The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: measurement of the BAO and growth rate of structure of the luminous red galaxy sample from the anisotropic power spectrum between redshifts 0.6 and 1.0 Free

ABSTRACT

1 INTRODUCTION

The large-scale structure (LSS) of the Universe contains valuable information of how the Universe has been evolving in the last ∼7 × 10⁹ yr, when the dark-energy-domination era started. The current state-of-the-art spectroscopic LSS observations allow to utilize the standard ruler baryon acoustic oscillations (BAO), first detected in Eisenstein et al. (2005) on the Sloan Digital Sky Survey (SDSS) data set and Cole et al. (2005) on the Two-Degree Field Survey (2dF, Colless et al. 2003), to determine with precision the background expansion history of the Universe at late-time. During the last decade, the BAO technique has evolved in both precision and accuracy becoming mature. Consequently, a plethora of measurements has been performed on spectroscopic galaxy surveys at different epochs: 6-degree Field Survey (6dF; Jones et al. 2009; Beutler et al. 2011) at z = 0.106, WiggleZ (Drinkwater et al. 2010; Blake et al. 2011b; Kazin et al. 2014) at |$z=0.44,\, 0.6,\, 0.73$|⁠, and Baryon Oscillation Spectroscopic Survey (BOSS) galaxies (Anderson et al. 2012, 2014a, b; Dawson et al. 2013; Alam et al. 2017) at |$z=0.38,\, 0.51,\, 0.61$|⁠, and BOSS Lyman α forests (Bautista et al. 2017; du Mas des Bourboux et al. 2017) at z = 2.40. Additionally, if we want to obtain a direct measurement of the growth of structures from these same spectroscopic surveys we need to measure the effect of redshift space distortions (RSD; Kaiser 1987). Consequently, we obtain both an expansion history and a growth of structure measurement from the same data set. Parallel to the BAO technique development, RSD analyses have also matured both in modelling and observational systematics treatment during the last decade: RSD in 2dF (Percival et al. 2004), in 6dF (Beutler et al. 2012), in WiggleZ (Blake et al. 2011a), in VIPERS (de la Torre et al. 2013; Guzzo et al. 2014; Pezzotta et al. 2017), in FastSound (Okumura et al. 2016), as well in BOSS galaxies (Alam et al. 2017).

Anisotropic BAO studies provide a direct measurement of the background expansion at the epoch of the observed galaxies, z, through the absolute and relative BAO peak position in the anisotropic multipoles of the power spectrum or correlation function. Under the assumption of a functional form of the background expansion, H(z; Ω_m) one can obtain a direct measurement of the density of matter in the Universe Ω_m. Note that the BAO peak position is not directly sensitive to H(z), but to H(z)r_drag, and to the comoving angular diameter distance over the comoving sound horizon at the epoch where the baryon-drag optical depth equals unity, D_M(z)/r_drag. From these measurements, one can either infer Ω_m from the product of the two (which is independent of r_drag), or assume an extra prior on r_drag, which can either come from cosmic microwave background (CMB) measurements, or from a functional form of r_drag given by priors on the baryon, Ω_b and radiation density |$\boldsymbol{\Theta }_{\rm rad}$| (which are typically not measured by LSS), and infer H₀ (see for e.g. Addison et al. 2018). Within the SDSS collaboration, we opt to analyse these results under the less restrictive set of priors, and thus only assume a functional form for H(z), but no restriction on r_drag as a function of cosmology. The motivation for proceeding this way is the robustness of the cosmological interpretation under potential changes of the cosmological paradigm if, for example, the state-of-the-art value of r_drag changes significantly in the future or ΛCDM is ruled out, as one would just only need to re-interpret the quantities H(z)r_drag and D_M(z)/r_drag rather than reanalysing the data. In this paper, we choose to work with the ‘Hubble distance’, D_H, defined as D_H(z) ≡ c/H(z), where c is the speed of light. The parameter D_H(z)/r_drag has the advantage of being dimensionless, of the order of unity and directly proportional to the scale factor, which is actually measured.

RSD are a measurement of the peculiar velocity field of the galaxies along the line-of-sight (LOS). As this velocity field is only detected along the LOS, it generates an anisotropic signal in the power spectrum expansion as a function of the cosine of the LOS with the vector separation of the galaxy pair. This velocity field is generated by overdensities of matter, and therefore is coherent with the growth of these density perturbations. Thus, by measuring the redshift space distortion effect on the power spectrum of galaxies one can set constraints on the logarithmic growth of structure parameter, f. For the two-point statistics, this parameter is degenerate with the parameter σ₈, the amplitude of dark matter fluctuations at the scale of |$8\, \, h^{-1}\, {\rm Mpc}$|⁠. For this reason power spectrum or correlation function redshift space distortion analyses are sensitive to the combination, f times σ₈, which we just refer as fσ₈.

In this paper, we perform two complementary analyses, BAO and full shape analyses in order to extract D_M(z)/r_drag, D_H(z)/r_drag and fσ₈ from the power spectrum of the final Data Release 16 (DR16) SDSS-IV eBOSS LRG catalogue in combination with the high redshift tail of the Data Release 12 (DR12) SDSS-III BOSS LRG catalogue (for simplicity we refer to this combined catalogue as the DR16 CMASS + eBOSS LRG catalogue). The catalogue consists of 377 458 galaxies between redshifts 0.6 and 1.0, with effective redshift of z_eff = 0.698 and effective comoving volume of |$2.72\, {\rm Gpc}^3$|⁠. The BAO analysis is focused exclusively on identifying the position of the BAO features in the power spectrum, whereas the full shape analysis models the anisotropic power spectrum shape to extract information. In order to enhance the BAO detection, we utilize the standard reconstruction algorithm (Eisenstein et al. 2007; Burden et al. 2014). Thanks to reconstruction we are able to remove most of the non-linear bulk flow effect and enhance the significance of the BAO features. For the BAO analysis, we therefore perform the standard analysis on the reconstructed catalogues, whereas the full shape analysis is performed on the original, pre-reconstructed catalogues. The results extracted from the analysis of the same sample in configuration space are presented in the companion paper (Bautista et al. 2020). Since these two results are expected to be highly correlated (as they are both extracted from the exact same catalogue) we perform a consensus results which is presented at the end of both papers.

The cosmological implication is presented instead in the companion paper (eBOSS Collaboration et al. 2020) along with the measurements of the rest of the galaxy and Lyman α samples of BOSS and eBOSS. These samples correspond to the following:¹

• luminous red galaxy sample (LRG), 0.6 < z < 1.0, power spectrum analysis (this paper) and correlation function analysis (Bautista et al. 2020)

• emission-line galaxy sample (ELG), 0.6 < z < 1.1, power spectrum analysis (de Mattia et al. 2020), correlation function analysis (Tamone et al. 2020) and catalogue description (Raichoor et al. 2020)

• quasar-clustering sample (QSO), 0.8 < z < 2.2, power spectrum analysis (Neveux et al. 2020), and correlation function analysis (Hou et al. 2020)

• lyman α cross-correlation and autocorrelation analysis (des Mas du Bourboux et al. 2020) with quasars z > 2.1.

In addition, eBOSS Collaboration et al. (2020) include as well the results from the two low- and middle-redshift overlapping bins from SDSS-III BOSS (Alam et al. 2017), as they do not overlap with any of the eBOSS samples. An essential component of these studies is the generation of data catalogues (Lyke et al. 2020; Ross et al. 2020), mock catalogues (Lin et al. 2020; Zhao et al. 2020a), and N-body simulations for assessing systematic errors on the LRG (Rossi et al. 2020; Smith et al. 2020) and ELG samples (Alam et al. 2020; Avila et al. 2020). Additionally, in Wang et al. (2020) and Zhao et al. (2020b), the cross-correlation signal between LRG and ELG samples is presented and studied.

Previous to the final DR16 analysis, these samples were already studied for the two-year observation catalogues Data Release 14 (DR14): DR14 eBOSS LRG BAO (Bautista et al. 2018), DR14 eBOSS LRG RSD (Icaza-Lizaola et al. 2019), DR14 eBOSS quasar BAO (Ata et al. 2018), DR14 eBOSS quasar RSD (Hou et al. (Gil-Marín et al. 2018; Hou et al. 2018; Zarrouk et al. 2018) and DR14 Lyman α (de Sainte Agathe (Blomqvist et al. 2019; de Sainte Agathe et al. 2019). Other studies that included redshift-weighting techniques of the DR14 quasar sample were also presented by Ruggeri et al. (2019), Wang et al. (2018), Zhao et al. (2019), and Zhu et al. (2018).

This paper is organized as follows. In Section 2, we briefly present the actual and synthetic galaxy catalogues used in this paper. In Section 3, we describe the methodology followed for performing the power spectrum estimation and the models used for both BAO and full shape analysis. In Section 4, we present the results of this paper as well as the consensus along with the complementary configuration space analysis. In Section 5, we perform an exhaustive systematic study to quantify the potential systematic effect that could affect the inferred cosmological parameters. In Section 6, we present the Fourier and configuration space consensus results and in Section 7 we compare our findings with the standard Lambda cold dark matter (ΛCDM) model predictions. Finally, in Section 8, we present the conclusions of this work.

2 DATA SET

We briefly describe the DR16 LRG data set along with the synthetic mock catalogues we use. A detailed description of the DR16 data set is presented in Ross et al. (2020); the synthetic fast EZmocks used for estimating the covariance are fully described in Zhao et al. (2020a); and the mocks based on OuterRimN-body simulation used for validating the pipeline are described in Rossi et al. (2020). Additionally, we make use of a series of N-body simulations used for previous BOSS analyses (Alam et al. 2017), which we refer as Nseries mocks.

2.1 LRG galaxy sample

The SDSS fourth-generation spectroscopic observations (SDSS-IV, Blanton et al. 2017) employ two multi-object spectrographs (Smee et al. 2013) installed on the Apache Point Observatory 2.5-m telescope located in New Mexico, USA (Gunn et al. 2006), to carry out spectroscopic measurements from a photometrically selected eBOSS LRGs sample (Dawson et al. 2016). Such LRGs were previously selected from the optical SDSS photometry DR13 (Albareti et al. 2017) with the supplementary infrared photometry from the WISE satellite (Lang, Hogg & Schlegel 2016). The same instrument was already used for the previous BOSS program.

A description of the final targeting algorithm is presented in Prakash et al. (2016), which produced 60 LRG targets per square-degree over a sky footprint of 7500 deg², of which ∼50 deg^-2 were spectroscopically observed. Such observations returned mainly objects between 0.6 ≤ z ≤ 1.0 as tested by The Sloan Extended Quasar, ELG and LRG Survey (Dawson et al. 2016).

The estimation of the redshift of each LRG spectrum was performed using the publicly available RedRock algorithm,² which improved the redshift efficiency of its predecessor, RedMonster (Hutchinson et al. 2016), from 90 up to 96.5 per cent in terms of objects with a confident redshift estimate, with less than 1 per cent catastrophic redshift errors.

The density of objects with spectroscopic information per sky-area in the galaxy catalogues is not constant over the eBOSS sky footprint, due to both observational systematics (varying observational features across the imaging survey) and geometrical effects (for example whether a region has been simultaneously observed by more than one plate). We refer to this whole effect as completeness, without separating the observational and geometrical contributions. Qualitatively, the completeness generates spurious signals, we need to filter out in order to measure the intrinsic clustering. Within the eBOSS collaboration, we define the completeness as the ratio of the number of weighted spectra (including also objects classified as stars and quasars) to the number of targets, which is computed per sky sector, this is, a connected region of the sky observed by a unique set of plates. In order to account for the effect of completeness, we downsample each object of the random catalogue by the completeness of its corresponding sky sector. In this way, the definition of completeness includes the systematic weight, w_sys, as well as other effects, including the variation of the mean density as a function of stellar density and galactic extinction. For further details on the catalogue creation, we refer the reader to Ross et al. (2020).

Additionally, a minimum variance weight is also applied, the FKP weight (Feldman, Kaiser & Peacock 1994). This accounts for the radial mean density dependence, w_FKP(z) = 1/[1 + n(z)P₀], where P₀ is chosen to be the amplitude of the power spectrum P(k) at the scales of BAO, |$k\sim 0.1\, \, h\, {\rm Mpc}^{-1}$|⁠, |$P_0=10\, 000\, (\, h^{-1}\, {\rm Mpc})^3$|⁠.

Fig. 1 displays the mean density of objects as a function of redshift for the eBOSS-only LRG (blue) and CMASS (red) galaxies, and the combined CMASS + eBOSS LRG catalogue (black). The solid lines stand for the density of the north galactic cap (NGC) and the dashed lines for the south galactic cap (SGC).

Figure 1.

Number density of objects with spectroscopic observations for DR12 BOSS CMASS LRGs (in blue) and DR16 eBOSS LRGs (in orange), for the NGC (solid lines) and SGC (dashed lines). In black is shown the addition of CMASS and eBOSS densities. Note that such additions only correspond to those regions with overlapping area between eBOSS and BOSS CMASS galaxies, which approximately correspond to the whole eBOSS LRG area. The effective redshift of the combined sample corresponds to z_eff = 0.698 according to the definition of equation (4).

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We have quantified the difference between the NGC and SGC using the mocks to infer the errors and covariance among redshift bins. Unlike the CMASS sample, we find that CMASS + eBOSS LRG n(z) distribution between NGC and SGC is significantly different, which we have imprinted in the EZmocks.

2.2 Synthetic catalogues

In this paper, we employ several type of mocks in order to estimate the covariance, quantify the impact of systematic errors and to validate the pipeline and methods employed on the data.

2.2.1 EZMOCKS

The EZmocks consist of a set of 1000 independent realizations using the fast approximative method based on Zeldovich approximation (Chuang et al. 2015) with the main purpose of estimating the covariance of the data. Such mocks consist of light-cones with the radial and angular geometry of the CMASS + eBOSS LRG data set, with observational effects, such as fibre collision, redshift failures, and completeness. These light-cones are drawn from four and five snapshots at different redshifts, for CMASS and eBOSS galaxies, respectively. A full description of these mocks is presented in Zhao et al. (2020a). These mocks are generated using fast-techniques, which are a good approximation of an actual N-body simulation at large scales, but which eventually fail to reproduce the complex gravity interaction and peculiar motions at small scales. Because of this, we use them to estimate the covariance matrix of the data, but their performance for reproducing physical effects such as BAO and RSD is not guaranteed at sub- per cent precision level. Thus, we do not estimate the potential modelling systematics based on these mocks, but on full N-body mocks. However, these mocks are useful to estimate the relative change on cosmological parameters when applying each of these observational features. We use them to quantify the potential impact of observational systematics in the final data results. In order to analyse these mocks, we use the covariance drawn from themselves.

2.2.2 Nseries mocks

The Nseries mocks are full N-body mocks populated with a fixed halo occupation distribution (HOD) model similar to the one corresponding to the DR12 BOSS NGC CMASS LRGs. Their effective redshift, z_eff = 0.56 is slightly smaller compared to the effective redshift of the DR16 CMASS + eBOSS LRG sample, z_eff = 0.698, as they were initially designed to test the potential systematics on the modelling used for the BOSS CMASS sample. They were generated out of seven independent periodic boxes of |$2.6\, h^{-1}{\rm Gpc}$| side, projected through 12 different orientations and cuts, per box. In total, after these projections and cuts 84 pseudo-independent realizations were produced. The mass resolution of these boxes is |$1.5\times 10^{11}\, {\rm M}_\odot {\it \,h}^{-1}$| and with 2048³ particles per box. The large effective volume, |$84\times 3.67\, [{\rm Gpc}]^3$| makes them ideal to test potential BAO and RSD systematics generated by the analysis pipeline, as to test the response of the arbitrary choice of reference cosmology on the BAO and full shape model templates, in the galaxy catalogues when converting redshifts into distances, and its impact on the inferred cosmological parameters. We use the NGC MD-Patchy mocks (Kitaura et al. 2016) to describe the covariance of these mocks. We rescale the covariance terms by 10 per cent based on the ratio of particles, as the MD-Patchy mocks have fewer particles than the Nseries mocks due to veto effects on DR12 CMASS data, which was also imprinted into the MD-Patchy mocks but not into Nseries mocks. When we run reconstruction on the Nseries mocks, we consistently also use the covariance from reconstructed MD-Patchy mocks.

2.2.3 OuterRim–HOD mocks

The OuterRim-HOD mocks are drawn from the OuterRimN-body simulation (Heitmann et al. 2019) and populated with different types of HOD models (see Rossi et al. 2020 for a full description), some of them similar to the LRG sample, but also others having different properties. The original simulation corresponds to a single cubic box realization with periodic boundary conditions whose size is |$3\, h^{-1}{\rm Gpc}$|⁠. This box is divided into 27 cubic sub-boxes of |$1\, h^{-1}{\rm Gpc}$| per side, without the periodicity of cubic-boxes. For those galaxy catalogues, whose HOD models are close to the actual data sample studied here (those labelled ‘Hearin-Threshold-2’, ‘Leauthaud-Threshold-2’, and ‘Tinker-Threshold-2’, see Rossi et al. 2020 for a description of all models), we place the galaxies in a larger box of |$3\, h^{-1}{\rm Gpc}$| per side with empty space between the galaxies and the box edges, and generate a random catalogue with the same distribution but with no clustering. In this way, when performing the discrete Fourier transform (FT) the non-periodicity conditions do not impact the results. We refer to this process as padding. Additionally, we also apply reconstruction on these padded catalogues.

The effective volume of each sub-box of the ‘Hearin-Threshold-2’, ‘Leauthaud-Threshold-2’, and ‘Tinker-Threshold-2’, corresponds to |$\sim 1.1\, {\rm Gpc}^3$|⁠. For the rest of the HOD-models, the effective volume varies between 2.1 and |$2.7\, {\rm Gpc}^3$|⁠, as the number density of objects, and consequently |$\bar{n}P$|⁠, is much higher.

In order to deal with the covariance of these mocks, we have used the covariance derived from the EZmocks and re-scaled by the difference in particle number. These rescalings correspond to the factors of 1.0, 0.64, and 9 for ‘Standard’, ‘Threshold-1’, and ‘Threshold-2’, respectively, for Hearin, Leauthaud and Tinker HOD-types. For Zheng HOD-type, we use 0.60, 2.37, and 0.60, for ‘Standard’, ‘Threshold 1’, and ‘Threshold 2’, respectively.

2.3 Reference cosmology

In this paper, we choose a set of cosmological parameters within the flat ΛCDM model to define a reference cosmology, which is used to (i) transform the redshifts of galaxies into comoving distances; and (ii) produce a linear template used to build a fitting model. We use as our main baseline analysis the fiducial set of parameters, |$\boldsymbol{\Theta }_{\rm fid}$|⁠, listed in the first row of Table 1 as a reference cosmology. In addition, we also analyse the mocks and data using other sets of reference cosmologies to check the impact of this arbitrary choice. Among these cosmologies, we choose to use as reference cosmology the underlying cosmology of the Nseries mocks, |$\boldsymbol{\Theta }_{\rm Nseries}$|⁠, the OuterRim derived mocks, |$\boldsymbol{\Theta }_{\rm OR}$| and three high-Ω_m cosmologies, |$\boldsymbol{\Theta }_X$|⁠, |$\boldsymbol{\Theta }_Y$|⁠, and |$\boldsymbol{\Theta }_Z$|⁠, whose properties are listed in Table 1. In particular, |$\boldsymbol{\Theta }_{Y}$| and |$\boldsymbol{\Theta }_{Z}$| have a very different r_drag value compared to the one inferred from the usual CMB-anisotropy experiments (Hinshaw et al. 2013; Aghanim et al. 2018). In case of |$\boldsymbol{\Theta }_Y$|⁠, this is driven by a large value of the total number of neutrino species, and for |$\boldsymbol{\Theta }_Z$| by a high value of the baryon density. The |$\boldsymbol{\Theta }_Y$| and |$\boldsymbol{\Theta }_Z$| correspond to a very disfavoured cosmologies compared to the state-of-the art CMB observations. However, our LSS results are presented in a compressed set of variables which do not depend on these CMB priors. Consequently, the results inferred from LSS observations by assuming any of the tested cosmologies as ‘reference-cosmology’ are valid, as we will demonstrate in Section 5.

2.4 Reconstruction

The BAO peak detection significance can be enhanced by applying the reconstruction technique (Eisenstein et al. 2007). We use the algorithm described by Burden et al. (2014), Burden, Percival & Howlett (2015) in which the underlying dark matter density field is inferred from the actual galaxy field by assuming a value of the growth of structure and bias, which can be estimated from a full shape-analysis on the pre-reconstruction catalogue, and used to remove both the non-linear motions and the redshift-space distortions of galaxies.

We make use of the publicly available code⁴ employed for performing reconstruction of the DR14 LRG sample (Bautista et al. 2018). In this paper, we apply this code to the combined CMASS + eBOSS sample, by assuming a bias value of b = 2.3 and a growth rate consistent with |$f(z)=\Omega _{\rm m}^\gamma (z)$|⁠, which in this case is f = 0.82, and using a smoothing scale of |$15\, \, h^{-1}\, {\rm Mpc}$|⁠. Recently, Carter et al. (2019) showed how the inferred cosmological parameters were not sensitive to these arbitrary choices. Potential systematics arising from reconstruction are checked in Section 5.

2.5 Power Spectrum estimator

In this work, we use 50 times more density for the random catalogue of the actual LRG data set and 20 times more for the randoms of the EZmocks. The difference in the estimated power spectrum using the 20 and 50 times random catalogue is smaller than 0.5 per cent per k-bin in the power spectrum monopole with no systematic offset. As described previously, both data and mocks catalogues total weight w_tot is made by the product of the systematic weight, w_sys which contains both completeness and imaging weight, the collision weight w_col, which contains both failures and close pairs collisions, and the FKP-weight. Further details of how these weights were constructed are given in Ross et al. (2020). The normalization factor |$I_2^{1/2}$| normalizes the amplitude of the observed power spectrum and is defined as, |$I_2\equiv \int {\rm d}{\boldsymbol r}\, [n_{\rm gal}w_{\rm tot}({\boldsymbol r})]^2$|⁠. Later in this section, we will comment on how this parameter is inferred and its impact on the final results.

In order to measure the power spectrum multipoles of the galaxy distribution, we follow the same procedure described in previous works (Gil-Marín et al. 2017). Briefly, we assign the objects of the data and random catalogues to a regular Cartesian grid, which allows the use of FT-based algorithms. We embed the full survey volume into a cubic box of side |$L_b=5000\, \, h^{-1}\, {\rm Mpc}$|⁠, and subdivide it into |$N_g^3=512^3$| cubic cells, whose resolution and Nyqvist frequency are |$9.8\, \, h^{-1}\, {\rm Mpc}$| and |$k_{\rm Ny}=0.322\, \, h\, {\rm Mpc}^{-1}$|⁠, respectively. We assign the particles to the cubic grid cells using a third-order B-spline mass interpolation scheme, usually referred to as Piecewise cubic shape (PCS), where each data or random particle is distributed among 5³ grid-cells. Additionally, we interlace two identical grid-cells schemes displaced by 1/2 of the size of the grid-cell; this allows us to reduce the aliasing effect below 0.1 per cent at scales below the Nyqvist frequency (Hockney & Eastwood 1981, Sefusatti et al. 2016).

Unless stated otherwise, we perform the measurement of the power spectrum linearly binning k in bins of |$\Delta k=0.01\, \, h\, {\rm Mpc}^{-1}$| up to |$k_{\rm max}=0.32\, \, h\, {\rm Mpc}^{-1}$|⁠, although not all the k-elements will be necessarily used in the final analysis. The resulting power spectrum multipoles for the combined CMASS + eBOSS LRG sample are displayed in Fig. 2. We observe a significant mismatch between the amplitude of the mocks and data. This difference is caused by an early version of the mocks (with no completeness) being fitted to reproduce an early version of the data (with completeness). The normalization of the data was initially set in such a way that the overall amplitude depended on the value of the overall completeness. As a consequence, when the completeness was applied in the final version of the mocks, mocks, and data did not match. This mismatching only appears to be evident in Fourier space, but not in configuration space (see e.g. fig. 2 of Bautista et al. 2020). Therefore, this effect must correspond to a mismatch at scales of around |$s\sim 1-5\, \mathrm{Mpc}\, {h}^{-1}$| in configuration space. We conclude that this effect has no impact on the final covariance of the data. On the other hand, the overall normalization of the data has no impact on the cosmological signal extracted, as it is appropriately modelled by the window function as we describe below.

Figure 2.

Power spectrum multipoles measured from the DR16 CMASS + eBOSS LRG sample, monopole (orange symbols), quadrupole (green symbols) and hexadecapole (purple symbols). The filled and empty symbols correspond to measurements from the NGC and SGC, respectively. The empty symbols are displaced horizontally for visibility. The black dashed and dotted lines correspond to the clustering of the mean of the 1000 realizations of the EZmocks with all the systematics applied, for NGC and SGC, respectively. The amplitude mismatch, more evident for the monopole, is due to the effect of completeness on the normalization factor of the power for data and mocks.

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In Appendix  D, we explicitly write how the selection effect is included in the power spectrum model.

3 METHODOLOGY

In this paper, we perform two parallel analyses: the analysis of the position of the BAO peak in the anisotropic power spectrum (hereafter BAO analysis), and on the RSD and Alcock–Paczynski effect (AP effect) using the full shape information in the power spectrum (hereafter Full Shape analysis or simply FS analysis).

• The BAO analysis consists of using a fixed and arbitrary template to compare the relative BAO peak positions in the power spectrum multipoles. Such analysis can be performed on both pre- and post-reconstruction catalogues. The analysis performed on the reconstructed catalogue measurements has a higher probability of providing a larger significance detection, and consequently, smaller error bars than the pre-reconstruction measurement. The BAO peak position along and across the LOS direction is then linked to the expansion history and angular diameter distance at the redshift-bin of the measurement.

• The FS analysis consists of a full modelling of the shape and amplitude of the power spectrum multipoles, taking into account non-linear dark matter effects, galaxy bias, and RSD, and is only performed over the pre-reconstructed catalogues. In order to do so, we choose an underlying linear power spectrum template at fixed cosmological parameters and infer the scale dilations and the amplitudes of the power spectrum multipoles. With this we are able to infer not only the expansion history and angular diameter distance, but as well the logarithmic growth of structure times the fluctuations of the dark matter field filtered by a top-hat function of |$8\, \, h^{-1}\, {\rm Mpc}$|⁠, fσ₈.

Unlike ΛCDM-model based analyses, the previously described FS and BAO analyses do not guarantee a consistent relation between the expansion history and the angular diameter distance within a ΛCDM model. In this sense, our analysis goes beyond such assumption and can be used to actually test the validity of the model.

Pre- and post-reconstruction catalogues are considered to contain independent, although correlated, cosmological information. In this fashion, we maximize the amount of cosmological information if we combine them with the appropriate covariance.

3.1 Modelling the BAO signal

We fit the data by considering independent NGC and SGC broad-band and bias parameters, both on the power spectra monopole and quadrupole. Thus, in the standard fit we consider two physical parameters, {α_∥, α_⊥} and 15 nuisance parameters, |$\lbrace \beta , B_{\rm N}, {A_i^{(0)}}_{\rm N}, {A_i^{(2)}}_{\rm N}, B_{\rm S}, {A_i^{(0)}}_{\rm S}, {A_i^{(2)}}_{\rm S} \rbrace$|⁠, where i = 1, …, n, and |${\rm N},\, {\rm S}$| stand for NGC and SGC, respectively.

The main difference between the above isotropic template and the anisotropic template of Equations (20) and (21) is the effect of the BAO damping parameter Σ_nl. In the anisotropic template, the exponential argument has an explicit μ-dependence through the damping terms along and across the LOS, Σ_⊥ and Σ_∥. In this case, the monopole and quadrupole contain an effective weighted-averaged damping parameter, Σ₀, Σ₂, and Σ₄. The main advantage of the isotropic template is that (i) it is faster to evaluate, as it does not require an integration over the LOS, and (ii) the broad-band parameters are in linear combination and therefore an analytical solver can be applied without the need of running an Markov chain Monte Carlo (MCMC) solver to explore the likelihood. The drawback is that the BAO damping is not as accurately described as in the anisotropic BAO template, especially for the anisotropic signal.

In Section 4, we test this effect on the mean of the mocks and in Table 3, we present an alternative analysis using this template. We show that the differences observed among these templates are sufficiently small to not be relevant for the precision of the measurements of this paper.

3.2 Modelling the RSD and galaxy bias

The FS analysis model employed to describe the power spectrum multipoles is the same 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. 2016a) and for DR14 eBOSS quasars 0.8 ≤ z ≤ 1.2 (Gil-Marín et al. 2018), so we briefly present it here to avoid repetition.

3.2.1 Galaxy bias model

We follow the Eulerian non-linear bias model presented by McDonald & Roy (2009). The model consists of four bias parameters: the linear galaxy bias b₁, the non-linear galaxy bias b₂, and two non-local galaxy bias parameters, b_s2 and b_3nl. We always consider the local biases b₁ and b₂ as nuisance and free parameters of the model. Unless stated otherwise, the non-local bias parameters are constrained by assuming the local bias relations from Lagrangian space, |$b_{s^2}=-4/7\, (b_1-1)$| (Baldauf et al. 2012) and |$b_{3{\rm nl}}=32/315\, (b_1-1)$| (Saito et al. 2014).

3.2.2 Real-space spectra

3.2.3 Redshift space distortions

3.3 Parameter inference

In this paper, we infer the covariance matrix from 1000 realizations of the EZmocks (Zhao et al. 2020a). Due to the finite number of mock catalogues when estimating the covariance, we expect a noise term arising when inverting the covariance. We apply the corrections described in Hartlap, Simon & Schneider (2007) which for the current sample is |$\sim 6{{\ \rm per\ cent}}$| factor in the χ² values for BAO analysis and |$4{{\ \rm per\ cent}}$| for FS when the hexadecapole is used. Extra corrections, such as the ones described in Percival et al. (2014), have a minor contribution to the final errors. They represent a |$2$| and |$1.4{{\ \rm per\ cent}}$| increase for the BAO and FS analyses, respectively. We include them only on the last stage of the analysis along with other systematic contributions.

In order to explore the full likelihood surface of a given set of parameters, we run Markov-chains (mcmc-chains). We use Brass¹¹ based on the Metropolis–Hasting algorithm with a proposal covariance and ensure its convergence performing the Gelman–Rubin convergence test, R − 1 < 0.005, on each parameter. We apply the flat priors listed in Table 2 otherwise stated.

Since σ₈ is very degenerate with f and b₁ this is equivalent to treat the terms |$f\sigma _8^{n+1}$| in the Equations (27)–(30), as two independent parameters:¹²fσ₈ and |$\sigma _8^n$|⁠, where fσ₈ is freely fit, whereas |$\sigma _8^n$| is kept fixed. For large scales n = 0, so this approach is exact. At smaller scales n > 0 terms arise, but the systematic effect of fixing this part to a constant is very small. We have checked that varying σ₈ on the |$\sigma _8^n$| terms by 15 per cent, only shifts fσ₈ by 0.2 per cent. In Section 4, we present a fit to the data where both f and σ₈ are varied freely and we show how this has no effect on the final results, although the convergence time for such runs is larger.

For the standard BAO case, we apply equation (21) and leave free |$\lbrace \alpha _\parallel ,\, \alpha _\perp ,\beta \rbrace$| and the broadband parameters |$\lbrace B,A_i^{(\ell)} \rbrace$|⁠, which we fit separately for NGC and SGC. This corresponds to 17 free parameters. In some cases, we also leave the damping terms, Σ_∥ and Σ_⊥, free, and treat them as independent.

For both BAO and FS cases the covariances from NGC and SGC are drawn from two independent sets of mocks and are assumed to be fully independent, as these are two disconnected patches of the Universe. In this fashion, the total likelihood is just the product of NGC and SGC likelihoods: |$\mathcal {L}=\mathcal {L}_{\rm NGC}\times \mathcal {L}_{\rm SGC}$|⁠. We expect that only for very large modes (k much smaller than |$0.02\, \, h^{-1}\, {\rm Mpc}$|⁠) this assumption loses validity.

In this paper, we report the mean of the mcmc chain when converged, R − 1 < 0.005, except for the burn-in part which we discard (the first 10⁴ steps), and report its rms as the 1σ error. This matches the 68 per cent confident level in case of having a Gaussian distribution. For the mocks, we run six independent sub-chains where after convergence, we concatenate and treat as a single chain when calculating the mean and rms. We also test that running different set of chains on the same data set report the same values within the statistical precision required, which indicates that the chain noise is below the statistical precision of the sample. In Appendix  B, we show how the contours drawn from the mcmc chain of the data are in very good agreement with the inferred Gaussian contours.

The isotropic BAO template described by equations (22) and (23) can be solved analytically for most of its parameters using the least-squares method. Given a fixed α_∥, α_⊥, and Σ_{nl, ℓ}, the rest of variables, B, and |$A_i^{(\ell)}$| can be solved analytically so a full MCMC run is not required. One therefore only needs to perform subsequent fits changing α_∥, α_⊥, and Σ_{nl, ℓ} within a fixed array, in order to resolve the likelihood shape, and then interpolate to find the best fit and its error.

4 RESULTS

In this section, we describe the results obtained when applying the BAO and FS pipeline described in the previous section. We perform these two analyses separately, and later in Section 6, we discuss how to combine them. The error bars reported in this section only contain the statistical error budget. Later, in Section 5, we discuss qualitatively and quantitatively the systematic error budget of such approaches.

4.1 BAO analysis

Fig. 3 displays the BAO oscillatory features measured from the CMASS + eBOSS LRG data with respect to the broadband, for the isotropic signal, in orange symbols, and the anisotropic μ²-moment, in green symbols. The black solid lines represent the best fit and the lower panel the model-data deviations in units of statistical 1σ error. The left-hand panel displays the pre-reconstructed results and the right-hand panel the post-reconstructed results. Reconstruction enhances significantly the BAO signal both in the isotropic and anisotropic power spectrum signal. Note that the actual BAO analysis is performed on the monopole and quadrupole, although we visually report the μ²-moment, as defined by equation (25) instead of the quadrupole, as the BAO feature is more evident there.

Figure 3.

DR16 CMASS + eBOSS LRG power spectrum measurements for the pre- (left-hand panel) and post-reconstructed catalogue (right-hand panel). The orange points display the power spectrum monopole and the green points the μ²-moment (see equation 25 for definition). The associated errors are drawn from the covariance of 1000 mocks and the black solid line represent the best-fitting solution (quoted in Table 3 using the anisotropic templated at the fixed values of |$\Sigma _\parallel =7.0\, \, h^{-1}\, {\rm Mpc}$| and |$\Sigma _\perp =2.0\, \, h^{-1}\, {\rm Mpc}$| for post-recon and |$\Sigma _\parallel =9.4\, \, h^{-1}\, {\rm Mpc}$| and |$\Sigma _\perp =4.8\, \, h^{-1}\, {\rm Mpc}$| for pre-recon). The bottom sub-panels show the difference between model and measurement divided by the 1σ errors.

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Table 3 presents the main results from the BAO analysis of the data in terms of the scaling parameters, α_∥ and α_⊥. We perform the BAO analysis keeping the Σ_∥ and Σ_⊥ variables fixed at their best-fitting values on the mean of the pre- and post-reconstructed mocks. These values are |$\Sigma _\parallel =9.4\, \, h^{-1}\, {\rm Mpc}$| and |$\Sigma _\perp =4.8\, \, h^{-1}\, {\rm Mpc}$| for the pre-reconstructed and |$\Sigma _\parallel =7.0$| and |$\Sigma _\perp =2.0\, \, h^{-1}\, {\rm Mpc}$| for the post-reconstructed catalogues.¹³ The first two rows of Table 3 report the BAO analysis on the pre- and post-reconstructed data in the Fourier space (matching the performance displayed by Fig. 3) and in configuration space of the same data set (presented in Bautista et al. 2020). Along with those the consensus between Fourier and gonfiguration space is also presented. The technique used to infer this value is described later in Section 6. The rest of the rows represent the values obtained from the pre- or post-reconstructed analysis on Fourier space with variations of the standard pipeline analysis, to show the sensitivity of the results under certain assumptions. Among these cases, we present analyses when: NGC and SGC are the only-fitted regions, ignoring the effect of the selection function in the modelling (no-mask case), turning off the systematic and collision weights on the data (no-w_sysw_col), using the isotropic template of equation (22) with three- (Isotropic template) and five-parameter broad-band (Isotropic template order-5), using the anisotropic template of equation (20) with five parameters (Order-5), allowing Σ_∥ and Σ_⊥ to be free parameters (⁠|$\Sigma _{\parallel ,\, \perp }$| Free), or free but with a Gaussian prior, |$\bar{x}\pm \sigma _x$|⁠,¹⁴ Σ_∥ = 7 ± 3, and Σ_⊥ = 2 ± 3 (⁠|$\Sigma _{\parallel ,\, \perp }$| Gaussian prior), using the hexadecapole along with the monopole and quadrupole on the BAO fit (+ hexadecapole), using a different reference cosmology for the BAO fitting template (⁠|$\boldsymbol{\Theta }_{\rm OR}$|⁠, |$\boldsymbol{\Theta }_X$|⁠, |$\boldsymbol{\Theta }_Y$|⁠, and |$\boldsymbol{\Theta }_{Z}$|⁠; see Table 1 and the top panel of Fig 10 for a description of these cosmologies) and finally using only 500 realizations of the EZmocks to estimate the covariance (500 real.). When using different x-reference cosmologies, we re-scale the obtained α-parameters by the appropriate factor, |$(D_{\rm H,M}^x/r_{\rm drag})/(D_{\rm H,M}^{\rm fid}/r_{\rm drag}^{\rm fid})$|⁠, to match the results one would have obtained if a fiducial cosmology would have been used as reference cosmology instead. In this way, all the α-parameters of the different rows are comparable, regardless of the template cosmology used.

In general, we see that most of these arbitrary choices produce no significant variation (<0.5σ) with respect to the standard pipeline, demonstrating a strong robustness on the BAO results. Some exceptions are when the data vector is different (pre- versus post-) or when the NGC and SGC are analysed independently. However, in these cases the cosmic variance has a much larger impact and therefore a larger shift is expected. The highest shift we observe (when the data vector is unchanged) is on the variable α_⊥ when the reference cosmology is varied from |$\boldsymbol{\Theta }_{\rm fid}$| to |$\boldsymbol{\Theta }_{\rm OR}$|⁠. In such case, α_⊥ changes by 0.85σ. Note that these α-values have been re-scaled after the actual fit to be both with respect to the same reference cosmology, so in the absence of noise and systematics both α-value should be the same. Later, in Section 5.1.2 and in Table 5, we investigate such effect using the EZmocks and the Nseries mocks, and find no strong shift when the template cosmology is changed, concluding that the difference we observe for the data is exclusively due to a statistical fluctuation.

The reader could think that the results obtained by adding the hexadecapole to the standard monopole plus quadrupole analysis should have reported larger BAO information and smaller statistical error component. Previously, Ross, Percival & Manera (2015a) demonstrated that the amount of BAO information that higher-than-quadrupole moments add in terms of anisotropic BAO is very small. Indeed, we report as well such findings later in Table 5 when we apply our analysis to the mocks. However, this is not the case for an FS analysis, where the hexadecapole is key to break degeneracies between the anisotropy generated by AP and RSD. We therefore conclude that the difference between the BAO analyses with and without hexadecapole are exclusively due to noise fluctuations, and do not correspond to any significant extra BAO information. Because of this, we take as our main BAO results those in which only the monopole and quadrupole are analysed.

Fig. 4 displays the likelihood posteriors for 1σ and 2σ for the BAO analysis using the pre- (orange) and post-reconstruction (blue) catalogues, in terms of the physical variables, D_M/r_drag – D_H/r_drag. In both cases, the agreement is very good. The statistical errors on the cosmological parameters inferred from the post-reconstructed catalogues present are a factor of 1.5 smaller than those obtained from the pre-reconstructed catalogues. Later, in Section 5, we study how typical this gain factor is by using the results from individual mocks.

Figure 4.

Likelihood posterior for 1σ and 2σ contours (only statistical contribution), from the BAO type of analysis on the DR16 CMASS + eBOSS LRG data for the pre-reconstructed catalogues (in orange) and the post-reconstructed catalogues (in blue) in terms of D_M(z_eff)/r_drag and D_H(z_eff)/r_drag variables, at z_eff = 0.698. Results corresponding to the first two rows of Table 3.

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4.2 Full shape analysis

We run the FS pipeline on the power spectrum monopole, quadrupole, and hexadecapole measured from the CMASS + eBOSS galaxies for the k-range |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}] \le 0.15$|⁠, as described in Section 3.2. The covariance among k-bins is estimated from the analysis of 1000 EZmocks.

Fig. 5 displays the monopole (round orange symbols), quadrupole (square green symbols), and hexadecapole (triangle purple symbols) for the results of the CMASS + eBOSS LRG data used for the FS analysis. The black solid lines display the performance of the best-fitting model when the three multipoles are simultaneously fitted, whereas the black-dashed lines when only the monopole and quadrupole are used.

Figure 5.

Power spectrum multipoles measured from the DR16 CMASS + eBOSS LRG sample (weight-averaged between NGC and SGC), monopole (circular orange symbols), quadrupole (square green symbols), and hexadecapole (triangle purple symbols), along with the error bars predicted by the rms of the 1000 EZmocks. The solid and dashed black lines represent the FS best-fitting model (weight-averaged between NGC and SGC) when the monopole and quadrupole only are fitting (black dashed lines) and when the hexadecapole is also used (black solid line). In the bottom sub-panel, the differences between the measurement and the model, relative to the value of 1σ error bar, are also displayed using the same colour notation. The results for the best-fitting parameters are reported in Table F3 for the narrow prior on the amplitude of shot noise, 0.5 ≤ A_noise ≤ 1.5.

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Table 4 displays the results for FS analysis under different cases. The first two rows represent the case where the monopole (M), quadrupole (Q), and hexadecapole (H) are fitted up to a |$k_{\rm max}=0.15\, \, h\, {\rm Mpc}^{-1}$| with a wide and flat uninformative prior on the amplitude of shot noise (first row) and with a more restrictive prior allowing for such amplitude to vary within |$50{{\ \rm per\ cent}}$| of its Poisson prediction (second row). The third row displays the result from the configuration space analysis reported in Bautista et al. (2020) and the fourth row the consensus between Fourier and configuration space, as described in Section 6. The rest of the rows are variations of the above pipeline (with a wide uninformative prior on the amplitude of shot noise as a default option): using only the monopole and quadrupole (M+Q), fitting to NGC- and SGC-only (NGC, SGC M+Q + H), fitting to the weighted mean signal of NGC and SGC (NGC+SGC M+Q + H), setting a hard prior on b₂ to be positive (b₂ > 0 prior), using the monopole, quadrupole, and hexadecapole with a different k-range, computing the hexadecapole using a different decomposition on the LOS (hexadecapole as |$\mathcal {L}_4(\hat{\boldsymbol k}\cdot \hat{\boldsymbol r}_1)\mathcal {L}_0(\hat{\boldsymbol k}\cdot \hat{\boldsymbol r}_2)$|⁠), ignoring the A^TNS and B^TNS in the modelling of equation (31) (no-TNS terms), only using one-loop correction in the modelling of equation (27) (only one-loop terms), using SPT predictions instead of RPT for the terms of equation (27) (SPT 2-loop), using the Gaussian form of equation (33) for FoG (FoG Gaussian), setting the fiducial σ₈ to a 15 per cent higher value than the predicted by the reference cosmology at z = 0.70 (σ₈ 15 per cent high), treating f and σ₈ as free independent parameters (σ₈ free), using a different cosmology as a reference cosmology (⁠|$\boldsymbol{\Theta }_{\rm OR}$|⁠, |$\boldsymbol{\Theta }_{\rm X}$|⁠, |$\boldsymbol{\Theta }_{\rm Y}$|⁠, and |$\boldsymbol{\Theta }_{\rm Z}$|¹⁵), turning off the systematics and/or collision weights (w_sys off, w_col off, w_sysw_col off), and using only 500 EZmock realization to estimate the covariance (500 real. in covariance).

Except for some extreme cases, such as those when the systematic weights are not applied, we do not observe any strong dependence of the inferred cosmological parameters with any of the studied variations. In particular, we observe very mild changes when the underlying linear power spectrum template is changed. We also note the change in the size of errors of α_∥ and fσ₈ when the 50 per cent prior on A_noise is set (first versus second row). In this last case the 2σ, contours do hit the higher boundary of A_noise = 1.5, and therefore the reduction of error is a direct consequence of this. The amplitude of shot noise is very correlated with b₂ which is poorly constrained without higher order moments as the bispectrum (Gil-Marín et al. 2017). In particular, the A_noise ≃ 1 solution also corresponds to b₂ > 0, whereas A_noise > 1.5 corresponds to b₂ < 0 (see Fig. E2). From the power spectrum and bispectrum analysis of BOSS DR12 CMASS sample 0.43 < z < 0.70, we expect that b₂ for these type of galaxies is close to the value reported in Gil-Marín et al. (2017), b₂σ₈ = 0.606 ± 0.069, which agrees with the solution of shot noise being close to the Poisson prediction. Thus, we find plausible that the shot noise should not differ by more than 50 per cent (which is already quite a large amount) from the Poissonian prediction and we decide to take as the FS analysis main result of this paper the cosmological parameters inferred when this |$50{{\ \rm per\ cent}}$| prior is applied. In Appendix  E, we further comment on this effect (see also Fig. E1).

In Fig. 6, we display the derived D_H/r_drag, D_M/r_drag, and fσ₈ cosmological parameters from the FS and BAO analysis on the pre- and post-reconstructed catalogues, respectively. Both measurements rely on very correlated pre- and post-recon catalogues, and therefore it is not straightforward to resolve the level of agreement between them. Later in Section 6, we come back to this question and also compare these findings with the quantities inferred from configuration space. For now, we just note that the reconstructed-BAO analysis provides tighter constrains on both D_H/r_drag and D_M/r_drag than the FS analysis. This feature is actually expected due to the enhancement that reconstruction provides in the measurement of the BAO peak oscillatory features. We also note that the reconstructed-BAO analysis favours higher values of both D_M/r_drag and D_H/r_drag with respect to the FS analysis on pre-reconstructed data. In fact, if we were looking at the pair of variables α₀ and ϵ we would notice that such difference arises from the AP variable, ϵ or F_ϵ, where the results inferred from reconstructed data present an ∼2σ deviation from the null-AP behaviour, which is what we observe for the pre-reconstructed catalogue. We will fully discuss these differences later in Section 6.

Figure 6.

Comparison of the cosmological inferred parameters from the FS and BAO analysis (on post-reconstructed catalogues), respectively, from the power spectrum multipoles. In this case, the FS derived results have been computed from the power spectrum monopole, quadrupole, and hexadecapole under the analysis with |$50{{\ \rm per\ cent}}$| priors on A_noise reported by the second row of Table 4. The BAO reconstructed results come from the standard analysis on power spectrum monopole and quadrupole, as it is reported by the second row of Table 3. In all cases, the contours represent only the statistical contribution.

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5 SYSTEMATIC TESTS

In this section, we aim to run the BAO and FS pipeline analyses on different sets of mocks to check the performance and to identify potential systematic errors. In total, we use N_EZ = 1000 realizations of the EZmocks, N_Nseries = 84 realizations of the Nseries mocks and N_OR = 27 realizations of the OuterRim-HOD mocks.

5.1 BAO systematics

We start by running the BAO pipeline described in Section 3.1 on the pre- and post-reconstructed EZmocks, Nseries and OuterRim +‘Hearin-Threshold-2’, +‘Leauthaud-Threshold-2’ and + ‘Tinker-Threshold-2’ HOD mocks. We run the BAO pipeline on the power spectrum monopole and quadrupole for |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}]\le 0.30$|⁠. Smaller and larger scales do not contain relevant BAO information.

In this section we aim to

• check how typical the data is with respect to the EZmocks;

• determine the systematic budget of the pipeline;

• check whether the arbitrary choice of the BAO reference template has an impact on the inferred cosmological parameters; and

• determine whether the underlying galaxy HOD has an impact on the recovered parameters.

The top panels of Fig. 7 display the recovered α_∥ and α_⊥ scaling parameters on the pre- (left-hand panels) and post-reconstructed (right-hand panel) 1000 EZmocks realizations (green points). The corresponding bottom panels display the distribution of errors inferred from the rms of the individual MCMC chains. In addition, we represent with a red cross the values for the actual data catalogue, and with a black dot the values obtained when fitting the average power spectrum of 1000 EZmocks realizations. The error of this last case is expected to scale with the square root of the total volume, and therefore we re-scale it by the |$\sqrt{N_{\rm EZ}}$| factor in order to match the value of the error of a single realization. For all the cases, the results inferred from the mean of the mocks are in excellent agreement with the results of the individual cases, suggesting that the mean of the fits is close to the fit of the mean (shown later in Table 5). We find that the values of α_∥ and α_⊥ inferred from the data catalogue are also consistent with the intrinsic scatter observed from the mocks. However, we obtain atypically small errors when analysing the data catalogue, both for the pre- and post-reconstructed cases of α_∥. In particular, for the post-reconstruction case, we have only found a total of ∼10/1000 realizations whose error on α_∥ is comparable to the one found in the data, which certainly suggest a |$\sim 1{{\ \rm per\ cent}}$| probability of being in such situation. As we show later (see Fig. 15 in Section 6), this result is perfectly compatible with what we find in the complementary BAO analysis in configuration space performed in Bautista et al. (2020). The χ² value of the data is not small with respect to the typical value obtained by the mocks, which suggests that this small BAO error on α_∥ may be caused by noise fluctuations, which enhance the BAO signal in the data along the LOS, with respect to the typical noise level predicted by the mocks. Given the number of physical parameters (α_∥, α_⊥, pre- and post-reconstructed catalogues, fσ₈ and their corresponding errors: 10 variables in total) it is not very unlikely that at least one of them is atypical at 3σ level, which is known as ‘the look-elsewhere effect’.

Figure 7.

The top sub-panels display the distribution of best-fitting α_∥ and α_⊥, for the pre- (left-hand panels) and post-reconstructed (right-hand panels) power spectrum monopole and quadrupole for |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}] \le 0.30$|⁠, for the 1000 realizations of the EZmocks catalogues (green points), for the DR16 CMASS + eBOSS LRG data catalogue (red cross) and for the best-fit to the mean of the 1000 mock power spectra (black dot). The horizontal and vertical black dotted lines represent the expected α values for the EZmocks. The distribution of χ² values is also shown for each case. The bottom sub-panels display analogous plots for the errors of α_∥ and α_⊥, instead. In this case, the error of the mean has been re-scaled by the square root of number of realizations.

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The panels in Fig. 8 display how the reconstruction algorithm performs on the EZmocks, for the errors of α_∥ and α_⊥. Reconstruction significantly helps to improve the determination of the αs in almost all the realizations of the mocks, where the typical improvement (ratio between pre- and post-recon errors) is |$\sim 40{{\ \rm per\ cent}}$| for α_∥ and |$20{{\ \rm per\ cent}}$| on α_⊥. This behaviour is expected for cosmic variance limited samples, like the LRGs and ELGs, unlike other more sparse samples like the quasars. We find that for the data, the improvement on α_⊥ is expected and typical with respect to what is observed in the mocks, whereas for α_∥ the values are atypical as we have commented above, but the improvement ratio is typical.

Figure 8.

Performance of the reconstruction on the DR16 CMASS + eBOSS LRG catalogues, the green symbols represent the 1000 realizations of the EZmocks, the red cross the actual DR16 CMASS + eBOSS LRG data catalogue. The x-axis represents the pre-reconstructed quantity and the y-axis the post-reconstructed quantity. For reference, a black dashed line, x = y is also shown. The top and bottom panels display the 1σ errors of α_∥ and α_⊥, respectively. The black dot represents the mean of 1000 realizations re-scaled by the square root of the 1000 realizations.

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5.1.1 Performance of BAO template

We start by applying the BAO anisotropic template described by equation (21) on the different sets of mocks. Table 5 displays the results when fitting the mean power spectra of all available realizations for a given type of mock (rows labelled ‘Mean’); and the mean of the fits of individual realizations (rows labelled ‘Individual’).

Fig. 9 graphically represents the data contained in Table 5. For each sub-panel, the difference between the measured α_∥ and α_⊥, and their expected value are inferred for the pre- (left-hand side) and post-reconstructed (right-hand side) catalogues. The individual results on the mocks are shown in grey, the results of the mean of the mocks are represented by a △-symbol in green (for the pre-reconstructed catalogues) and in orange (for post-reconstructed catalogues). The associated errors are consistently the errors of the mean, obtained by re-scaling the covariance by a factor the number of realizations, N_EZ, N_Nseries, and N_OR. Therefore, these errors are a factor |$\sqrt{N_i}$| smaller than the error we obtain for a single realization of these mocks. The mean of the individual fits is represented by a \space-symbol in pink (for pre-reconstructed catalogues) and in purple (for post-reconstructed catalogues). In this case, the error associated is the rms of all the individual fits, which is |$\sim \sqrt{N_i}$| larger than the error associated to the mean. The sub-panels show the difference between the measured and the expected value of α_∥ and α_⊥ in terms of number of statisticalσ of the error of the mean, and the |${\rm rms}/\sqrt{N_i}$|⁠. Note that for the EZmocks the effective volume is so large (⁠|$\simeq 2,650\, {\rm Gpc}^3$|⁠) that in some cases the result is totally dominated by systematics and the symbols are off the scale of ±3σ.

Figure 9.

Performance of the BAO type of analysis for α_∥ (top panel) and α_⊥ (bottom panel) as indicated. Each of the five vertical sub-panels corresponds to the results on the EZmocks, Nseries, and the OuterRim-HOD mocks by Hearin+threshold2, Leauthaud+threshold2, and Tinker + threshold2, as indicated. A padding has been applied to the original non-periodic OuterRim-HOD cubic sub-box in order avoid spurious non-periodic effects. For each α, we display the panels Δα ≡ α − α^exp and Δα/σ. For each of these sub-panels, we display the pre- (left-hand panel) and post-recon results (right-hand panel). The grey points correspond to the individual best-fitting parameters. The green (for pre-recon) and orange (for post-recon) triangles symbols correspond to the results of fitting the mean of the mocks, and its error corresponds to the error of the mean. The pink (for pre-recon) and purple (for post-recon) squares symbols display the results of taking the mean of N_det individual fits, and the reported error is its rms. Consequently, the error of the mean of the fits is |$\sqrt{N_{\rm det}}$| times the error of the mean. In the Δα/σ sub-panels, σ represents the error of the mean for the fit to the mean (orange and green triangles), and the rms/|$\sqrt{N_{\rm det}}$| for the mean of the individual fits. The numerical results of this plot are also listed in Table 5.

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The EZmocks ‘Mean’ results on post-reconstruction catalogues reveal that the α_∥ variable is significantly shifted by |$0.6{{\ \rm per\ cent}}$| with respect to their expected quantity, which corresponds to 6σ deviation from the expected value; whereas for α_⊥, mocks and model agree to within |$0.068{{\ \rm per\ cent}}$| (1σ level). It is also worth mentioning that at this sub-per cent level of precision, we would require full N-body mocks to actually validate this kind of systematic shifts, as the EZmocks have not been designed to be accurate at this level of precision. Therefore, we cannot discern whether this observed |$0.6{{\ \rm per\ cent}}$| shift in α_∥ is due to a limitation of the model of equation (21), a limitation of the EZmocks themselves, or an effect arising from the reconstruction technique. From the remaining N-body mocks, we do not observe any significant BAO peak position shift with respect to their corresponding expected value in any of the post-reconstructed catalogues analysed. The BAO pipeline is able to deal with different kinds of HOD models. We do see some fluctuations, but these are always below ±2.5σ limit, so we do not take them as significant shifts. However, the statistical errors associated to these catalogues are not as small as those corresponding to the EZmocks, so we can only state that we have not detected any systematic above the statistical threshold of |$1{-}2{{\ \rm per\ cent}}$| for OuterRim-HOD, and |$0.1{-}0.5{{\ \rm per\ cent}}$| for Nseries. Such upper limits are below the statistical precision of our sample: for post-reconstructed catalogues we obtain a statistical precision of |$\sim 2.4{{\ \rm per\ cent}}$| for α_∥ and |$\sim 1.9{{\ \rm per\ cent}}$| for α_⊥. From the Nseries results, we conclude that there are no strong modelling-systematic errors associated when determining αs, which validates our modelling pipeline, including the reconstruction technique. From the OuterRim-HOD results, we conclude that we do not detect any relative systematic due to different HOD modelling, although the statistical precision reached on these mocks is comparable to the statistical precision of our sample.

5.1.2 Effect of reference cosmology on BAO

We are interested in testing the potential impact of the arbitrary choice of reference cosmology. For simplicity, we use the same cosmology to (i) produce the BAO template, and (ii) convert redshifts into distances in both random and galaxy catalogues. The BAO analysis measures relative differences between the BAO peak position in the power spectrum with respect to the template. Therefore, a priori the specific choice of reference cosmology should not impact this result. In this section, we explicitly check this by analysing the Nseries mocks in five different cosmologies: their own cosmology, |$\boldsymbol{\Theta }_{\rm Nseries}$|⁠; the fiducial cosmology which is used for the baseline analysis of the actual data catalogue, |$\boldsymbol{\Theta }_{\rm fid}$|⁠; and three extra cosmologies with a higher value of Ω_m: |$\boldsymbol{\Theta }_X$|⁠, |$\boldsymbol{\Theta }_Y$|⁠, and |$\boldsymbol{\Theta }_Z$|⁠, all listed in Table 1. The oscillatory features of these cosmologies are plotted in the top panel of Fig. 10. The expected values of α_∥ and α_⊥ are therefore different among these cosmologies. We do not analyse these mocks on the |$\boldsymbol{\Theta }_{\rm OR}$| cosmology, as this cosmology is very similar to the |$\boldsymbol{\Theta }_{\rm Nseries}$|⁠. Inputing the values of the true |$\boldsymbol{\Theta }_{\rm Nseries}$| and these four cosmologies into equations (14) and (15), we determine the expected values of the scaling factors. For the |$\boldsymbol{\Theta }_{\rm fid}$| cosmology, we find that |$\alpha ^{\rm exp}_\parallel =0.9875$| and |$\alpha _\perp ^{\rm exp}=0.9787$|⁠; for the |$\boldsymbol{\Theta }_X$| cosmology, we find that |$\alpha _\parallel ^{\rm exp} = 0.9846$| and |$\alpha _\perp ^{\rm exp} = 0.9620$|⁠; for the |$\boldsymbol{\Theta }_Y$| cosmology, we find that |$\alpha _\parallel ^{\rm exp} = 0.9543$| and |$\alpha _\perp ^{\rm exp} = 0.9325$|⁠; for the |$\boldsymbol{\Theta }_Z$| cosmology, we find that |$\alpha _\parallel ^{\rm exp} = 0.9557$| and |$\alpha _\perp ^{\rm exp} = 0.9291$|⁠; and of course when the mocks are analysed in their own cosmology the expected values are unity.

Figure 10.

Top panel: BAO signal, |$\mathcal {O}_{\rm lin}(k)$| of the cosmologies listed in Table 1 (except for |$\boldsymbol{\Theta }_{\rm EZ}$|⁠), used to test the cosmology dependence of the BAO and FS analyses with the reference cosmology displayed in the middle panel (BAO analysis on post-recon catalogues and listed in Table F4) and in the bottom panel (FS analysis on pre-recon catalogues listed in Table F5). Middle panel: BAO scale factors along (α_∥) and across (α_⊥) the LOS measured on the reconstructed Nseries mocks. The △ symbols display the fit on the mean of the 84 realizations, the grey symbols display the fit on the individual 84 realizations, and the ▽ symbols the mean of the 84 individual fits. The error associated is the error of the mean and the rms among the 84 realizations divided by |$\sqrt{84}$|⁠, respectively. Bottom panel: same notation than the middle panel applied to FS analysis on the same mocks. The x-axis shows the results for different cosmologies used for the template (and for mapping redshifts into comoving distances when computing the power spectrum). The horizontal dashed lines mark the expected value of |$\alpha _{\parallel ,\, \perp }$|⁠, and fσ₈.

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The results of the fits on the mean of the mocks and on the individual fits are reported in Table F4, for post-reconstruction analyses, where reconstruction has been performed using the true value of f and b₁. We follow this approach because in this test we aim to check the impact of the arbitrary choice of the reference cosmology when recovering the BAO parameters, rather than to test the efficiency of reconstruction as a function of the assumed parameters (see Carter et al. 2019 for such a study). The middle panel of Fig. 10 displays the post-reconstruction results from Table F4, where the horizontal dashed lines represent the expected values for αs in each cosmology. For each case, we display the fit to the mean of the N_Nseries realizations along with the error of the mean, the individual fit to these N_Nseries realizations, and mean of the fit of these N_Nseries realizations, along with the rms divided by |$N_{\rm Nseries}^{1/2}$|⁠. When studying the post-reconstructed catalogues, we find that for both α_∥ and α_⊥ the highest shift, relative to those parameters inferred from |$\boldsymbol{\Theta }_{\rm Nseries}$|⁠, are those inferred using the templates of |$\boldsymbol{\Theta }_Z$|⁠, which show a deviation of |$1{{\ \rm per\ cent}}$| in α_∥ and |$0.8{{\ \rm per\ cent}}$| for α_⊥. We note that |$\boldsymbol{\Theta }_Z$| represents a very distinct cosmology with respect to the true cosmology of Nseries, with shifts of ΔΩ_m = 0.08 and ΔΩ_b = 0.019, which are 10σ and 50σ away, respectively, from the results reported by Planck (Aghanim et al. 2018). On the other hand, if a closer-to-standard ΛCDM reference cosmology is used, such as |$\boldsymbol{\Theta }_{\rm fid}$|⁠, these shifts reduce to |$0.5{{\ \rm per\ cent}}$| on α_∥ and |$0.3{{\ \rm per\ cent}}$| for α_⊥.

5.1.3 Effect of non-periodicity on BAO measurements

The OuterRim-HOD mocks comes from a single OuterRim dark matter simulation, split into 27 non-periodic cubic sub-boxes and populated with different types of HOD models and flavours. In this section, we aim to quantify the impact of analysing the non-periodic sub-boxes when DFT algorithms are used to obtain the power spectra, which implicitly assume periodic boundary conditions. In this case, and across the paper we only consider wave numbers between |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}]\le 0.30$| for BAO analyses. In order to test this impact within such scale-range, we only focus on the set of HOD types and flavours closer to the LRG galaxy sample. We refer to such padded catalogues as ‘sky-cuts’. In Table F1, we display the results on the mean of the 27 mocks, for both cubic non-periodic and sky-cut case. Note that the non-periodic cubic and sky-cut contain the same galaxies, therefore, the information content should be the same in the absence of spurious non-periodic effects. We do not observe any significant changes in α_⊥ parameter, but a consistent shift of α_∥ by |$2{-}3{{\ \rm per\ cent}}$|⁠: for the non-periodic cubic box, we find an excess in the value of α_∥ with respect to what is expected, whereas for the sky-cut case the results are in agreement (within 2σ error bars) with the expected values. We conclude that (1) the non-periodic effects are important when determining α_∥, but not α_⊥ (2) we do not observe relative shifts in any of the α parameters when the HOD model or flavour is varied.

5.1.4 Impact of HOD modelling in BAO

In this section, we aim to explore the impact of different HODs and flavours when recovering the scaling parameters. Ideally, we should pad as well around these catalogues in order to remove the effect of non-periodicity. However, since the galaxy catalogues can be very large for some of the HODs studied (five million galaxies for threshold1), and the random catalogues need to be at least 20 to 50 times larger, this is not feasible. Since the spurious effects of non-periodicity have a geometrical origin, they should be independent of the intrinsic clustering, and we are only interested in the relative effect of the HOD modelling, we opt to analyse the OuterRim-HOD mocks as if they were periodic boxes, and compare only the relative recovered values among them, bearing in mind that we expect an |$\sim 2-3{{\ \rm per\ cent}}$| offset on α_∥. These results are listed in Table F2 for the mean of 27 mocks and graphically represented in Fig. 11, where the filled black symbols are the results for the cubic boxes (affected by the non-periodicity) and the empty black symbols the results where the boxes have been padded (not affected by non-periodicity). For all the HODs, we consistently see that there is the expected |$3{{\ \rm per\ cent}}$| offset on α_∥ as an effect of the non-periodicity of the box, whereas α_⊥ is well recovered in all the cases. We therefore conclude that within the statistical errors of |$1{-}2{{\ \rm per\ cent}}$| the AP parameters recovered from the BAO type of analysis are not affected by the HOD model for LRGs. These findings are in agreement with the recent study by Duan & Eisenstein (2019).

Figure 11.

Inferred scaling parameters, α_∥ and α_⊥, and fσ₈ using a BAO (black symbols) and FS (coloured symbols) type of analysis from the pre-reconstructed OuterRim-HOD mocks, for different types of HOD models and flavours (listed in the x-axis). Red, purple, orange, and blue colours correspond to Hearin, Leauthaud, Tinker and Zheng HOD models for the FS type of analysis, respectively. For each of these models three flavours have been implemented: standard (std), threshold 1 (th1), and threshold2 (th2), as labelled. The filled symbols correspond to |$1\, h^{-1}{\rm Gpc}$|-size cubic box without periodic boundary conditions. The empty symbols correspond to the results obtained from a padded and larger box corresponding to |$3\, h^{-1}{\rm Gpc}$|-size, where the non-periodicity impact is negligible. All results correspond to fitting the mean of 27 mocks, and the reported error is 1σ of the error of the mean.

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5.2 RSD systematics

We repeat the same strategy used for BAO systematics in the previous Section 5.1. We run the standard FS pipeline described previously in Section 3.2 on the pre-reconstructed mocks. We aim to check the typicality of the data with respect to the EZmocks under the FS analysis, how such analysis responses to change in the HOD of the mocks, the impact of the arbitrary choice of the reference template used to compute the FS model, and to calibrate the optimal k-range to be used on FS in order to maximize the statistical error and minimize the systematic budget.

The panels of Fig. 12 display the recovered parameters, α_∥, α_⊥, and fσ₈ (top panels), and their errors (bottom panels) from an FS analysis on the power spectrum monopole, quadrupole, and hexadecapole using |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}]\le 0.15$|⁠. Later, we justify the choice of these specific scales. As in Fig. 7, the red cross displays the performance of the DR16 CMASS + eBOSS LRG data catalogue, and the black dots the performance on the mean of the 1000 realizations of EZmocks. In the bottom panels, the error of the mean has been re-scaled by the factor |$\sqrt{1000}$| to match the typical error of one single realization. We note that the re-scaled error of the mean of the mocks is displaced from the centre of the cloud of errors from individual mocks. This behaviour is caused by the 50 per cent prior on A_noise on the individual mocks (this prior has no effect when fitting the mean of the mocks) which shrinks the distribution tail towards smaller errors, especially for α_∥ and fσ₈. The values and errors of α_∥, α_⊥, and fσ₈ inferred from the data catalogue are consistent with the intrinsic scatter observed from the mocks; and also the χ² value of the data is probable given the distribution of the mocks.

Figure 12.

The top sub-panels display the distribution of inferred α_∥, α_⊥, and fσ₈ parameters using an FS analysis of the power spectrum monopole, quadrupole, and hexadecapole for the 1000 realizations of the EZmocks catalogues (blue points), DR16 CMASS + eBOSS LRG data catalogue (red cross) and the best-fit to the mean of the 1000 mock power spectra. The horizontal and vertical black dashed lines represent the expected αs and fσ₈ values for the EZmocks. The scales fitted are in all cases |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}]\le 0.15$|⁠. In each panel, the distribution of χ² values is also shown. The bottom sub-panels display an analogous plots for the errors of α_∥, α_⊥, and fσ₈ instead. In this case, the error of the mean have been re-scaled by the square root of number of realizations.

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5.2.1 Optimal range of scales for the FS analysis

We aim to determine the range of scales, we should be using when performing an FS analysis. Ideally, the wider this range, the smaller the statistical uncertainty in the inferred cosmological parameters should be. We expect that at very small scales the amount of information on cosmological parameters saturates, although the state-of-the art FS techniques have not reached that limit yet (Hand et al. 2017). Thus, we set the small scale truncation limit based on the ability of the FS model to recover unbiased cosmological parameters. In order to determine this limit, we apply the FS analysis on the full N-body Nseries mocks truncating the model at different scales. Fig. 13 shows the response of α_∥, α_⊥, and fσ₈ on the truncation scale, k_max, when the monopole and quadrupole are used (denoted MQ) and when the hexadecapole is also used (MQH).

Figure 13.

Test of performance when recovering cosmological parameters for the FS analysis described in Section 3.2 as a function of the range of scales used in each power spectrum multipole, where Δx ≡ x − x^exp for α_∥, α_⊥, and fσ₈ for the three different panels. The symbols represent the response on the mean of the 84 realizations of the Nseries mocks when the fiducial cosmology is used as reference template (red △ symbol), when the |$\boldsymbol{\Theta }_Y$|–cosmology template is used (blue ▽ symbol), and when the true cosmology template is used (black • points). The x-axis labels indicate the combination of multipoles (monopole, quadrupole, hexadecapole) used, and the value of k_max in |$\, h\, {\rm Mpc}^{-1}$|⁠. The results do not show any particular trend with the template. Note that in some cases the value of k_max is different for the hexadecapole compared to that for the monopole and quadrupole. For all cases, the value of k_min is set to |$0.02\, \, h\, {\rm Mpc}^{-1}$|⁠. The horizontal dotted lines represent the |$1{{\ \rm per\ cent}}$| deviation for αs and |$2{{\ \rm per\ cent}}$| for fσ₈. The numerical results of this plot are displayed in Table F5.

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We study the following cases: using only the power spectrum monopole and quadrupole for |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}] \le k_{\rm max}$| (labelled MQ k_max); using monopole, quadrupole, and hexadecapole with the same-scale truncation |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}] \le k_{\rm max}$| for all three (labelled MQH k_max); and truncation of multipoles at different scales, |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}] \le k_{\rm max}^{(1)}$| for monopole and quadrupole and |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}] \le k_{\rm max}^{(2)}$| for hexadecapole (labelled MQ |$k_{\rm max}^{(1)}$| + H |$k_{\rm max}^{(2)}$|⁠). The hexadecapole is more sensitive to RSD than the other multipoles, as the modes parallel to the LOS are relatively more weighted than the transverse ones. Therefore, a potential μ-dependent systematic could appear as a parameter-shift when k_max grows for the hexadecapole, but not necessarily when it grows for lower multipoles.

For completeness, we perform such analysis using three different templates, corresponding to |$\boldsymbol{\Theta }_{\rm fid}$| at z = 0.70 (red symbols), |$\boldsymbol{\Theta }_Y$| at z = 0.55 (blue symbols) and to |$\boldsymbol{\Theta }_{\rm Nseries}$| at z = 0.55 (black symbols). The difference in redshift among these templates (and in particular for z = 0.70 which is different from the effective redshift of the Nseries mocks, z_eff = 0.55) only enters in the model through the fixed σ₈ value in the second-order terms of the model, and the difference between the σ₈ values at these two redshift is |$\sim 11{{\ \rm per\ cent}}$|⁠. From the results of Fig. 13, there is no significant difference among these templates, suggesting that, (i) the shape of the template for the reference cosmology has a negligible impact on the inferred cosmological parameters (we test this later in Section 5.2.3 for a wider range of templates); and (ii) the redshift at which this template is computed (which solely regulates its amplitude) has an impact on fσ₈ which is |$\lt 1{{\ \rm per\ cent}}$| when the truncation scale is |$\le 0.15\, \, h\, {\rm Mpc}^{-1}$| and |$\simeq 1{{\ \rm per\ cent}}$| when the truncation scale is |$0.20\, \, h\, {\rm Mpc}^{-1}$|⁠. This happens due to the non-linear terms proportional to |$f\times \sigma _8^n$| for n > 1 in the non-linear terms of the model as already discussed in Section 3.3. The statistical error on the fσ₈ measurement is about 10 per cent for this sample and consequently this effect is completely negligible.

The results reported in Fig. 13 suggest that the effect of the truncation scale on α_∥ is of the order of |$\le 1{{\ \rm per\ cent}}$| for the k-ranges studied here. For α_⊥, we find that the effect of increasing the value of k_max and including the hexadecapole tend to underpredict its value by |$1{-}1.5{{\ \rm per\ cent}}$| depending on the truncation option. On fσ₈ the effect of increasing k_max can be either overpredict and underpredict, depending on which template is being used, but the effects are always below |$2{-}3{{\ \rm per\ cent}}$|⁠.

As a fiducial choice, we opt to truncate the FS analysis at |$k_{\rm max}=0.15\, \, h\, {\rm Mpc}^{-1}$| for all three multipoles, (MHQ 0.15). This option shows no detected systematic on α_∥ and |$1{{\ \rm per\ cent}}$| systematic on α_⊥ and fσ₈. Alternatively, we could also have considered to use only the monopole and quadrupole at the same truncation scale (MQ 0.15), which shows no significant systematic in any of the three variables. However, not using the hexadecapole significantly increases the statistical errors (see Fig. 17), which does not compensate for the reduction of systematic errors. Therefore, our main analysis relies on the choice MQH 0.15.

We have not shown any results with k_max below |$0.15\, \, h\, {\rm Mpc}^{-1}$|⁠. The reason is that doing this actually also introduces systematics. This paradoxical effect is caused by worsening the BAO detection. If k_max is reduced down to |$0.10\, \, h\, {\rm Mpc}^{-1}$|⁠, although the power spectrum and RSD behave closer to linear physics (which is better modelled), the lost BAO information between |$0.10$| and |$0.15\, \, h\, {\rm Mpc}^{-1}$| makes it hard to distinguish the RSD signal from the AP signal, which ends up introducing very long degeneracy tails between fσ₈, α_∥, and α_⊥, which in the end introduce other type of systematics. This effect even appears in a BAO-only analysis, suggesting that there is an intrinsic effect of the data vector not related to the specific model, BAO or FS, used. On the other hand, we do not explore scales above |$k_{\rm min}=0.02\, \, h\, {\rm Mpc}^{-1}$|⁠, as they are usually more contaminated by large-scale systematics, and they barely contain extra information on α_∥, α_⊥, and fσ₈.

In Beutler et al. (2017) DR12 CMASS LRG galaxies, and in particular those between 0.5 ≤ z ≤ 0.75 were analysed under the truncation scheme ‘MQ 0.15 H 0.10’ using a very similar model. In this work, we have not found significant differences in terms of systematics between ‘MQ 0.15 H 0.10’ and ‘MQH 0.15’, and therefore we have decided to also include those hexadecapole modes between |$0.10\le k\, [\, h\, {\rm Mpc}^{-1}]\le 0.15$|⁠.

5.2.2 Performance of the RSD modelling

We apply the FS template pipeline described in Section 3.2 for |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}] \le 0.15$| on the different set of mocks: the results are displayed in Fig. 14 for EZmocks, Nseries, and OuterRim-HOD mocks (for threshold2 flavour only and with padding included), as labelled, when the monopole and quadrupole are used (MQ) and when the hexadecapole is also used (MQH). The green and orange symbols display the results when fitting the mean of the mocks and report the error of the mean. The pink and purple symbols display the average of the individual fits (displayed in grey) and report the rms in the error bars. Consequently, the error on the mean of the fits is a factor |$\sqrt{N_{i}}$| larger than the error on the fit of the mean. We stress that the error bars associated to both measurements should be the same, as they are both inferred from the same volume (except for large-scale modes and cases where the BAO is not detected), although the latter may suffer from extra systematic effects if the individual fits do not have a sufficiently high signal-to-noise ratio. In Fig. 14 (as in Fig. 9), we opt to display the rms for the mean of the individual fits as an error bar, along with the individual fits in grey, for visualization purposes. These results are also presented in Table 6, using the same notation and format of Table 5, where the error reported for the mean of individual fits is the rms divided by |$\sqrt{N_{\rm det}}$|⁠, matching the error of the mean. The results displayed in Fig. 14 for the EZmocks and Nseries present a good agreement between the fit of the mean and the mean of individual fits. For the Nseries mocks α_∥ and fσ₈ are recovered with no systematics, for both MQ and MQH. For the Nseries mocks, we are able to recover the expected α_⊥ when the monopole and quadrupole are used, whereas when we add the hexadecapole there is a systematic offset of |$\sim 1{{\ \rm per\ cent}}$|⁠, as was reported in the previous section and in Fig. 13. As we have already mentioned, the EZmocks are not full N-body mocks, and therefore they should not be used to validate our pipeline in terms of systematics of the model. However, as a general trend, we observe that the response these mocks have is very similar to the one observed for the Nseries, which serves as a validation of these mocks reproducing the clustering properties of full N-body mocks, and therefore for producing a reliable covariance.

Figure 14.

Performance of the FS type of analyses on different type of mocks, with a similar notation as in Fig. 9. In this case, we analyse only pre-reconstruction catalogues using monopole and quadrupole only (MQ) and also including the hexadecapole (MQH). In all cases, the range of scales used are |$0.02\le k\, [\, h\, {\rm Mpc}^{-1}]\le 0.15$|⁠, motivated the findings of Fig. 13. The corresponding numerical results for this figure can be found in Table 6.

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