The clustering of the SDSS-IV extended Baryon Oscillation Spectroscopic Survey DR14 quasar sample: structure growth rate measurement from the anisotropic quasar power spectrum in the redshift range 0.8 < z < 2.2

Abstract

1 INTRODUCTION

The large-scale structure of the Universe encodes a significant amount of information on how the late-time Universe has evolved since the accelerated expansion became the dominant component of the cosmos at z ≲ 2. One way to access this information is through spectroscopic observations of dark matter tracers, such as galaxies, quasars, or inter-galactic gas. Measuring the correlation function of these tracers allows to infer the distribution of dark matter of the Universe and to constrain cosmological parameters such as the matter density of the Universe, namely Ω_m, how gravity behaves at large scales, or to put constraints in the total neutrino masses and its effective number of species.

Two complementary approaches to extract such information are the baryon acoustic oscillations (BAOs) and redshift space distortions (RSDs). The BAO technique measures the BAO peak position of the observed tracer to infer the evolution of the Universe since the epoch of recombination, when the BAO peak was imprinted in the matter distribution. The BAO signal was detected on the galaxy distribution for the first time in the Sloan Digital Sky Survey (SDSS; Eisenstein et al. 2005) and in the 2-degree Field Galaxy Redshift Survey (2dFGRS; Cole et al. 2005a). The RSD technique (Kaiser 1987) examines the information of the radial component of the peculiar velocity field and the corresponding distortion in the position of tracers in redshift space. Such distortions contain information about how gravity behaves at intercluster scales (⁠|${\gtrsim }10\ \rm {Mpc}$|⁠) as well as the total matter content of the Universe. Since the distortions caused by the peculiar velocity field are coherent with the growth of structure, the RSD technique is sensitive to the matter content and to the model of gravity of the Universe.

The extended-Baryon Oscillation Spectroscopic Survey (eBOSS; Dawson et al. 2016), part of the SDSS-IV experiment (Blanton et al. 2017), has been constructed, in part, to measure redshifts for approximately 500 000 quasars at 0.8 < z < 2.2 (Myers et al. 2015, including spectroscopically confirmed quasars previously observed in the SDSS-I/II/III). Compared to previous SDSS large-scale projects, the eBOSS quasar sample presents relatively low number density of objects, which for the current data release 14 (DR14; Abolfathi et al. 2017), oscillates typically between 1 × and 2 × 10⁻⁵ [Mpc/h]³. However, eBOSS will compensate for this drawback by covering a large volume of the Universe in a redshift range, which has been barely unexplored to date by any spectroscopic survey.

The selection of quasars in eBOSS uses two different techniques: (i) a ‘CORE’ sample uses a Bayesian technique called XDQSOz (Bovy et al. 2012) that selects from the SDSS optical ugriz imaging combined with mid-IR imaging from the Wide-Field Infrared Survey Explorer (WISE) satellite; and (ii) a selection based on variability in the multi-epoch imaging from the Palomar Transient Factory (e.g. Palanque-Delabrouille et al. 2016). A full description of these selection techniques is presented in Myers et al. (2015), alongside the characterization of the final quasar sample, as determined by the early data. These early data were observed as a part of SEQUELS (Sloan Extended QUasars, ELG and LRG survey), part of SDSS-III and -IV, which acted as a pilot survey for eBOSS (Dawson et al. 2013; Ross et al. 2012a).

Recently, Ata et al. (2018) measured the isotropic BAO scale using the same DR14 quasar sample (DR14Q). In the present paper we describe a complementary analysis based on RSD which extends the anisotropic signal to the previous BAO analysis. In particular, we measure the power-spectrum monopole, quadrupole, and hexadecapole from the DR14Q sample in the redshift range 0.8 < z < 2.2. Pioneering works such as Blake et al. (2011) and Rota et al. (2017) have also studied the clustering of galaxies at high redshift using the power-spectrum statistics on the WiggleZ¹ and VIPERS² samples, respectively. We perform the following complementary analyses: (i) we examine the whole redshift bin and perform the measurement of parameters of cosmological interest at the effective redshift, z_eff = 1.52; (ii) we explore the cosmological constraints by setting the ratio of parameters α_∥/α_⊥ to be 1 (see equations 16 and 17 for definitions) or to leave them as free parameters; (iii) we use three different redshift estimates, based on different features of the quasar spectra; and (iv) we separate the full redshift range in three overlapping redshifts bins, lowz between 0.8 ≤ z ≤ 1.5; mid-z between 1.2 ≤ z ≤ 1.8 and high-z between 1.5 ≤ z ≤ 2.2, and measure cosmological parameters in each of these three redshift bins, where the correlation among the parameters at different redshift bins is also computed. In all cases, we focus on measuring the logarithmic growth of structure times the amplitude of dark matter density fluctuations, fσ₈(z). For those analyses where α_∥ and α_⊥ are treated as free independent parameters, we also measure the angular diameter distance, D_A(z), and Hubble parameter, H(z).

This paper is structured as follows. In Section 2, we describe the data set used in the paper, including how the actual quasars have been targeted, their redshifts estimated, and also the techniques to produce the quasar mocks used in this paper. In Section 3, we present the methodology of our analysis, how the power-spectrum multipoles have been measured, and the theoretical model used for measuring the cosmological parameters. In Section 4, we present the power-spectrum multipoles measurements and how they compare to the mocks and to the best-fitting models. In Section 5, we perform systematic and robustness tests, using mocks and N-body simulations, in order to evaluate the systematic error budget. Section 6 displays the final results in terms of cosmological parameters measured from the quasar sample using the analyses described above, and Section 7 displays the cosmological implications of our findings. This paper is presented alongside several companion papers that perform complementary and supporting analyses on the same DR14Q sample. Hou et al. (2018) and Zarrouk et al. (2018) perform a reciprocal RSD analysis to the one presented in this paper, but in configuration space instead of Fourier space. Zhao et al. (2018) and Ruggeri et al. (2018) perform RSD analyses using a redshift weighting technique, which accounts for a redshift evolution of the cosmological parameters across the considered redshift bin. A more detailed description of these works, along with a comparison on the predicted cosmological parameters, is presented in Section 8. Finally, in Section 9, we present the conclusions of this paper.

2 DATASET

We start by describing the DR14Q data set features in detail, along with the mock catalogues used in this work.

2.1 SDSS IV DR14 quasar sample

We review the imaging data that have been used to define the observed quasar sample, which is later selected for spectroscopic observation, how the spectroscopy for each quasar target is obtained, and how the quasar redshifts are measured.

All the eBOSS quasar targets selected for the DR14Q catalogue (Pâris et al. 2018) are based on the imaging from SDSS-I/II/III and the WISE (Wright et al. 2010). We briefly describe these data sets below. SDSS-I/II catalogues (York et al. 2000) imaged a 7606 deg² northern and 600 deg² southern parts of the sky in the ugriz photometric pass bands (Fukugita et al. 1996; Smith et al. 2002; Doi et al. 2010) and were released as part of the SDSS DR7 (Abazajian et al. 2009). The SDSS-III catalogues (Eisenstein et al. 2011) observed additional photometry in the SGC area, increasing the contiguous footprint up to 3172 deg², and were released as part of DR8 (Aihara et al. 2011). Further astrometry improvement of these data was presented in DR9 (Ahn et al. 2012). All the photometric data were collected on the 2.5-metre Sloan Telescope (Gunn et al. 2006), located at the Apache Point Observatory in New Mexico in the USA, using a drift-scanning mosaic CCD camera (Gunn et al. 1998). The eBOSS project does not add any extra imaging area to that released in DR8, although it takes advantages of upgraded photometric calibrations of these data, so-called ‘uber-calibration’ (Padmanabhan et al. 2008; Schlafly et al. 2012), released under the name of SDSS DR13 (Albareti et al. 2017). In addition, the WISE satellite (Wright et al. 2010) observed the full sky using four infrared channels centred at |$3.4\, \mu {\rm m}$| (W1), |$4.6\, \mu {\rm m}$| (W2), |$12 \, \mu {\rm m}$| (W3), and |$22 \, \mu {\rm m}$| (W4), and the eBOSS quasar sample makes use of W1 and W2 band for its targeting.

The quasar target selection criteria for eBOSS is presented in Myers et al. (2015). Objects that fulfill this criteria and without any previously known and secure redshift measurements are flagged as ‘QSO EBOSS CORE’ selected for spectroscopic observation and assigned an optical fibre. The spectroscopic observation are performed using the BOSS double-armed spectrographs (Smee et al. 2013), which cover the wavelength range of 3600 ≤ λ[Å] ≤ 10 000, with R = 1500 up to 2600. The description on how the pipelines process the data from a CCD level to a 1D spectrum level, and eventually to the measurement of the redshift are described in Albareti et al. (2017) and Bolton et al. (2012). The sources of redshifts are divided into three classes: (i) Legacy, where the quasar redshifts are obtained by SDSS I/II/III via non-eBOSS-related programmes; (ii) SEQUELS, where the redshifts are obtained from the Sloan Extended QUasar, ELG and LRG programme (SEQUELS; Pâris et al. 2017); and (iii) eBOSS, for those previously unknown quasar redshifts obtained by the eBOSS project. The eBOSS quasar redshifts represent more than 75 per cent of the redshifts in the current DR14Q catalogue. For further details on the imaging data, target selection criteria and the final construction of the DR14Q catalogues we refer the reader to Pâris et al. (2018).

2.1.1 Redshift measurements

One of the main challenges of using quasars as dark matter tracers is the reliability of their spectral classification and consequently their redshift estimation. Although the typical quasar spectrum has wide and prominent emission lines, the existence of quasars outflows may produce systematic shifts in the location of the broad emission lines, which may lead to uncorrected errors in the measurements of their redshifts (Shen et al. 2016). Therefore, having an accurate measurement of quasar redshift is key for achieving the scientific goals of SDSS-IV/eBOSS. For the present DR14Q catalogue, we use a number of different redshift estimates to test the impact of these potential systematics in the final scientific outcome.

The large number of quasar targets in the current DR14Q catalogue makes the systematic visually inspection procedure (used in the previous SDSSIII/BOSS Ly α analyses) unfeasible. However, the observations taken on the subprogramme SEQUELS were all visual inspected, which tested the performance of the automated classification used in the whole DR14Q. The automated pipeline was able to securely classify 91 per cent of the quasar spectra targeted for clustering studies; less than 0.5 per cent of these classifications were found to be false when visually examined (Dawson et al. 2016). Among the remaining 9 per cent of objects, which the automated pipeline failed to report a secure classification, approximately half were identified as quasars when they were visually inspected. As described in Pâris et al. (2018), the DR14Q combines automated pipeline together with visual inspections results, providing a variety of value-added information, containing three automated redshift estimates that we consider in this paper – z_PL, z_PCA, and |$z_\rm {{Mg}\,}{\rm {\small II}}$|⁠.

• The z_PL automated classification uses a Principle Component Analysis (PCA) decomposition of galaxy and quasar templates (Bolton et al. 2012), alongside a library of stellar templates, to fit a linear combination of four eigenspectra to each observed spectrum. The reference sample for these redshift estimates are visually inspected quasars from SEQUELS.

• The z_PCA automated classification uses a PCA decomposition of a sample of quasars with redshifts measurements at the location of the maximum of the Mg ii emission line, fitting a linear combination of four eigenvectors to each spectrum. In addition, this classification accounts for the potential presence of absorption lines, including broad ones, and it is trained to ignore them.

• The |$z_{\rm Mg\, {\small II}}$| automated classification uses the maximum of the Mg ii emission line at 2799 Å. This broad emission line is in principle less susceptible to the systematic shifts produced by astrophysical phenomena; when a robust measurement of this line is present, it offers a minimally biased estimate of the systemic redshift of the quasar. Consequently, this method produces an extremely low number of redshift failures (less than 0.5 per cent). On the other hand, this method is more susceptible to variations in the signal-to-noise ratio. When this emission line is not detected in the spectrum of the quasar, the |$z_{\rm Mg\, {\small II}}$| automated classification uses the z_PL prescription.

A comparison of the performance of these redshift estimates is presented in table 4 of Pâris et al. (2018) along with visually inspected redshifts. For the DR14Q, we adopt as a standard redshift estimate z_fid, which consist of any of the three options described above depending on the particular object (see Pâris et al. 2018 for further details), which provides the lowest rate of catastrophic failures. In order to test the robustness of the different redshift estimates, we run at the same time our science pipeline code on the DR14Q using z_fid, z_PCA, and |$z_{\rm Mg\, {\small II}}$|⁠, as we did for the BAO analysis in Ata et al. (2018).

2.1.2 DR14Q catalogue details

The DR14Q catalogue used in this paper (Pâris et al. 2018) comprises 158 757 objects between 0.8 ≤ z ≤ 2.2 that the automatic pipeline has classified as quasars. 20 641 of these objects were also visually inspected and confirmed to be quasars and their redshifts were also determined. 148 659 of these quasars have a secure spectroscopic redshift determination and are the objects used in this paper. The remaining objects 10 098, either did not received a spectroscopic fibre or the redshift could not be determined accurately, as we describe below in more detail.

5188 objects were photometrically identified as potential quasars, but did not receive a spectroscopic observation. The fibre allocation is designed to maximize the number of fibres placed on targets, considering the constraints of the physical size of the fibres, which correspond to an angle in the focal plane of 62 arcsec, which at z = 1.5 corresponds to 0.54 Mpc. The fibre-assignment algorithm is therefore sensitive to the target density of the sky, so highly populated regions tend to be covered by several tiles. This overlap of tiles locally resolves some collision (1015 quasars redshifts are identified at less than 62 arcsec angular separation, 677 in the northern Galactic hemisphere and 338 in the southern). In Section 2.2, we describe how the unobserved quasar due to fibre collisions are treated.

4910 objects were securely classified by the automated pipeline as quasars, but their redshifts could not be securely determined and did not receive a visual inspection. The distribution of these objects is not uniform across the plate position. We refer to these objects as ‘redshift failure quasars.’ In Section 2.2, we describe how we treat these objects in our analysis. Fig. 1 displays the success rate of securely measuring the redshift of a quasar (number of successfully identified redshifts over total number of objects) as a function of the fibre location in the plate. For each tile in the survey, the vertical axis is aligned to lines of constant declination. The top panels show the success rate produced by the automated pipeline (without any visual inspection), whereas the bottom panels display the success rate after a fraction of the objects were visually inspected. The non-uniform distribution of failure rates across the plate is produced by the non-uniform efficiency of the detectors that record the spectra. The fibres positioned on holes on the left and right edges of the plate are most frequently fibres on the edges of the fibre slit in the spectrographs, corresponding to edges of the spectrograph camera focal plane for which the optical aberrations are larger. The variation of the sensitivity of the spectrograph across its position can reach 5 per cent (Laurent et al. 2017).

Figure 1.

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Redshift success rate as a function of plate position for the DR14Q catalogue after and before visual inspections, top and bottom panels, respectively. The higher failure rate (lower success rate) in the edges of the plate across the x-axis is caused by the less sensitive areas of the detector associated with those plate regions. The higher failure rate in the SGC plates is associated with a poorer photometric conditions in the SGC compared to those in the NGC. For each tile of the survey, the x-axis of the plate is aligned along the iso-declination lines, and the y-axis along the iso-right ascension lines, in such a way that the top areas of the plate in the figure correspond to objects with higher declination than the lower areas of the plate.

The observed objects are distributed along an angular footprint (see Fig. 3) with an effective area of 2112.9 deg², with three disconnected regions: 1 in the Northern Galactic Cap (NGC) whose effective area is 1214.6 deg², and two in the Southern Galactic Cap (SGC), with a total area of 898.3 deg². The sub-region of the SGC with declinations <10 deg has an area of 412.2 deg², and the other one has an area of 486.1 deg².

In total, the DR14Q sample contains an effective volume³ of 0.246 Gpc³ that corresponds to an associated comoving volume of ∼32 Gpc³. The large difference between these two volumes is caused by the factor |$\lbrace P_0\bar{n}(r)/[1+P_0\bar{n}(r)]\rbrace ^2$| in the effective volume definition. In case we had a high density number of objects, |$P_0\bar{n}\gg 1$|⁠, both effective and comoving volume would be similar, as |$\lbrace P_0\bar{n}(r)/[1+P_0\bar{n}(r)]\rbrace ^2\rightarrow 1$|⁠. On the other hand, for the DR14Q sample we have P₀ ∼ 6 × 10³ [h⁻¹ Mpc]³ and |$\bar{n}\sim 10^{-5}\,[h\,{\rm Mpc}^{-1}]^3$|⁠, and therefore, |$P_0\bar{n}\ll 1$|⁠, indicating that we are in a shot noise dominated regime and the two definitions are substantially different. The effective volume should be interpreted as the fraction of the associated comoving volume utilized for measuring the power at the wavenumber whose P(k) is P₀. Therefore in terms of Fisher information, the covariance matrices scale according the effective volume.

All quoted distances in this work correspond to comoving and all quoted volumes are effective volumes according to Tegmark (1997) unless mentioned otherwise.

2.2 Weights

Figure 2.

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Mean density of 148 659 quasars in the DR14Q catalogue as a function of redshift, for the NGC and SGC regions in blue and yellow lines, respectively. The slight difference between the two regions is caused by differences in the target efficiency. The black dashed lines correspond to the adopted smoothed model for producing the mock catalogues. In grey, we have overplotted the mean density of 100 ez-mock realizations with the NGC selection function, which shows and excellent agreement with the data.

2.2.1 Spectroscopic weights

The spectroscopic completeness is mainly affected by two effects, the fibre collisions, and redshift failures. We have briefly described these processes above.

The physical size of the optical fibres prevents the observation of two quasars at an angular scale lower than 62 arcsec  angular separation using a single tile. This effect is partially mitigated by overlapping tiles in those regions of the sky, where the concentration of targets is high. However, we still miss a small fraction of quasars due to this effect. We account for this effect by up-weighting the lost target to the nearest neighbour with a valid redshift and spectroscopic classification (always within 62 arcsec unless it has been flagged as a redshift failure). This weight is denoted as w_cp, which is 1 by default for all those quasars that have not been up-weighted, and an integer >1 for the cases of fibre collisions. In total, 4 per cent and 3 per cent of the eBOSS quasar targets are flagged as fibre close pairs in the NGC and SGC, respectively. A fraction of these up-weighted quasars are true companions of the lost target in physical distance. In these cases, the up-weight is physically motivated: we displace the lost target by a small cosmological distance (few Mpc) along the line of sight (LOS), which barely distorts the clustering signal. However, for the cases where two targets are not true companions, and the LOS projected distance is large (hundreds of Mpc), moving targets along the LOS does produce a spurious clustering signal along the LOS with respect to the clustering across the LOS. More complex prescriptions based on the probability distribution of the close-pairs along the LOS have been recently presented in the literature (Bianchi & Percival 2017; Hahn et al. 2017). In this work, we do not implement these techniques, which may have a subdominant contribution with respect to the statistical errors, and leave their implementation for future data releases.

We have shown above that the efficiency in which the redshift of a quasar is inferred depends on its position in the plate. In previous data releases of the BOSS survey, the fraction of objects that were classified as redshift failures was less than |$1\hbox{ per cent}$|⁠. For the DR14Q, the percentage of failures has increased up to 3.4 per cent and 3.6 per cent in the NGC and SGC, respectively, due to the more challenging task of measuring the redshift of a quasar at z ≃ 1.5, compared to, e.g., an LRG at z ≃ 0.5. In the recent BAO analysis of the DR14Q data (Ata et al. 2018), we opted to correct the redshift failures with a similar procedure as the one used to correct for fibre collisions: up-weighting the lost target to the nearest neighbour with a valid redshift and spectroscopic classification, what we designate w_noz. However, later we will show that this prescription produces a spurious signal in the LOS-dependent quantities, such as the quadrupole and hexadecapole, which are later transmitted to systematic shifts on the fσ₈ value.

2.2.2 Imaging weights

We make use of the imaging weights defined in Laurent et al. (2017) and applied to the DR14Q catalogues in Ata et al. (2018). These weights are required in order to remove the spurious dependency on the 5σ depth magnitude, known as ‘depth’, and Galactic extinction. Laurent et al. (2017) found that quasars are more securely identified where the value of the depth larger, and Galactic extinction is the variable that most affects differences in depth among the SDSS imaging bands, as they were almost simultaneously observed. The most important observational systematics identified in Laurent et al. (2017) were those related to the depth in the g-band magnitude and Galactic extinction, which used the map determined by Schlegel, Finkbeiner & Davis (1998).

Figure 3.

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Angular footprint of the DR14Q sample for the NGC (top panels) and SGC (bottom panels), where the colour mapping indicates the completeness, C_eBOSS (see equation 6), and the imaging weight, w_sys, (see equation 4), in the left-hand and right-hand panels, respectively.

2.2.3 Targeting completeness and veto mask

We apply a veto mask to the DR14Q catalogue in order to exclude sectors in potentially problematic regions. For the DR14Q catalogues, we veto areas under the same conditions than those in BOSS DR12 (Reid et al. 2016). These veto conditions include bad photometric fields, cuts on seeing and on Galactic extinction. Further details on the veto mask areas are described in section of 3.2 of Ata et al. (2018), we do not repeat them here.

2.3 DR14Q synthetic catalogues

In this paper, we employ three types of synthetic catalogues, constructed to reproduce the observed DR14 quasar sample. We generically refer to them as ‘mocks’, although they are generated with different techniques and are thus characterized by distinct properties. The first two types of mock catalogues are indicated as the ‘Extended Zel'dovich mocks’ (or ‘ez mocks’; Chuang et al. 2015) and the ‘Quick Particle Mesh’ mocks (or ‘qpm mocks’; White, Tinker & McBride 2014). They both consist of hundreds of realizations and are constructed with approximate methods to avoid performing computationally expensive N-body simulations. We use these mocks to estimate the covariance matrix of measured quantities from actual data catalogues, to test our pipeline codes that extract cosmological parameters from the data, and to compute the correlation among parameters inferred at different redshift bins. Tests on our pipeline codes are further refined by a third set of high-fidelity mocks, constructed instead from a high-resolution N-body simulation (the OuterRim simulation; Habib et al. 2016). In what follows, we provide a brief description of the main features of all of these mock catalogues.

2.3.1 qpm mocks

The qpm mocks follow the procedure described in White et al. (2014). Briefly, a low-resolution particle mesh gravity solver is used to evolve a density field in time, partially capturing the non-linear evolution of the field, but with insufficient spatial resolution to resolve virialized dark matter haloes. Particles are sampled from the field to approximate the distribution of the small-scale densities of haloes, mimicking the one-point and two-point distribution of haloes and their mass and bias functions. We have adjusted the parameters of White et al. (2014) that map the local density into the halo mass in order to account for the actual redshift range of the catalogue, also extending this mapping to lower mass haloes, required by the halo occupation distribution (HOD) of quasars.

We parametrize the HOD of quasars through the five-parameter HOD presented in Tinker et al. (2012), which divides objects into central and satellite quasars. The HOD parameters are determined by matching (i) the peak of the n(z) curve observed (see Fig. 2) and (ii) the measured large-scale quasar bias, b_Q = 2.45, in Laurent et al. (2017). This approach also allows the estimation of the fraction of haloes with a quasar object in their centres, usually named the duty cycle. The best-fitting parameters suggest that the satellite fraction is around 0.15 (see fig. 9 of Ata et al. 2018), although there is some expected degeneracy between the satellite fraction and the duty cycle, which remains unknown.

We simulated 100 cubic boxes of side L_b = 5120 h⁻¹ Mpc, which we remapped to fit the volume of the full-planned survey using the code make survey (Carlson & White 2010; White et al. 2014). Since the DR14Q catalogues correspond to a smaller volume than the mocks, we can use different parts of the qpm cubic box to produce different realizations. We identify four configurations with less than 1.5 per cent overlap, which allow us to generate 400 qpm realizations per Galactic cap. Since the same 100 cubic boxes are used for the NGC and SGC, we need to combine them by shifting the indices of the four realizations produced out of each cubic box. After this action, the overlap among NGC and SGC could be as high as 10 per cent, although we identified pairs of configurations where the overlap is less than |$2\hbox{ per cent}$|⁠. The veto mask and the survey geometry of both Galactic caps are applied also using the code make survey, which downsamples the redshift distributions to match a smoothed distribution in agreement with the observed one (black dashed lines of Fig. 2). Finally, we apply a Gaussian smearing that accounts for the spectroscopic redshift errors (Dawson et al. 2016), whose Gaussian width is, σ_z = 300 km s⁻¹ for z < 1.5 and σ_z = [400 × (z − 1.5) + 300] km s⁻¹ for z ≥ 1.5. Comparisons among qpm mocks and DR14Q measurements are displayed later in the bottom panel of Fig. 6.

The underlying cosmological model in which the density field has been generated and evolved follows a flat ΛCDM with the following parameters, |$\mathbf {\Delta }^{\rm {\small QPM}}=\lbrace \Omega _m, \Omega _bh^2,h, \sum m_\nu , \, \sigma _8,n_s\rbrace =\lbrace 0.31,0.022,0.676,0,0.8,0.97\rbrace$|⁠, where the subscripts m, b, and ν stand for the matter, baryon, and neutrino, respectively, h is the standard dimensionless Hubble parameter, σ₈ is the amplitude of dark matter perturbations, and n_s is the spectral index. Additionally, other derived parameters, such as the Hubble parameter, the angular and isotropic-BAO diameter distances, and the sound horizon at drag redshift, are displayed in Table 2.

2.3.2 ez mocks

Following the methodology described by Chuang et al. (2015), we generated 1000 ez-mock realizations for each Galactic cap, matching the DR14Q footprint and redshift evolution. These mocks are produced via the Zel'dovich approximation of the density field, which is able to account for non-linear effects and also halo bias. In particular, non-linearities and halo bias are modelled through effective free parameters directly calibrated from DR14Q measurements, independently treating the NGC and SGC regions. Using this technique we are able to rapidly generate catalogues that reproduce the two- and three-point correlation functions of the desired sample. Each light-cone mock is constructed from seven redshift shells generated from ez mock cubic volumes of L_b = 5000h⁻¹ Mpc at different epochs using make survey. Each of these cubic boxes is computed using different internal parameters, but they share the same initial Gaussian density field, making the background density field continuous. More details on the generation of the ez mocks can be found in section 5.1 of Ata et al. (2018). Comparisons among ez mocks and DR14Q measurements are displayed later in the top panel of Fig. 6.

The underlying cosmological model of the ez mocks follows a flat ΛCDM with the following parameters, |$\mathbf {\Delta }^{\rm {\small EZ}}= \lbrace \Omega _m, \Omega _bh^2,h, \sum m_\nu , \, \sigma _8,n_s\rbrace\, =\,\lbrace 0.307115,0.02214,0.6777,0, 0.8288,0.96\rbrace$|⁠. Other derived parameters, such as the Hubble parameter, the angular and isotropic-BAO diameter distances, and the sound horizon at drag redshift, are displayed in Table 2.

2.3.3 OuterRimN-body mock

We perform an accurate systematic test of our pipeline code using a small set of high-fidelity mocks, constructed from a high-resolution N-body simulation. Unlike ez and qpm mocks, synthetic catalogues directly constructed from N-body simulations fully capture the non-linear signal of the clustering at all scales of interest, and are thus more reliable to assess the validity of our pipeline. Clearly, N-body simulations are expensive to run, but they do contain the correct non-linear dark matter evolution field, and they may be able to resolve dark matter haloes with sufficiently small mass to host quasars, depending on their actual resolution power. In this work, we use the OuterRimN-body simulation (or; Habib et al. 2016), a cubic box of size L_b = 3000 h⁻¹ Mpc with 10 240³ dark matter particles with a force resolution of 6 h⁻¹ kp, implying a mass resolution per particle m_part = 1.82 × 10⁹ h⁻¹M_⊙; hence, dark matter haloes with sufficient mass to host quasars (i.e. M = 10^12.5M_⊙) are well resolved.

We construct the or-skycut from a single snapshot at z = 1.433, applying the same HOD parametrization used in the qpm mocks (Rodríguez-Torres et al. 2017), except for the fraction of satellite quasars, which we fix at distinct values to test its effect. The concentration of each halo is determined from its mass using the Ludlow et al. (2014) prescription. The positions and velocities of the satellites are drawn from an NFW profile (Navarro, Frenk & White 1996). Finally, the fraction of satellites is chosen to be 0 per cent (f_no-sat), 13 per cent (f_std), and 22 per cent (f_high) and the fraction used on the qpm mocks HOD is 15 per cent. Finally, sky geometry cuts are applied so that the final or-skycut derived from the or cubic box covers an angular area of 1888 deg², and the downsampling of objects is performed to match the redshift distribution of the data. Taking advantage that the DR14Q measurements are shot noise dominated due to the low density of objects and that the duty cycle for quasars is low, we draw 20 realizations out of the same single parent box, which we consider to be independent. Additionally, to each configuration, we do and do not apply a Gaussian smearing in order to mimic the effect of spectroscopic redshift errors, in the same manner done for the qpm mocks. Using this procedure, we generate 20 independent realizations for 3 × 2 cases, although the realizations are not independent across the different HOD or smearing parameters. Fig. 4 displays the mean of the 20 measurements of the monopole and quadrupole signal of the or-skycut, for the different satellite fractions, and for the smeared for the f_std case.

Figure 4.

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OuterRimN-body simulation power-spectrum monopole (solid lines) and quadrupole (dashed) lines, computed as the mean of 20 realizations. The colours represent different satellite fractions used, no-sat with f = 0 (orange lines), std with f = 0.13 (dark-blue lines), and high with f = 0.22 (red lines), with no smearing. The light blue lines correspond to the smearing case for the f_std satellite fraction. At large scales increasing the satellite fraction increases the amplitude of the monopole, consistent with an enhancement of the linear bias parameter. At small scales, the satellites induce a non-linear damping term consistent with the expected by a intra-halo velocity dispersion. This effect is saturated when the redshift smearing effect is included, making it difficult to distinguish among the cases with different fractions at small scales (not plotted for clarity).

The underlying cosmological model of the or simulations follows a flat ΛCDM with the following parameters, |$\mathbf {\Delta }^{\rm {\small OR}}=\lbrace \Omega _m, \Omega _bh^2,h, \sum m_\nu , \, \sigma _8,n_s\rbrace =\lbrace 0.26479,0.02258,0.71, 0,0.8,0.963\rbrace$|⁠, which is consistent with the WMAP7 cosmology (Komatsu et al. 2011).

Bear in mind that the or-skycuts are derived from a single snapshot at z = 1.433, which does not match the effective redshift⁵ derived from the DR14Q range (0.8 ≤ z ≤ 2.2). Since here we are interested in using the or just to perform systematic tests on the model, it is not really important that DR14Q and or match perfectly the redshift range. Because of this freedom, we reduce the redshift range of the or-skycut to be 0.8 < z < 2.0, which has an effective redshift of z_eff = 1.43, matching the cubic snapshot epoch. With these parameters, the expected value for fσ₈ is 0.38216, and the expected values for the αs are 1, as we analyse the or-skycut using the simulated cosmology as fiducial cosmology.

2.3.4 Synthetic observational features

We include the fibre collision and redshift failures in the ez and qpm mocks in order to (i) have a more realistic covariance matrices that match the actual number of observed targets and (ii) quantify the systematic shifts (if any) that the weights described in the Section 2.2.1 produce in the cosmological parameters of interest.

We start by imprinting the same tile distribution of the data in the mocks. In practice, the tile distribution of the data is applied in order to minimize the number of untargeted objects by overlapping the tiles in the densest regions of the survey, which makes the tiling process cluster dependent. We do not follow the same procedure on the mocks, which would require us to run the same algorithm for every mock, producing a different tiling pattern each time. For simplicity, we apply the DR14Q tiling distribution.

We start by assigning each mock particle to a specific plate. In case the mock particle falls in an overlap region, it is randomly assigned to an overlapping plate, but with higher probability of falling to the plates whose centre is closer. The collision pair effect is applied to those particles within 62 arcsec and which both fall into non-overlapping regions (and to those particles that have not been already removed by the close pair selection algorithm). One particle is removed and the other is assigned a +1 w_cp weight. The redshift failure effect is applied following the pattern of bottom panels of Fig. 1. We assign the plate coordinates (x_foc,  y_foc) to each mock particles and from those a probability of failing (1 − P_success). The particles tagged as failure are removed from the catalogue. At the end of these two processes, the remaining particles are assigned a w_foc weight according to the same pattern, as it is done for the DR14Q catalogue.

These two processes do not change the effective number of particles (N_eff = ∑_iw_cpw_foc), although they remove actual particles from the mocks. Since the covariance matrix of the DR14Q sample is dominated by shot noise, by producing the mocks with the same number of particles that the DR14Q catalogue, the covariances derived from the mocks contain the same level of shot noise.

3 METHODOLOGY

3.1 Fiducial cosmology

We analyse all the ez, qpm mocks and data in a flat, ΛCDM cosmological model with |$\mathbf {\Delta }^{\rm fid}=\lbrace \Omega _m, \Omega _bh^2,h, \sum m_\nu , \, \sigma _8,n_s\rbrace =\lbrace 0.31, 0.022, 0.676, 0.06eV, 0.8, 0.97\rbrace$|⁠, which matches the fiducial cosmology used for the BOSS DR12 analysis (Alam et al. 2017) and for the eBOSS DR14Q BAO analysis (Ata et al. 2018). The cosmology of the mocks is similar to the chosen fiducial cosmology and, as a consequence, the expected shift in the dilation scale factors parameter is |$\le {}1\hbox{ per cent}$|⁠.

Table 1 displays the expected values for the dilation scale factors and fσ₈ for qpm and ez mocks when analysed under the fiducial cosmology model. Since the ez mocks light-cone is produced using snapshots at different epochs, we display the expected parameters at the different redshift ranges that are later used in the analysis of the data.

3.2 Power-spectrum estimator

The radial distribution, n(z) of the random catalogue associated with the actual data catalogue, matches the observed NGC and SGC n(z) distributions, plotted in blue and yellow lines in Fig. 2. On the other hand, the n(z) of the random catalogue associated with the quasar mock catalogues matches the smoothed n(z) represented by black dashed lines in Fig. 2. The potential impact on the choice of the degree of smoothing of the n(z) distribution has been previously studied within the BOSS survey (Wang, Guo & Cai 2017 in the context of BAO detection) and within the 2 deg Field Galaxy Redshift Survey (see section 3.1 of Cole et al. 2005b). We have tested the impact of using both set of randoms on the actual DR14Q data catalogue, without observing any significant deviation with respect the typical size of the error-bars. We leave for a future work a more detailed study on the optimal choice of the n(z) of the random catalogue.

In order to measure the power-spectrum multipoles of the quasar distribution, we begin by assigning the objects of the data and random catalogues to a regular Cartesian grid. This approach allows the use of Fourier Transform (FT)-based algorithms. In order to avoid spurious effects of the Cartesian grid, we developed a convenient interpolation scheme to convert particle position in grid overdensity field.

We embed the full survey volume into a cubic box of side L_b = 7200 h⁻¹ Mpc, and subdivide it into |$N_g^3=1024^3$| cubic cells, whose resolution and Nyqvist frequency are 7 h⁻¹ Mpc and k_Ny = 0.447 h Mpc⁻¹, respectively. We assign the particles to the cubic grid cells using a 5th-order B-spline mass interpolation scheme, where each data/random particle is distributed among 6³ surrounding grid cells. Additionally, we interlace two identical grid cells schemes displaced by half of the size of the grid cell; this allow us to reduce the aliasing effect below 0.1 per cent at scales below the Nyqvist frequency (Hockney & Eastwood 1981; Sefusatti et al. 2016).

Figure 5.

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Top panel: The DR14 quasar power-spectrum monopole (green), quadrupole (orange), and hexadecapole (purple) in the redshift range 0.8 ≤ z ≤ 2.2, including both NGC and SGC sky patches. The displayed error-bars are the rms of 1000 realizations of the ez mocks. The dashed black lines represent the best-fitting model for the k-range 0.02 ≤ k [h Mpc⁻¹] ≤ 0.30. The bottom sub-panel shows the differences between the model and data, divided by the diagonal errors, using the same colour scheme. The 2σ and 3σ confidence levels are marked with dashed and dotted black lines, respectively. Bottom panel: Monopole and quadrupole measurement of the data for the different redshift estimates described in Section 2.1.1: z_fid in black symbols, |$z_{\rm Mg\, {{\small II}}}$| in red symbols, and z_PCA in blue symbols. The bottom sub-panels display the difference with respect to the fiducial redshift estimate relative to the statistical errors for the monopole and quadrupole. For clarity, we do not display the results for the hexadecapole, where the degree of agreement is similar to the other two multipoles.

3.3 Modelling

The theoretical model used in this paper to describe the power-spectrum multipoles is identical to the one used in previous analyses of the BOSS survey for the redshift range 0.15 < z < 0.70 (Gil-Marín et al. 2015; Gil-Marín et al. 2016), so we briefly present the model without details to avoid repetition. We refer the reader to the references of this section for a further description.

3.3.1 Bias model

We assume the Eulerian non-linear bias model presented by McDonald & Roy (2009). The model has four bias parameters: the linear bias b₁, the non-linear bias b₂, and two non-local bias parameters, |$b_{s^2}$| and b_3nl. As in previous works, we assume b₁ and b₂ to be free parameters of the model. The remaining two non-local bias parameters can be constrained by assuming that the bias model is local in Lagrangian space, which sets |$b_{s^2}$| and b_3nl as a function of b₁: |$b_{s^2}=-4/7\,(b_1-1)$| (Baldauf et al. 2012) and b_3nl = 32/315 (b₁ − 1) (Saito et al. 2014).

3.3.2 Redshift space distortions

We model the redshift space distortions in the power-spectrum multipoles following the approach presented by Taruya, Nishimichi & Saito (2010) (TNS model). We assume that there is no velocity bias between the galaxy field and the underlying dark matter field, at least on the scale of interest for this paper. The TNS model provides a prescription for the redshift space power spectrum in terms of the real space quantities: the matter–matter, velocity–velocity, and the cross-matter–velocity non-linear power spectra. These non-linear quantities are computed using the resumed perturbation theory at two-loop order as described in Gil-Marín et al. (2012). All these non-linear power-spectrum quantities are fuelled with the linear matter power spectrum computed using camb (Lewis, Challinor & Lasenby 2000). The power-spectrum multipoles encode the coherent velocity field through the redshift space displacement and the logarithmic growth of structure parameter, |$f\equiv \frac{\text{d}\log D(z)}{\text{d}\log a(z)}$|⁠. The effect of this parameter is to increase the clustering along the LOS with respect to the transverse direction, boosting the amplitude of the isotropic power spectrum and generating an anisotropic component.

3.3.3 The Alcock–Paczynski effect

The dilation scale factors α_∥ and α_⊥ describe how the true frequencies k΄ have been distorted into the observed ones k, (⁠|$k_\parallel =\alpha _\parallel k^{\prime }_\parallel$| and |$k_\perp =\alpha _\perp k^{\prime }_\perp$|⁠), by the effect of assuming an incorrect cosmological model.

Table 2 displays the fiducial values for D_A, H, and D_V for the different cosmologies used in this paper at their effective redshifts.

3.3.4 Survey geometry

The last step to be included in the model is the effect of the window function produced by the non-uniform distribution of quasars, both angularly and radially. We account this effect by following the procedure described by Wilson et al. (2017). In practise, the window has two main effects: (i) to reduce the observed power at large scales, being more severe as one increases the value of the ℓ-multipole and (ii) to increase the covariance among adjacent k-modes, specially at large scales. We follow the same formalism used by Beutler et al. (2017), which is fully described in Appendix B.

3.3.5 Free parameters of the model

In addition to the free parameters of the model described above, we also marginalize over the amplitude of the linear power spectrum through the amplitude of the dark matter fluctuations filtered with a top-hat filter of 8 Mpc, σ₈(z). This parameter is highly degenerate with other parameters such as the bias parameters and the logarithmic growth of structure. Thus, we set σ₈ to the fiducial value of our cosmology in the non-linear terms of the model, and constrain the combination of σ₈ times the bias parameters or the logarithmic growth of structure from the large-scale modes.

To summarize, the full power-spectrum model described in the sections above has seven free parameters: two bias parameters, b₁σ₈ and b₂σ₈; two nuisance parameters, A_noise and σ_P; and three cosmological parameters, fσ₈, α_∥, and α_⊥. For those cases where ε is set to 0, α_⊥ = α_∥ ≡ α_iso, and the number of free parameters is reduced by one.

3.4 Parameter estimation

The covariance matrix is computed using a large number of mock quasar samples described in Section 2.3. For our fiducial results, we use the 1000 realizations of the ez mocks, unless otherwise noted. Due to the finite number of mock realizations when estimating the covariance, we expect a noise term to be present that requires a correction to the final χ² values. We apply the corrections described in Hartlap, Simon & Schneider (2007). Such corrections represent a |${\sim }15\hbox{ per cent}$| factor in the χ² values; we use 1000 mock realizations to estimate the full covariance of 84 k-bins, including monopole quadrupole and hexadecapole. We do not apply any extra corrections, such the ones described in Percival et al. (2014), which have a minor contribution to the final errors.

Using a simplex minimization algorithm (Nelder & Mead 1965; Press et al. 2002), we explore the surface of the likelihood function to find the best-fitting value for each of the p-parameters and its 1σ marginalized error. We ensure that the minima found are global and not local by running the algorithm multiple times with different starting points and different variation ranges.

As mentioned above, the value of σ₈ is set constant to its fiducial value in the non-linear terms of the model, so the parameter fσ₈ is effectively fitted. We have checked that due to the high degree of degeneracy between f and σ₈, the impact of following this procedure does not change our results for physical values of σ₈.

In order to compute the full likelihood surface of a set of parameters, we also run Markov chains (mcmc chains). We use a simple Metropolis–Hasting algorithm with a proposal covariance and ensure its convergence performing the Gelman–Rubin convergence test, R − 1 < 10⁻³, on each parameter. We apply the flat priors listed in Table 3 otherwise stated.

4 MEASUREMENTS

In this section, we present the measurement of the power-spectrum multipoles of the DR14 quasar sample, as well as the performance of the model and the mocks. We start by discussing the measurements in the whole redshift bin, 0.8 ≤ z ≤ 2.2; and we later divide the full redshift range into three overlapping redshift bins: lowz, 0.8 ≤ z ≤ 1.5; midz, 1.2 ≤ z ≤ 1.8; highz, 1.5 ≤ z ≤ 2.2. Following the second approach, we are in principle sensitive to redshift-evolution quantities, such as b₁(z)σ₈(z) and f(z)σ₈(z). In Section 7, we will explore how the two different approaches, the single and multiple redshift bins, performed when constraining cosmological parameters.

4.1 Single redshift bin

The top panel of Fig. 5 displays the DR14Q measured power-spectrum monopole (green circles), quadrupole (orange squares), and hexadecapole (purple triangles) in the redshift range of 0.8 ≤ z ≤ 2.2. The error-bars correspond to the diagonal elements of the covariance matrix estimated from the rms of the 1000 realizations of the ez mocks. The black dashed lines represent the best-fitting theoretical model. Although the power spectrum has been measured in the range 0 < k [h Mpc⁻¹] < 0.40, only those k-bins in the range 0.02 < k [h Mpc⁻¹] < 0.30 have been used to fit the theoretical model; therefore, the black dashed lines only cover this specific range. The lower sub-panels display the difference between the measured power spectrum and the best-fitting theoretical model divided by the 1σ error. The associated χ² with this fit is 84.0/(84 − 7). The contribution from the monopole-only data points is |$\chi ^2_{P^{(0)}}=20.1/(28-7)$|⁠, from the quadrupole |$\chi ^2_{P^{(2)}}=30.2/(28-6)$|⁠, and from the hexadecapole |$\chi ^2_{P^{(4)}}=34.6/(28-4)$|⁠.⁸ Ignoring the covariance between the multipoles would reduce the χ² by just 0.9, suggesting that the three power-spectrum multipoles are barely correlated (see Fig. D1 in Appendix D for a further description of the correlation among k-bins).

The bottom panel of Fig. 5 displays the impact of changing the redshift estimate of the DR14Q sample from its fiducial methodology, z_fid to the one based on the maximum of the Mg ii line, |$z_{\rm Mg\, {\small II}}$| (red symbols) and to the one based on the PCA decomposition technique that also uses the position of the Mg ii line, z_PCA (blue symbols). The three redshift estimates display a consistent behaviour for both the monopole and quadrupole, not revealing any specific systematic trend, and the differences on specific k-modes are always below 2σ. We must bear in mind that these measurements must be correlated up to some extent, and therefore we cannot quantify in terms of χ² the agreement among them, nor their correlation, as we lack different redshift estimates for the mocks. Producing mocks that capture such behaviour would require a simulation of realistic quasar spectra at a given redshift for each particle in the mocks, which is beyond the scope of this paper. For simplicity, we do not show the hexadecapole measurements, as the degree of agreement is similar to the one found in the monopole and quadrupole. In Section 6, we will present the cosmological derived parameters based on these three redshift estimates.

The top panel of Fig. 6 shows the power-spectrum monopole and quadrupole measured on the NGC and SGC separately. Since the two samples are well disconnected they can be considered fully independent. The different colours and symbols distinguish between the NGC and SGC region and the power-spectrum multipole, as indicated. The best-fitting theoretical model is indicated by a solid black line for NGC, and a dashed black line for SGC. The middle and lower sub-panels display the difference between the model and the data divided by the corresponding 1σ error. For clarity, we do not induce the results on the hexadecapole.

Figure 6.

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Top panel: DR14 quasar sample measurement for the power-spectrum monopole (circle symbols) and quadrupole (square symbols), for the NGC region (filled symbols) and SGC region (empty symbols) using the full redshift range, 0.8 ≤ z ≤ 2.2. The solid and dashed lines indicate the best-fitting model for the NGC and SGC, respectively. The error-bars display the rms from the ez mocks. The middle and bottom sub-panels display the difference between the model and the measurement divided by the errors, for the quadrupole and monopole, respectively. The horizontal dashed and dotted lines represent the 2 and 3σ confidence levels. Bottom panel: The measured DR14Q data power-spectrum monopole, quadrupole, and hexadecapole are shown for the NGC+SGC region in green, orange, and purple, respectively. The solid and dotted curves are the measured multipoles from the mean of the qpm (400 realizations) and ez mocks (1000 realizations), respectively, using the same colour notation. Since the error-bar for the mean of the mocks is small, it is not plotted. At large scales, both mocks and data show a good agreement, but at small scales, both qpm and ez mocks fail to accurately reproduce the data. However, this behaviour can be partially fixed in the monopole by modifying the shot noise value of the mocks, as it demonstrated in fig. 6 of Ata et al. (2018).

We observe that the data from the SGC presents a slightly higher amplitude than in the NGC, especially for k ≳ 0.15 h Mpc⁻¹ and in the quadrupole. This effect translates into a best-fitting theoretical model with higher bias in the SGC quadrupole. In Section 6, we will quantify this discrepancy and conclude that these differences are not statistically significant. As for the combined NGC+SGC sample, there is no k-bin that deviates more than 3σ with respect to the prediction of the model, and just three points at more than 2σ. The χ² values for NGC and SGC are 64.5/(84 − 7) and 76.6/(84 − 7), respectively. For the NGC, the separate contribution for the monopole, quadrupole, and hexadecapole are |$\chi ^2_{P^{(0)}}=19.8/(28-7)$|⁠, |$\chi ^2_{P^{(2)}}=22.0/(28-6),$| and |$\chi ^2_{P^{(4)}}=25.0/(28-4)$|⁠; and for the SGC |$\chi ^2_{P^{(0)}}=25.6/(28-7)$|⁠, |$\chi ^2_{P^{(2)}}=24.0/(28-6),$| and |$\chi ^2_{P^{(4)}}=27.6/(28-6)$|⁠.

The bottom panel of Fig. 6 displays the performance of the mean of the 1000 realizations of the ez mocks (dashed lines) and the mean of the 400 realizations of the qpm mocks (solid lines) along with the DR14Q measurements. For the monopole, the ez and qpm underestimate and overestimate, respectively, the measurement from the data. This behaviour was also reported the fig. 6 of Ata et al. (2018).⁹ When a constant value is added to the mocks, the agreement improves significantly. We also observe differences in the behaviour of the quadrupole at small scales, k > 0.20 h Mpc⁻¹, where the ez mocks tend to overestimate the DR14Q measurements. The hexadecapole measurements are similar for both ez and qpm mocks, and consistent along with the data. In Section 5.3, we quantify the impact of these two covariance matrices in the cosmological parameters of interest.

4.2 Multiple redshift bins

In the previous section, we have presented the measured power-spectrum multipoles for the entire redshift bin, 0.8 ≤ z ≤ 2.2, with an effective redshift of z_eff = 1.52. However, the size of this redshift bin is large, which covers a wide range of epochs. During these epochs, we expect that the cosmological parameters, such as fσ₈ and b₁σ₈, will significantly evolve with redshift. Constraining the evolution of these parameters with redshift will better constrain potential departures from the standard cosmological model than just the average measurements of the whole redshift bin.

Following this approach, we divide the DR14Q NGC+SGC sample in three overlapping redshift bins with similar effective volumes: lowz, which covers 0.8 ≤ z ≤ 1.5 and whose effective volume is |$V_{\rm eff}^{lowz}=0.126\,{\rm Gpc}^3$|⁠; midz, which covers 1.2 ≤ z ≤ 1.8 and whose effective volume is |$V_{\rm eff}^{midz}=0.131\,{\rm Gpc}^3$|⁠; and highz, which covers 1.5 ≤ z ≤ 2.2 and whose effective volume is |$V_{\rm eff}^{highz}=0.119\,{\rm Gpc}^3$|⁠. Since the midz range overlaps with both lowz and highz, we expect a significant correlation among these measurements, and their derived cosmological parameters. Using the ez mocks, which contain an intrinsic evolution of the bias and cosmological parameters with redshift, we can compute the cross-correlation coefficients among the parameters of the different redshift bins.

The three panels of Fig. 7 display the power-spectrum monopole (left-hand panel), quadrupole (middle panel), and hexadecapole (right-hand panel) for the DR14Q, for the different redshift bins, as indicated, in the coloured symbols. The coloured dashed lines indicate the best-fitting model, with the same colour notation.

Figure 7.

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Power-spectrum monopole (left-hand panel), quadrupole (middle panel), and hexadecapole (right-hand panel), for different z-bins: lowz 0.8 ≤ z ≤ 1.5 (green), midz 1.2 ≤ z ≤ 1.8 (purple), highz 1.5 ≤ z ≤ 2.2 (orange). The coloured symbols display the measurements from the NGC+SGC DR14Q sample, whereas the dashed lines indicate the best-fitting model. The lower sub-panels show the differences between the model and the data in terms of 1σ confidence levels.

For the power-spectrum monopole, at large scales the amplitude of the power spectrum increases with redshift. The Kaiser boost factor for the monopole is ([b₁σ₈]² + 2/3[fσ₈][b₁σ₈] + 1/5[fσ₈]²). The quantity b₁σ₈(z) is a slightly increasing function with redshift in 0.8 ≤ z ≤ 2.2 (b₁ increases with z and σ₈ decreases), whereas fσ₈(z) is a decreasing function with redshift. However, the bias has a dominant effect over the logarithmic growth factor, and the overall effect is an increase of the amplitude, as observed on the data. At small scales, we observe the opposite effect: the highz redshift bin presents a more important damping factor than the lowz bin. This behaviour is also expected, as the parameter σ_P in our model accounts not only for the damping caused by the intra-halo velocity dispersion of the satellite quasars but also for the effect of spectroscopic redshift errors. It is expected that these errors will increase with redshift, as the more distant objects tend to have lower signal-to-noise ratio spectra, which can lead to significant errors in the measurement of their radial distance (see the Gaussian smearing redshift error model at the end of Section 2.3.1).

For the power-spectrum quadrupole, we observe the opposite behaviour at large scales. In this case, the Kaiser boost factor becomes, (4/3[b₁σ₈][fσ₈] + 4/7[fσ₈]²), where the dominant component is fσ₈(z), which drives the whole factor to decrease with redshift, as seen in the data and best-fitting model. Indeed, the importance of the bias parameters decreases for high-order multipoles, which causes the Kaiser boost factor to be dominated by the z-evolution of the fσ₈(z) parameter. At small scales, we observe the same behaviour in the monopole. The redshift failure effects that produce that the damping factor strongly increase with redshift.

Finally, in the hexadecapole the Kaiser boost factor is ∝ [fσ₈]², with no bias contribution at large scales. Because of the large statistical errors, we do not observe any particular trend of the hexadecapole as a function of the redshift bin. In addition, in the midz redshift bin, the hexadecapole at k ∼ 0.11 h Mpc⁻¹ is a 4σ outlier with respect to the expected model. This tension is reduced to the 3σ discrepancy when the full redshift range is considered, as shown in the top panel of Fig. 5. In Section 6.3, we will discuss the impact of this frequency in the total χ² and in the cosmological parameters.

5 SYSTEMATIC TESTS

We aim to identify potential systematic errors of our model, as well as potential systematics on the data, and quantify their effect on the measurements of cosmological interest. We start by using the ez and qpm mocks to recover the expected cosmological parameters. Although these mocks are not a proper N-body simulation, they can provide a first approximation on the performance of how the model works at these redshifts, and, more importantly, can test the effect of the potential systematics introduced by the spectroscopic weights. We will later use the OuterRimN-body simulations to test the performance of the model using different prescriptions for the fraction of satellite quasars.

5.1 Isotropic fits on mocks

Table 4 displays the shifts between the measured α_iso and fσ₈ parameters, and the expected value from the known cosmology of the mocks and or-skycut, when ε is fixed to 0. The expected values for both qpm and ez mocks can be found in Table 1. For or, the expected α_iso is 1 as explained in Section 2.3.3. The 〈x〉_i rows contain those quantities obtained by fitting the mean of all available realizations. In this case, the errors represent the errors of the mean, where all the elements of the covariance matrix have been re-scaled by the inverse of the total number of realizations 1000, 400, and 20 for the ez, qpm, and or-skycut mocks, respectively. The 〈x_i〉 rows are the average of the best-fitting parameters individually on each realization. In this case, the errors represent the average of the errors of each individual fit. Additionally, the S columns display the rms among best-fitting values of α_iso and fσ₈ of each realization. The N_det column is the number of mocks whose best-fitting values for α_iso lie between 0.8 and 1.2. We consider those realizations as mocks with detection of α_iso (this is the same definition in table 4 of Ata et al. 2018). The average quantities 〈fσ_8i〉 and 〈α_isoi〉 (as well as the respective S_i) are computed only using these ‘detection’ realizations, and discarding the rest. The Δα_iso, Δfσ₈, and S values are expressed in terms of 10⁻² units. Thus, Δx = 1 corresponds, for instance, to a shift of 0.01 with respect to the true expected value.

For the ez and qpm mock cases, the rows labelled with ‘raw’ correspond to those results obtained when no spectroscopic effects, such redshift failures and fibre collision, are added (and therefore there is no need for these corrections). The rows labelled with z_f correspond to those mocks with the redshift failure effect applied and corrected using the w_foc weight (according to equation 2). Those rows labelled as w_focw_cp refer to mocks where both fibre collisions and redshift failures are applied and corrected following the prescription described in Section 2.2.1. The rows labelled as w_nozw_cp also contain both fibre collisions and redshift failures, but in this case, the redshift failures have been corrected using the near-neighbour technique. Those rows noted as +P⁽⁴⁾ represent the analysis including the hexadecapole signal. The ez and qpm mocks are analysed using their own covariance. However, the ez mocks also include the case where the qpm-derived covariance is used in order to test its impact.

In general, there is a concordance between the shifts observed for the 〈x_i〉 and 〈x〉_i variables. The α_iso variable is robust under the different analysis methods, and we observe a consistent |$1\text{--}2\hbox{ per cent}$| shift for the ez mocks to systematically lower values than expected, and |$<{}1\hbox{ per cent}$| shift for the qpm mocks. The systematic |$1\text{--}2\hbox{ per cent}$| shift observed on the ez mocks (and not observed in the qpm) may be due to either some systematic of the model or an intrinsic systematic of the mocks.

The fσ₈ variable is more sensitive to the spectroscopic weights, in particular, to the redshift failures when they are corrected through the w_noz prescription. In this case, the systematic shifts on fσ₈ can reach |${\sim }6\hbox{ per cent}$|⁠, whereas the correction through w_foc does not produce any measurable systematic shift. Moreover, the impact of fibre collisions through the w_cp weight correction is |$<{}2\hbox{ per cent}$|⁠. Adding the hexadecapole does not produce any significant change on the fits. This is because we are performing fits with ε set to 0, whereas the main extra information of the hexadecapole comes for breaking the degeneracy among α_∥ and α_⊥. By adding the spectroscopic weights, the errors and the rms values increase because of the increase of the shot noise component in the covariance matrix, which is the dominant term. Finally, the fits on the ez mocks with either the ez-derived covariance matrix or the qpm-derived covariance matrix do not produce any significant shift.

The or-skycut mock results are also displayed in the lower columns of Table 4 for three different satellite fraction, f_no-sat = 0, f_std = 0.13 and f_high = 0.22, and with and without a redshift smearing effect, which would mimic the expected uncertainty in the model determination. The qpm mocks, which are also based on a HOD technique, have a satellite fraction similar to f_std. For the or-skycut mock analyses, no fibre collision or redshift failure effect has been included. The or-skycut mock results show that neither the satellite fraction nor the redshift error smearing has an important impact on fσ₈; the systematic shifts are below |${\sim } 4\hbox{ per cent}$| (shift of ≤0.02). Similarly, the value of the α_iso parameter does not produce shifts higher than |$2\hbox{ per cent}$|⁠, although the systematic shift increases as the satellite fraction increases.

The upper panel of Fig. 8 displays the distribution of the best-fitting solution corresponding to the 1000 realizations of the ez mocks with the w_focw_cp weights applied (blue circles) and DR14Q data (for the same weighting scheme) for the different redshift estimates: z_fid (orange cross), |$z_{\rm Mg \,{{\small II}}}$| (red cross), and z_PCA (black cross) for the parameters α_iso, fσ₈, and b₁σ₈; as well as the corresponding histograms including the χ² distribution. The figure shows the degeneracies among parameters, in particular the strong correlation between α_iso and b₁σ₈, as both parameters are sensitive to the amplitude of the power spectrum at all scales. Given these results, the data accordingly fit into a typical realization of the mocks, and all the redshift estimates provide similar results on the studied parameters. The bottom panel of Fig. 8 displays the distribution of the 1σ error for α_iso in the x-axis and fσ₈ in the y-axis. The mocks show a positive correlation between the error of fσ₈ and α_iso, as expected, since α_iso and fσ₈ are correlated as well, i.e. α_iso is well determined in a particular realization, the probability that fσ₈ is well determined in that particular realization is high. The errors measured from the data when z_fid and z_PCA are used are consistent with the mocks. The measured errors on the data with redshift estimate |$z_{\rm Mg\, {\small II}}$| are ∼2σ larger than the distribution of the mocks. This behaviour may be caused by the broadening on spectroscopic redshift errors when estimating the redshifts using the Mg ii line with respect to the other two methodologies.

Figure 8.

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Top panel: Best-fitting solution for the 1000 realizations of the ez mocks (in blue circles) and DR14Q data set: orange cross for z_fid, red for |$z_{\rm Mg\, {\small II}}$|⁠, and black for z_PCA, under the constraint ε = 0 for the parameters b₁σ₈, fσ₈, α_iso when the power-spectrum monopole and quadrupole are used. The histograms show the distribution of parameters, as well as the χ² values. The panels display the correlation among parameters. Bottom panel: Distribution of the 1σ errors of the fσ₈ and α_iso parameters, for the ez mocks in blue circles, and DR14Q data using the same colour notation. The uncertainty of fσ₈ and α_iso parameters have a positive correlation. In both panels, the results from the DR14Q data, when the z_fid and z_PCA estimates are used, are a typical individual case of the mocks. The errors obtained from the |$z_{\rm Mg\, {\small II}}$| redshift estimates are ∼2σ larger than those expected from the mocks.

5.2 Anisotropic fits on mocks

In this section, we extend the above tests by relaxing the ε = 0 prior. We refer to the fit under these conditions as ‘full-AP’ fit. The results are shown in Table 5 for the ez, qpm, and or-skycut mocks, similarly as it was presented in the previous section.

We start by describing the results on the ez and qpm mocks. When the power-spectrum monopole and quadrupole are used, the results on the ‘raw’ mocks present less than |$2\text{--}3\hbox{ per cent}$| systematic shifts, similar to what was observed in the isotropic case. By including the hexadecapole on the ez mocks, there is a slight increase on the systematics of the α_∥ parameter, which reaches a |$2.5\hbox{ per cent}$| shift, but it does not have any effect on the rest of parameters or on the qpm mocks, other than reducing the statistical uncertainty by |${\sim }20\hbox{ per cent}$|⁠.

We compare the observed shifts on the ‘raw’ mocks with the rest of the mocks using different weigh-based prescriptions. In the z_f case, the w_foc weight perfectly accounts for the redshift failure effects (see also in Appendix C the effect on the signal from the power-spectrum multipoles), even when the hexadecapole signal is considered. For the w_focw_cp case, when only the power-spectrum monopole and quadrupole are considered the inferred cosmological parameters shift slightly more than |$1\hbox{ per cent}$| both for ez and qpm mocks. When the hexadecapole is added, we observe a systematic |$2\hbox{ per cent}$| for both ez and qpm mocks towards lower values of α_∥; |${\lesssim }1\hbox{ per cent}$| shift on α_⊥ towards higher values; and ∼0.02 shift towards higher values of fσ₈, which represent a |${\sim }5\hbox{ per cent}$| shift. We interpret these changes as uncorrected systematics caused by the LOS distortion by the nearest-neighbour correction for the fibre collisions.

Similarly to Fig. 8 for the isotropic case, Fig. 9 displays the scatter of the ez mocks with respect to the DR14Q for the three different redshift estimates studied in this paper. As for the isotropic case, the main values drawn from these three redshifts are similar and are consistent with typical realization of the mocks. Similarly, the errors associated with the DR14Q for |$z_{\rm Mg\, {\small II}}$| and z_PCA estimates are larger than those corresponding to z_fid for α_∥, but in any case within the expected range from the mocks.

Figure 9.

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Same format as in Fig. 8, but for the full-AP fits, for the power-spectrum monopole, quadrupole, and hexadecapole. Unlike for the ε = 0 case of Fig. 8, for the full-AP case with the monopole, quadrupole, and hexadecapole are considered. The main values drawn from the different redshift estimates of the DR14Q data set (top panels) agree very well among themselves and represent a typical single example from the mocks. On the error panels (bottom panels), the data sets corresponding to the three redshift estimates present a typical behaviour with respect to the mocks.

As a summary, the observed shifts due to redshift failures are totally negligible for both the P⁽⁰⁾ + P⁽²⁾ and P⁽⁰⁾ + P⁽²⁾ + P⁽⁴⁾ cases. The systematic shifts produced by fibre collisions on the P⁽⁰⁾ + P⁽²⁾ case are ≲ 0.01 for α_∥, α_⊥, and fσ₈. Finally, the systematic shifts produced by fibre collisions when the hexadecapole is added are ∼0.02 for α_∥ towards lower values, ∼0.01 for α_⊥ towards higher values; and 0.027 on fσ₈ towards higher values. These systematic shifts are below the statistical budget we expect from the data, given in the errors of the 〈x_i〉 rows. However, we will add in quadrature these errors in Sections 6 and 7 when the covariances and cosmological results are presented.

We are also interested in testing the potential systematics of the theoretical modelling using full N-body mocks. We employ the same or-skycut mocks used in the section above, but now we perform a full-AP analysis on all three sets of mocks including the hexadecapole signal. The results are presented in Table 5 for the different satellite fractions, with and without the redshift smearing effect. When the fraction of satellite quasars is kept to 0, the |$f_{\rm no\text{-}sat}$| case, the systematic shifts observed for all the parameters are ≲ 0.01 for both with and without redshift smearing, and the model is able reproduce the expected signal. Introducing a satellite fraction produces some systematic shifts in some parameters. When the satellite fraction is kept to a value close to that used on the qpm mocks, f_std = 0.13, we observe shifts above 0.01 on all the parameters, which reaches deviations between 0.02 and 0.03 for all variables. When the fraction of satellites is high and no smearing is applied, the shifts on α_∥ are ∼0.03, whereas on the rest of the cases are ≲ 0.02. We believe that these shifts are entirely due to a limitation of the theoretical model used to describe the clustering of quasars either by the adopted bias model or by the redshift space distortions. In this work, we do not investigate further the origin of these discrepancies, which in all cases are ≲ 1/3 of the expected statistical errors.

The top panel of Fig. 10 summarizes the systematic shifts of Table 5 along with the expected statistical errors (horizontal dashed lines) computed from the average errors of the 〈x〉_i w_cpw_foc row of ez mocks. For all the cases, the shifts are computed using the 〈x_i〉 rows. The shifts obtained with the ez (dark blue) and qpm (light blue) mocks are driven by our treatment of observational inefficiencies with respect to their corresponding measured raw value. Therefore, these shifts display only the effects of fibre collisions and redshift failures on the measurements (any other systematic cancels). The effects are consistent among ez and qpm mocks, as expected, since they contain the same spectroscopic systematic effects. Examining the highest deviation on either ez or qpm mocks reveals that (i) for fσ₈ the observational weights tend to produce a systematic shift towards higher values of 0.027; (ii) for α_∥ these weights tend to reduce the measured quantity by 0.018; and (iii) for α_⊥ the observational systematics tend to increase the measured quantity by 0.013. These values are summarized in Table 6 in the systematic observational budget column, σ_obs. The shifts obtained with the or-skycuts mocks and various satellite fractions, however, indicate modelling errors. We adopt as a systematic error the largest shift obtained with among the different satellite fractions and smearing effects, with respect to the true underlying value. Orange circle, red square, and green triangle symbols represent the results for |$f_{\rm no\text{-}sat}$|⁠, f_std, and f_high, respectively. The information in Fig. 10 does not display a clear correlation between the modelling systematic shift and the satellite fraction value. Although the true satellite fraction of the data is unknown, as a conservative choice we expect the upper limit to be less than f_high, so we adopt the highest deviation among the six possible combinations as the modelling systematic contribution. These shifts are summarized in Table 6 under the modelling systematic column, σ_mod. As σ_obs and σ_mod are different sources of uncertainties, we add them in quadrature to our uncertainty budget |$\sigma ^2_{\rm systot}\equiv \sigma _{\rm obs}^2+\sigma _{\rm mod}^2$|⁠. As both shifts tend to be of opposite signs and its origin is uncertain, we do not apply any shift to the best-fitting values of the measured quantities.

Figure 10.

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Top panel: Systematic error shifts for fσ₈, α_∥, and α_⊥ computed from the results presented in Table 5. The observational systematics represent the residual effects of the fibre collision + redshift failures. These shifts are computed as, [Δx]_obs ≡ 〈x_i〉 w_cpw_foc − 〈x_i〉 raw on both the ez and qpm mocks, as indicated; both sets have consistent systematic offsets. The modelling systematic results are computed only from the or-skycut mocks and they represent the uncertainties when recovering the true cosmological parameters due only to modelling limitations, [Δx]_mod ≡ 〈x_i〉 − x_true. The different colours and symbols represent the satellite fractions, |$f_{\rm no\text{-}sat}$|⁠, f_std, and f_high for orange circles, red triangles, and green squares, respectively. The dashed black horizontal lines represent the expected statistical error for the data, computed as the average errors over the ez mocks. Bottom panel: The quantities of the top panel plotted as a function of the maximum k-wavenumber chosen to perform the fit. For simplicity, we only show the or-skycut mocks with smearing and we do not display the qpm mock results. The same colour code displayed in the top panel is used. Note that for the or-skycut mocks we display the deviation with respect to their expected value, whereas for the ez mocks the deviation is plotted with respect to the ‘raw’ case.

In addition, the bottom panel of Fig. 10 displays the same quantities of the top panel as a function of the maximum wavenumber used for the fit, k_max. For simplicity, we only show the results for the ez and or-skycut mocks with smearing. The systematic shifts reported in the top panel and Table 6 do not present any significant dependence with k_max, except for the extreme case of k_max = 0.37 h Mpc⁻¹, which is outside the range of validity of the theoretical modelling at this redshift.

5.3 Isotropic fits on data

As an additional sanity check on our analysis procedure we perform a set of fits on the actual DR14Q data set changing different specifications in order to examine how strong the results are with respect to the data parametrization. For simplicity, we only perform these fits fixing ε to 0. We call the ‘standard’ (std) choice of specifications: (i) z_fid as redshift estimates, (ii) the power-spectrum monopole and quadrupole as observables, (iii) of both NGC and SGC regions, (iv) sampled with a linear binning of Δk = 0.01 with k-centres at k_i = (i + 0.5)Δk, (v) scale range considered between 0.02 ≤ k [h Mpc⁻¹] ≤ 0.30, (vi) where the fibre collisions and redshift failures are corrected through w_cpw_foc, and (vii) where the covariance matrix of the power-spectrum multipoles is inferred from 1000 ez mocks.

Table 7 and Fig. 11 display the measured fσ₈, α_iso, and b₁σ₈ measurements when some of these seven conditions are relaxed or modified. The +1/4, +2/4, and +3/4 rows correspond to the effect of shifting the centres of the k-bins by the respective fraction (with respect to the standard k-bin centre positions), and the ‘comb’ rows correspond to the result of combining these four results into a single measurement.¹⁰ Hereafter, we will refer to the ‘standard’ specification with the 4 k-bin centres combined as ‘comb’ or ‘combined’. Likewise, we refer to |$z_{\rm Mg\, {\small II}}$|-comb and z_PCA-comb as the ‘combined’ specification with the redshift estimate changed from the fiducial case to the |$z_{\rm Mg\, {\small II}}$| and z_PCA cases, respectively. The horizontal dashed lines in Fig. 11 correspond to the ±1σ errors of the ‘comb’ case. Along with the fiducial combined case, we present the combined measurements for the two other redshift estimates, |$z_{\rm Mg\, {\small II}}$| and z_PCA. The NGC and SGC rows correspond to the measurements when only one Galactic Cap is used. Those rows labelled with log k correspond to changing the linear binning to logarithmic binning in the same k-range and with similar number of k-bins. Consequently, the log k cases have more k-measurements at large scales (low k-values) with respect the linear binning (and the covariance accounts for an extra statistical correlation because of this effect). The qpm row displays the result when the covariance matrix is changed by that inferred by the 400 realizations of the qpm mocks. For comparison, the results inferred from only 400 realizations of the ez mocks is also displayed. Finally, the effect of changing or turning off the imaging and spectroscopic weights is also presented. Bear in mind that all the results are highly correlated, as they are based on an identical data set, with the exception of those NGC and SGC cases, which can be considered as totally uncorrelated.

Figure 11.

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Impact of different parameters (see the text for definitions) on the measured b₁σ₈, fσ₈, and α_iso quantities, corresponding to those results presented in Fig. 7. For reference, the horizontal dashed lines correspond to the fiducial choice of parameters when the different k-bin centres results have been combined (comb option), and the horizontal dotted lines to the 1σ deviation for the same case.

We observe a ∼1σ deviation between NGC and SGC on the α_iso and b₁σ₈ (both parameters are strongly correlated as is shown in Fig. 8), but since these two regions are statistically independent a 1σ shift is expected. The higher value of b₁σ₈ in the SGC is related to the observed excess of power in the power-spectrum monopole that is visible on top panel of Fig. 6. As stated previously, we conclude that this shift is not statistically significant.

For the rest of parameters studied, none have a strong effect on the cosmological parameters, with the exception of the redshift estimates (which we have already commented above) and the qpm covariance matrix, which produce a ∼1σ shift on b₁σ₈ and α_iso, but no significant effect on fσ₈. However, this 1σ shift appears only when we compare the ‘comb’ results with the qpm-cov results, which is drawn from the same k-bin centre condition as in ‘std’ case. Therefore, a fair comparison between ‘std’ and qpm-cov case yields only a 0.67σ offset. Further investigation on the potential impact of the choice of the covariance matrix suggests that this 0.67σ shift is not caused by the limited amount of realizations (400 on qpm versus 1000 on the ez mocks) nor by the differences in their off-diagonal elements that are sub-dominant in the total χ² contribution. Therefore, the origin of this shift is located in the diagonal elements of the covariance, which present variations from 5–|$10\hbox{ per cent}$|⁠.

Fig. D2 shows the ratio between the diagonal errors of ez and qpm mocks. In general, qpm-derived errors are |$5\hbox{ per cent}$| larger than ez-derived errors, but for the monopole at large scales, this tendency is inverted. Examining at the shifts on the ez mocks results presented in Table 4 when the ez- and qpm-derived covariances were applied, we find that 134/962¹¹ of the mocks have shifts >0.67σ on α_iso when the covariance matrix is changed, 75 towards lower values and 59 towards higher values, and consequently the average mean value of the inferred α_iso is not significantly affected by the choice of the covariance (see Table 4). We conclude that the observed behaviour on the data is consistent with the behaviour of the mocks (occurs 14 per cent of times on the mocks) and that the origin of this effect is the 5–|$10\hbox{ per cent}$| differences in the diagonal terms of the two covariances, which in combination with the intrinsic statistical noise of the data can produce the observed ∼0.67σ fluctuation. In any case, we believe that the ez mocks are a better representation of the actual DR14Q data set, as they have been produced using different epoch snapshots, whereas the qpm is generated from a single one. Therefore, we assign a higher level of likelihood to those results derived from the ez-mock covariance, over those from the qpm.

We conclude that none of the studied specifications produce major systematic shifts in the studied parameters, and we consider that our results are robust under the change of the specifications presented in this section.

6 RESULTS

In this section, we describe the results on the DR14Q data sample. We present the results for both isotropic fits (keeping ε = 0) and full-AP fits. Additionally, we perform two parallel analyses, (i) we consider the full 0.8 ≤ z ≤ 2.2 redshift range as a single redshift bin, (ii) we divide this redshift range into three overlapping redshift bins, as described above in Section 4.2.

6.1 Isotropic fits

We start by performing an isotropic fit to the power-spectrum monopole and quadrupole of the DR14Q data set, keeping ε = 0. Here, we do not consider to use the hexadecapole because, as demonstrated in Section 5.1, it does not add any extra information when ε is set to a constant value. Table 8 displays the mean value between the 1σ errors, x_mean ≡ [(x_bf + σ⁺) + (x_bf − σ⁻)]/2¹² for the free parameters of the model. The different columns display the results using the fiducial redshift estimate, z_fid, applied to the individual NGC and SGC, along with the combination of both. In addition, the results for the two extra redshift estimates, |$z_{\rm Mg\, {\small II}}$| and z_PCA are presented. For all the cases, the rest of the specifications correspond to the ‘standard’ case defined in Section 5.3, unless otherwise stated. Fig. 8 has previously shown the results on the three redshift estimates corresponding to the entire NGC+SGC sample for the parameters b₁σ₈, fσ₈, and α_iso, along with the ez mocks. As described in Section 5.3, the results of the NGC and SGC are consistent within 1σ for all the parameters. The results on the different redshift estimates are also consistent and there is no observed evident tension in any of the parameters. The errors associated with the parameters inferred from the |$z_{\rm Mg\, {\small II}}$| redshifts are in general higher, as already discussed in Section 5.1.

These results are affected by the prior condition of ε = 0. These results imply that H(z)D_A(z) is equal to |$H^{\rm fid}(z)D_A^{\rm fid}(z)$|⁠, which is similar to the best-fitting Planck cosmology. Therefore, the isotropic derived results are not independent from Planck results, and thus, Planck-cosmic microwave background (CMB) data should not be added to these results as an extra uncorrelated data set.

6.2 Anisotropic fits

We consider the power-spectrum monopole, quadrupole, and hexadecapole, and perform a full-AP fit, relaxing the ε = 0 condition of the previous section. Thus, α_∥ and α_⊥ can freely vary, which enable constraints to be placed on H(z)r_s(z_d) and D_A(z)/r_s(z_d) independent of the CMB data.¹³ Table 9 displays these results using the same notation as in Table 8. The first three columns show the results for the different redshift estimates when only the power-spectrum monopole and quadrupole have been fitted. The following three columns present the results when the power-spectrum hexadecapole is added to the analysis. For all the cases, we fix the rest of specifications to ‘combined’, as they are described in Section 5.3, which consist on the ‘standard’ specifications combining the four shifts on the k-bin centres.