arXiv subjects

Mehdi Rezaie

Publications and source records attributed to Mehdi Rezaie.

At least 37 records · Page 2Linked to original sources

arXiv2024

Constraining primordial non-Gaussianity from DESI quasar targets and Planck CMB lensing

We detect the cross-correlation between 2.7 million DESI quasar targets across 14,700 deg$^2$ (180 quasars deg$^{-2}$) and Planck 2018 CMB lensing at $\sim$30$σ$. We use the cross-correlation on very large scales to constrain local primordial non-Gaussianity via the scale dependence of quasar bias. The DESI quasar targets lie at an effective redshift of 1.51 and are separated into four imaging regions of varying depth and image quality. We select quasar targets from Legacy Survey DR9 imaging, apply additional flux and photometric redshift cuts to improve the purity and reduce the fraction of unclassified redshifts, and use early DESI spectroscopy of 194,000 quasar targets to determine their redshift distribution and stellar contamination fraction (2.6%). Due to significant excess large-scale power in the quasar autocorrelation, we apply weights to mitigate contamination from imaging systematics such as depth, extinction, and stellar density. We use realistic contaminated mocks to determine the greatest number of systematic modes that we can fit, before we are biased by overfitting and spuriously remove real power. We find that linear regression with one to seven imaging templates removed per region accurately recovers the input cross-power, $f_{\textrm{NL}}$ and linear bias. As in previous analyses, our $f_{\textrm{NL}}$ constraint depends on the linear primordial non-Gaussianity bias parameter, $b_ϕ = 2(b - p)δ_c$ assuming universality of the halo mass function. We measure $f_{\textrm{NL}} = -26^{+45}_{-40}$ with $p=1.6$ $(f_{\textrm{NL}} = -18^{+29}_{-27}$ with $p=1.0$), and find that this result is robust under several systematics tests. Future spectroscopic quasar cross-correlations with Planck lensing lensing can tighten the $f_{\textrm{NL}}$ constraints by a factor of 2 if they can remove the excess power on large scales in the quasar auto power spectrum.

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arXiv2024

Long-term follow-up observations of extreme coronal line emitting galaxies

We present new spectroscopic and photometric follow-up observations of the known sample of extreme coronal line emitting galaxies (ECLEs) identified in the Sloan Digital Sky Survey (SDSS). With these new data, observations of the ECLE sample now span a period of two decades following their initial SDSS detections. We confirm the nonrecurrence of the iron coronal line signatures in five of the seven objects, further supporting their identification as the transient light echoes of tidal disruption events (TDEs). Photometric observations of these objects in optical bands show little overall evolution. In contrast, mid-infrared (MIR) observations show ongoing long-term declines. The remaining two objects had been classified as active galactic nuclei (AGN) with unusually strong coronal lines rather than being TDE related, given the persistence of the coronal lines in earlier follow-up spectra. We confirm this classification, with our spectra continuing to show the presence of strong, unchanged coronal-line features and AGN-like MIR colours and behaviour. We have constructed spectral templates of both subtypes of ECLE to aid in distinguishing the likely origin of newly discovered ECLEs. We highlight the need for higher cadence, and more rapid, follow-up observations of such objects to better constrain their properties and evolution. We also discuss the relationships between ECLEs, TDEs, and other identified transients having significant MIR variability.

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arXiv2024

Redshift evolution and covariances for joint lensing and clustering studies with DESI Y1

Galaxy-galaxy lensing (GGL) and clustering measurements from the Dark Energy Spectroscopic Instrument Year 1 (DESI Y1) dataset promise to yield unprecedented combined-probe tests of cosmology and the galaxy-halo connection. In such analyses, it is essential to identify and characterise all relevant statistical and systematic errors. In this paper, we forecast the covariances of DESI Y1 GGL+clustering measurements and characterise the systematic bias due to redshift evolution in the lens samples. Focusing on the projected clustering and galaxy-galaxy lensing correlations, we compute a Gaussian analytical covariance, using a suite of N-body and log-normal simulations to characterise the effect of the survey footprint. Using the DESI One Percent Survey data, we measure the evolution of galaxy bias parameters for the DESI Luminous Red Galaxy (LRG) and Bright Galaxy Survey (BGS) samples. We find mild evolution in the LRGs in 0.4 < z < 0.8, subdominant compared to the expected statistical errors. For BGS, we find less evolution effects for brighter absolute magnitude cuts, at the cost of reduced sample size. We find that with a fiducial redshift bin width delta z = 0.1, evolution effects on GGL is negligible across all scales, all fiducial selection cuts, all fiducial redshift bins, given DESI Y1 sample size. Galaxy clustering is more sensitive to evolution due to the bias squared scaling. Nevertheless the redshift evolution effect is insignificant for clustering above the 1-halo scale of 0.1Mpc/h. For studies that wish to reliably access smaller scales, additional treatment of redshift evolution is likely needed. This study serves as a reference for GGL and clustering studies using the DESI Y1 sample

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arXiv2024

Redshift-dependent RSD bias from Intrinsic Alignment with DESI Year 1 Spectra

We estimate the redshift-dependent, anisotropic clustering signal in DESI's Year 1 Survey created by tidal alignments of Luminous Red Galaxies (LRGs) and a selection-induced galaxy orientation bias. To this end, we measured the correlation between LRG shapes and the tidal field with DESI's Year 1 redshifts, as traced by LRGs and Emission-Line Galaxies (ELGs). We also estimate the galaxy orientation bias of LRGs caused by DESI's aperture-based selection, and find it to increase by a factor of seven between redshifts 0.4 - 1.1 due to redder, fainter galaxies falling closer to DESI's imaging selection cuts. These effects combine to dampen measurements of the quadrupole of the correlation function caused by structure growth on scales of 10 - 80 Mpc/h by about 0.15% for low redshifts (0.4<z<0.6) and 0.8% for high (0.8<z<1.1). We provide estimates of the quadrupole signal created by intrinsic alignments that can be used to correct this effect, which is necessary to meet DESI's forecasted precision on measuring the growth rate of structure. While imaging quality varies across DESI's footprint, we find no significant difference in this effect between imaging regions in the Legacy Imaging Survey.

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arXiv2023

DESI luminous red galaxy samples for cross-correlations

We present two galaxy samples, selected from DESI Legacy Imaging Surveys (LS) DR9, with approximately 20,000 square degrees of coverage and spectroscopic redshift distributions designed for cross-correlations such as with CMB lensing, galaxy lensing, and the Sunyaev-Zel'dovich effect. The first sample is identical to the DESI Luminous Red Galaxy (LRG) sample, and the second sample is an extended LRG sample with 2-3 times the DESI LRG density. We present the improved photometric redshifts, tomographic binning and their spectroscopic redshift distributions and imaging systematics weights, and magnification bias coefficients. The catalogs and related data products will be made publicly available. The cosmological constraints using this sample and Planck lensing maps are presented in a companion paper. We also make public the new set of general-purpose photometric redshifts trained using DESI spectroscopic redshifts, which are used in this work, for all galaxies in LS DR9.

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arXiv2023

DESI Mock Challenge: Constructing DESI galaxy catalogues based on FastPM simulations

Together with larger spectroscopic surveys such as the Dark Energy Spectroscopic Instrument (DESI), the precision of large scale structure studies and thus the constraints on the cosmological parameters are rapidly improving. Therefore, one must build realistic simulations and robust covariance matrices. We build galaxy catalogues by applying a Halo Occupation Distribution (HOD) model upon the \textsc{FastPM} simulations, such that the resulting galaxy clustering reproduces high resolution $N$-body simulations. While the resolution and halo finder are different from the reference simulations, we reproduce the reference galaxy two-point clustering measurements -- monopole and quadrupole -- to a precision required by the DESI Year 1 Emission Line Galaxy sample down to non-linear scales, i.e. $k<0.5\,h\mathrm{Mpc}$ or $s>10\,\mathrm{Mpc}/h$. Furthermore, we compute covariance matrices based on the resulting \textsc{FastPM} galaxy clustering -- monopole and quadrupole. We study for the first time the effect of fitting on Fourier conjugate [e.g. power spectrum] on the covariance matrix of the Fourier counterpart [e.g. correlation function]. We estimate the uncertainties of the two parameters of a simple clustering model and observe a maximum variation of 20 per cent for the different covariance matrices. Nevertheless, for most studied scales the scatter is between two to ten per cent Consequently, using the current pipeline we can precisely reproduce the clustering of $N$-body simulations and the resulting covariance matrices provide robust uncertainty estimations against HOD fitting scenarios. We expect our methodology will be useful for the coming DESI data analyses and their extension for other studies.

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arXiv2023

The DESI One-Percent Survey: A concise model for galactic conformity of ELGs

Galactic conformity is the phenomenon in which a galaxy of a certain physical property is correlated with its neighbors of the same property, implying a possible causal relationship. The observed auto correlations of emission line galaxies (ELGs) from the highly complete DESI One-Percent survey exhibit a strong clustering signal on small scales, providing clear evidence for the conformity effect of ELGs. Building upon the original subhalo abundance matching (SHAM) method developed by Gao et al. (2022, 2023), we propose a concise conformity model to improve the ELG-halo connection. In this model, the number of satellite ELGs is boosted by a factor of $\sim 5$ in the halos whose central galaxies are ELGs. We show that the mean ELG satellite number in such central halos is still smaller than 1, and the model does not significantly increase the overall satellite fraction. With this model, we can well recover the ELG auto correlations to the smallest scales explored with the current data (i.e. $r_{\mathrm{p}} > 0.03$ $\mathrm{Mpc}\,h^{-1}$ in real space and at $s > 0.3$ $\mathrm{Mpc}\,h^{-1}$ in redshift space), while the cross correlations between luminous red galaxies (LRGs) and ELGs are nearly unchanged. Although our SHAM model has only 8 parameters, we further verify that it can accurately describe the ELG clustering in the entire redshift range from $z = 0.8$ to $1.6$. We therefore expect that this method can be used to generate high-quality ELG lightcone mocks for DESI.

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arXiv2023

GTC Follow-up Observations of Very Metal-Poor Star Candidates from DESI

The observations from the Dark Energy Spectroscopic Instrument (DESI) will significantly increase the numbers of known extremely metal-poor stars by a factor of ~ 10, improving the sample statistics to study the early chemical evolution of the Milky Way and the nature of the first stars. In this paper we report high signal-to-noise follow-up observations of 9 metal-poor stars identified during the DESI commissioning with the Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy (OSIRIS) instrument on the 10.4m Gran Telescopio Canarias (GTC). The analysis of the data using a well-vetted methodology confirms the quality of the DESI spectra and the performance of the pipelines developed for the data reduction and analysis of DESI data.

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arXiv2023

Mitigating the noise of DESI mocks using analytic control variates

In order to address fundamental questions related to the expansion history of the Universe and its primordial nature with the next generation of galaxy experiments, we need to model reliably large-scale structure observables such as the correlation function and the power spectrum. Cosmological $N$-body simulations provide a reference through which we can test our models, but their output suffers from sample variance on large scales. Fortunately, this is the regime where accurate analytic approximations exist. To reduce the variance, which is key to making optimal use of these simulations, we can leverage the accuracy and precision of such analytic descriptions using Control Variates (CV). The power of control variates stems from utilizing inexpensive but highly correlated surrogates of the statistics one wishes to measure. The stronger the correlation between the surrogate and the statistic of interest, the larger the variance reduction delivered by the method. We apply two control variate formulations to mock catalogs generated in anticipation of upcoming data from the Dark Energy Spectroscopic Instrument (DESI) to test the robustness of its analysis pipeline. Our CV-reduced measurements offer a factor of 5-10 improvement in the measurement error compared with the raw measurements. We explore the relevant properties of the galaxy samples that dictate this reduction and comment on the improvements we find on some of the derived quantities relevant to Baryon Acoustic Oscillation (BAO) analysis. We also provide an optimized package for computing the power spectra and other two-point statistics of an arbitrary galaxy catalog as well as a pipeline for obtaining CV-reduced measurements on any of the AbacusSummit cubic box outputs. We make our scripts publicly available and report a speed improvement of $\sim$10 for a grid size of $N_{\rm mesh} = 256^3$ compared with \texttt{nbodykit}.

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arXiv2023

Unraveling emission line galaxy conformity at z~1 with DESI early data

Emission line galaxies (ELGs) are now the preeminent tracers of large-scale structure at z>0.8 due to their high density and strong emission lines, which enable accurate redshift measurements. However, relatively little is known about ELG evolution and the ELG-halo connection, exposing us to potential modeling systematics in cosmology inference using these sources. In this paper, we propose a physical picture of ELGs and improve ELG-halo connection modeling using a variety of observations and simulated galaxy models. We investigate DESI-selected ELGs in COSMOS data, and infer that ELGs are rapidly star-forming galaxies with a large fraction exhibiting disturbed morphology, implying that many of them are likely to be merger-driven starbursts. We further postulate that the tidal interactions from mergers lead to correlated star formation in central-satellite ELG pairs, a phenomenon dubbed "conformity." We argue for the need to include conformity in the ELG-halo connection using galaxy models such as IllustrisTNG, and by combining observations such as the DESI ELG auto-correlation, ELG cross-correlation with Luminous Red Galaxies (LRGs), and ELG-cluster cross-correlation. We also explore the origin of conformity using the UniverseMachine model and elucidate the difference between conformity and the well-known galaxy assembly bias effect.

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arXiv2023

The DESI One-Percent Survey: Evidence for Assembly Bias from Low-Redshift Counts-in-Cylinders Measurements

We explore the galaxy-halo connection information that is available in low-redshift samples from the early data release of the Dark Energy Spectroscopic Instrument (DESI). We model the halo occupation distribution (HOD) from z=0.1-0.3 using Survey Validation 3 (SV3; a.k.a., the One-Percent Survey) data of the DESI Bright Galaxy Survey (BGS). In addition to more commonly used metrics, we incorporate counts-in-cylinders (CiC) measurements, which drastically tighten HOD constraints. Our analysis is aided by the Python package, galtab, which enables the rapid, precise prediction of CiC for any HOD model available in halotools. This methodology allows our Markov chains to converge with much fewer trial points, and enables even more drastic speedups due to its GPU portability. Our HOD fits constrain characteristic halo masses tightly and provide statistical evidence for assembly bias, especially at lower luminosity thresholds: the HOD of central galaxies in $z\sim0.15$ samples with limiting absolute magnitude $M_r < -20.0$ and $M_r < -20.5$ samples is positively correlated with halo concentration with a significance of 99.9% and 99.5%, respectively. Our models also favor positive central assembly bias for the brighter $M_r < -21.0$ sample at $z\sim0.25$ (94.8% significance), but there is no significant evidence for assembly bias with the same luminosity threshold at $z\sim0.15$. We provide our constraints for each threshold sample's characteristic halo masses, assembly bias, and other HOD parameters. These constraints are expected to be significantly tightened with future DESI data, which will span an area 100 times larger than that of SV3.

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arXiv2023

On the impact of the galaxy window function on cosmological parameter estimation

One important source of systematics in galaxy redshift surveys comes from the estimation of the galaxy window function. Up until now, the impact of the uncertainty in estimating the galaxy window function on parameter inference has not been properly studied. In this paper, we show that the uncertainty and the bias in estimating the galaxy window function will be salient for ongoing and next-generation galaxy surveys using a simulation-based approach. With a specific case study of cross-correlating Emission-line galaxies from the DESI Legacy Imaging Surveys and the Planck CMB lensing map, we show that neural network-based regression approaches to modelling the window function are superior in comparison to linear regression-based models. We additionally show that the definition of the galaxy overdensity estimator can impact the overall signal-to-noise of observed power spectra. Finally, we show that the additive biases coming from the window functions can significantly bias the modes of the inferred parameters and also degrade their precision. Thus, a careful understanding of the window functions will be essential to conduct cosmological experiments.

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arXiv2022

Angular clustering properties of the DESI QSO target selection using DR9 Legacy Imaging Surveys

The quasar target selection for the upcoming survey of the Dark Energy Spectroscopic Instrument (DESI) will be fixed for the next five years. The aim of this work is to validate the quasar selection by studying the impact of imaging systematics as well as stellar and galactic contaminants, and to develop a procedure to mitigate them. Density fluctuations of quasar targets are found to be related to photometric properties such as seeing and depth of the Data Release 9 of the DESI Legacy Imaging Surveys. To model this complex relation, we explore machine learning algorithms (Random Forest and Multi-Layer Perceptron) as an alternative to the standard linear regression. Splitting the footprint of the Legacy Imaging Surveys into three regions according to photometric properties, we perform an independent analysis in each region, validating our method using eBOSS EZ-mocks. The mitigation procedure is tested by comparing the angular correlation of the corrected target selection on each photometric region to the angular correlation function obtained using quasars from the Sloan Digital Sky Survey (SDSS)Data Release 16. With our procedure, we recover a similar level of correlation between DESI quasar targets and SDSS quasars in two thirds of the total footprint and we show that the excess of correlation in the remaining area is due to a stellar contamination which should be removed with DESI spectroscopic data. We derive the Limber parameters in our three imaging regions and compare them to previous measurements from SDSS and the 2dF QSO Redshift Survey.

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arXiv2021

The clustering of galaxies in the completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Primordial non-Gaussianity in Fourier Space

We present measurements of the local primordial non-Gaussianity parameter \fNLloc from the clustering of 343,708 quasars with redshifts 0.8 < z < 2.2 distributed over 4808 square degrees from the final data release (DR16) of the extended Baryon acoustic Oscillation Spectroscopic Survey (eBOSS), the largest volume spectroscopic survey up to date. Our analysis is performed in Fourier space, using the power spectrum monopole at very large scales to constrain the scale dependent halo bias. We carefully assess the impact of systematics on our measurement and test multiple contamination removal methods. We demonstrate the robustness of our analysis pipeline with EZ-mock catalogues that simulate the eBOSS DR16 target selection. We find $f_\mathrm{NL}=-12\pm 21$ (68\% confidence) for the main clustering sample including quasars with redshifts between 0.8 and 2.2, after exploiting a novel neural network scheme for cleaning the DR16 sample and in particular after applying redshift weighting techniques, designed for non-Gaussianity measurement from large scales structure, to optimize our analysis, which improve our results by 37\%.

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arXiv2021

Primordial non-Gaussianity from the Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey I: Catalogue Preparation and Systematic Mitigation

We investigate the large-scale clustering of the final spectroscopic sample of quasars from the recently completed extended Baryon Oscillation Spectroscopic Survey (eBOSS). The sample contains $343708$ objects in the redshift range $0.8<z<2.2$ and $72667$ objects with redshifts $2.2<z<3.5$, covering an effective area of $4699~{\rm deg}^{2}$. We develop a neural network-based approach to mitigate spurious fluctuations in the density field caused by spatial variations in the quality of the imaging data used to select targets for follow-up spectroscopy. Simulations are used with the same angular and radial distributions as the real data to estimate covariance matrices, perform error analyses, and assess residual systematic uncertainties. We measure the mean density contrast and cross-correlations of the eBOSS quasars against maps of potential sources of imaging systematics to address algorithm effectiveness, finding that the neural network-based approach outperforms standard linear regression. Stellar density is one of the most important sources of spurious fluctuations, and a new template constructed using data from the Gaia spacecraft provides the best match to the observed quasar clustering. The end-product from this work is a new value-added quasar catalogue with the improved weights to correct for nonlinear imaging systematic effects, which will be made public. Our quasar catalogue is used to measure the local-type primordial non-Gaussianity in our companion paper, Mueller et al. in preparation.

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arXiv2021

The clustering of the SDSS-IV extended Baryon Oscillation Spectroscopic Survey quasar sample: Testing observational systematics on the Baryon Acoustic Oscillation measurement

Baryon Acoustic Oscillations are considered to be a very robust standard ruler against various systematics. This premise has been tested against observational systematics, but not to the level required for the next generation of galaxy surveys such as the Dark Energy Spectroscopic Instrument (DESI) and Euclid. In this paper, we investigate the effect of observational systematics on the BAO measurement of the final sample of quasars from the extended Baryon Oscillation Spectroscopic Survey Data Release 16 in order to prepare and hone a similar analysis for upcoming surveys. We employ catalogues with various treatments of imaging systematic effects using linear and neural network-based nonlinear approaches and consider how the BAO measurement changes. We also test how the variations to the BAO fitting model respond to the observational systematics. As expected, we confirm that the BAO measurements obtained from the DR16 quasar sample are robust against imaging systematics well within the statistical error, while reporting slightly modified constraints that shift the line-of-sight BAO signal by less than 1.1% . We use realistic simulations with similar redshift and angular distributions as the DR16 sample to conduct statistical tests for validating the pipeline, quantifying the significance of differences, and estimating the expected bias on the BAO scale in future high-precision data sets. Although we find a marginal impact for the eBOSS QSO data, the work presented here is of vital importance for constraining the nature of dark energy with the BAO feature in the new era of big data cosmology with DESI and Euclid.

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arXiv2021

The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: measurement of the BAO and growth rate of structure of the emission line galaxy sample from the anisotropic power spectrum between redshift 0.6 and 1.1

We analyse the large-scale clustering in Fourier space of emission line galaxies (ELG) from the Data Release 16 of the Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey. The ELG sample contains 173,736 galaxies covering 1,170 square degrees in the redshift range $0.6 < z < 1.1$. We perform a BAO measurement from the post-reconstruction power spectrum monopole, and study redshift space distortions (RSD) in the first three even multipoles. Photometric variations yield fluctuations of both the angular and radial survey selection functions. Those are directly inferred from data, imposing integral constraints which we model consistently. The full data set has only a weak preference for a BAO feature ($1.4σ$). At the effective redshift $z_{\rm eff} = 0.845$ we measure $D_{\rm V}(z_{\rm eff})/r_{\rm drag} = 18.33_{-0.62}^{+0.57}$, with $D_{\rm V}$ the volume-averaged distance and $r_{\rm drag}$ the comoving sound horizon at the drag epoch. In combination with the RSD measurement, at $z_{\rm eff} = 0.85$ we find $fσ_8(z_{\rm eff}) = 0.289_{-0.096}^{+0.085}$, with $f$ the growth rate of structure and $σ_8$ the normalisation of the linear power spectrum, $D_{\rm H}(z_{\rm eff})/r_{\rm drag} = 20.0_{-2.2}^{+2.4}$ and $D_{\rm M}(z_{\rm eff})/r_{\rm drag} = 19.17 \pm 0.99$ with $D_{\rm H}$ and $D_{\rm M}$ the Hubble and comoving angular distances, respectively. These results are in agreement with those obtained in configuration space, thus allowing a consensus measurement of $fσ_8(z_{\rm eff}) = 0.315 \pm 0.095$, $D_{\rm H}(z_{\rm eff})/r_{\rm drag} = 19.6_{-2.1}^{+2.2}$ and $D_{\rm M}(z_{\rm eff})/r_{\rm drag} = 19.5 \pm 1.0$. This measurement is consistent with a flat $Λ$CDM model with Planck parameters.

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arXiv2020

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

We analyse the clustering of the Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey Data Release 16 luminous red galaxy sample (DR16 eBOSS LRG) in combination with the high redshift tail of the Sloan Digital Sky Survey III Baryon Oscillation Spectroscopic Survey Data Release 12 (DR12 BOSS CMASS). We measure the redshift space distortions (RSD) and also extract the longitudinal and transverse baryonic acoustic oscillation (BAO) scale from the anisotropic power spectrum signal inferred from 377,458 galaxies between redshifts 0.6 and 1.0, with effective redshift of $z_{\rm eff}=0.698$ and effective comoving volume of $2.72\,{\rm Gpc}^3$. After applying reconstruction we measure the BAO scale and infer $D_H(z_{\rm eff})/r_{\rm drag} = 19.30\pm 0.56$ and $D_M(z_{\rm eff})/r_{\rm drag} =17.86 \pm 0.37$. When we perform a redshift space distortions analysis on the pre-reconstructed catalogue on the monopole, quadrupole and hexadecapole we find, $D_H(z_{\rm eff})/r_{\rm drag} = 20.18\pm 0.78$, $D_M(z_{\rm eff})/r_{\rm drag} =17.49 \pm 0.52$ and $fσ_8(z_{\rm eff})=0.454\pm0.046$. We combine both sets of results along with the measurements in configuration space of \cite{LRG_corr} and report the following consensus values: $D_H(z_{\rm eff})/r_{\rm drag} = 19.77\pm 0.47$, $D_M(z_{\rm eff})/r_{\rm drag} = 17.65\pm 0.30$ and $fσ_8(z_{\rm eff})=0.473\pm 0.044$, which are in full agreement with the standard $Λ$CDM and GR predictions. These results represent the most precise measurements within the redshift range $0.6\leq z \leq 1.0$ and are the culmination of more than 8 years of SDSS observations.

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