© ESO, 2017

1. Introduction

The discovery of the accelerated expansion of the Universe in the late stages of the twentieth century has given us a self-consistent standard cosmological model, which is in close agreement with virtually all current cosmological observations. Multiple lines of evidence, such as cosmic microwave background anisotropies (Hinshaw et al. 2013; Planck Collaboration XIII 2016), baryon acoustic oscillations in the galaxy distribution (Beutler et al. 2011; Blake et al. 2011; Anderson et al. 2012), and SNe Ia luminosity distances (Riess et al. 1998; Perlmutter et al. 1999), require most of the energy content of the Universe to be in the form of a repulsive dark energy that is empirically close in behaviour to the classical cosmological constant (see e.g. Weinberg et al.2013, for some history and a review of current constraints). The nature of dark energy is naturally a question of huge interest, with possibilities ranging from a fixed vacuum energy density with equation of state w = P/ρc² = − 1 to dynamical models based on evolving scalar fields varying both in space and time. Such models motivate an effort to measure w and its evolution. But independent of the outcome of this exercise, a puzzle remains that a very large vacuum density seems to be necessary, and hence the much smaller observed value therefore requires a challenging degree of fine-tuning (Weinberg 1989).

A more radical explanation for the observed acceleration could be that the theory of gravity itself is modified on cosmological scales (Carroll et al. 2004; Jain & Khoury 2010; Clifton 2011). Commonly discussed alternatives include f(R) gravity, where the gravitational Lagrangian is made more complicated than a simple Ricci scalar R; chameleon models that invoke a fifth fundamental force to drive the acceleration; and DGP (Dvali-Gabadadze-Porrati) models, which postulate a higher dimensional Minkowski space-time, within which the ordinary 3+1 space-time is embedded. For an appropriate choice of model parameters, dark energy and modified gravity can both reproduce the observed expansion history H(z). In principle this degeneracy can be lifted by measuring the growth rate of cosmic structure. Modifications of gravity involve a variation in the strength of the gravitational force with scale or environment, and thus a key question is whether density fluctuations are growing at the rate predicted by models involving general relativity and a homogeneous dark energy.

Among observational methods to estimate the growth rate of structure, redshift-space distortions (RSD) in the clustering pattern of galaxies (Kaiser 1987) have assumed a growing importance in the last decade (e.g. Guzzo et al. 2008). These distortions arise when the Doppler effect of galaxy peculiar velocities supplements the isotropic Hubble expansion. Peculiar velocities are inevitably associated with gravitational growth of inhomogeneities, which can be described by the logarithmic growth rate of density perturbations, $\mathit{f}\mathrm{\equiv }\frac{\mathrm{d}\ln \mathit{\delta }}{\mathrm{d}\ln \mathit{a}}\mathit{,}$(1)where δ is the fractional density fluctuation and a is the cosmic scale factor. For many (but not all) theories of gravity, this growth rate can be well approximated by an empirical relation as f(z) = [ Ω(z) ] ^γ (Peebles 1980; Lahav et al. 1991), provided the fluctuations are in the linear regime and in the growing mode. For Einstein gravity, γ ≃ 0.55, but this parameter can vary by around 0.1 between different commonly discussed models of late-time dark energy and modified gravity (Dvali et al. 2000; Linder & Cahn 2007). Measurements of linear RSD from galaxy redshift surveys constrain the combination β = f/b, where b is an unknown linear galaxy bias parameter. But the real-space galaxy autocorrelation function, b²ξ_mass, is observable, so the combined parameter β can be split to yield an estimate of a quantity that purely concerns dark matter: fσ₈, where σ₈ is the rms linear matter fluctuations within spheres of radius 8 h^-1 Mpc.

Unfortunately, extracting the linear RSD signal from galaxy redshift surveys is non-trivial because much of the RSD signal lies on quasi-linear and non-linear scales. A simple and widely used extension of the linear Kaiser model is the dispersion model (Peacock & Dodds 1994), which accounts for radial convolution by a random velocity dispersion plus non-linear corrections to the real-space power spectrum. This model was successfully applied to several galaxy surveys in the past (Peacock et al. 2001; Guzzo et al. 2008), but is insufficiently accurate to be trusted when the precision allowed by the data goes below 10% (Okumura & Jing2011; Bianchi et al.2012; see also the companion paper by Wilson et al. 2017). There have been a number of attempts to derive improved RSD models. As shown by Scoccimarro (2004), the dispersion model is a simplification of the original streaming model (Peebles 1980; Fisher 1995), in which the full redshift-space correlation function is obtained by convolution with a proper scale-dependent pairwise velocity distribution. But predicting this distribution function is hard (e.g. Bianchi et al. 2015, 2016; Uhlemann et al. 2015) and typical applications simplify the problem by adopting a scale-dependent Gaussian pairwise distribution function (e.g. Reid et al. 2012). Scoccimarro (2004) proposed an influential alternative, in which the linear Kaiser term is generalised by including the velocity and velocity-density power spectra. This concept was extended by the TNS model (Taruya et al. 2010), which takes better into account the non-linear coupling between the density and the velocity field. This model is currently considered one of the best descriptions of RSD down to the quasi-linear regime.

These theoretical developments have been stimulated by a growing number of new measurements from larger datasets. These included in particular the 6dfGS (Beutler et al. 2012), WiggleZ (e.g. Blake et al. 2011; Contreras et al. 2013), and BOSS (e.g. Reid et al. 2014; Beutler et al. 2017; Satpathy et al. 2017; Sanchez et al. 2017; Grieb et al. 2017). The present paper is one in a series aimed at extending this RSD work to higher redshifts by analysing the final PDR-2 release of the VIMOS Public Extragalactic Redshift Survey (VIPERS; Guzzo et al. 2014; Garilli et al. 2014; Scodeggio et al. 2017). This survey has collected redshifts for about 90 000 galaxies in the range 0.4 ≲ z ≲ 1.2 with sampling and volume comparable to those of local surveys, such as the Two-degree Field Galaxy Redshift Survey (2dFGRS) at z ≃ 0.1 (Colless et al. 2001). The prime goal of VIPERS was an accurate measurement of the growth rate of structure at a redshift around unity. An early measurement was performed using the Public Data Release 1 (PDR-1; Garilli et al.2014), setting a reference measurement of fσ₈ at z = 0.8 (de la Torre et al. 2013). Having nearly doubled the sample, this analysis is now revisited and expanded in a number of ways. de la Torre et al. (2017) performs a configuration space joint analysis involving RSD and galaxy-galaxy lensing, while Wilson et al. (2017) develops a direct Fourier-space approach coupled with the so-called “clipping” linearisation of the density field. With a similar aim, Mohammad et al. (in prep.) identifies optimal subclasses of RSD tracers, focusing on luminous blue galaxies. The analysis we present here uses the configuration-space information contained in the first two even multipole moments of the anisotropic correlation function, implementing the currently most advanced non-linear corrections and testing their performances on VIPERS-like mocks.

The paper is organised as follows. In Sect. 2 we give a description of the final VIPERS dataset and of the corresponding mock catalogues used throughout the analysis, while in Sect. 3 we describe the estimation of the two-point correlation function of galaxies in redshift space. Section 4 describes the target selection biases and how these are mitigated. In Sect. 5 we present the VIPERS measurements. The error estimates are described in Sect. 6 along with the fitting procedure. Section 7 gives a description of the RSD models that are used in Sect. 8 to understand the level of systematics in the recovery of the growth rate of structure. The results are presented in Sect. 9 and discussed in Sect. 10 with our conclusions.

Throughout this analysis, if not specified otherwise, we assume a fiducial flat ΛCDM cosmological model with (Ω_m,Ω_b,n_s) = (0.30,0.045,0.96) and parametrise the Hubble constant as H₀ = 100 h km s^-1 Mpc^-1.

2. The VIPERS survey

2.1. Observations

The VIPERS survey covers an overall area of 23.5 deg² over the W1 and W4 fields of the Canada-France-Hawaii Telescope Legacy Survey Wide (CFHTLS-Wide). The VIMOS multi-object spectrograph (Le Fèvre et al. 2003) was used to cover these two fields with a mosaic of 288 pointings, 192 in W1 and 96 in W4 (see Fig. 1). Galaxies are selected from the CFHTLS catalogue to a faint limit of i_AB = 22.5, applying an additional (r − i) vs. (u − g) colour preselection that efficiently and robustly removes galaxies at z< 0.5. Coupled with a highly optimised observing strategy (Scodeggio et al. 2009), this doubles the mean galaxy sampling efficiency in the redshift range of interest, compared to a purely magnitude-limited sample, bringing it to 47%.

Spectra are collected at moderate resolution (R ≃ 220) using the LR Red grism, providing a wavelength coverage of 5500−9500 Å. The typical redshift error for the sample of reliable redshifts is σ_z = 0.00054(1 + z), which corresponds to an error on a galaxy peculiar velocity at any redshift of 163 km s^-1. These and other details are given in the full PDR-2 release accompanying paper (Scodeggio et al. 2017). A discussion of the data reduction and management infrastructure was presented in Garilli et al. (2014), while a complete description of the survey design and target selection was given in the survey description paper (Guzzo et al. 2014). The dataset used in this paper is an early version of the PDR-2 data, 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 flags 2 to 9 inclusive are used, corresponding to a sample with a redshift confirmation rate of 96.1% (for a description of the quality flag scheme, see Scodeggio et al.2017).

In the analysis presented here we analyse two redshift subsamples of the whole survey (W1 + W4) in the ranges 0.5 <z< 0.7 and 0.7 <z< 1.2, including 30 764 and 35 734 galaxies, respectively, for a total of 66 498 high-quality redshifts (out of the total number of 76 552 galaxies of this quality, i.e. with flag ≥2, in the full survey, see Scodeggio et al. 2017).

2.2. Redshift distribution

The redshift distribution of the galaxy sample is shown in Fig. 2. At z> 0.6, it follows the typical decay in the number of objects expected for a magnitude-limited survey, while the rapid fall of the counts at z< 0.5 is the result of the colour-colour preselection. In de la Torre et al. (2013) it was shown that this histogram can be modelled analytically by the functional form $\mathit{N}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathrm{=}\mathit{A}{\left(\frac{\mathit{z}}{{\mathit{z}}_{\mathrm{0}}}\right)}^{\mathit{\alpha }}\exp \left(\mathrm{-}{\left(\frac{\mathit{z}}{{\mathit{z}}_{\mathrm{0}}}\right)}^{\mathit{\beta }}\right)\hspace{0.17em}\mathrm{CSR}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathit{,}$(2)where A, z₀, α, and β are fitting parameters. The term CSR(z) (colour sampling rate) describes the colour−colour preselection in terms of an error function transitioning between 0 and 1 around redshift z = 0.5, i.e. CSR(z) = (1 − erf[b(z_t − z)]) / 2 where the transition redshift z_t and the transition width b are free parameters. As shown in Scodeggio et al. (2017), CSR(z) is unity for z ≥ 0.6, corresponding to a purely magnitude-limited selection.

The best fit of Eq. (2) to the final VIPERS data is shown by the red curve in Fig. 2. Such modelling of the redshift distribution is an important and sensitive ingredient when estimating galaxy clustering, as we discuss in Sect. 3 and in de la Torre et al. (2013). We compare it with the V_max technique (e.g. Cole 2011; de la Torre et al. 2013) shown in Fig. 2 with the purple curve. Although we find no significant difference in the resulting clustering between the two methods, here we chose to use the V_max method, as in the companion paper of de la Torre et al. (2017). A further method often used in the literature is that of smoothing the observed redshift distribution with a Gaussian kernel (as for instance in the parallel papers by Rota et al.2017; and Wilson et al. 2017).

2.3. Mock galaxy samples

In order to test the details of the analysis and the modelling of RSD, we make use of a suite of mock galaxy catalogues designed to be a realistic match to the VIPERS final dataset. These were constructed from the Big MultiDark N-body simulation (Klypin et al. 2016), which assumes a flat ΛCDM cosmology with (Ω_m,Ω_Λ,Ω_b,h,n_s,σ₈) = (0.307,0.693,0.0482,0.678,0.960,0.823) and covers a volume of 15.625 h^-3 Gpc³. The construction of the mock samples is described in de la Torre et al. (2017) and is based on the method detailed in de la Torre et al. (2013). These papers provide for further detail; we only give a brief overview of the adopted method in the following.

We extracted 153 independent light cones from the simulation volume, which follows the geometry of the VIPERS W1+W4 fields. The dark matter haloes identified in the simulation were populated with galaxies using the halo occupation distribution (HOD) technique. Because of the halo mass resolution of the simulation which is too large to host the faintest galaxies observed in VIPERS, the method of de la Torre & Peacock (2013) was applied to reconstruct haloes below the resolution limit. Each halo was then populated with galaxies according to its mass as described by the HOD. The HOD was calibrated directly on the VIPERS data as presented in de la Torre et al. (2013). To obtain fully realistic VIPERS mocks one needs to reproduce the VIPERS survey selection function. This was carried out following several steps. First, the magnitude cut i_AB< 22.5 and the effect of the colour selection on the radial distribution of the mocks were applied. The mock catalogues thus obtained are similar to the parent photometric sample used as target galaxy sample for spectroscopy in VIPERS. The slit-positioning algorithm with the same setting as for the data was further applied to parent mock catalogues. This allows us to reproduce the VIPERS footprint on the sky, the small-scale angular pair incompleteness, and the variation of TSR across the fields. Finally, random redshift errors were added to mock galaxy redshifts, which are similar to those present in the data. This procedure allows us to produce realistic mock galaxy catalogues that contain the detailed survey completeness function and observational biases of VIPERS.

3. Galaxy clustering estimation

We quantify galaxy clustering in redshift space by estimating the anisotropic two-point correlation function ξ(s,μ), where s is the redshift-space separation of galaxy pairs and μ is the cosine of the angle between the separation vector and the line of sight. We generate a catalogue of randomly distributed objects subject to the same angular and radial selection as the true data and use the Landy & Szalay (1993) estimator $\mathit{\xi }\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}\mathrm{=}\frac{\mathit{GG}\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}\mathrm{-}\mathrm{2}\mathit{GR}\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}\mathrm{+}\mathit{RR}\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}}{\mathit{RR}\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}}\mathit{,}$(3)where GG(s,μ), GR(s,μ), and RR(s,μ) are the normalised galaxy-galaxy, galaxy-random, and random-random pair counts in bins of s (Δ(log ₁₀s) = 0.1) and μ (Δμ = 0.01), respectively. This estimator has been shown to provide a nearly unbiased estimate of the two-point correlation function, while minimising its variance (Landy & Szalay 1993). We typically use random samples with 30 times more objects than in the true data to reduce their shot noise contribution to a negligible amount.

In this work we estimate the growth rate by fitting RSD models not to the full shape of ξ(s,μ), but rather to its first two even multipole moments, ξ⁽⁰⁾(s) and ξ⁽²⁾(s), defined as ${\mathit{\xi }}^{\mathrm{\left(}\mathit{\ell }\mathrm{\right)}}\mathrm{\left(}\mathit{s}\mathrm{\right)}\mathrm{=}\frac{\mathrm{2}\mathit{\ell }\mathrm{+}\mathrm{1}}{\mathrm{2}}{\mathrm{\int }}_{-1}^{\mathrm{+}\mathrm{1}}\mathit{\xi }\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}{\mathrm{ℒ}}_{\mathit{\ell }}\mathrm{\left(}\mathit{\mu }\mathrm{\right)}\mathrm{d}\mathit{\mu ,}$(4)where ℒ_ℓ is the ℓth order Legendre polynomials. Such an approach is normally preferred to prevent the size of data vectors and the resulting covariance matrix from becoming too large for practical computation (but see Mohammad et al.2016, for discussion of some drawbacks of this choice). We do not include in this analysis the extra information potentially provided by the hexadecapole ξ⁽⁴⁾. In addition to being noisier than the lower order moments, we found that our corrections of ξ⁽⁴⁾ for observational effects (see next chapter) do not fully recover the correct shape, thus there is a risk of introducing a further systematic bias in the final measurements.

4. Systematic selection effects

The VIPERS angular selection function is the result of combining several different angular completeness functions. Two of these are binary masks, i.e. describing areas that are fully used or fully lost. The first mask is related to defects in the parent photometric sample, which are mostly areas masked by bright stars, and the other mask is associated with the specific footprint of VIMOS and how the different pointings are tailored together to mosaic the VIPERS area. Working in configuration space, these masks are easily accounted for when defining the area and the auxiliary random samples for clustering measurements.

A more complex selection is related to the incomplete target sampling of VIPERS; on average 47% of the targets satisfying the VIPERS selection criteria can be placed behind a slit and observed, defining what we call the average target sampling rate (TSR). In principle, we should also account for the colour-colour preselection of the target sample, which introduces a colour sampling rate (CSR; see Scodeggio et al.2017). In practice, since the CSR can be safely assumed to be constant over the survey area thanks to the particularly careful homogenisation of the parent sample photometry (see Guzzo et al.2014), its effect is absorbed into the fit or model describing the smoothed redshift distribution, as in Eq. (2). In any case, the CSR is consistent with being unity for z ≥ 0.6. Finally, we also have to take into account how the probability of measuring the redshift of a targeted galaxy depends on observational conditions or technical issues, which can be location-dependent, which we call the spectroscopic success rate (SSR). The relative relevance, modelling, and overall impact of all these effects is described in more detail the following sections.

4.1. Slit collisions

A multi-object spectrograph survey must inevitably face the limitations imposed by the mechanics of how light from the targets is collected on the focal plane. Either fibres or slitlets (as in the case of VIMOS) impose a minimum physical size below which the spectrum of two adjacent galaxies on the sky cannot be collected at the same time. This completely suppresses the small-scale clustering amplitude, unless multiple telescope visits of the same field are performed, which is not the case with VIPERS. Furthermore, the same limit on close pairs causes high-density regions on the sky to be more poorly sampled with respect to low-density regions; this introduces a mismatch that, as we show here, affects the amplitude of clustering on all scales. For VIMOS, this effect is further enhanced by the slit-positioning optimisation software (SPOC; Bottini et al. 2005), which attempts to maximise the number of slits observed in each quadrant and as such tends to homogenise the angular distribution of targets.

Furthermore, in a multi-slit spectrograph such as VIMOS the dispersed spectrum is imaged directly onto the detector. As is evident from Fig. 3, this creates another “forbidden zone” perpendicular to the slit, where no other target can be observed without causing two spectra to overlap; this unlike in fibre spectrographs, where fibres are typically taken away from the telescope to a standing spectrograph and the spectra are conveniently aligned and packed on the CCD. Since the projected length of the spectrum on the detector is much larger than the corresponding size of the slit, this introduces another typical scale below which the number of measured angular pairs is reduced, again limiting the sampling of overdensities on the sky. In VIPERS, the spectral dispersion is always oriented along the north-south direction, so the depletion of galaxy pairs is anisotropic on the sky and larger along the declination direction.

The impact of these effects on angular clustering is quantified in Fig. 4, where in the top panel we plotted, for both the average of 153 mocks (solid lines) and the VIPERS data (filled points), the angular correlation function of the parent and spectroscopic samples (w_p(θ) and w_s(θ), respectively). The bottom panel shows instead the ratio of the corresponding numbers of pairs (bottom panel), defined as $\mathit{C}\mathrm{\left(}\mathit{\theta }\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}\mathrm{+}{\mathit{w}}_{\mathrm{s}}\mathrm{\left(}\mathit{\theta }\mathrm{\right)}}{\mathrm{1}\mathrm{+}{\mathit{w}}_{\mathrm{p}}\mathrm{\left(}\mathit{\theta }\mathrm{\right)}}\mathrm{·}$(5)In this figure we find clear evidence of the two angular scales discussed earlier, which are related to the width and length of the spectra; these are identified in the figure by the vertical dashed lines. The origin of this effect can be better identified if we split the separation angle θ into its components along the right ascension and declination directions, Δ_RA and Δ_DEC. The angular completeness map C(Δ_RA,Δ_DEC), corresponding to Eq. (5) is shown in Fig. 5. Here the ‘shadow’ of the target spectra is recognisable as the rectangular region with nearly zero counts at small separations. The few residual counts in this area are produced by the small variations in the slit length, together with the effect of the few serendipitous targets observed by chance within the slit of a primary target.

Translated to spatial scales, this angular selection function results in a strong suppression of the clustering amplitude below 1 h^-1 Mpc, as shown by the dotted line in Fig. 6. In de la Torre et al. (2013), we corrected for this effect by up-weighting each galaxy-galaxy pair at a given angular separation θ_ij by the inverse of the corresponding value of C(θ_ij), i.e. ${\mathit{w}}^{\mathit{A}}\mathrm{\left(}\mathit{\theta }\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathit{C}\mathrm{\left(}{\mathit{\theta }}_{\mathit{ij}}\mathrm{\right)}}\mathrm{·}$(6)We discuss the effectiveness of this weight together with the correction of the large-scale effect of the TSR at the end of the next section.

4.2. Larger-scale effects

Along with the drastic suppression at small separations, the physical size of the slits is responsible for the inhomogeneous sampling between high- and low-density regions across a single VIMOS quadrant. This translates in an almost constant suppression of the clustering amplitude on scales above 1 h^-1 Mpc. The correcting scheme we discuss here builds upon the original approach of de la Torre et al. (2013), in which galaxies are assigned a further weight ${\mathit{w}}_{\mathit{i}}\mathrm{=}\frac{\mathrm{1}}{{\mathrm{TSR}}_{\mathit{i}}}\mathrm{·}$(7)In that paper, however, the TSR used for each galaxy was simply the average value over the corresponding VIMOS quadrant; in this way, all target galaxies in a quadrant were up-weighted by the same factor. As shown by the dot-dashed curve in Fig. 6, when considering the real-space correlation function ξ(r) this procedure has limited effect; however when combined with the w^A(θ) = 1 /C(θ_ij) small-scale boost, the TSR up-weighting strategy provides a better correction (see Fig. 8 of de la Torre et al.2013).

The improved correction adopted here uses instead a local estimate of the TSR_i, which is defined as the ratio of the local surface densities of target and parent galaxies (i.e. before and after applying the target selection); these are estimated as detailed below and then averaged within an aperture of a given shape and size. If we call these quantities ${\mathit{\delta }}_{\mathit{i}}^{\mathrm{p}}$ and ${\mathit{\delta }}_{\mathit{i}}^{\mathrm{s}}$, the TSR_i is defined as ${\mathrm{TSR}}_{\mathit{i}}\mathrm{=}\frac{{\mathit{\delta }}_{\mathit{i}}^{\mathrm{s}}}{{\mathit{\delta }}_{\mathit{i}}^{\mathrm{p}}}\mathrm{·}$(8)The continuous δ fields are obtained, starting from the discrete distributions of parent and target galaxies, using a Delaunay tessellation (Delaunay 1934) to estimate the density at the position of each galaxy, and then linearly interpolating. These two continuous fields are then used to compute the values of ${\mathit{\delta }}_{\mathit{i}}^{\mathrm{p}}$ and ${\mathit{\delta }}_{\mathit{i}}^{\mathrm{s}}$ within an aperture of a given shape and size.

We identified the best-performing geometry for this aperture through the tests shown in Fig. 6. The overall correction is remarkable, since we are able to accurately recover the parent ξ(r) at large separations, both with a circular and rectangular aperture. The rectangular aperture provides the best correction to real-space clustering, which can be understood in terms of the anisotropy of the spectral “shadows” discussed earlier. The optimal size of the rectangular aperture is found to be 60 × 100 arcsec². The resulting distribution of the TSR_i values over the survey regions is shown in Fig. 1. One may expect the size of the optimal aperture for estimating the TSR to match to some extent the typical shadow of the VIPERS spectra on the focal plane. By direct test, this is clearly not the case because the typical size of a spectrum defines an aperture that is too small, if one is to make shot noise negligible.

4.3. Redshift dependence of angular corrections

Some of the corrections for angular selection biases also have an effect on the redshift distribution. Figure 7 shows the effect of correcting for the TSR and SSR on the observed redshift distribution of the VIPERS data. While the TSR does not introduce a significant redshift dependence, the application of the SSR boosts the expected number of galaxies in the distant (z> 1) part of the sample. This clearly reflects the increased inefficiency to measure redshifts for more and more distant objects. To be fully consistent with the data, the random samples used for the clustering analyses have to be weighted accordingly.

5. Two-point correlations from the VIPERS data

We thus proceed to estimate the redshift space correlation function and its moments for the VIPERS survey, adopting the weighting scheme discussed in the previous sections, which we recap for convenience:

• Each galaxy is up-weighted by the inverse of its TSR defined byEqs. (7) and (8), ${\mathit{w}}_{\mathit{i}}^{\mathrm{TSR}}$, as well as by the inverse of its SSR, ${\mathit{w}}_{\mathit{i}}^{\mathrm{SSR}}$.

• Each galaxy-galaxy pair with angular separation θ is up-weighted by the angular weight w^A(θ) defined in Eqs. (5) and (6).

Pair counts in the two-point correlation function estimator of Eq. (3) are then expressed as $\begin{array}{ccc}\mathit{GG}\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}& \mathrm{=}& \sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{{\mathit{N}}_{\mathit{G}}}\sum _{\mathit{j}\mathrm{=}\mathit{i}\mathrm{+}\mathrm{1}}^{{\mathit{N}}_{\mathit{G}}}{\mathit{w}}^{\mathit{A}}\mathrm{\left(}{\mathit{\theta }}_{\mathit{ij}}\mathrm{\right)}{\mathit{w}}_{\mathit{i}}^{\mathrm{TSR}}{\mathit{w}}_{\mathit{j}}^{\mathrm{TSR}}{\mathit{w}}_{\mathit{i}}^{\mathrm{SSR}}{\mathit{w}}_{\mathit{j}}^{\mathrm{SSR}}{\mathrm{\Theta }}_{\mathit{ij}}\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}\mathit{,}\\ \mathit{GR}\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}& \mathrm{=}& \sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{{\mathit{N}}_{\mathit{G}}}\sum _{\mathit{j}\mathrm{=}\mathrm{1}}^{{\mathit{N}}_{\mathit{R}}}{\mathit{w}}_{\mathit{i}}^{\mathrm{TSR}}{\mathit{w}}_{\mathit{i}}^{\mathrm{SSR}}{\mathrm{\Theta }}_{\mathit{ij}}\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}\mathit{,}\\ \mathit{RR}\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}& \mathrm{=}& \sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{{\mathit{N}}_{\mathit{R}}}\sum _{\mathit{j}\mathrm{=}\mathit{i}\mathrm{+}\mathrm{1}}^{{\mathit{N}}_{\mathit{R}}}{\mathrm{\Theta }}_{\mathit{ij}}\mathrm{\left(}\mathit{s,\mu }\mathrm{\right)}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{,}\end{array}$where Θ_ij(s,μ) is equal to unity for log (s_ij) in [ log (s) − Δlog (s) / 2,log (s) + Δlog (s) / 2 ] and μ_ij in [ μ − Δμ/ 2,μ + Δμ/ 2 ], and null otherwise.

The final performance of this weighting scheme on the recovered monopole and quadrupole of the redshift space correlation function are shown in Fig. 8 for the two redshift ranges considered in the analysis. The combined correction recovers the amplitude of the monopole at the 2% level, down to the Mpc scale, yielding a quasi-unbiased estimate of ξ⁽⁰⁾(s) on all comoving scales that are used for the RSD fitting. As for the quadrupole, we are able to have a reliable measurement of ξ⁽²⁾(s) (<5% deviation from the fiducial value) down to a few Mpc. This is an encouraging result; any uncorrected anisotropy from selection effects would be in danger of inducing a spurious contribution to the quadrupole, since this is our main measure of anisotropy.

Figure 9 shows the measurement of the anisotropic correlation function ξ(r_p,π) obtained from the full VIPERS data at 0.5 <z< 0.7 and 0.7 <z< 1.2. A bin size Δs = 0.5h^-1 Mpc was used in both r_p and π directions. We combine the results coming from the two VIPERS fields W1 and W4 simply by summing up the pair counts in each bin of separation and normalising for the total number of objects.

6. Covariance matrix and error estimation

Given the intrinsic correlation among different bins of the two-point correlation function (and consequently of its multipoles), it is essential to obtain a reliable estimate of the covariance matrix to be used during the fitting procedure. The fit is carried out by performing a maximum likelihood analysis of the data given the RSD model, which can be more easily described as the search throughout the parameter space of the position maximizing the likelihood function ℒ defined as $\mathrm{-}\mathrm{2}\ln \mathrm{ℒ}\mathrm{=}\sum _{\mathit{i}\mathrm{=}\mathrm{0}}^{{\mathit{N}}_{\mathrm{b}}\mathrm{-}\mathrm{1}}\sum _{\mathit{j}\mathrm{=}\mathrm{0}}^{{\mathit{N}}_{\mathrm{b}}\mathrm{-}\mathrm{1}}\mathrm{(}{\mathit{y}}_{\mathit{i}}^{\mathit{d}}\mathrm{-}{{\mathit{y}}_{\mathit{i}}^{\mathit{m}}}^{\mathrm{)}}{\mathrm{\Psi }}_{\mathit{ij}}\mathrm{(}{\mathit{y}}_{\mathit{j}}^{\mathit{d}}\mathrm{-}{{\mathit{y}}_{\mathit{j}}^{\mathit{m}}}^{\mathrm{)}}\mathit{.}$(12)Here the observable y = (ξ⁰,ξ²) is the monopole-quadrupole combined vector, Ψ ≡ C^-1 is the precision matrix (the inverse of the covariance matrix), N_b is the total number of data points, and indices d and m stand for data and model, respectively.

The covariance matrix C is organised in four blocks corresponding to the monopole-monopole, quadrupole-quadrupole and monopole-quadrupole cross covariance (two identical blocks in the latter case). The full monopole-quadrupole covariance matrix is estimated from the 153 mock realisations as $\mathit{Ĉ}\mathit{ij}\mathrm{=}\frac{\mathrm{1}}{{\mathit{N}}_{\mathrm{s}}\mathrm{-}\mathrm{1}}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{{\mathit{N}}_{\mathrm{s}}}\mathrm{(}{\mathit{y}}_{\mathit{i}}^{\mathit{k}}\mathrm{-}\mathit{y̅}{\mathit{i}}^{\mathrm{)}}\mathrm{(}{\mathit{y}}_{\mathit{j}}^{\mathit{k}}\mathrm{-}\mathit{y̅}{\mathit{j}}^{\mathrm{)}}\mathit{,}$(13)where N_s is the number of independent realisations used to estimate the covariance, y is the monopole-quadrupole vector, indices i,j run over the data points, and index k runs over different realisations. The mean value is estimated by averaging the measured values from different realisations, namely $\mathit{y̅}\mathrm{=}\frac{\mathrm{1}}{{\mathit{N}}_{\mathrm{s}}}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{{\mathit{N}}_{\mathrm{s}}}{\mathit{y}}^{\mathit{k}}\mathit{.}$(14)The corresponding correlation matrices obtained in this way for the two redshift subsamples are shown in Fig. 10.

Given the large number of mock samples, the estimate and the inversion of the covariance matrices can be achieved with good accuracy. However, the use of a finite number of mocks has two implications. Firstly, the estimated precision matrix obtained by taking the inverse of Ĉ is biased with respect to the true one, Ψ, where the difference is well-represented by an inverse Wishart distribution. Furthermore, the precision matrix Ψ contains statistical errors that propagate to the parameter space, affecting the derived errors on the cosmological parameters. We follow Percival et al. (2014) and correct for these effects by applying two correction factors. In the first case, we can remove the systematic bias of the precision matrix by rescaling Ĉ^-1 as $\mathrm{\Psi }\mathrm{=}\left(\mathrm{1}\mathrm{-}\frac{{\mathit{N}}_{\mathit{b}}\mathrm{+}\mathrm{1}}{{\mathit{N}}_{\mathit{s}}\mathrm{-}\mathrm{1}}\right)\mathit{Ĉ}-1\mathit{.}$(15)The latter correction factor involves the total number of data points N_b and realisations N_s. It takes into account the typical skewness characterising an inverse Wishart distribution and is capable of providing an unbiased estimate of the precision matrix (Hartlap et al. 2007). In the second case, the propagation of errors from the precision matrix to the derived parameters can be corrected by defining $\begin{array}{ccc}\mathit{A}& \mathrm{=}& \frac{\mathrm{2}}{\mathrm{(}{\mathit{N}}_{\mathrm{s}}\mathrm{-}{\mathit{N}}_{\mathrm{b}}\mathrm{-}\mathrm{1}\mathrm{)}\mathrm{(}{\mathit{N}}_{\mathrm{s}}\mathrm{-}{\mathit{N}}_{\mathrm{b}}\mathrm{-}\mathrm{4}\mathrm{)}}\mathit{,}\\ \mathit{B}& \mathrm{=}& \frac{\mathrm{(}{\mathit{N}}_{\mathrm{s}}\mathrm{-}{\mathit{N}}_{\mathrm{b}}\mathrm{-}\mathrm{2}\mathrm{)}}{\mathrm{(}{\mathit{N}}_{\mathrm{s}}\mathrm{-}{\mathit{N}}_{\mathrm{b}}\mathrm{-}\mathrm{1}\mathrm{)}\mathrm{(}{\mathit{N}}_{\mathrm{s}}\mathrm{-}{\mathit{N}}_{\mathrm{b}}\mathrm{-}\mathrm{4}\mathrm{)}}\mathit{,}\end{array}$(16)and applying the correction factor ${\mathit{m}}_{\mathrm{1}}\mathrm{=}\frac{\mathrm{1}\mathrm{+}\mathit{B}\mathrm{(}{\mathit{N}}_{\mathrm{b}}\mathrm{-}{\mathit{N}}_{\mathrm{p}}\mathrm{)}}{\mathrm{1}\mathrm{+}\mathit{A}\mathrm{+}\mathit{B}\mathrm{(}{\mathit{N}}_{\mathrm{p}}\mathrm{+}\mathrm{1}\mathrm{)}}$(17)to the estimated parameter covariance. In the previous equation, N_p is the total number of free parameters.

7. Modelling redshift-space distortions

Redshift-space distortions arise because the apparent position of galaxies is modified by the Doppler effect of their peculiar velocity v. In this way, the redshift-space position s of galaxies located at r becomes ${s}\mathrm{=}{r}\mathrm{+}\frac{{\mathit{v}}_{\mathrm{\parallel }}}{\mathit{aH}\mathrm{\left(}\mathit{a}\mathrm{\right)}}{{ê}}_{\mathrm{\parallel }}\mathit{,}$(18)where a is the scale factor, H(a) is the expansion rate and v_∥ = v·ê_∥ is the component of the galaxy peculiar velocity along the line of sight. Invoking mass conservation, the redshift-space density field δ^s(s) can be expressed as a function of its real-space counterpart δ(r) as ${\mathit{\delta }}^{\mathrm{s}}\mathrm{\left(}{s}\mathrm{\right)}\mathrm{=}\mathrm{[}\mathrm{1}\mathrm{+}\mathit{\delta }\mathrm{(}{r}\mathrm{\left)}\mathrm{\right]}|\frac{{\mathrm{d}}^{\mathrm{3}}{s}}{{\mathrm{d}}^{\mathrm{3}}{r}}{|}^{-1}\mathrm{-}\mathrm{1.}$(19)The targeting of high-redshift galaxies in VIPERS means that the largest pair separations are much smaller than the distance from the observer, so we can use the small-angle plane-parallel approximation; the Jacobian of the real-to-redshift space transformation then reduces to $|\frac{{\mathrm{d}}^{\mathrm{3}}{s}}{{\mathrm{d}}^{\mathrm{3}}{r}}|\mathrm{=}\mathrm{1}\mathrm{-}\mathit{f}{\mathit{\partial }}_{\mathrm{\parallel }}{\mathit{u}}_{\mathrm{\parallel }}\mathit{,}$(20)where the normalised velocity field is defined as u(r) = − v(r) / [ faH(a) ]. Substituting this expression inside Eq. (19) it follows that ${\mathit{\delta }}^{\mathrm{s}}\mathrm{\left(}{s}\mathrm{\right)}\mathrm{=}\frac{\mathit{\delta }\mathrm{\left(}{r}\mathrm{\right)}\mathrm{+}\mathit{f}{\mathit{\partial }}_{\mathrm{\parallel }}{\mathit{u}}_{\mathrm{\parallel }}}{\mathrm{1}\mathrm{-}\mathit{f}{\mathit{\delta }}_{\mathrm{\parallel }}{\mathit{u}}_{\mathrm{\parallel }}}\mathrm{·}$(21)Taking the Fourier transform of this equation and making explicit the dependence on , we obtain $\begin{array}{ccc}{\mathit{\delta }}^{\mathrm{s}}\mathrm{\left(}\mathit{k,\mu }\mathrm{\right)}& \mathrm{=}& \int \frac{{\mathrm{d}}^{\mathrm{3}}{s}}{\mathrm{(}\mathrm{2}\mathit{\pi }{\mathrm{)}}^{\mathrm{3}}}{\mathrm{e}}^{\mathrm{-}\mathrm{i}{k}\mathrm{·}{s}}{\mathit{\delta }}^{\mathit{s}}\mathrm{\left(}{s}\mathrm{\right)}\\ & \mathrm{=}& \mathrm{\int }\frac{{\mathrm{d}}^{\mathrm{3}}{r}}{\mathrm{(}\mathrm{2}\mathit{\pi }{\mathrm{)}}^{\mathrm{3}}}{\mathrm{e}}^{\mathrm{-}\mathrm{i}{k}\mathrm{·}{r}}{\mathrm{e}}^{\mathrm{i}\mathit{k\mu f}{\mathit{u}}_{\mathrm{\parallel }}}[\mathit{\delta }\mathrm{\left(}{r}\mathrm{\right)}\mathrm{+}\mathit{f}{\mathit{\partial }}_{\mathrm{\parallel }}{\mathit{u}}_{\mathrm{\parallel }}]\mathit{.}\end{array}$(22)The redshift-space power spectrum can thus be written as (Scoccimarro et al. 1999) $\begin{array}{ccc}{\mathit{P}}^{\mathrm{s}}\mathrm{\left(}\mathit{k,\mu }\mathrm{\right)}& \mathrm{=}& \int \frac{{\mathrm{d}}^{\mathrm{3}}{r}}{\mathrm{(}\mathrm{2}\mathit{\pi }{\mathrm{)}}^{\mathrm{3}}}{\mathrm{e}}^{\mathrm{-}\mathrm{i}{k}\mathrm{·}{r}}⟨{\mathrm{e}}^{\mathrm{-}\mathrm{i}\mathit{k\mu f}\mathrm{\Delta }{\mathit{u}}_{\mathrm{\parallel }}}\\ & & \end{array}$(23)with Δu_∥ = u_∥(x) − u_∥(x^′) and r = x − x^′. This last equation completely describes the anisotropies produced by peculiar velocities on the clustering of matter particles at each separation. Here, the only assumption is the plane-parallel approximation limit.

It is possible to identify two main regimes within which distortions manifest themselves. At large separations, matter has a coherent flow towards overdense regions. In this regime, the velocity field is mainly irrotational (Bernardeau et al. 2002) and can thus be described by its divergence θ(x) = ∇·u(x). These motions produce a systematic distortion of the large-scale distribution along the line of sight. This “Kaiser effect” (Kaiser 1987) is basically produced by the terms inside the square brackets in Eq. (23).

In contrast, within the typical scale of haloes, galaxy orbits cross each other: there is a random dispersion in velocities at a given point, which convolves the redshift-space structure in the radial direction. The clustering amplitude is thus suppressed on small scales, and structures appear stretched along the line of sight in the so-called “Fingers of God” (Jackson 1972). This effect is mainly generated by the exponential pre-factor involving the moment generating function of the velocity field.

Equation (23) is hard to use in its given form, because we lack an analytic formula for the ensemble average term inside the integral, particularly in the strongly non-linear regime. But a number of simpler approximate forms have been suggested, which aim to provide a satisfactory representation of the redshift-space power spectrum measured from galaxy surveys:

• Kaiser model (Kaiser 1987): within the linear theory approximation, the exponential pre-factor can be suppressed since its impact on the largest scales is negligible and θ ∝ δ. If the galaxy-matter bias relation is also assumed to be linear (δ_g = bδ), it follows that ${\mathit{P}}^{\mathrm{s}}\mathrm{\left(}\mathit{k,\mu }\mathrm{\right)}\mathrm{=}(\mathrm{1}\mathrm{+}\frac{\mathit{f}}{\mathit{b}}{\mathit{\mu }}^{\mathrm{2}}{)}^{\mathrm{2}}{\mathit{b}}^{\mathrm{2}}{\mathit{P}}_{\mathit{\delta \delta }}\mathrm{\left(}\mathit{k}\mathrm{\right)}\mathit{,}$(24)where P_δδ = P is the linear real-space matter power spectrum and b is the linear galaxy bias.

• Dispersion model (Peacock & Dodds 1994): although the previous model can reproduce the apparent enhancement of clustering at large separations, it fails in the description of the non-linear regime. The latter can be treated in a phenomenological way by artificially suppressing the linear theory predictions to account for the effect of the Fingers of God. Eq. (24) can thus be written as ${\mathit{P}}^{\mathrm{s}}\mathrm{\left(}\mathit{k,\mu }\mathrm{\right)}\mathrm{=}\mathit{D}(\mathit{k\mu }{\mathit{\sigma }}_{\mathrm{12}})\hspace{0.17em}(\mathrm{1}\mathrm{+}\frac{\mathit{f}}{\mathit{b}}{\mathit{\mu }}^{\mathrm{2}}{)}^{\mathrm{2}}{\mathit{b}}^{\mathrm{2}}{\mathit{P}}_{\mathit{\delta \delta }}\mathrm{\left(}\mathit{k}\mathrm{\right)}\mathit{,}$(25)where D(kμσ₁₂) is an analytical damping factor. This term depends on a nuisance parameter σ₁₂, which plays the role of a pairwise velocity dispersion. The basic assumption of the dispersion model is that σ₁₂ is not scale-dependent, but rather can be fitted as a free parameter. An useful extension of this model is to replace the linear P_δδ by a non-linear version (using an analytic approximation such as HALOFIT). This then allows the dispersion model to give the correct prediction for μ = 0: such modes run transverse to the line of sight and undergo no RSD effect. Some of the alternatives discussed here fail to match the real-space power exactly at μ = 0; this is because they are attempting the harder task of predicting the non-linearities, rather than taking them from a fit to N-body simulation data.

• Scoccimarro model (Scoccimarro 2004): as soon as the mildly non-linear regime is entered, the density and velocity divergence fields must be treated separately to account for the non-linear mode coupling between them. The ansatz proposed by Scoccimarro is that the exponential pre-factor inside Eq. (23) can be decoupled from the Kaiser term, so that its impact on the clustering is limited only to the smallest scales. In this case, it can be replaced with a damping factor similar to that already used in the dispersion model, leading to ${\mathit{P}}^{\mathrm{s}}\mathrm{\left(}\mathit{k,\mu }\mathrm{\right)}\mathrm{=}\mathit{D}(\mathit{k\mu }{\mathit{\sigma }}_{\mathrm{12}})\hspace{0.17em}({\mathit{b}}^{\mathrm{2}}{\mathit{P}}_{\mathit{\delta \delta }}\mathrm{\left(}\mathit{k}\mathrm{\right)}\mathrm{+}\mathrm{2}\mathit{fb}{\mathit{\mu }}^{\mathrm{2}}{\mathit{P}}_{\mathit{\delta \theta }}\mathrm{\left(}\mathit{k}\mathrm{\right)}\mathrm{+}{\mathit{f}}^{\mathrm{2}}{\mathit{\mu }}^{\mathrm{4}}{\mathit{P}}_{\mathit{\theta \theta }}\mathrm{\left(}\mathit{k}\mathrm{\right)})\mathit{,}$(26)where P_δθ and P_θθ are the density-velocity divergence cross-spectrum and the velocity divergence auto-spectrum, respectively. When applying this (and the following) model to real data, these quantities cannot be obtained from the data under analysis. As such, applications of this (and the following) model have used empirical fitting functions calibrated using numerical simulations (Jennings et al. 2011). In a parallel paper (Bel et al., in prep.), we used a large set of N-body simulations in different cosmologies (the DEMNUni simulations; Carbone et al. 2016), to derive the following, more general set of fitting formulae: $\begin{array}{ccc}{\mathit{P}}_{\mathit{\delta \theta }}\mathrm{\left(}\mathit{k}\mathrm{\right)}& \mathrm{=}& \\ {\mathit{P}}_{\mathit{\theta \theta }}\mathrm{\left(}\mathit{k}\mathrm{\right)}& \mathrm{=}& \end{array}$where P^lin(k) is the linear matter power spectrum and k^∗ is a parameter representing the typical damping scale of the velocity power spectra. The latter is well described as $\frac{\mathrm{1}}{{\mathit{k}}^{\mathrm{\ast }}}\mathrm{=}{\mathit{p}}_{\mathrm{1}}{\mathit{\sigma }}_{\mathrm{8}}^{{\mathit{p}}_{\mathrm{2}}}\mathit{,}$(29)where p₁,p₂ are the only free parameters of the fit. These forms for P_δθ and P_θθ have valuable, physically motivated properties: they naturally converge to P_δδ(k) in the linear regime, including a dependence on redshift through σ₈(z). Full details on the derivation and performances of these fitting formulas are presented in Bel et al. (in prep.). Their use in the analysis presented in the following sections is a significant improvement over previous applications of the Scoccimarro and TNS (Taruya et al. 2010) models, as it allows us to extend our tests to smaller scales and apply the models to a higher redshift, as sampled by VIPERS.

• Taruya (or TNS) model (Taruya et al. 2010): the non-linear mode coupling between the density and velocity divergence fields is responsible for a systematic bias between measurements of the power spectrum and its prediction using the previous RSD model. The origin of this deviation is the additional terms inside Eq. (26), which are not accounted for within the previous ansatz. The corrected model can be written as $\begin{array}{ccc}{\mathit{P}}^{\mathrm{s}}\mathrm{\left(}\mathit{k,\mu }\mathrm{\right)}& \mathrm{=}& \mathit{D}(\mathit{k\mu }{\mathit{\sigma }}_{\mathrm{12}})\hspace{0.17em}({\mathit{b}}^{\mathrm{2}}{\mathit{P}}_{\mathit{\delta \delta }}\mathrm{\left(}\mathit{k}\mathrm{\right)}\mathrm{+}\mathrm{2}\mathit{fb}{\mathit{\mu }}^{\mathrm{2}}{\mathit{P}}_{\mathit{\delta \theta }}\mathrm{\left(}\mathit{k}\mathrm{\right)}\mathrm{+}{\mathit{f}}^{\mathrm{2}}{\mathit{\mu }}^{\mathrm{4}}{\mathit{P}}_{\mathit{\theta \theta }}\mathrm{\left(}\mathit{k}\mathrm{\right)}\\ & & \end{array}$(30)where C_A and C_B are terms derived using perturbation theory, which aim to account for the density and velocity divergence couplings with the exponential pre-factor in Eq. (23). This model bears strong advantages, as it potentially can break the f − σ₈ degeneracy at the expense that it is intrinsically more difficult to implement. See de la Torre & Guzzo (2012) for the details of its application to biased tracers.

All the tested RSD models feature a phenomenological damping factor D(kμσ₁₂). The function D(kμσ₁₂) damps the power spectra in the Kaiser term but also partially mimics the effects of the pairwise velocity distribution in virialised systems. The expected analytic form of the damping factor on large enough scales assuming the Scoccimarro ansatz is Gaussian (Scoccimarro 2004); but analyses of simulated galaxy samples (de la Torre & Guzzo 2012) have shown that a Lorentzian template provides a better practical fit.

Models in Eqs. (25), (26), and (30) are all tested in the next sections to understand their impact on the recovery of the growth rate. In all cases, at each step of our Monte Carlo Markov chains we generate the full anisotropic redshift-space power spectrum. For this we make use of CAMB with the latest HALOFIT prescription for the non-linear P_δδ (Takahashi et al. 2012), and Eqs. (27) and (28) to generate the P_δθ and P_θθ power spectra. The normalisation of the latter real-space power spectra, which can be set by σ₈, is degenerate with f and b. This is why one generally parametrises RSD models in terms fσ₈ and bσ₈ parameters. In the case of the TNS model, however, this is not possible directly since the C_A term involves sub-terms that are not multiples of the fσ₈ or bσ₈ parameters (e.g. Taruya et al. 2010; de la Torre & Guzzo 2012). Therefore for the TNS model, and for the others for consistency, we decide to treat f, b, σ₈, σ₁₂ as independent parameters in the fit, and provide derived constraints on fσ₈ a posteriori from the MCMC chains.

In contrast to the implementation discussed in de la Torre & Guzzo (2012), a linear bias is assumed here in the modelling. Given the galaxy population sampled by VIPERS, the redshift range and the scales that are included in the final fits (s> 5 h^-1 Mpc), this is a reasonable assumption (see Fig. 8 in de la Torre & Guzzo 2012). This assumption is relaxed in the parallel complementary RSD analysis of the same VIPERS data by de la Torre et al. (2017), where a full non-linear bias model is adopted to describe simultaneously clustering and galaxy-galaxy lensing down to small separations.

It is important to emphasise that σ₈(z) not only plays a role in shaping the C_A term, it also controls the level of non-linearity in P_δδ, P_δθ, and P_θθ. In particular for P_δδ, the HALOFIT non-linear correction to the linear matter power spectrum is computed at each step of the MCMC according to the tested value of σ₈(z). This represents a significant improvement over what is usually done in RSD analyses, where σ₈(z) is fixed to its fiducial value for the description of P_δδ.

In the end, we measure the Fourier-space multipole moments as${\mathit{P}}^{\mathrm{\left(}\mathit{\ell }\mathrm{\right)}}\mathrm{\left(}\mathit{k}\mathrm{\right)}\mathrm{=}\frac{\mathrm{2}\mathit{\ell }\mathrm{+}\mathrm{1}}{\mathrm{2}}{\mathrm{\int }}_{-1}^{\mathrm{+}\mathrm{1}}{\mathit{P}}^{\mathrm{s}}\mathrm{\left(}\mathit{k,\mu }\mathrm{\right)}{\mathrm{ℒ}}_{\mathit{\ell }}\mathrm{\left(}\mathit{\mu }\mathrm{\right)}\mathrm{d}\mathit{\mu ,}$(31)and convert them to their configuration space counterparts as ${\mathit{\xi }}^{\mathrm{\left(}\mathit{\ell }\mathrm{\right)}}\mathrm{\left(}\mathit{s}\mathrm{\right)}\mathrm{=}{\mathit{i}}^{\mathit{\ell }}\mathrm{\int }\frac{\mathrm{d}\mathit{k}}{\mathrm{2}{\mathit{\pi }}^{\mathrm{2}}}{\mathit{k}}^{\mathrm{2}}{\mathit{P}}^{\mathrm{\left(}\mathit{\ell }\mathrm{\right)}}\mathrm{\left(}\mathit{k}\mathrm{\right)}{\mathit{j}}_{\mathit{\ell }}\mathrm{\left(}\mathit{ks}\mathrm{\right)}\mathit{,}$(32)where j_ℓ denotes the spherical Bessel functions.

Finally, we do not attempt to account in our modelling for the Alcock-Paczynski (AP) effect (Alcock & Paczynski 1979), i.e. the geometrical distortion introduced on the measured statistical quantities by the choice of a wrong expansion history when transforming angles and redshifts into comoving coordinates. This would require the inclusion of two extra parameters, allowing us in principle to constrain the expansion rate H(z) and the angular diameter distance D_A(z). This is explored in two other parallel RSD analyses using the VIPERS data. In Wilson et al. (2017) joint constraints on fσ₈ and the AP parameter F_AP ≡ (1 + z)D_AH(z) /c are presented, clearly at the expense of a larger uncertainty on the recovered value of fσ₈. In de la Torre et al. (2017) we also explore the impact of the AP-RSD degeneracy, but adding the extra constraint of galaxy-galaxy lensing. Here we instead work with observed quantities computed in the fiducial cosmology, if not for the single parameter σ₈, which determines the shape of the non-linear density and velocity spectra to some extent. In this respect, changing the fiducial cosmological model has a very small impact on the recovered growth rate, for example less than 1% when moving from a WMAP9-like to a Planck-like cosmology (de la Torre et al. 2013). The bottom line is that dynamical distortions dominate over the AP effect, as directly shown in earlier works (Marulli et al. 2012; Guzzo et al. 2008).

8. Tests of RSD models

We test in this section the RSD models introduced previously on our set of N_s = 153 mocks. In practice, analysing each mock and averaging the measurements would be computationally infeasible, considering the large number of configurations to be tested. We thus chose to average the monopole and quadrupole measurements over the mocks, scale the covariance matrix properly, and fit the models to these average measurements. The aim is to reach a statistical uncertainty that is a factor $\mathrm{1}\mathit{/}\sqrt{{\mathit{N}}_{\mathrm{s}}}$ smaller than a single VIPERS survey to be able to detect potential systematics as small as 1%. This process is more revealing and can show how well a given model performs in recovering the detailed shapes of the quadrupole and monopole correlation function.

We perform likelihood analyses of the mock mean measurements in different configurations, starting with the ideal case and moving on to the case in fully realistic conditions. All likelihood analyses are carried out using an MCMC code, whose output has been cross-checked with the independent MCMC code used in de la Torre et al. (2017). We select flat priors for the full set of free parameters, using boundaries that allow a large set of late-time evolution cosmological models to be considered as possible alternatives to standard ΛCDM. The full list of priors is shown in Table 1, while the best-fit values for the parameters are listed in Table 2. We vary the minimum scale s_min of the fit to understand how to select the best-fitting range for the VIPERS data – we expect all RSD models to fail at sufficiently small and non-linear scales. The maximum scale of the fit is fixed at s_max = 50 h^-1 Mpc, above which errors on the VIPERS measured monopole and quadrupole become too large.

8.1. Ideal case

We first study the ideal case that neglects the complex VIPERS angular selection function by using the parent mocks. Redshift errors are also not considered here, to understand how different RSD models behave in the absence of any observational bias. Furthermore, here we consider mock measurements from the full redshift range probed by VIPERS, i.e. 0.5 <z< 1.2, to maximise the available volume for the test; however, we include, in this way, a range of redshift over which we know that galaxies and large-scale structures do evolve. The relative difference of the recovered fσ₈ with respect to the fiducial one is shown in the top panel of Fig. 11. Two types of small-scale damping factor D(kμσ₁₂) are tested: the Lorentzian (filled symbols) and Gaussian (dashed lines) forms.

The overall trend of models using Lorentzian damping favours the TNS model, which yields almost unbiased measurements of the growth rate down to s_min = 5 h^-1 Mpc. Some overestimation is however seen for minimum scales close to s_min = 8 h^-1 Mpc, which in fact corresponds to the zero-crossing scale of the quadrupole ξ⁽²⁾(s). In contrast, both dispersion and Scoccimarro models consistently underestimate the growth rate with an error that fluctuates between 5–10%.

We also check (bottom panel) how the models perform when we fix the shape of the non-linear density and velocity divergence power spectra to the fiducial cosmology of the mocks through a fixed value for σ₈. Reducing the degrees of freedom does reduce the statistical error bars, but has a negative impact on the performance of the TNS model; indeed this model shows a stable positive systematic error of ~5% also when fitting large scales (s_min> 25 h^-1 Mpc) only.

Returning to the top panel, we also see that in all the cases the choice of a Lorentzian damping yields smaller systematic deviations than with a Gaussian damping. This is reflected by the trend of the dashed lines, which are close to the corresponding markers only when the minimum fitting separation s_min is larger than ~15 h^-1 Mpc, while rapidly deteriorating when smaller separations are included in the fit. This is highlighted in Fig. 12, where the different best-fitting models for the monopole and quadrupole using s_min = 5 h^-1 Mpc are directly compared to the mock data. Using a Gaussian damping, the model is no longer able to provide a good description of ξ⁽⁰⁾ and ξ⁽²⁾. Actually, the fit is mostly dominated by the small scales, whose data points have the smallest errors, and this explains why separations above 7 h^-1 Mpc are apparently those giving the strongest deviation between model and data. This result is in close agreement with previous work on the subject (e.g. Bianchi et al. 2012; de la Torre & Guzzo 2012).

8.2. Case with redshift errors

So far no redshift error was assumed in the mock samples. However, real VIPERS redshifts have significant uncertainties, which clearly impact observed redshift-space distortions. We know from the multiple redshift measurements (Scodeggio et al. 2017) that the redshift error probability distribution for the VIPERS sample of reliable redshifts used here is well described by a Gaussian with standard deviation σ_z = 0.00054(1 + z). This corresponds to a dispersion in galaxy peculiar velocity of 163 km s^-1.

By applying random errors to mock galaxy redshifts following the VIPERS observed distribution, we can effectively see additional distortions. These are shown in Fig. 8.2, where one can see how the shapes of the monopole ξ⁽⁰⁾(s) and the quadrupole ξ⁽²⁾(s) are affected. The imprint of redshift errors is similar to that of a small-scale damping of the power spectrum. While the monopole is damped below 4 h^-1 Mpc, the quadrupole is corrupted over a range extending out to ~20 h^-1 Mpc. Clearly, this effect needs to be carefully handled or modelled, if one hopes to recover an unbiased value for the growth rate. The consequences of not correcting for this effect are shown by the dashed lines in Fig. 14, where we repeat the tests of Fig. 11, but now include redshift errors. As feared, there is a significant deviation from the values of fσ₈ previously measured with the models in the best configuration, i.e. with the Lorentzian damping.

Rather than correcting for redshift errors in the measurements, as performed for the angular selection selection, it is better to include it in the modelling. It is intuitive to supplement the models with a convolution by an extra Gaussian distribution with standard deviation fixed to the VIPERS rms value of σ_z = 163 km s^-1, which corresponds to ${\mathit{\sigma }}_{\mathit{\pi }}\mathrm{=}\frac{\mathit{c}{\mathit{\sigma }}_{\mathit{z}}}{\mathit{H}\mathrm{\left(}\mathit{z}\mathrm{\right)}}\mathit{,}$(33)in terms of line-of-sight comoving separation. The filled symbols in Fig. 14 show how with this extra damping term we recover a performance similar to the more idealised case of Fig. 11.

We therefore adopt the TNS model with Lorentzian damping and Gaussian error damping, as our reference model for the final RSD analysis of the VIPERS data. However, we also test, for consistency, the behaviour of the other two models on the actual data to verify whether the trends seen in the mocks are confirmed.

8.3. Fully realistic case

We now turn to the case including fully realistic observing conditions. This means considering the target selection (masks, TSR, SSR, etc.) and limiting the samples to the same redshift ranges covered by the data and including redshift errors. The results that we obtain are shown in Fig. 15. The trends of the systematic error as a function of s_min are less stable than in the previous case, although the general behaviour and relative performances of the models are the same. The variations give us an idea of the impact of the selection function on samples this size. Again, we see some instability in the TNS model (at least in the bin 0.7 <z< 1.2) when the minimum scale of the fit is chosen around s_min ≃ 8 h^-1 Mpc. When we look into the MCMC results in more detail, we see that in this case the Markov chain tends to drift towards unrealistic values of σ₈, which are outside of the prior range defined in Table 2. This seems to be related to the difficulty of the TNS model to reach a stable fit in the region where the quadrupole crosses zero. As soon as we include smaller scales (or we move to larger scales), the regular trend is recovered. Nevertheless, we confirm the TNS model as the best-performing model with systematics ≲5% down to the smallest probed minimum scales.

Overall, the different tests performed on the mock catalogues indicate that we can safely use the TNS model with the appropriate damping functions as well as with a minimum fitting scale of s_min = 5 h^-1 Mpc. This is the configuration that we adopt for the analysis of the data.

9. VIPERS RSD results

We present in this section the results of the RSD analysis of the VIPERS final dataset. We apply the methodology described in the previous sections to the VIPERS galaxy sample, including the priors on the sampling parameters reported in Table 1. Since f and σ₈ are treated as separate parameters in the modelling, despite their intrinsic degeneracy the choice of priors on these two parameters is particularly important. Specifically, the most sensitive prior is that on σ₈, as this is the main parameter entering the non-linear modelling of RSD.

To define a sensible and realistic prior, while allowing room for deviations from GR, we based our choice on the effective field theory of dark energy formalism (Gubitosi et al. 2013; Bloomfield et al. 2013; Gleyzes et al. 2013), which allows a description of various kinds of dark energy models and modifications of gravity to be expressed in a self-consistent framework that includes the growth rate of structure (Piazza et al. 2014; Perenon et al. 2015). The latter works show that the range spanned by σ₈(z) for stable theories can vary significantly, suggesting a range [0.2, 0.65] as appropriate to account for early- and late-time dark energy models at the redshifts covered by VIPERS (for definitions, see Perenon et al.2017). Although this choice might exclude some more extreme modified gravity models from being selected by the fit, at the same time it avoids non-physical degeneracies that arise in the likelihood for some particular values of σ₈ outside of this range. This choice is corroborated by our parallel complementary analysis using the same data by de la Torre et al. (2017), in which the combination of RSD with galaxy-galaxy lensing constrains directly σ₈(z), allowing a broader prior at the outset. The best-fit values of σ₈ that we obtain for the different values of s_min are all well inside the adopted prior range with the wings of the posterior distribution pushed against the boundaries only for the largest values of s_min, where the signal from the data is the weakest.

Thus, finally combining the f and σ₈ measurements obtained using our standard configuration (s_min = 5 h^-1 Mpc), we obtain fσ₈(z = 0.6) = 0.55 ± 0.12 and fσ₈(z = 0.86) = 0.40 ± 0.11. We consider these as our reference measurements in this work and discuss their cosmological implications in the next section. The measurements and best-fitting model monopole and quadrupole correlation functions obtained in the two considered redshift bins are shown in Fig. 16. The corresponding best-fit values for the derived parameters are reported in Table 2.

It is interesting to verify a posteriori whether the trends and relative RSD model performances as a function of s_min established from the mock catalogues are similar to those seen in the real data. It is of course clear that any trend is less significant, as the data are statistically equivalent to considering just one of the 153 mock catalogues. In the left panel of Fig. 17, we show the result of this exercise, where the measured values of fσ₈ as a function of s_min are shown for the different tested models. To ease comparison, we reported in the right panel using the same scale, the corresponding results from the mock test for the realistic case (i.e. those of Fig. 15). Apart from the different statistical errors, the three tested RSD models provide virtually identical results in the real data, as opposed to the behaviour seen in the mock catalogues.

Moreover, it seems that in the data the variation of the fσ₈ measurements with minimum scale are not driven by the adequacy of the model down to those scales, but rather by statistical uncertainties in the measured galaxy correlation functions. The similarity in the results obtained from the different models is confirmed directly by the values of the reduced χ², which turn out to be very similar.

By directly looking at the posterior likelihood distributions of the parameters obtained with the three models in Fig. 18 (for the high-redshift bin), we can see that each model provides slightly different parameter degeneracies, although after marginalisation, fσ₈ posterior likelihood distributions are almost identical for the three RSD models, with only a slightly larger statistical uncertainty with the TNS model. However, some trends seen in the mock results are recognised in the data, as for example the preference of the TNS model in the high-redshift sample to deliver higher values of fσ₈ when s_min is close to the zero-crossing scale of the quadrupole.

Finally, it is important to emphasise the global non-linear approach to RSD that has been used in this analysis. We used rather small non-linear scales in the fit and by adopting consistent modelling for the non-linearities in the real-space density and velocity divergence power spectra, we can obtain further cosmological insight. The level of non-linearity in our analysis is controlled by one single parameter, σ₈(z), and we find that by letting this parameter vary, we can partly break the standard degeneracy that dominates on linear scales between f, σ₈, and b parameters. If we marginalise the posterior likelihood function over the σ₁₂, σ₈, b parameters, we obtain the following direct growth rate and σ₈ constraints: [ f(z = 0.6),σ₈(z = 0.6) ] = [ 1.048 ± 0.199,0.528 ± 0.076 ] and [ f(z = 0.86),σ₈(z = 0.86) ] = [ 0.742 ± 0.179,0.539 ± 0.068 ]. A similar approach has been adopted in de la Torre et al. (2017), where this is strengthened by additional constraints from galaxy-galaxy lensing. In particular, the latter allows the improvement of σ₈ constraints while keeping similar uncertainties on f. A detailed discussion of these results is given in de la Torre et al. (2017). Overall, these findings demonstrate the additional constraining power encapsulated in quasi-linear scales, which can be used to break degeneracies and further improve the precision of measurement of the growth rate of structure.

10. Discussion and conclusions

The measurements of the growth rate of structure times σ₈ that we obtained are

$\begin{array}{ccc}& & \mathit{f}{\mathit{\sigma }}_{\mathrm{8}}\mathrm{(}\mathit{z}\mathrm{=}\mathrm{0.6}\mathrm{)}\hspace{0.17em}\mathrm{=}\hspace{0.17em}\mathrm{0.55}\mathrm{\pm }\mathrm{0.12}\\ & & \mathit{f}{\mathit{\sigma }}_{\mathrm{8}}\mathrm{(}\mathit{z}\mathrm{=}\mathrm{0.86}\mathrm{)}\hspace{0.17em}\mathrm{=}\hspace{0.17em}\mathrm{0.40}\mathrm{\pm }\mathrm{0.11.}\end{array}$These values are confronted in Fig. 19 with different measurements, including results from other surveys, the VIPERS earlier PDR-1 dataset, and parallel works analysing analogous subsets of the VIPERS PDR-2 dataset with complementary techniques. It may look surprising that there is no appreciable improvement in the error bars between the former measurement from the PDR-1 (de la Torre et al. 2013, red circle) and the new PDR-2 estimate in a comparable redshift bin despite a ~30% increase in the sample size. As discussed in de la Torre & Guzzo (2012), this is essentially a price to pay for the more sophisticated treatment of non-linear effects through the TNS model, which increases the degrees of freedom.

The parallel PDR-2 results include measurements obtained from the combination of RSD with galaxy-galaxy lensing (de la Torre et al. 2017) or using the void-galaxy cross-correlation (Hawken et al. 2017). In forthcoming papers, we shall additionally present further pieces of this combined approach, using specific colour-selected subsamples (Mohammad et al., in prep.) or the linearised density field in Fourier space (Wilson et al. 2017), to minimise the need for non-linear corrections. All these papers represent complementary approaches towards understanding the current limitations we face in our ability to extract in practice the value of these parameters from the modelling of RSD.

The values measured by these different techniques on the same VIPERS data as well as from other surveys at similar redshifts are virtually all compatible within 1σ and agree with the predictions of a ΛCDM model governed by Einstein gravity. But on a larger sample, with much smaller statistical errors, greater care would be needed to test for possible systematic biases that might still be hidden in one or more of the analyses. The application of such a variety of approaches to VIPERS has been made possible by the specific properties of the survey, in particular its dense sampling and rich content of information. With a sparse sampled survey, which is the approach of most of the cosmologically oriented surveys, it would have been impossible to characterise accurately the density field and apply the clipping linearisation technique of Wilson et al. (2017) or to detect reliably cosmic voids such as those used in Hawken et al. (2017). At the same time, a survey with limited imaging information would not permit the investigation of the selection of optimal sub-populations (or the combination of different sub-populations), as we are pursuing in Mohammad et al. (in prep.) or to exploit the combination of RSD with lensing, as we have carried out in de la Torre et al. (2017) and which should be exploited to the fullest by Euclid mission (Laureijs et al. 2011) in the next decade. We therefore believe that the detailed investigation of the properties of RSD within VIPERS should serve as a valuable foundation for next-generation studies of greater statistical power.

Acknowledgments

We thank M. Wilson for useful comments. We acknowledge the crucial contribution of the ESO staff for the management of service observations. In particular, we are deeply grateful to M. Hilker for his constant help and support of this programme. Italian participation in VIPERS has been funded by INAF through PRIN 2008, 2010, 2014 and 2015 programmes. L.G. and B.R.G. acknowledge support from the European Research Council through grant No. 291521. O.L.F. acknowledges support from the European Research Council through grant No. 268107. J.A.P. acknowledges support of the European Research Council through the COSFORM ERC Advanced Research Grant (# 670193). G.D.L. acknowledges financial support from the European Research Council through grant No. 202781. R.T. acknowledges financial support from the European Research Council through grant No. 202686. A.P., K.M., and J.K. have been supported by the National Science Centre (grants UMO-2012/07/B/ST9/04425 and UMO-2013/09/D/ST9/04030). E.B., F.M. and L.M. acknowledge the support from grants ASI-INAF I/023/12/0 and PRIN MIUR 2010-2011. L.M. also acknowledges financial support from PRIN INAF 2012. S.D.L.T. and M.P. acknowledge 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 program managed by the ANR. Research conducted within the scope of the HECOLS International Associated Laboratory, supported in part by the Polish NCN grant DEC-2013/08/M/ST9/00664. T.M. and S.A. acknowledge financial support from the ANR Spin(e) through the French grant ANR-13-BS05-0005.

References

All Tables

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