}r^+)_{\rm {temp}} < 1.2$ leads to a morphologically clean sample of late-type spiral and irregular galaxies with template SED-based SSFRs that are clearly separated from the passive population (see Section 2.4). This color threshold is somewhat arbitrary (as it is less well motivated than the cut we applied to select SF systems) but by virtue of being substantially bluer than our original choice it minimizes contamination by passive galaxies.

We derived SSFRs as a function of redshift and mass in the same way as before for galaxies with (${\rm {NUV}}\hbox{--}r^+)_{\rm {temp}} < 1.2$. Although the exclusion of systems with intermediate SF activity has reduced the sample size considerably, it was still possible to cover the same dynamic ranges. Only the binning scheme has been slightly modified for this strongly star formation population (see Figure 13). Its SSFRs usually are significantly higher compared to our original choice of SF galaxies, the slope β of the SSFR–M_* relation is shallower at low to intermediate redshifts and thus in excellent agreement with those literature results for SF galaxies discussed in Section 4.4 that report an almost flat SSFR sequence. At the high-z end we see a steepening of the slope β (i.e., an "upsizing" trend), similar or even more evident to what was found by Rodighiero et al. (2010a) and, at lower significance, also by Oliver et al. (2010). The evolutionary exponent n is consistent with previous measurements as well (see, e.g., Pannella et al. 2009).

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The bluer color threshold hence is able to reproduce most literature findings albeit with SSFRs that tend to be comparatively high, especially at low redshift. However, it has yet to be confirmed that the galaxy population selected in this way is representative of the entire SF population.

We cross-matched our 3.6 μm selected catalog with the K-band selected catalog for the COSMOS field (McCracken et al. 2010) and thus obtain a magnitude calibration in the crucial wavebands that allows us to apply the BzK selection criterion of Daddi et al. (2004). Figure 14 shows the BzK diagram for our sample, with galaxies in six redshift slices color coded according to their (${\rm {NUV}}\hbox{--}r^+)_{\rm {temp}}$ color as described in Section 2.4. Our "star-forming" sample is the union of all galaxies plotted in blue and green.

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At z>1.5, the sBzK criterion (all galaxies to the left of the diagonal line in each panel) is established to efficiently select normal SF systems. Figure 14 illustrates that the selection window for sBzK galaxies is populated by both the most actively SF sources (blue dots) and the majority of the sources with intermediate SFRs (plotted in green; i.e., the rest of the SF sample used throughout this article). Only a small number of objects with moderate SF activity fall into the passive BzK region in the upper right of each panel. This is the reason for the aforementioned excellent agreement of our results with Pannella et al. (2009) at z ≈ 2.1; their and our sample are virtually indistinguishable and both studies rely on the same radio data. More importantly, however, the BzK diagram strongly supports our original selection of SF objects in the crucial redshift regime z>1.5, where we have just shown that previously reported changes in the slope β can be mimicked by simply selecting only very blue objects, hence, most actively forming systems.