arXiv subjects

Martin Eriksen

Publications and source records attributed to Martin Eriksen.

At least 19 recordsLinked to original sources

arXiv2026

Joint probabilistic inference of galaxy redshifts and rest-frame spectra from photometric fluxes with latent diffusion

Wide-field imaging surveys now provide photometry for billions of sources, while spectroscopic observations remain limited, motivating methods that can extract spectroscopic information from photometric data. We present a generative framework for the joint probabilistic inference of galaxy redshifts and rest-frame spectra from broadband photometric fluxes. The model provides a sampling-based estimate of the photometric-redshift probability density function (PDF) for each galaxy, from which accurate point estimates are derived, and reconstructs rest-frame spectra that preserve key spectral properties. We pre-train a spectral autoencoder, SPENDER, on 5 million DESI DR1 spectra to learn a low-dimensional latent space that represents rest-frame spectra. Conditioned on galaxy broadband photometric fluxes, a diffusion model jointly infers the corresponding spectral latent representation and photometric redshift. The inferred latent representation is decoded into a high-resolution rest-frame spectrum, which can be transformed to the observed frame by redshifting and resampling. Sampling from the conditional diffusion model yields a full photometric-redshift PDF for each galaxy, with the resulting point estimates showing a precision comparable to that of a gradient-boosted decision tree model. In most cases, the reconstructed rest-frame spectra reproduce the overall continuum shape and capture the presence of prominent spectral features. For galaxies with sufficiently high signal-to-noise ratios in their observed spectra, the Dn4000 index shows good agreement between the reconstructed spectra and the observed spectra. On average, the spectral reconstruction residuals are close to the noise level of the observed spectra. Latent-diffusion generative modeling enables joint inference of galaxy photometric-redshift PDFs and rest-frame spectra from photometric fluxes.

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arXiv2025

Halo Lightcones with Optimised Orientation and Interpolation in Cosmological Simulations -- an application to mock H$\alpha$ selected galaxies

A critical step to create realistic mock catalogs that support large-scale photometric and spectroscopic sky surveys is the production of cosmological simulations that accurately model the survey observables taking into account the redshift-dependent galaxy formation and evolution processes. Here we develop an efficient framework, HOLCon (Halo Optimised Lightcone Constructor), for post-facto construction of dark matter halo lightcones from simulations and use them to generate a mock galaxy catalogue. HOLCon includes a module to optimise the lightcone's orientation within the simulation box, minimising repeated structures when the survey volume exceeds a single box -- a common challenge in modern surveys. A linear interpolation scheme tracks the evolution of halo properties across snapshots. Applied to the publicly available Uchuu simulation, we construct a lightcone of 50 ${\rm deg}^2$ and extending up to $z = 10$, providing representative coverage of deep fields of Stage IV surveys. We validate the lightcone for cosmological applications by comparing the dark matter halo clustering in the lightcone with those from the original simulation snapshots. Subsequently, we make the galaxy-halo connection on the lightcone with a redshift extended version of the SciPIC algorithm producing a comprehensive set of descriptive galaxy attributes. HOLCon leverages Dask, a scalable parallel computing pythonic framework for fast construction of dark matter halo lightcones enabling rapid creation of multiple statistical realizations essential for robust cosmological inference. The produced galaxy mock makes predictions for clustering of H$\alpha$ emitters, making it a useful cosmology resource.

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arXiv2024

ANNZ+: an enhanced photometric redshift estimation algorithm with applications on the PAU Survey

ANNZ is a fast and simple algorithm which utilises artificial neural networks (ANNs), it was known as one of the pioneers of machine learning approaches to photometric redshift estimation decades ago. We enhanced the algorithm by introducing new activation functions like tanh, softplus, SiLU, Mish and ReLU variants; its new performance is then vigorously tested on legacy samples like the Luminous Red Galaxy (LRG) and Stripe-82 samples from SDSS, as well as modern galaxy samples like the Physics of the Accelerating Universe Survey (PAUS). This work focuses on testing the robustness of activation functions with respect to the choice of ANN architectures, particularly on its depth and width, in the context of galaxy photometric redshift estimation. Our upgraded algorithm, which we named ANNZ+, shows that the tanh and Leaky ReLU activation functions provide more consistent and stable results across deeper and wider architectures with > 1 per cent improvement in root-mean-square error ($\sigma_{\textrm{RMS}}$) and 68th percentile error ($\sigma_{68}$) when tested on SDSS data sets. While assessing its capabilities in handling high dimensional inputs, we achieved an improvement of 11 per cent in $\sigma_{\textrm{RMS}}$ and 6 per cent in $\sigma_{68}$ with the tanh activation function when tested on the 40-narrowband PAUS dataset; it even outperformed ANNZ2, its supposed successor, by 44 per cent in $\sigma_{\textrm{RMS}}$. This justifies the effort to upgrade the 20-year-old ANNZ, allowing it to remain viable and competitive within the photo-z community today. The updated algorithm ANNZ+ is publicly available at https://github.com/imdadmpt/ANNzPlus.

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arXiv2024

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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arXiv2024

The PAU Survey: galaxy stellar population properties estimates with narrowband data

Narrowband galaxy surveys have recently gained interest as a promising method to achieve the necessary accuracy on the photometric redshift estimate of individual galaxies for stage-IV cosmological surveys. One key advantage is the ability to provide higher spectral resolution information about galaxies that should allow a more accurate and precise estimation of galaxy stellar population properties. However, the impact of adding narrow-band photometry on the stellar population properties estimate is largely unexplored. The scope of this work is two-fold: on one side, leveraging the predictive power of broad-band and narrow-band data to infer galaxy physical properties such as stellar masses, ages, star formation rates and metallicities. On the other hand, evaluating the improvement of performance in estimating galaxy properties when we use narrow-band data instead of broad-band. In this work we measure the stellar population properties of a sample of galaxies in the COSMOS field for which both narrowband and broadband data are available. In particular, we employ narrowband data from PAUS and broad-band data from CFHTLS. We use two different spectral energy distribution fitting codes to measure galaxy properties, namely CIGALE and Prospector. We find that the increased spectral resolution of narrow-band photometry does not yield a substantial improvement on constraining galaxy properties using spectral energy distribution fitting. Still we find that we obtain a more diverse distribution of metallicities and dust optical depths with cigale when employing the narrowband data. The effect is not as prominent as expected, which we relate this to the low narrowband SNR of a majority of the galaxies, the respective drawbacks of both codes as well as the coverage only in the optical regime. The measured properties are afterwards compared to the COSMOS2020 catalogue, showing good agreement.

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arXiv2021

The PAU Survey: Measurement of Narrow-band galaxy properties with Approximate Bayesian Computation

Narrow-band imaging surveys allow the study of the spectral characteristics of galaxies without the need of performing their spectroscopic follow-up. In this work, we forward-model the Physics of the Accelerating Universe Survey (PAUS) narrow-band data. The aim is to improve the constraints on the spectral coefficients used to create the galaxy spectral energy distributions (SED) of the galaxy population model in Tortorelli et al. 2020. In that work, the model parameters were inferred from the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) data using Approximate Bayesian Computation (ABC). This led to stringent constraints on the B-band galaxy luminosity function parameters, but left the spectral coefficients only broadly constrained. To address that, we perform an ABC inference using CFHTLS and PAUS data. This is the first time our approach combining forward-modelling and ABC is applied simultaneously to multiple datasets. We test the results of the ABC inference by comparing the narrow-band magnitudes of the observed and simulated galaxies using Principal Component Analysis, finding a very good agreement. Furthermore, we prove the scientific potential of the constrained galaxy population model to provide realistic stellar population properties by measuring them with the SED fitting code \textsc{CIGALE}. We use CFHTLS broad-band and PAUS narrow-band photometry for a flux-limited ($\mathrm{i}<22.5$) sample of galaxies up to redshift $\mathrm{z \sim 0.8}$. We find that properties like stellar masses, star-formation rates, mass-weighted stellar ages and metallicities are in agreement within errors between observations and simulations. Overall, this work shows the ability of our galaxy population model to correctly forward-model a complex dataset such as PAUS and the ability to reproduce the diversity of galaxy properties at the redshift range spanned by CFHTLS and PAUS.

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arXiv2021

The PAU survey: Estimating galaxy photometry with deep learning

With the dramatic rise in high-quality galaxy data expected from Euclid and Vera C. Rubin Observatory, there will be increasing demand for fast high-precision methods for measuring galaxy fluxes. These will be essential for inferring the redshifts of the galaxies. In this paper, we introduce Lumos, a deep learning method to measure photometry from galaxy images. Lumos builds on BKGnet, an algorithm to predict the background and its associated error, and predicts the background-subtracted flux probability density function. We have developed Lumos for data from the Physics of the Accelerating Universe Survey (PAUS), an imaging survey using a 40 narrow-band filter camera (PAUCam). PAUCam images are affected by scattered light, displaying a background noise pattern that can be predicted and corrected for. On average, Lumos increases the SNR of the observations by a factor of 2 compared to an aperture photometry algorithm. It also incorporates other advantages like robustness towards distorting artefacts, e.g. cosmic rays or scattered light, the ability of deblending and less sensitivity to uncertainties in the galaxy profile parameters used to infer the photometry. Indeed, the number of flagged photometry outlier observations is reduced from 10% to 2%, comparing to aperture photometry. Furthermore, with Lumos photometry, the photo-z scatter is reduced by ~10% with the Deepz machine learning photo-z code and the photo-z outlier rate by 20%. The photo-z improvement is lower than expected from the SNR increment, however currently the photometric calibration and outliers in the photometry seem to be its limiting factor.

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arXiv2021

The PAU Survey: narrowband photometric redshifts using Gaussian processes

We study the performance of the hybrid template-machine-learning photometric redshift (photo-$z$) algorithm Delight, which uses Gaussian processes, on a subset of the early data release of the Physics of the Accelerating Universe Survey (PAUS). We calibrate the fluxes of the $40$ PAUS narrow bands with $6$ broadband fluxes ($uBVriz$) in the COSMOS field using three different methods, including a new method which utilises the correlation between the apparent size and overall flux of the galaxy. We use a rich set of empirically derived galaxy spectral templates as guides to train the Gaussian process, and we show that our results are competitive with other standard photometric redshift algorithms. Delight achieves a photo-$z$ $68$th percentile error of $σ_{68}=0.0081(1+z)$ without any quality cut for galaxies with $i_\mathrm{auto}<22.5$ as compared to $0.0089(1+z)$ and $0.0202(1+z)$ for the BPz and ANNz2 codes, respectively. Delight is also shown to produce more accurate probability distribution functions for individual redshift estimates than BPz and ANNz2. Common photo-$z$ outliers of Delight and BCNz2 (previously applied to PAUS) are found to be primarily caused by outliers in the narrowband fluxes, with a small number of cases potentially indicating spectroscopic redshift failures in the reference sample. In the process, we introduce performance metrics derived from the results of BCNz2 and Delight, allowing us to achieve a photo-$z$ quality of $σ_{68}<0.0035(1+z)$ at a magnitude of $i_\mathrm{auto}<22.5$ while keeping $50$ per cent objects of the galaxy sample.

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arXiv2021

The PAU Survey: Intrinsic alignments and clustering of narrow-band photometric galaxies

We present the first measurements of the projected clustering and intrinsic alignments (IA) of galaxies observed by the Physics of the Accelerating Universe Survey (PAUS). With photometry in 40 narrow optical passbands ($450\rm{nm}-850\rm{nm}$), the quality of photometric redshift estimation is $σ_{z} \sim 0.01(1 + z)$ for galaxies in the $19\,\rm{deg}^{2}$ Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) W3 field, allowing us to measure the projected 3D clustering and IA for flux-limited, faint galaxies ($i < 22.5$) out to $z\sim0.8$. To measure two-point statistics, we developed, and tested with mock photometric redshift samples, `cloned' random galaxy catalogues which can reproduce data selection functions in 3D and account for photometric redshift errors. In our fiducial colour-split analysis, we made robust null detections of IA for blue galaxies and tentative detections of radial alignments for red galaxies ($\sim1-3σ$), over scales of $0.1-18\,h^{-1}\rm{Mpc}$. The galaxy clustering correlation functions in the PAUS samples are comparable to their counterparts in a spectroscopic population from the Galaxy and Mass Assembly survey, modulo the impact of photometric redshift uncertainty which tends to flatten the blue galaxy correlation function, whilst steepening that of red galaxies. We investigate the sensitivity of our correlation function measurements to choices in the random catalogue creation and the galaxy pair-binning along the line of sight, in preparation for an optimised analysis over the full PAUS area.

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arXiv2021

The PAU survey: Ly$α$ intensity mapping forecast

In this work, we explore the application of intensity mapping to detect extended Ly$α$ emission from the IGM via cross-correlation of PAUS images with Ly$α$ forest data from eBOSS and DESI. Seven narrow-band (FWHM=13nm) PAUS filters have been considered, ranging from 455 to 515 nm in steps of 10 nm, which allows the observation of Ly$α$ emission in a range $2.7<z<3.3$. The cross-correlation is simulated first in an area of 100 deg$^2$ (PAUS projected coverage), and second in two hypothetical scenarios: a deeper PAUS (complete up to $i_{\rm AB}<24$ instead of $i_{\rm AB}<23$, observation time x6), and an extended PAUS coverage of 225 deg$^2$ (observation time x2.25). A hydrodynamic simulation of size 400 Mpc/h is used to simulate both extended Ly$α$ emission and absorption, while the foregrounds in PAUS images have been simulated using a lightcone mock catalogue. Using an optimistic estimation of uncorrelated PAUS noise, the total probability of a non-spurious detection is estimated to be 1.8\% and 4.5\% for PAUS-eBOSS and PAUS-DESI , from a run of 1000 simulated cross-correlations with different realisations of instrumental noise and quasar positions. The hypothetical PAUS scenarios increase this probability to 15.3\% (deeper PAUS) and 9.0\% (extended PAUS). With realistic correlated noise directly measured from PAUS images, these probabilities become negligible. Despite these negative results, some evidences suggest that this methodology may be more suitable to broad-band surveys.

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arXiv2020

The PAU Survey: An improved photo-$z$ sample in the COSMOS field

We present -- and make publicly available -- accurate and precise photometric redshifts in the ACS footprint from the COSMOS field for objects with $i_{\mathrm{AB}}\leq 23$. The redshifts are computed using a combination of narrow band photometry from PAUS, a survey with 40 narrow bands spaced at $100Å$ intervals covering the range from $4500Å$ to $8500Å$, and 26 broad, intermediate, and narrow bands covering the UV, visible and near infrared spectrum from the COSMOS2015 catalogue. We introduce a new method that models the spectral energy distributions (SEDs) as a linear combination of continuum and emission line templates and computes its Bayes evidence, integrating over the linear combinations. The correlation between the UV luminosity and the OII line is measured using the 66 available bands with the zCOSMOS spectroscopic sample, and used as a prior which constrains the relative flux between continuum and emission line templates. The flux ratios between the OII line and $\mathrm{H}_α$, $\mathrm{H}_β$ and $\mathrm{OIII}$ are similarly measured and used to generate the emission line templates. Comparing to public spectroscopic surveys via the quantity $Δ_z\equiv(z_{\mathrm{photo}}-z_{\mathrm{spec}})/(1+z_{\mathrm{spec}})$, we find the photometric redshifts to be more precise than previous estimates, with $σ_{68}(Δ_z) \approx (0.003, 0.009)$ for galaxies at magnitude $i_{\mathrm{AB}}\sim18$ and $i_{\mathrm{AB}}\sim23$, respectively, which is $3\times$ and $1.66\times$ tighter than COSMOS2015. Additionally, we find the redshifts to be very accurate on average, yielding a median of the $Δ_z$ distribution compatible with $|\mathrm{median}(Δ_z)|\leq0.001$ at all redshifts and magnitudes considered. Both the added PAUS data and new methodology contribute significantly to the improved results.

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arXiv2020

CosmoHub: Interactive exploration and distribution of astronomical data on Hadoop

We present CosmoHub (https://cosmohub.pic.es), a web application based on Hadoop to perform interactive exploration and distribution of massive cosmological datasets. Recent Cosmology seeks to unveil the nature of both dark matter and dark energy mapping the large-scale structure of the Universe, through the analysis of massive amounts of astronomical data, progressively increasing during the last (and future) decades with the digitization and automation of the experimental techniques. CosmoHub, hosted and developed at the Port d'Informació Científica (PIC), provides support to a worldwide community of scientists, without requiring the end user to know any Structured Query Language (SQL). It is serving data of several large international collaborations such as the Euclid space mission, the Dark Energy Survey (DES), the Physics of the Accelerating Universe Survey (PAUS) and the Marenostrum Institut de Ciències de l'Espai (MICE) numerical simulations. While originally developed as a PostgreSQL relational database web frontend, this work describes the current version of CosmoHub, built on top of Apache Hive, which facilitates scalable reading, writing and managing huge datasets. As CosmoHub's datasets are seldomly modified, Hive it is a better fit. Over 60 TiB of catalogued information and $50 \times 10^9$ astronomical objects can be interactively explored using an integrated visualization tool which includes 1D histogram and 2D heatmap plots. In our current implementation, online exploration of datasets of $10^9$ objects can be done in a timescale of tens of seconds. Users can also download customized subsets of data in standard formats generated in few minutes.

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arXiv2019

The Physics of the Accelerating Universe Camera

The PAU (Physics of the Accelerating Universe) Survey goal is to obtain photometric redshifts (photo-z) and Spectral Energy Distribution (SED) of astronomical objects with a resolution roughly one order of magnitude better than current broad band photometric surveys. To accomplish this, a new large field of view camera (PAUCam) has been designed, built, commissioned and is now operated at the William Herschel Telescope (WHT). With the current WHT Prime Focus corrector, the camera covers ~1-degree diameter Field of View (FoV), of which, only the inner ~40 arcmin diameter are unvignetted. The focal plane consists of a mosaic of 18 2k$x4k Hamamatsu fully depleted CCDs, with high quantum efficiency up to 1 micrometers in wavelength. To maximize the detector coverage within the FoV, filters are placed in front of the CCDs inside the camera cryostat (made out of carbon fiber) using a challenging movable tray system. The camera uses a set of 40 narrow band filters ranging from ~4500 to ~8500 Angstroms complemented with six standard broad-band filters, ugrizY. The PAU Survey aims to cover roughly 100 square degrees over fields with existing deep photometry and galaxy shapes to obtain accurate photometric redshifts for galaxies down to i_AB~22.5, detecting also galaxies down to i_AB~24 with less precision in redshift. With this data set we will be able to measure intrinsic alignments, galaxy clustering and perform galaxy evolution studies in a new range of densities and redshifts. Here, we describe the PAU camera, its first commissioning results and performance.

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arXiv2018

The PAU Survey: star-galaxy classification with multi narrow-band data

Classification of stars and galaxies is a well-known astronomical problem that has been treated using different approaches, most of them relying on morphological information. In this paper, we tackle this issue using the low-resolution spectra from narrow band photometry, provided by the PAUS (Physics of the Accelerating Universe) survey. We find that, with the photometric fluxes from the 40 narrow band filters and without including morphological information, it is possible to separate stars and galaxies to very high precision, 98.4% purity with a completeness of 98.8% for objects brighter than I = 22.5. This precision is obtained with a Convolutional Neural Network as a classification algorithm, applied to the objects' spectra. We have also applied the method to the ALHAMBRA photometric survey and we provide an updated classification for its Gold sample.

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arXiv2018

The PAU Survey: Operation and orchestration of multi-band survey data

The Physics of the Accelerating Universe (PAU) Survey is an international project for the study of cosmological parameters associated with Dark Energy. PAU's 18-CCD camera (PAUCam), installed at the prime focus of the William Herschel Telescope at the Roque de los Muchachos Observatory (La Palma, Canary Islands), scans part of the northern sky, to collect low resolution spectral information of millions of galaxies with its unique set of 40 narrow-band filters in the optical range from 450 nm to 850 nm, and a set of 6 standard broad band filters. The PAU data management (PAUdm) team is in charge of treating the data, including data transfer from the observatory to the PAU Survey data center, hosted at Port d'Informació Científica (PIC). PAUdm is also in charge of the storage, data reduction and, finally, of making the results available to the scientific community. We describe the technical solutions adopted to cover different aspects of the PAU Survey data management, from the computing infrastructure to support the operations, to the software tools and web services for the data process orchestration and exploration. In particular we will focus on the PAU database, developed for the coordination of the different PAUdm tasks, and to preserve and guarantee the consistency of data and metadata.

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arXiv2018

The PAU Survey: A Forward Modeling Approach for Narrow-band Imaging

Weak gravitational lensing is a powerful probe of the dark sector, once measurement systematic errors can be controlled. In Refregier & Amara (2014), a calibration method based on forward modeling, called MCCL, was proposed. This relies on fast image simulations (e.g., UFig; Berge et al. 2013) that capture the key features of galaxy populations and measurement effects. The MCCL approach has been used in Herbel et al. (2017) to determine the redshift distribution of cosmological galaxy samples and, in the process, the authors derived a model for the galaxy population mainly based on broad-band photometry. Here, we test this model by forward modeling the 40 narrow-band photometry given by the novel PAU Survey (PAUS). For this purpose, we apply the same forced photometric pipeline on data and simulations using Source Extractor (Bertin & Arnouts 1996). The image simulation scheme performance is assessed at the image and at the catalogues level. We find good agreement for the distribution of pixel values, the magnitudes, in the magnitude-size relation and the interband correlations. A principal component analysis is then performed, in order to derive a global comparison of the narrow-band photometry between the data and the simulations. We use a `mixing' matrix to quantify the agreement between the observed and simulated sets of Principal Components (PCs). We find good agreement, especially for the first three most significant PCs. We also compare the coefficients of the PCs decomposition. While there are slight differences for some coefficients, we find that the distributions are in good agreement. Together, our results show that the galaxy population model derived from broad-band photometry is in good overall agreement with the PAUS data. This offers good prospect for incorporating spectral information to the galaxy model by adjusting it to the PAUS narrow-band data using forward modeling.

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arXiv2018

Cosmological constraints from multiple tracers in spectroscopic surveys

We use the Fisher matrix formalism to study the expansion and growth history of the Universe using galaxy clustering with 2D angular cross-correlation tomography in spectroscopic or high resolution photometric redshift surveys. The radial information is contained in the cross correlations between narrow redshift bins. We show how multiple tracers with redshift space distortions cancel sample variance and arbitrarily improve the constraints on the dark energy equation of state $ω(z)$ and the growth parameter $γ$ in the noiseless limit. The improvement for multiple tracers quickly increases with the bias difference between the tracers, up to a factor $\sim4$ in $\text{FoM}_{γω}$. We model a magnitude limited survey with realistic density and bias using a conditional luminosity function, finding a factor 1.3-9.0 improvement in $\text{FoM}_{γω}$ -- depending on global density -- with a split in a halo mass proxy. Partly overlapping redshift bins improve the constraints in multiple tracer surveys a factor $\sim1.3$ in $\text{FoM}_{γω}$. This findings also apply to photometric surveys, where the effect of using multiple tracers is magnified. We also show large improvement on the FoM with increasing density, which could be used as a trade-off to compensate some possible loss with radial resolution.

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arXiv2017

Implications of a wavelength dependent PSF for weak lensing measurements

The convolution of galaxy images by the point-spread function (PSF) is the dominant source of bias for weak gravitational lensing studies, and an accurate estimate of the PSF is required to obtain unbiased shape measurements. The PSF estimate for a galaxy depends on its spectral energy distribution (SED), because the instrumental PSF is generally a function of the wavelength. In this paper we explore various approaches to determine the resulting `effective' PSF using broad-band data. Considering the Euclid mission as a reference, we find that standard SED template fitting methods result in biases that depend on source redshift, although this may be remedied if the algorithms can be optimised for this purpose. Using a machine-learning algorithm we show that, at least in principle, the required accuracy can be achieved with the current survey parameters. It is also possible to account for the correlations between photometric redshift and PSF estimates that arise from the use of the same photometry. We explore the impact of errors in photometric calibration, errors in the assumed wavelength dependence of the PSF model and limitations of the adopted template libraries. Our results indicate that the required accuracy for Euclid can be achieved using the data that are planned to determine photometric redshifts.

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