Two-Point Correlation Function as a Cosmological Probe

The near-future of observational cosmology lies in extensive galaxy surveys that will map the dynamic evolution of the Universe for large part of its history. An example of such a survey is the European Space Agencys (ESA) Euclid mission. It is a space telescope scheduled to launch in 2023. The Euclid wide survey will cover 15000 square degrees, which is more than one third of the sky. It is expected to observe photometrically roughly 1.5 billion galaxies for the purposes of weak lensing analysis. In addition, 50 million galaxies will be observed spectroscopically, which will allow accurate determination of their three-dimensional distribution. The spectroscopic measurements will reach above a redshift of 2 - covering over 75% of the history of the Universe -- and photometric measurements to even higher redshifts. An important aspect of measuring the galaxy distribution is that its evolution reflects the expansion history of the Universe. It is thus a powerful tool for understanding the processes that drive the expansion. To be able to compare the evolution of the distribution with predictions from, for example, models of dark energy, we need to extract various statistics of the galaxy distribution. An important quantity is the two-point correlation function, which is a measure of amount of clustering at different spatial scales. Estimators for the two-point correlation functions rely on counting pairs of objects, usually galaxies or their clusters. The large number of galaxies Euclid is going to observe will make the clustering analysis a non-trivial computational task. The situation becomes even more challenging when one needs to estimate the covariance of the two-point correlation function estimates. This is usually done by measuring clustering of mock galaxy samples. To reach the parameter accuracy Euclid is aiming for, the number of samples needed is of the order of many thousands, thus in principle increasing the cost of a two-point correlation function estimate by the same factor. It is therefore of paramount importance to perform this analysis step as efficiently as possible. Two of the three publications included in this thesis deal with this need for efficiency. A central part of methods introduced in them is the so-called random catalog. It is a counterpart to the measured galaxy catalog that is used to model the survey geometry and various selection effects. We show that it is possible to use the random sample in clever ways to significantly speed up the computation with negligible loss of accuracy. We present one method for a single two-point correlation function estimate and another one for its covariance. We show that in a situation representative of a spectroscopic galaxy sample we can reach speed-up by a factor of order of ten -- for each method. The combined factor of hundred could already be decisive in whether the analysis is computationally feasible or not. In the end, the computational efficiency is only a tool for reaching more and more precise scientific results. The third paper in this thesis offers an example of a clustering-based cosmological analysis. Within the current theoretical understanding of the large scale structure of the Universe it is possible to obtain relations between the clustering of clusters of galaxies and the masses of their hosting dark matter halos. Generally, the more massive the halos, the more strongly clustered they are. In the paper we study this connection in detail using an X-ray selected, already-published sample of galaxy clusters. We find that the cosmological predictions for the clustering of clusters gives results that are compatible with the observations and the cosmological parameters inferred from the measurements are consistent with those obtained from the cluster mass function and the Cosmic Microwave Background.Havaitsevan kosmologian tulevaisuus on laajoissa galaksikartoituksissa, joiden avulla saadaan tarkka kuva maailmankaikkeuden laajenemishistoriasta. Yksi esimerkki tällaisesta galaksikartoitushankkeesta on Euroopan avaruusjärjestön ESA:n Euclid avaruusteleskooppi, joka laukaistaan 2023. Euclid tulee kuuden vuoden aikana kuvaamaan noin kolmanneksen taivaasta ja havainnot kattavat yli 75 % maailmankaikkeuden laajenemishistoriasta. Tämän historian puolestaan määrää maailmankaikkeuden aine-energiasisältö, mikä mahdollistaa esimerkiksi pimeän energian luonteen tutkimisen. Oleellisia ovat erilaiset galaksien avaruudellista jakaumaa kuvaavat tilastolliset suureet, joita voidaan verrata pimeän energian malleista saataviin ennusteisiin. Tärkeä esimerkki tällaisesta suureesta on galaksien kaksipistekorrelaatiofunktio, joka kuvaa galaksien klusteroitumisen voimakkuutta. Tämä funktio voidaan määrittää havaitusta galaksiaineistosta numeerisesti laskemalla galaksiparien määriä eri etäisyysskaaloilla. Esimerkiksi Euclidin tuottama suuri havaintoaineisto tekee tästä laskennallisesti raskasta. Tilanne muuttu yhä haastavammaksi, kun ryhdytään laskemaan korrelaatiofunktion hajontaa eli kovarianssia. Tämä tehdään tyypillisesti laskemalla korrelaatiofunktio suuresta määrästä, mahdollisesti useista tuhansista simuloiduista galaksikatalogeista, jolloin laskennallinen vaativuus kasvaa periaatteessa samalla kertoimella. Kaksi väitöskirjaan sisältyvästä kolmesta artikkelista pyrkii vastaamaan tähän haasteeseen tehostamalla korrelaatiofunktion ja sen kovarianssin laskemisessa käytettäviä algoritmeja. Oleellinen osa kehitettyjä menetelmiä on niin sanottu satunnaiskatalogi, jonka avulla mallinnetaan kunkin galaksikartoituksen havaitseman alueen geometriaa ja erilaisia valintaefektejä. Julkaisuissa osoitamme, että muokkaamalla satunnaiskatalogin käyttöä sopivasti voidaan sekä yksittäisen korrelaatiofunktion että sen kovarianssimatriisin määrittämiseen tarvittava laskenta-aika laskea alle kymmenesosaan tarkkuuden kärsimättä. Yhdistettynä nämä menetelmät vähentävät kovarianssin laskentavaativuutta noin tekijällä sata, mikä on ratkaisevan tärkeää esimerkiksi Euclidin data-analyysissä. Väitöskirjaan sisältyvä kolmas julkaisu on esimerkki korrelaatiofunktion käytöstä analyysissä, jossa pyritään määrittämään kosmologisen mallin parametrien arvoja. Analyysissä havaittiin, että galaksijoukkojen (jotka eivät klusteroitumisen puolesta periaatteessa eroa yksittäisistä galakseista) korrelaatiofunktiosta johdetut parametrien arvot ovat yhteensopivia esimerkiksi kosmisesta mikroaaltotaustasta saatujen tulosten kanssa

Instrumenttikohina taustasäteilyanalyysissä

Tässä työssä esitellään kosmisen mikroaaltotaustan analysointiin liittyviä menetelmiä ja erityisesti sitä, miten nämä menetelmät huomioivat instrumenttikohinan vaikutuksia. Päähuomio on Euroopan avaruusjärjestön 2009 laukaiseman Planck-satelliitin data-analyysissä tarvittavissa menetelmissä. Noin 380 000 vuoden ikäisesssä maailmankaikkeudessa syntynyt kosminen mikroaaltotausta on tärkein yksittäinen kosmologinen havaintoaineisto. Tämä säteily on suorin ikkuna hyvin varhaiseen maailmankaikkeuteen ja sen merkitys maailmankaikkeuden historian ja ominaisuuksien kartoittamisessa on ollut valtava siitä lähtien, kun se 1960-luvulla ensi kerran havaittiin. Tämän merkityksen korostamiseksi tutkielman alussa käydään läpi kosmologian teoreettista perustaa. Planck-satelliitin tuottama valtava datamäärä ja mittaustarkkuus muodostaa huomattavan haasteen datan käsittelylle. Rajallisten laskentaresurrsien vuoksi on kehitettävä tehokas, mutta samaan aikaan informaatiota mahdollisimman vähän hukkaava analyysiketju. Tärkeä huomioon otettava tekijä tässä kehitystyössä on mittalaitteissa esiintyvä kohina. Tämä kohina on luontevaa käsitellä samalla kun taustasäteilymittauksista muodostetaan säteilyä kuvaavia taivaankarttoja. Kohinan poistoon on kehitetty monia tehokkaita algoritmeja. Tästä huolimatta osa siitä jää väistämättä karttoihin ja tämän odotettavissa oleva vaikutus on otettava jatkoanalyysissä huomioon tieteellisten tulosten luotettavuuden varmistamiseksi. Tässä tutkielmassa käsitellään yksityiskohtaisesti sekä kohinan poiston että karttoihin jäävän kohinan kuvaamisen kannalta oleellisia seikkoja. Karttoihin jäävän residuaalikohinan kuvaamisessa tärkein työkälu ovat niin sanotut kohinan kovarianssimatriisit. Tätä tutkielmaa varten tuotettiin esimerkkejä näistä matriiseista ja tutkittiin näiden onnistumista kohinan kuvaamisessa numeeristen simulaatoiden avulla. Tässä havaittiin, että kovarianssimatriisien laatuun voidaan vaikuttaa merkittävästi valitsemalla näiden numeeriseen laskemiseen vaikuttavat parametrit oikein. Lopuksi tutkielmassa esitellään kohinan kovarianssimatriisien merkitystä taustasäteilyhavaintojen ja teoreettisten ennusteiden vertailussa. Ne kuvaavat kuinka suuren epävarmuustekijän instrumenttikohina tuo tähän vertailuun ja tämä osaltaan parantaa arviota siitä kuinka luotettavasti erilaiset teoreettiset mallit ovat mahdollisia tai pois suljettuja

Euclid preparation : XIX. Impact of magnification on photometric galaxy clustering

Aims. We investigate the importance of lensing magnification for estimates of galaxy clustering and its cross-correlation with shear for the photometric sample of Euclid. Using updated specifications, we study the impact of lensing magnification on the constraints and the shift in the estimation of the best fitting cosmological parameters that we expect if this effect is neglected. Methods. We follow the prescriptions of the official Euclid Fisher matrix forecast for the photometric galaxy clustering analysis and the combination of photometric clustering and cosmic shear. The slope of the luminosity function (local count slope), which regulates the amplitude of the lensing magnification, and the galaxy bias have been estimated from the Euclid Flagship simulation. Results. We find that magnification significantly affects both the best-fit estimation of cosmological parameters and the constraints in the galaxy clustering analysis of the photometric sample. In particular, including magnification in the analysis reduces the 1 sigma errors on Omega(m,0), w(0), w(a) at the level of 20-35%, depending on how well we will be able to independently measure the local count slope. In addition, we find that neglecting magnification in the clustering analysis leads to shifts of up to 1.6 sigma in the best-fit parameters. In the joint analysis of galaxy clustering, cosmic shear, and galaxy-galaxy lensing, magnification does not improve precision, but it leads to an up to 6 sigma bias if neglected. Therefore, for all models considered in this work, magnification has to be included in the analysis of galaxy clustering and its cross-correlation with the shear signal (3 x 2pt analysis) for an accurate parameter estimation.Peer reviewe

Euclid preparation : XV. Forecasting cosmological constraints for the Euclid and CMB joint analysis

The combination and cross-correlation of the upcoming Euclid data with cosmic microwave background (CMB) measurements is a source of great expectation since it will provide the largest lever arm of epochs, ranging from recombination to structure formation across the entire past light cone. In this work, we present forecasts for the joint analysis of Euclid and CMB data on the cosmological parameters of the standard cosmological model and some of its extensions. This work expands and complements the recently published forecasts based on Euclid-specific probes, namely galaxy clustering, weak lensing, and their cross-correlation. With some assumptions on the specifications of current and future CMB experiments, the predicted constraints are obtained from both a standard Fisher formalism and a posterior-fitting approach based on actual CMB data. Compared to a Euclid-only analysis, the addition of CMB data leads to a substantial impact on constraints for all cosmological parameters of the standard ?-cold-dark-matter model, with improvements reaching up to a factor of ten. For the parameters of extended models, which include a redshift-dependent dark energy equation of state, non-zero curvature, and a phenomenological modification of gravity, improvements can be of the order of two to three, reaching higher than ten in some cases. The results highlight the crucial importance for cosmological constraints of the combination and cross-correlation of Euclid probes with CMB data.Peer reviewe

Euclid preparation : I. The Euclid Wide Survey

Euclid is a mission of the European Space Agency that is designed to constrain the properties of dark energy and gravity via weak gravitational lensing and galaxy clustering. It will carry out a wide area imaging and spectroscopy survey (the Euclid Wide Survey: EWS) in visible and near-infrared bands, covering approximately 15 000 deg(2) of extragalactic sky in six years. The wide-field telescope and instruments are optimised for pristine point spread function and reduced stray light, producing very crisp images. This paper presents the building of the Euclid reference survey: the sequence of pointings of EWS, deep fields, and calibration fields, as well as spacecraft movements followed by Euclid as it operates in a step-and-stare mode from its orbit around the Lagrange point L2. Each EWS pointing has four dithered frames; we simulated the dither pattern at the pixel level to analyse the effective coverage. We used up-to-date models for the sky background to define the Euclid region-of-interest (RoI). The building of the reference survey is highly constrained from calibration cadences, spacecraft constraints, and background levels; synergies with ground-based coverage were also considered. Via purposely built software, we first generated a schedule for the calibrations and deep fields observations. On a second stage, the RoI was tiled and scheduled with EWS observations, using an algorithm optimised to prioritise the best sky areas, produce a compact coverage, and ensure thermal stability. The result is the optimised reference survey RSD_2021A, which fulfils all constraints and is a good proxy for the final solution. The current EWS covers approximate to 14 & x2006;500 deg(2). The limiting AB magnitudes (5 sigma point-like source) achieved in its footprint are estimated to be 26.2 (visible band I-E) and 24.5 (for near infrared bands Y-E, J(E), H-E); for spectroscopy, the H alpha line flux limit is 2 x 10(-16) erg(-1) cm(-2) s(-1) at 1600 nm; and for diffuse emission, the surface brightness limits are 29.8 (visible band) and 28.4 (near infrared bands) mag arcsec(-2).Peer reviewe

Euclid: Fast two-point correlation function covariance through linear construction

We present a method for fast evaluation of the covariance matrix for a two-point galaxy correlation function (2PCF) measured with the Landy-Szalay estimator. The standard way of evaluating the covariance matrix consists in running the estimator on a large number of mock catalogs, and evaluating their sample covariance. With large random catalog sizes (random-to-data objects' ratio M >> 1) the computational cost of the standard method is dominated by that of counting the data-random and random-random pairs, while the uncertainty of the estimate is dominated by that of data-data pairs. We present a method called Linear Construction (LC), where the covariance is estimated for small random catalogs with a size of M = 1 and M = 2, and the covariance for arbitrary M is constructed as a linear combination of the two. We show that the LC covariance estimate is unbiased. We validated the method with PINOCCHIO simulations in the range r = 20-200 h(-1) Mpc. With M = 50 and with 2h(-1) Mpc bins, the theoretical speedup of the method is a factor of 14. We discuss the impact on the precision matrix and parameter estimation, and present a formula for the covariance of covariance.Peer reviewe

Euclid preparation : XVIII. The NISP photometric system

Euclid will be the first space mission to survey most of the extragalactic sky in the 0.95-2.02 mu m range, to a 5 sigma point-source median depth of 24.4 AB mag. This unique photometric dataset will find wide use beyond Euclid's core science. In this paper, we present accurate computations of the Euclid Y-E, J(E), and H-E passbands used by the Near-Infrared Spectrometer and Photometer (NISP), and the associated photometric system. We pay particular attention to passband variations in the field of view, accounting for, among other factors, spatially variable filter transmission and variations in the angle of incidence on the filter substrate using optical ray tracing. The response curves' cut-on and cut-off wavelengths - and their variation in the field of view - are determined with similar to 0.8 nm accuracy, essential for the photometric redshift accuracy required by Euclid. After computing the photometric zero points in the AB mag system, we present linear transformations from and to common ground-based near-infrared photometric systems, for normal stars, red and brown dwarfs, and galaxies separately. A Python tool to compute accurate magnitudes for arbitrary passbands and spectral energy distributions is provided. We discuss various factors, from space weathering to material outgassing, that may slowly alter Euclid's spectral response. At the absolute flux scale, the Euclid in-flight calibration program connects the NISP photometric system to Hubble Space Telescope spectrophotometric white dwarf standards; at the relative flux scale, the chromatic evolution of the response is tracked at the milli-mag level. In this way, we establish an accurate photometric system that is fully controlled throughout Euclid's lifetime.Peer reviewe

Euclid preparation : XIII. Forecasts for galaxy morphology with the Euclid Survey using deep generative models

We present a machine learning framework to simulate realistic galaxies for the Euclid Survey, producing more complex and realistic galaxies than the analytical simulations currently used in Euclid. The proposed method combines a control on galaxy shape parameters offered by analytic models with realistic surface brightness distributions learned from real Hubble Space Telescope observations by deep generative models. We simulate a galaxy field of 0.4x2006;deg(2) as it will be seen by the Euclid visible imager VIS, and we show that galaxy structural parameters are recovered to an accuracy similar to that for pure analytic Sersic profiles. Based on these simulations, we estimate that the Euclid Wide Survey (EWS) will be able to resolve the internal morphological structure of galaxies down to a surface brightness of 22.5x2006;magx2006;arcsec(-2), and the Euclid Deep Survey (EDS) down to 24.9x2006;magx2006;arcsec(-2). This corresponds to approximately 250 million galaxies at the end of the mission and a 50% complete sample for stellar masses above 10(10.6)M(circle dot) (resp. 10(9.6)M(circle dot)) at a redshift zx2004;similar to 0.5 for the EWS (resp. EDS). The approach presented in this work can contribute to improving the preparation of future high-precision cosmological imaging surveys by allowing simulations to incorporate more realistic galaxies.Peer reviewe

Euclid preparation : XVII. Cosmic Dawn Survey: Spitzer Space Telescope observations of the Euclid deep fields and calibration fields

We present a new infrared survey covering the three Euclid deep fields and four other Euclid calibration fields using Spitzer Space Telescope's Infrared Array Camera (IRAC). We combined these new observations with all relevant IRAC archival data of these fields in order to produce the deepest possible mosaics of these regions. In total, these observations represent nearly 11% of the total Spitzer Space Telescope mission time. The resulting mosaics cover a total of approximately 71.5 deg(2) in the 3.6 and 4.5 mu m bands, and approximately 21.8 deg(2) in the 5.8 and 8 mu m bands. They reach at least 24 AB magnitude (measured to 5 sigma, in a 2 ''.5 aperture) in the 3.6 mu m band and up to similar to 5 mag deeper in the deepest regions. The astrometry is tied to the Gaia astrometric reference system, and the typical astrometric uncertainty for sources with 16 < [3.6] < 19 is less than or similar to 0 ''.15. The photometric calibration is in excellent agreement with previous WISE measurements. We extracted source number counts from the 3.6 mu m band mosaics, and they are in excellent agreement with previous measurements. Given that the Spitzer Space Telescope has now been decommissioned, these mosaics are likely to be the definitive reduction of these IRAC data. This survey therefore represents an essential first step in assembling multi-wavelength data on the Euclid deep fields, which are set to become some of the premier fields for extragalactic astronomy in the 2020s.Peer reviewe

Euclid preparation : XVI. Exploring the ultra-low surface brightness Universe with Euclid/VIS

Context. While Euclid is an ESA mission specifically designed to investigate the nature of dark energy and dark matter, the planned unprecedented combination of survey area (similar to 15000 deg(2)), spatial resolution, low sky-background, and depth also make Euclid an excellent space observatory for the study of the low surface brightness Universe. Scientific exploitation of the extended low surface brightness structures requires dedicated calibration procedures that are yet to be tested. Aims. We investigate the capabilities of Euclid to detect extended low surface brightness structure by identifying and quantifying sky-background sources and stray-light contamination. We test the feasibility of generating sky flat-fields to reduce large-scale residual gradients in order to reveal the extended emission of galaxies observed in the Euclid survey. Methods. We simulated a realistic set of Euclid/VIS observations, taking into account both instrumental and astronomical sources of contamination, including cosmic rays, stray-light, zodiacal light, interstellar medium, and the cosmic infrared background, while simulating the effects of background sources in the field of view. Results. We demonstrate that a combination of calibration lamps, sky flats, and self-calibration would enable recovery of emission at a limiting surface brightness magnitude of mu(lim) = 29.5(-0.27)(+0.08) mag arcsec(-2) (3 sigma, 10 x 10 arcsec(2)) in theWide Survey, and it would reach regions deeper by 2 mag in the Deep Surveys. Conclusions. Euclid/VIS has the potential to be an excellent low surface brightness observatory. Covering the gap between pixel-to-pixel calibration lamp flats and self-calibration observations for large scales, the application of sky flat-fielding will enhance the sensitivity of the VIS detector at scales larger than 1 '', up to the size of the field of view, enabling Euclid to detect extended surface brightness structures below mu(lim) = 31 mag arcsec(-2) and beyond.Peer reviewe