A&A 619, A17 (2018) 
https://doi.org/10.1051/0004-6361/201833853 
 cESO 2018 
Astronomy&Astrophysics The VIMOS Public Extragalactic Redshift Survey(VIPERS) 
Unbiased clustering estimate with VIPERS slit assignment? <BR>
F.G. Mohammad1,2,D. Bianchi3,W.J. Percival4,5,3,S.delaTorre10,L. Guzzo2,1,B.R. Granett1,2,E. Branchini7,8,9, M. Bolzonella13, B. Garilli14, M. Scodeggio14, U. Abbas15, C. Adami10, J. Bel6,1, D. Bottini14, A. Cappi13,16, O. Cucciati11,13,I.Davidzon10,13,P. Franzetti14,A. Fritz14,A.Iovino1,J. Krywult17,V.Le Brun10,O.Le Fèvre10, K. Ma ek18,10,F. Marulli11,12,13,M. Polletta14,19,20,A. Pollo18,21,L.A.M.Tasca10,R.Tojeiro22,D.Vergani23, A. Zanichelli24, S. Arnouts10,25, J. Coupon26, G. De Lucia27, O. Ilbert10, L. Moscardini11,12,13, andT. Moutard28 
(Affiliations can be found after the references) 
Received 13 July 2018 / Accpeted6August 2018 
ABSTRACT 
The VIPERSgalaxy survey has measured the clustering of0.5 < z < 1.2galaxies, enablinga numberof measurementsofgalaxy properties and cosmological redshift-space distortions (RSD). Because the measurements were made using one-passof the VIMOS instrument on theVery LargeTelescope(VLT),thegalaxies observedonly represent approximately47%ofthe parenttarget sample,witha distribution imprintedwith the patternof the VIMOS slitmask. Correcting for theeffect on clustering has previously been achieved using an approximate approach developed using mock catalogues.Pairwise inverse probability (PIP) weighting has recently been proposed to correct for missinggalaxies, and we apply it to mock VIPERS catalogues to show that it accurately corrects the clustering for the VIMOS effects, matching the clustering measured from the observed sample to that of the parent.We then apply PIP-weighting to the VIPERS data, and ft the resulting monopole and quadrupole moments of thegalaxy two-point correlation function with respect to the line-of-sight, making measurements of RSD. The results are close to previous measurements, showing that the previous approximate methods used by the VIPERS team are sufficient given the errors obtained on the RSD 
parameter. 
Keywords. cosmology: observations – large-scale structure of Universe 
1. Introduction 
The clustering of galaxies within galaxy surveys provides a wealth of astrophysical information, allowing measurements of galaxy formation,galaxyevolution, and cosmological parameters. Missing galaxies within surveys can however distort the clustering compared to that of the full population of the type of objects to be observed if the missed galaxies are not ran-domly chosenbut instead clusterinadifferent way to the full population. Such a situation is often induced by the mechanics of the experimental apparatus, which, given a parent population of targets, limits what can actually be observed. In this paper we consider missinggalaxiesin the VIPERS survey(Guzzoetal. 2014; Scodeggio et al. 2018). VIPERS collected 89022galaxy redshifts over an overall area of 23.5 deg2, covering the W1 and W4 felds of the Canada-France-HawaiiTelescope Legacy 
? Based on observations collected at the European Southern Observatory, Cerro Paranal, Chile, using the Very Large Telescope under programs 182.A-0886 and partly 070.A-9007. Also based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-HawaiiTelescope (CFHT), which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifque (CNRS) of France, and the University of Hawaii. This work is based in part on data products pro-duced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-HawaiiTelescopeLegacySurvey,a collaborative project of NRC and CNRS. The VIPERS web site is http://www. <BR>vipers.inaf.it/ <BR>
–galaxies: high-redshift–galaxies: statistics 
SurveyWide (CFHTLS-Wide)1.Acolour pre-selectionwas used to remove galaxies at z < 0.5, helping to bring the sampling efficiency to 47%. VIPERS conducted observations using the VIMOS multi-object spectrograph(Le Fèvre et al. 2003), which applies a slit-mask to select targets for follow-up spectroscopy. Abrief descriptionof VIPERSisprovidedin Sect.2. 
The requirement that spectra taken with VIMOS should not overlap on the focal plane limits the placement of slits, and consequently thegalaxies that canbe observed. Thiseffect is stronger along the dispersion direction compared to across it, because of the rectangular nature of the projected spectra. The occulted region around each galaxy is imprinted on the statistical distribution of the observed galaxies. There are no overlapping observations, such as those present in the Baryon Oscillation Spectroscopic Survey (BOSS, Dawson et al. 2016), meaning that the lost information cannot be recovered: we sim-ply do not have clustering information on scales smaller than the minimum separation perpendicular to the dispersion direction. On larger scales, the slitmask still impacts on the measured clustering through the large-scale pattern imprinted on the sky, and the density dependence of the selection. 
Bianchi&Percival (2017) and Percival&Bianchi (2017) presented a new method to correct for missing galaxies in surveys. This builds up a probability for each pair of galaxies in the observed sample to have been observed in a set of 
1 http://www.cfht.hawaii.edu/Science/CFHTLS/ <BR>
Article publishedby EDP Sciences A17, page1of 14 
realisations of the survey2. These realisations, drawn from the same underlying parent catalogue, are all equally likely. Each sample can be obtained by simply re-running the targeting algo-rithm after moving or rotating the parent sample, or changing anyrandom selection performedby the selection algorithm.We observe oneof these setsofgalaxies, andbyinverse weighting bythe pairwise probability of observation we force the clustering of the one realisations to match that of the set as a whole. Pro-vided there are no pairs of zero weight, this weighting leads to a clustering estimate of the observed sample that is unbiased com-pared to that of the full parent sample. The method is described in more detail in Sect. 4.1. 
In this paper we apply this method to remove the effects of the VIMOS slitmask from the VIPERS survey. The slitmask has a strong effect, leading to an observed clustering signal that is very different from thatexpected(delaTorreetal. 2013). In previous VIPERS papers this was approximately corrected using a target sampling rate (TSR) given by the fraction of potential targets placed behind a slit in a rectangular region around each targeted galaxy (Pezzotta et al. 2017).Afurther correction that up-weightsgalaxy pairsby the ratio [1 + ws(θ)]/[1 + wp(θ)] of the angular clustering of the observed ws and parent wp samples(delaTorreetal. 2013)was also used to improve the small-scale clustering measurements. While similar in principle to the method of Bianchi&Percival (2017)andPercival&Bianchi (2017), this relies on the missed pairs being statistically identical to the population as a whole. This is not the case in VIPERS as galaxies are more likely to be missed in denser regions where they have different properties. The TSR up-weighting method was extensively tested in past VIPERS analyses to provide a sub-percentagelevel accuracy on the clustering measurements in mock catalogues. However, the TSR weighting is a parametric method that was calibrated on mock catalogues to minimise the systematic bias of the clustering measurements. It does not take into account possible differences in the clustering of simulated and observed datasets. The pairwise inverse probability (PIP) weighting scheme uses the data themselves to infer the selection probabilities providing the same level of accuracy. In this sense the new correction method is self-contained and more robust than the method based on using the TSR. 
To optimise the design of the slitmasks, VIPERS uses the socalled SPOC algorithm (Slit Positioning and Optimisation Code), within the ESO VIMOS mask preparation software VMMPS (Bottini et al. 2005).SPOC was designed to obtain the most spectra possible given an input parent sample. Rather than trying to change the internal properties of SPOC to make our set of real-isations of the survey, we instead rely on spatially moving the surveymask and rotating the sample.We still miss all pairs that have a separation that is less than the minimum slit separation scale,but this is not an issue as we only consider larger scales here. 
We use mock catalogues of VIPERS to test the new algo-rithmin Sect. 6, showing thatitworks asexpected.Having cor<BR>rected for the slit-mask effects, we consider how this changes the redshift-space distortions (RSD) signal within the sample. VIPERS was designed with RSD as one of the key measurements to be made: RSD are caused by the peculiar velocities of galaxies, which systematically distort redshifts leaving an 
Withthe term “surveyrealisation”we indicatea possible outcomeof the spectroscopic observation given an underlying parent sample. It is nottobe confusedwiththeterm“surveymocks”thatarebuiltfroman ensembleof parent catalogueskeeping the observational setup fxed. 
A17, page2of 14 
enhanced clustering signal along the line-of-sight(Kaiser 1987). By measuring the clustering anisotropyaround the line-of-sight through observations of the multipole moments of the correlation function one can constrain the growth rate of cosmological structure parameterisedby f σ8, which constitutes the frst-order contribution to the RSD signal. 
Early RSD measurements from VIPERS were based on the Public Data Release1 sample(Garillietal. 2014), measuring f σ8(z = 0.8)(delaTorreetal. 2013). Subsequent measurements from the fnal data sample, Public Data Release2(PDR2, Scodeggio et al. 2018), were presentedbyPezzotta et al. (2017). Extensions to these measurements include a confguration space joint analysisofRSDand weak-lensing(delaTorreetal.2017), andananalysis splittingthesamplebasedongalaxytypeinorder to extract extra information by comparing samples that trace the dark matter feld in differentways(Mohammadetal. 2018). 
We presentRSD measurementsmadebythe “standard”two-point correlation function-based method in Sect. 8. These are compared to the previous VIPERS measurements, and we show that previous slit-mask-correction techniques were sufficient to make these measurements from VIPERS. This is discussed fur-therin Sect. 9. 
To analyse the VIPERS-PDR2 data we used the same fducial cosmology adopted in previous VIPERS clustering analyses, that is, a fat ΛCDM cosmology with parameters (Ωb, Ωm, h, ns,σ8)= (0.045, 0.3, 0.7, 0.96, 0.80). 
2. The VIPERS survey 
The VIPERS surveyextendsoveran areaof23.5deg2 within the W1 and W4 felds of the CFHTLS-Wide. The VIMOS multiobject spectrograph(Le Fèvre et al. 2003) was used to cover these two felds with a mosaic of 288 pointings, 192 in W1 and 96 in W4. Given the VIMOS footprint, which consists of four distinct quadrants separated by an empty “cross” of about 2arcmin width (see Fig. 1), the survey area includes a regu-lar grid ofgaps where nogalaxies were observed (see following section).Targetgalaxies were selected from the CFHTLS-Wide cataloguetoafaint limitofiAB = 22.5, applying an additional(r − i)versus (u − g)colour preselection that efficiently and robustly removesgalaxies at z < 0.5. Coupled with a highly optimised observing strategy(Scodeggio et al. 2009), this doublesthe meangalaxy samplingefficiencyin the redshift range of interest compared to a purely magnitude-limited sample, bringing it to 47%. 
Spectra were collected at moderate resolution(R ' 220) using the LR Red grism, providing a wavelength coverage of5500–9500Å. The typical redshift error for the sample of reliable redshifts is σz = 0.00054(1 + z), which corresponds to an erroronagalaxy peculiarvelocityatanyredshiftof163kms−1. These and other details are given in the PDR-2 release paper (Scodeggio et al. 2018).Adiscussion of the data reduction and management infrastructurewas presentedin Garillietal. (2014), while a complete description of the survey design and target selection was given in Guzzo et al. (2014). The dataset used here is the same early version of the PDR-2 catalogue used in Pezzottaetal. (2017)anddelaTorreetal. (2017), from which it differs by a few hundred redshifts revised during the very last period before the release. In total it includes 89 022 objects with measured redshifts. As in all statistical analyses of the VIPERS data, only measurements with quality fags 2–9 (inclusive) are used, corresponding to a sample with a redshift confrmation rate of 96.1% (for a description of the quality fag scheme, see Scodeggio et al. 2018). 
F. G. Mohammad et al.: Redshift-space distortions in VIPERS 

Fig. <BR>1. <BR>Example of the slit/spectrum distribution over a full VIMOS pointing, showing the disposition of the four quadrants and the “cross” among them. The circles identify the targets selected by the SPOC optimisation algorithm. The elongated blue rectangles reproduce the “shadow” of the 2D spectrum that will result from each target in the fnal spectroscopic exposure. The thin red lines show the boundary of the actual spectroscopic mask, traced pointing-by-pointing through an automatic detection algorithm that follows the borders of the illuminated area (see Guzzo et al. 2014, for details) 
The procedures for defning the target list within the VIMOS spectroscopic masks were described in detail in Bottini et al. (2005). Within the VMMPS environment, the SPOC algorithm is used to optimise the position, size and, total number of slits. The fnal solution is derived by cross-correlating the user target catalogue with the corresponding object positions in a VIMOS direct exposure of the feld (“pre-image”), observed beforehand. This operation matches the astrometric coordinates to the actual instrument coordinate system, selecting the subset of objects that will eventually deliver a spectrum and, potentially, a redshift measurement. 
SPOC aims at fnding an optimal disposition of the slits, packing the largest possible number of spectra over each quadrant (see Bottini et al. 2005 for a detailed description of SPOC). This happens irrespectively of the parent sample angular clustering. As such,it will tendtobuilda distribution thatis more homo-geneous on the sky compared to the full galaxy population at the corresponding magnitude limit. The denser the parentgalaxy sample, the stronger the bias. If the number density ofgalaxies on the skyis much larger than the maximum density of slits that can be packed, SPOC will essentially pickgalaxiesina regular grid, packing the spectra in regular rows on top of each other. This is not quite the case for VIPERS, for which the relatively bright magnitude limit allows for targeting, on average, about one half of the available galaxies, as shown in Fig. 1. In this way,the measured sample still preservesa signifcant fractionof the original angular clustering. Still, a bias is inevitably intro-duced and needs to be properly accounted for in anyclustering measurement, which is the subject of this paper. In addition, the fnite size of slits introduces a proximity effect that also needs to be corrected for when computinggalaxy clustering. 
Figure1shows anexample VIMOS observation. Theover-all mosaic of such pointings composing the full VIPERS sur-vey is shown in Fig. 2 for the two survey areas, W1 and W4. The boundaries of each single observation are described by the black polygons. In this fgure,galaxies in the photometric par-ent sample and in the fnal VIPERS-PDR2 redshift catalogue are over-plotted as red and blue dots, respectively. The gaps of the VIMOS footprint are clearly visible as vertical and horizontal stripes, in which only unobserved objects, marked in red, are present. In addition, the overall survey mask includes: 
(a)gaps in the photometric sample due to bright star or photometric problems (small irregular empty regions); (b) fullyfailed quadrants due to mechanicalfailure in the VIMOS metal mask insertion before the spectroscopic observation (white regular rectangles, mostly in W4); and (c) specifc details in the spectroscopic observations,suchas,forexample, vignettingbytheVLT guideprobe (describedbytheredlineinFig. 1;seeGuzzoetal. 2014;Scodeggio et al. 2018, for details). 
Throughout this work we have defned, as parent catalogue, the photometric catalogue selected according to the VIPERS target selection function(Guzzo et al. 2014), including allgalaxies matching the external boundaries of the VIPERS-PDR2 sam-ple,but with no mask applied.We have also ascribed the empty pointings and quadrants in the VIPERS-PDR2 sample to the photometric mask to avoid unnecessary complications in the implementation of the pipeline used for this analysis. 
3. VIPERS Mocks 
VIPERS mocks are based on the Big MultiDark Planck (BigMDPL, Klypin et al. 2016) dark matter N-body simulation. The simulation was carried out in the fat ΛCDM cosmological model with parameters: (Ωm, Ωb, h, ns,σ8) = (0.307, 0.048, 0.678, 0.96, 0.823). Since the resolution is not suffcient to match the typical halo masses probed by VIPERS, low-mass haloes were added following the recipe proposed by delaTorre&Peacock (2013). 
Dark-matter halos were populated withgalaxies using halo occupation distribution prescriptions with parameters calibrated using luminosity-dependent clustering measurements from early VIPERS data.We refer the reader todelaTorreetal. (2013, 2017)for a detailed description of the procedure. 
We used a set of 153 independent realistic parent and VIPERS-like mocks for each of the two VIPERS felds, W1 and W4. VIPERS-like mocks were obtained from the corresponding set of parent mocks in two steps: frst, VIPERS targeting algo-rithm was applied by means of SPOC using the grid of VIPERS pointings; afterwards the footprint of VIPERS spectroscopic and photometric masks was imprinted to include the effect of obscured sky regions and quadrant vignetting (see Sect. 2).We also included the effect of VIPERS redshift error in the mock catalogues by blurring the cosmological redshifts using a Gaus-sian distribution of width σz/(1 + z)= 0.00047. Although different from the latest estimate from the PDR2 data, we used this valueto performafair comparisonof our results with thosein Pezzotta et al. (2017). 
We used this set of mock samples to test the reliability of the weighting schemes proposed in Bianchi&Percival (2017) and Percival&Bianchi (2017). The same setof mockswas also 
A17, page3of 14 Fig. <BR>2. <BR>Scatter plot in the (RA, Dec) plane forgalaxiesin the parent sample (red dots) and VIPERS-PDR2 catalogue (blue dots). Top and bottom panels: W1 and W4 felds, respectively. Portions of the skyunobserved in the spectroscopic samples due to defects in the photometric sample, bright stars, or missing quadrants have been ascribed to the photometric mask. 

employed to estimate the data covariance matrix and quantify the systematic bias on estimates of the growth rate of structure. 
4. Measurements 
We measured the anisotropic two-point correlation function ξ(s,µ)as a function of the angle-averaged pair separation s and the cosine µ of the angle between the pair separation and the lineofsight.Weemployedthe minimumvariance Landy–Szalay estimator(Landy&Szalay 1993), 
DD(s,µ)− 2DR(s,µ) 
ξ (s,µ)=+ 1, (1) 
RR(s,µ) 
where DD, DR, and RR are the data-data, data-random, and random-random normalized pair counts, respectively.Webinned µ in 200 linear binsin the range0 ≤ µ ≤ 1taking the mid-point of each bin as reference. The pair separation s is instead binned using logarithmic bins, 
log si+1 = log si +Δslog, (2) 
with Δslog = 0.1. The measurement in each pair separation bin is referenced to the logarithmic mean, 
log si + log si+1 
loghsii = · (3) 
2 
The multipole moments ξs,(`)(s)of the two-point correlation function are defned as its projection on the Legendre polynomials L` (µ). Since we deal with discrete bins of the variable µ, we replaced the integralby the Riemann sum such that, 
200
X 
ξs,(`)(si)= (2` + 1) ξs(si,µj)L`(µ j)Δµ· (4) j=1 
When performing the angular pair counts DDa(θ)andDRa(θ)we used 100 linear bins within0◦≤ θ ≤ 8◦. This range is sufficiently large to cover a transverse pair separation of ∼185h−1Mpc at z = 0.5in VIPERS fducial cosmology. 
Following Pezzotta et al. (2017) we divided the redshift range 0.5 < z < 1.2 covered by VIPERS into two bins spanning0.5 < z < 0.7and0.7 < z < 1.2with effective redshifts of zeff = 0.60 and zeff = 0.86, respectively. The subsample at low redshifts contains30910galaxies whilethe oneathigh redshifts includes33679galaxies. These parameters are listedinTable 1. Since VIPERS targeting over W1 and W4 felds was performed using the same observational setup we treated them as a single survey and performed the pair counts simultaneously on both felds rather than combining the measurements of the correlation function from each feld. 
4.1. Mitigating for missing targets 
The PIP approach provides us with unbiased estimates of the galaxy pair counts in the presence of missing observations, with the only formal requirement being that no pair has zero probabilityof being observed(Bianchi&Percival 2017). 
At each separation s, the data-datapair counts are obtained as 
X 
DD(ap)(θ) 
DD(s)= wmn , (5) 
DDa(θ)
xm−xn≈s 
where wmn = 1/pmn is the inverse of the selection probability of the pair formedby thegalaxies m and n, whereas DD(ap) and DDa represent the angular pair counts of parent and observed sample, respectively. The observed angular pair counts are, in 
A17, page4of 14 
F. G. Mohammad et al.: Redshift-space distortions in VIPERS 
Table 1. Parameters characterising the two VIPERS subsamples split by redshift as used in this work. 
Redshift Ngal V [h−3Gpc3] zeff 
0.5< z < 0.7 30, 910 1.76 × 10−30.60 0.7< z < 1.2 33, 679 7.34 × 10−30.86 
Notes. “Redshift” denotes the redshift range, Ngal is the number of galaxies, V stands for the volume in the VIPERS fducial cosmology (see Sect. 1)andzeff is theeffective redshift of the sample computed as the median of the mean redshifts ofgalaxy-galaxy pairs with separations5h−1Mpc < s < 50h−1Mpc. All fgures refer to the full VIPERS, that is, both W1 and W4 felds. 
turn, computed via the same wmn weights, 
X 
DDa(θ)= wmn · (6) 
um ·un≈cos(θ) 
P 
For brevity, we have adopted the notation xm−xn≈s and 
P 
·un≈cos(θ), with ui = xi/|xi|, to indicate that the sum is per-
um 
formed in bins of s and θ, respectively. Similarly, for the data-random pair counts, 
X 
DR(ap)(θ) 
DR(s)= wm , (7) 
DRa(θ)
xm −y≈s 
n
where wm = 1/pm is the inverse of the selection probability of thegalaxy m, and 
X 
DRa(θ)= wm · (8) 
um ·un≈cos(θ) 
We evaluate the selection probabilities pmn and pm empirically,by creatingan ensembleof possible outcomesofthetarget selection given an underlying parentcatalogue; that is, we rerun the slit-assignment algorithm on the same parent sample several times (see Sect. 5). As discussed in Bianchi&Percival (2017), rather than storing all the PIP weights (one for each pair), it is convenient to compress the information in the form of individual bitwise weights (one for eachgalaxy). The bitwise weight of agalaxy w(b) is defned as a binary array, of length Nruns, in 
i 
which the n-th bit equals1if thegalaxy has been selectedin the n-th targeting realisation and 0 otherwise. Nruns represents, by construction, the total number of realisations.For convenience, we use base-ten integers to encode the bitwise weights. The PIP weights are obtained “on the fy”, while doing pair counts, as 
Nruns 
wmn = hi, (9) popcnt wm (b)&w(nb) 
where&and popcnt arefast bitwise operators, which multiply twointegersbitbybitand returnthesumofthebitsofthe resulting integer, respectively. Similarly, for individual weights, we have 
Nruns 
wm = hi· (10) 
(b) 
popcnt wm 
The requirement that all pairs are observable (they can be observed in at least one VIPERS realisation) means that the expectation value of the PIP estimator (excluding angular up-weighting) matches the clustering of all of the pairs within the parent sample -those targeted for possible VIPERS observation.Pairsin the parent sample that cannotbe observedwould formallyhave infnite weightbut, practically, they would never appear in the pair counts in a particular realisation of VIPERS3. If we have some pairs that are not observable (they have zero probability of observation), angular up-weighting can serve two different purposes: 
(i) 
The number of unobservable pairs is not negligible,but the clustering of these pairs is statistically equivalent to that of the observable ones. This happens when being observable or not is a property that does not depend on cluster-ing; forexample, whengalaxiesfallina blind spotof the instrument’s focal plane (see Bianchi et al. 2018 for a more detailed discussion). In this case it is formally correct to use the full set of observable plus unobservable pairs to perform angular upweighting to recover unbiased estimates of the three-dimensional clustering.Wenote that here these regions wouldnotbeexcludedinthemaskusedto createthe random catalogue. 

(ii) 
The unobservable pairs are such because of their clustering butthetotal numberissmallenoughthattheireffect is negligible, at least on the scales of interest. In this second scenario angular upweighting is simply a way to reduce the variance and the more self-consistent approach is to use only the set of observable pairs. Using the full set of pairs could potentially increase the effect of the unobservable pairs. As discussed in Sect. 6, the VIPERS survey is compati-ble with category (ii). Interestingly, we fnd that the mean fraction of unobservable pairs in mock samples is about a factor of two larger than what is shown in Fig. 3 for the VIPERS-PDR2galaxy sample. This points to some difference between mocks and data in terms of galaxy cluster-ing. Unlike the weighting schemes calibrated on simulated datasets (e.g. TSR weights), PIP weights are built to be insensitive to this difference. We use mocks just to verify that the effect of unobservable pairs is confned to the small-est scales. The above mentionedfactor two guaranties that the same conclusion holds for real data. 


4.2. Correcting for redshift failures 
The reliability of each VIPERS redshift measurement is quanti-fed by a quality fag. Spectroscopic redshift measurements with a quality fag 2–9 (inclusive) have a redshift confrmation rate of 96.1% and are regraded as reliable.We label all objects that do not satisfy this condition as “redshiftfailures”. The reliability of a redshift measurement depends on a number of factors such as the feld-to-feld observational conditions and the presence of clear spectral features and presents a correlation with somegalaxy properties such as colour and luminosity. Theeffect of redshiftfailuresis quantifedby meansof the spectroscopic success rate (SSR) defned as the ratio between the number of objects with a reliable redshift measurement (in our case the ones witha quality fag between2and9) and the total number of targets placed behind a slit in a given VIMOS quadrant. It is computedasa functionofthegalaxy rest-frame U–V colour and B-band luminosityandis assignedtoeachgalaxywitha reliable redshift measurement. 
To correct the clustering measurementsagainst redshiftfailures, we have up-weighted each galaxy by the corresponding 
3 For the sake of clarity, we note that S pairs ⊆ S observable pairs ⊆ S observed pairs , where Sx stands for set of x.We also note that, in general, it is not possible to infer S observable pairs from S observedgalaxies . 
A17, page5of 14 Fig. <BR>3. <BR>Fraction of unobservablegalaxies and pairs ofgalaxies in the VIPERS parent catalogue as a function of the number of targeting runs Nruns. Points show the case when multiple surveyrealisations are gen-erated only spatially moving the spectroscopic mask, while lines result from also rotating the parent catalogue by θRot.For the latter case,vertical dashed lines delimit the subset of targeting runs sharing the same θRot. Blue flled points and continuous line show the fraction of unobservablegalaxy-galaxy pairs while red empty markers and dashed line correspond to individualgalaxies. 

weight wSSR = SSR−1. Equations(5)and(6)are therefore mod-ifed as 
X 
DD(ap)(θ)(m)(n) 
DD(s)= × w(11) 
wmn SSRwSSR, 
DDa(θ) 
xm−xn≈s 
and 
X 
(m)(n) 
DDa(θ)= wmn × w(12) 
SSRwSSR · 
um ·un≈cos(θ) 
Data-random cross-pair counts in Eqs.(7)and(8)now become, 
X 
DR(ap)(θ)(m) 
DR(s)= wm × w(13) 
DRa(θ) SSR, 
xm −yn≈s 
and 
X 
(m) 
DRa(θ)= wm × w(14) 
SSR, um ·un≈cos(θ) 
respectively. 
The effect of redshiftfailures is not reproduced in the mock catalogues.Wethereforemakeuseof spectroscopic success rates only when dealing with the VIPERS-PDR2galaxy catalogue. 
5. Pipeline 
The weighting scheme presented in Sect. 4.1 relies on generating multiple surveyrealisations to assign selection probabilities and correct the pair counts. In principle, for a slit or fber assignment scheme that randomly selects targets in the presence of collided objects, this can be achieved by simply re-running the targeting algorithm Nruns times on the parent catalogue, with different random selection choices each time. 
As describedin Sect. 2, SPOC appliesa deterministic algo-rithm to maximise the number of slits assigned to potential tar-gets, with no free parameters. Re-running the targeting algorithm with the same confguration of parent sample and spectroscopic mask would produce exactly the same outcome. We therefore generated multiple realisations of the spectroscopic observations from a given parent catalogue by spatially moving the spectroscopic mask in the (RA, Dec) plane. As the VIPERS felds are equatorial, we can accurately quantify small shifts in the survey position using ΔRA and ΔDec. Given the periodicity in the pat-tern of pointings in the VIPERS spectroscopic mask, the amount of this shift, with respect to the original VIPERS confguration, was taken as being smaller than the size of a single VIMOS pointing.We generated Nruns = 2170 VIPERS target realisations on each parent sample. The frst of these 2170 such runs was kept fxed to the actual VIPERS-PDR2 position. 
The VIPERS spectroscopic mask is defned only over the area coveredby the actual VIPERS observations.Ashiftwould therefore inevitably yieldgalaxies at the edges of the sample to be covered by a lower number of targeting runs with respect to those located nearthe centre(Fig. 5). Ratherthanhavingtokeep track of this, we replicated the grid of VIPERS pointings beyond the survey area such that in each run, all portions of the parent catalogue are covered by a VIMOS pointing. However, unlike the pointings in the original spectroscopic mask, we do not know the exact shapes of the quadrants belonging to the “artifcial” pointings outside the surveyarea, so we used the shapes of the quadrants in the original VIPERS spectroscopic mask as templatesand randomly assignedthemtothe artifcial pointings.We henceforth refer to the new mask as the “extended spectroscopic mask”. 
Only shifting the extended spectroscopic mask by small off-sets with respect to the parent sample would require a very large number of targeting runs to accurately infer the selection probabilities and reach sub-percent level accuracy on the mea-surements of the two-point correlation function (see Fig. 3). In particular, after Nruns = 2170 targeting runs obtained by only shifting the extended spectroscopic mask, ∼0.6% of parent galaxies remain unobserved in any of these realisations and therefore cannot be assigned a targeting probability (red empty circles in Fig. 3). This fraction increases to ∼5.5% forgalaxy-galaxy pairs (blue flled points in Fig. 3). This is due to the fact that under particular conditions such as in very close pairs, SPOC systematically selects the same objects in different tar-geting runs. This effect is quantifed by the 2D angular com-pleteness function of the sub-sample of observable pairs (i.e. the ones that are observed in anyof the 2170 targeting runs) in the (RA, Dec) plane, 
1+ wtarg (RA, Dec) 
C (RA, Dec)= , (15) 
1+ wpar(RA, Dec)
where wpar and wtarg are the 2D angular correlation functions of the parent catalogue and its sub-sample of observable pairs, respectively.We are unable to assign selection probabilities to a signifcant fraction of pairs at separations ΔRA . 
500 
and ΔDec . 13000 due to a combination of “slit-” and “spectra-collision” as illustratedin the left panelof Fig. 4. 
Given the geometry of the problem, we were able to reduce the fractionof unobservablegalaxiesandgalaxy-galaxypairsby rotating the parent catalogue by 90◦, 180◦ and 270◦ around an axis that passes through the sample, together with random shifts of the extended spectroscopic mask. Each of the 2170 survey realisations is now characterised by a rotation angle of the corresponding parent catalogue (namely 0◦, 90◦, 180◦ and 270◦) and a shift of the extended spectroscopic mask in the (RA, Dec) plane.We stress here that we only rotate the parent sample while keeping the orientation of quadrants and dispersion direction of the galaxy spectra fxed; that is, the larger side of the quad-rants is always aligned along the declination axis. In this way 
A17, page6of 14 
F. G. Mohammad et al.: Redshift-space distortions in VIPERS 

Fig. <BR>4. <BR>2Dangular completeness functionofgalaxy-galaxypairs(seeEq.(15))observedin2170surveyrealisationswithrespecttotheVIPERS parent catalogue. Left panel:surveyrealisations are obtained spatially moving the spectroscopic mask over the parent catalogue. The rectangular “shadow” at small separations is the typical footprint of VIMOS spectra. Right panel:asin the left panelbut when also the rotationsof the parent catalogue are added to make multiple surveyrealisations. The size of the square shadow at small pair separations in the right panel is the typical length of VIMOS slits and is produced by the slit collisions only. 

Fig. <BR>5. <BR>Top <BR>panel: sketch showing the border effects when multiple surveyrealisations are generated shifting the original VIPERS spectroscopic mask over the underlying parent catalogue (red dots). The area covered by the actual VIPERS spectroscopic mask is delimited by the blue continuous line while the corresponding VIMOS pointings are displayed as blue flled dots.Arandom shift of(ΔRA, ΔDec) is then applied to obtain a new surveyrealisation. The area covered in the new realisation is shown as black dashed contour with black empty circles being the new positionsof VIMOS pointings.We highlightthe portionofthe parent cataloguesatlowRAandlowDecthatisnotcoveredbythe shifted mask. The effect is even more severe for the realisations obtained rotating the underlying parent catalogue. Bottom panel:asinthetoppanelbut herethe new surveyrealisation is generated shifting the “extended” spectroscopic mask (see Sect. 5). Black dots show the shifted position of the pointings in the original VIPERS spectroscopic mask while the black crosses represent the “artifcial” pointings in the extended spectroscopic mask. The extended mask is large enough to fully cover also the parent catalogue rotated by 90◦, 180◦ , or 270◦. In both panels a number of pointings in the shifted mask are located outside the boundaries of the parent sample. These are the pointings that only partially overlap with the parent catalogue. 
A17, page7of 14 Fig. <BR>6. <BR>Normalised distributions of the observed fraction of VIPERS parentgalaxies among 2170 targeting runs.Different colour coding and line styles differentiate runs with different rotation angles of the parent catalogue. Thevertical dashed line shows the fractionsofgalaxiesin the VIPERS-PDR2galaxy catalogue. 

wewereabletoassign selection probabilitiestoallparentgalaxiesandlowerthe fractionof unobservablegalaxy-galaxypairsto ∼0.06%, respectively (red dashed and blue solid linesin Fig. 3). The price to pay is that realisations with different rotation angle of the parent catalogue are not equivalent to each other in terms of the fraction of observedgalaxies. In particular, rotating the parent catalogue by (90◦ , 270◦)provides, on average, a number of observedgalaxies that is ∼1% lower than the confgurations with a rotation of (0◦ , 180◦)as shownin Fig. 6. Thisisa con-sequence of the rectangular nature of the projected spectra and their alignment with the surveyboundaries. This produces a different normalisation factor between these two sets of confgu-rations that can be mitigated by angular up-weighting the pair counts. 
Given the limited number of survey realisations we used to infer selection probabilities a small fraction of pairs remain unobserved in any realisation (we refer to them as unobservable). This introduces a systematic bias on small scales, which we do not use for RSD ftting. As discussed in Sect. 4.1, given the nature of the unobservable pairs, it is appropriate to replace DD(p)(θ)andDD(p)(θ)in Eqs.(5)and(7)withDD(targ.)(θ)and DR(targ.)(θ),the numberof observablegalaxy-galaxyandgalaxy-random pairs, respectively. In the following part, we use these quantities to compute the angular weights. Unless specifed otherwise,we usethe parent catalogueasa referenceto estimatethe systematic biases. 
We treated the unobserved pointings and individual quad-rantsasapropertyofthe photometric mask. Finally,weregarded theskyregions obscuredbythe photometricmaskasa featureof the parent catalogues and imprinted the emptygaps accordingly. In particular, not imprinting the emptygaps due to unobserved pointings and quadrants in the parent catalogue would introduce a difference in the mean number of observedgalaxies in different subsetsoftargeting runs. Indeedthegapsdueto unobserved pointings in the uppermost row in the W1 feld or in general those locatedfar from the rotation axiswould notbe presentin the confgurations characterised by a rotation of the parent catalogueby 90◦ or 270◦ . 
Finally, we constructed the random sample by matching the radial distribution of the VIPERS sample and imprinting the angular selection function of the parentgalaxy sample, that is, applying the photometric mask. The correction scheme based on up-weighting individual galaxies according to the local densities of parent and targetedgalaxies such as the TSR weighting usedin delaTorreetal. (2013)wouldhave required including also the effect of the VIPERS spectroscopic mask. However, in our case this is not necessary, as this effect is already accounted for by using the PIP weighting. Including such a selection effect alsointhe random cataloguewouldhave resultedinoverweighting the pair counts. 
6.Validation on mockcatalogues 
6.1. Consistency tests 
We measured the multipole moments of the two-point correlation function from each of the 2170 surveyrealisations, obtained by rotating the parent catalogue and shifts of the extended spec-troscopic mask, using the weighted pair counts. Each of 2170 measurements was then compared to the reference estimate obtained from the mock parent catalogue to assess the mean systematic bias and related error. These measurements are shown in the top large panels of Fig. 7 for the two redshift bins, while the bottom smaller panels show the corresponding fractional systematic bias with respect to the reference measurement. In particular, the PIP weighting scheme performs well over all scaleswithasystematicbias confnedtothe sub-percentagelevel on scales s > 1h−1Mpc for all multipole moments in both red-shift bins. These results are confrmed also when including the angular weights that however improve the statistical precision of the measurements. 
The very small residual offset between the reference and the mean estimate among the corresponding 2170 realisations obtained using weighted pair counts is produced by the fnite number of targeting runs that are used to sample the selection probabilities.Asmall fraction ofgalaxy-galaxy pairs is not observedin anyof the targeting runs as shownin Fig. 3.We are therefore unable to assign selection probabilities to these objects. In particular, we can split the correlation function into two sum-mands, 
" #"# 
DDobs − 2DRobs DDunobs − 2DRunobs 
ξ (r)=+ 1 + , (16) 
RR RR 
where the frst bracket represents the contribution from the sub-set of observable pairs while the second one results from the unobservable pairs.We measured these quantities froma mock sample using the set of corresponding bitwise weights. It is clear fromFig. 8thatthe unobservablepairs clusterinaverydifferent way with respect to thegalaxies in the full parent sample. They provide a non-negligible contribution to the overall clustering signal such that the expectation value of the estimator becomes different from that of the underlying parent sample. Indeed, the mean estimate of the two-point correlation function among 2170 surveyrealisation is unbiased if we limit the reference sample to only observable pairs. 
6.2. Observational systematic bias 
Wequantifed the observational systematic bias in the case where a set of 153 independent parent mocks is available and we have access to only one realisation of the spectroscopic observations for each parent sample, namely the one that matches the VIPERS-PDR2 observational confguration.We refer to this particular realisation as the VIPERS-like mock catalogue. We implemented the pipeline described in Sect. 5 for each of the 
A17, page8of 14 
F. G. Mohammad et al.: Redshift-space distortions in VIPERS 

Fig. <BR>7. <BR>Top <BR>large <BR>panels: measurements of the frst three even multipoles of the two-point correlation function from one reference mock parent sample (lines). Points with error bars show the mean and related errors among 2170 measurements obtained using the PIP weighting scheme alone (empty markers, dashed error bars) and when supplemented with an angular up-weighting (flled markers, continuous error bars) on independent surveyrealisations drawn from the same mock parent sample. Bottom small panels:empty and flled markers display the fractional systematic bias of the corresponding measurements in the top large panels with respect to that from the reference mock parent sample. The horizontal continuous coloured lines and the shaded bands show the equivalent of the empty markers in the same panelsbut when the reference sample is limited to thegalaxiesandgalaxypairsthataretargetedatleastonceinthe2170surveyrealisations.Errorbarsinthe bottom panels are obtained using the standard error propagation formula. Left and right panels show results from the lower-and higher-redshift bins, respectively. All measurements use data from W1 and W4 (mock) felds. 
153 mock parent samples to assign selection probabilities.We measured the multipole moments from each VIPERS-like mock using the angular up-weighting and compared to the reference measurement from the corresponding parent mock to assess the observational systematic bias. The mean and related errors on such systematic biases among 153 mocks are displayed in the bottomsmallpanelsofFig. 9whilethe correspondingmeanestimates and errors among 153 parent and VIPERS-like mocks are shown in the top large panels of the same fgure. The measurements from the low-and high-redshift bins are shown in the left and right panels, respectively. 
The new weighting scheme provides clustering measurements accurate at the sub-percentage level down to very small scales(∼0.6h−1Mpc) in both redshift ranges. The systematic bias increases at scales of &40h−1Mpc remaining within 2σ of the reference estimates. The residual systematic offset on scales of interest(.50h−1Mpc) results from a combination of two effects:a)in each mock samplea small fractionofgalaxy pairs remain unobserved in the ensemble of 2170 surveyrealisations (see Figs. 3 and 8);b) the VIPERS-like confguration is not a random realisation but rather a particular case among the 2170 survey realisations used to infer selection probabilities, namely the one characterised by a rotation angle of the parent sample of θ = 0◦ and no shifts (see Fig. 6). Figure 9 also shows results obtained up-weighting each galaxy by the corresponding TSR. This technique, used in previous analyses of VIPERS-PDR2 data, performs similarly to the new method tested in this work. It is important to recall here that the TSR weighting schemewas calibratedto minimisethe systematicbias on clustering estimates in mock catalogues. As such it does not assure a similar performance on real data due to possible differences between the clusteringof real and simulatedgalaxies. 
7. RSD ftting 
7.1. Theoretical modelling 
We modelled the anisotropic clustering in the monopoleξ(0) and quadrupole ξ(2) two-point correlation functions as described in Pezzottaetal. (2017).We used the TNS model(Taruyaetal. 2010)that reads in the case of biased tracers, 
24 
Ps (k,µk)= D (kµkσv)[b2Pδδ (k)+ 2µ k fbPδθ + µ kf 2Pθθ (k)+ A (k,µk, f, b)+ B (k,µk, f, b)], (17) 
with f and b being the growth rate and linear galaxy bias, respectively. In Eq.(17), Pδδ is the non-linear matter power spectrum and Pδθ and Pθθ are density-velocity divergence and velocity divergence-velocity divergence power spectra, respectively. The correctionfactors A (k,µk, f, b)and B (k,µk, f, b)are derived using perturbation theory and provided in Taruya et al. (2010) and delaTorre&Guzzo (2012), and account for the 
A17, page9of 14 

Fig. <BR>8. <BR>Top <BR>panels:multipole moments measured from one mock parent catalogue (black continuous lines). The contribution to theoverall cluster-ing from the sub-samples of observable (blue dashed lines) and unobservable (red dash-dotted lines) pairs (defned respectively as those targeted at least onceandthe onesnevertargetedinthe ensembleof2170targetingruns)as writteninEq.(16)arealsoshown.The combinationof these two contributions is plotted as green flled markers. Bottom panels: fractional offset of the contribution from observable pairs and the unobservable/observable combination with respect to the reference measurement from the parent mock. This measurement refers to the low-redshift bin 0.5< z < 0.7. The measurementin the high-redshift bin0.7< z < 1.2shows a very similar behaviour. 
mode coupling between density and velocity felds. The phenomenological dampingfactor D(kµkσv)mimics theeffect of the small-scale pairwisevelocity dispersionbysuppressing the clusteringpower predictedbythe “Kaiserfactor”and dependsonthe nuisance parameter σv.WeusedaLorentzian functional form for D(kµkσv)as it is found to better describe the observations with respect to the theoretically predicted Gaussian damping factor 
(e.g. Pezzottaetal.2017).The modelinEq.(17)isalso supplemented witha second Gaussian dampingfactor with fxed dis-persion σz to account for theeffect of VIPERS redshift errors on clustering measurements. 
The model in Eq.(17)depends on four ftting parameters (f, b,σ8,σv). However, we provide measurements of the derived parameters f σ8 and bσ8 as σ8, the normalisation of the linear matter power spectrum Plin, is degenerate with the growth rate 
δδ 
parameter f and the linear biasfactor b. Plin 
The linear matter power spectrum is obtained using 
δδ 
the Code for Anisotropies in the Microwave Background (Lewis et al. 2000, CAMB) that is combined with HALOFIT (Smith et al. 2003; Takahashi et al. 2012) to predict the non-linear matter power spectrum Pδδ. The density-velocity divergence Pδθ and velocity divergence-velocity divergence Pθθ power spectra cannot be measured from data directly. They can be predicted by either using perturbation theory or by means of empirical ftting functions calibrated on numerical simulations 
(e.g. Jennings et al. 2011). Perturbation theory however breaks down at scales accessible in VIPERS. We therefore used the improved ftting functions described in Bel et al. (in prep.), 
" !#1/2 
k 
Plin 
Pδθ (k)= (k)Pδδ (k)exp − , (18a) 
δδ kcut δθ 
" !# 
k 
Plin 
Pθθ (k)= δδ (k)exp −· (18b) 
kcut 
θθ 
In Eq.(18), kcut and kcut are defned as 
δθ θθ 
1 
kcut σ−2.034 
δθ = 8 , (19a) 
2.972 
1 
kcut σ−2.163 
= 8 , (19b) 
θθ 1.906 with σ8 being the amplitude of the linear matter power spectrum. We note that in our model, σ8 controls the level of non-linearity (within HALOFIT) in the matter non-linear density-velocity divergence and velocity divergence-velocity divergencepower spectra that enter the RSD modelofEq.(17). 
7.2. Fitting method and data covariance matrix 
The measured monopole and quadrupole are simultaneously ftted with the TNS model to estimate the ftting parameters using the Monte-Carlo Markov chain (MCMC) technique. The MCMC algorithm explores the posterior distribution in the parameter space constrained by the data likelihood and parame-ter priors. The data likelihood is, 
&#16;&#17;X&#16;&#17; &#16;&#17; 
−2lnL = χ2 θp =Δi θpC−1 Δ j θp , (20) 
ij i, j 
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F. G. Mohammad et al.: Redshift-space distortions in VIPERS 

Fig. <BR>9. <BR>Top <BR>panels: mean estimates and related errors of multipole moments of the two-point correlation function from the set of 153 mock parent samples (lines with shaded bands) and the corresponding VIPERS-like mocks obtained using the PIP and angular up-weighting method (points with error-bars). Bottom panels:mean fractional systematic bias of measurements from VIPERS-like mocks with respect to the ones from the underlying parent samples (flled points with solid error-bars). In the bottom small panels we also display the case when the sub-sample of observable pairs is used as reference (empty markers with dashed error-bars). Measurements obtained using the TSR weighting scheme are plotted for comparison (dash-dotted lines with hatched areas). Error bars in the bottom panels quantify the scatter of the systematic offsets among 153 mocks. Left and right panels show the measurementsin thelow0.5< z < 0.7and high0.7< z < 1.2redshift bins, respectively. 
where θp denotes the set of ftting parameters, Δi is the discrepancybetween the data and model prediction in bin i and Ci j −1 is the precision matrix, that is, the inverse of the data covariance matrix Ci j.We ft the monopole s2ξ(0) and quadrupole s2ξ(2) of the two-
point correlation functions simultaneously and accounted for their cross-covariance in the data covariance matrix. 
The covariance matrices Ci j were estimated using the set of 153 VIPERS-like mocks. Noise in the covariance matrix is amplifed when inferring the precision matrix using Ci j and leads toabiased estimateofthe precision matrix.Wecorrectedforthis biasby meansofthe correctivefactorprovidedin Percivaletal. (2014). The correlation matrices, that is, Ri j = Ci j/(CiiC j j)1/2, for the two redshift bins and restricted to the range of ftting scales used here are shown in Fig. 10. 
The robustness of the data analysis method has already been tested in Pezzotta et al. (2017).We therefore focus on repeating the analysis using only the range of ftting scales adopted in Pezzotta et al. (2017)to obtain the reference estimates of the f σ8 parameter, that is, minimum and maximum scales fxed at smin = 5h−1Mpc and smax = 50h−1Mpc, respectively. In particular, we ft the mean estimates of s2ξ(`) for the monopole ` = 0 and quadrupole ` = 2from the mock catalogues with the TNS model and obtained a systematic offset, with respect to the fducial values, of 
Δ (f σ8)(z = 0.60) = 0.009 ± 0.015 Δ (f σ8)(z = 0.86) = −0.006 ± 0.012· 
These estimates are un-biased compared to the expected values of f σ8(z)in the mock fducial cosmology. Moreover our measurements are also compatible with estimates obtained in Pezzotta et al. (2017), Δ (f σ8)(z = 0.60) = 0.019 ± 0.012 and Δ (f σ8)(z = 0.86) = −0.018 ± 0.011, within 1σ. The marginalised one-and two-dimensional posterior likelihoods are shown in Fig. 11. For comparison we also show, in the same fgure, the results obtained by Pezzotta et al. (2017)using the same set of mocks. 
8. Growth rate measurements 
To correct the measurements of the two-point correlation function from the VIPERS-PDR2galaxy catalogue we followed the same procedure adopted on mock catalogues, including calcu-lating the PIP weights using both rotations to the parent catalogueand shiftsoftheextended spectroscopic mask.As VIPERS parent catalogue we used the photometric catalogue from the CFHTLS W1 and W4 felds, from which VIPERS targets were drawn, restricted to the area covered by the VIPERS observations. However, unlike mock samples the VIPERS parent catalogue contains Nc = 449 compulsory targets that do not enter the maximisation of the number of slits. Although a negligible fraction, we accounted for these objects when generating multiple survey realisations unless they fall inside the emptygaps between VIMOS quadrants. As anticipated in Sect. 2, we used onlygalaxies with quality fags 2–9 (inclusive) corresponding to 
A17, page 11 of 14 


p
a sample with a redshift confrmation rate of 96.1%. The effect of redshiftfailuresis not accounted for when computing the PIP weights.We therefore correctedfortheeffectof redshiftfailures byup-weightingeachgalaxyinthe VIPERS-PDR2 catalogueby the corresponding SSR as described in Sect. 4.2. 
We ft the monopoles2ξ(0) and quadrupole s2ξ(2) of the two-point correlation function between smin = 5h−1Mpc and smax = 50h−1MpcwiththeTNS modelinEq.(17)supplementedwitha second GaussiandampingfactorwithwidthfxedtotheVIPERS spectroscopic redshift error σz/(1+ z)= 0.00054. The measured values for the derived parameter f σ8 are, 
f σ8(z = 0.60) = 0.49 ± 0.12 
f σ8(z = 0.86) = 0.46 ± 0.09· 
These values are compatible within 1-σ with estimates from Pezzotta et al. (2017), namely f σ8(z = 0.60) = 0.55 ± 0.12 and f σ8(z = 0.86) = 0.40±0.11, who used the same datasets and the-oretical prescriptions for RSD modelling. Furthermore our mea-surements are also consistent within the error bars with the ones obtainedwith alternative methodssuchasa combinationofRSD andgalaxy-galaxy lensingin delaTorreetal. (2017)orthe one usinga sampleof luminous bluegalaxiesin VIPERS as donein Mohammad et al. (2018). The best-ft models corresponding to the results in Fig. 12 are displayed in Fig. 13 along with the mea-surements of the monopole s2ξ(0) and quadrupole s2ξ(2) moments of the two-point correlation function using the VIPERS-PDR2 
Fig. <BR>11. <BR>One-and two-dimensional marginalised posterior likelihoods of the derived parameter f σ8,bσ8 and the nuisance parameter σv resulting from the analysis of the mean clustering estimates obtained from 153 VIPERS-like mock catalogues using the method in Sect. 4.1. Fits are performed with TNS model between a minimum ftting scale of smin = 5h−1 Mpc up to a maximum scale of smax = 50h−1Mpc. For comparisonwehavealsoover-plotted results obtainedin Pezzottaetal. (2017)using the same setof mock samples and ftting method.Vertical dash-dotted and solid lines correspond to the expected values of f σ8 at z = 0.6andz = 0.86, respectively. 
galaxy sample (points with error-bars) and VIPERS-like mocks (cyan lines). 
9. Summaryand conclusions 
We corrected the clustering estimates from the VIPERSPDR2 galaxy sample using the PIP method described in Bianchi&Percival (2017). This technique was supplemented with the angular up-weighting scheme proposed in Percival&Bianchi (2017)to improve the statistical precision of the measurements. The PIP method relies on up-weighting thepair-counts based on the corresponding selection probabilities. These probabilities were inferred empiricallyby generating multiple surveyrealisations from a parent catalogue and count-ing the number of times a given pair is observed. To compare the performance of this new technique with the results obtained in Pezzotta et al. (2017)we split the redshift range probed by VIPERS into two bins spanning0.5< z < 0.7and0.7< z < 1.2. The following considerations equally apply to both redshift bins. 
Given the features of the VIPERS targeting algorithm and the limited extension of the VIPERS parent mocks, we generated multiple (2170) VIPERS realisations from each parent sample by spatially moving the spectroscopic mask.To assign selection probabilitiestogalaxy pairs withareasonable amountof computation time we also rotated the parent catalogue in each targeting run. The price to pay is that survey realisations with different rotations of the parent sample are not fully equivalent to each other producinga “normalisation problem” for the weighted pair 
A17, page 12 of 14 
F. G. Mohammad et al.: Redshift-space distortions in VIPERS 


technique with the angular up-weighting method.A negligible mean systematic bias was found comparing clustering measurements from each of 2170 surveyrealisations with the reference measurement from the parent catalogue. Nevertheless, we have shown that this bias is produced by the very small fraction of galaxy pairs unobserved in Nruns = 2170 survey realisations. Indeed these pairs are not randomly distributedbut ratherexhibit a small-scale clustering. 
Toassess the observational systematic bias on clustering mea-surements, we selected, for each parent mock, only the survey realisation obtained with actual VIPERS observational setup, that is no rotation of the parent sample and no shift in the spectroscopic mask.We founda mean fractional systematic bias among 153mocksamplestobebelowthe percentagelevel.Wearguethat such a small offset results from a combination of two effects: a) we are unableto assign selection probabilitiestoa small fraction of pairs that cannot be observed using only 2170 surveyrealisations, referred to as unobservable pairs; and b) the VIPERS-like mock is a particular confguration among the 2170 realisations used to infer the selection probabilities. Our tests using mocks catalogues have shown the new method to be a valid and robust way to correct for missing targets in VIMOS observations. 
We tested the impact of these corrections on estimates of the growth rate of structure times the amplitude of dark matter density fuctuations f σ8. In particular we ftted the mean estimates of the corrected monopole and quadrupole among 153 VIPERS-like mocks with the TNS model on scales5h−1Mpc < s < 50h−1Mpc. The analysis provided un-biased estimates of the ftting parameter f σ8 that are fully consistent with those obtained in Pezzotta et al. (2017)using the same confguration of ftting scales and theoretical model. The measurements made using the new technique are slightly closer to the expected val-ues,but the difference is within the expected errors. This pro-vides further confrmation of the robustness of previous RSD 
Fig. <BR>13. <BR>Monopole s2ξ(0) and quadrupole s2ξ(2) moments of the two-point correlation function measured from VIPERS-PDR2galaxy sample using the weighted pair counts as described in Sect. 4.1 (points with error-bars). Diagonal errors are estimated using the set of 153 VIPERS-like mocks. Cyan lines show the measurements from individual VIPERS-like mocks. The best-ft models corresponding to the results in Fig. 12 are also displayed as solid blue and dashed red lines. Top and bottom panels show results from the low-and high-redshift bins, respectively. 
analysesin VIPERS.However we stress here thefact that while the correction scheme adopted in previous VIPERS works (e.g. delaTorreetal. 2017; Pezzottaetal. 2017; Mohammadetal. 2018)relied on a fne-tuned parametric approach calibratedon mock catalogues to minimise the observational systematic bias, the new technique proposed in Bianchi&Percival (2017)and Percival&Bianchi (2017)isexact andis self-contained, using only the data itself. 
Finally, we applied this method to correct the measurements of the two-point correlation function using the VIPERS-PDR2 galaxy catalogue. When dealing with data we have accounted for theeffectof redshiftfailuresby meansoftheso called “spectroscopic success rate” (SSR). We also took into account the presence of a small fraction of compulsory targets in the parent sample. Both these features were not reproduced in the mock samples. The measured monopole and quadrupole moments of the two-point correlation functions were ftted with the TNS model to estimate the f σ8 parameter at the effective red-shifts of the two redshift bins. Our measurements are in agreement within 1-σ with previous measurements by Pezzotta et al. (2017),delaTorreetal. (2017)andMohammadetal. (2018)at the same redshifts. 
In future work, we will improve upon this analysis using the method of Percival&Bianchi (2017) to include angular 
A17, page 13 of 14 
clustering measurements from the full CFHTLS sample. By using a combination of the angular and 3D clustering measurements, we hope to observe baryon acoustic oscillations, as well as to improve on the current RSD measurements. 
Acknowledgements. We thank Michael Wilson for useful comments on this work.We acknowledge the crucial contribution of the ESO staff for the man-agementof service observations through which the VIPERS surveywasbuilt.In particular, we are deeply grateful to M. Hilker for his constant help and support of this programme. Italian participation to VIPERS has been funded by INAF through PRIN 2008, 2010, 2014 and 2015 programs. LG, FGM, BRG and JB acknowledge support from the European Research Council through grant 
n. 291521. DB and WJP acknowledge support from the European Research Council through grant n. 614030. OLF acknowledges support from the European Research Council through grant n. 268107. SDLTacknowledges the support of the OCEVU Labex (ANR-11-LABX-0060) and the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French government programme managed by the ANR. RT acknowledges fnancial support from the European Research Council through grant n. 202686. AP, KM, and JK have been supported by the National Science Centre (grants UMO-2012/07/B/ST9/04425 and UMO-2013/09/D/ST9/04030). EB, FM and LM acknowledge the support from grants ASI-INAFI/023/12/0and PRIN MIUR 2010-2011. TM and SA acknowledge fnancial support from the ANR Spin(e) through the French grant ANR-13-BS05-0005. The Big MultiDark Database used in this paper and the web application providing online access to it were constructed as part of the activities of the German AstrophysicalVirtual Observatoryas resultofa collaboration betweenthe Leibniz-Institutefor Astrophysics Potsdam (AIP) and the Spanish MultiDark Consolider Project CSD2009-00064. The Bolshoi and MultiDark simulations were run on theNASA’s Pleiades super-computer at theNASA Ames Research Center. 
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1 INAF – Osservatorio Astronomico di Brera,Via Brera 28, 20122 Milano – via E. Bianchi 46, 23807 Merate, Italy e-mail: faizan.mohammad@brera.inaf.it 
2 Università degli Studi di Milano, via G. Celoria 16, 20133 Milano, Italy 3 Institute for Cosmology and Gravitation, University of Portsmouth, 1-8 Burnaby Rd, Portsmouth PO1 3, UK 4 DepartmentofPhysics and Astronomy, UniversityofWaterloo, 200 UniversityAveW,Waterloo,ON N2L 3G1, Canada 5 Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada 6 Aix-Marseille Université, Université Toulon, CNRS, CPT, Mar-seille, France 7 Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre,via dellaVascaNavale84, 00146Roma,Italy 8 INFN, Sezione di Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy 9 INAF – Osservatorio Astronomico di Roma, via Frascati 33, 00040 
Monte Porzio Catone (RM), Italy 
10 Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France 
11 Dipartimento di Fisica e Astronomia -Alma Mater Studiorum Università di Bologna, via Gobetti 93/2, 40129 Bologna, Italy 
12 INFN, Sezione di Bologna, viale Berti Pichat6/2, 40127 Bologna, Italy 
13 INAF – Osservatorio Astronomico di Bologna, via Gobetti 93/3, 40129 Bologna, Italy 
14 INAF – Istitutodi Astrofsica SpazialeeFisica Cosmica Milano,via Bassini 15, 20133 Milano, Italy 
15 INAF – Osservatorio Astrofsico di Torino, 10025 Pino Torinese, Italy 
16 Laboratoire Lagrange, UMR7293, Université de Nice Sophia Antipolis, CNRS, Observatoire de la Ce d’Azur, 06300 Nice, France 
17 InstituteofPhysics,JanKochanowskiUniversity,ul. Swietokrzyska 15, 25-406 Kielce, Poland 
18 National Centre for Nuclear Research, ul. Hoza 69, 00-681 Warszawa, Poland 
19 Aix-Marseille Université, Jardin du Pharo, 58 bd Charles Livon, 13284 Marseille Cedex 7, France 
20 IRAP,9 av.du colonel Roche,BP 44346, 31028Toulouse Cedex4, France 
21 Astronomical Observatory of the Jagiellonian University, Orla 171, 30-001 Cracow, Poland 
22 School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, UK 
23 INAF – Istituto di Astrofsica Spaziale e Fisica Cosmica Bologna, via Gobetti 101, 40129 Bologna, Italy 
24 INAF – Istituto di Radioastronomia, via Gobetti 101, 40129 Bologna, Italy 
25 Canada-France-Hawaii Telescope, 65–1238 Mamalahoa Highway, Kamuela, HI 96743, USA 
26 Department of Astronomy, University of Geneva, ch. d’Ecogia 16, 1290Versoix, Switzerland 
27 INAF – Osservatorio AstronomicodiTrieste,viaG.B.Tiepolo11, 34143Trieste, Italy 
28 Departmentof Astronomy&Physics, Saint Mary’s University, 923 Robie Street, Halifax Nova Scotia B3H 3C3, Canada 
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