Galaxy Two-point Correlation Function Research Articles

Cosmological constraints from density-split clustering in the BOSS CMASS galaxy sample

ABSTRACT We present a clustering analysis of the BOSS DR12 CMASS galaxy sample, combining measurements of the galaxy two-point correlation function and density-split clustering down to a scale of $1 \, h^{-1}\, \text{Mpc}$. Our theoretical framework is based on emulators trained on high-fidelity mock galaxy catalogues that forward model the cosmological dependence of the clustering statistics within an extended-ΛCDM framework, including redshift-space and Alcock–Paczynski distortions. Our base-ΛCDM analysis finds ωcdm = 0.1201 ± 0.0022, σ8 = 0.792 ± 0.034, and ns = 0.970 ± 0.018, corresponding to fσ8 = 0.462 ± 0.020 at z ≈ 0.525, which is in agreement with Planck 2018 predictions and various clustering studies in the literature. We test single-parameter extensions to base-ΛCDM, varying the running of the spectral index, the dark energy equation of state, and the density of mass-less relic neutrinos, finding no compelling evidence for deviations from the base model. We model the galaxy–halo connection using a halo occupation distribution framework, finding signatures of environment-based assembly bias in the data. We validate our pipeline against mock catalogues that match the clustering and selection properties of CMASS, showing that we can recover unbiased cosmological constraints even with a volume 84 times larger than the one used in this study.

Galaxy clustering measurements out to redshift z ˜ 8 from Hubble Legacy Fields

ABSTRACT We present a novel approach for measuring the two-point correlation function of galaxies in narrow pencil beam surveys with varying depths. Our methodology is utilized to expand high-redshift galaxy clustering investigations up to z ∼ 8 by analysing a comprehensive sample consisting of Ng = 160 Lyman break galaxy candidates obtained through optical and near-infrared photometric data within the CANDELS GOODS data sets from the Hubble Space Telescope Legacy Fields. For bright sources with MUV < −19.8, we determine a galaxy bias of b = 9.33 ± 4.90 at $\overline{z} = 7.7$ and a correlation length of r0 = 10.74 ± 7.06 $h^{-1}\, \mathrm{Mpc}$. We obtain similar results for the XDF, with a galaxy bias measurement of b = 8.26 ± 3.41 at the same redshift for a slightly fainter sample with a median luminosity of MUV = −18.4. By comparing with dark-matter halo bias and employing abundance matching, we deduce a characteristic halo mass of Mh ∼ 1011.5 M⊙ and a duty cycle close to unity. To validate our approach for variable-depth data sets, we replicate the analysis in a region with near-uniform depth using a standard two-point correlation function estimator, yielding consistent outcomes. Our study not only provides a valuable tool for future utilization in JWST data sets but also suggests that the clustering of early galaxies continues to increase with redshift beyond z ≳ 8, potentially contributing to the existence of protocluster structures observed in early JWST imaging and spectroscopic surveys at z ≳ 8.

The Galaxy Number Density Profile of Halos

More precise measurements of galaxy clustering will be provided by the next generation of galaxy surveys, such as DESI, WALLABY, and the Square Kilometre Array. To utilize this information to improve our understanding of the Universe, we need to accurately model the distribution of galaxies in their host dark matter halos. In this work, we present a new galaxy number density profile of halos, which makes predictions for the positions of galaxies in the host halo, different to the widely adopted Navarro–Frenk–White (NFW) profile, since galaxies tend to be found more in the outskirts of halos (nearer the virial radius) than an NFW profile. The parameterized galaxy number density profile model of halos is fit and tested using the Dark Sage semi-analytic model of galaxy formation. We find that our galaxy number density profile model of halos can accurately reproduce the halo occupation distribution and galaxy two-point correlation function of the Dark Sage simulation. We also derive the analytic expressions for the circular velocity and gravitational potential energy for this profile model. We use the SDSS Data Release 10 galaxy group catalog to validate this galaxy number density profile model of halos. Compared to the NFW profile, we find that our model more accurately predicts the positions of galaxies in their host halo and the galaxy two-point correlation function.

Cosmology from LOFAR Two-metre Sky Survey Data Release 2: angular clustering of radio sources

ABSTRACT Covering $\sim 5600\, \deg ^2$ to rms sensitivities of ∼70−100 $\mu$Jy beam−1, the LOFAR Two-metre Sky Survey Data Release 2 (LoTSS-DR2) provides the largest low-frequency (∼150 MHz) radio catalogue to date, making it an excellent tool for large-area radio cosmology studies. In this work, we use LoTSS-DR2 sources to investigate the angular two-point correlation function of galaxies within the survey. We discuss systematics in the data and an improved methodology for generating random catalogues, compared to that used for LoTSS-DR1, before presenting the angular clustering for ∼900 000 sources ≥1.5 mJy and a peak signal-to-noise ≥ 7.5 across ∼80 per cent of the observed area. Using the clustering, we infer the bias assuming two evolutionary models. When fitting angular scales of $0.5 \le \theta \lt 5{^\circ }$, using a linear bias model, we find LoTSS-DR2 sources are biased tracers of the underlying matter, with a bias of $b_{\rm C}= 2.14^{+0.22}_{-0.20}$ (assuming constant bias) and $b_{\rm E}(z=0)= 1.79^{+0.15}_{-0.14}$ (for an evolving model, inversely proportional to the growth factor), corresponding to $b_{\rm E}= 2.81^{+0.24}_{-0.22}$ at the median redshift of our sample, assuming the LoTSS Deep Fields redshift distribution is representative of our data. This reduces to $b_{\rm C}= 2.02^{+0.17}_{-0.16}$ and $b_{\rm E}(z=0)= 1.67^{+0.12}_{-0.12}$ when allowing preferential redshift distributions from the Deep Fields to model our data. Whilst the clustering amplitude is slightly lower than LoTSS-DR1 (≥2 mJy), our study benefits from larger samples and improved redshift estimates.

Clustering of red sequence galaxies in the fourth data release of the Kilo-Degree Survey

We present a sample of luminous red sequence galaxies as the basis for a study of the large-scale structure in the fourth data release of the Kilo-Degree Survey. The selected galaxies are defined by a red sequence template, in the form of a data-driven model of the colour-magnitude relation conditioned on redshift. In this work, the red sequence template was built using the broad-band optical+near infrared photometry of KiDS-VIKING and the overlapping spectroscopic data sets. The selection process involved estimating the red sequence redshifts, assessing the purity of the sample and estimating the underlying redshift distributions of redshift bins. After performing the selection, we mitigated the impact of survey properties on the observed number density of galaxies by assigning photometric weights to the galaxies. We measured the angular two-point correlation function of the red galaxies in four redshift bins and constrain the large-scale bias of our red sequence sample assuming a fixed ΛCDM cosmology. We find consistent linear biases for two luminosity-threshold samples (‘dense’ and ‘luminous’). We find that our constraints are well characterised by the passive evolution model.

Cosmological forecasts for future galaxy surveys with the linear point standard ruler: Toward consistent BAO analyses far from a fiducial cosmology

The linear point (LP) standard ruler was identified as the basis of a purely geometric method for exploiting the baryon acoustic oscillations (BAO). The LP exploits the BAO feature imprinted in the galaxy two-point correlation function to measure cosmological distances independent of any specific cosmological model. We forecast the expected precision of future and ongoing spectroscopic galaxy surveys to measure distances leveraging the linear point. We investigate the cosmological implications of our forecasted results. We focus in particular on a relevant working example: the detection of the late-time cosmic acceleration independent of other cosmological probes. Our findings show that, even within the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ standard cosmological paradigm, estimated distances need to be reliable over a very wide parameter range in order to realize their maximum utility. This is particularly relevant if we aim to properly characterize cosmological tensions. The LP is a promising candidate approach to achieve this reliability. In contrast, widely employed procedures in BAO analysis estimate distances keeping fixed cosmological parameters to fiducial values close to cosmic-microwave-background constraints in flat-$\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$. It is unclear whether they are purely geometric methods. Moreover, they rely on untested extrapolations to explore the parameter space away from those fiducial flat-$\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ values. We recommend that all BAO methodologies be validated across the full range of models and parameters over which their results are quoted, first by means of linear predictions and then $N$-body simulations.

New constraints on primordial features from the galaxy two-point correlation function

Features in the primordial power spectrum represent the imprinted signal in the density perturbations of the physics and evolution of the early Universe. A measurement of such signals will represents the need to go beyond the minimal assumption made for the initial conditions of the cosmological perturbations. For the first time, we study different templates with undamped oscillations or a bump from the two-point correlation function measured from BOSS DR12 galaxies constraining the amplitude of the features to be at most a few percent. Constraints are competitive to the ones obtained with {\em Planck} DR3.

Artificial neural networks for galaxy clustering: Learning from the two-point correlation function of BOSS galaxies

The increasingly large amount of cosmological data coming from ground-based and space-borne telescopes requires highly efficient and fast enough data analysis techniques to maximise the scientific exploitation. In this work, we explore the capabilities of supervised machine learning algorithms to learn the properties of the large-scale structure of the Universe, aiming at constraining the matter density parameter, Ωm. We implement a new Artificial Neural Network for a regression data analysis, and train it on a large set of galaxy two-point correlation functions in standard cosmologies with different values of Ωm. The training set is constructed from log-normal mock catalogues which reproduce the clustering of the Baryon Oscillation Spectroscopic Survey (BOSS) galaxies. The presented statistical method requires no specific analytical model to construct the likelihood function, and runs with negligible computational cost, after training. We test this new Artificial Neural Network on real BOSS data, finding Ωm=0.309±0.008, which is consistent with standard analysis results.

Galaxy and Mass Assembly (GAMA)

Aims. We investigate how different mid-infrared (mid-IR) properties of galaxies are correlated with the environment in which the galaxies are located. For this purpose, we first study the dependence of galaxy clustering on the absolute magnitude at 3.4 μm and redshift. Then, we look into the environmental dependence of mid-IR luminosities and the galaxy properties derived from these luminosities. We also explore how various IR galaxy luminosity selections influence the galaxy clustering measurements. Methods. We used a set of W1 (3.4 μm) absolute magnitude (MW1) selected samples from the Galaxy and Mass Assembly (GAMA) survey matched with mid-IR properties from the Wide-field Infrared Survey Explorer (WISE) in the redshift range 0.07 ≤ z < 0.43. We computed the galaxy two-point correlation function (2pCF) and compared the clustering lengths between subsamples binned in MW1 and in redshift. We also measured the marked correlation function (MCF), in which the galaxies are weighted by marks when measuring clustering statistics, using the luminosities in the WISE W1 to W4 (3.4 to 22 μm) bands as marks. Additionally, we compared the measurements of MCFs with different estimates of stellar mass and star formation rate (SFR) used as marks. Finally, we checked how different selections applied to the sample affect the clustering measurements. Results. We show strong clustering dependence on the W1 absolute magnitude: galaxies brighter in the W1 band are more strongly clustered than their fainter counterparts. We also observe a lack of significant redshift dependence of clustering in the redshift range 0.07 ≤ z < 0.43. We show that although the W1 and W2 bands are direct indicators of stellar mass, a galaxy sample selected based on W1 or W2 bands does not perfectly show the clustering behaviour of a stellar mass-selected sample. The proxy relation between W3 and W4 bands and SFR is similar. We also demonstrate the influence of estimation techniques of stellar mass and SFR on the clustering measurements.

Accelerating BAO scale fitting using Taylor series

ABSTRACT The Universe is currently undergoing accelerated expansion driven by dark energy. Dark energy’s essential nature remains mysterious: one means of revealing it is by measuring the Universe’s size at different redshifts. This may be done using the baryon acoustic oscillation (BAO) feature, a standard ruler in the galaxy two-point correlation function (2PCF). In order to measure the distance scale, one dilates and contracts a template for the 2PCF in a fiducial cosmology, using a scaling factor α. The standard method for finding the best-fitting α is to compute the likelihood over a grid of roughly 100 values of it. This approach is slow; in this work, we propose a significantly faster way. Our method writes the 2PCF as a polynomial in α by Taylor-expanding it about α = 1, exploiting that we know the fiducial cosmology sufficiently well that α is within a few per cent of unity. The likelihood resulting from this expansion may then be analytically solved for the best-fitting α. Our method is 48–85× faster than a directly comparable approach in which we numerically minimize α, and ∼12 000× faster than the standard iterative method. Our work will be highly enabling for upcoming large-scale structure redshift surveys such as that by Dark Energy Spectroscopic Instrument.

Euclid: Fast two-point correlation function covariance through linear construction

We present a method for fast evaluation of the covariance matrix for a two-point galaxy correlation function (2PCF) measured with the Landy–Szalay estimator. The standard way of evaluating the covariance matrix consists in running the estimator on a large number of mock catalogs, and evaluating their sample covariance. With large random catalog sizes (random-to-data objects’ ratio M ≫ 1) the computational cost of the standard method is dominated by that of counting the data-random and random-random pairs, while the uncertainty of the estimate is dominated by that of data-data pairs. We present a method called Linear Construction (LC), where the covariance is estimated for small random catalogs with a size of M = 1 and M = 2, and the covariance for arbitrary M is constructed as a linear combination of the two. We show that the LC covariance estimate is unbiased. We validated the method with PINOCCHIO simulations in the range r = 20 − 200 h−1 Mpc. With M = 50 and with 2 h−1 Mpc bins, the theoretical speedup of the method is a factor of 14. We discuss the impact on the precision matrix and parameter estimation, and present a formula for the covariance of covariance.

Angular systematics-free cosmological analysis of galaxy clustering in configuration space

ABSTRACT Galaxy redshift surveys are subject to incompleteness and inhomogeneous sampling due to the various constraints inherent to spectroscopic observations. This can introduce systematic errors on the summary statistics of interest, which need to be mitigated in cosmological analysis to achieve high accuracy. Standard practices involve applying weighting schemes based on completeness estimates across the survey footprint, possibly supplemented with additional weighting schemes accounting for density-dependent effects. In this work, we concentrate on pure angular systematics and describe an alternative approach consisting in analysing the galaxy two-point correlation function where angular modes are nulled. By construction, this procedure removes all possible known and unknown sources of angular observational systematics, but also part of the cosmological signal. We use a modified Landy–Szalay estimator for the two-point correlation function that relies on an additional random catalogue where angular positions are randomly drawn from the galaxy catalogue, and provide an analytical model to describe this modified statistic. We test the model by performing an analysis of the full anisotropic clustering in mock catalogues of luminous red and emission-line galaxies at 0.43 < z < 1.1. We find that the model fully accounts for the modified correlation function in redshift space, without introducing new nuisance parameters. The derived cosmological parameters from the analysis of baryon acoustic oscillations and redshift-space distortions display slightly larger statistical uncertainties, mostly for the growth rate of structure parameter fσ8 that exhibits a $50{{\ \rm per\ cent}}$ statistical error increase, but free from angular systematic error.

Observational measures of halo properties beyond mass

ABSTRACT Different properties of dark matter haloes, including growth rate, concentration, interaction history, and spin, correlate with environment in unique, scale-dependent ways. While these halo properties are not directly observable, galaxies will inherit their host haloes’ correlations with environment. In this paper, we show how these characteristic environmental signatures allow using measurements of galaxy environment to constrain which dark matter halo properties are most tightly connected to observable galaxy properties. We show that different halo properties beyond mass imprint distinct scale-dependent signatures in both the galaxy two-point correlation function and the distribution of distances to galaxies’ kth nearest neighbours, with features strong enough to be accessible even with low-resolution (e.g. grism) spectroscopy at higher redshifts. As an application, we compute observed two-point correlation functions for galaxies binned by half-mass radius at $z$ = 0 from the Sloan Digital Sky Survey, showing that classic galaxy size models (i.e. galaxy size being proportional to halo spin) as well as other recent proposals show significant tensions with observational data. We show that the agreement with observed clustering can be improved with a simple empirical model in which galaxy size correlates with halo growth.

A Fast and Accurate Analytic Method of Calculating Galaxy Two-point Correlation Functions

Abstract We have developed a new analytic method to calculate the galaxy two-point correlation functions accurately and efficiently, applicable to surveys with finite, regular, and mask-free geometries. We have derived simple, accurate formulas of the normalized random–random pair counts RR as functions of the survey area dimensions. We have also suggested algorithms to compute the normalized data-random pair counts DR analytically. With all edge corrections fully accounted for analytically, our method computes RR and DR with perfect accuracy and zero variance in O(1) and O(N g) time, respectively. We test our method on a galaxy catalog from the Evolution and Assembly of GaLaxies and their Environments (EAGLE) simulation. Our method calculates RR + DR at a speed 3–6 orders of magnitude faster than the brute-force Monte Carlo method and 2.5 orders of magnitude faster than tree-based algorithms. For a galaxy catalog with 10 million data points in a cube, this reduces the computation time to under 1 minute on a laptop. Our analytic method is favored over the traditional Monte Carlo method whenever applicable. Some applications in the study of correlation functions and power spectra in cosmological simulations and galaxy surveys are discussed. However, we recognize that its applicability is very limited for realistic surveys with masks, irregular shapes, and/or weighted patterns.

Subhalo abundance matching through the lens of a hydrodynamical simulation

ABSTRACT We use the IllustrisTNG100 hydrodynamical simulation to study the dependence of the galaxy two-point correlation function on a broad range of secondary subhalo and galactic properties. We construct galaxy mock catalogues adopting a standard subhalo abundance matching scheme coupled with a secondary assignment between galaxy colour or specific star formation rate and the following subhalo properties: starvation redshift zstarve, concentration at infall, overdensity $\delta _R^{\rm env}$, tidal anisotropy αR, and tidal overdensity δR. The last two quantities allow us to fully characterize the tidal field of our subhaloes, acting as mediators between their internal and large-scale properties. The resulting mock catalogues overall return good agreement with the IllustrisTNG100 measurements. The accuracy of each model strongly depends on the correlation between the secondary galaxy and subhalo properties employed. Among all the subhalo proxies tested, we find that zstarve and cinfall are the ones that best trace the large-scale structure, producing robust clustering predictions for different samples of red/blue and quenched/star-forming galaxies.

The dependence of subhalo abundance matching on galaxy photometry and selection criteria

ABSTRACT Subhalo abundance matching (SHAM) is a popular technique for assigning galaxy mass or luminosity to haloes produced in N-body simulations. The method works by matching the cumulative number functions of the galaxy and halo properties, and is therefore sensitive both to the precise definitions of those properties and to the selection criteria used to define the samples. Further dependence follows when SHAM parameters are calibrated with galaxy clustering, which is known to depend strongly on the manner in which galaxies are selected. In this paper we introduce a new parametrization for SHAM and derive the best-fitting SHAM parameters as a function of various properties of the selection of the galaxy sample and of the photometric definition, including Sérsic versus Petrosian magnitudes, stellar masses versus r-band magnitudes, and optical (Sloan Digital Sky Survey) versus $\mathrm{H}\, \small {\rm I}$ (ALFALFA) selection. In each case we calculate the models’ goodness-of-fit to measurements of the projected two-point galaxy correlation function. In the optically selected samples we find strong evidence that the scatter in the galaxy–halo connection increases towards the faint end, and that AM performs better with luminosity than stellar mass. The SHAM parameters of optically and $\mathrm{H}\, \small {\rm I}$-selected galaxies are mutually exclusive, with the latter suggesting the importance of properties beyond halo mass. We provide best-fitting parameters for the SHAM galaxy–halo connection as a function of each of our input choices, extending the domain of validity of the model while reducing potential systematic error in its use.

Improving estimates of the growth rate using galaxy–velocity correlations: a simulation study

ABSTRACT We present an improved framework for estimating the growth rate of large-scale structure, using measurements of the galaxy–velocity cross-correlation in configuration space. We consider standard estimators of the velocity autocorrelation function, ψ1 and ψ2, the two-point galaxy correlation function, ξgg, and introduce a new estimator of the galaxy–velocity cross-correlation function, ψ3. By including pair counts measured from random catalogues of velocities and positions sampled from distributions characteristic of the true data, we find that the variance in the galaxy–velocity cross-correlation function is significantly reduced. Applying a covariance analysis and χ2 minimization procedure to these statistics, we determine estimates and errors for the normalized growth rate fσ8 and the parameter β = f/b, where b is the galaxy bias factor. We test this framework on mock hemisphere data sets for redshift z < 0.1 with realistic velocity noise constructed from the l-picola simulation code, and find that we are able to recover the fiducial value of fσ8 from the joint combination of ψ1 + ψ2 + ψ3 + ξgg, with 15 per cent accuracy from individual mocks. We also recover the fiducial fσ8 to within 1σ regardless of the combination of correlation statistics used. When we consider all four statistics together we find that the statistical uncertainty in our measurement of the growth rate is reduced by $59{{\ \rm per\ cent}}$ compared to the same analysis only considering ψ2, by $53{{\ \rm per\ cent}}$ compared to the same analysis only considering ψ1, and by $52{{\ \rm per\ cent}}$ compared to the same analysis jointly considering ψ1 and ψ2.

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

ABSTRACT We present the cosmological analysis of the configuration-space anisotropic clustering in the completed Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey (eBOSS) Data Release 16 galaxy sample. This sample consists of luminous red galaxies (LRGs) spanning the redshift range 0.6 < $z$ < 1, at an effective redshift of $z$eff = 0.698. It combines 174 816 eBOSS and 202 642 BOSS LRGs. We extract and model the baryon acoustic oscillation (BAO) and redshift-space distortion (RSD) features from the galaxy two-point correlation function to infer geometrical and dynamical cosmological constraints. The adopted methodology is extensively tested on a set of realistic simulations. The correlations between the inferred parameters from the BAO and full-shape correlation function analyses are estimated. This allows us to derive joint constraints on the three cosmological parameter combinations: DM($z$)/rd, DH($z$)/rd, and fσ8($z$), where DM is the comoving angular diameter distance, DH is the Hubble distance, rd is the comoving BAO scale, f is the linear growth rate of structure, and σ8 is the amplitude of linear matter perturbations. After combining the results with those from the parallel power spectrum analysis of Gil-Marin et al., we obtain the constraints: DM/rd = 17.65 ± 0.30, DH/rd = 19.77 ± 0.47, and fσ8 = 0.473 ± 0.044. These measurements are consistent with a flat Lambda cold dark matter model with standard gravity.

Toward accurate measurement of property-dependent galaxy clustering: I. Comparison of the Vmax method and the “shuffled” method

Galaxy clustering provides insightful clues to our understanding of galaxy formation and evolution, as well as the universe. The redshift assignment for the random sample is one of the key steps to accurately measure galaxy clustering. In this paper, by virtue of the mock galaxy catalogs, we investigate the effect of two redshift assignment methods on the measurement of galaxy two-point correlation functions (hereafter 2PCFs), the Vmax method and the “shuffled” method. We have found that the shuffled method significantly underestimates both of the projected 2PCFs and the two-dimensional 2PCFs in redshift space, while the Vmax method does not show any notable bias on the 2PCFs for volume-limited samples. For flux-limited samples, the bias produced by the Vmax method is less than half of the shuffled method on large scales. Therefore, we strongly recommend the Vmax method to assign redshifts to random samples in the future galaxy clustering analysis.

Validating the methodology for constraining the linear growth rate from clustering anisotropies

ABSTRACT Redshift-space clustering distortions provide one of the most powerful probes to test the gravity theory on the largest cosmological scales. We perform a systematic validation study of the state-of-the-art statistical methods currently used to constrain the linear growth rate from redshift-space distortions in the galaxy two-point correlation function. The numerical pipelines are tested on mock halo catalogues extracted from large N-body simulations of the standard cosmological framework. We consider both the monopole and quadrupole multipole moments of the redshift-space two-point correlation function, as well as the radial and transverse clustering wedges, in the comoving scale range 10 < r[$h^{-1}\, \mbox{Mpc}$] < 55. Moreover, we investigate the impact of redshift measurement errors on the growth rate and linear bias measurements due to the assumptions in the redshift-space distortion model. Considering both the dispersion model and two widely used models based on perturbation theory, we find that the linear growth rate is underestimated by about $5\!-\! 10\, {\rm {per\ cent}}$ at $z$ < 1, while limiting the analysis at larger scales, r > 30 $h^{-1}\, \mbox{Mpc}$, the discrepancy is reduced below $5\, {\rm {per\ cent}}$. At higher redshifts, we find instead an overall good agreement between measurements and model predictions. Though this accuracy is good enough for clustering analyses in current redshift surveys, the models have to be further improved not to introduce significant systematics in RSD constraints from next-generation galaxy surveys. The effect of redshift errors is degenerate with the one of small-scale random motions, and can be marginalized over in the statistical analysis, not introducing any statistically significant bias in the linear growth constraints, especially at $z$ ≥ 1.