arXiv subjects

G. Lavaux

Publications and source records attributed to G. Lavaux.

At least 19 recordsLinked to original sources

arXiv2026

Euclid preparation. First investigation of the impact of cross-contamination on spectroscopic redshift measurements with pixel-level simulations

We present a study on simulated data focused on understanding the performance of the spectroscopic redshift measurements with the Near-Infrared Spectrometer and Photometer (NISP) instrument on Euclid. Simulations include scenarios with different levels of cross-contamination arising from overlapping spectra of nearby sources, which represents one of the main drawbacks of slitless spectroscopy. We present a new analysis based on pixel-level simulations of the NISP images, with the data processed using the Euclid spectroscopic pipeline. We first consider an idealised case with non-overlapping spectra to assess the accuracy and reliability of the redshift measurement as a function of the flux of the H-alpha emission line and galaxy size. We then introduce more realistic contamination scenarios, distinguishing between two contributions: contamination from H-alpha emitters, which are the Euclid targets for cosmological analyses, and contamination from all other galaxies. In the second case, we analyse the impact of cross-contamination with an increasing number of contaminants, from the brighter to the fainter galaxies. Given that our results show no clear evidence that sources fainter than magnitude 20 degrade redshift measurements, we conservatively restrict our analysis to galaxies with magnitudes up to 24. In particular, we provide a preliminary estimate that contamination from galaxies within the same redshift range as the target sample contributes to about 4% of the total degradation due to cross-contamination from all galaxies.

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arXiv2025

Euclid preparation. XCII. Controlling angular systematics in the Euclid spectroscopic galaxy sample

We present the strategy used to identify and mitigate potential sources of angular systematics in the \textit{Euclid} spectroscopic galaxy survey, and we quantify their impact on galaxy clustering measurements and cosmological parameter estimation. We first surveyed the \textit{Euclid} processing pipeline to identify all evident, potential sources of systematics, and classified them into two broad classes: angular systematics, which modulate the galaxy number density across the sky, and catastrophic redshift errors, which lead to interlopers in the galaxy sample. We then used simulated spectroscopic surveys to test our ability to mitigate angular systematics by constructing a random catalogue that represents the `visibility mask' of the survey; this is a dense set of intrinsically unclustered objects, subject to the same selection effects as the data catalogue. The construction of this random catalogue relies on a detection model, which gives the probability of reliably measuring the galaxy redshift as a function of the signal-to-noise ratio (S/N) of its emission lines. We demonstrate that, in the ideal case of a perfect knowledge of the visibility mask, the galaxy power spectrum in the presence of systematics is recovered, to within sub-per cent accuracy, by convolving a theory power spectrum with a window function obtained from the random catalogue itself. In the case of only approximate knowledge of the visibility mask, we test the stability of power spectrum measurements and cosmological parameter posteriors by using perturbed versions of the random catalogue. We find that significant effects are limited to very large scales, and parameter estimation remains robust; the most impacting effects are connected to the calibration of the detection model.

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arXiv2025

Euclid preparation: Expected constraints on initial conditions

The Euclid mission of the European Space Agency will deliver galaxy and cosmic shear surveys, which will be used to constrain initial conditions and statistics of primordial fluctuations. We present highlights for the Euclid scientific capability to test initial conditions beyond LCDM with the main probes, i.e. 3D galaxy clustering from the spectroscopic survey, the tomographic approach to 3x2pt statistics from photometric galaxy survey, and their combination. We provide Fisher forecasts from the combination of Euclid spectroscopic and photometric surveys for spatial curvature, running of the spectral index of the power spectrum of curvature perturbations, isocurvature perturbations, and primordial features. For the parameters of these models we also provide the combination of Euclid forecasts (pessimistic and optimistic) with current and future measurements of the cosmic microwave background (CMB) anisotropies., i.e. Planck, the Simons Observatory (SO), and CMB-S4. We provide Fisher forecasts for how the power spectrum and bispectrum from the Euclid spectroscopic survey will constrain the local, equilateral, and orthogonal shapes of primordial non-Gaussianity. We also review how Bayesian field-level inference of primordial non-Gaussianity can constrain local primordial non-Gaussianity. We show how Euclid, with its unique combination of the main probes, will provide the tightest constraints on low redshift to date. By targeting a markedly different range in redshift and scale, Euclid's expected uncertainties are complementary to those obtained by CMB primary anisotropy, returning the tightest combined constraints on the physics of the early Universe.

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arXiv2025

Euclid preparation. Simulating thousands of Euclid spectroscopic skies

We present two extensive sets of 3500+1000 simulations of dark matter haloes on the past light cone, and two corresponding sets of simulated (`mock') galaxy catalogues that represent the Euclid spectroscopic sample. The simulations were produced with the latest version of the PINOCCHIO code, and provide the largest, public set of simulated skies. Mock galaxy catalogues were obtained by populating haloes with galaxies using an halo occupation distribution (HOD) model extracted from the Flagship galaxy catalogue provided by Euclid Collaboration. The Geppetto set of 3500 simulated skies was obtained by tiling a 1.2 Gpc/h box to cover a light-cone whose sky footprint is a circle of 30 deg radius, for an area of 2763 deg$^2$ and a minimum halo mass of $1.5\times10^{11}$ Msun/h. The relatively small box size makes this set unfit for measuring very large scales. The EuclidLargeBox set consists of 1000 simulations of 3.38 Gpc/h, with the same mass resolution and a footprint that covers half of the sky, excluding the Milky Way zone of avoidance. From this we produced a set of 1000 EuclidLargeMocks on the 30 deg radius footprint, whose comoving volume is fully contained in the simulation box. We validated the two sets of catalogues by analysing number densities, power spectra, and 2-point correlation functions, showing that the Flagship spectroscopic catalogue is consistent with being one of the realisations of the simulated sets, although we noticed small deviations limited to the quadrupole at k>0.2 h/Mpc. We show cosmological parameter inference from these catalogues and demonstrate that using one realisation of EuclidLargeMocks in place of the Flagship mock produces the same posteriors, to within the expected shift given by sample variance. These simulated skies will be used for the galaxy clustering analysis of Euclid's Data Release 1 (DR1).

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arXiv2025

Euclid preparation. The impact of redshift interlopers on the two-point correlation function analysis

The Euclid survey aims to measure the spectroscopic redshift of emission-line galaxies by identifying the H$\,{\alpha}$ line in their slitless spectra. This method is sensitive to the signal-to-noise ratio of the line, as noise fluctuations or other strong emission lines can be misidentified as H$\,{\alpha}$, depending on redshift. These effects lead to catastrophic redshift errors and the inclusion of interlopers in the sample. We forecast the impact of such redshift errors on galaxy clustering measurements. In particular, we study the effect of interloper contamination on the two-point correlation function (2PCF), the growth rate of structures, and the Alcock-Paczynski (AP) parameters. We analyze 1000 synthetic spectroscopic catalogues, the EuclidLargeMocks, designed to match the area and selection function of the Data Release 1 (DR1) sample. We estimate the 2PCF of the contaminated catalogues, isolating contributions from correctly identified galaxies and from interlopers. We explore different models with increasing complexity to describe the measured 2PCF at fixed cosmology. Finally, we perform a cosmological inference and evaluate the systematic error on the inferred $f\sigma_8$, $\alpha_{\parallel}$ and $\alpha_{\perp}$ values associated with different models. Our results demonstrate that a minimal modelling approach, which only accounts for an attenuation of the clustering signal regardless of the type of contaminants, is sufficient to recover the correct values of $f\sigma_8$, $\alpha_{\parallel}$, and $\alpha_{\perp}$ at DR1. The accuracy and precision of the estimated AP parameters are largely insensitive to the presence of interlopers. The adoption of a minimal model induces a 1%-3% systematic error on the growth rate of structure estimation, depending on the redshift. However, this error remains smaller than the statistical error expected for the Euclid DR1 analysis.

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arXiv2024

Euclid: Field-level inference of primordial non-Gaussianity and cosmic initial conditions

A primary target of the \Euclid space mission is to constrain early-universe physics by searching for deviations from a primordial Gaussian random field. A significant detection of primordial non-Gaussianity would rule out the simplest models of cosmic inflation and transform our understanding of the origin of the Universe. This paper forecasts how well field-level inference of galaxy redshift surveys can constrain the amplitude of local primordial non-Gaussianity ($f_{NL}$), within a Bayesian hierarchical framework, in the upcoming \Euclid data. We design and simulate mock data sets and perform Markov chain Monte Carlo analyses using a full-field forward modelling approach. By including the formation history of the cosmic matter field in the analysis, the method takes into account all available probes of primordial non-Gaussianity, and goes beyond statistical summary estimators of $f_{NL}$. Probes include, for example, two-point and higher-order statistics, peculiar velocity fields, and scale-dependent galaxy biases. Furthermore, the method simultaneously handles systematic survey effects, such as selection effects, survey geometries, and galaxy biases. The forecast shows that the method can reach precision levels of up to $σ(f_{NL}) = 2.3$ (68.3\% CI, and at the grid resolution $ΔL = 62.5\,h^{-1}$Mpc) with \Euclid data. We also provide data products, including realistic $N$-body simulations with nonzero values of $f_{NL}$ and maps of adiabatic curvature fluctuations. The results underscore the feasibility and advantages of field-level inference to constrain $f_{NL}$ in galaxy redshift surveys. Our approach consistently captures all the information available in the large-scale structure to constrain $f_{NL}$, and resolves the degeneracy between early-universe physics and late-time gravitational effects, while mitigating the impact of systematic and observational effects.

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arXiv2024

Euclid: The search for primordial features

Primordial features, in particular oscillatory signals, imprinted in the primordial power spectrum of density perturbations represent a clear window of opportunity for detecting new physics at high-energy scales. Future spectroscopic and photometric measurements from the $Euclid$ space mission will provide unique constraints on the primordial power spectrum, thanks to the redshift coverage and high-accuracy measurement of nonlinear scales, thus allowing us to investigate deviations from the standard power-law primordial power spectrum. We consider two models with primordial undamped oscillations superimposed on the matter power spectrum, one linearly spaced in $k$-space the other logarithmically spaced in $k$-space. We forecast uncertainties applying a Fisher matrix method to spectroscopic galaxy clustering, weak lensing, photometric galaxy clustering, cross correlation between photometric probes, spectroscopic galaxy clustering bispectrum, CMB temperature and $E$-mode polarization, temperature-polarization cross correlation, and CMB weak lensing. We also study a nonlinear density reconstruction method to retrieve the oscillatory signals in the primordial power spectrum. We find the following percentage relative errors in the feature amplitude with $Euclid$ primary probes for the linear (logarithmic) feature model: 21% (22%) in the pessimistic settings and 18% (18%) in the optimistic settings at 68.3% confidence level (CL) using GC$_{\rm sp}$+WL+GC$_{\rm ph}$+XC. Combining all the sources of information explored expected from $Euclid$ in combination with future SO-like CMB experiment, we forecast ${\cal A}_{\rm lin} \simeq 0.010 \pm 0.001$ at 68.3% CL and ${\cal A}_{\rm log} \simeq 0.010 \pm 0.001$ for GC$_{\rm sp}$(PS rec + BS)+WL+GC$_{\rm ph}$+XC+SO-like both for the optimistic and pessimistic settings over the frequency range $(1,\,10^{2.1})$.

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arXiv2022

Euclid: Cosmological forecasts from the void size function

The Euclid mission $-$ with its spectroscopic galaxy survey covering a sky area over $15\,000 \ \mathrm{deg}^2$ in the redshift range $0.9<z<1.8\ -$ will provide a sample of tens of thousands of cosmic voids. This paper explores for the first time the constraining power of the void size function on the properties of dark energy (DE) from a survey mock catalogue, the official Euclid Flagship simulation. We identify voids in the Flagship light-cone, which closely matches the features of the upcoming Euclid spectroscopic data set. We model the void size function considering a state-of-the art methodology: we rely on the volume conserving (Vdn) model, a modification of the popular Sheth & van de Weygaert model for void number counts, extended by means of a linear function of the large-scale galaxy bias. We find an excellent agreement between model predictions and measured mock void number counts. We compute updated forecasts for the Euclid mission on DE from the void size function and provide reliable void number estimates to serve as a basis for further forecasts of cosmological applications using voids. We analyse two different cosmological models for DE: the first described by a constant DE equation of state parameter, $w$, and the second by a dynamic equation of state with coefficients $w_0$ and $w_a$. We forecast $1σ$ errors on $w$ lower than $10\%$, and we estimate an expected figure of merit (FoM) for the dynamical DE scenario $\mathrm{FoM}_{w_0,w_a} = 17$ when considering only the neutrino mass as additional free parameter of the model. The analysis is based on conservative assumptions to ensure full robustness, and is a pathfinder for future enhancements of the technique. Our results showcase the impressive constraining power of the void size function from the Euclid spectroscopic sample, both as a stand-alone probe, and to be combined with other Euclid cosmological probes.

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arXiv2022

GLADE+: An Extended Galaxy Catalogue for Multimessenger Searches with Advanced Gravitational-wave Detectors

We present GLADE+, an extended version of the GLADE galaxy catalogue introduced in our previous paper for multimessenger searches with advanced gravitational-wave detectors. GLADE+ combines data from six separate but not independent astronomical catalogues: the GWGC, 2MPZ, 2MASS XSC, HyperLEDA, and WISExSCOSPZ galaxy catalogues, and the SDSS-DR16Q quasar catalogue. To allow corrections of CMB-frame redshifts for peculiar motions, we calculated peculiar velocities along with their standard deviations of all galaxies having $B$-band magnitude data within redshift $z=0.05$ using the "Bayesian Origin Reconstruction from Galaxies" formalism. GLADE+ is complete up to luminosity distance $d_L=47^{+4}_{-2}$ Mpc in terms of the total expected $B$-band luminosity of galaxies, and contains all of the brightest galaxies giving 90\% of the total $B$-band and $K$-band luminosity up to $d_L\simeq 130$ Mpc. We include estimations of stellar masses and individual binary neutron star merger rates for galaxies with $W1$ magnitudes. These parameters can help in ranking galaxies in a given gravitational wave localization volume in terms of their likelihood of being hosts, thereby possibly reducing the number of pointings and total integration time needed to find the electromagnetic counterpart.

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arXiv2022

LyMAS reloaded: improving the predictions of the large-scale Lyman-α forest statistics from dark matter density and velocity fields

We present LyMAS2, an improved version of the "Lyman-α Mass Association Scheme" aiming at predicting the large-scale 3d clustering statistics of the Lyman-α forest (Ly-α) from moderate resolution simulations of the dark matter (DM) distribution, with prior calibrations from high resolution hydrodynamical simulations of smaller volumes. In this study, calibrations are derived from the Horizon-AGN suite simulations, (100 Mpc/h)^3 comoving volume, using Wiener filtering, combining information from dark matter density and velocity fields (i.e. velocity dispersion, vorticity, line of sight 1d-divergence and 3d-divergence). All new predictions have been done at z=2.5 in redshift-space, while considering the spectral resolution of the SDSS-III BOSS Survey and different dark matter smoothing (0.3, 0.5 and 1.0 Mpc/h comoving). We have tried different combinations of dark matter fields and found that LyMAS2, applied to the Horizon-noAGN dark matter fields, significantly improves the predictions of the Ly-α 3d clustering statistics, especially when the DM overdensity is associated with the velocity dispersion or the vorticity fields. Compared to the hydrodynamical simulation trends, the 2-point correlation functions of pseudo-spectra generated with LyMAS2 can be recovered with relative differences of ~5% even for high angles, the flux 1d power spectrum (along the light of sight) with ~2% and the flux 1d probability distribution function exactly. Finally, we have produced several large mock BOSS spectra (1.0 and 1.5 Gpc/h) expected to lead to much more reliable and accurate theoretical predictions.

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arXiv2021

Euclid: Forecasts from redshift-space distortions and the Alcock-Paczynski test with cosmic voids

Euclid is poised to survey galaxies across a cosmological volume of unprecedented size, providing observations of more than a billion objects distributed over a third of the full sky. Approximately 20 million of these galaxies will have their spectroscopy available, allowing us to map the 3D large-scale structure of the Universe in great detail. This paper investigates prospects for the detection of cosmic voids therein and the unique benefit they provide for cosmology. In particular, we study the imprints of dynamic and geometric distortions of average void shapes and their constraining power on the growth of structure and cosmological distance ratios. To this end, we made use of the Flagship mock catalog, a state-of-the-art simulation of the data expected to be observed with Euclid. We arranged the data into four adjacent redshift bins, each of which contains about 11000 voids and estimated the stacked void-galaxy cross-correlation function in every bin. Fitting a linear-theory model to the data, we obtained constraints on $f/b$ and $D_M H$, where $f$ is the linear growth rate of density fluctuations, $b$ the galaxy bias, $D_M$ the comoving angular diameter distance, and $H$ the Hubble rate. In addition, we marginalized over two nuisance parameters included in our model to account for unknown systematic effects. With this approach, Euclid will be able to reach a relative precision of about 4% on measurements of $f/b$ and 0.5% on $D_M H$ in each redshift bin. Better modeling or calibration of the nuisance parameters may further increase this precision to 1% and 0.4%, respectively. Our results show that the exploitation of cosmic voids in Euclid will provide competitive constraints on cosmology even as a stand-alone probe. For example, the equation-of-state parameter $w$ for dark energy will be measured with a precision of about 10%, consistent with previous more approximate forecasts.

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arXiv2019

The peculiar velocity field up to $z \sim 0.05$ by forward-modeling Cosmicflows-3 data

A hierarchical Bayesian model is applied to the Cosmicflows-3 catalog of galaxy distances in order to derive the peculiar velocity field and distribution of matter within $z \sim 0.054$. The model assumes the $Λ$CDM model within the linear regime and includes the fit of the galaxy distances together with the underlying density field. By forward modeling the data, the method is able to mitigate biases inherent to peculiar velocity analyses, such as the Homogeneous Malmquist bias or the log-normal distribution of peculiar velocities. The statistical uncertainty on the recovered velocity field is about 150 km/s depending on the location, and we study systematics coming from the selection function and calibration of distance indicators. The resulting velocity field and related density fields recover the cosmography of the Local Universe which is presented in an unprecedented volume of universe 10 times larger than previously reached. This methodology open the doors to reconstruction of initial conditions for larger and more accurate constrained cosmological simulations. This work is also preparatory to larger peculiar velocity datasets coming from Wallaby, TAIPAN or LSST.

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arXiv2015

Bayesian 3d velocity field reconstruction with VIRBIuS

I describe a new Bayesian based algorithm to infer the full three dimensional velocity field from observed distances and spectroscopic galaxy catalogues. In addition to the velocity field itself, the algorithm reconstructs true distances, some cosmological parameters and specific non-linearities in the velocity field. The algorithm takes care of selection effects, miscalibration issues and can be easily extended to handle direct fitting of, e.g., the inverse Tully-Fisher relation. I first describe the algorithm in details alongside its performances. This algorithm is implemented in the VIRBIuS (VelocIty Reconstruction using Bayesian Inference Software) software package. I then test it on different mock distance catalogues with a varying complexity of observational issues. The model proved to give robust measurement of velocities for mock catalogues of 3,000 galaxies. I expect the core of the algorithm to scale to tens of thousands galaxies. It holds the promises of giving a better handle on future large and deep distance surveys for which individual errors on distance would impede velocity field inference.

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arXiv2014

Answers from the void: VIDE and its applications

We discuss various applications of VIDE, the Void IDentification and Examination toolkit, an open-source Python/C++ code for finding cosmic voids in galaxy redshift surveys and N-body simulations. Based on a substantially enhanced version of ZOBOV, VIDE not only finds voids, but also summarizes their properties, extracts statistical information, and provides a Python-based platform for more detailed analysis, such as manipulating void catalogs and particle members, filtering, plotting, computing clustering statistics, stacking, comparing catalogs, and fitting density profiles. VIDE also provides significant additional functionality for pre-processing inputs: for example, VIDE can work with volume- or magnitude-limited galaxy samples with arbitrary survey geometries, or dark matter particles or halo catalogs in a variety of common formats. It can also randomly subsample inputs and includes a Halo Occupation Distribution model for constructing mock galaxy populations. VIDE has been used for a wide variety of applications, from discovering a universal density profile to estimating primordial magnetic fields, and is publicly available at http://bitbucket.org/cosmicvoids/vide_public and http://www.cosmicvoids.net.

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arXiv2012

Cosmic flow from 2MASS redshift survey: The origin of CMB dipole and implications for LCDM cosmology

We generate the peculiar velocity field for the 2MASS Redshift Survey (2MRS) catalog using an orbit-reconstruction algorithm. The reconstructed velocities of individual objects in 2MRS are well-correlated with the peculiar velocities obtained from high-precision observed distances within 3,000 km/s. We estimate the mean matter density to be 0.31 +/- 0.05 by comparing observed to reconstructed velocities in this volume. The reconstructed motion of the Local Group in the rest frame established by distances within 3,000 km/s agrees with the observed motion and is generated by fluctuations within this volume, in agreement with observations. Then, we reconstruct the velocity field of 2MRS in successively larger radii, to study the problem of convergence towards the CMB dipole. We find that less than half of the amplitude of the CMB dipole is generated within a volume enclosing the Hydra-Centaurus-Norma supercluster at around 40 Mpc/h. Although most of the amplitude of the CMB dipole seems to be recovered by 120 Mpc/h, the direction does not agree and hence we observe no convergence up to this scale. We develop a statistical model which allows us to estimate cosmological para meters from the reconstructed growth of convergence of the velocity of the Local Group towards the CMB dipole motion. For scales up to 60 Mpc/h, assuming a Local Group velocity of 627 km/s, we estimate Omega_m h^2 = 0.11 +/- 0.06 and sigma_8=0.9 +/- 0.4, in agreement with WMAP5 measurements at the 1-sigma level. However, for scales up to 100 Mpc/h, we obtain Omega_m h^2 = 0.08 +/- 0.03 and sigma_8=1.0 +/- 0.4, which agrees at the 1 to 2-sigma level with WMAP5 results. (abridged)

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arXiv2009

Dynamics of the Local Universe: cosmic velocity flows and voids

A valuable amount of information is available in peculiar velocities of galaxies. Peculiar velocity surveys have recently allowed the discovery of potential problems with LCDM. Nonetheless, their direct observation through distance measurements remains a daunting task. Another way of considering the problem is to use orbit reconstruction methods assuming some mass-to-light assignment for galaxies. We give here two applications of this procedure for the study of large-scale bulk flows and the dynamics of voids in the Local Universe. We concentrate our study on the use of the Monge-Ampere-Kantorovitch reconstruction technique. Using peculiar velocities reconstructed from the 2MASS Galaxy Redshift Survey, and after comparison with the NGB-3k peculiar velocity catalog, we look in the details of these peculiar velocities. More particularly, we estimate the constraints that the peculiar velocities put on the cosmology. The information on dynamics that is included in reconstructed orbits of galaxies also allows us to have a much better prescription for defining and identifying voids in simulations and redshift catalogs. We present this new technique and how voids may give us additional constraints on cosmology with current and future surveys.

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arXiv2008

Lagrangian reconstruction of cosmic velocity fields

We discuss a Lagrangian reconstruction method of the velocity field from galaxy redshift catalog that takes its root in the Euler equation. This results in a ``functional'' of the velocity field which must be minimized. This is helped by an algorithm solving the minimization of cost-flow problems. The results obtained by applying this method to cosmological problems are shown and boundary effects happening in real observational cases are then discussed. Finally, a statistical model of the errors made by the reconstruction method is proposed.

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arXiv2007

Observational biases in Lagrangian reconstructions of cosmic velocity fields

Lagrangian reconstruction of large-scale peculiar velocity fields can be strongly affected by observational biases. We develop a thorough analysis of these systematic effects by relying on specially selected mock catalogues. For the purpose of this paper, we use the MAK reconstruction method, although any other Lagrangian reconstruction method should be sensitive to the same problems. We extensively study the uncertainty in the mass-to-light assignment due to luminosity incompleteness, and the poorly-determined relation between mass and luminosity. The impact of redshift distortion corrections is analyzed in the context of MAK and we check the importance of edge and finite-volume effects on the reconstructed velocities. Using three mock catalogues with different average densities, we also study the effect of cosmic variance. In particular, one of them presents the same global features as found in observational catalogues that extend to 80 Mpc/h scales. We give recipes, checked using the aforementioned mock catalogues, to handle these particular observational effects, after having introduced them into the mock catalogues so as to quantitatively mimic the most densely sampled currently available galaxy catalogue of the nearby universe. Once biases have been taken care of, the typical resulting error in reconstructed velocities is typically about a quarter of the overall velocity dispersion, and without significant bias. We finally model our reconstruction errors to propose an improved Bayesian approach to measure Omega_m in an unbiased way by comparing the reconstructed velocities to the measured ones in distance space, even though they may be plagued by large errors. We show that, in the context of observational data, a nearly unbiased estimator of Omega_m may be built using MAK reconstruction.

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