The cross correlation between the thermal Sunyaev-Zeldovich (tSZ) effect and gravitational lensing in wide field has recently been measured. It can be used to probe the distribution of the diffuse gas in large scale structure, as well as inform us about the missing baryons. As for any lensing-based quantity, higher order lensing effects can potentially affect the signal. Here, we extend previous higher order lensing calculations to the case of tSZ-lensing cross correlations. We derive terms analogous to corrections due to the Born approximation, lens-lens coupling, and reduced shear up to fourth order in the Newtonian potential. Redshift distortions and vector modes are shown to be negligible at this order. We find that the dominant correction due to the reduced shear exceeds percent-level only at multipoles of $\ell \gtrsim 3000$.
Residual foreground contamination by thermal Sunyaev-Zeldovich (tSZ) effect from galaxy clusters in cosmic microwave background (CMB) maps propagates into the reconstructed CMB lensing field, and thus biases the intrinsic cross-correlation between CMB lensing and large-scale structure (LSS). Through stacking analysis, we show that residual tSZ contamination causes an increment of lensing convergence in the central part of the clusters and a decrement of lensing convergence in the cluster outskirts. We quantify the impact of residual tSZ contamination on cross-correlations between the Planck 20…
We search for the potential contamination in the Planck thermal Sunyaev-Zeldovich (tSZ) map by calculating the cross-correlation between the tSZ signal and weak lensing by large scale structure and the Cosmic Microwave Background (CMB). The lensing data we use is the convergence map from the Red Sequence Cluster Lensing Survey (RCSLens) and the Planck CMB lensing map. We reconstruct the tSZ $y$ map with a Needlet Internal Linear Combination method using the HFI sky maps from the Planck satellite. We remove the CMB signal while minimizing the residual noise. The cross correlation signal from ou…
The thermal Sunyaev-Zel'dovich (tSZ) effect signal is widely recognized as a robust mass proxy of galaxy clusters with small intrinsic scatter. However, recent observational calibration of the tSZ scaling relation using weak lensing (WL) mass exhibits considerably larger scatter than the intrinsic scatter predicted from numerical simulations. This raises a question as to whether we can realize the full statistical power of ongoing and upcoming tSZ-WL observations of galaxy clusters. In this work, we investigate the origin of observed scatter in the tSZ-WL scaling relation, using mock maps of g…
Cross-correlation analysis of the thermal Sunyaev-Zel'dovich (tSZ) effect and weak gravitational lensing (WL) provides a powerful probe of cosmology and astrophysics of the intra-cluster medium. We present the measurement of the cross-correlation of tSZ and WL from Planck and Subaru Hyper-Suprime Cam. The combination enables us to study cluster astrophysics at high redshift. We use the tSZ-WL cross-correlation and the tSZ auto-power spectrum measurements to place a tight constraint on the hydrostatic mass bias, which is a measure of the degree of non-thermal pressure support in galaxy clusters…
We constrain the redshift dependence of gas pressure bias $\left\langle b_{y} P_{\mathrm{e}}\right\rangle$ (bias-weighted average electron pressure), which characterises the thermodynamics of intergalactic gas, through a combination of cross-correlations between galaxy positions and the thermal Sunyaev-Zeldovich (tSZ) effect, as well as galaxy positions and the gravitational lensing of the cosmic microwave background (CMB). The galaxy sample is from the fourth data release of the Kilo-Degree Survey (KiDS). The tSZ $y$ map and the CMB lensing map are from the {\textit{Planck}} 2015 and 2018 dat…
Approximately half of the Universe's baryons are in a form that has been hard to detect directly. However, the missing component can be traced through the cross-correlation of the thermal Sunyaev-Zeldovich (tSZ) effect with weak gravitational lensing. We build a model for this correlation and use it to constrain the extended baryon component, employing data from the Canada France Hawaii Lensing Survey and the {\it Planck\/} satellite. The measured correlation function is consistent with an isothermal $β$-model for the halo gas pressure profile, and the 1- and 2-halo terms are both detected at …
Cross correlation with thermal Sunyaev-Zel'dovich (tSZ) effect in cosmic microwave background observation and weak gravitational lensing effect in galaxy imaging survey opens a new window on constraining matter contents in the Universe at redshift less than 1. In this paper, we study the impact of radio sources and cosmic infrared background (CIB) on observed tSZ-lensing correlation. Assuming the best-fit model of CIB by the Planck satellite, we estimate that the residual correlation of CIB with large-scale structures will be of an order of $2\%$ of expected tSZ-lensing correlation from intrac…
Recent detections of the cross-correlation of the thermal Sunyaev-Zel'dovich (tSZ) effect and weak gravitational lensing (WL) enable unique studies of cluster astrophysics and cosmology. In this work, we present constraints on the amplitude of the non-thermal pressure fraction in galaxy clusters, $α_0$, and the amplitude of the matter power spectrum, $σ_8$, using measurements of the tSZ power spectrum from Planck, and the tSZ-WL cross-correlation from Planck and the Red Cluster Sequence Lensing Survey. We fit the data to a semi-analytic model with the covariance matrix using $N$-body simulatio…
We present novel statistical tools to cross-correlate frequency cleaned thermal Sunyaev-Zel'dovich (tSZ) maps and tomographic weak lensing (wl) convergence maps. Moving beyond the lowest order cross-correlation, we introduce a hierarchy of mixed higher-order statistics, the cumulants and cumulant correlators, to analyze non-Gaussianity in real space, as well as corresponding polyspectra in the harmonic domain. Using these moments, we derive analytical expressions for the joint two-point probability distribution function (2PDF) for smoothed tSZ (y_s) and convergence (κ_s) maps. The presence of …
The thermal Sunyaev-Zel'dovich (tSZ) effect is produced by the interaction of cosmic microwave background (CMB) photons with the hot (a few keV) and diffuse gas of electrons inside galaxy clusters integrated along the line of sight. This effect produces a distortion of CMB blackbody emission law. This distortion law depends on the electronic temperature of the intra-cluster hot gas, $T_{e}$, through the so-called tSZ relativistic corrections. In this work, we have performed a statistical analysis of the tSZ spectral distortion on large galaxy cluster samples. We performed a stacking analysis f…
The hot electrons in the intra-cluster medium produce a spectral distorsion of the cosmic microwave background (CMB) black body emission, the thermal Sunyaev-Zel'dovich effect (tSZ). This characteristic spectral distorsion is now commonly used to detect and characterize the properties of galaxy clusters. The tSZ effect spectral distorsion does not depends on the redshift, and is only slightly affected by the galaxy cluster properties via the relativistic corrections, when the electrons reach relativistic velocities. The present work proposes a linear component separation approach to extract th…
Mass calibration uncertainty is the largest systematic effect for using clusters of galaxies to constrain cosmological parameters. We present weak lensing mass measurements from the Canada-France-Hawaii Telescope Stripe 82 Survey for galaxy clusters selected through their high signal-to-noise thermal Sunyaev-Zeldovich (tSZ) signal measured with the Atacama Cosmology Telescope (ACT). For a sample of 9 ACT clusters with a tSZ signal-to-noise greater than five the average weak lensing mass is $\left(4.8\pm0.8\right)\,\times10^{14}\,\mathrm{M}_\odot$, consistent with the tSZ mass estimate of $\lef…
The cosmological parameters prefered by the cosmic microwave background (CMB) primary anisotropies predict many more galaxy clusters than those that have been detected via the thermal Sunyaev-Zeldovich (tSZ) effect. This tension has attracted considerable attention since it could be evidence of physics beyond the simplest $Λ$CDM model. However, an accurate and robust calibration of the mass-observable relation for clusters is necessary for the comparison, which has been proven difficult to obtain so far. Here, we present new contraints on the mass-pressure relation by combining tSZ and CMB len…
The nominal mission maps from the Planck satellite contain a wealth of information about secondary anisotropies in the cosmic microwave background (CMB), including those induced by the thermal Sunyaev-Zel'dovich (tSZ) effect and gravitational lensing. As both the tSZ and CMB lensing signals trace the large-scale matter density field, the anisotropies sourced by these processes are expected to be correlated. We report the first detection of this cross-correlation signal, which we measure at 6.2 sigma significance using the Planck data. We take advantage of Planck's multifrequency coverage to co…
We use the cosmo-OWLS suite of cosmological hydrodynamical simulations, which includes different galactic feedback models, to predict the cross-correlation signal between weak gravitational lensing and the thermal Sunyaev-Zeldovich (tSZ) $y$-parameter. The predictions are compared to the recent detection reported by van Waerbeke and collaborators. The simulations reproduce the weak lensing-tSZ cross-correlation, $ξ_{yκ}(θ)$, well. The uncertainty arising from different possible feedback models appears to be important on small scales only ($θ\lesssim 10$ arcmin), while the amplitude of the corr…
We conduct a pseudo-$C_\ell$ analysis of the tomographic cross-correlation between 1000 deg$^2$ of weak lensing data from the Kilo-Degree Survey (KiDS-1000) and the thermal Sunyaev-Zeldovich (tSZ) effect measured by Planck and the Atacama Cosmology Telescope (ACT). Using HMx, a halo-model-based approach that consistently models the gas, star, and dark matter components, we are able to derive constraints on both cosmology and baryon feedback for the first time from these data, marginalising over redshift uncertainties, intrinsic alignment of galaxies, and contamination by the cosmic infrared ba…
We present a new method for joint cosmological parameter inference and cluster mass calibration from a combination of weak lensing measurements and the abundance of thermal Sunyaev-Zel'dovich (tSZ) selected galaxy clusters. We combine cluster counts with the spherical harmonic cosmic shear power spectrum and the cross-correlation between cluster overdensity and cosmic shear. These correlations constrain the cluster mass-observable relation. We model the observables using a halo model framework, including their full non-Gaussian covariance. Forecasting constraints on cosmological and mass calib…
The thermal Sunyaev-Zel'dovich (tSZ) effect is the inverse-Compton scattering of cosmic microwave background (CMB) photons off hot, ionized electrons, primarily located in galaxy groups and clusters. Recent years have seen immense improvement in our ability to probe cosmology and the astrophysics of the intracluster medium using the tSZ signal. Here, I describe cross-correlations of the tSZ effect measured in Planck data with gravitational lensing maps from Planck and the Canada-France-Hawaii Telescope Lensing Survey, as well as hydrodynamical simulations which show that such measurements do n…
We present measurements of the spatial mapping between (hot) baryons and the total matter in the Universe, via the cross-correlation between the thermal Sunyaev-Zeldovich (tSZ) map from Planck and the weak gravitational lensing maps from the Red Sequence Cluster Survey (RCSLenS). The cross-correlations are performed on the map level where all the sources (including diffuse intergalactic gas) contribute to the signal. We consider two configuration-space correlation function estimators, $ξ^{ y-κ}$ and $ξ^ {y-γ_{t}}$, and a Fourier space estimator, $C_{\ell}^{y-κ}$, in our analysis. We detect a s…
Recent first detections of the cross-correlation of the thermal Sunyaev-Zel'dovich (tSZ) signal in Planck cosmic microwave background (CMB) temperature maps with gravitational lensing maps inferred from the Planck CMB data and the CFHTLenS galaxy survey provide new probes of the relationship between baryons and dark matter. Using cosmological hydrodynamics simulations, we show that these cross-correlation signals are dominated by contributions from hot gas in the intracluster medium (ICM), rather than diffuse, unbound gas located beyond the virial radius (the "missing baryons"). Thus, these cr…
Tracing Missing Baryons in the Cosmic Filaments with tSZ and CMB-Lensing Stacking
School of Physics and Astronomy, Sun Yat-sen University（中山大学物理学院）；CSST Science Center for the Guangdong-Hong kong-Macau Greater Bay Area, SYSU（中山大学粤港澳大湾区深空探测科学中心）
We report the first detection of a correlation between gravitational lensing by large scale structure and the thermal Sunyaev-Zeldovich (tSZ) effect. Using the mass map from the Canada France Hawaii Telescope Lensing Survey (CFHTLenS) and a newly-constructed tSZ map from Planck, we measure a non-zero correlation between the two maps out to one degree angular separation on the sky, with an overall significance of 6 sigma. The tSZ maps are formed in a manner that removes primary cosmic microwave background fluctuations and minimizes residual contamination by galactic and extragalactic dust emiss…
The recent release of {\it Planck} data gives access to a full sky coverage of the thermal Sunyaev-Zel'dovich (tSZ) effect and of the cosmic microwave background (CMB) lensing potential ($ϕ$). The cross-correlation of these two probes of the large-scale structures in the Universe is a powerful tool for testing cosmological models, especially in the context of the difference between galaxy clusters and CMB for the best-fitting cosmological parameters. However, the tSZ effect maps are highly contaminated by cosmic infra-red background (CIB) fluctuations. Unlike other astrophysical components, th…
Evidence for a thermal pressure deficit in galaxy groups from the tSZ effect and weak lensing
Princeton University（普林斯顿大学）；Liverpool John Moores University（利物浦约翰摩尔斯大学）；DAMTP, Centre for Mathematical Sciences, University of Cambridge（剑桥大学数学科学中心应用数学与理论物理系）
Reconstructed maps of the lensing convergence of the cosmic microwave background (CMB) will play a major role in precision cosmology in coming years. CMB lensing maps will enable calibration of the masses of high-redshift galaxy clusters and will yield precise measurements of the growth of cosmic structure through cross-correlations with galaxy surveys. During the next decade, CMB lensing reconstruction will rely heavily on temperature data, rather than polarization, thus necessitating a detailed understanding of biases due to extragalactic foregrounds. In the near term, the most significant b…
Measurements of the thermal Sunyaev-Zel'dovich (tSZ) effect have long been recognized as a powerful cosmological probe. Here we assess the importance of relativistic temperature corrections to the tSZ signal on the power spectrum analysis of the Planck Compton-$y$ map, developing a novel formalism to account for the associated effects. The amplitude of the tSZ power spectrum is found to be sensitive to the effective electron temperature, $\bar{T}_e$, of the cluster sample. Omitting the corresponding modifications leads to an underestimation of the $yy$-power spectrum amplitude. Relativistic co…
The thermal Sunyaev-Zel'dovich (tSZ) effect measures the line-of-sight projection of the thermal pressure of free electrons and lacks any redshift information. By cross correlating the tSZ effect with an external cosmological tracer we can recover a good fraction of this lost information. Weak lensing (WL) is thought to provide an unbiased probe of the dark Universe, with many WL surveys having sky coverage that overlaps with tSZ surveys. Generalising the tomographic approach, we advocate the use of the spherical Fourier-Bessel (sFB) expansion to perform an analysis of the cross-correlation be…
The thermal Sunyaev-Zel'dovich (tSZ) effect directly measures the thermal pressure of free electrons integrated along the line of sight and thus contains valuable information on the thermal history of the universe. However, the redshift information is entangled in the projection along the line of sight. This projection effect severely degrades the power of the tSZ effect to reconstruct the thermal history. We investigate the tSZ tomography technique to recover this otherwise lost redshift information by cross correlating the tSZ effect with galaxies of known redshifts, or alternatively with ma…
Most secondary sources of cosmic microwave background anisotropy (radio sources, dusty galaxies -DSFGs-, thermal Sunyaev Zel'dovich -tSZ- distortions from hot gas, and gravitational lensing) are highly non-Gaussian. Statistics beyond the power spectrum are therefore potentially important sources of information about the physics of these processes. We combine data from the Atacama Cosmology Telescope and with data from the Planck satellite (only using Planck data in the overlapping region) to constrain the amplitudes of a set of theoretical bispectrum templates from the tSZ effect, DSFGs, gravi…
On the scale of galactic haloes, the distribution of matter in the cosmos is affected by energetic, non-gravitational processes; so-called baryonic feedback. A lack of knowledge about the details of how feedback processes redistribute matter is a source of uncertainty for weak-lensing surveys, which accurately probe the clustering of matter in the Universe over a wide range of scales. We develop a cosmology-dependent model for the matter distribution that simultaneously accounts for the clustering of dark matter, gas and stars. We inform our model by comparing it to power spectra measured from…
Galaxy clusters observed through the thermal Sunyaev-Zeldovich (tSZ) effect are a recent cosmological probe. The precision on the cosmological constraints is affected mainly by the current knowledge of cluster physics, which enters the analysis through the scaling relations. Here we aim to study one of the most important sources of systematic uncertainties, the mass bias, $b$. We have analysed the effects of a mass-redshift dependence, adopting a power-law parametrisation. We applied this parametrisation to the combination of tSZ number counts and power spectrum, finding a hint of redshift dep…
Secondary contributions to the anisotropy of the Cosmic Microwave Background (CMB), such as the integrated Sachs-Wolfe (ISW) effect, the thermal Sunyaev-Zel'dovich effect (tSZ), and the effect of gravitational lensing, have distinctive non-Gaussian signatures, and full descriptions therefore require information beyond that contained in their power spectra. In this paper we use the recently introduced skew-spectra associated with the Minkowski Functionals (MF) to probe the topology of CMB maps to probe the secondary non-Gaussianity as a function of beam-smoothing in order to separate various co…
We present a first joint analysis of the power spectra of the thermal Sunyaev-Zeldovich (tSZ) effect measured by the Planck and the number density fluctuations of galaxies in the 2MASS redshift survey (2MRS) catalog, including their cross-correlation. Combining these measurements with the cosmic microwave background (CMB) data and CMB lensing of Planck assuming a flat $Λ$CDM model, we constrain the mass bias parameter as $B = 1.54 \pm 0.098\;(1σ)$ [$(1-b) = 0.649 \pm 0.041$, where $(1-b) \equiv B^{-1}$], i.e., the Planck cluster mass should be $35\%$ lower than the true mass. The mass bias det…
Future high-resolution measurements of the cosmic microwave background (CMB) will produce catalogs of tens of thousands of galaxy clusters through the thermal Sunyaev-Zel'dovich (tSZ) effect. We forecast how well different configurations of a CMB Stage-4 experiment can constrain cosmological parameters, in particular the amplitude of structure as a function of redshift $σ_8(z)$, the sum of neutrino masses $Σm_ν$, and the dark energy equation of state $w(z)$. A key element of this effort is calibrating the tSZ scaling relation by measuring the lensing signal around clusters. We examine how the …
Thermal Sunyaev-Zeldovich (tSZ) power spectrum is a powerful probe of the present-day amplitude of matter density fluctuations, and has been measured up to $\ell\approx 10^3$ from the {\it Planck} data. The largest systematic uncertainty in the interpretation of this data is the so-called "mass bias" parameter $B$, which relates the true halo mass to the mass proxy used by the {\it Planck} team as $M_{\rm 500c}^{Planck}=M_{\rm 500c}^{\rm true}/B$. Since the power spectrum of the cosmic weak lensing shear is also sensitive to the amplitude of matter density fluctuations via $S_8\equiv σ_8 Ω_m^α…
We cross-correlate maps of the thermal Sunyaev-Zeldovich (tSZ) Compton-$y$ parameter published by Planck with the projected distribution of galaxies in a set of low-redshift tomographic bins. We use the nearly full-sky 2MASS Photometric Redshift and WISE $\times$ SuperCOSMOS public catalogues, covering the redshift range $z\lesssim0.4$. Our measurements allow us to place constraints on the redshift dependence of the mass-observable relation for tSZ cluster count analyses in terms of the so-called 'hydrostatic mass bias' parameter $1-b_{\rm H}$. These results can also be interpreted as measurem…
Running hydrodynamical simulations to produce mock data of large-scale structure and baryonic probes, such as the thermal Sunyaev-Zeldovich (tSZ) effect, at cosmological scales is computationally challenging. We propose to leverage the expressive power of deep generative models to find an effective description of the large-scale gas distribution and temperature. We train two deep generative models, a variational auto-encoder and a generative adversarial network, on pairs of matter density and pressure slices from the BAHAMAS hydrodynamical simulation. The trained models are able to successfull…
We constructed a new all-sky Compton parameter map (y-map) of the thermal Sunyaev-Zel'dovich (tSZ) effect from the 100 to 857 GHz frequency channel maps delivered within the Planck data release 4. The improvements in terms of noise and systematic effects translated into a y-map with a noise level smaller by ~7% compared to the maps released in 2015, and with significantly reduced survey stripes. The produced 2020 y-map is also characterized by residual foreground contamination, mainly due to thermal dust emission at large angular scales and to CIB and extragalactic point sources at small angul…
We study the scaling relations between the baryonic content and total mass of groups of galaxies, as these systems provide a unique way to examine the role of non-gravitational processes in structure formation. Using Planck and ROSAT data, we conduct detailed comparisons of the stacked thermal Sunyaev-Zel'dovich (tSZ) and X-ray scaling relations of galaxy groups found in the Galaxy And Mass Assembly (GAMA) survey and the BAHAMAS hydrodynamical simulation. We use weak gravitational lensing data from the Kilo Degree Survey (KiDS) to determine the average halo mass of the studied systems. We anal…
CMB Stage-4 experiments will reduce the uncertainties on the gravitational lensing potential by an order of magnitude compared to current measurements, and will also produce a Sunyaev-Zel'dovich (SZ) cluster catalog containing $\sim10^{5}$ objects, two orders of magnitudes higher than what is currently available. In this paper we propose to combine these two observables and show that it is possible to calibrate the masses of the full Stage-4 cluster catalog internally owing to the high signal to noise measurement of the CMB lensing convergence field. We find that a CMB Stage-4 experiment will …
We confront the universal pressure profile (UPP) proposed by~\citet{Arnaud10} with the recent measurement of the cross-correlation function of the thermal Sunyaev-Zeldovich (tSZ) effect from Planck and weak gravitational lensing measurement from the Red Cluster Sequence lensing survey (RCSLenS). By using the halo model, we calculate the prediction of $ξ^{y-κ}$ (lensing convergence and Compton-$y$ parameter) and $ξ^{y-γ_{\rm t}}$ (lensing shear and Compton-$y$ parameter) and fit the UPP parameters by using the observational data. We find consistent UPP parameters when fixing the cosmology to ei…
Observations of galaxies and galaxy clusters in the local universe can account for only $\sim\,10\%$ of the total baryon content. Cosmological simulations predict that the `missing baryons' are spread throughout filamentary structures in the cosmic web, forming a low-density gas with temperatures of $10^5-10^7\,\!$K. We search for this warm-hot intergalactic medium (WHIM) by stacking the Planck Compton $y$-parameter map of the thermal Sunyaev-Zel'dovich (tSZ) effect for 1,002,334 pairs of CMASS galaxies from the Sloan Digital Sky Survey. We model the contribution from the galaxy halo pairs ass…
We present a tomographic measurement of the cross-correlation between thermal Sunyaev-Zeldovich (tSZ) maps from ${\it Planck}$ and the Atacama Cosmology Telescope (ACT) and weak galaxy lensing shears measured during the first three years of observations of the Dark Energy Survey (DES Y3). This correlation is sensitive to the thermal energy in baryons over a wide redshift range, and is therefore a powerful probe of astrophysical feedback. We detect the correlation at a statistical significance of $21σ$, the highest significance to date. We examine the tSZ maps for potential contaminants, includ…
We study the impact of lensing corrections on modeling cross correlations between CMB lensing and galaxies, cosmic shear and galaxies, and galaxies in different redshift bins. Estimating the importance of these corrections becomes necessary in the light of anticipated high-accuracy measurements of these observables. While higher order lensing corrections (sometimes also referred to as post Born corrections) have been shown to be negligibly small for lensing auto correlations, they have not been studied for cross correlations. We evaluate the contributing four-point functions without making use…
We construct the mean thermal Sunyaev-Zel'dovich (tSZ) Comptonization y profile around Luminous Red Galaxies (LRGs) in the redshift range 0.16 < z < 0.47 from the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7) using the Planck y map. The mean central tSZ signal for the full sample is y ~ 1.8 * 10^(-7) and we detect tSZ emission out to ~30 arcmin, which is well beyond the 10 arcmin angular resolution of the y map and well beyond the virial radii of the LRGs. We compare the measured profile with predictions from the cosmo-OWLS suite of cosmological hydrodynamical simulations. This comparis…
It is usually assumed that the ellipticity power spectrum measured in weak lensing observations can be expressed as an integral over the underlying matter power spectrum. This is true at second order in the gravitational potential. We extend the standard calculation, constructing all corrections to fourth order in the gravitational potential. There are four types of corrections: corrections to the lensing shear due to multiple-deflections; corrections due to the fact that shape distortions probe the reduced shear $γ/(1-κ)$ rather than the shear itself; corrections associated with the non-linea…
We measure the cross-correlation between galaxy groups constructed from DESI Legacy Imaging Survey DR8 and \emph{Planck} CMB lensing, over overlapping sky area of 16876 $\rm deg^2$. The detections are significant and consistent with the expected signal of the large-scale structure of the universe, over group samples of various redshift, mass, richness $N_{\rm g}$ and over various scale cuts. The overall S/N is 40 for a conservative sample with $N_{\rm g}\geq 5$, and increases to $50$ for the sample with $N_{\rm g}\geq 2$. Adopting the \emph{Planck} 2018 cosmology, we constrain the density bias…
