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Cosmological and Astrophysical Structures in the Einstein-De Sitter Universe

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Cosmological and Astrophysical Structures in the Einstein-De Sitter Universe Cosmological and astrophysical structures in the Einstein-de Sitter universe A Thesis Submitted to the Faculty of Sciences of Universidad de los Andes In Partial Fulfillment of the Requirements For the Degree of Master of Sciences-Physics by Andres´ Balaguera Antol´ınez Bogot´a D.C Colombia 2006 Universidad de los Andes Cosmological and astrophysical structures in the Einstein-de Sitter universe A thesis submitted to the Faculty of Sciences of Universidad de los Andes In Partial Fulfillment of the Requirements For the Degree of Master of Sciences-Physics c Copyright All rights reserved by Andr´es Balaguera Antol´ınez Bogot´a D.C Colombia 2006 LATEX2006 La tesis escrita por Andres´ Balaguera Antol´ınez cumple con los requerimientos exigidos para aplicar al titulo de Maestr´ıa en Ciencias-F´ısica y ha sido aprobada por el profesor Marek Nowakowski Ph.D, como director y por los profesores Juan Manuel Tejeiro Ph.D. y Rolando Roldan,´ Ph.D., como evaluadores. Marek Nowakowski Ph.D., Director Universidad de los Andes Juan Manuel Tejeiro, Ph.D., Evaluador Universidad Nacional de Colombia Rolando Roldan,´ Ph.D., Evaluador Universidad de los Andes Cosmological and astrophysical structures in the Einstein-de Sitter universe Andr´es Balaguera Antol´ınez, Physicist Director: Marek Nowakowski Ph.D Abstract In this work we explore the consequences of a non zero cosmological constant on cosmological and as- trophysical structures. We find that the effects are associated to the density of the configurations as well as to the geometry. Homogeneous and spherical configurations are slightly affected. For non ho- mogeneous configurations, we calculate the effects on a polytropic configurations and on the isothermal sphere, making special emphasis on the fact that the cosmological constant sets certain scales of length, time, mass and density. Sizable effects are determined for non spherical systems, as elliptical galaxies or galactic clusters, where the effects of Λ are increased as long as the configurations deviates from spherical symmetry, i.e, for flat systems. The equilibrium of rotating ellipsoids are modified and the cosmological constant allows new configurations in equilibrium. Finally we explore the motion of a test particle in the Schwarszchild- de Sitter space time and set astrophysical bounds for the cosmological constant, not only from the Newtonian limit, but also from a full general relativistic analysis. Agradecimientos Durante los dos anos˜ y medio en que este trabajo fue pensado, moldeado y producido, son muchas las personas que han puesto su granos de arena (enormes cantidades de apoyo y carino)˜ para que esta labor pudiese ser terminada con buenos resultados. Es casi el mismo tiempo que llevo unido a Pilarcita, de modo que es ella (quien me ha elegido por esposo) a quien agradezco en primera instancia, porque fue ella quien estuvo a mi lado cuando las cosas no parecian tan claras y por supuesto, cuando las fueron, al menos, lo suficiente para poder escribir este texto. A ella mi infinito agradecimiento por aguantar, tolerar, maniobrar y calmar el genio de urano˜ que adquiremos algunos cuando de hacer f´ısica se trata. En seguida debo agradecer a mis padres Socorro y Fredy, por quienes soy quien soy. A mis suegros Oliva y Alfonso, que mas que suegros, son la versi´on dual de mis padres y a´un mas. Sin su apoyo, la confianza y el ambiente de familia que han tendido a nuestro alrededor, estoy totalmente seguro de que este trabajo no ser´ıa ni la mitad del que hoy presento. A mi maestro Marek Nowakowski, quien me di´o el coraje para creer en mi y confi´o en mis palabras y mis actos, enormemente agradecido me siento con ´el. Igualmente agradezco al profesor Juan Carlos Sanabria, por la ayuda prestada cuando de poner los pies sobre la tierra se trataba, por las preguntas cruciales y su buen humor. A Alonso Botero por presentarme a las formas diferenciales. Al departamento de F´ısica de la Universidad de los Andes (desde las senoras˜ que preparan los tintos hasta el director, Profesor Bernardo G´omez, y con ellos Julieta, Janneth y Elsy) por el gran apoyo y el excelente ambiente de trabajo que han puesto a mi disposici´on y a la de mis companeros.˜ Es claro que quedan muchisimas personas por fuera de este leve intento que, no por falta de voluntad, sino por falta de palabras, hago para agradecer. Pero procuro recuperar con estas lineas las providenciales presencias de Claudia, Juan David, Manuel Esteban, Edgar, Diana, Olga Yavanna, Gonzalo, Lina Mar´ıa, Carolina (la nana), Ben- jam´ın (Calvo, mi buen maestro-padrino y Balaguera, mi abuelo); mi tia Susana y Catalina, mi tia Mariela y su familia, mi tio Gilberto y su familia; los esp´ıritus de mis abuelitas Merceditas y Ernestina, de mi abuelo F´elix y de quienes ellos provienen. Mis amig@s Marthica, Germ´an, Juan Harvey; mis companeros˜ Juli´an, Jaime (todos los Jaimes), Nicol´as, Alberto, y a los que de una u otra forma me ayudaron a ver la luz, no solo frente a unas ecuaciones, sino a´un mas importante, frente a la vida misma. A los seres que ambientaron nuestra est´ancia en la caracas con cuarta: Domingo de Ramos, Charlotte Manotas, Blaze Alexandra y Grau Maria por su felina compania.˜ Todo hombre necesita un esp´ıritu-impulso-´ımpetu que le inspire a vivir, sonar˜ y luchar: yo encontr´e el esp´ıritu Gai- tanista, que me invadi´o si previo aviso y para siempre. Agradezco a la providencia haber podido encontrarlo entre los oscuros rincones de la historia pasada pare ver con mayor claridad los oscuros parajes de la historia presente. S´e que quienes m´as meritos tienen en este trabajo no lo entiender´an, ya que no son f´ısic@s, ni astr´onom@s; en realidad tampoco hace falta: la vida les ha puesto en mejores situaciones: son madres, hermanas, tias, abuelas: son las personas que cuya sola presencia se convirti´o en un enorme motor que me impuls´o hacia los garabatos que est´an aqui contenidos en este texto, y a quienes tenia siempre presente en alma y coraz´on cuando, aparentemente ausente, en ellos trabajaba. Bogot´a D.D., Enero 27 2006 A las fuentes de mi vida, Pilar, Socorro y Oliva Contents Table of contents ix List of figures x List of tables xi 1 Introduction 1 1.1 Why a cosmological constant? . 1 1.2 Where does it come from? . 2 1.3 Where do we stand? . 3 1.4 Where do we go from here? . 4 1.5 Description . 4 1.6 Publications and related works . 5 1.7 Units and constants . 5 2 Background cosmology 7 2.1 Standard cosmology with Λ . 7 2.1.1 Evolution of the universe . 8 2.1.2 Relevant epochs in standard cosmology . 10 2.2 Dynamical Dark Energy and cosmological constant . 14 2.2.1 Scalar fields . 15 2.2.2 Dark Energy . 16 2.2.3 Chaplygin Gas . 16 2.3 The Newtonian limit with cosmological constant . 18 2.4 Structure formation in expanding universe . 20 2.4.1 Nonlinear evolution: the spherical collapse model . 23 2.4.2 Collapse with Dark Energy . 24 2.5 Remarks . 28 3 Spherical configurations with cosmological constant 31 3.1 Description of equilibrium . 31 3.1.1 General consequences of ΛVT . 33 3.2 Spherical configurations . 34 3.2.1 Homogeneous sphere . 34 3.2.2 Radius at virial equilibrium . 35 3.3 Non-constant density: Polytropic configurations . 37 3.3.1 Parameters of the configuration . 38 ix 3.3.2 The ΛVT for polytropes . 41 3.4 Stability in the Newton-Hooke spacetime . 47 3.5 Remarks . 50 4 Ellipsoidal configurations 51 4.1 Rotation of non spherical configurations . 51 4.2 Allowed configurations in equilibrium . 55 4.2.1 Bifurcation point: From oblate to triaxial . 55 4.2.2 Minor axis rotation: from prolate to triaxial . 56 4.3 Other effects of Λ for non-rotating configurations . 58 4.3.1 Mean mass-weighted rotational velocity . 59 4.3.2 Critical mass . 61 4.3.3 Mean velocity and Mass-Temperature relation . 61 4.4 Remarks . 62 5 Scales set by the cosmological constant 63 5.1 Dynamics in the Schwarszchild-de Sitter spacetime . 63 5.2 Radial motion . 64 5.2.1 Astrophysical scale set by Λ . 65 5.2.2 Constrains in the velocities of test particles . 66 5.3 Motion with angular momentum . 69 5.4 Remarks . 72 6 Astrophysical bounds on the cosmological constant 73 6.1 Bounds from the Newtonian limit . 73 6.2 Bounds from general relativity . 74 6.2.1 General solutions with equation of state . 76 7 Conclusions 77 A The virial equation 81 B An effect with generalized vacuum energy density 93 C Schwarszchild-de Sitter metric 97 D Roots of third and fourth order equation 101 Bibliography 103 List of Figures 1.1 Constrains on cosmological parameters . 3 2.1 Scale factor as a function of time . 10 2.2 Evolution of the density parameter . 11 2.3 Scale factors for different values of Ω with k = 0 . 13 vac 6 2.4 Age of a flat universe for different !x . 14 2.5 Scale factor and age of the universe for the Chaplygin Gas model . 18 2.6 Growing factor in the linear approximation . 23 2.7 Ratio between initial radius and radius at virialisation. 25 2.8 Radius at virialisation . 26 2.9 Evolution of the scale factor . 27 3.1 Ratio between Rλ,vir(η) and Rλ,vir(0). 36 3.2 Effects of Λ on the behavior of the density of a polytropic configuration .
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