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Appendix A: Comparison of the GSMFs obtained with different recipes

Fig. A.1 Comparison of the GSMF obtained with different photometric redshift recipes. The stellar mass function of galaxies at 3.5 ≤ z ≤ 7.5 in the CANDELS GOODS-South and UDS fields with the Bayesian photometric redshifts is shown by the black triangles and the solid continuous curves. The red triangles, green circles, cyan triangles, magenta and pink asterisks show the GSMFs obtained using the individual photometric redshifts of five different groups that have been used to derive the Bayesian photo-z described in Dahlen el al. (2013).

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Fig. A.2 Comparison of the GSMF obtained with different star formation histories. The stellar mass function of galaxies at z ≥ 3.5 in the CANDELS GOODS-South and UDS fields with our standard BC03 fit (exponentially declining SFHs) is shown by the black triangles and the solid continuous curves. The red squares and the blue stars show the GSMFs derived using exponentially increasing and a t2/τ × exp( − t/τ) SFH, respectively. All these star formation histories have been tested without the nebular contribution.

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Fig. A.3 Comparison of the GSMF with and without an allowed nebular contribution. The stellar mass function of galaxies at 3.5 ≤ z ≤ 7.5 in the CANDELS GOODS-South and UDS fields with our standard BC03 fit (exponentially declining SFHs and no nebular contribution) is shown by the black triangles and the solid continuous curves. The blue asterisks show the GSMF derived using BC03 models and exponentially declining SFHs, but this time including the contribution of nebular lines and nebular continuum.

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	Fig. A.4 Stellar mass function of galaxies at 3.5 ≤ z ≤ 7.5 in the CANDELS UDS field (blue squares) is compared with the one derived from the GOODS-South data (black circles). At 6.5 <z< 7.5 the fit to the data point was not derived due to the small range in mass of the observed data in the individual fields.
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	Fig. A.5 Comparison of the GSMF with and without AGN. The stellar mass function of galaxies at 3.5 ≤ z ≤ 7.5 in the CANDELS GOODS-South field considering only normal galaxies is shown by the black triangles. The red asterisks show the derived GSMF including AGN.
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In this section we show the comparison of the GSMF derived with our standard approach (GOODS-South and UDS fields, Bayesian photometric redshifts, BC03 library (Bruzual & Charlot 2003), exponential declining SFHs, no nebular contribution, no AGN) against the different GSMFs obtained varying only one ingredient at a time. Figure A.1 shows the GSMF at 3.5 <z< 7.5 obtained with different photometric redshift recipes available within the CANDELS team (Dahlen el al. 2013). Figure A.2 investigates the impact of different SFHs, Fig. A.3 shows the effect of the nebular contribution on the mass estimates. Finally, Figs. A.4 and A.5 illustrate the cosmic variance effect (or field-to-field variation) and the contribution of AGN to the GSMF. In this case we must point out that the information available for the UDS field (X-ray coverage, deep spectroscopy, variability) is not equivalent to the rich data set in GOODS-South, so at the present stage only some type-1 AGN can be found in the current UDS galaxy sample adopted in this paper. For this reason only the GOODS-South field has been used to carry out the comparison in Fig. A.5.

Appendix B: The correction of the Eddington bias

As already discussed in the main text, the stellar mass of a single galaxy is not unequivocally determined due to the uncertainties on the photometric redshift determination and on the mass-to-light conversion adopted to derive the stellar mass from the observed SED. These uncertainties must be properly taken into account when the observed data points of the mass function are fitted with a parametric function. This is the so-called Eddington bias (Eddington 1913), and as we will show here it could systematically affect the derivation of the Schechter parameters for the observed mass function.

To correct for this effect we used the probability distribution function in mass PDF(M | z) that we derived for each individual galaxy during our SED fitting procedure, as described in Sect. 4.1.1. For each galaxy in a given redshift and mass bin (i.e. 3.5 ≤ z ≤ 4.5 and M₁ ≤ M ≤ M₂), we summed up all the individual PDF(M | z) in the given redshift interval, after normalizing them to unit probability. This procedure gives the probability P(M_j,M_i) for a galaxy with an observed mass M_j (resulting from the best fit of its SED) to have a mass M_i still compatible with its photometry and photometric redshifts PDF(z) derived from the Bayesian analysis, as described in Sect. 2.3.

Figure B.1 shows the probability distribution functions for the GOODS-South and UDS fields in different mass bins, both at 3.5 ≤ z ≤ 4.5 (top) and at 5.5 ≤ z ≤ 6.5 (bottom). The dark-green curves are associated with galaxies in the bin centred at log (M/M_⊙) = 8.9, while red, blue, green, cyan and black curves are associated with galaxies with masses log (M/M_⊙) of 9.3, 9.7, 10.1, 10.7 and 11.1 respectively. For comparison, we plot on the same figure also the PDF(M) adopted by Ilbert et al. (2013) at z = 4, which is the product of a Gaussian with σ = 0.5 and a Lorentzian distribution with τ = 0.04 ∗ (1 + z). The CANDELS PDFs shown in Fig. B.1 are based on the BC03 stellar library and on the choice of physical parameters (star formation histories, age, metallicity, dust extinction, IMF) adopted in this paper. The PDFs could vary slightly by adopting different ingredients. However, as shown in the main text, the uncertainties due to photometric scatter and photometric redshift uncertainties are much larger than those due to different choices of the physical ingredients.

Fig. B.1 Probability distribution functions (PDFs) of stellar mass for galaxies with measured stellar mass M at different masses and redshifts, resulting from the Monte Carlo simulations described in the text. The PDFs were averaged over all galaxies contained in contiguous bins with separation in mass of 0.2 in log  (M), although we plot here only a few examples. The upper panel presents the PDFs at z = 4 ± 0.5, the lower panel at z = 6 ± 0.5. In both panels the dark-green curves are associated with galaxies in the bin centred at log (M/M⊙) = 8.9, while red, blue, green, cyan, and black curves are associated with galaxies with masses log (M/M⊙) of 9.3, 9.7, 10.1, 10.7 and 11.1 respectively. Solid lines refer to GOODS-South, dashed to UDS. The dotted green line shows for comparison the PDF adopted at all masses by Ilbert et al. (2013) at z = 4.

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From this plot we can draw some conclusions. First, and most important, the error in the mass estimation is not constant at all masses, as usually assumed (e.g. Ilbert et al. 2013): it is indeed smaller for higher-mass galaxies than for lower ones. This is expected since larger photometric errors lead to wider ranges of acceptable photometric redshifts and spectral models. We also note that the error is not symmetric and, especially at M ≥ 10^{9 − 10} M_⊙ starts to show a tail towards lower masses. The second important aspect is that at higher redshifts the combined PDFs are wider and the asymmetry is more pronounced than at z = 4, due to a combination of larger photometric uncertainties and progressive dimming of the galaxies for a given stellar mass. Thus, this plot shows the relative enhancement of the uncertainties in PDFs towards low-mass objects, independent of the adopted method for the stellar mass derivation. We finally note that the PDF(M | z) adopted by Ilbert et al. (2013) is smaller than our own, at the same mass and redshift, and substantially smaller than our own for faint galaxies. Given the higher S/N and quality of our photometric data, this may likely reflect a more conservative estimate of the implied errors in our computation, which have been derived adopting a different technique w.r.t. Ilbert et al. (2013).

Armed with this full characterization of the error on the estimated mass, we can evaluate the impact of such errors on the estimate of the (binned) GSMF. This is accomplished by convolving any input GSMF with the error distribution of Fig. B.1, at the corresponding redshift. For any given input GSMF we compute its expected values Φ(M_j) in the same mass bins (with a step of 0.2 in log (M)) used to derive the various PDF(M | z). The expected mass function in output is , where N is the number of bins adopted to compute the PDF(M | z).

To illustrate the effect of this procedure, and the differences with respect to the previous analysis, we first take a representative input mass function (with α ≃ − 1.6 and M_∗ ≃ 11) and convolve it with the PDF of Ilbert et al. (2013) at z = 4, namely a constant function at all masses. Figure B.2 shows the convolved mass function (empty blue squares) resulting from the intrinsic mass function (solid line) after applying the convolution process described above. We obtain a behaviour for the convolved GSMF similar to what has been found by Ilbert et al. (2013; their Fig. A.2), namely that the low-mass side is unaffected by the Eddington bias while the density at the high-mass end is enhanced.

Fig. B.2 Effect of different prescription of the Eddington bias on the observed GSMF. The black line shows a Schechter function representing a GSMF with α ≃ − 1.6 and M∗ ≃ 11. The open blue squares show the resulting GSMF in bins of 0.2 in log M after convolution with a constant PDF at all masses, as adopted by Ilbert et al. (2013) at z = 4. The green filled squares represent the GSMF after convolution with the more realistic mass-dependent PDF that widens when mass decreases as we find in CANDELS (Fig. B.1).

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Then, we consider an error distribution similar to that found in CANDELS, i.e. with larger uncertainties at lower masses (Fig. B.1). We adopt the same functional form of Ilbert et al. (2013), namely the product of a Gaussian and a Lorentzian distributions, but we let the σ of the Gaussian change as a function of mass in order to coarsely reproduce the observed PDFs at z = 4 (hence smaller than the Ilbert et al. (2013) one at log (M/M_⊙) > 10.3 and larger at smaller masses). Figure B.2 shows with filled green squares the GSMF resulting from the same intrinsic mass function (solid line) adopted in Fig. B.2 after the convolution process with a variable PDF in mass.

The combination of wider PDFs at lower masses and asymmetric distributions has an interesting behaviour on the GSMF shape: in the low-mass regime, M ≤ 10⁹ M_⊙, the error is symmetric and contributes to a slight enhancement in the number density of galaxies at the low-mass end, typically fitted with a power law. This results into a steepening of the GSMF. According to Eddington (1913), the effect is higher for steeper distributions and for larger errors. At the high-mass side, instead, the errors in mass are smaller and show an asymmetry towards lower masses. As a consequence, the exponential tail of the mass function is less affected by this scatter, and the observed data points are a good representation of the intrinsic GSMF, at least at z = 4 for the CANDELS GOODS-South and UDS fields. This behaviour is markedly different from what is derived assuming instead a constant error on PDF(M | z), even if the average error is adopted.

Moving to higher redshift, we find that at z = 5 the situation is similar to z = 4, while at z = 6 the effect of noise in the mass estimate becomes more severe and hence the correction for the Eddington bias is larger and more uncertain, as shown in Fig. B.1 (lower panel). Figure B.3 shows the effect of the Eddington bias correction at z = 6. The wide uncertainties in mass, both for faint and for bright objects, affect the GSMF at all scales, producing a steepening of the low-mass side and a pronounced increase of the exponential tail at high masses.

	Fig. B.3 Effect of the Eddington bias on the observed GSMF at z = 6 (black dots). The black line shows a Schechter function representing the resulting best fit GSMF at z = 6. The open blue squares show the resulting GSMF after convolution with the observed PDFs at z = 6 (Fig. B.1, bottom panel).
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For instance, we find that there is a small but non-negligible probability (~10^-3) for a galaxy at z = 6 to be scattered from a mass of 10^9.3 M_⊙ to 10^10.5 M_⊙, while it is ~10^-6 at z = 4.

At z ≃ 7 these effects become so large that a proper treatment of the Eddington bias is simply impossible with the present data. For instance, the probability that the same galaxy at a mass of 10^9.3 M_⊙ is scattered to 10^10.5 M_⊙ is as large as ~10^-2. Thus if we observe a density of galaxies of Φ = 10^-5 Mpc^-3 Mag^-1 at M = 10^10.5 M_⊙, this can be entirely due to galaxies at M = 10^9.3 M_⊙ (with Φ = 10^-3 Mpc^-3 Mag^-1) that are scattered to higher masses due to uncertainties in their stellar mass estimation. Since this correction is so important at the high-mass end of the z> 6 GSMF and the derivation of the PDFs at such low levels of probability depends critically on the details (photometric redshifts, stellar libraries adopted, star formation histories, grid of age, dust, metallicity), we can conclude that at the present stage the Eddington bias correction at z = 7 is highly uncertain. Moreover, at z = 7 the PDFs are very noisy due also to the low number statistics (at mass greater than 10^10.3 M_⊙ we have only 4 galaxies in the whole GOODS-South and UDS fields).

The derivation of the best-fitting Schechter functions have been carried out using the formalism described above. For any possible combination of the Schechter parameters α, M^∗, and Φ^∗, we compute the convolved GSMF using the observed PDFs in mass and we compare it with the observed mass function. We scan the three parameters of the Schechter function to find the best fit solution by a χ² minimization. The GSMFs presented in the main text have been computed accordingly.

We note that, for the reasons described above, we decided not to apply the proper correction for the Eddington bias in the z = 7 GSMF. We adopt at z = 7 the same PDFs derived at z = 6, which are less noisy, as a conservative assumption. The small range in masses sampled by our GSMF at z = 7 results in large uncertainties in the best fit Schechter function parameters due to degeneracies between α, M^∗, and Φ^∗, as shown in Fig. 11. We have verified that due to these degeneracies the parameter space allowed at 1σ by the present data is wide and it does not depend strongly on whether we adopt the PDFs at z = 6 or the ones determined at z = 7 to correct the Eddington bias in the redshift range 6.5 <z< 7.5.

Appendix C: The impact of neglecting to correct for Eddington bias at high redshift

In Sect. 6 we have shown that the uncertainties in the measurement of the galaxy stellar mass do produce a systematic effect in the output GSMF, that must be taken into account when one derives the best-fitting Schechter parameters. However, we have also shown that accurate corrections for this effect are difficult to estimate, partly because they are model-dependent and partly because of the limited statistics available, especially at the highest redshifts.

For the sake of completeness, we report here the results on the GSMF fitting without the corrections for the Eddington bias. This is useful to understand what are the uncertainties at work when one is dealing with the mass function, and to warn the reader about the consequences of neglecting or underestimating this correction. We would like to stress that, given the very high quality of the data used here, other surveys with data of lower S/N or narrower wavelength range are even more affected.

Following the technique described in the previous sections, but removing any correction for the Eddington bias, we have obtained the GSMF that is shown in Fig. C.1. The derived uncertainties on the Schechter function parameters α, M^∗ and Φ^∗, are also shown in Fig. C.2.

Fig. C.1 GSMFs from z = 4 to z = 7 in the CANDELS UDS and GOODS-South fields. At variance with the main paper, we have neglected here the effects of the uncertainties in the stellar mass (the so-called Eddington bias). The error bars take into account the Poissonian statistics and the uncertainties derived through the Monte Carlo simulations. The solid continuous curves show the best-fitting Schechter function.

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It is immediately clear that the results are significantly different from our main analysis. The slope α is steeper than our best fit and further steepens with redshift moving from α = −1.8 at z = 4 to α ≃ − 2 at z = 6. The characteristic mass log (M^∗), on the contrary, evolves only marginally from z = 4 to z = 6. This is exactly what is predicted by our analysis of the Eddington bias, that is expected to artificially steepen the slope (at all redshifts) and progressively increase the GSMF in our higher redshift bins.

These changes are also reflected in the derived evolution of the stellar mass density that is reported in Fig. C.3. As a result of the steeper slope and higher M^∗, the resulting ρ_M is significantly above the integrated evolution of the star formation rate density.

These results underline the importance of a careful description of the Eddington bias in the estimate of the GSMF.

Fig. C.2 Evolution of the three parameters (α, M∗, Φ∗) of the GSMF with redshift, neglecting the effect of the Eddington bias. The bottom-right panel shows the dependencies of the Stellar Mass Density (ρM in unit of  M⊙ Mpc-3) from the parameter α. The stars mark the position of the best fit of the observed GSMF with a Schechter function, while the triangles indicate the position of the best fit from the maximum likelihood procedure.

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Fig. C.3 Redshift evolution of the stellar mass density (SMD) at 3.5 <z< 7.5 obtained in the CANDELS UDS and GOODS-South fields presented in this paper (black points), when the effect of uncertainties in the stellar mass (the so-called Eddington bias) are not taken into account. The SMD is compared to the lower redshift data from different surveys. ρM is in units of  M⊙ Mpc-3 and has been obtained by integrating the best fit mass functions from Mmin = 108 M⊙ to Mmax = 1013 M⊙. All the SMDs have been converted to a Salpeter IMF for comparison. The error bars of the CANDELS data have been computed using the same Monte Carlo simulations developed to derive the uncertainties on the Schechter function parameters. The short-dashed line is the stellar mass density obtained integrating over cosmic time the star formation rate density (SFRD) of Hopkins & Beacom (2006). The long-dashed line is the SMD from the SFRD of Reddy & Steidel (2009). The solid line is the SMD obtained by integrating the SFRD of Behroozi et al. (2013), while the dotted line is the SMD derived by integrating the new fit of the Hopkins & Beacom (2006) carried out by Behroozi et al. (2013). The dotted-dashed line shows the SMD derived from the SFRD given by of Madau & Dickinson (2014). All the stellar mass densities obtained by integrating the different SFRDs assume a constant recycling fraction of 28%.

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