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Spherical Collapse Model and Cosmic Acceleration UNIVERSIDADE FEDERAL DO RIO DE JANEIRO INSTITUTO DE FISICA Spherical Collapse Model and Cosmic Acceleration Duvan Ricardo Herrera Herrera Ph.D. Thesis presented to the Graduate Program in Physics of the Institute of Physics of the Federal Uni- versity of Rio de Janeiro - UFRJ, in partial fulfillment of the requirements for the degree of Doctor of Philoso- phy(Physics). Advisor: Ioav Waga Co-Advisor: Sergio E. Jor´as Rio de Janeiro April, 2019 ii iii CIP - Catalogação na Publicação Herrera Herrera, Duvan Ricardo H 565s Spherical Collapse Model and Cosmic Acceleration / Duvan Ricardo Herrera Herrera. -- Rio de Janeiro, 2019. 106 f. Orientador: Ioav Waga. Coorientador: Sergio E Jorás. Tese (doutorado) - Universidade Federal do Rio de Janeiro, Instituto de Física, Programa de Pós Graduação em Física, 2019. 1. Dark energy. 2. Spherical collapse. 3. f(R) theories. 4. Structure Formation. 5. Cosmology. I. Waga, Ioav, orient. II. Jorás, Sergio E, coorient. III. Título. Elaborado pelo Sistema de Geração Automática da UFRJ com os dados fornecidos pelo(a) autor(a), sob a responsabilidade de Miguel Romeu Amorim Neto - CRB-7/6283. iv Resumo Modelo de Colapso Esf´ericoe Acelera¸c~aoC´osmica Duvan Ricardo Herrera Herrera Orientador: Ioav Waga Coorientador: Sergio E. Jor´as Resumo da Tese de Doutorado apresentada ao Programa de P´osGradua¸c~ao em F´ısicado Instituto de F´ısicada Universidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necess´arios`aobten¸c~aodo t´ıtulode Doutor em Ci^encias(F´ısica). Compreender a influ^enciada acelera¸c~aoc´osmica na forma¸c~aode estruturas ´eatual- mente um dos grandes desafios em Cosmologia, j´aque ela pode ajudar distinguir os diferentes modelos cosmol´ogicosvi´aveis. Neste trabalho estudamos o modelo de colapso esf´ericoe determinamos seus principais resultados (como o contraste de densidade cr´ıtico, a evolu¸c~aodo raio e densidade do n´umerode halos) quando a acelera¸c~ao c´osmica´emod- elada de duas formas. Na primeira, consideramos os modelos de gravidade modificada f(R) na abordagem m´etrica;em particular, usamos os chamados limite de campo forte (F = 1=3, onde a gravidade ´emodificada com uma constante de Newton maior) e limite de campo fraco (F = 0, onde a gravidade n~ao´emodificada devido ao efeito camale~ao). No ´ultimocaso, contemplamos modelos de energia escura \phantom" e \non-phantom", onde ela pode se aglomerar parcialmente (ou totalmente), de acordo com um par^ametro livre γ. Palavras-chave: Energia escura. Teorias f(R). Colapso esf´erico. L´ımitede campo fraco. L´ımitede campo \forte". Cosmologia. Forma¸c~aode estruturas. v Abstract Spherical Collapse Model and Cosmic Acceleration Duvan Ricardo Herrera Herrera Advisor: Ioav Waga Co-Advisor: Sergio E. Jor´as Abstract da Tese de Doutorado apresentada ao Programa de P´osGradua¸c~ao em F´ısicado Instituto de F´ısicada Universidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necess´arios`aobten¸c~aodo t´ıtulode Doutor em Ci^encias(F´ısica) Understanding the influence of cosmic acceleration on the formation of structures is currently a major challenge in Cosmology, since it can distinguish otherwise degenerated viable models. In this work we study the Spherical collapse model and determinate its outcomes (such as critical density contrast, radius evolution and number density of halos) when the cosmic acceleration is modeled of two ways. In the former we consider f(R) modified-gravity models in the metric approach; in particular, we use the so-called large (F = 1=3, where the gravity is modified with a larger Newton's constant) and small-field (F = 0, where the gravity is not modified due chameleon effect) limits. In the latter we contemplate phantom and non-phantom dark energy models, which can partially (or totally) cluster, according to a free parameter γ. Keywords: Dark energy. f(R) theories . Spherical collapse. Small field limit. Large field limit. Cosmology. Structure Formation. vi Acknowledgments My deepest gratitude to Professors Ioav Waga and Sergio Jor´asfor the great help, dedication, and time spent for me could conclude this work. Their knowledge in both cos- mology and program were fundamental to solve the theoretical and computational prob- lems presented throughout this thesis. Also thank to my friends Dalma,Daniel, Leonardo, Guarumo, Omar, Matu and Rhonald for the discussions on physics, politics, philoso- phy, among others, and my mother for her infinite and unconditional support. Finally, I thank to CAPES for the financial support that made possible the accomplishment of this Ph.D, and to officials (especially them of the Postgraduate) and professors of the Physics Institute. vii Contents Contents vii List of Figuresx 1 Introduction1 2 Relativistic Cosmology and Standard ΛCDM Model5 2.1 The Cosmological Principle..........................5 2.2 Background Evolution.............................7 2.3 Energy content of the Universe........................ 11 2.3.1 Baryonic matter............................. 11 2.3.2 Dark matter............................... 11 2.3.3 Relativistic components (rc)...................... 15 2.3.4 Dark energy............................... 19 2.4 ΛCDM model.................................. 24 3 Dark Energy and cosmic acceleration 26 3.1 f(R) gravity theories.............................. 26 3.1.1 Hu-Sawicki model............................ 33 3.1.2 γ gravity model............................. 36 4 Spherical collapse model 39 4.1 Spherical Model in a Universe dominated by matter............. 39 viii 4.1.1 Virialization............................... 42 4.2 Spherical collapse in the ΛCDM model.................... 43 4.2.1 Virialization............................... 44 5 Calculation of the critical overdensity in the spherical-collapse approx- imation 46 5.1 Spherical collapse in f(R) theories....................... 48 5.2 Calculating the critical density contrast δc .................. 50 5.2.1 The differential-radius method..................... 51 5.2.2 The constant-infinity method and its mending............ 52 5.3 Comparing the results from different approaches............... 54 6 Number density of halos in f(R) theories 60 6.1 Comparing the f(R) models with ΛCDM................... 60 7 Top-Hat Spherical Collapse with Clustering Dark Energy: Radius Evo- lution and Critical Density Contrast 67 7.1 Spherical collapse with dark energy perturbations.............. 68 7.2 Bubble Evolution................................ 73 7.2.1 Radius.................................. 74 7.2.2 The critical contrast density...................... 76 8 Conclusions and future works 84 Bibliography 86 A SC model equations in f(R) theories 97 B Solution of the differential equation for the density contrast in the linear regime 100 ix C Converting from Mv to Mh 103 D Solution of linear equations for δm and δde 105 x List of Figures 2.1 Full sky map of the CMB temperature anisotropies observed by Planck satellite. Figure taken from Ref. [1]......................6 2.2 Hubble diagram for a compilation of data. Type Ia supernovae (squares), Tully-Fisher clusters (solid circles), fundamental plane clusters (triangles), surface brightness fluctuation galaxies (diamonds) and Type II supernovae (open squares). Figure taken from [2].....................7 2.3 Representation the spiral galaxy inside a dark matter halo......... 12 2.4 Angular Temperature spectrum. Figure taken from Ref[1].......... 17 3.1 Effective Dark Energy Equation of State for the Hu-Sawicki f(R) model 0 as a function of redshift z. Figure taken from [3] where fR ≡ fR0...... 35 3.2 Effective equation-of-state parameter wde as a function of z for n = 1; 2 and 3 and different values of α. Figure taken from [4]............ 38 5.1 (a,left panel) Critical density contrast δc as a function of Ωm0 for F = 0 (solid blue line) and F = 1=3 (dotted red line) when the collapse occurs at redshift zc = 0, following our approach. (b,right panel) Critical density contrast δc as function of redshift collapse zc for F = 0 (solid blue line) and F = 1=3 (dotted red line) with Ωm0 = 0:3, following our approach...... 54 xi 5.2 Relative errors for δc (see text for definition) between our method and (a) the differential-radius one (dotted black line), the constant-infinity one, with (b) Inff = 105 (dashed red line) and (c) Inff = 108 (solid blue line) for F = 0 (left panel) and F = 1=3 (right panel). For all models, we assumed Ωm0 = 0:3.................................... 55 5.3 Relative errors for nln M (see text for definition) between our method and the constant-infinity one, with (a) Inff = 108 (solid blue line) and (b) Inff = 105 (dashed red line) for redshift z = 0 in the cases F = 0 (left panel) and F = 1=3 (right panel). For all models, we assumed Ωm0 = 0:3 and h = 0:6774.................................. 57 5.4 Relative errors for nln M (see text for definition) between our method and the constant-infinity one, with (a) Inff = 108 (solid blue line) and (b) Inff = 105 (dashed red line) as a function of redshift for F = 0, F = 1=3 13 −1 and virial mass of 10 h M (upper panels). The same plots, are shown 15 −1 for 10 h M (lower panels). For all models, we assumed Ωm0 = 0:3 and h = 0:6774.................................... 58 5.5 Nbin (see text for definition) as a function of redshift for F = 0 and virial 13 14 −1 14 15 −1 masses between 10 and 10 h M (left panel) and 10 and 10 h M (right panel). As always, we assumed Ωm0 = 0:3 and h = 0:6774....... 58 5.6 Relative errors for Nbin (see text for definition) between our method and the constant-infinity one, with (a) Inff = 105 (dashed red line) and (b) Inff = 108 (solid blue line) as a function of redshift for virial masses between 13 14 −1 10 and 10 h M and F = 0 (left panel) and F = 1=3 (right panel). For all models, we assumed Ωm0 = 0:3 and h = 0:6774...........
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