Alma Mater Studiorum | Universita` di Bologna SCUOLA DI SCIENZE Corso di Laurea Magistrale in Astrofisica e Cosmologia Dipartimento di Fisica e Astronomia Modelling the size distribution of Cosmic Voids Tesi di Laurea Magistrale Candidato: Relatore: Tommaso Ronconi Lauro Moscardini Correlatori: Marco Baldi Federico Marulli Sessione I Anno Accademico 2015-2016 Contents Contentsi Introduction vii 1 The cosmological framework1 1.1 The Friedmann-Robertson-Walker metric...............2 1.1.1 The Hubble Law and the reddening of distant objects....4 1.1.2 Friedmann Equations......................5 1.1.3 Friedmann Models........................6 1.1.4 Flat VS Curved Universes....................8 1.2 The Standard Cosmological Model...................9 1.3 Jeans Theory and Expanding Universes................ 12 1.3.1 Instability in a Static Universe................. 13 1.3.2 Instability in an Expanding Universe.............. 14 1.3.3 Primordial Density Fluctuations................ 15 1.3.4 Non-Linear Evolution...................... 17 2 Cosmic Voids in the large scale structure 19 2.1 Spherical evolution............................ 20 2.1.1 Overdensities........................... 23 2.1.2 Underdensities.......................... 24 2.2 Excursion set formalism......................... 26 2.2.1 Press-Schechter halo mass function and exstension...... 29 2.3 Void size function............................. 31 2.3.1 Sheth and van de Weygaert model............... 33 2.3.2 Volume conserving model (Vdn)................ 35 2.4 Halo bias................................. 36 3 Methods 38 3.1 CosmoBolognaLib: C++ libraries for cosmological calculations... 38 3.2 N-body simulations............................ 39 3.2.1 The set of cosmological simulations............... 40 3.3 Void Finders............................... 42 3.4 From centres to spherical non-overlapped voids............ 43 3.4.1 Void centres............................ 44 3.4.2 The cleaning algorithm..................... 45 3.4.3 The rescaling algorithm..................... 46 i CONTENTS ii 4 Vdn model validation 48 4.1 Void size function in different cosmologies............... 48 4.2 Extending the model validation over a larger range of radii..... 58 4.3 Problems related to resolution and subsampling............ 60 4.4 Cosmological dependencies of the model................ 63 5 Voids in the halo distribution 68 5.1 Stacked Profiles.............................. 70 5.2 Towards a model for the VSF in the halo distribution........ 72 5.2.1 The analytical tracer-dependent threshold........... 72 5.2.2 Shell-crossing fixed threshold.................. 74 5.3 Scaling relations............................. 78 5.3.1 Halo-radius vs DM-radius.................... 78 5.3.2 Halo-radius VS DM-density................... 80 5.4 Building a semi-analytical model for halo voids............ 83 6 Discussion and conclusions 86 Bibliography 91 CONTENTS iii Abstract The currently accepted flat-ΛCDM Standard Cosmological Model accounts for the existence of two major energy components, Cold Dark Matter and Dark Energy, and for the flatness of the metric. Despite the theoretical framework of the Standard Cosmological Model is well outlined and generally accepted by the scientific community, there are still an amount of unresolved issues, which span from the definition of the initial conditions, to the physical nature of the major components of the Universe. The appearance of today's large scale structure depends on the initial conditions and on the metric and energy properties of the Universe. Thus, from the observable characteristics of these structures, it should be possible to impose constraints on the cosmological model's defining parameters. In this context, Cosmic Voids are a major component of the large- scale distribution of matter in the Universe. They emerge out of the density troughs on the primordial Gaussian field of density fluctuations and are between the largest observable structures in the Universe (Guzzo and The Vipers Team, 2013; Nadathur, 2016). With their relatively simple structure and shape, voids are useful probes of a variety of cosmic parameters and have indeed been shown to possess great potential for constraining Dark Energy and testing theories of gravity (Krause et al., 2013; Sutter et al., 2014b; Pollina et al., 2016). In spite of their popularity as cosmological tools, a gap of knowledge between Cosmic Void observations, simulations and theory still persists. Voids are for their nature only mildly non-linear and tend to become more spherical as they evolve (Icke, 1984), which suggests that their evolution should be easier to reconstruct than that of positive perturba- tions. Therefore, the spherical evolution approximation is particularly suited for describing the expansion of cosmic voids (Blumenthal et al., 1992). This approximation was used to build a theoretical model for the prediction of the void statistical distribution as a function of size (Sheth and van de Weygaert, 2004, hereafter SvdW), with the same approach used to derive the halo mass function. Since the Dark Matter halo distribution provides a large amount of cosmological information, the negative of this information should also be found in the Cosmic Void distribution. However, the SvdW model of void size function has been proven unsuccesful in reproducing the results obtained from cosmologi- cal simulations: previous works could only fit the observed distributions with a SVdW size function in relaxed conditions (i.e. letting the lower threshold δv vary to better represent the data), since voids found in simulations are usually under-predicted by models (Colberg et al., 2005; Pisani et al., 2015; Nadathur and Hotchkiss, 2015). This undermines the possibility of using voids as cosmological probes. The goal of our thesis work is to cover the gap between theoretical predictions and measured distributions of cosmic voids. To this end we develop an algorithm to identify voids in simulated tracer distributions, CONTENTS iv consistently with theory. We reconsider the accepted SvdW model, by inspecting the possibilities offered by a recently proposed refinement (the Vdn model, Jennings et al., 2013). Comparing the void catalogues obtained applying our algorithm to a set of simulations which differ in resolution, total size and cosmology, we validate the Vdn model, finding out that it is reliable over a large range of radii, at all the redshifts considered and for all the cosmological models inspected. We have then extended our discussion to more realistic cases, search- ing for a size function model for voids identified in a distribution of biased tracers. We find that, naively applying the same procedure used for the unbiased tracers to a halo mock distribution does not provide success- full results, suggesting that the Vdn model requires to be reconsidered when dealing with biased samples. Thus, we test two alternative exten- sions of the model and find that two scaling relations exist: both the Dark Matter void radii and the underlying Dark Matter density contrast scale with the halo-defined void radii. We use these findings to develop a semi-analytical model which gives promising results. CONTENTS v Sommario Il modello cosmologico attualmente accettato, chiamato flat-ΛCDM, prevede che l'Universo sia composto principalmente da due componenti energetiche principali, Materia Oscura Fredda ed Energia Oscura, e che abbia una metrica piatta. Nonostante la struttura teorica del Mod- ello Cosmologico Standard sia ben delineata e generalmente accettata dalla comunit`ascientifica, sono ancora presenti svariati problemi non risolti. Tali problemi spaziano dalla definizione delle condizioni iniziali, alla natura fisica delle componenti principali dell'Universo. L'aspetto delle strutture su grande scala dipende dalle condizioni iniziali, dalla metrica e dalle propriet`aenergetiche dell'Universo. Perci`o,dalle carat- teristiche di queste strutture dovrebbe essere possibile imporre dei limiti ai parametri che definiscono il modello cosmologico. In questo contesto, i vuoti cosmici sono una componente rilevante della distribuzione su grande scala delle materia. Si formano dalle de- pressioni del campo Gaussiano di fluttuazioni in densit`ae sono tra le pi`u grandi strutture osservabili nell'Universo (Guzzo and The Vipers Team, 2013; Nadathur, 2016). Grazie alla loro forma relativamente semplice, i vuoti sono utili sonde di una variet`adi parametri cosmologici ed `estato infatti dimostrato che hanno un grande potenziale nel porre limiti sull'Energia Oscura e per testare le teorie di gravitazione (Krause et al., 2013; Sutter et al., 2014b; Pollina et al., 2016). A dispetto della loro popolarit`acome sonde cosmologiche, le osservazioni, i risultati delle simulazioni e i modelli teorici di vuoti non sono tuttora in totale accordo. I vuoti sono per loro natura solo lievemente non lineari e tendono a diventare sferici mentre evolvono (Icke, 1984). Questo suggerisce che la loro evoluzione dovrebbe essere pi`usemplice da modellare rispetto alle perturbazioni di densit`apositive. Per tale motivo, l'approssimazione di evoluzione sferica `eparticolarmente adatta a descrivere l'espansione dei Vuoti Cosmici (Blumenthal et al., 1992). Questa approssimazione `estata infatti utilizzata per costruire un modello teorico in grado di predire la distribuzione statistica dei vuoti in funzione delle loro dimensioni (Sheth and van de Weygaert, 2004, da qu`ı in avanti SvdW), con lo stesso approccio utilizzato per derivare la funzione di massa degli aloni. Dato che la distribuzione