Submanifolds with constant Moebius curvature and flat normal bundle
We classify isometric immersions $f\colon M^{n}\to \mathbb{R}^{n+p}$, $n \geq 5$ and $2p \leq n$, with constant Moebius curvature and flat normal bundle.
math.DG↗
arXiv subjects
R. Tojeiro
Publications and source records attributed to R. Tojeiro.
Infinitesimally Moebius bendable hypersurfaces
Li, Ma and Wang have provided in [\emph{Deformations of hypersurfaces preserving the M\"obius metric and a reduction theorem}, Adv. Math. 256 (2014), 156--205] a partial classification of the so-called Moebius deformable hypersurfaces, that is, the umbilic-free Euclidean hypersurfaces $f\colon M^n\to \mathbb{R}^{n+1}$ that admit non-trivial deformations preserving the Moebius metric. For $n\geq 5$, the classification was completed by the authors in \cite{JT2}. In this article we obtain an infinitesimal version of that classification. Namely, we introduce the notion of an infinitesimal Moebius variation of an umbilic-free immersion $f\colon M^n\to \mathbb{R}^m$ into Euclidean space as a one-parameter family of immersions $f_t\colon M^n\to \mathbb{R}^m$, with $t\in (-\epsilon, \epsilon)$ and $f_0=f$, such that the Moebius metrics determined by $f_t$ coincide up to the first order. Then we characterize isometric immersions $f\colon M^n\to \mathbb{R}^m$ of arbitrary codimension that admit a non-trivial infinitesimal Moebius variation among those that admit a non-trivial conformal infinitesimal variation, and use such characterization to classify the umbilic-free Euclidean hypersurfaces of dimension $n\geq 5$ that admit non-trivial infinitesimal Moebius variations.
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The DESI Survey Validation: Results from Visual Inspection of Bright Galaxies, Luminous Red Galaxies, and Emission Line Galaxies
The Dark Energy Spectroscopic Instrument (DESI) Survey has obtained a set of spectroscopic measurements of galaxies to validate the final survey design and target selections. To assist in these tasks, we visually inspect (VI) DESI spectra of approximately 2,500 bright galaxies, 3,500 luminous red galaxies (LRGs), and 10,000 emission line galaxies (ELGs), to obtain robust redshift identifications. We then utilize the VI redshift information to characterize the performance of the DESI operation. Based on the VI catalogs, our results show that the final survey design yields samples of bright galaxies, LRGs, and ELGs with purity greater than $99\%$. Moreover, we demonstrate that the precision of the redshift measurements is approximately 10 km/s for bright galaxies and ELGs and approximately 40 km/s for LRGs. The average redshift accuracy is within 10 km/s for the three types of galaxies. The VI process also helps improve the quality of the DESI data by identifying spurious spectral features introduced by the pipeline. Finally, we show examples of unexpected real astronomical objects, such as Ly$\alpha$ emitters and strong lensing candidates, identified by VI. These results demonstrate the importance and utility of visually inspecting data from incoming and upcoming surveys, especially during their early operation phases.
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On the Moebius deformable hypersurfaces
In the article [\emph{Deformations of hypersurfaces preserving the M\"obius metric and a reduction theorem}, Adv. Math. 256 (2014), 156--205], Li, Ma and Wang investigated the interesting class of Moebius deformable hypersurfaces, that is, the umbilic-free Euclidean hypersurfaces $f\colon M^n\to \mathbb{R}^{n+1}$ that admit non-trivial deformations preserving the Moebius metric. The classification of Moebius deformable hypersurfaces of dimension $n\geq 4$ stated in the aforementioned article, however, misses a large class of examples. In this article we complete that classification for $n\geq 5$.
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Ribaucour partial tubes and hypersurfaces of Enneper type
In this article we introduce the notion of a Ribaucour partial tube and use it to derive several applications. These are based on a characterization of Ribaucour partial tubes as the immersions of a product of two manifolds into a space form such that the distributions given by the tangent spaces of the factors are orthogonal to each other with respect to the induced metric, are invariant under all shape operators, and one of them is spherical. Our first application is a classification of all hypersurfaces with dimension at least three of a space form that carry a spherical foliation of codimension one, extending previous results by Dajczer, Rovenski and the second author for the totally geodesic case. We proceed to prove a general decomposition theorem for immersions of product manifolds, which extends several related results. Other main applications concern the class of hypersurfaces of $\mathbb{R}^{n+1}$ that are of Enneper type, that is, hypersurfaces that carry a family of lines of curvature, correspondent to a simple principal curvature, whose orthogonal $(n-1)$-dimensional distribution is integrable and whose leaves are contained in hyperspheres or affine hyperplanes of $\mathbb{R}^{n+1}$. We show how Ribaucour partial tubes in the sphere can be used to describe all $n$-dimensional hypersurfaces of Enneper type for which the leaves of the $(n-1)$-dimensional distribution are contained in affine hyperplanes of $\mathbb{R}^{n+1}$, and then show how a general hypersurface of Enneper type can be constructed in terms of a hypersurface in the latter class. We give an explicit description of some special hypersurfaces of Enneper type, among which are natural generalizations of the so called Joachimsthal surfaces.
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Umbilical submanifolds of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$
In this article we complete the classification of the umbilical submanifolds of a Riemannian product of space forms, addressing the case of a conformally flat product $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$, which has not been covered in previous works on the subject. We show that there exists precisely a $p$-parameter family of congruence classes of umbilical submanifolds of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$ with substantial codimension~$p$, which we prove to be at most $\mbox{min}\,\{k+1, n-k+2\}$. We study more carefully the cases of codimensions one and two and exhibit, respectively, a one-parameter family and a two-parameter family (together with three extra one-parameter families) that contain precisely one representative of each congruence class of such submanifolds. In particular, this yields another proof of the classification of all (congruence classes of) umbilical submanifolds of $\mathbb{S}^n\times \mathbb{R}$, and provides a similar classification for the case of $\mathbb{H}^n\times \mathbb{R}$. We determine all possible topological types, actually, diffeomorphism types, of a complete umbilical submanifold of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$. We also show that umbilical submanifolds of the product model of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$ can be regarded as rotational submanifolds in a suitable sense, and explicitly describe their profile curves when $k=n$. As a consequence of our investigations, we prove that every conformal diffeomorphism of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$ onto itself is an isometry.
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The VIMOS Public Extragalactic Redshift Survey (VIPERS) -- Environment-size relation of massive passive galaxies at 0.5 < z < 0.8
We use the statistics of the VIPERS survey to investigate the relation between the surface mean stellar mass density Sigma=Mstar/(2*pi*Re^2) of massive passive galaxies (MPGs, Mstar>10^11 Msun) and their environment in the redshift range 0.5 2*10^11 Msun, we find an excess of MPGs with low Sigma and a deficit of high-Sigma MPGs in the densest regions wrt other environments. We interpret this result as due to the migration of some high-Sigma MPGs (<1% of the total population of MPGs) into low-Sigma MPGs, probably through mergers or cannibalism of small satellites. In summary, our results imply that the accretion of satellite galaxies has a marginal role in the mass-assembly history of most MPGs. We have previously found that the number density of VIPERS massive star-forming galaxies (MSFGs) declines rapidily from z=0.8 to z=0.5, which mirrors the rapid increase in the number density of MPGs. This indicates that the MSFGs at z>0.8 migrate to the MPG population. Here, we investigate the Sigma-delta relation of MSFGs at z>0.8 and find that it is consistent within 1 sigma with that of low-Sigma MPGs at z<0.8. Thus, the results of this and our previous paper show that MSFGs at z>0.8 are consistent in terms of number and environment with being the progenitors of low-Sigma MPGs at z<0.8.
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The VIMOS Public Extragalactic Redshift Survey (VIPERS). Unsupervised classification with photometric redshifts: a method to accurately classify large galaxy samples without spectroscopic information
Techniques to classify galaxies solely based on photometry will be necessary for future large cosmology missions, such as Euclid or LSST. However, the precision of classification is always lower in photometric surveys and can be systematically biased with respect to classifications based upon spectroscopic data. We verified how precisely the detailed classification scheme introduced by Siudek et al. (2018, hereafter: S1) for galaxies at z~0.7 could be reproduced if only photometric data are available. We applied the Fisher Expectation-Maximization (FEM) unsupervised clustering algorithm to 54,293 VIPERS galaxies working in a parameter space of reliable photometric redshifts and 12 corresponding rest-frame magnitudes. The FEM algorithm distinguishes four main groups: (1) red, (2) green, (3) blue, and (4) outliers. Each group is further divided into 3, 3, 4, and 2 subclasses, respectively. The accuracy of reproducing galaxy classes using spectroscopic data is high: 92%, 84%, 96% for red, green, and blue classes, respectively, except for dusty star-forming galaxies. The presented verification of the photometric classification demonstrates that large photometric samples can be used to distinguish different galaxy classes at z > 0.5 with an accuracy provided so far only by spectroscopic data except for particular galaxy classes.
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The VIMOS Public Extragalactic Redshift Survey (VIPERS). The complexity of galaxy populations at 0.4< z<1.3 revealed with unsupervised machine-learning algorithms
Various galaxy classification schemes have been developed so far to constrain the main physical processes regulating evolution of different galaxy types. In the era of a deluge of astrophysical information and recent progress in machine learning, a new approach to galaxy classification becomes imperative. We employ a Fisher Expectation-Maximization unsupervised algorithm working in a parameter space of 12 rest-frame magnitudes and spectroscopic redshift. The model (DBk) and the number of classes (12) were established based on the joint analysis of standard statistical criteria and confirmed by the analysis of the galaxy distribution with respect to a number of classes and their properties. This new approach allows us to classify galaxies based just on their redshifts and UV-NIR spectral energy distributions. The FEM unsupervised algorithm has automatically distinguished 12 classes: 11 classes of VIPERS galaxies and an additional class of broad-line AGNs. After a first broad division into blue, green and red categories we obtained a further sub-division into three red, three green, and five blue galaxy classes. The FEM classes follow the galaxy sequence from the earliest to the latest types that is reflected in their colours (which are constructed from rest-frame magnitudes used in classification procedure) but also their morphological, physical, and spectroscopic properties (not included in the classification scheme). We demonstrate that the members of each class share similar physical and spectral properties. In particular, we are able to find three different classes of red passive galaxy populations. Thus, we demonstrate the potential of an unsupervised approach to galaxy classification and we retrieve the complexity of galaxy populations at z~0.7, a task that usual simpler colour-based approaches cannot fulfil.
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The VIMOS Public Extragalactic Redshift Survey (VIPERS): Unbiased clustering estimate with VIPERS slit assignment
The VIPERS galaxy survey has measured the clustering of $0.5<z<1.2$ galaxies, enabling a number of measurements of galaxy properties and cosmological redshift-space distortions (RSD). Because the measurements were made using one-pass of the VIMOS instrument on the Very Large Telescope (VLT), the galaxies observed only represent approximately 47\% of the parent target sample, with a distribution imprinted with the pattern of the VIMOS slitmask. Correcting for the effect on clustering has previously been achieved using an approximate approach developed using mock catalogues. Pairwise inverse probability (PIP) weighting has recently been proposed by Bianchi & Percival to correct for missing galaxies, and we apply it to mock VIPERS catalogues to show that it accurately corrects the clustering for the VIMOS effects, matching the clustering measured from the observed sample to that of the parent. We then apply PIP-weighting to the VIPERS data, and fit the resulting monopole and quadrupole moments of the galaxy two-point correlation function with respect to the line-of-sight, making measurements of RSD. The results are close to previous measurements, showing that the previous approximate methods used by the VIPERS team are sufficient given the errors obtained on the RSD parameter.
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The Sloan Digital Sky Survey Quasar Catalog: Fourteenth Data Release
We present the Data Release 14 Quasar catalog (DR14Q) from the extended Baryon Oscillation Spectroscopic Survey (eBOSS) of the Sloan Digital Sky Survey IV (SDSS-IV). This catalog includes all SDSS-IV/eBOSS objects that were spectroscopically targeted as quasar candidates and that are confirmed as quasars via a new automated procedure combined with a partial visual inspection of spectra, have luminosities $M_{\rm i} \left[ z=2 \right] < -20.5$ (in a $Λ$CDM cosmology with $H_0 = 70 \ {\rm km \ s^{-1} \ Mpc ^{-1}}$, $Ω_{\rm M} = 0.3$, and $Ω_{\rm Λ} = 0.7$), and either display at least one emission line with a full width at half maximum (FWHM) larger than $500 \ {\rm km \ s^{-1}}$ or, if not, have interesting/complex absorption features. The catalog also includes previously spectroscopically-confirmed quasars from SDSS-I, II and III. The catalog contains 526,356 quasars 144,046 are new discoveries since the beginning of SDSS-IV) detected over 9,376 deg$^2$ (2,044 deg$^2$ having new spectroscopic data available) with robust identification and redshift measured by a combination of principal component eigenspectra. The catalog is estimated to have about 0.5% contamination. The catalog identifies 21,877 broad absorption line quasars and lists their characteristics. For each object, the catalog presents SDSS five-band CCD-based photometry with typical accuracy of 0.03 mag. The catalog also contains X-ray, ultraviolet, near-infrared, and radio emission properties of the quasars, when available, from other large-area surveys.
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The VIMOS Public Extragalactic Redshift Survey (VIPERS). AGN feedback in [NeV] emitters
Using an unconventional single line diagnostic that unambiguously identifies AGNs in composite galaxies we report statistical differences in the properties (stellar age, [OII] luminosity, colour) between active and inactive galaxies at 0.62<z<1.2 extracted from the VIMOS Public Extragalactic Redshift Survey (VIPERS). The nuclear activity is probed by the high-ionization [NeV] emission line and along with their parent samples, the galaxies are properly selected according to their stellar mass, redshift, and colour distributions. We report younger underlying stellar ages and higher [OII] luminosities of active galaxies in the green valley and in the blue cloud compared to control samples. We observe higher fractions of green galaxies hosting AGN activity at progressively bluer (r-K) colours. Depending on the location of the host galaxy in the NUVrK colour diagram we find higher AGN fractions in massive blue galaxies and in the least massive red galaxies, in agreement with the picture that black holes vary their properties when hosted in either star-forming or passive galaxies. Exactly where the fast quenching processes are expected to play a role, we identify a novel class of active galaxies in the blue cloud with signatures typical for a suddenly suppression of their star formation activity after a burst happening in the recent past. Their optical spectra resemble those of post-starburst galaxies, that would never be identified in a spectroscopic search using classical selection techniques. Broadly, these active galaxies selected on the [NeV] line are not commonly represented in shallow X-ray, mid-IR, or classical line diagnostics. If we consider that our results are limited by the shallow observational limits and rapid AGN variability, the impact of AGN feedback on galaxy formation and evolution may represent an important channel of fast-transiting galaxies moving to the red sequence.
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The VIMOS Public Extragalactic Redshift Survey (VIPERS): An unbiased estimate of the growth rate of structure at $\mathbf{\left =0.85}$ using the clustering of luminous blue galaxies
We use the VIPERS final data release to investigate the performance of colour-selected populations of galaxies as tracers of linear large-scale motions. We empirically select volume-limited samples of blue and red galaxies as to minimise the systematic error on the estimate of the growth rate $fσ_8$ from the anisotropy of the two-point correlation function. To this end, rather than rigidly splitting the sample into two colour classes we define the red/blue fractional contribution of each object through a weight based on the $(U-V)$ colour distribution. Using mock surveys that are designed to reproduce the observed properties of VIPERS galaxies, we find the systematic error in recovering the fiducial value of $fσ_8$ to be minimized when using a volume-limited sample of luminous blue galaxies. We model non-linear corrections via the Scoccimarro extension of the Kaiser model, finding systematic errors on $fσ_8$ of below $1-2\%$, using scales as small as 5 $h^{-1}\mathrm{Mpc}$. We interpret this result as indicating that selection of luminous blue galaxies maximises the fraction that are central objects in their dark matter haloes; this in turn minimises the contribution to the measured $ξ(r_p,π)$ from the 1-halo term, which is dominated by non-linear motions. The gain is inferior if one uses the full magnitude-limited sample of blue objects, consistent with the presence of a significant fraction of blue, fainter satellites dominated by non-streaming, orbital velocities. We measure a value of $fσ_8=0.45 \pm 0.11$ over the single redshift range $0.6\le z\le 1.0$, corresponding to an effective redshift for the blue galaxies $\left =0.85$. Including in the likelihood the potential extra information contained in the blue-red galaxy cross-correlation function does not lead to an appreciable improvement in the error bars, while it increases the systematic error.
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The VIMOS Public Extragalactic Redshift Survey (VIPERS). Gravity test from the combination of redshift-space distortions and galaxy-galaxy lensing at $0.5 < z < 1.2$
We carry out a joint analysis of redshift-space distortions and galaxy-galaxy lensing, with the aim of measuring the growth rate of structure; this is a key quantity for understanding the nature of gravity on cosmological scales and late-time cosmic acceleration. We make use of the final VIPERS redshift survey dataset, which maps a portion of the Universe at a redshift of $z \simeq 0.8$, and the lensing data from the CFHTLenS survey over the same area of the sky. We build a consistent theoretical model that combines non-linear galaxy biasing and redshift-space distortion models, and confront it with observations. The two probes are combined in a Bayesian maximum likelihood analysis to determine the growth rate of structure at two redshifts $z=0.6$ and $z=0.86$. We obtain measurements of $fσ_8(0.6) = 0.48 \pm 0.12$ and $fσ_8(0.86) = 0.48 \pm 0.10$. The additional galaxy-galaxylensing constraint alleviates galaxy bias and $σ_8$ degeneracies, providing direct measurements of $[f(0.6),σ_8(0.6)] = [0.93 \pm 0.22, 0.52 \pm 0.06]$ and $f(0.86),σ_8(0.86)] = [0.99 \pm 0.19, 0.48 \pm 0.04]$. These measurements are statistically consistent with a Universe where the gravitational interactions can be described by General Relativity, although they are not yet accurate enough to rule out some commonly considered alternatives. Finally, as a complementary test we measure the gravitational slip parameter, $E_G$ , for the first time at $z>0.6$. We find values of $\smash{\overline{E}_G}(0.6) = 0.16 \pm 0.09$ and $\smash{\overline{E}_G}(0.86) = 0.09 \pm 0.07$, when $E_G$ is averaged over scales above $3 h^{-1} \rm{Mpc}$. We find that our $E_G$ measurements exhibit slightly lower values than expected for standard relativistic gravity in a ΛCDM background, although the results are consistent within $1-2σ$.
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The VIMOS Public Extragalactic Redshift Survey (VIPERS). The matter density and baryon fraction from the galaxy power spectrum at redshift $0.6<z<1.1$
We use the final catalogue of the VIMOS Public Extragalactic Redshift Survey (VIPERS) to measure the power spectrum of the galaxy distribution at high redshift, presenting results that extend beyond $z=1$ for the first time. We apply an FFT technique to four independent sub-volumes comprising a total of $51,728$ galaxies at $0.6<z<1.1$ (out of the nearly $90,000$ included in the whole survey). We concentrate here on the shape of the direction-averaged power spectrum in redshift space, explaining the level of modelling of redshift-space anisotropies and the anisotropic survey window function that are needed to deduce this in a robust fashion. We then use covariance matrices derived from a large ensemble of mock datasets in order to fit the spectral data. The results are well matched by a standard $Λ$CDM model, with density parameter $Ω_M h =\smash{0.227^{+0.063}_{-0.050}}$ and baryon fraction $\smash{f_B=Ω_B/Ω_M=0.220^{+0.058}_{-0.072}}$. These inferences from the high-$z$ galaxy distribution are consistent with results from local galaxy surveys, and also with the Cosmic Microwave Background. Thus the $Λ$CDM model gives a good match to cosmic structure at all redshifts so far accessible to observational study.
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The VIMOS Public Extragalactic Redshift Survey (VIPERS). Exploring the dependence of the three-point correlation function on stellar mass and luminosity at 0.5<z<1.1
The three-point correlation function (3PCF) is a powerful probe to investigate the clustering of matter in the Universe in a complementary way with respect to lower-order statistics, providing additional information with respect to the two-point correlation function and allowing us to shed light on biasing, nonlinear processes, and deviations from Gaussian statistics. In this paper, we analyse the first data release of the VIMOS Public Extragalactic Redshift Survey (VIPERS), determining the dependence of the three-point correlation function on luminosity and stellar mass at $z=[0.5,1.1]$. We exploit the VIPERS Public Data Release 1, consisting of more than 50,000 galaxies with B-band magnitudes in the range $-21.6\lesssim M_{\rm B}-5\log(h)\lesssim-19.9$ and stellar masses in the range $9.8\lesssim\log(M_\star[h^{-2}\,M_\odot])\lesssim 10.7$. We measure both the connected 3PCF and the reduced 3PCF in redshift space, probing different configurations and scales, in the range $2.5<r\,$[Mpc/h]$<20$. We find a significant dependence of the reduced 3PCF on scales and triangle shapes, with stronger anisotropy at larger scales ($r\sim10$ Mpc/h) and an almost flat trend at smaller scales, $r\sim2.5$ Mpc/h. Massive and luminous galaxies present a larger connected 3PCF, while the reduced 3PCF is remarkably insensitive to magnitude and stellar masses in the range we explored. These trends, already observed at low redshifts, are confirmed for the first time to be still valid up to $z=1.1$, providing support to the hierarchical scenario for which massive and bright systems are expected to be more clustered. The possibility of using the measured 3PCF to provide independent constraints on the linear galaxy bias $b$ has also been explored, showing promising results in agreement with other probes.
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The VIMOS Public Extragalactic Redshift Survey (VIPERS). The decline of cosmic star formation: quenching, mass, and environment connections
[Abridged] We use the final data of the VIMOS Public Extragalactic Redshift Survey (VIPERS) to investigate the effect of environment on the evolution of galaxies between $z=0.5$ and $z=0.9$. We characterise local environment in terms of the density contrast smoothed over a cylindrical kernel, the scale of which is defined by the distance to the $5^{th}$ nearest neighbour. We find that more massive galaxies tend to reside in higher-density environments over the full redshift range explored. Defining star-forming and passive galaxies through their (NUV$-r$) vs ($r-K$) colours, we then quantify the fraction of star-forming over passive galaxies, $f_{\rm ap}$, as a function of environment at fixed stellar mass. $f_{\rm ap}$ is higher in low-density regions for galaxies with masses ranging from $\log(\mathcal{M}/\mathcal{M}_\odot)=10.38$ (the lowest value explored) to at least $\log(\mathcal{M}/\mathcal{M}_\odot)\sim11.3$, although with decreasing significance going from lower to higher masses. This is the first time that environmental effects on high-mass galaxies are clearly detected at redshifts as high as $z\sim0.9$. We compared these results to VIPERS-like galaxy mock catalogues based on the galaxy formation model of De Lucia & Blaizot. The model correctly reproduces $f_{\rm ap}$ in low-density environments, but underpredicts it at high densities. The discrepancy is particularly strong for the lowest-mass bins. We find that this discrepancy is driven by an excess of low-mass passive satellite galaxies in the model. Looking at the accretion history of these model galaxies, i.e. the times when they become satellites, a better (yet not perfect) agreement with observations can be obtained in high density regions by assuming either that a not-negligible fraction of satellites is destroyed, or that their quenching time-scale is longer than $\sim 2$ Gyr.
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The VIMOS Public Extragalactic Redshift Survey (VIPERS): The growth of structures at $0.5<z<1.2$ from redshift-space distortions in the clustering of the PDR-2 final sample
We present measurements of the growth rate of cosmological structure from the modelling of the anisotropic galaxy clustering measured in the final data release of the VIPERS survey. The analysis is carried out in configuration space and based on measurements of the first two even multipole moments of the anisotropic galaxy auto-correlation function, in two redshift bins spanning the range $0.5 < z < 1.2$. We provide robust and cosmology-independent corrections for the VIPERS angular selection function, allowing recovery of the underlying clustering amplitude at the percent level down to the Mpc scale. We discuss several improvements on the non-linear modelling of redshift-space distortions (RSD) and perform detailed tests of a variety of approaches against a set of realistic VIPERS-like mock realisations. This includes using novel fitting functions to describe the velocity divergence and density power spectra $P_{θθ}$ and $P_{δθ}$ that appear in RSD models. These tests show that we are able to measure the growth rate with negligible bias down to separations of $5h^{-1}Mpc$. Interestingly, the application to real data shows a weaker sensitivity to the details of non-linear RSD corrections compared to mock results. We obtain consistent values for the growth rate times the matter power spectrum normalisation parameter of $fσ_8=0.55\pm 0.12$ and $0.40\pm0.11$ at effective redshifts of $z = 0.6$ and $z=0.86$ respectively. These results are in agreement with standard cosmology predictions assuming Einstein gravity in a $Λ\rm{CDM}$ background.
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