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Appendix A: Correlation function corrections scheme

Fig. A.1 Projected two point correlation function wp(rp) measured for 66 VUDS mock catalogs in three redshift ranges. The points correspond to the mean measurement of all 66 mock catalogues, while the errors are computed as their standard deviations. The true wp(rp) computed for the whole parent sample (open circles) is compared to that measured from the observed sample (filled circles). Upper panel: without corrections (2) and (3). Lower panel: after our full correction scheme was applied.

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In the galaxy surveys, there exists a number of observational biases which may influence the clustering measurements. These biases are related to the construction of the survey. Both the parent photometric catalogue, from which target galaxies are selected, and the final spectroscopic catalogue may be the source of these biases. Below, we list the most important survey features that cause the biases. Next, we describe the corrections applied to minimize their effects. Since the VUDS survey is performed on the same multi-slit spectrograph VIMOS as the VVDS, zCOSMOS and VIPERS surveys, the correction scheme adopted here is based on the same methodology and the ones used for the other VIMOS-based surveys in the past. In particular, the main corrections used here were proposed and fully described by Pollo et al. (2005) and de la Torre et al. (2011a).

The main observational biases in the VUDS data can be listed as follows.

• 1.

  As shown in Fig. 1, some of the areas are excisedfrom the observations, due to the VIMOS layout (see A.2). Thefield of view of this instrument consists of four quadrantsseparated by 2′ gaps. No galaxies are observed in these gapsbetween quadrants, which influences the pair counts.

• 2.

  Obviously, not all galaxies from the photometric target candidate sample can be spectroscopically observed. Each galaxy spectrum occupies a certain area on the CCD detector, calculated from the spatial extent of the slit and the length of the spectrum due to spectral dispersion, which imposes geometrical constraints in target selection as galaxies too close to each other cannot be targeted simultaneously. It means that the galaxies from the parent photometric sample must be chosen is specific way to effectively allocate as many as possible spectral slits for each observation. In the case of the VIMOS observations, the slit allocation is performed automatically by the Super-SPOC code (Bottini et al. 2005). As a result, the spectroscopically observed sample is not a random representation of the parent photometric sample, and the introduced bias is relatively complex, especially on small scales.

These biases, as well as a few other, less important, effects are described in detail by Pollo et al. (2005).

However, these biases can be minimized by a combination of corrections, which were presented by Pollo et al. (2005) and de la Torre et al. (2011a), and which we found the best working for the VUDS data:

• 1.

  Random sample construction. The first, most basic part of thecorrection scheme is the appropriate construction of the randomcatalogue which needs to be geometrically identical (with theexception of the small-scale non-linear biases introduced by theSSPOC) with the spectroscopic sample. Generating randomobjects in this catalogue, we take into account the shape of thesingle pointings and quadrants, which are related to the shape ofthe VIMOS spectrograph. Additionally, we exclude the regionsremoved from the parent photometric sample (e.g. due to thepresence of a bright star), by applying the same photometric maskto the random sample. These first-order corrections reduce mostof the negative effects on the correlation function.

• 2.

  A global correction. In order to account for the missing pairs (due to the VIMOS limitations and the SSPOC strategy) we assign a global weight to each galaxy-galaxy pair. Assuming that the parent catalogue is free from angular incompleteness, we define a weighting function f(θ) as a ratio between the mean number of pairs in the parent photometric catalogue and the main number of pairs in the spectroscopic sample, as a function of angular separation (de la Torre et al. 2011a): (A.1)where w_parθ and w_spec(θ) are the angular correlation functions of the parent photometric and spectroscopic samples, respectively. Then, each pair from the spectroscopic sample separated by the angular distance θ is weighted by this ratio f(θ).

• 3.

  Small-scale corrections. As it comes out, even this strategy does not fully account for the small-scale angular effects of the SSPOC target selection strategy. In order to account for these local small-scale biases we additionally use the local weighting scheme which is also using the parent photometric catalogue as a reference. In this case, we count how many galaxies around the targeted galaxy we are missing due to the limited space for spectroscopic slits. Each targeted galaxy pair is then weighted proportionally to its representativeness of the surrounding density field. Similarly to Pollo et al. (2005) we found that the optimal size of the weighting area is the circle with a radius ~ 40′′.

The correcting scheme including all these ingredients was confirmed to be optimal after a series of tests performed using all 66 VUDS mock catalogues. During these tests we have been switching on and off different combinations of corrections listed in Sect. A1 (i.e. changing parameters controlling random catalogue properties, global correction scheme, weighting on small scales, etc.). All of these changes naturally influence the correlation function measured on different scales r_p. Hence, for each case we measured the deviations with respect to the real correlation function computed for the parent mock catalogues. As a result of this procedure we choose the set of corrections which – when combined together – minimized the differences between real and observed correlation functions at each separation r_p most efficiently. In Fig. A.1 we present the results of the correlation function measurements for the mock catalogues in our three redshift ranges. All the points correspond to the means of the measurements from 66 VUDS mock catalogues, while the errors are computed as the standard deviations between them. The three upper panels show the comparison between (1) the true (computed for whole mock parent sample) correlation function and (2) the observed correlation function computed only with the most obvious correction number (1), i.e. the appropriate geometrically cut random sample. The lower panels present the comparison of the same true correlation function and (3) the observed correlation function when the full correcting scheme described above was applied.

It is clearly seen that the introduced corrections significantly improve the retrieved signal from the observed sample at all the spatial scales, and in all three redshift bins considered. However, it can be seen that on small scales (r_p< 3 h^-1 Mpc), there is still a signal missing, even after introducing the full optimal correcting scheme (see lower panels of Fig. A.1). For these separations, our measurement is on average underestimated by 10% ± 3% with respect to the true value of w_p(r_p) computed from the parent sample. The possible influence of this effect on the final results is discussed in Sect. 3.3. At separations larger than r_p ~ 3 h^-1 Mpc the value of w_p(r_p) is also slightly underestimated, but only by about 1.6% ± 0.9%.

Our correction scheme provides a significant improvement in comparison to the average 28% ± 5% bias observed without applying it. However, the remaining systematic underestimation of the w_p(r_p) has to be taken into account during the correlation function measurements (see Sect. 3.3).

	Fig. A.2 Layout of the VIMOS field of view. Picture from Bottini et al. (2005).
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Appendix B: Impact of the reliability of redshift measurements on the correlation function accuracy

The precision and reliability of the galaxy correlation function measurement strongly depends on the reliability and accuracy of the measurement of their radial distance, i.e. on the quality of the redshift measurement in the survey. In this section we investigate the effect that a small fraction of the VUDS data with incorrect redshifts could have on galaxy clustering measurements. In this paper we use the VUDS galaxies with the best quality flags 2,3,4 and 9 (see Sect. 2.1 for the description of the quality flags), which in turn should result in a robust estimation of the correlation function. As the reliability of each flag has been independently estimated (Le Fèvre et al. 2015) the confidence level of the whole sample can be computed using the numbers of galaxies with different flags and their corresponding reliability estimates (95% at worst for flags 3+4, 75% for flag 2 and 80% for flag 9): (B.1)where N_fi is the number of galaxies with a given i flag and N_g is the total number of galaxies within the sample. From this formula, for our selected VUDS sample the total confidence level is as high as ~ 86%, which is unprecedented at these redshifts and at this depth i_AB = 25.

How could the remaining ~ 14% fraction of galaxy with wrong redshifts possibly affect the correlation function measurement? To clarify this we performed a set of tests, as described below.

The presence of galaxies with wrongly measured redshift in the data sample can distort the correlation function w_p(r_p) at each r_p. In particular, we expect a decrease in the measured correlation length r₀ as the correlation amplitude will be diluted by a number of pairs of incorrect separations. In the mock catalogues, we can simulate these distortions by changing the spatial distribution of a corresponding fraction of mock galaxies. For our simulations we assume that galaxies with reliability flag 3 and 4 do not introduce any significant distortions to the correlation function measurements, thanks to the 95−100% confidence level. We also assume that the possible influence of flag 9 galaxies is negligible, due to their low number in the sample (see Table 1). Thus, the largest fraction of galaxies with possibly wrongly measured redshift is expected to come from flag 2 galaxies. Hence, we focused our attention on these galaxies.

	Fig. B.1 Influence of the possible incorrect redshift measurements on the correlation function. Solid lines represent the projected two point correlation function wp(rp). Colours indicate the different levels of sample randomization (as labelled). The black solid line represents the true undistorted correlation function.
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We measured how strongly the correlation function would be affected if 15%, 25%, 30% or 50% (highly unlikely case) of flag 2 galaxies would have wrongly measured redshifts. Within the VUDS galaxy sample selected for this study, galaxies with flag 2 represent 48% of the total sample (see Table 1). Therefore, in the mock catalogues we randomly selected a corresponding fraction of objects to play the role of flag 2 galaxies (i.e. ). Then, in order to mimic their possibly wrong redshift measurements, we randomized the spatial coordinates (α and δ) of 15%, 25%, 30% and 50% of these objects, to obtain differently distorted samples. We then computed the projected correlation function w_p(r_p) and fitted it with a two parameter power-law.

The procedure was repeated for 10 independent VUDS mock catalogues (see Sect. 2.3) and the final estimation of w_p(r_p) was computed as the average. The results are presented in Fig. B.1. As expected, the amplitudes of the estimated projected correlation functions for the samples with wrongly measured redshifts are underestimated with respect to the unaffected one (black line), and the underestimation increases with the increasing fraction of wrongly estimated redshifts. However, the changes are not large except of the unrealistic 50% case: the correlation length for the fraction of 30% randomized galaxies differ form the real r₀ only by Δr₀ = 0.13 ± 0.04.

Based on these simple tests we conclude that the correlation function measurements performed in this study are robust, in spite of a possible contamination from wrong redshift measurements. Assuming the worst possible case, i.e. the lower limit for flag 2 galaxies confidence level being 70%, the resultant clustering strength r₀ will be underestimated only by Δr₀ = 0.13, which is ~ 3.3% of the measured amplitude. Therefore, the upper error bar of the measured r0 has been rescaled by a corresponding value, to fully account for the uncertainties introduced by possible wrongly measured redshifts in the data.

Appendix C: Projected correlation function measurements

For every galaxy subsample used in this work we present the tables of the projected correlation function measurements w_p(r_p) with associated 1σ errors at different separations r_p (in units h^-1 Mpc).

© ESO, 2015