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Henk Hoekstra

Publications and source records attributed to Henk Hoekstra.

At least 19 recordsLinked to original sources

Cosmological constraining power of the redshifts, heights, and angular clustering of weak gravitational lensing peaks

Weak gravitational lensing (WL) peaks probe non-Gaussian information of the large-scale distribution of matter that is not captured by two-point lensing statistics. We study the cosmological potential of the height distribution, redshift distribution, and angular clustering of high-valued WL peaks using a Bayesian inference approach that mimics a $\textit{Euclid}$ analysis. We use a forthcoming dark-matter-only hypercube, which varies cosmology in a ten-dimensional space, including evolving dark energy, neutrino mass, decaying dark matter, and the running of the scalar spectral index. We find the individual WL peak statistics to be complementary, as the redshift distribution best constrains the matter density, $\Omega_\mathrm{m}$, and the dark energy equation-of-state parameters, $w_0$, and $w_a$; the height distribution and angular clustering are most sensitive to the amplitude of the primordial power spectrum, $\ln(10^{10}A_\mathrm{s})$; while combining the three statistics allows us to also probe the baryon density, $\Omega_\mathrm{b}$, and the Hubble parameter, $h$. A comparison to the shear two-point correlation function demonstrates that WL peaks alone outperform the commonly used statistics, while even tighter constraints are obtained when combining all. Considering the 2-dimensional $\Omega_\mathrm{m}-\ln(10^{10}A_\mathrm{s})$ and $w_0-w_a$ parameter planes, we find that the redshift distribution outperforms the angular clustering and peak-height distributions. We study the impact of the smoothing scale and find that, typically, the smallest scales yield the best results and the figure of merit improves by a factor $\approx2$ when combining multiple scales.

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Three-point intrinsic alignments of galaxies and haloes in the FLAMINGO simulations

Third-order statistics provide information beyond two-point measures, but extracting this information requires accurate and consistent modelling. We measure and detect the three-point correlation function and third-order aperture mass statistics of intrinsic alignments (IA) for galaxies and for haloes with $M_{\rm halo} > 10^{13}\,{\rm M}_\odot$ in the $(2.8\,\mathrm{Gpc})^3$ simulation volume of the FLAMINGO hydrodynamical simulation suite. We model the third-order aperture mass statistics and show that on large scales both the galaxy and halo samples are well described by the tree-level effective field theory (EFT) of IA across the three dark matter density-shape combinations and a wide range of triangle configurations, with the alignment amplitude consistent with that inferred from two-point statistics. We compare the full EFT to several other models: a version neglecting the velocity-shear term, the non-linear alignment model, and a reduced EFT assuming co-evolution relations that follow from the assumption that alignment is linear in Lagrangian space. The first two models yield biased constraints on the alignment amplitude, but the reduced EFT performs remarkably well, achieving a low reduced chi-squared and minimal bias. We examine the redshift and mass dependence of the higher-order bias parameters, finding that the linear Lagrangian bias assumption is approximately satisfied across the explored halo mass and redshift ranges for both galaxies and haloes. These co-evolution relations can be valuable for photometric shear surveys, where limited constraining power on IA parameters favours models with fewer free parameters.

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KiDS-Legacy: Constraining dark energy, neutrino mass, and curvature

We constrained minimally extended cosmological models with the cosmic shear analysis of the final data release from the Kilo-Degree Survey (KiDS-Legacy) in combination with external probes. Due to the consistency of the KiDS-Legacy analysis with the cosmic microwave background (CMB), we could combine these datasets reliably for the first time. Additionally, we used CMB lensing, galaxy redshift-space distortions, and baryon acoustic oscillations. We assessed, in turn, the effects of spatial curvature, varying neutrino masses, and an evolving dark energy component on cosmological constraints from KiDS-Legacy alone and from KiDS-Legacy combined with external probes. We find KiDS-Legacy to be consistent with the fiducial flat $\Lambda$-cold dark matter ($\Lambda$CDM) analysis with $c^2 \sum m_\nu\leq 1.5\,$eV, $w_0 = -1.0\pm 0.7$, and $w_a = -1.3^{+1.9}_{-2.0}$ while $\Omega_K = 0.08^{+0.16}_{-0.17}$ (1$\sigma$ bounds) with an almost equal goodness of fit. The $w_0w_a$CDM model is not a significant improvement over $\Lambda$CDM when cosmic shear and CMB lensing are combined, yielding a Bayes factor $B = 0.07$. If all probes are combined, however, $B$ increases to 2.73, corresponding to a $2.6\sigma$ suspiciousness tension. The constraint on $S_8 = \sigma_8\sqrt{\Omega_\mathrm{m}/0.3}$ is robust to opening up the parameter space for cosmic shear. Adding all external datasets to KiDS-Legacy, we find $S_8 = 0.816 \pm 0.006$ in $\Lambda$CDM and $S_8 = 0.837 \pm 0.008$ in $w_0 w_a$CDM for all probes combined.

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KiDS-1000 cosmic shear reanalysis using MetaCalibration

A number of cosmic shear studies have reported results that are in mild tension with the Planck cosmic microwave measurement. To explore if this can be caused by biases in the shear estimation, we revisit the analysis of data from the Kilo-Degree Survey (KiDS) using an alternative shape measurement pipeline that is more robust to uncertainties in the calibration. To this end, we present an implementation of MetaCalibration, and compare its performance to that of lensfit, which has been used in previous analyses of these data. We find that the multiplicative bias is reduced, especially for the most distant redshifts, as derived from multi-band image simulations designed to match the KiDS data (SURFS-based KiDS-Legacy-Like Simulations: SKiLLS). For all tomographic bins we obtain a multiplicative bias $|m|<0.017$, with negligible additive bias. Importantly, the calibration has a negligible sensitivity to key galaxy properties. The resulting robust shear estimates were used to obtain cosmological parameter constraints. We find that the parameter $S_8\equiv \sigma_8 \sqrt{\Omega_\mathrm{m}/0.3} =0.789_{-0.024}^{+0.020}$ is consistent with the previous KiDS-1000 lensfit constraint of $S_8=0.776^{+0.029 +0.002}_{-0.027-0.003}$ (statistical + systematic errors). Thanks to the higher effective source density, the constraining power is improved by about 28%. The difference in $S_8$ with the Planck value remains at a similar level, 1.8$\sigma$, implying that it is not caused by the shear measurements.

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Matter power spectrum reconstruction with KiDS-Legacy: Improved internal $\Lambda$CDM consistency and preference for strong baryonic feedback

Direct measurements of the matter power spectrum, $P_\mathrm{m}(k,z)$, provide a powerful tool to investigate observed tensions between models of structure growth while also testing the internal consistency of cosmological probes. We analyse cosmic shear data from the final data release of the Kilo-Degree Survey (KiDS), presenting a deprojected $P_\mathrm{m}(k,z)$, measured in up to three redshift bins. Compared to analyses using previous KiDS releases, we find improved internal consistency in the $z\lesssim0.7$ regime. At large scales, $k\lesssim0.1\,h\,\rm Mpc^{-1}$, our power spectrum reconstruction aligns with $\Lambda$CDM predictions with a density fluctuation amplitude $\sigma_8=0.81$. Furthermore, at small scales, $k=3$-$20\,h\,\rm Mpc^{-1}$, the average matter power spectrum is suppressed by $30\%\pm10\%\,{\rm (stat.)}\pm4\%\,{\rm (sys.)}$ with $2.8\sigma$ significance relative to a dark-matter-only model, consistent with expectations of strong baryonic feedback.

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Intrinsic alignments of galaxies in multiple projections

Intrinsic alignments of galaxies are measured and modelled to gain cosmological information, to further understand the interactions between galaxies and to mitigate their effects on gravitational weak lensing studies. Hydrodynamical simulations are often used to constrain priors or calibrate models. Therefore, obtaining the maximum amount of information possible from these simulations is imperative. In this work, we have combined the information of shapes projected over two or three axes ($x,y,z$), for intrinsic alignment signals ($w_{g+},\ \tilde{\xi}_{g+,2}$), showing a consistent gain in signal-to-noise ratio (SNR) for all cases studied using TNG300-1. The gain in SNR is found to be higher for the addition of the second projection than for the third, and higher for shapes calculated using the reduced inertia tensor rather than the simple one. The two shape samples studied, $n_\star>300$ and $\mathrm{log}(M_\star \ h/\mathrm{M_\odot})>10.5$, where the latter has a much higher signal amplitude, show similar gains in SNR when more projections are added. We also model the correlation functions with the non-linear alignment model. The SNR gains from the measurements are higher but consistent with the constraints on the non-linear alignment amplitude $A_{\rm IA}$ and galaxy bias $b_{\rm g}$. Using multiple projection axes increases SNR overall, enabling more efficient use of numerically expensive hydrodynamical simulations.

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Improved photometric redshift estimations through self-organising map-based data augmentation

We introduce a framework for the enhanced estimation of photometric redshifts using Self-Organising Maps (SOMs). Our method projects galaxy Spectral Energy Distributions (SEDs) onto a two-dimensional map, identifying regions that are sparsely sampled by existing spectroscopic observations. These under-sampled areas are then augmented with simulated galaxies, yielding a more representative spectroscopic training dataset. To assess the efficacy of this SOM-based data augmentation in the context of the forthcoming Legacy Survey of Space and Time (LSST), we employ mock galaxy catalogues from the OpenUniverse2024 project and generate synthetic datasets that mimic the expected photometric selections of LSST after one (Y1) and ten (Y10) years of observation. We construct 501 degraded realisations by sampling galaxy colours, magnitudes, redshifts and spectroscopic success rates, in order to emulate the compilation of a wide array of realistic spectroscopic surveys. Augmenting the degraded mock datasets with simulated galaxies from the independent CosmoDC2 catalogues has markedly improved the performance of our photometric redshift estimates compared to models lacking this augmentation, particularly for high-redshift galaxies ($z_\mathrm{true} \gtrsim 1.5$). This improvement is manifested in notably reduced systematic biases and a decrease in catastrophic failures by up to approximately a factor of 2, along with a reduction in information loss in the conditional density estimations. These results underscore the effectiveness of SOM-based augmentation in refining photometric redshift estimation, thereby enabling more robust analyses in cosmology and astrophysics for the NSF-DOE Vera C. Rubin Observatory.

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Quantitative Morphology of Galactic Cirrus in Deep Optical Imaging

Imaging of optical Galactic cirrus, the spatially resolved form of diffuse Galactic light, provides important insights into the properties of the diffuse interstellar medium (ISM) in the Milky Way. While previous investigations have focused mainly on the intensity characteristics of optical cirrus, their morphological properties remain largely unexplored. In this study, we employ several complementary statistical approaches -- local intensity statistics, angular power spectrum / $\Delta$-variance analysis, and wavelet scattering transform analysis -- to characterize the morphology of cirrus in deep optical imaging data. We place our investigation of optical cirrus into a multi-wavelength context by comparing the morphology of cirrus seen with the Dragonfly Telephoto Array to that seen with space-based facilities working at longer wavelengths (Herschel 250 $\mu m$, WISE 12 $\mu m$, and Planck radiance), as well as with structures seen in the DHIGLS HI column density map. Our statistical methods quantify the similarities and the differences of cirrus morphology in all these datasets. The morphology of cirrus at visible wavelengths resembles that of far-infrared cirrus more closely than that of mid-infrared cirrus; on small scales, anisotropies in the cosmic infrared background and systematics may lead to differences. Across all dust tracers, cirrus morphology can be well described by a power spectrum with a common power-law index $\gamma\sim-2.9$. We demonstrate quantitatively that optical cirrus exhibits filamentary, coherent structures across a broad range of angular scales. Our results offer promising avenues for linking the analysis of coherent structures in optical cirrus to the underlying physical processes in the ISM that shape them. Furthermore, we demonstrate that these morphological signatures can be leveraged to distinguish and disentangle cirrus from extragalactic light.

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Blending effects on shear measurement synergy between Euclid-like and LSST-like surveys

Weak gravitational lensing is a powerful probe for constraining cosmological parameters, but its success relies on accurate shear measurements. In this paper, we use image simulations to investigate how a joint analysis of high-resolution space-based and deep ground-based imaging can improve shear estimation. We simulate two scenarios: a grid-based setup, where galaxies are placed on a regular grid to mimic an idealised, blending-free scenario, and a random setup, where galaxies are randomly distributed to capture the impact of blending. Comparing these cases, we find that blending introduces significant biases, particularly in LSST-like data due to its larger point spread function. This highlights the importance of including realistic blending effects when evaluating the performance of joint analyses. Using simulations that account for blending, we find that the most effective catalogue-level synergy is achieved by combining all galaxies detected in either survey. This approach yields an effective galaxy number density of $44.08~\rm arcmin^{-2}$ over the magnitude range of 20.0 to 27.5, compared to $39.17~\rm arcmin^{-2}$ for LSST-like data alone and $30.31~\rm arcmin^{-2}$ for \textit{Euclid}-like data alone. Restricting the analysis to only the overlapping sources detected in both surveys results in a gain of ${\sim}12\%$ compared to using either survey alone. While this joint-object approach is suboptimal at the catalogue level, it may become more effective in pixel-level analyses, where a joint fit to individual galaxy shapes can better leverage the complementary strengths of both data sets. Studying such pixel-level combinations, with realistic blending effects properly accounted for, remains a promising direction for future work.

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The fifth data release of the Kilo Degree Survey: Multi-epoch optical/NIR imaging covering wide and legacy-calibration fields

We present the final data release of the Kilo-Degree Survey (KiDS-DR5), a public European Southern Observatory (ESO) wide-field imaging survey optimised for weak gravitational lensing studies. We combined matched-depth multi-wavelength observations from the VLT Survey Telescope and the VISTA Kilo-degree INfrared Galaxy (VIKING) survey to create a nine-band optical-to-near-infrared survey spanning $1347$ deg$^2$. The median $r$-band $5\sigma$ limiting magnitude is 24.8 with median seeing $0.7^{\prime\prime}$. The main survey footprint includes $4$ deg$^2$ of overlap with existing deep spectroscopic surveys. We complemented these data in DR5 with a targeted campaign to secure an additional $23$ deg$^2$ of KiDS- and VIKING-like imaging over a range of additional deep spectroscopic survey fields. From these fields, we extracted a catalogue of $126\,085$ sources with both spectroscopic and photometric redshift information, which enables the robust calibration of photometric redshifts across the full survey footprint. In comparison to previous releases, DR5 represents a $34\%$ areal extension and includes an $i$-band re-observation of the full footprint, thereby increasing the effective $i$-band depth by $0.4$ magnitudes and enabling multi-epoch science. Our processed nine-band imaging, single- and multi-band catalogues with masks, and homogenised photometry and photometric redshifts can be accessed through the ESO Archive Science Portal.

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KiDS-Legacy: Consistency of cosmic shear measurements and joint cosmological constraints with external probes

We present a cosmic shear consistency analysis of the final data release from the Kilo-Degree Survey (KiDS-Legacy). By adopting three tiers of consistency metrics, we compare cosmological constraints between subsets of the KiDS-Legacy dataset split by redshift, angular scale, galaxy colour and spatial region. We also review a range of two-point cosmic shear statistics. With the data passing all our consistency metric tests, we demonstrate that KiDS-Legacy is the most internally consistent KiDS catalogue to date. In a joint cosmological analysis of KiDS-Legacy and DES Y3 cosmic shear, combined with data from the Pantheon+ Type Ia supernovae compilation and baryon acoustic oscillations from DESI Y1, we find constraints consistent with Planck measurements of the cosmic microwave background with $S_8\equiv \sigma_8\sqrt{\Omega_{\rm m}/0.3} = 0.814^{+0.011}_{-0.012}$ and $\sigma_8 = 0.802^{+0.022}_{-0.018}$.

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KiDS-Legacy: Cosmological constraints from cosmic shear with the complete Kilo-Degree Survey

We present cosmic shear constraints from the completed Kilo-Degree Survey (KiDS), where the cosmological parameter $S_8\equiv\sigma_8\sqrt{\Omega_{\rm m}/0.3} = 0.815^{+0.016}_{-0.021}$, is found to be in agreement ($0.73\sigma$) with results from the Planck Legacy cosmic microwave background experiment. The final KiDS footprint spans $1347$ square degrees of deep nine-band imaging across the optical and near-infrared, along with an extra $23$ square degrees of KiDS-like calibration observations of deep spectroscopic surveys. Improvements in our redshift distribution estimation methodology, combined with our enhanced calibration data and multi-band image simulations, allow us to extend our lensed sample out to a photometric redshift of $z_{\rm B}\leq2.0$. Compared to previous KiDS analyses, the increased survey area and redshift depth results in a $\sim32\%$ improvement in constraining power in terms of $\Sigma_8\equiv\sigma_8\left(\Omega_{\rm m}/0.3\right)^\alpha = 0.821^{+0.014}_{-0.016}$, where $\alpha = 0.58$ has been optimised to match the revised degeneracy direction of $\sigma_8$ and $\Omega_{\rm m}$. We adopt a new physically motivated intrinsic alignment model that depends jointly on the galaxy sample's halo mass and spectral type distributions, and that is informed by previous direct alignment measurements. We also marginalise over our uncertainty on the impact of baryon feedback on the non-linear matter power spectrum. Comparing to previous KiDS analyses, we conclude that the increase seen in $S_8$ primarily results from our improved redshift distribution estimation and calibration, as well as new survey area and improved image reduction. Our companion paper St\"olzner et al. (submitted) presents a full suite of internal and external consistency tests, finding the KiDS-Legacy data set to be the most internally robust sample produced by KiDS to date.

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KiDS-Legacy: Redshift distributions and their calibration

We present the redshift calibration methodology and bias estimates for the cosmic shear analysis of the fifth and final data release (DR5) of the Kilo-Degree Survey (KiDS). KiDS-DR5 includes a greatly expanded compilation of calibrating spectra, drawn from $27$ square degrees of dedicated optical and near-IR imaging taken over deep spectroscopic fields. The redshift distribution calibration leverages a range of new methods and updated simulations to produce the most precise $N(z)$ bias estimates used by KiDS to date. Improvements to our colour-based redshift distribution measurement method (SOM) mean that we are able to use many more sources per tomographic bin for our cosmological analyses, and better estimate the representation of our source sample given the available spec-$z$. We validate our colour-based redshift distribution estimates with spectroscopic cross-correlations (CC). We find that improvements to our cross-correlation redshift distribution measurement methods mean that redshift distribution biases estimated between the SOM and CC methods are fully consistent on simulations, and the data calibration is consistent to better than $2\sigma$ in all tomographic bins.

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The FLAMINGO project: cosmology with the redshift dependence of weak gravitational lensing peaks

Weak gravitational lensing (WL) convergence peaks contain valuable cosmological information in the regime of non-linear collapse. Using the FLAMINGO suite of cosmological hydrodynamical simulations, we study the physical origin and redshift distributions of the objects generating WL peaks selected from a WL convergence map mimicking a $\textit{Euclid}$ signal. We match peaks to individual haloes and show that the high signal-to-noise ratio (SNR$~>~5$) WL peaks measured by Stage IV WL surveys primarily trace $M_{\mathrm{200c}} > 10^{14}~\mathrm{M_\odot}$ haloes. We find that the WL peak sample can compete with the purity and completeness of state-of-the-art X-ray and Sunyaev-Zel'dovich cluster abundance inferences. By comparing the distributions predicted by simulation variations that have been calibrated to the observed gas fractions of local clusters and the present-day galaxy stellar mass function, or shifted versions of these, we illustrate that the shape of the redshift distribution of SNR$~>~5$ peaks is insensitive to baryonic physics while it does change with cosmology. The difference highlights the potential of using WL peaks to constrain cosmology. As the WL convergence and redshift number densities of WL peaks scale differently with cosmology and baryonic feedback, WL peak statistics can simultaneously calibrate baryonic feedback and constrain cosmology.

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Weak lensing constraints on the stellar-to-halo mass relation of galaxy groups with simulation-informed scatter

Understanding the scaling relation between baryonic observables and dark matter halo properties is crucial not only for studying galaxy formation and evolution, but also for deriving accurate cosmological constraints from galaxy surveys. In this paper, we constrain the stellar-to-halo mass relation of galaxy groups identified by the Galaxy and Mass Assembly survey, using weak lensing signals measured by the Kilo-Degree Survey. We compare our measured scaling relation with predictions from the FLAMINGO hydrodynamical simulations and the L-Galaxies semi-analytical model. We find a general agreement between our measurements and simulation predictions for halos with masses ${\gtrsim}10^{13.5}\ h_{70}^{-1}{\rm M}_{\odot}$, but observe slight discrepancies with the FLAMINGO simulations at lower halo masses. We explore improvements to the current halo model framework by incorporating simulation-informed scatter in the group stellar mass distribution as a function of halo mass. We find that including a simulation-informed scatter model tightens the constraints on scaling relations, despite the current data statistics being insufficient to directly constrain the variable scatter. We also test the robustness of our results against different statistical models of miscentring effects from selected central galaxies. We find that accounting for miscentring is essential, but our current measurements do not distinguish among different miscentring models.

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KiDS-1000: Weak lensing and intrinsic alignment around luminous red galaxies

We study the properties of luminous red galaxies (LRGs) selected from the fourth data release of the Kilo Degree Survey (KiDS-1000) via galaxy-galaxy lensing of the background galaxies from KiDS-1000. We used a halo model formalism to interpret our measurements and obtain estimates of the halo masses as well as the satellite fractions of the LRGs, resulting in halo masses of $2.7 \times 10^{12} h^{-1} {\rm M}_{\odot}<M_{\rm h}< 2.6 \times 10^{13} h^{-1} {\rm M}_{\odot}$. We studied the strength of intrinsic alignments (IA) using the position-shape correlations as a function of LRG luminosity, where we used a double power law to describe the relation between luminosity and halo mass to allow for a comparison with previous works. Here, we directly linked the observed IA of the (central) galaxy to the mass of the hosting halo, which is expected to be a fundamental quantity in establishing the alignment. We find that the dependence of the IA amplitude on halo mass is described well by a single power law, with an amplitude of $A = 5.74\pm{0.32}$ and slope of $\beta_M = 0.44\pm{0.04}$, in the range of $1.9 \times 10^{12}h^{-1} {\rm M}_{\odot}<M_{\rm h}<3.7 \times 10^{14} h^{-1} {\rm M}_{\odot}$. We also find that both red and blue galaxies from the source sample associated with the LRGs are randomly oriented, with respect to the LRGs, although our detection significance is limited by the uncertainty in our photometric redshifts.

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The PAU Survey: The quasar UV and Ly$\alpha$ luminosity functions at $2.7<z<5.3$

We present the Lyman-$\alpha$ (Ly$\alpha$) and ultraviolet (UV) luminosity function (LF), in bins of redshift, of quasars selected in the Physics of the Accelerating Universe Survey (PAUS). A sample of 915 objects was selected at $2.7 10^{44}$ ${\rm erg\,s}$$^{-1}$. In order to obtain corrections for the LF estimation, and assess the accuracy of our selection method, we produced mock catalogs of $0<z<4.3$ quasars and galaxies that mimic our target population and their main contaminants. Our results show a clear evolution of the Ly$\alpha$ and UV LFs, with a declining tendency in the number density of quasars towards increasing redshifts. In addition, the faint-end power-law slope of the Ly$\alpha$ LF becomes steeper with redshift, suggesting that the number density of Ly$\alpha$-bright quasars declines faster than that of fainter emitters. By integrating the Ly$\alpha$ LF we find that the total Ly$\alpha$ emitted by bright quasars per unit volume rapidly declines with increasing redshift, being sub-dominant to that of star-forming galaxies by several orders of magnitude by $z\sim 4$. Finally, we stack the NB pseudo-spectra of a visually selected "golden sample" of 591 quasars to obtain photometric composite SEDs in bins of redshift, enabling to measure the mean IGM absorption by the Lyman-$\alpha$ forest as a function of redshift.

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Constraining galaxy properties with complete samples of lenses

The statistics of Einstein radii for a sample of strong lenses can provide valuable constraints on the underlying mass distribution. The correct interpretation, however, relies critically on the modelling of the selection of the sample, which has proven to be a limiting factor. This may change thanks to upcoming uniform high-resolution imaging surveys that cover a large fraction of the sky, because they can provide complete lens samples, with well understood selection criteria. To explore how the observed distribution of Einstein radii depends on the galaxy properties, we simulated a realistic complete sample of strong lenses, predicting a number density of lenses of about 2.5 deg$^{-2}$ for a \Euclid-like setup. Such data can break the degeneracy between the stellar initial mass function (IMF) and the inner slope of the density profile of dark matter, without having to rely on additional information from stellar dynamics. We found that a survey covering only 50 deg$^2$ can already provide tight constraints: assuming that the cosmology is known, the dark matter slope is recovered with an uncertainty of $3.5\%$, while the uncertainty in the ratio between the true stellar mass and that inferred from stellar population modelling was found to be $10\%$. These findings highlight the potential of this method when applied to samples of lenses with well-understood selection functions.

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