We use a set of N-body simulations employing a modified gravity (MG) model with Vainshtein screening to study matter and halo hierarchical clustering. As test-case scenarios we consider two normal branch Dvali-Gabadadze-Porrati (nDGP) gravity models with mild and strong growth rate enhancement. We study higher-order correlation functions $ξ_n(R)$ up to $n=9$ and associated hierarchical amplitudes $S_n(R)\equivξ_n(R)/σ(R)^{2n-2}$. We find that the matter PDFs are strongly affected by the fifth-force on scales up to $50h^{-1}$Mpc, and the deviations from GR are maximised at $z=0$. For reduced cumulants $S_n$, we find that at small scales $R\leq10h^{-1}$Mpc the MG is characterised by lower values, with the deviation growing from $7\%$ in the reduced skewness up to even $40\%$ in $S_5$. To study the halo clustering we use a simple abundance matching and divide haloes into thee fixed number density samples. The halo two-point functions are weakly affected, with a relative boost of the order of a few percent appearing only at the smallest pair separations ($r\leq 5h^{-1}$Mpc). In contrast, we find a strong MG signal in $S_n(R)$'s, which are enhanced compared to GR. The strong model exhibits a $>3σ$ level signal at various scales for all halo samples and in all cumulants. In this context, we find that the reduced kurtosis to be an especially promising cosmological probe of MG. Even the mild nDGP model leaves a $3σ$ imprint at small scales $R\leq3h^{-1}$Mpc, while the stronger model deviates from a GR-signature at nearly all scales with a significance of $>5σ$. Since the signal is persistent in all halo samples and over a range of scales, we advocate that the reduced kurtosis estimated from galaxy catalogues can potentially constitute a strong MG-model discriminatory as well as GR self-consistency test.
Halo bias is the one of the key ingredients of the halo models. It was shown at a given redshift to be only dependent, to the first order, on the halo mass. In this study, four types of cosmic web environments: clusters, filaments, sheets and voids are defined within a state of the art high resolution $N$-body simulation. Within those environments, we use both halo-dark matter cross-correlation and halo-halo auto correlation functions to probe the clustering properties of halos. The nature of the halo bias differs strongly among the four different cosmic web environments we describe. With resp…
Interpreting the small-scale clustering of galaxies with halo models can elucidate the connection between galaxies and dark matter halos. Unfortunately, the modelling is typically not sufficiently accurate for ruling out models statistically. It is thus difficult to use the information encoded in small scales to test cosmological models or probe subtle features of the galaxy-halo connection. In this paper, we attempt to push halo modelling into the "accurate" regime with a fully numerical mock-based methodology and careful treatment of statistical and systematic errors. With our forward-modell…
We present a quantitative measurement of the amount of clustering present in the inner $\sim30$ kpc of the stellar halo of the Andromeda galaxy (M31). For this we analyse the angular positions and radial velocities of the carefully selected Planetary Nebulae (PNe) in the M31 stellar halo. We study the cumulative distribution of pair-wise distances in angular position and line of sight velocity space, and find that the M31 stellar halo contains substantially more stars in the form of close pairs as compared to that of a featureless smooth halo. In comparison to a smoothed/scrambled distribution…
The simplest analyses of halo bias assume that halo mass alone determines halo clustering. However, if the large scale environment is fixed, then halo clustering is almost entirely determined by environment, and is almost completely independent of halo mass. We show why. Our analysis is useful for studies which use the environmental dependence of clustering to constrain cosmological and galaxy formation models. It also shows why many correlations between galaxy properties and environment are merely consequences of the underlying correlations between halos and their environments, and provides a…
Understanding the impact of halo properties beyond halo mass on the clustering of galaxies (namely galaxy assembly bias) remains a challenge for contemporary models of galaxy clustering. We explore the use of machine learning to predict the halo occupations and recover galaxy clustering and assembly bias in a semi-analytic galaxy formation model. For stellar-mass selected samples, we train a Random Forest algorithm on the number of central and satellite galaxies in each dark matter halo. With the predicted occupations, we create mock galaxy catalogues and measure the clustering and assembly bi…
We present a new theory for the hierarchical clustering of dark matter (DM) halos based on stochastic differential equations, that constitutes a change of perspective with respect to existing frameworks (e.g., the excursion set approach); this work is specifically focused on the halo mass function. First, we present a stochastic differential equation that describes fluctuations in the mass growth of DM halos, as driven by a multiplicative white (Gaussian) noise dependent on the spherical collapse threshold and on the power spectrum of DM perturbations. We demonstrate that such a noise yields a…
We extend the results of previous analyses towards constraining the abundance and clustering of post-reionization ($z \sim 0-5$) neutral hydrogen (HI) systems using a halo model framework. We work with a comprehensive HI dataset including the small-scale clustering, column density and mass function of HI galaxies at low redshifts, intensity mapping measurements at intermediate redshifts and the UV/optical observations of Damped Lyman Alpha (DLA) systems at higher redshifts. We use a Markov Chain Monte Carlo (MCMC) approach to constrain the parameters of the best-fitting models, both for the HI…
We study the dependence of galaxy clustering on atomic gas mass using a sample of $\sim$16,000 galaxies with redshift in the range of $0.0025<z<0.05$ and HI mass of $M_{\rm HI}>10^8M_{\odot}$, drawn from the 70% complete sample of the Arecibo Legacy Fast ALFA survey. We construct subsamples of galaxies with $M_{\rm HI}$ above different thresholds, and make volume-limited clustering measurements in terms of three statistics: the projected two-point correlation function, the projected cross-correlation function with respect to a reference sample selected from the Sloan Digital Sky Survey, and th…
Applying halo models to analyze the small-scale clustering of galaxies is a proven method for characterizing the connection between galaxies and their host halos. Such works are often plagued by systematic errors or are limited to clustering statistics which can be predicted analytically. In this work, we employ a numerical mock-based modeling procedure to examine the clustering of SDSS DR7 galaxies. We apply a standard halo occupation distribution (HOD) model to dark-matter-only simulations with a LCDM cosmology. To constrain the theoretical models, we utilize a combination of galaxy number d…
We present a fully analytical halo model of colour-dependent clustering that incorporates the effects of galactic conformity in a halo occupation distribution (HOD) framework. The model, based on our previous numerical work, describes conformity through a correlation between the colour of a galaxy and the concentration of its parent halo, leading to a correlation between central and satellite galaxy colours at fixed halo mass. The strength of the correlation is set by a tunable `group quenching efficiency', and the model can separately describe group-level correlations between galaxy colour (1…
We present a study of the clustering and halo occupation distribution of BOSS CMASS galaxies in the redshift range 0.43 < z < 0.7 drawn from the Final SDSS-III Data Release. We compare the BOSS results with the predictions of a Halo Abundance Matching (HAM) clustering model that assigns galaxies to dark matter halos selected from the large BigMultiDark $N$-body simulation of a flat $Λ$CDM Planck cosmology. We compare the observational data with the simulated ones on a light-cone constructed from 20 subsequent outputs of the simulation. Observational effects such as incompleteness, geometry, ve…
We measure the angular clustering of galaxies from the Sloan Digital Sky Survey Data Release 7 in order to probe the spatial distribution of satellite galaxies within their dark matter halos. Specifically, we measure the angular correlation function on very small scales (7-320") in a range of luminosity threshold samples (absolute r-band magnitudes of -18 up to -21) that are constructed from the subset of SDSS that has been spectroscopically observed more than once (the so-called plate overlap region). We choose to measure angular clustering in this reduced survey footprint in order to minimiz…
We use the Millennium simulation to show that halo clustering varies significantly with cosmic web type. Halos are classified as node, filament, sheet and void halos based on the eigenvalue decomposition of the velocity shear tensor. The velocity field is sampled by the peculiar velocities of a fixed number of neighbouring halos and spatial derivatives are computed using a kernel borrowed from smoothed particle hydrodynamics. The classification scheme is used to examine the clustering of halos as a function of web type for halos with masses larger than $10^{11}$. We find that node halos show p…
We use the IllustrisTNG (TNG) simulations to explore the galaxy-halo connection as inferred from state-of-the-art cosmological, magnetohydrodynamical simulations. With the high mass resolution and large volume achieved by combining the 100 Mpc (TNG100) and 300 Mpc (TNG300) volumes, we establish the mean occupancy of central and satellite galaxies and their dependence on the properties of the dark matter haloes hosting them. We derive best-fitting HOD parameters from TNG100 and TNG300 for target galaxy number densities of $\bar{n}_g = 0.032\,h^3$Mpc$^{-3}$ and $\bar{n}_g = 0.016\,h^3$Mpc$^{-3}$…
In the framework of the stochastic theory for hierarchical clustering, we investigate the time-dependent solutions of the Fokker-Planck equation describing the statistics of dark matter halos, and discuss the typical timescales needed for these to converge toward stationary states, far away enough from initial conditions. Although we show that the stationary solutions can reproduce the outcomes of state-of-the-art $N-$body simulations at $z\approx 0$ to a great accuracy, one needs to go beyond to fully account for the cosmic evolution of the simulated halo mass function toward high-redshift. S…
We utilize $Λ$CDM halo occupation models of galaxy clustering to investigate the evolving stellar mass dependent clustering of galaxies in the PRIsm MUlti-object Survey (PRIMUS) and DEEP2 Redshift Survey over the past eight billion years of cosmic time, between $0.2<z<1.2$. These clustering measurements provide new constraints on the connections between dark matter halo properties and galaxy properties in the context of the evolving large-scale structure of the universe. Using both an analytic model and a set of mock galaxy catalogs, we find a strong correlation between central galaxy stellar …
Halo assembly bias is the secondary dependence of the clustering of dark-matter haloes on their assembly histories at fixed halo mass. This established dependence is expected to manifest itself on the clustering of the galaxy population, a potential effect commonly known as galaxy assembly bias. Using the IllustrisTNG300 magnetohydrodynamical simulation, we analyse the dependence of the properties and clustering of galaxies on the shape of the specific mass accretion history of their hosting haloes (sMAH). We first show that several halo and galaxy properties strongly correlate with the slope …
Halo models provide a simple and computationally inexpensive way to investigate the connection between galaxies and their dark matter haloes. However, these models rely on the assumption that the role of baryons can be easily parametrized in the modelling procedure. We aim to examine the ability of halo occupation distribution (HOD) modelling to reproduce the galaxy clustering found in two different hydrodynamic simulations, Illustris and EAGLE. For each simulation, we measure several galaxy clustering statistics on two different luminosity threshold samples. We then apply a simple five parame…
We present a Halo Occupation Distribution (HOD) analysis of the luminosity- and colour-dependent galaxy clustering in the Sloan Digital Sky Survey. A novelty of our technique is that it uses a combination of clustering measurements in luminosity bins to perform a global likelihood analysis, simultaneously constraining the HOD parameters for a range of luminosity thresholds. We present simple, smooth fitting functions which accurately describe the resulting luminosity dependence of the best-fit HOD parameters. To minimise systematic halo modelling effects, we use theoretical halo 2-point correl…
Luminosity-Dependent Assembly Bias of Central Galaxies from Weak Lensing and Clustering
Shanghai Jiao Tong University（上海交通大学）；Graduate School of Science, Nagoya University（名古屋大学理学研究科）；Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya University（名古屋大学小林-益川粒子宇宙起源研究所）；Institute for Advanced Research, Nagoya University（名古屋大学高等研究院）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo（东京大学理化学研究所 Kavli 宇宙粒子物理学研究所）；Leiden Observatory, Leiden University（莱顿大学莱顿天文台）；Lorentz Institute for Theoretical Physics, Leiden University（莱顿大学洛伦兹理论物理研究所）；Academia Sinica Institute of Astronomy and Astrophysics (ASIAA)（中央研究院天文及天文物理研究所）
Understanding the connections between galaxy stellar mass, star formation rate, and dark matter halo mass represents a key goal of the theory of galaxy formation. Cosmological simulations that include hydrodynamics, physical treatments of star formation, feedback from supernovae, and the radiative transfer of ionizing photons can capture the processes relevant for establishing these connections. The complexity of these physics can prove difficult to disentangle and obfuscate how mass-dependent trends in the galaxy population originate. Here, we train a machine learning method called Explainabl…
The clustering of dark matter halos depends on the assembly history of halos at fixed halo mass; a phenomenon referred to as \textit{halo assembly bias}. Halo assembly bias is readily observed in cosmological simulations of dark matter. However, it is difficult to detect it in observations. The identification of galaxy or cluster properties correlated with the formation time of the halo at fixed halo mass and the ability to select galaxy clusters free from projection effects are the two most significant hurdles in the observational detection of halo assembly bias. The latter, in particular, ca…
Clustering of dark matter halos has been shown to depend on halo properties beyond mass such as halo concentration, a phenomenon referred to as halo assembly bias. Standard halo occupation models (HOD) in large scale structure studies assume that halo mass alone is sufficient in characterizing the connection between galaxies and halos. Modeling of galaxy clustering can face systematic effects if the number of galaxies within a halo is correlated with other halo properties. Using the Small MultiDark-Planck high resolution $N$-body simulation and the clustering measurements of the Sloan Digital …
We investigate a conceptual modification of the halo occupation distribution approach, using the halos' present-day maximal circular velocity, $\vmax$, as an alternative to halo mass. In particular, using a semi-analytic galaxy formation model applied to the Millennium WMAP7 simulation, we explore the extent that switching to $\vmax$ as the primary halo property incorporates the effects of assembly bias into the formalism. We consider fixed number density galaxy samples ranked by stellar mass and examine the variations in the halo occupation functions with either halo concentration or formatio…
One of the main predictions of excursion set theory is that the clustering of dark matter haloes only depends on halo mass. However, it has been long established that the clustering of haloes also depends on other properties, including formation time, concentration, and spin; this effect is commonly known as halo assembly bias. We use a suite of gravity-only simulations to study the dependence of halo assembly bias on cosmology; these simulations cover cosmological parameters spanning 10$σ$ around state-of-the-art best-fitting values, including standard extensions of the $Λ$CDM paradigm such a…
A promising route for revealing the existence of dark matter structures on mass scales smaller than the faintest galaxies is through their effect on strong gravitational lenses. We examine the role of local, lens-proximate clustering in boosting the lensing probability relative to contributions from substructure and unclustered line-of-sight (LOS) halos. Using two cosmological simulations that can resolve halo masses of $M_{\rm halo} \simeq 10^{9}\ M_{\odot}$ (in a simulation box of length $L_{\rm box}{\sim}100\,{\rm Mpc}$) and $10^{7}\ M_{\odot}$ ($L_{\rm box}\sim20\,{\rm Mpc}$), we demonstra…
We study the dependence of angular two-point correlation functions on stellar mass ($M_{*}$) and specific star formation rate (sSFR) of $M_{*}>10^{10}M_{\odot}$ galaxies at $z\sim1$. The data from UKIDSS DXS and CFHTLS covering 8.2 deg$^{2}$ sample scales larger than 100 $h^{-1}$Mpc at $z\sim1$, allowing us to investigate the correlation between clustering, $M_{*}$, and star formation through halo modeling. Based on halo occupation distributions (HODs) of $M_{*}$ threshold samples, we derive HODs for $M_{*}$ binned galaxies, and then calculate the $M_{*}/M_{\rm halo}$ ratio. The ratio for cent…
We investigate the clustering of Lyman-break galaxies (LBGs) at $z\sim4$. Using the hierarchical galaxy formation model GALFORM, we predict, for the first time using a semi-analytical model with feedback from active galactic nuclei (AGN), the angular correlation function (ACF) of LBGs and find agreement within $3\,σ$ with new measurements of the ACF from surveys including the Hubble eXtreme Deep Field (XDF) and CANDELS field. Our simulations confirm the conclusion reached using independent models that although the predicted ACFs reproduce the trend of increased clustering with luminosity, the …
Solving Milky Way-sized Systems with Haskap Pie: A Halo finding Algorithm with efficient Sampling, K-means clustering, tree-Assembly, Particle tracking, Python modules, Inter-code applicability, and Energy solving
We present the clustering properties of low-$z$ $(z\leq1.4)$ galaxies selected by the Hyper Suprime-Cam Subaru Strategic Program Wide layer over $145$ deg$^{2}$. The wide-field and multi-wavelength observation yields $5,064,770$ galaxies at $0.3\leq z\leq1.4$ with photometric redshifts and physical properties. This enables the accurate measurement of angular correlation functions and subsequent halo occupation distribution (HOD) analysis allows the connection between baryonic properties and dark halo properties. The fraction of less-massive satellite galaxies at $z\lesssim1$ is found to be alm…
We model the luminosity-dependent projected and redshift-space two-point correlation functions (2PCFs) of the Sloan Digital Sky Survey (SDSS) DR7 Main galaxy sample, using the halo occupation distribution (HOD) model and the subhalo abundance matching (SHAM) model and its extension. All the models are built on the same high-resolution $N$-body simulations. We find that the HOD model generally provides the best performance in reproducing the clustering measurements in both projected and redshift spaces. The SHAM model with the same halo-galaxy relation for central and satellite galaxies (or dis…
The AIDA-TNG project. Abundance, radial distribution, and clustering properties of halos in alternative dark matter models
Alma Mater Studiorum Università di Bologna（博洛尼亚大学）；INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna（意大利国家天体物理研究所博洛尼亚天文台）；INFN - Sezione di Bologna（意大利国家核物理研究所博洛尼亚分部）
We perform an ensemble of $N$-body simulations with $2048^3$ particles for 101 flat $w$CDM cosmological models sampled based on a maximin-distance Sliced Latin Hypercube Design. By using the halo catalogs extracted at multiple redshifts in the range of $z=[0,1.48]$, we develop Dark Emulator, which enables fast and accurate computations of the halo mass function, halo-matter cross-correlation, and halo auto-correlation as a function of halo masses, redshift, separations and cosmological models, based on the Principal Component Analysis and the Gaussian Process Regression for the large-dimension…
It is widely reported, based on clustering measurements of observed active galactic nuclei (AGN) samples, that AGN reside in similar mass host dark matter halos across the bulk of cosmic time, with log $M/M_\odot$~12.5-13.0 to z~2.5. We show that this is due in part to the AGN fraction in galaxies rising with increasing stellar mass, combined with AGN observational selection effects that exacerbate this trend. Here, we use AGN specific accretion rate distribution functions determined as a function of stellar mass and redshift for star-forming and quiescent galaxies separately, combined with th…
We present clustering analysis results from 10,381 Lyman break galaxies (LBGs) at z~ 4-7, identified in the Hubble legacy deep imaging and new complimentary large-area Subaru/Hyper Suprime-Cam data. We measure the angular correlation functions (ACFs) of these LBGs at z~4, 5, 6, and 7, and fit these measurements using halo occupation distribution (HOD) models that provide an estimate of halo masses, M_h~(1-20)x10^11 Msun. Our M_h estimates agree with those obtained by previous clustering studies in a UV-magnitude vs. M_h plane, and allow us to calculate stellar-to-halo mass ratios (SHMRs) of LB…
We determine the error introduced in a joint halo model analysis of galaxy-galaxy lensing and galaxy clustering observables when adopting the standard approximation of linear halo bias. Considering the Kilo-Degree Survey, we forecast that ignoring the non-linear halo bias would result in up to 5$σ$ offsets in the recovered cosmological parameters describing structure growth, $S_8$, and the matter density parameter, $Ω_{\mathrm{m}}$. We include the scales $10^{-1.3}<r_{\rm{p}} \ / h^{-1}\, \mathrm{Mpc}<10$ in the data vector, and the direction of these offsets are shown to depend on the freedom…
The analysis of cosmological galaxy surveys requires realistic simulations for their interpretation. Forward modelling is a powerful method to simulate galaxy clustering without the need for an underlying complex model. This approach requires fast cosmological simulations with a high resolution and large volume, to resolve small dark matter halos associated to single galaxies. In this work, we present fast halo and subhalo clustering simulations based on the Lagrangian perturbation theory code PINOCCHIO, which generates halos and merger trees. The subhalo progenitors are extracted from the mer…
Halo assembly bias in the early Universe: a clustering probe of the origin of the Little Red Dots
Peking University（北京大学）；Kavli Institute for Astronomy and Astrophysics, Peking University（北京大学科维理天文与天体物理研究所）；Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology（麻省理工学院科维理天体物理与空间研究）
We explore the phenomenon commonly known as halo assembly bias, whereby dark matter halos of the same mass are found to be more or less clustered when a second halo property is considered, for halos in the mass range $3.7 \times 10^{11} \; h^{-1} \mathrm{M_{\odot}} - 5.0 \times 10^{13} \; h^{-1} \mathrm{M_{\odot}}$. Using the Large Suite of Dark Matter Simulations (LasDamas) we consider nine commonly used halo properties and find that a clustering bias exists if halos are binned by mass or by any other halo property. This secondary bias implies that no single halo property encompasses all the …
We incorporate the non-linear clustering of dark matter halos, as modelled by Jose et al. (2016) into the halo model to better understand the clustering of Lyman break galaxies (LBGs) in the redshift range $z=3-5$. We find that, with this change, the predicted LBG clustering increases significantly on quasi-linear scales ($0.1 \leq r\,/\,h^{-1} \,{\rm Mpc} \leq 10$) compared to that in the linear halo bias model. This in turn results in an increase in the clustering of LBGs by an order of magnitude on angular scales $5" \leq θ\leq 100"$. Remarkably, the predictions of our new model on the whol…
Dark matter halo clustering depends not only on halo mass, but also on other properties such as concentration and shape. This phenomenon is known broadly as assembly bias. We explore the dependence of assembly bias on halo definition, parametrized by spherical overdensity parameter, $Δ$. We summarize the strength of concentration-, shape-, and spin-dependent halo clustering as a function of halo mass and halo definition. Concentration-dependent clustering depends strongly on mass at all $Δ$. For conventional halo definitions ($Δ\sim 200\mathrm{m}-600\mathrm{m}$), concentration-dependent cluste…
Using dark matter simulations we show how halo bias is determined by local density and not by halo mass. This is not totally surprising, as according to the peak-background split model, local density is the property that constraints bias at large scales. Massive haloes have a high clustering because they reside in high density regions. Small haloes can be found in a wide range of environments which determine their clustering amplitudes differently. This contradicts the assumption of standard Halo Occupation Distribution (HOD) models that the bias and occupation of haloes is determined solely b…
Associating the formation sites of haloes with the maxima of the smoothed linear density field, we present non-perturbative predictions for the Lagrangian and evolved halo correlation functions that are valid at all separations. In Lagrangian space, we find significant deviations from the perturbative bias calculation at small scales, in particular, a pronounced exclusion region where $ξ=-1$ for maxima of unequal height. Our predictions are in good agreement with the Lagrangian clustering of dark matter proto-haloes reconstructed from N-body simulations. Our predictions for the mean infall and…
High-resolution cosmological N-body simulations are excellent tools for modelling the formation and clustering of dark matter haloes. These simulations suggest complex physical theories of halo formation governed by a set of effective physical parameters. Our goal is to extract these parameters and their uncertainties in a Bayesian context. We make a step towards automatising this process by directly comparing dark matter density projection maps extracted from cosmological simulations, with density projections generated from an analytical halo model. The model is based on a toy implementation …
The anisotropy or triaxiality of massive dark matter haloes largely defines the structure of the cosmic web, in particular the filaments that join the haloes together. Here we investigate such oriented correlations in mass-Peak Patch halo catalogues by using the initial strain tensor of spherical proto-halo regions to orient the haloes. To go beyond the spherically averaged two-point correlation function of haloes we use oriented stacks to compute oriented two-point correlations: we explicitly break isotropy by imposing a local frame set by the strain tensor of the reference halo before stacki…
We combine the most precise small scale ($< 100\, \rm h^{-1}kpc$) quasar clustering constraintsto date with recent measurements at large scales ($> 1\, \rm h^{-1}Mpc$) from the extended Baryon Oscillation Spectroscopic Survey (eBOSS) to better constrain the satellite fraction of quasars at $z\sim 1.5$ in the halo occupation formalism. We build our Halo Occupation Distribution (HOD) framework based on commonly used analytic forms for the one and two-halo terms with two free parameters: the minimum halo mass that hosts a central quasar and the fraction of satellite quasars that are within one ha…
The mapping of galaxy clustering from real space to redshift space introduces the anisotropic property to the measured galaxy density power spectrum in redshift space, known as the redshift space distortion (RSD) effect. The mapping formula is intrinsically non-linear, which is complicated by the higher order polynomials due to indefinite orders of cross correlations between density and velocity fields, and the Finger--of--God (FoG) effect due to the randomness of the galaxy peculiar velocity field. In previous works, we have verified the robustness of advanced TNS mapping formula in our hybri…
