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 correlation function between redshifts 0.6 and 1 Free

ABSTRACT

1 INTRODUCTION

The large-scale structure (LSS) in the late Universe is a fundamental probe of the cosmological model, sensitive to both universal expansion and structure growth, and complementary to early Universe observations from the cosmic microwave background. The LSS can be mapped by large redshift surveys through systematic measurements of the three-dimensional positions of matter tracers such as galaxies or quasars. Because the observed LSS is the result of the growth of initial matter perturbations through gravity in an expanding universe, it gives the possibility of both testing the expansion and structure growth histories,which in turn puts us in a unique position to solve the question of the origin of the late acceleration of the expansion and dark energy (Clifton et al. 2012; Weinberg et al. 2013; Zhai et al. 2017b; Ferreira 2019).

Over the last two decades, redshift surveys have explored increasingly larger volumes of the Universe at different cosmic times. The methodology to extract the cosmological information from those redshift surveys has evolved and has now reached maturity. Particularly, the baryon acoustic oscillations (BAOs) and the redshift-space distortions (RSDs) in the two-point and three-point statistics of the galaxy spatial distribution are now key observables to constrain cosmological models. The BAO horizon scale imprinted in the matter distribution was frozen in the LSS at the drag epoch, slightly after matter–radiation decoupling. This characteristic scale can still be seen in the large-scale distribution of galaxies at late times and be used as a standard ruler to measure the expansion history. At the same time, the galaxy peculiar velocities distorting the line-of-sight cosmological distances based on observed redshifts are sensitive on large scales to the coherent motions induced by the growth rate of structure, which in turn depends on the strength of gravity. BAO and RSD are highly complementary, as they allow both geometrical and dynamical cosmological constraints from the same observations.

The signature of baryons in the clustering of galaxies was first detected in the Sloan Digital Sky Survey (SDSS; Eisenstein et al. 2005) and 2dF Galaxy Redshift Survey (2dFGRS; Percival et al. 2001; Cole et al. 2005). Since then, further measurements using the 2dFGRS, SDSS, and additional surveys have improved the accuracy of BAO measurements and extended the range of redshifts covered from |$z$| = 0 to 1. Examples of analyses include those of the SDSS-II (Percival et al. 2010), 6dFGS (Beutler et al. 2011), WiggleZ (Kazin et al. 2014), and SDSS-MGS (Ross et al. 2015a) galaxy surveys. An important milestone was achieved with the Baryon Oscillation Spectroscopic Survey (BOSS; Dawson et al. 2013), part of the third generation of the SDSS (Eisenstein et al. 2011). This allowed the most precise measurements of BAO using galaxies achieved to date using galaxies as direct tracers (Alam et al. 2017) and Lyman-α forest measurements (Bautista et al. 2017; du Mas des Bourboux et al. 2017), reaching a relative precision of 1 per cent on the distance relative to the sound horizon at the drag epoch.

Although RSDs have been understood and measured since the late 1980s (Kaiser 1987), it is only in the last decade, when there has been significant interest in deviations from standard gravity that would explain the apparent late-time acceleration of the expansion of the Universe, that the ability of RSD measurements to provide such tests has been explored (Guzzo et al. 2008; Song & Percival 2009). This has resulted in renewed interest in RSD with examples of RSD measurement from the WiggleZ (Blake et al. 2011), 6dFGRS (Beutler et al. 2012), SDSS-II (Samushia, Percival & Raccanelli 2012), SDSS-MGS (Howlett et al. 2015), FastSound (Okumura et al. 2016), and VIPERS (Pezzotta et al. 2017) galaxy surveys, with BOSS achieving the best precision of ∼6 per cent on the parameter combination fσ₈ (Beutler et al. 2017; Grieb et al. 2017; Sánchez et al. 2017; Satpathy et al. 2017), which is commonly used to quantify the amplitude of the velocity power spectrum.

The extended Baryon Oscillation Spectroscopic Survey (eBOSS; Dawson et al. 2016) programme is the successor of BOSS in the fourth generation of the SDSS (Blanton et al. 2017). It maps the LSS using four main tracers: luminous red galaxies (LRGs), emission line galaxies (ELGs), quasars used as direct tracers of the density field, and quasars from whose spectra we can measure the Ly α forest. With respect to BOSS, it explores galaxies at higher redshifts, covering the range 0.6 < |$z$| < 2.2. Using the first 2 yr of data from Data Release 14 (DR14), BAO and RSD measurements have been performed using different tracers and methods: LRG BAO (Bautista et al. 2018), LRG RSD (Icaza-Lizaola et al. 2020), quasar BAO (Ata et al. 2018), quasar BAO with redshift weights (Zhu et al. 2018), quasar BAO Fourier space (Wang et al. 2018), quasar RSD Fourier space (Gil-Marín et al. 2018), quasar RSD Fourier space with redshift weights (Ruggeri et al. 2017, 2019), quasar RSD in configuration space (Hou et al. 2018; Zarrouk et al. 2018), and quasar tomographic RSD in Fourier space with redshift weights (Zhao et al. 2019).

In this paper, we perform the BAO and RSD analyses in configuration space of the completed eBOSS LRG sample, part of DR16. This work is part of a series of papers using different tracers and methods.¹ The official SDSS-IV DR16 quasar catalogue is described in Lyke et al. (2020). The production of the catalogues specific for large-scale clustering measurements of the quasar and LRG sample (input for this work) is described in Ross et al. (2020), while the analogous work for the ELG sample is described in Raichoor et al. (2020). From the same LRG catalogue, Gil-Marín et al. (2020) report the BAO and RSD analyses in Fourier space. The BAO and RSD constraints from the quasar sample are presented by Hou et al. (2020) in configuration space and by Neveux & Burtin (2020) in Fourier space. The clustering from the ELG sample is described by de Mattia et al. (2020) in Fourier space and by Amelie et al. (2020) in configuration space. Finally, a series of articles describes the simulations used to test the different methodologies for each tracer. The approximate mocks used to estimate covariance matrices and assess observational systematics for the LRG, ELG, and quasar samples are described in Zhao et al. (2020) (see also Lin et al. 2020, for an alternative method for ELGs), while realistic N-body simulations were produced by Rossi et al. (2020) for the LRG sample, by Smith et al. (2020) for the quasar sample, and by Alam et al. (2020) for the ELG sample. In Ávila et al. (2020), halo occupation models for ELGs are studied. A machine-learning method to remove systematics caused by photometry was applied to the ELG sample (Kong et al. 2020) and a new method to account for fibre collisions in the eBOSS sample is described in Mohammad et al. (2020). The BAO analysis of the Lyman-α forest sample is presented by du Mas des Bourboux et al. (2020). The final cosmological implications from all these clustering analyses are presented in eBOSS Collaboration (2020).

The paper is organized as follows: Section 2 describes the LRG data set and simulations used in this analysis. Section 3 presents the adopted methodology and particularly BAO and RSD theoretical models. We estimate biases and systematic errors from different sources in Section 4. We present BAO and RSD results in Section 5 and finally conclude in Section 6.

2 DATA SET

In this section, we summarize the observations, catalogues, and mock data sets that are used to test our methodology, as well as the clustering statistics used in this work.

2.1 Spectroscopic observations and reductions

The fourth generation of the SDSS (SDSS-IV; Blanton et al. 2017) employed the two multi-object BOSS spectrographs (Smee et al. 2013) installed on the 2.5-m telescope (Gunn et al. 2006) at the Apache Point Observatory in New Mexico, USA, to carry out spectroscopic observations for eBOSS. The target sample of LRGs, the analysis of which is our focus, was selected from the optical SDSS photometry from DR13 (Albareti et al. 2017), with additional infrared information from the Wide-field Infrared Survey Explorer satellite (Lang, Hogg & Schlegel 2014). The final targeting algorithm is described in detail in Prakash et al. (2016) and produced about 60 deg⁻² LRG targets over the 7500 deg² of the eBOSS footprint, of which 50 deg⁻² were observed spectroscopically. The selection was tested over 466 deg² covered during the Sloan Extended Quasar, ELG, and LRG Survey (SEQUELS), confirming that more than 41 deg⁻² LRGs have 0.6 < |$z$| < 1.0 (Dawson et al. 2016).

The raw CCD images were converted to one-dimensional, wavelength and flux calibrated spectra using version v5_13_0 of the SDSS spectroscopic pipeline idlspec2d.² Two main improvements of this pipeline since its previous release (DR14; Abolfathi et al. 2018) include a new library of stellar templates for flux calibration and a more stable extraction procedure. Ahumada et al. (2020) provide a summary of all improvements of the spectroscopic pipeline since SDSS-III.

The redshift of each LRG was estimated with the redrock algorithm.³ This algorithm improves classification rates with respect to its predecessor redmonster (Hutchinson et al. 2016). redrock uses templates derived from principal component analysis of the SDSS data to classify spectra, which is followed by a redshift refinement procedure that uses stellar population models for galaxies. On average, 96.5 per cent of spectra yield a confident redshift estimate with redrock compared to 90 per cent with redmonster, with less than 1 per cent of catastrophic redshift errors (details can be found in Ross et al. 2020).

2.2 Survey geometry and observational features

The full procedure to model the survey geometry and correct for observational features is described in detail in the companion paper Ross et al. (2020). We summarize it in the following.

The random catalogue allows estimating the survey geometry and number density of galaxies in the observed sample. It contains a random population of objects with the same radial and angular selection functions as the data. A random uniform sample of points is drawn over the angular footprint of eBOSS targets to model its geometry. We use random samples with 50 times more objects than in the data to minimize the shot noise contribution in the estimated correlation function, and redshifts are randomly taken from galaxy redshifts in the data. A series of masks are then applied to both data and random samples in order to eliminate regions with bad photometric properties, targets that collide with quasar spectra (which had priority in fibre assignment), and the central region of the plates where it is physically impossible to put a fibre. All masks combined cover 17 per cent of the initial footprint, with the quasar collision mask accounting for 11 per cent. The spectroscopic information is finally matched to the remaining targets.

About 4 per cent of the LRG targets were not observed due to fibre collisions; i.e. when a group of two or more galaxies are closer than 62 arcsec, they cannot all receive a fibre. On regions of the sky observed more than once, some collisions could be resolved. These collisions can bias the clustering measurements, so we applied the following correction: N_targ objects in a given collision group for which N_spec have a spectrum, all objects are up-weighted by |$w$|_cp = N_targ/N_spec. This is different compared to Bautista et al. (2018), where the weight of the collided object without spectrum was transferred to its nearest neighbour with valid spectrum. Both corrections are only approximations valid on scales larger than 62 arcsec. An unbiased correction method is described in Bianchi & Percival (2017) and applied to eBOSS samples in Mohammad et al. (2020). We show in Appendix  B that our results are insensitive to the correction method since it affects mostly the smallest scales.

A similar procedure as in Bautista et al. (2018) was used to account for the 3.5 per cent of LRG targets without reliable redshift estimate. The redshift-failure weight |$w$|_noz acts as an inverse probability weight, boosting galaxies with good redshifts such that this weighted sample is an unbiased sampling of the full population. This assumes that the probability of a given galaxy being selected is a function of both its trace position on the CCD and the overall signal-to-noise ratio of the spectrograph in which this target was observed, and that the galaxies not observed are statistically equivalent to the observed galaxies. Spurious fluctuations in the target selection caused by the photometry are corrected by weighting each galaxy by |$w$|_sys. These weights are computed with a multilinear regression on the observed relations between the angular overdensities of galaxies versus stellar density, seeing, and galactic extinction. Fitting all quantities simultaneously automatically accounts for their correlations. The weights |$w$|_noz and |$w$|_sys are computed independently.

The observational completeness creates artificial angular variations of the density that are accounted for using the random catalogue. The completeness is defined as the ratio of the number of weighted spectra (including those classified as stars or quasars) to the number of targets (equation 11 in Ross et al. 2020). This quantity is computed per sky sector, i.e. a connected region observed by a unique set of plates. We downweight each point in the random catalogue by the completeness of its corresponding sky sector.

Optimal weights for large-scale correlations, known as FKP weights (Feldman, Kaiser & Peacock 1994), are computed with the estimated comoving density of tracers |$\bar{n}(z)$| as a function of redshift using our fiducial cosmology in Table 1. The final weight for each galaxy is defined⁴ as |$w$| = |$w$|_noz|$w$|_cp|$w$|_syst|$w$|_FKP. The weight for each galaxy from the random catalogue is the same, with the completeness information already included in |$w$|_sys.

The eBOSS sample of LRGs overlaps in area and redshift range with the highest redshift bin of the CMASS sample (0.5 < |$z$| < 0.75). We combine the eBOSS LRG sample with all the |$z$| > 0.6 BOSS CMASS galaxies and their corresponding random catalogue (including the non-overlapping with eBOSS), making sure that the data-to-random number ratio is the same for both samples. This combination is beneficial for two reasons. First, the combined sample supersedes the last redshift bin of BOSS measurements while being completely independent of the first two lower redshift bins. Secondly, the reconstruction technique applied to this sample (see the next section) benefits from a higher density of tracers, reducing potential noise introduced by the procedure. The new eBOSS LRG sample covers 4242 deg² of the total BOSS CMASS footprint of 9494 deg² (NGC and SGC combined). Considering their spectroscopic weights, the new eBOSS sample has 185 295 new redshifts over 0.6 < |$z$| < 1.0 while CMASS contributes with 104 865 redshifts in the overlapping area and 111 892 in the non-overlapping area. A total of 402 052 LRGs over 0.6 < |$z$| < 1.0 contribute to this measurement, with a total effective comoving volume of 2.72 Gpc³ (1.43 Gpc³ from the CMASS sample and 1.28 Gpc³ from the new eBOSS sample). A detailed description of these numbers is given in Ross et al. (2020). In the following, we simply refer to the combined CMASS+LRG sample as the eBOSS LRG sample. The number density of CMASS galaxies, LRGs, and combined CMASS+LRG sample are presented in Fig. 1.

Figure 1.

The observed number density of eBOSS LRGs (dashed curve), BOSS CMASS galaxies (dotted curve), and combined CMASS+LRG sample galaxies (solid curve) at 0.6 < |$z$| < 1. This combines NGC and SGC fields.

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2.3 Reconstruction

While constraints on the growth rate of structure are obtained using the information from the full shape of the correlation function, BAO analyses extract the cosmological information only from the position of the BAO peak. In our BAO analysis, we applied the reconstruction technique of Burden et al. (2014) and Burden, Percival & Howlett (2015) to the observed galaxy density field in order to remove a fraction of the RSDs, as well as non-linear motions of galaxies that smeared out the BAO peak. This technique sharpens the BAO feature in the two-point statistics in Fourier and configuration space, increasing the precision of the measurement of the acoustic scale. Reconstruction is applied on actual data and on mock catalogues using a publicly available⁵ code (Bautista et al. 2018). Our final BAO results are solely based on reconstructed catalogues, while full-shape results use the pre-reconstruction sample.

We apply reconstruction to the full eBOSS+CMASS final LRG catalogue. We use our fiducial cosmology from Table 1 to convert redshifts to comoving distances. For the reconstruction, we fix the bias value to b = 2.3 and assume the standard gravity relation between the growth rate of structure and Ω_m, i.e. |$f = \Omega _\mathrm{ m}^{6/11}(z=0.7) = 0.815$|⁠. We use a smoothing scale of 15 h⁻¹ Mpc. The BAO results are not sensitive to small variations of those parameter choices as studied in Carter et al. (2020).

2.4 Mocks

In order to test the overall methodology and study the impact of systematic effects, we have constructed several sets of mock samples. Approximate methods are considered to be sufficient for covariance matrix estimates and to derive systematic biases in BAO measurements. However, the full-shape analysis of the correlation function requires more realistic N-body simulations, particularly in order to test the modelling. In this study, our synthetic data sets are the following:

• 1000 realizations of the LRG eBOSS+CMASS survey geometry using the EZmock method (Chuang et al. 2015), which employs the Zel’dovich approximation to compute the density field at a given redshift and populates it with galaxies. This method is fast and has been calibrated to reproduce the two- and three-point statistics of the given galaxy sample, to a good approximation and up to mildly non-linear scales. The angular and redshift distributions of the eBOSS LRG sample in combination with the |$z$| > 0.6 CMASS sample were reproduced in these mock catalogues. The full description of the EZmock LRG samples can be found in the companion paper Zhao et al. (2020). We use these mocks in several steps of our analysis: to infer the error covariance matrix of our clustering measurements in the data, to study the impact of observational systematic effects on cosmology, and to estimate the correlations between different methods for the calculation of the consensus results.

• 84 realizations of the Nseries mocks, which are N-body simulation snapshots populated with a single halo occupation distribution (HOD) model. These mock catalogues reproduce the angular and redshift distributions of the North Galactic Cap of the BOSS CMASS sample within the redshift range 0.43 < |$z$| < 0.70 (Alam et al. 2017). While this data set is not fully representative of the eBOSS LRG sample, we use these N-body mocks to test the RSD models down to the non-linear regime. The number of available realizations and their large volume are ideal to test model accuracy in the high-precision regime. The covariance matrix for these mocks is computed from 2048 realizations of the same volume with the MD-Patchy approximated method (Kitaura, Yepes & Prada 2014). The redshift of those mocks is |$z$| = 0.55.

• 27 realizations extracted from the OuterRim  N-body simulation (Heitmann et al. 2019), and corresponding to cubical mocks of 1 h⁻³ Gpc³ each. The dark matter haloes have been populated with galaxies using four different HODs (Zheng, Coil & Zehavi 2007; Leauthaud et al. 2011; Tinker et al. 2013; Hearin, Watson & van den Bosch 2015) at three different luminosity thresholds to cover a large range of galaxy populations. These mocks are part of our internal MockChallenge and aimed at quantifying potential systematic errors originating from the HODs. A detailed description of these simulations and the MockChallenge can be found in the companion paper Rossi et al. (2020). The redshift of those mocks is |$z$| = 0.695.

2.5 Fiducial cosmologies

The redshift of each galaxy is converted into radial comoving distances for clustering measurements by means of a fiducial cosmology. The fiducial cosmologies employed in this work are shown in Table 1. Our baseline choice, named ‘Base’, is a flat Lambda cold dark matter (ΛCDM) model matching the cosmology used in previous BOSS analyses (Alam et al. 2017) with parameters within 1σ of Planck best-fitting parameters (Planck Collaboration I 2018a). Some of these cosmologies were used to produce the mock data sets described in Section 2.4. A choice of fiducial cosmology is also needed when computing the linear power spectrum P_lin(k), input for all our correlation function models in this work (see Sections 3.1 and 3.2). In Sections 4.1 and 4.2, we study the dependence of our results on the choice of fiducial cosmology.

2.6 Galaxy clustering estimation

The measured anisotropic correlation function, where the galaxy separation vector |$\boldsymbol{r}$| has been decomposed into line-of-sight and transverse separations (r_⊥, r_∥), is presented in the left-hand panel of Fig. 2. A clear BAO feature is seen at r ≈ 100 h⁻¹ Mpc as well as the impact of RSD, which squashes the contours along the line of sight on large scales. In the right-hand panel of Fig. 2, we show the post-reconstruction correlation function where some of the isotropy is recovered and the BAO feature is sharpened.

Figure 2.

Anisotropic two-point correlation function of eBOSS LRG+CMASS galaxies at 0.6 < |$z$| < 1. The left-hand (right) panel shows the pre-reconstruction (post-reconstruction) two-point correlation function in bins of r_⊥ and r_∥. Bins of size 1.25 h⁻¹ Mpc and a bi-cubic spline interpolation have been used to produce the contours.

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The red points with error bars in Fig. 3 show the even multipoles of the correlation function from the eBOSS LRG sample. The solid, dashed, and dotted black curves display the average multipoles in the different mock data sets used in this study: EZmocks, Nseries, and MockChallenge. The error bars are obtained from the dispersion of the 1000 EZmocks multipoles around their mean. By construction, the amplitude of the EZmock multipoles matches the data at separations s < 70 h⁻¹ Mpc. A slight mismatch in the BAO peak amplitudes between data and EZmocks is visible. This mismatch does not impact cosmological results from the data since the covariance matrix dependence on the peak amplitude is small. However, the comparison of the precision of BAO peak measurements between mocks and data needs to account for this mismatch: The expected errors of our BAO measurement are smaller for data than those for the ensemble of EZmocks. For comparison, the average multipoles of the Nseries mocks, also shown in Fig. 3, are a better match to the peak amplitude seen in the data.

Figure 3.

Multipoles of the correlation function of data compared to the mock catalogues. The data are the combined eBOSS LRG+CMASS (NGC+SGC) samples and the mocks are the average multipoles of 1000 EZmocks realizations (solid line), 84 Nseries realizations (dashed line), and 27 MockChallenge mocks populated with L11 HOD model (dotted lines). The top panels show the monopole, quadrupole, and hexadecapole of the pre-reconstruction samples while the bottom panels show the same for the post-reconstruction case.

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3 METHODOLOGY

In this section, we describe the BAO and RSD modelling, fitting procedure, and how errors on cosmological parameters are estimated.

3.1 BAO modelling

We employ the standard approach used in previous SDSS publications for measuring the BAO scale in configuration space (e.g. Anderson et al. 2014; Alam et al. 2017; Ross et al. 2017; Bautista et al. 2018). The code that produces the model and performs the fitting to the data is publicly available.⁶

We follow the procedure from Kirkby et al. (2013) to decompose the BAO peak component P_peak from the linear power spectrum P_lin. We start by computing the correlation function by Fourier transforming P_lin, and then we replace the correlations over the peak region by a polynomial function fitted using information outside the peak region (50 < r < 80 and 160 < r < 190 h⁻¹ Mpc). The resulting correlation function is then Fourier transformed back to get P_no peak. The linear power spectrum P_lin is computed using the code camb  ⁷ (Lewis, Challinor & Lasenby 2000) with cosmological parameters of our fiducial cosmology (Table 1). The analysis in Fourier space uses the same procedure (see Gil-Marín et al. 2020). Previous BOSS and eBOSS analyses making BAO measurements from direct tracer galaxies used the approximate formulae from Eisenstein, Hu & Tegmark (1998) for decomposing the peak. We have checked that both methods yield only negligibly different results.

Our baseline BAO analysis uses the monopole ξ₀ and the quadrupole ξ₂ of the correlation function. We performed fits on mock multipoles including the hexadecapole ξ₄, finding that it does not add information (see Table 6). We fix β = 0.35 and fit b with a flat prior between b = 1.0 and 4. For all fits, the broad-band parameters are free, while both dilation parameters are allowed to vary between 0.5 and 1.5. A total of nine parameters are fitted simultaneously.

3.2 RSD modelling

We describe the apparent distortions introduced by galaxy peculiar velocities in the redshift-space galaxy clustering pattern using two different analytical models: the combined Gaussian streaming (GS) and Convolutional Lagrangian Perturbation Theory (CLPT) formalism developed by Reid & White (2011), Carlson, Reid & White (2013), Wang, Reid & White (2014), and the Taruya, Nishimichi & Saito (2010) model (TNS) supplemented with a non-linear galaxy bias prescription. These two models, frequently used in the literature, partially account for RSD non-linearities and describe the anisotropic clustering down to the quasi-linear regime. We use both models to fit the multipoles of the correlation function and later combine their results to provide more robust estimates of the growth rate of structure and geometrical parameters. This procedure should reduce the residual theoretical systematic errors. In this section, we briefly describe the two models and assess in Section 4.2 their performance in the recovery of unbiased cosmological parameters using mock data sets.

3.2.1 CLPT with GS

The kernels |$M_{1,2}(\boldsymbol{r}, \boldsymbol{q})$| also depend on the first two non-local derivatives of the Lagrangian bias 〈F^′〉 and 〈F^″〉, which are free parameters in addition to the linear growth rate f in our model. Hereafter, we eliminate the angle brackets around the Lagrangian bias terms to simplify the notation in the following sections.

Although CLPT is more accurate than Lagrangian Resummation Theory from Matsubara (2008) in real space, we still have to improve the small-scale modelling in order to study RSDs. This is particularly important considering that part of peculiar velocities is generated by interactions that occur at the typical scales of clusters of galaxies (∼1 Mpc). This is achieved by mapping the real-space CLPT model of the two-point statistics into redshift space with the GS model proposed by Reid & White (2011). The pairwise velocity distribution of tracers is assumed to have a Gaussian distribution that depends on both the separation r and the angle between the separation vector and the line of sight μ.

3.2.2 TNS model

The input linear power spectrum P_lin is obtained with camb, while the non-linear power spectrum P_δδ is calculated from the respresso code (Nishimichi, Bernardeau & Taruya 2017). This non-linear power spectrum prediction does agree very well with successful perturbation theory-based predictions such as RegPT, but extend their validity to k ≃ 0.4 (Nishimichi et al. 2017). This is very relevant for configuration-space analysis, where one needs to have both a correct BAO amplitude and a non-vanishing signal at high k to avoid aliasing in the transformation from Fourier to configuration space.

In total, this model has either four or five free parameters, [f, b₁, b₂, σ_|$v$|] or |$[f,b_1,b_2,b_{\Gamma _{3}},\sigma _v]$|⁠, depending on the number of bias parameters that are let free. Finally, the multipole moments of the anisotropic correlation function are obtained by performing the Hankel transform of the model |$P_\ell ^s(k)$|⁠.

3.2.3 Alcock–Paczynski effect

3.2.4 The fiducial scale at which σ₈ is measured

3.3 Parameter inference

In the BAO analysis, the best-fitting parameters (α_⊥, α_∥) are found by minimizing |$-2\ln \mathcal {L} = \chi ^2$| using a quasi-Newton minimum finder algorithm iMinuit.⁹ The errors in α_∥ and α_⊥ are found by computing the intervals where χ² increases by unity. Gaussianity is not assumed in the error calculation, but we find that on average, errors are symmetric and correctly described by a Gaussian. The 2D errors in (α_⊥, α_∥), such as those presented in Fig. 13, are found by scanning χ² values in a regular grid in α_⊥ and α_∥. In the case of the full-shape analysis, we explore the likelihood with the Markov chain Monte Carlo (MCMC) ensemble sampler emcee.¹⁰ The input power spectrum shape parameters are fixed at the fiducial cosmology and any deviations are accounted for through the Alcock–Paczynski parameters α_⊥ and α_∥. We assume the uniform priors on model parameters given in Table 3.

The final parameter constraints are obtained by marginalizing the full posterior likelihood over the nuisance parameters. The marginal posterior is approximated by a multivariate Gaussian distribution with central values given by best-fitting parameter values θ* = (α_⊥, α_∥, fσ₈) and parameter covariance matrix C_θ. Since the covariance matrix is computed from a finite number of mock realizations, we need to apply correction factors to the obtained C_θ. These factors are equations (18) and (22) from Percival et al. (2014) to be applied to uncertainties and to the scatter over best-fitting values, respectively. These factors, which depend on the number of mocks, parameters, and bins in the data vectors, are presented in Table 4. The final parameter constraints from this work are available to the public in this format.¹¹

3.4 Combining BAO and RSD constraints

From the same input LRG catalogue, we produced BAO-only and full-shape RSD constraints, both in configuration and Fourier space (Gil-Marín et al. 2020). Each measurement yields a marginal posterior on (α_⊥, α_∥) for BAO-only or (α_⊥, α_∥, fσ₈) for the full-shape RSD analyses. In the following, we describe the procedure to combine all these posteriors into a single consensus constraint, while correctly accounting for their covariances. This consensus result is the one used for the final cosmological constraints described in eBOSS Collaboration (2020).

We use the method described above to perform all the constraint combinations, except for the combination of results from CLPT-GS and TNS RSD models, which use the same input data vector (pre-reconstruction multipoles in configuration space). For this particular combination, we simply assume that |$C_c^{-1} = 0.5(C_{mm}^{-1}+C_{nn}^{-1})$| and |$x_c = 2 C_c^{-1}\left(C_{mm}^{-1} x_{m} + C_{nn}^{-1} x_{n}\right)$|⁠. For all combinations, we chose to use the results from at most two methods at once (M = 2) in order to reduce the potential noise introduced by the procedure.

Denoting ξ_ℓ the results from the configuration-space analysis and P_ℓ that from the Fourier-space analysis, our recipe to obtain the consensus result for the LRG sample is as follows:

• Combine RSD ξ_ℓ TNS and RSD ξ_ℓ CLPT-GS results into RSD ξ_ℓ,

• Combine BAO ξ_ℓ with BAO P_ℓ into BAO (ξ_ℓ + P_ℓ),

• Combine RSD ξ_ℓ with RSD P_ℓ into RSD (ξ_ℓ + P_ℓ),

• Combine BAO (ξ_ℓ + P_ℓ) with RSD (ξ_ℓ + P_ℓ) into BAO+RSD (ξ_ℓ + P_ℓ).

Alternatively, we can proceed as

• Combine BAO ξ_ℓ with RSD ξ_ℓ into (BAO+RSD) ξ_ℓ,

• Combine BAO P_ℓ with RSD P_ℓ into (BAO+RSD) P_ℓ,

• Combine BAO+RSD ξ_ℓ with BAO+RSD P_ℓ into (BAO+RSD) ξ_ℓ + P_ℓ.

In Section 4.3, we test this procedure on the mock catalogues.

4 ROBUSTNESS OF THE ANALYSIS AND SYSTEMATIC ERRORS

4.1 Systematics in the BAO analysis

The methodology described in Section 3.1 was tested using the 1000 EZmocks mock survey realizations and 84 Nseries realizations. For each realization, we compute the correlation function and its multipoles, and fit for the BAO peak position to determine the dilation parameters α_∥, α_⊥ and associated errors. We compare the best-fitting α_⊥, α_∥ to their expected values, which are obtained from the cosmological models described in Table 1. The effective redshift of the EZmocks is |$z$|_eff = 0.698 and |$z$|_eff = 0.56 for Nseries.

In Fig. 4, we summarize the systematic biases from pre- and post-reconstruction mocks for a few choices of fiducial cosmology, parametrized by |$\Omega _\mathrm{ m}^{\rm fid}$|⁠. In pre-reconstruction mocks, biases in the recovered α values reach up to 0.5 per cent in α_⊥ and 1.0 per cent in α_∥. These biases are expected due to the impact of non-linear effects on the position of the peak that cannot be correctly accounted for with the Gaussian damping terms in equation (4) at this level of precision (Seo et al. 2016). We recall that we are fitting the average of all realizations. The reconstruction procedure removes in part the non-linear effects and this is seen as a reduction of the biases to less than 0.2 per cent. The bias reduction is also seen in the Nseries mocks, particularly on α_⊥, confirming that the bias reduction is not related to a feature of the mocks induced by the approximate method used to build them.

Figure 4.

Impact of choice of fiducial cosmology in the recovered values of α_∥ and α_⊥ from the stacks of 1000 multipoles from the EZmocks (blue) and 84 Nseries mocks (orange), for pre- (top panels) and post-reconstruction (bottom panels). Associated error bars correspond to the error on the mean of the mocks. The grey shaded areas correspond to 1 per cent errors. For comparison, the error on real data is near 1.9 per cent for α_⊥ and 2.6 per cent for α_∥ in the post-reconstruction case.

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Table 5 shows results from Fig. 4 for the post-reconstruction case only, including the fits with the hexadecapole ξ_{ℓ = 4}. The impact of the hexadecapole is negligible even in this very low-noise regime, for both types of mocks. The reported dilation parameters for almost all cases are consistent with expected value within 2σ. We see a 2.6σ deviation on α_⊥ for the Nseries case analysed with |$\Omega _\mathrm{ m}^{\rm fid} = 0.35$|⁠. However, this choice of |$\Omega _\mathrm{ m}^{\rm fid}$| is the most distant from the true value of the simulation and its observed bias is still less than half a per cent, which is small compared to the statistical power of our sample. For the EZmocks, which have smaller errors, the biases are up to 0.13 per cent for α_⊥ and 0.18 per cent for α_∥. These biases are much smaller than the expected statistical errors in our data, i.e. ∼1.9 per cent for α_⊥ and ∼2.6 per cent for α_∥, showing that our methodology is robust at this statistical level. In these fits, all parameters except Σ_rec = 15 h⁻¹ Mpc were left free. The best-fitting values of Σ_⊥, Σ_∥, and Σ_s were used and held fixed in the fits of individual realizations.

Fig. 5 displays the distribution of recovered α_⊥, α_∥ and their respective errors measured from each of the individual EZmocks. The error distribution shows that reconstruction improves the constraints on α_⊥ or α_∥ in 94 per cent of the realizations (89 per cent have both errors improved). As expected, realizations with smaller errors generally exhibit larger values of |$\Delta \chi ^2 = \chi ^2_{\rm no \ peak} - \chi ^2_{\rm peak}$|⁠, meaning a more pronounced BAO peak and higher detection significance. We see no particular trend in the best-fitting α values with Δχ² in the two top panels. The red stars in Fig. 5 indicate the values obtained in real data. The error in α_⊥ in the data is typical of what is found in mocks, although for α_∥ it is found at the extreme of the mocks distribution. As discussed in Section 2.4 and displayed in Fig. 3, the BAO peak amplitude in the data multipoles is slightly larger than the one seen in this EZmock sample. A similar behaviour is observed in the eBOSS QSO sample (Hou et al. 2020; Neveux & Burtin 2020), who also use EZmocks from Zhao et al. (2020), and in the BOSS DR12 CMASS sample (see fig. 12 of Ross et al. 2017).

Figure 5.

Distribution of dilation parameters α_⊥ and α_∥ and its estimated errors for pre- and post-reconstruction EZmock catalogues with systematic effects. The colour scale indicates the difference in χ² values between a model with and without BAO peak. The red stars show results with real data. There is a known mismatch in the BAO peak amplitude between data and EZmocks causing the accuracy of the data point to be slightly smaller than the error distribution in the EZmocks (see Section 2.4).

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Table 6 presents a statistical summary of the fits performed on the EZmocks. We tested several changes to our baseline analysis: include the hexadecapole, change the separation range [r_min, r_max], allow BAO damping parameters Σ_⊥ and Σ_∥ to vary within a Gaussian prior (5.5 ± 2 h⁻¹ Mpc), and fit the pre-reconstruction multipoles. We remove realizations with fits that did not converge or with extreme error values (more than 5σ of their distribution, where σ is defined as the half the range covered by 68 per cent of values). The total number of valid realizations is given by N_good in Table 6. In most cases studied, the observed standard deviation of the best-fitting parameters σ(α) is consistent with the average per-mock error estimates 〈σ_α〉, indicating that our errors are correctly estimated. We also see that the dispersion of dilation parameters is not significantly reduced when adding the hexadecapole ξ₄ to the BAO fits, showing that most of the BAO information is contained in the monopole and quadrupole at this level of precision. The mean and dispersion of the pull parameter, defined as Z_α = (α − 〈α〉)/σ_α, are consistent with a unit Gaussian for almost all cases, which further validates our error estimates.

All the tests performed in this section show that our BAO analysis is unbiased and provides correct error estimates. We apply our baseline analysis to the real data and report results in Section 5.1.

4.2 Systematics in the RSD analysis

We present in this section the systematic error budget of the full-shape RSD analysis. Particularly, we discuss the impact of the choice of scales used in the fit, the bias introduced by each model, the bias introduced by varying the fiducial cosmology, the bias associated with the choice of the LRG HOD model, and the impact of observational effects. These are quantified through the analysis of the various sets of mocks with both TNS and CLPT-GS models, which are described in Section 3.2.

4.2.1 Optimal fitting range of scales

We first study the optimal range of scales in the fit for the two RSD models considered in this work (see Section 3). It is worth noting that the optimal range of scales is not necessarily the same for the two models. Generally, full-shape RSD analyses use scales going from tens of h⁻¹ Mpc to about |$130\!-\!150\,$|h⁻¹ Mpc. Including smaller scales potentially increases the precision of the constraints but at the expense of stronger biases on the recovered parameters. This is related to the limitations of current RSD models to fully describe the non-linear regime. On the other hand, including scales larger than ∼130 h⁻¹ Mpc does not significantly improve the precision, since the variations of the model on those scales are small.

In order to determine the optimal range of scales for our RSD models, we performed fits to the mean correlation function of the Nseries mocks, which are those that most accurately predict the expected RSD in the data. Fig. 6 shows the best-fitting values of fσ₈, α_∥, and α_⊥ as a function of the minimum scale used in the fit, r_min. In each panel, the grey bands show 1 per cent errors in α_⊥, α_∥ and 3 per cent errors in fσ₈ for reference. The top panels present the measurements from the TNS model when the parameter b_Γ3 is fixed to the value given by equation (26), while in the mid-panels this parameter is let free. The bottom panels show best-fitting values for the CLPT-GS model as studied in Icaza-Lizaola et al. (2020). As noted in Zarrouk et al. (2018), the hexadecapole is more sensitive to the difference between the true and fiducial cosmologies and is generally less well modelled on small scales compared to the monopole and quadrupole. We therefore consider the possibility of having a different minimum fitting scale for the hexadecapole with respect to the monopole and quadrupole that share the same r_min. For consistency with the other systematic tests, we performed this analysis using two choices of fiducial cosmologies, |$\Omega ^{\rm fid}_\mathrm{ m} = 0.286$| (blue) and |$\Omega ^{\rm fid}_\mathrm{ m} = 0.31$| (red). The maximum separation in all cases is r_max = 130 h⁻¹ Mpc, as we find that using larger r_max has a negligible impact on the recovered parameter values and associated errors.

Figure 6.

Biases in the measurement of fσ₈, α_∥, α_⊥ obtained from full-shape fits to the average of 84 multipoles from the Nseries mocks as a function of the separation range used. The y-axis displays the value of the minimal separation r_min used in fits of the monopole, quadrupole (MQ), and hexadecapole (H). The top and middle rows display results for the TNS model when fixing or letting free the parameter bΓ₃, respectively. The bottom row presents results for the CLPT-GS model. The blue circles correspond to the analysis using |$\Omega _\mathrm{ m}^{\rm fid} = 0.286$| (the true value of simulations) while the red squares correspond to |$\Omega _\mathrm{ m}^{\rm fid} = 0.31$|⁠. The grey shaded areas correspond to 1 per cent errors in α_⊥, α_∥ and to 3 per cent in fσ₈. The green shared area shows our choice for baseline analysis for TNS and CLPT-GS models.

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In the case of the TNS model, we consider two different cases that correspond to when b_Γ3 is fixed to its Lagrangian prediction and when b_Γ3 is allowed to vary. In the case of |$\Omega ^{\rm fid}_\mathrm{ m} = 0.286$| and when b_Γ3 is fixed, in the top panels of Fig. 6, we can see that fσ₈ is overestimated by 1.5 per cent when using scales above 25 h⁻¹ Mpc and by 2 per cent below. Using r_min > 25 h⁻¹ Mpc reduces the bias to about 1 per cent on fσ₈. For α_∥ and α_⊥ parameters, biases range from 0.3 to 0.5 per cent and are all statistically consistent with zero. When b_Γ3 is let free, in the middle panels of Fig. 6, the model provides more robust measurements of fσ₈ at all tested ranges. The biases in fσ₈ over all ranges do not exceed 0.6σ, compared to approx. 2.5σ for the fixed b_Γ3 case. We also remark that letting |$b_{\Gamma _3}$| free also provides a better fit to the BAO amplitude and the hexadecapole on the scales of 20–25 h⁻¹ Mpc. We see a 1 per cent bias on α_∥ when r_min = 20 h⁻¹ Mpc for all three multipoles. This bias is, however, reduced by increasing the hexadecapole minimum scale to r_min = 25 h⁻¹ Mpc. The most optimal configuration for the TNS model is to let bΓ₃ free and fit the monopole and quadrupole in the range 20 ≤ r ≤ 130 h⁻¹ Mpc and the hexadecapole in the range 25 ≤ r ≤ 130 h⁻¹ Mpc, as marked by the green band in Fig. 6. If we use |$\Omega ^{\rm fid}_\mathrm{ m}=0.31$|⁠, the trends and quantitative results are similar to the case with |$\Omega ^{\rm fid}_\mathrm{ m}=0.286$|⁠.

For the CLPT-GS model, an exploration of the optimal fitting range was done in Icaza-Lizaola et al. (2020). Two sets of tests have been performed. The first set consisted of fitting the mean of the mocks when varying r_min and the second, fitting the 84 individual mocks and measuring the bias and variance of the best fits when varying r_min. We revisit the first set of tests, but this time performing a full MCMC analysis to determine best fits and errors. The bottom panels of Fig. 6 summarize the results. In the case of |$\Omega ^{\rm fid}_\mathrm{ m} = 0.286$|⁠, we see that using r_min = 25 h⁻¹ Mpc for all multipoles yields biases of 0.1, 1.1, and 1.6 per cent in α_⊥, α_∥, and fσ₈, respectively. Increasing r_min for the hexadecapole while fixing r_min = 25 h⁻¹ Mpc for the monopole and quadrupole does not change the results significantly; the biases are 0.1 per cent for all ranges in α_⊥, and 1 per cent also for all ranges in α_∥. For fσ₈ variation of 0.1–0.2 per cent arises when varying the range, but this variation in statistically consistent with zero. In the case of |$\Omega ^{\rm fid}_\mathrm{ m} = 0.31$|⁠, we find very similar trends. Using r_min = 25 h⁻¹ Mpc for all multipoles yields biases of 0.2, 0.9, and 1.6 in α_⊥, α_∥, and fσ₈, respectively. When we decrease the range of the fits, the biases on (α_⊥, α_∥, fσ₈) vary by (0.1–0.2, 0.2–0.3, 0.3–0.4) per cent. These variations are not significant and we decide to keep the lowest considered minimum scales on the hexadecapole in the fits.

Compared with previous BOSS full-shape RSD analysis in configuration space, we used for CLPT-GS model the same minimum scale for the monopole and quadrupole (Alam et al. 2017; Satpathy et al. 2017). The hexadecapole was not included in BOSS analyses. The exploration for the optimal minimum scale to be used for the hexadecapole was done in Icaza-Lizaola et al. (2020) and revisited in this work. The systematic error associated with the adopted fitting range is also consistent with previous results for the case where only the monopole and quadrupole are used, as reported in Icaza-Lizaola et al. (2020). The TNS model was not used in configuration space for analysing previous SDSS samples. However, as we describe in Section 4.2.2, the bias associated with both models when using their optimal fitting range is consistent between them, as well as consistent with previous BOSS results.

Overall, these tests performed on the Nseries mocks allow us to define the optimal fitting ranges of scales for both RSD models. Minimizing the bias of the models while keeping r_min as small as possible, we eventually adopt the following optimal ranges:

• TNS model: 20 < r < 130 h⁻¹ Mpc for ξ₀ and ξ₂, and 25 < r < 130 h⁻¹ Mpc for ξ₄,

• CLPT-GS model: 25 < r < 130 h⁻¹ Mpc for all multipoles,

which serve as baseline in the following. We compare the performance of the two models using these ranges in the following sections.

4.2.2 Systematic errors from RSD modelling and adopted fiducial cosmology

We quantify in this section the systematic error introduced by the RSD modelling and the choice of fiducial cosmology. For this, we used the Nseries mocks.¹² The measurements of α_⊥, α_∥, and fσ₈ from fits to the average multipoles are given in Table 7 and shown in Fig. 7. The shaded area in the figure corresponds to 1 per cent deviation for α_⊥, α_∥ expected values and 3 per cent for fσ₈ expected value. We used both TNS (red) and CLPT-GS (blue) models and consider three choices of fiducial cosmologies parametrized by their value of |$\Omega _\mathrm{ m}^{\rm fid}$|⁠. Note that, as for the BAO analysis, we only test flat ΛCDM models close to the most probable one. We expect the full-shape analysis to be biased if the fiducial cosmology is too different from the truth (the parametrization with α_⊥ and α_∥ would not fully account for the distortions and the template power spectrum would differ significantly).

Figure 7.

Biases in best-fitting parameters for both CLPT-GS (blue) and TNS (red) models from fits to the average multipoles of 84 Nseries mocks. Shaded grey areas show the equivalent of 1 per cent error for α_⊥, α_∥ and 3 per cent for fσ₈. In the right-hand panel, crosses indicate fσ₈ values when σ₈ is not recomputed as described in Section 3.2.4. The true cosmology of the mocks is Ω_m = 0.286. For reference, the errors on our data sample are ∼2, 3, and 10 per cent for α_⊥, α_∥, and fσ₈, respectively.

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The biases on the recovered parameters shown in Fig. 7 induced by the choice of fiducial cosmology remain within 1, 1, and 3 per cent for α_⊥, α_∥, and fσ₈, respectively. For α_⊥, both CLPT-GS and TNS models produce biases lower than 2σ for all cosmologies except |$\Omega _\mathrm{ m}^{\rm fid}=0.35$|⁠, which is the most distant value from the true cosmology of the simulation Ω_m = 0.286. For α_∥, all biases are consistent with zero at 2σ level for the TNS model, while CLPT-GS shows biases slightly larger than 2σ for all |$\Omega _\mathrm{ m}^{\rm fid}$|⁠.

4.2.3 Systematic errors from HOD

We quantify in this section the potential systematic errors introduced by the models with respect to how LRGs occupy dark matter haloes. This is done by analysing mock catalogues produced with different HOD models that mimic different underlying galaxy clustering properties. The same input dark matter field is used when varying the HOD model. We use the OuterRim mocks described in Section 2.4 and in Rossi et al. (2020). Specifically, we analysed the mocks constructed using the ‘Threshold 2’ for the HOD models from Leauthaud et al. (2011), Tinker et al. (2013), and Hearin et al. (2015) and performed fits to the average multipoles over the 27 realizations available for each HOD model.

Fig. 8 and Table 8 show the results. In this figure, each best-fitting parameter is compared to the average best fit over all HOD models in order to quantify the relative impact of each HOD (instead of comparing with their true value). The biases with respect to the true values were quantified in the previous section. The shaded regions represent 1 per cent error for α_⊥ and α_∥, and 3 per cent error for fσ₈.

Figure 8.

Best-fitting values of α_⊥, α_∥, and fσ₈ from fitting the average multipoles of the OuterRim mocks compared to their average over all HOD models. Blue points show results for the CLPT-GS model and red points show results for the TNS model. The shaded area shows 1 per cent error for α_⊥, α_∥ and 3 per cent for fσ₈.

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We find that the biases for both RSD models are all within 1σ from the mean, although statistical errors are quite large (around 1 per cent for α_⊥, α_∥) compared to Nseries mocks for instance. Also, the observed shifts are all smaller than the systematic errors estimated in the previous section. If we were to use the same definition for the systematic error introduced in Section 4, the relatively large errors from these measurements would produce a significant contribution to the error budget. Therefore, we consider that HOD has a negligible contribution to the total systematic error budget.

4.2.4 Systematic errors from observational effects

We investigate in this section the observational systematics. We used a set of 100 EZmocks to quantify their impact on our measurements. From the same set, we added different observational effects. For simplicity, those samples were made from mocks reproducing only the eBOSS component of the survey, neglecting the CMASS component. We consider that the systematic errors estimated this way can be extrapolated to the full eBOSS+CMASS sample by assuming that their contribution is the same over the CMASS volume. We thus produced the following samples:

1. no observational effects included, which we use as reference,

2. including the effect of the radial integral constraint (RIC; de Mattia & Ruhlmann-Kleider 2019), where the redshifts of the random catalogue are randomly chosen from the redshifts of the data catalogue,

3. including RIC and all angular features: fibre collisions, redshift failures, and photometric corrections.

For each set, we computed the average multipoles and fitted them using our two RSD models. The covariance matrix is held fixed between cases. Table 9 summarizes the biases in α_⊥, α_∥, and fσ₈ caused by the different observational effects. The shifts are relative to results of mocks without observational effects. We find that the RIC produces the greatest effect, particularly for the CLPT-GS model for which the deviation on fσ₈ is slightly larger than 2σ. Indeed, the quadrupole for mocks with RIC has smaller absolute amplitude, which translates into small fσ₈ values. However, when adding angular observational effects the shifts are all broadly consistent with zero, which indicates that the two effects partially cancel each other.

4.2.5 Total systematic error of the full-shape RSD analysis

Table 10 summarizes all systematic error contributions to the full-shape measurements discussed in the previous sections. We show the results for our two configuration-space RSD models TNS and CLPT-GS and for the Fourier-space analysis of Gil-Marín et al. (2020). We compute the total systematic error σ_syst by summing up all the contributions in quadrature, assuming that they are all independent. By comparing the systematic errors with the statistical error from the baseline fits to the data (see Section 5.2), we find that the systematic errors are far from being negligible: more than 50 per cent of the statistical errors for all parameters. The systematic errors are in quadrature to the diagonal of the covariance of each measurement. We do not attempt to compute the covariance between systematic errors and this approach is more conservative (it does not underestimate errors).

4.3 Statistical properties of the LRG sample

We can also use the EZmocks for evaluating the statistical properties of the LRG sample, in particular to quantify how typical is our data compared with EZmocks, but also for measuring the correlations among the different methods and globally validating our error estimation.

The left-hand panel of the Fig. 9 presents a comparison between the best fit (α_⊥, α_∥, fσ₈) and their estimated errors from fits of the TNS and CLPT-GS models. The confidence contours contain approximately 68 per cent and 95 per cent of the results around the mean. The contours and histograms reveal a good agreement for the two models. Stars indicate the corresponding best-fitting values obtained from the data. The correlations between best-fitting parameters of both models are 86, 83, and 93 per cent for α_⊥, α_∥, and fσ₈, respectively. A similar comparison for the errors is presented in the right-hand panel of the Fig. 9. The errors inferred from the data analysis, shown as stars, are in good agreement with the 2D distributions from the mocks, lying within the 68 per cent contours. The histograms comparing the distributions of errors for both methods also show a good agreement, in particular for α_∥ and fσ₈. For α_⊥, we observe that the distribution from CLPT-GS is slightly peaked towards smaller errors, while for TNS the error distribution has a larger dispersion for this parameter. The correlation coefficients between estimated errors from the two models are: 56, 38, and 39 per cent for α_⊥, α_∥, and fσ₈, respectively.

Figure 9.

Comparison between best-fitting values (left-hand panels) and estimated errors (right-hand panel) for (α_⊥, α_∥, fσ₈) using 1000 realizations of EZmocks fitted with the TNS and CLPT-GS models. The values obtained with real data are indicated by stars in each panel or as coloured vertical lines in the histograms. The thin black dashed line on the 2D plots refers to the true values of each parameter in the EZmocks.

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Table 11 summarizes the statistical properties of errors for α_⊥, α_∥, and fσ₈ for both BAO and full-shape RSD analysis in configuration space (noted ξ_ℓ). We also include for reference the results from Fourier-space analysis of Gil-Marín et al. (2020), noted P_ℓ. For each parameter, we show the standard deviation of the best-fitting values, σ, the mean estimated error 〈σ〉, the mean of the pull, Z_i = (x_i − 〈x〉)/σ_x, where x = α_⊥, α_∥, fσ₈, and its standard deviation σ(Z). If errors are correctly estimated and follow a Gaussian distribution, we expect that σ = 〈σ_i〉, 〈Z_i〉 = 0, and σ(Z) = 1. For method, we remove results from non-converged chains and 5σ outliers in both best-fitting values and errors (with σ defined as half of the range covered by the central 68 per cent values). Table 11 also shows the results from combining different methods employing the procedure described in Section 3.4. For each combination, we create the covariance matrix C (equation 40) from the correlation coefficients obtained from 1000 EZmocks fits, with small adjustments to account for the observed errors of a given realization. The correlation coefficients (before this adjustment) are shown in Fig. 10 for all five methods. The BAO measurements from configuration and Fourier spaces are 87 and 88 per cent correlated for α_⊥ and α_∥, respectively. In RSD analyses, these correlations reduce to slightly less than 80 per cent between α_⊥ and α_∥ of both spaces, while fσ₈ correlations reach 84 per cent. The fact that these correlations are not exactly 100 per cent indicates that there is potential gain combining them.