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

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

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 astrophysical 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 homogeneous 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 versión dual de mis padres y aún 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 dió el coraje para creer en mi y confió en mis palabras y mis actos, enormemente agradecido me siento con él. 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 Gómez, y con ellos Julieta, Janneth y Elsy) por el gran apoyo y el excelente ambiente de trabajo que han puesto a mi disposición 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 Félix y de quienes ellos provienen. Mis amig@s Marthica, Germán, Juan Harvey; mis companeros˜ Julián, Jaime (todos los Jaimes), Nicolás, Alberto, y a los que de una u otra forma me ayudaron a ver la luz, no solo frente a unas ecuaciones, sino aún mas importante, frente a la vida misma. A los seres que ambientaron nuestra estáncia 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.

Sé que quienes más meritos tienen en este trabajo no lo entienderán, ya que no son f´ısic@s, ni astrónom@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 convirtió en un enorme motor que me impulsó hacia los garabatos que están aqui contenidos en este texto, y a quienes tenia siempre presente en alma y corazón 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 ωx ...... 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 ...... 40 3.3 Effects of Λ on the behavior of the density of a isothermal sphere ...... 45 3.4 Isothermal sphere ...... 47 3.5 Critical adiabatic index ...... 48

4.1 Effects of Λ on the angular velocity ...... 52 4.2 Function g(e) and geometrical factor (e) for prolate and oblate ellipsoids...... 54 A 4.3 Critical eccentricity ...... 55 4.4 Maclaurin and Jacobi solution ...... 56 4.5 Maclaurin and Jacobi solutions for different ζc ...... 57 4.6 Maclaurin and Jacobi solutions for different ratios ζ ...... 58 4.7 Bifurcation point prolate-triaxial ...... 59 4.8 Angular velocity for prolate and triaxial solution ...... 60

5.1 Eﬀective potential for Λ = 0 ...... 64 6 2 1/2 5.2 The relativistic factor γ = (1 v )− ...... 68 − max 5.3 Maximum and minimum values for velocity ...... 69

B.1 Critical scale factor as a function of the eccentricity for vanishing angular velocity . . . . 94

List of Tables

2.1 Cosmological parameters...... 29 2.2 Dark energy models ...... 29

3.1 Fraction ξ (Λ = 0)/ξ (Λ = 0) for diﬀerent ζ polytropic index...... 41 1 1 6 c 3.2 Numerical values for the enhancement factors in the polytropic model ...... 44

4.1 Parameters for the oblate solition q2(q3)...... 56 4.2 Fits for the bifurcation between triaxial-prolate and prolate ellipsoids ...... 58

5.1 Scales set by the cosmological constant in combination with the Schwarszchild’s radius. . 66 5.2 Maximum angular momentum for stable orbits ...... 71

7.1 Equilibrium condition % > ρ for diﬀerent conﬁgurations ...... 78 A vac

xiii

CHAPTER 1

Introduction

Several books, reviews and papers have been dedicated to the cosmological constant. Hence here we will be brief. Detailed explanations and calculations can be found in such references [1, 2, 3, 4, 5, 6, 7], even though the author took care in reproducing the majority of them. In order to introduce the reader to the context in which this work has been developed, it worths to mention some relevant details found in the references mentioned before. This will also help us to justify the relevance of the present work.

1.1 Why a cosmological constant?

It is well known that the cosmological constant Λ was first introduced by Einstein. The origin for this term came with the necessity to obtain an static universe with positive curvature and dominated by matter with positive pressure, inspired by the Mach’s principle (i.e, the idea that matter dominates the inertia) and in order to mix the theory of general relativity with the astronomical context of the first two decades of the twentieth century. The cosmological constant acts then as a balance to the energy density. Otherwise, the universe would be dynamical. However, small departures from this perfect balance would lead the universe into a dynamical behavior. It can be understood simply from a Newtonian picture by identifying the parameter of curvature as minus the energy of a gravitational bounded test particle. The static universe is then analog to circular orbits and for positive energies, such orbits are unstable. Such unstable behavior was one of the major criticism that the Einstein’s universe received. The discovery of the red-shift of distant galaxies made by Hubble eliminated then the requirement of an static universe and hence the cosmological constant generating of an static universe was by rejected Einstein himself. Nowadays, the cosmological constant plays a relevant role in modern cosmology. Although astronomical data implies a dynamical universe, the cosmological constant has been brought again into the cosmological context not to reach static universe, but to explain the observed acceleration and to account for the total constant of energy density in the universe. The main physical phenomena used to reveal the most important features of the universe are the cosmic microwave background radiation (CMBR), the large scale structure and the distance-luminosity relation for type I supernova (SnI). From the CMBR it has been determined that the universe has a flat geometry [8]. This flatness implies some questions, since the observed and estimated amount conventional matter (barionic, nonbarionic massive neutrinos) and radiation is not enough to reach the critical density needed to have a flat geometry. Hence a new kind of component has been included to account for the missing energy. Such component is called dark energy (DE), whose most special feature is that it displays a negative

1 2 CHAPTER 1. INTRODUCTION pressure. This negative pressure is needed to account for the results inferred from the measurements of luminosities of distant SnI: the universe is accelerated at the present time [9, 10]. Such acceleration can be reproduced from the theory of relativity by considering that the dark energy has a very special equation of state p = ωρ, with ω < 0. The SnI observations then implies that the dominant component in a ﬂat and accelerated universe is the dark energy with an equation of state close to ω 1, at least at the ∼ − present epoch. This is precisely the case of a cosmological constant.

1.2 Where does it come from?

As commented before, original version of the cosmological constant had a clear goal: to generate an static matter dominated universe with non zero curvature. This solution can be reached by introducing a (bared) cosmological constant Λb in the Einstein-Hilbert ﬁeld equations as 1 g + Λ g = 8πG . (1.1) Rµν − 2 µν R b µν NTµν

Formally, these equations can be derived by applying a variational principle on the action S = Sg + Sm, which is a contribution of gravity (curvature) and it’s source (matter). These contributions are written as [11] 1 S = √ g d4x, S = √ gf( ) d4x, m − Lm g 16πG − R Z N Z where is the Lagrangian density associated to matter (energy) and f( ) is a function of the scalar Lm R of curvature. Equation (1.1) is then derived with f( ) = 2Λ , which implies that the introduction R R − b of Λb is understood geometrical factor or simply a constant in the Lagrangian density of the theory. Nevertheless, generalizations of the function f( ), the so called higher order gravity [12], are often taken R into account to reproduce the effects of a cosmological constant in the evolution of the universe. The cosmological constant can be also interpreted as a modification of the energy-momentum tensor by writing the field equations as 1 g = 8πG ˜ , ˜ = + vac, vac = ρ g . (1.2) Rµν − 2 µν R NTµν Tµν Tµν Tµν Tµν − vac µν The energy momentum tensor ˜ is now the contributions from two components: the component with Tµν energy density ρ representing matter and/or radiation, and a new component interpreted as a vacuum energy density ρvac defined as

Λ 8 3 ρvac 10− erg/cm . (1.3) ≡ 8πGN ∼ For a perfect fluid the energy momentum tensor is written as = pg + (p + ρ)u u , with ρ the Tµν µν µ ν energy density and p the pressure. Hence the field equations for a perfect fluid can be written with an energy-momentum tensor which has the same structure of that of a perfect fluid with the modifications ρ ρ˜ = ρ + ρ and p p˜ = p p . → vac → − vac The cosmological constant can be introduced in the theory of general relativity from a more elaborated theoretical framework through the concepts of scalar fields, widely used in particle physics with great (theoretical) success. In this context, it is assumed the existence of an scalar field whose dynamics is ruled by a potential V (φ), which displays a non zero vacuum expectation value. These are the so called Quintessence models. The action for the scalar field is written as [11]

1 S = √ gd4x gµν ∂ φ∂ φ V (φ) , φ − 2 µ ν − Z and the associated energy-momentum tensor is written as 1 1 φ = ∂ φ∂ φ + gρσ∂ φ∂ φ V (φ) g . (1.4) Tµν 2 µ ν 2 ρ σ − µν 1.3. WHERE DO WE STAND? 3

Figure 1.1: Left: Conﬁdence regions for Ωvac vs. Ωmatter from galaxy clusters, CMBR and SnI measurements, from the Supernova Cosmology Project. Right: constrains on the equation of state for dark energy vs. Ωmatter. Taken from [10].

The variational principle δ(S + S + S ) = 0 would lead to an effective vacuum energy density ρ m g φ vac ∼ V (φ ) (and = V (φ )g as in (1.2)), where φ is the field in the lowest energy configuration. 0 Tµν − 0 µν 0 Actually, this view is what give rise to the interpretation of the cosmological constant as a vacuum energy density, i.e, as is written in equation (1.3). A final contribution for the vacuum energy density comes from quantum field theory. Briefly speaking, since a quantum field behaves as a collection of harmonic oscillators each one with a non zero vacuum energy (1/2)~ω, the total contribution of all these vacuum energies (without taking small wavelengths that make the sum infinite) lead to ρQFT 10112erg/cm3, which is 120 orders of magnitude greater vac ∼ ∼ than the measured value in (1.3)[1]. Hence, an effective cosmological constant can then be written as a contributions from the bared term, the Quintessence models and the quantum field theory Λ = Λb + Λφ + ΛQFT.

1.3 Where do we stand?

Several models have been proposed for the dominant component of the universe. The simplest one is the cosmological constant associated to a negative pressure fluid with equation of state p = ρ. This − picture has been improved with the introduction of dynamical dark energy models and scalar fields. The common factor between these models is that they mimic a vacuum energy density at the present time, and display a dynamical behavior at early epochs. This dynamical behavior is used, specially in Quintessence models, to explain successfully the curvature problem, the Ω problem and the homogeneity observed at cosmological scales. In figure 1.1 it is shown the analysis made from measurements of SnI, CMBR and galaxy clusters [10] in order to constrain the confidence regions for cosmological parameters Ωvac, Ωmatt and the equation of state for dark energy. Clearly this astronomical data is favoring a pure cosmological constant ω = 1 and a universe dominated by the vacuum energy density. − 4 CHAPTER 1. INTRODUCTION

1.4 Where do we go from here?

Since astronomical data has shown that we live in a flat universe dominated by a vacuum energy density associated to the cosmological constant, the next step concerned with Λ consists in determine the origin of such dominant component, to explain why the vacuum energy density associated is much smaller than the value predated from particle physics and why such contribution is comparable today to the contribution from matter (the so called coincidence problem). A crucial step is to identify it through its effects of astronomical phenomena, as gravitational lensing, structure formation, and the equilibrium of astrophysical structures, where this work is actually placed. Such effects can be also used to discriminate between different models of dark energy.

1.5 Description

The main goal of this work is to determine the equilibrium and stability conditions of astrophysical and cosmological conﬁgurations in a vacuum dominated universes. The title of this work, Cosmological and astrophysical structures in the Einstein - de Sitter universe refers to the fact that we will consider a cosmological background dominated by vacuum (de Sitter) and matter (Einstein), which individual contributions are taken from latest astronomical observations. The chapters of the work are described as follows: Chapter 2. We review the basics of the standard cosmology model with dark energy. We show standard results (most found in references) and show the possible generalizations for a dynamical vacuum energy density. The Newtonian limit derived from ﬁeld equations with cosmological constant is also considered. From this we review the mechanism of structure formation in expanding universe.

Chapter 3. We set the equilibrium condition for such structures via the virial equation. Relevant results we derive cosmological constant on the equilibrium and stability of cosmological as well as astrophysical configurations with spherical symmetry. We consider configurations with constant and varying density. For the latter, we use a polytropic equation of state to model the interior of the configuration. We show that for these model the cosmological constant may play a relevant role in determining the possible existence of a configuration. The Chandrasekhar’s limit with cosmological constant is considered. Finally, the isothermal sphere is also considered.

Chapter 4. Based on the theoretical framework described in chapter 3, we explore the equilibrium of homogeneous non spherical configurations. We show that the cosmological constant modifies the allowed shapes for rotating ellipsoids in virial equilibrium. In particular, we show that a minor axis rotation of triaxial configurations are allow only for Λ = 0. 6

Chapter 5. We study the motion of test particles in de Sitter Schwarszchild space-time. We show how the non-arbitrary combination of length scales lead to a new astrophysical scale which is only possible with Λ = 0. This scale corresponds to the last circular orbit allowed by Λ, which is located between as- 6 trophysical and cosmological scales. Constrains on the velocities and angular momentum of test particles both in radial and circular motions are derived.

Chapter 6. This chapter is devoted to set astrophysical bounds on the cosmological constant. Although some of these bounds were derived in previous chapters, they were understood as constrains on the parameters of the structures under analysis. Here we also derive bounds from the hydrostatic equilibrium in the context of general relativity. 1.6. PUBLICATIONS AND RELATED WORKS 5

1.6 Publications and related works

A. Balaguera-Antol´ınez , M. Nowakowski, C. B¨ohmer, Int. J. Mod. Phys. D14 (2005), Astrophysical • bounds on the cosmological constant; arXiv: gr-qc/0409004.

A. Balaguera-Antol´ınez , M. Nowakowski, Astron. & Astrophys. 441. 23-29 (2005), Equilibrium of • large astrophysical structures in the Newton-Hooke spacetime; arXiv: astro-ph/0511738.

A. Balaguera-Antol´ınez , M. Nowakowski, C. B¨ohmer, Class. Quant. Grav. 23, 485-496 (2006), • Scales set by the cosmological constant; arXiv gr-qc/0511057.

A. Balaguera-Antol´ınez, D. Mota, M. Nowakowski, Ellipsoidal conﬁgurations in the Einstein-de • Sitter universe. Submitted to Class. Quant. Grav.

A. Balaguera-Antol´ınez, M. Nowakowski, Polytropic conﬁgurations with cosmological background. In • preparation.

A. Balaguera-Antol´ınez, M. Nowakowski, Eﬀects of generalized dark energy on cosmological struc- • tures. In preparation.

This work has been presented in the following scenarios:

A. Balaguera-Antol´ınez , M. Nowakowski, Virial equilibrium of large structures, Cosmology Meeting, • Granada (Spain), April 2005. (oral presentation by A.B.A).

A. Balaguera-Antol´ınez , M. Nowakowski, Estructuras cosmologicas en un universo en expansi´on, • Congreso Nacional de F´ısica, Barranquilla (Colombia), October 2005 (oral presentation by A.B.A)

A. Balaguera-Antol´ınez , M. Nowakowski, Cosmological and astrophysical structures in a vacuum • dominated universe , 11th Latin American IAU (International Astronomical Union) regional Meeting, Puc´on (Chile), December 2005 (poster presentation).

A. Balaguera-Antol´ınez , M. Nowakowski, Dark energy in an astrophysical context, Albert Einsten • Century conference at Unesco, Paris (France), July 2005 (oral presentation by M.N).

A. Balaguera-Antol´ınez , M. Nowakowski, C. B¨ohmer. Astrophysical Facets of the Cosmological • Constant, Beyond Einstein: Physics for the 21th century, Bern (Switzerland), July 2005 (oral presentation by M.N).

1.7 Units and constants

During this work we have worked with geometrized units where GN = c = 1. The gravitational constant GN may appear in some expressions in order to compare relevant quantities. Useful conversions are

30 30 1Km = 1.3469 10 Kg, 1s = 299792458m, 1MeV = 1.782 10− Kg. × × Some relevant parameters are the mass of the sun, the mean density of the sun, the Planck’s constant and Boltzman’s constant:

18 2 16 2 40 1 M = 1.4766Km ρ¯ = 1.04512 10− Km− , ~ = 4.7350 10− Kg , kB = 1.5361 10− KgK− × × × Other useful conversions are

12 19 20 1 L.Y = 9.467 10 Km, 1Mpc = 3.0863 10 Km, M = 4.7843 10− Mpc × × × 6 CHAPTER 1. INTRODUCTION CHAPTER 2

Background cosmology

2.1 Standard cosmology with Λ

The cosmological standard model is based in the assumption of a spatially isotropic and homogeneous universe evolving in time. This means that the spacetime can be foliated in maximally symmetric three- dimensional slides, which evolves according to a scale factor. This symmetry allows us to write the metric in spherical comoving coordinates (r, θ, φ) as

dr2 ds2 = dt2 + a2(t) + r2dΩ2 , (2.1) − 1 kr2 − where a(t) is the dimensionless scale factor, which determines the time evolution of the spatial slides, and k is the parameter of curvature normalized as k 1, 0, +1 for negative (open), null (flat) or positive ∈ {− } (closed) curvature (universe) respectively, and dΩ2 = dθ2 + sin2 θdφ2 as the metric on a two-sphere. The line element (2.1) models the Friedmann-Robertson-Walker (FRW) universe. Equation (2.1) implies that proper distances r between two points will change proportionally to a(t) as r = a(t) x , where x the | | | | | | | | comoving separation. The rate of change of the comoving positions is given as v(t) = r˙ H(t)r, where − v(t) is the peculiar velocity (associated to deviations from the Hubble’s expansion due to local processes) and H(t) a/a˙ is the Hubble’s parameter. Hubble’s law follows by neglecting such peculiar velocities as ≡ r˙ = Hr. Hence, the acceleration measured by an observer moving along with the expansion is written in proper coordinates as ¨r = (a/a¨ )r. That is, two fixed (comoving) points seems to be pulled apart from each other because of a force proportional to the distance between them and modulated by the acceleration of 1 the universe through the term a/a¨ . Peculiar velocities are effected as v a˙ − . This force has also effect ∝ in the propagation of light through the an expanding universe, which is represented in the cosmological red-shift. Associated to the Hubble parameter it is defined the Hubble distance dH and the Hubble’s time TH 1 as TH = H− = dH (in natural units). The Hubble’s time is a measure of the age of the universe as understood from the Hubble’s law, i.e, the time elapsed from the moment when r˙ = 0 until the actual recession velocities. The Hubble distance is the distance that a ray of light can travel in one Hubble time, i.e, the horizon which encloses a TH -old universe. In general, the horizon dhor at some time t can be t 2 1 ∞ 1 calculated from (2.1) with ds = 0. For a flat universe one has dhor(t) = 0 a− (t0)dt0 = (1+z)−1 H(z)− dz 1 where z (1 + a)− is the red-shift. In terms of z, the Hubble law can be written (for z 1) as z = Hr. ≡ R R

7 8 CHAPTER 2. BACKGROUND COSMOLOGY

2.1.1 Evolution of the universe By applying the conservation law µν = 0, one obtains the conservation of mass (energy) for ν = 0, ∇µT and the conservation of momentum for ν = i, for which we simply have Euler’s equation ∂ip = 0. For the first case we have a˙ ρ˙(t) + 3 [ρ(t) + p(t)] = 0. (2.2) a If we consider interaction between the components (matter-radiation, matter-Dark Energy, radiation-Dark Energy e.t.c.), then the r.h.s of (2.2) acquires a term associated to the interchange of energy between these components. Different components of the total amount of energy density have different equation of state p = p(ρ). Some of the most relevant are

1 ωρ(t) Cold matter (ω = 0), radiation (ω = 3 ), cosmological constant (ω = 1), Dark Energy ( 1 < ω < 0)  − −  γ p(ρ) = ωρ (t) Polytropic equation of state,  ωρ(t) + κρ(t)γ Chaplygin Gas, ω(t)ρ(t) Dynamical Dark Energy.   By assumingthat each of these components evolve independently (Γ = 0), the integration of the continuity equation yields respectively for these cases

n ρ?aˆ(t)− , 3 m m m m ρ? [ωρ? (aˆ(t) 1) + aˆ(t) ]− , ρ(t) =  − 1 (2.3)  n(γ+1) γ+1  A + Baˆ(t)− ,  a(t) 1+ω(a) ρ? exp 3 0 da0 , − a? a   h R i where aˆ a(t)/a(t ), m = 3(γ 1), n = 3(1 + ω) and ρ? as a? are the the energy density and scale factor ≡ ∗ − respectively, at some fixed instant of cosmic time t?. The constant factors A and B are function of the Chaplygin Gas parameters and will be specified when necessary. The equations of evolution for the universe are found from the spatial components of the field equations with the FRW line element. From the t t component of (1.1) one gets the acceleration equation: − a¨(t) 4 = π [ρ (t) + 3p (t)] . (2.4) a(t) −3 tot tot The spatial components of the field equations (1.1) yield

2 a¨(t) a(˙t) k + 2 + 2 = 4π [ρ (t) p (t)] . a(t) a(t) a2(t) tot − tot " # Combining these expressions we get the Friedmann equation

a˙(t) 2 8 k = H(t)2 = πρ (t) . (2.5) a(t) 3 tot − a2(t) The solution for the coupled set of diﬀerential equations (2.2) and (2.5) yield evolution of the universe. Friedmann equation can be regarded as an equation for the curvature parameter as k = a2H2 (Ω (t) 1) tot − or as a normalization condition Ωcurvat + Ωtot = 1, where density parameter Ω(t) and the critical energy density are deﬁned respectively by (reincorporating Newton’s constant):

2 8πGNρtot(t) ρtot(t) 3H(t) k Ωtot(t) 2 , ρc(t) , Ωcurvat 2 2 . (2.6) ≡ 3H (t) ≡ ρc(t) ≡ 8πGN ≡ −a H The dynamics of the universe is ruled by its content of matter-energy. One simple (and useful) model is to assume the universe as filled with a cold dark matter CDM corresponding to non relativistic matter 2.1. STANDARD COSMOLOGY WITH Λ 9 which partook in the formation of galaxy clusters and galaxies. Some of the proposed components of CDM are the so-called WIMPS (weakly interacting massive particles) and particles predicted by extensions of the particle standard model as SUSY (axions, neutralinos among others) [13]. Although the structure formation is an strong evidence in favor of this model, the CDM models cannot give a complete picture of the dynamics of the universe as astronomical observation suggest, where the dominant component of the universe is given by the cosmological constant. This is the ΛCDM model, where contributions from CDM and Λ at the present time are given as Ω = 0.27 0.04 and Ω = 0.73 0.04 [9]. These measured mat vac values generated the coincidence problem, which wonders why the contribution from matter and vacuum are comparable today, the Λ-problem, which wonders why is the contribution from vacuum much smaller than the expected from one theoretical grounds (mentioned before), and the Dark Energy problem which wonders what is the origin of such dominant contribution. Other models consider a time-varying vacuum energy density: those with a constant equation of state are referred to Dark Energy (DECDM) and with a time dependent e.o.s are refereed to dynamical Dark Energy and Quintessence models QCDM. Although photons are the more abundant particles in the universe, the contribution through the CMBR are 4 very small compared to the contribution from the vacuum energy density and CDM, with Ω 10− [14]. rad ≈ Massless neutrinos contribute with the same order of magnitude while massive neutrinos may contribute 3 as Ων > 10− [15]. On the other hand, the theory of nucleosynthesys of light elements yields a contribution 2 of barionic matter of Ω 10− [1, 2, 6]. bar ∼ Together with the problems associated to the cosmological constant, standard cosmology faces other complex situations. The observed homogeneity in the temperature of the CMBR implies that two causally disconnected objects at the present time had have been within each other’s horizon (i.e, they were causally connected) in order to reach thermodynamical equilibrium through some physical mechanism. It implies that at some time the proper distances grew faster than the horizon, and standard cosmology does not supply a mechanism capable to reproduce a fast expansion at early times. This is the so called horizon problem. Another problem of standard cosmology is the so called Ω-problem, related to the observed flatness of the universe [8]: from Friedmann equation one can shown that we live in a flat universe only if at early times the total density parameter was shifted from 1 in approximately fifty two decimal places (a fine tuning problem) [14]. This imposes very constrained set of initial conditions for standard cosmology and any small deviation from such initial conditions would lead to a very different universe.

Cosmological parameters

The evaluation of the Friedmann equation at the present time deﬁnes the so-called Hubble constant H0. It’s value has been measured as [9, 14]

1 1 10 1 H = 100 h km s− Mpc− 1.02 10− h years− , h = 0.7 0.1. (2.7) 0 70 ≈ × 70 70 This parameter encodes the information of how fast is the universe expanding or contracting. The Hubble’s time and the Hubble’s horizon are then T 13.9 Gyr and d = 4.2 Gpc. On the other hand, the geometry H ≈ H of the universe is ruled by the parameter k which in turn depends on the content of matter modulated by the critical density given in (2.6). Its present value of ρc is found to be [14] 26 2 3 11 2 3 ρc = 9.2 10− h70kg m− = 1.4 10 h70M Mpc− . (2.8) × × The rate of acceleration of the universe is measured by the deceleration parameter q, which is defined 2 from a series expansion about a fixed instant t0 of the scale factor as q aa/¨ a˙ . For the ΛCDM ≡ − t0 model, q reduces to q = 1 (1 + 3ω Ω ). Since q < 0 implies an accelerated universe, then the e.o.s for 2 x x 0 1 1 Dark Energy in a ΛCDM model should be such that ω < Ω− . For the current values of Ω 0.7, we x − 3 x x ≈ get ω < 10/21. For the vacuum energy density one has q = 0.5(1 3Ω ) which implies that Λ > H 2 x − − vac 0 agrees with an accelerated universe. The parametrization of the deceleration parameter according with astronomical observations can be found in [16]. Other parameters as the so-called statefinder parameters are introduced in order to study Dark Energy models. These are defined as [17] ... a 2 r 1 r , s − , ≡ aH3 ≡ 3 2q 1 − 10 CHAPTER 2. BACKGROUND COSMOLOGY

2 5 Ω =0 mat k = 0 Λ = 0 0.2 Ω 0.7 mat=1.0 3.0 0.7 10 4 0.5 1.5 0.3 0

Scale Factor Scale Factor 2

0.5 1

0 0 −1 0 1 2 3 −3 −2 −1 0 1 2 3 4 τ τ = H0 t = H0 t

Figure 2.1: Left: scale factor for diﬀerent curvature k = Sgn(Ωmat − 1) (Λ = 0). Right: Comparison between the standard case (Λ = 0) with the case Λ =6 0 for a ﬂat universe, Ωmat = 1 − Ωvac

For a ﬂat DECDM model, these parameters can be written as

9 1 1 d 1 1 d r = (1 + ω ) H− ln ω Ω ω + 1, s = 1 + ω H− ln ω . 2 x − 2 dt x x x x − 9 dt x For ω constant, the DECDM model is in general represented through a parabola in the r s plane given x − as r = 4.5s(s 1)Ω + 1. The ΛCDM model is represented by a fixed point in the r s plane located at − x − s = 0 and r = 1. The contribution from Dark Energy with Ω is represented by the point r = 1 Ω and x − x s = 2. For Quintessence models or Chaplygin Gas r and s become function of the scale factor since ωx is not constant anymore. As pointed before, recent measurements of cosmological parameters determined that the major contribution to the total energy density comes in form of Dark Energy associated to the cosmological constant +0.094 with ωx = 0.94 0.096 [18]. The value of the cosmological constant can be written from (2.8) as − − 2 7 2 2 Λ = 3H Ω = 1.6837 10− Ω h Mpc− . (2.9) 0 vac × vac 70 As well as the Hubble parameter defines scales of length and time, the cosmological constant also sets relevant scales. In our units this scales of time and length are defined as

1 TH 3 1 1/2 10 TΛ = RΛ = = 2.4 10 h70− Ωv−ac Mpc 1 10 ly, (2.10) ≡ rΛ √3Ωvac × ≈ × 3 1/2 while the scale of mass can be written by writing ρvac = 3MΛ/4πRΛ, i.e, MΛ = (1/6)Λ− . The coincidence problem is enhanced by checking numerically this expression and concluding that the length scale is approximately equal to the Hubble horizon, that the mass scale is approximately of the same order as the mass of the universe and that the time scale is of the same order of magnitude of the age of the universe, as can be seen in ﬁgure 2.4.

2.1.2 Relevant epochs in standard cosmology The solution of the Friedmann equations implies the knowledge of the diﬀerent kinds energy density as function of the scale factor, which are given in (2.3). From those expressions one deﬁnes the constants 2.1. STANDARD COSMOLOGY WITH Λ 11

0.8

Ω 0.6 mat ω Ω (a) =−1 rad i x Ω Ω 0.4 x

0.2

0 10−5 10−4 10−3 10−2 10−1 100 101 scale factor 1

0.8

0.6 ω (a) =−1/3 i x Ω 0.4

0.2

0 10−5 10−4 10−3 10−2 10−1 100 101

Figure 2.2: Evolution of the density parameter Ω(a) in a ﬂat universe for cold dark matter and x-component with −4 ωx = −1, −1/3 with Ωmat = 0.3, Ωx = 0.7, and Ωrad = 10 .

C 8 πρ ani (i = radiation, matter, x-component) so that Friedmann equation takes the form of an i ≡ 3 i energy equation

2 1 2 (1+3ω) k = a˙ C a− C a− C a− . (2.11) − − mat − rad − x

In figure 2.1 we plot the scale factor a(t) as a function of τ = H0t for different values of Ωmat and Ωvac a the present time. In the left-hand figure, we find the standard cosmological model with Λ = 0, from where we observe the different fates of the universe as a function of Ωmat. In the right-hand side of the same figure, we compare the standard model for a flat universe with a flat universe where Λ = 0. We see 6 that even if k = 0 in both cases, the cosmological constant contributes in the form of a monotonically increasing velocity of expansion of the universe, while the standard model predicts a slow deceleration. Figure 2.2 shows the density parameter for radiation, CDM and the x-component as a function of then scale factor for two different values of ωx. Although a complete analytical solution for a(t) cannot be derived, one usually write some approximate solutions as follows: Radiation dominated era. Early epochs which follows from a Hot Big-Bang are dominated by radiation. Friedman equation is written as a˙ 2a2 = C ka2, and it solution reads as rad − 1/2 a(t; Ω ) = 2 Ω τ kt2 . rad rad − h p i Hence in a k = +1 universe, radiation can produce a turning point in a(t) leading the universe to a future singularity, taking a time t = 4√C from the big-bang (t = 0) until it reach a = 0. For k = 1 the rad − universe expands forever, faster than the corresponding case for a flat universe (k = 0), where a(t) t1/2. ∝ Matter dominated era. The transition from a radiation dominated era to a matter dominated era 3 occurs at a scale factor given by a Ωrad/Ωmat 3 10− where the contribution from radiation and ∼ ≈ × 2 1 matter are equal. For higher values of the scale factor, the equation (2.11) is written as a˙ = C a− k. mat − This resembles the equation of motion for a test particle under Newtonian gravity, which implies that its solution depends on the sign of the constant k, which plays the role of E in the Newtonian picture. − 12 CHAPTER 2. BACKGROUND COSMOLOGY

The solution for k = 0 can be written through the parametric angle θ as a(θ) = Ak(1 C (θ)) and 6 − k t(θ) = Bk(θ Sk(θ)), where Sk(Ck)(θ) = sin(cos)θ for k = +1 and sinh(cosh)θ for k = 1, together 3− 1 2 − with with A = 2 CmatB . For a closed universe, the case θ = 0 shows a big-bang at t = 0 and a = 0. For θ = π, the scale factor reaches its maximum value at the turn-around a = ata = 2A with tta = πB. Finally, collapse is reached at θ = 2π with tcol = 2πB. For a matter dominated universe, Ωmat = 1, 2 k = 0, we obtain the Einstein-de Sitter universe, which scales as a(t) t 3 . The matter dominated era is ∝ important because in this regime structure formation took place. As written above, this starts at a scale 3 factor 10− and ends when the contribution of matter starts to be comparable with the contribution ∼ 1/3ω from the x-component. It occurs at a scale factor a (Ω /Ω ) x 0.75 for ω = 1 and 0.42 for ∼ x matt ∼ x − ∼ ω = 1/3. x − Vacuum dominated era. For the age dominated by the x-component, we cannot write explicitly the time dependence of the scale factor for a non-ﬂat universe. If k = 0, we have for ω = 1 and Ω = 1: x 6 − x 2 3 3(1+ωx) a(τ; Ω ) = (1 + ω )τ . (2.12) x 2 x Dark Energy then yields a t (ω = 1/3), a t2 (ω = 2/3). If we set ω = 1 in (2.11) we ∝ x − ∝ x − − have a˙ 2 = C a2 k, which yields the solutions a = exp [H t] (for k = 0 and Λ > 0, de Sitter model), vac − 0 a = cos [H t] (for k = 0 and Λ < 0, anti de Sitter model) and for Λ = 0: 0 6

k 1 2 a(τ; Ωvac) = 2 cosh k− τ , (2.13) sH0 For a negative cosmological constant, the de Sitter universe evolves periodically reaching future singular- 2 ities in a time T = √12π TΛ. In accordance to (2.5), the last contribution may come from the curvature term. In such case, an open universe evolves with a t. ∝ In ﬂat ΛCDM model we can integrate analytically the Friedmann equation and solve for the scale factor:

1 3 Ωcdm 2 3 a(τ; Ωcdm) = sinh 3 τ 1 Ωcdm . (2.14) 1 Ωcdm 2 − − p From this expression we can immediately determine the age of the universe in the ΛCDM model, which will be written down at the end of this section. Quasistatic evolution. This solution refers to a time interval when the scale factor took (or will take) a constant value aqs, but was (or will) changing at any other time constrained to the present acceleration. This is the so called Lemaitre’s coasting model. To ﬁnd the value of aqs, we can write the Friedmann (2.5) equation and the acceleration equation (2.4) as

2 2 da 1 a 1 a 2 n = Ω − + Ω − + Ω a − 1 + 1, (2.15) dτ mat a rad a2 x − 2 d a 1 1 n Ωmat Ωrad = (2 n)Ω a − , dτ 2 2 − vac − a2 − a3 Explicit solutions can be found in a ΛCDM model. By requiring that a˙ = a¨ = 0 we get the following polynomial equation:

1 3−n 3 n 2 n Ωmat 1 (2 n)Ωx (n 2)/(n 3) x − αx + − − = 0, x − , α (3 n)(8) − − . − 8 Ω ≡ 8Ω ≡ − mat mat

One interesting solution appears when we consider Ωx = Ωvac (n = 3). The resulting expression is written as a third order equation [19]

1 3 1 Ω 1 Ω 3 x3 x + mat − = 0, x x . (2.16) − 4 4 Ω ≡ 4Ω mat mat 2.1. STANDARD COSMOLOGY WITH Λ 13

Ω Ω mat = 0.1 mat = 0.3 2.5 2.5

2 2 Ω vac = −1.0 0 1.709 1.71346 1.5 1.5 1.718 2

Ω 1 vac = −0.3 1

Scale Factor 0 Scale Factor 1.3 1.35 0.5 7 0.5 1.5

0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 τ τ = H 0 t = H 0 t

Ω Ω mat = 0.7 mat = 3 2.5 2.5 Ω vac = 0 0.16 0.5 2 2 4.479 Ω vac = −1.0 4.4809 0 4.48 2.24 1.5 2.2533 1.5 2.26 2.8

1 1 Scale Factor Scale Factor

0.5 0.5

0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 τ τ = H 0 t = H 0 t

Figure 2.3: Scale factors for diﬀerent values of ΩΛ0 with k =6 0. The present epoch is at τ = 0.

The real and positive roots for this cubic equation are written as a function of the behavior of Ωmat. We have (see appendix D)

1 1 1 Ωmat 1 cosh cosh− − 0 < Ωmat < 3 Ωmat 2  h1 1 1Ωmat i 1 x = cos 3 cos− −Ω 2 Ωmat 1, Ωmat > 1  mat ≤ ≤  1 1 1 Ωmat 4 cos h cos− − i+ π Ωmat > 1. 3 Ωmat 3  h i The valueof the scale factor at the quasistatic time interval is given again by the condition a¨ = 0 as 1 1/2 aqs = 2 x− . The global behavior of the scale factor as a function of the dimensionless time τ is shown in figure 2.3 for different values of Ωmat. As seen in the plots, the static solution is only reached when crit 3 the cosmological constant takes certain specific value, given from a˙ = a¨ = 0 as Ωvac = 4Ωmatx . For Ω > 1 we find two solutions for the cubic equation (2.16). The second solution occurs for τ > 0. For a flat space-time (k = 0), it is easy to show that the condition a˙ = a¨ = 0 implies a unique solution with ω = 1 and a ( Ω /Ω )1/4. That is, a flat and accelerated universe with Λ > 0 doesn’t admits − qs → − mat vac quasistatic solutions. On the other hand, by setting n = 3, corresponding to ω = 1/3, the contribution − from Ω vanishes and the value a is again undetermined, just as in the case k = 0 and ω = 1. x qs − Turn-around. The condition for a turn around is represented by a˙ = 0 in equation (2.15). One sees that for n = 2 (ω = 1/3) and if Ω > 1, the scale factor reaches a turning point located at x − mat a = Ωmat/(Ωmat 1) (k = +1). For n = 0 (ωx = 1), we obtain from equation (2.15) a cubic equation ∗ − − in the form 1 Ω Ω Ω a3 + − mat − vac a + mat = 0. ∗ Ω ∗ Ω vac vac 14 CHAPTER 2. BACKGROUND COSMOLOGY

2.5

k= 0 ω x = −1 H 2 ω x = −0.8 ω x = −0.6 ω x = −0.3 T / T 1.5 Λ H

Age of the Universe / T 1

0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ω mat

−1 Figure 2.4: Age of a DECDM ﬂat universe for diﬀerent e.o.s ωx in units of the Hubbles’s time TH = H0 . The scale of time introduced by the cosmological constant is also shown (see equation (2.9).)

For k = +1 the solution is written as

1/2 1 1 Ωmat 1 Ωmat Ωvac a = α cosh cosh− , α − − . (2.17) ∗ 3 2α3Ω ≡ 3Ω vac vac For k = 1, the solution lies in the complex plane and its real part is same as (2.17) with the change − cosh sinh. On the other hand, flat universes doesn’t admits turning points, since the Friedmann → 1/(3 n) equation yields for a = ( Ωmat/Ωx) − (k = 0). ∗ − Finally, integration of Friedmann equation (or (2.1)) allow us to determine the age of the universe as 1 1 1 T = T (xE(x))− dx, where E(a) H(a)/H . For a flat EdS universe we have T = (2/3)T 9h− H 0 ≡ 0 H ≈ 70 Gyr. For a flat x-dominated universe T = 2TH /(2n 1), i.e, only equations of state ωx > 5/6 have a R − 1 − definite age (big bang). A radiation dominated universe has T = 0.5T 4h− Gyr. In figure 2.4 we H ≈ 70 show the age of a flat universe as a function of the content of matter Ωmat. For the flat ΛCDM model, integration of Friedmann equation yields analytical expression

2 √Ωvac + 1 TΛCDM = ln 0.96TH 13.5Gyr. 3H √Ω √1 Ω ≈ ≈ 0 vac − vac This can also be obtained from (2.14) by setting a = 1 and solve for t. The last equality is valid for Ωvac = 0.7. From this expression we see (as can be seen also from the plot) that a vacuum dominated universe has not a deﬁnite age. The coincidence problem can be visualized with this example since for the value of Ω we get T 0.7T . That is, T /T 1.3. This age is in accordance with the values vac Λ ≈ H ΛCDM Λ ≈ determined from SnIa analysis with T = 14.2 1.7 Gyr [9]. Other estimates are determined from the age of low metallicity stars with T = 15.6 4.6 Gyr [20] and the age of globular clusters, with T = 14.6 1.6Gyr [21].

2.2 Dynamical Dark Energy and cosmological constant

The fact that the universe is accelerated has been supported by observations of the luminosity of distant SnI [9]. This implies that the strong energy condition ρ + 3p 0 [22] must be violated by the total ≥ amount of energy density in the universe, following (2.4). On the other hand, the flatness of the universe 2.2. DYNAMICAL DARK ENERGY AND COSMOLOGICAL CONSTANT 15 has been also indicated by measurements in the anisotropies of the CMBR [8], together with the recent measurements of cosmological parameters [18]. This compels us to find a kind of energy density which shows negative pressure and in a great abundance such that Ω 0.7. The cosmological constant is x ≈ the immediate option and it is favored by astronomical measurements [9, 18]. Nevertheless, the horizon problem cannot be solved only with Λ, since the primordial vacuum energy density must be able to generate the initial conditions of standard cosmology, i.e, a radiation dominated universe followed by a matter dominated era and then to a vacuum dominated era. The classical problems described before are often attacked with the idea of Inflation [23, 24]. The main assumption of inflationary models is that at early times the universe was dominated by a vacuum-like energy density which led the universe to a fast expansion (similar to the evolution of a EdS universe with Λ > 0 given in (2.13)). This would account not only for the horizon problem (since such expansion could make proper distances to grow faster than the Hubble’s horizon), but also for the fine tuning problem (since this rapid expansion would lead quickly the density parameter Ωtot to 1 even before inflation ends), yielding the initial conditions of standard cosmology. Together with this, models of a dynamical Dark Energy and time dependent Λ [25] have been also proposed to account for the dominant component of the total energy density if the universe. Here we will briefly describe the most relevant models of Dark Energy.

2.2.1 Scalar fields The dynamics of the universe in the inflationary epoch is assumed to be ruled by a constant energy density. Nevertheless, such energy density cannot be taken as the one associated to the cosmological constant since at the end of inflation that energy density must give rise (via some decaying mechanism) to a radiation dominated epoch, i.e, must produce photons and massive particles (CDM), and hence, reproduce the initial conditions for the Hot Big Bang universe. This mechanism is done by introducing scalar field φ(x, t) with a potential energy V (φ). The energy density and the pressure associated to the scalar field can be taken from (1.4) as

1 1 1 1 ρ = (∂ φ)2 + ( φ)2 + V (φ), p = (∂ φ)2 ( φ)2 V (φ). (2.18) φ 2 t 2 ∇ φ 2 t − 6 ∇ − For homogeneous scalar ﬁeld, the equation of motion for φ can be derived by using the conservation equation (2.2) with the density given in (2.18):

Γ dV φ¨ + 3Hφ˙ + V 0 = , V 0 , (2.19) φ˙ ≡ dφ where the factor Γ represents the loss or gain of energy due to the interaction between the scalar ﬁeld and the rest of components of the universe. On the other hand, since it is assumed that the scalar ﬁeld dominates at early times, the Friedmann equation and the acceleration equation both can be written as

8 1 φ˙2 8 a¨ 8 φ˙2 8 H = πV (φ) 1 + πV (φ), = πV (φ) 1 πV (φ), (2.20) 3 " 2 V (φ) # ≈ 3 a 3 " − V (φ) # ≈ 3 where the last equalities are valid in the slow roll approximation, summarized by the introduction of 1 2 1 the slow roll parameters (16π)− (V 0/V ) 1 and η (8πV )− V 00 1 [23], so that equation ≡ 1 ≡ (2.19) is simplified to 3Hφ˙ V 0 + Γφ˙− . These conditions guaranties that the homogeneous scalar ≈ − field violates the strong energy condition ρφ + 3pφ < 0 and hence inflation takes place during the time when the approximations in (2.20) are valid, i.e, when the scale factor has an exponential growth as a(t) = a exp [H (t t )] with t < t < t and H = H(t = t ). The time t marks the time when inflation 1 1 − 1 1 2 1 1 1 starts, while t2 is refereed to the end of inflation. Inflation starts with an initial displacement of the field from the minimum of V (φ), where it is assumed to fulfill , η 1 (V (φ) constant ), and ends when the ∼ slow rolling parameters are comparable to 1, i.e, when the field falls into the minimum of V (φ), where the creation of particles (radiation) generates the radiation-dominated era. Hence the duration of the inflationary epoch depends on the behavior of potential V (φ) and the scalar field φ(t). A model in which 16 CHAPTER 2. BACKGROUND COSMOLOGY the slow rolling condition is satisfied and the slow rolling parameter are independent of φ, inflation never ends. The choice of the potential is not unique, but there are different models leading to an inflationary epoch. The most simple example is the harmonic potential V (φ) = (1/2)m2φ2, where m is the mass of 1/2 the Inflaton. For this potential the slow rolling condition reads as φ (4π)− while (2.19) requires m < 1.5H0. Some of the most known potentials which are valid in the slow rolling condition are the power law inflation V (φ) = α exp [ βφ], the double exponential potential V (φ) = α (exp [βφ] + exp [γφ]), − 1 the exponential with inverse power law V (φ) = α exp βφ− 1 , where α, β and γ are constants [26]. − − A detailed analysis of cosmological models with Quintessence and astronomical constrains can be found in [27]. Dynamics of dark energy modeled as a polytropic gas is also explored in [28].

2.2.2 Dark Energy Dark Energy is a generalization of the vacuum energy density. It introduces an equation of state of 3(1+ω ) the form p = ω p with ω < 0. This energy density evolves as ρ a− x , and we recover the x x x x x ∝ cosmological constant for ω = 1. From (2.4) we see that e.o.s is constrained as ω < 1/3 in order x − x − make a¨ > 0. Particular features can be described for the specific value ω = 1/3: in the first place, x − note that in this case Dark Energy contributes in the same way as the curvature term in Friedmann equation (2.4). This implies that Dark Energy with ω = 1/3 only contributes at late times maintaining x − a well definite rate of expansion if ρx(t?) > 3k/8π; in the same context, note also that the evolution of a flat universe dominated by this Dark Energy (ω = 1/3, k = 0) is the same evolution followed by an x − open universe dominated by matter(k = 0, ω = 0), as seen from Friedmann equation (2.5) by replacing 6 x k 8πρ (t )/3H2. Hence it is often usual to consider the violation of the strong energy condition as a → − x 0 0 consequence not of dark energy but of curvature effects [29]. Secondly, this contribution vanishes in the acceleration equation and hence it does not play a role in determining whether the universe is accelerated. Finally we see from the a-dependence of the energy density that the bound ω > 1 is required in order x − for ρx to be a decreasing function of the scale factor, which is natural to expect when no interactions are taken into account. Models with ω < 1 are referred as phantom Dark Energy, where the density x − increases leading to future singularities [30]. Dark Energy with time dependent equation of state px(t) = ωx(t)ρx(t) are also considered. In this case, the energy density scales as a f(a) 3 ωx(a0) + 1 ρ = ρ a− f(a) da0, (2.21) x vac ≡ ln a a Z1 0 where the e.o.s is constrained to ω (1) = 1. Phenomenological parameterization for ω are proposed x − x in [31]. A linear parameterization is given as ω = ω + ω (a 1) [32] , and its currently used in the x 0 1 − context of structure formation with Dark Energy [33]. The time dependent vacuum energy density can be related to the homogeneous scalar field potential V (φ) with ωx(a) = pφ/ρφ and using (2.18) [34]. More parameterization can be found in [35].

2.2.3 Chaplygin Gas The diﬀerent epochs through which the universe had passed can be associated to an e.o.s of an ideal gas 1 given as p = κρ− [36] (originally proposed to model the lifting force on wings of airplanes), with ch − ch κ > 0. This is the so-called Chaplygin Gas, and it relevance lies in the fact that it violates the strong 2 2 energy condition while generates a well deﬁnite speed of sound cs = κρc−h . A generalized equation of state is often written with a second component with equation of state p = ωρ. The generalized Chaplygin Gas (GCG) reads [37]: κ pch = ωchρch γ , (2.22) − ρch which is constrained to violate strong energy condition with ρ > ((n 2)/(3κ))1/β, where β = 1 + γ and ch − n = 3(1 + ωch). The evolution of the energy density is given in equation (2.3) as

1 nβ β β 3κ ρ (a) = A + Ba− , B ρ A, A . (2.23) ch ≡ ch − ≡ n 2.2. DYNAMICAL DARK ENERGY AND COSMOLOGICAL CONSTANT 17

For early times, the generalized Chaplygin Gas behaves like matter (n = 3, ωch = 0) or radiation (n = 4, n ωch = 1/3), since ρch a− . As long as the scale factor grows, the GCG passes from a matter(radiation)- ∝ 1/β like behavior to a vacuum-like behavior for large a with ρch A = constant. We can then identify an ≈ 1/β eﬀective cosmological constant induced in a universe dominated by the Chaplygin Gas as ρvac = (3κ/n) . The acceleration equation (2.4) reads

4 3κ a¨ = πaρ (a) 1 + 3ω . −3 ch ch − A + Ba nβ − One then sees again that at early times the rate of acceleration was decreasing. At some value of the scale factor atran, the vacuum-like behavior dominates and the universe passes to an accelerated epoch. The acceleration changes sign at a scale factor a = (B(n 2)/2A)1/nβ. Hence, if Chaplygin Gas is expected tran − to reproduce the observed acceleration at the present time, the parameters κ and β should be related as

1 3 β κ (n 2)H2β, (2.24) ≤ 3 8π − 0 which follows from a 1. This inspires us to write κ = α H2β so that the parameters A and B are tran ≤ n 0 written as

3 3Ω β 3 A = α H2β, B = H2β ch α . (2.25) n n 0 0 8π − n n " # For a flat universe, we can integrate Friedmann equation and obtain an explicit expression for the age of the universe as function of the parameters αn, n and γ: 1 3 q 1 2nβ + n + 4 A T = 1 2F1 , , , , 2β 2π 2nβ 2β 2nβ −B qB r 1 1 1 3 β 3 − 2β 3 q 1 2nβ + n + 4 3 3 β 3 − = α F , , , α α , qH 8π − n n 2π 2 1 2nβ 2β 2nβ −n n 8π − n n  0 " # r " #   with q 3ω + 7. From this expression we see that if the parameter α takes the critical value α = ≡ n n (n/3)(3/8π)β (i.e, B = 0), then the universe dominated by Chaplygin Gas does not allow a Big-Bang since its age becomes undetermined. In this case Chaplygin Gas behaves for all times as a vacuum energy density and then T as is shown in Fig 2.4. Definite ages are then reached for α < (n/3)(3/8π)β. → ∞ n For the pure Chaplygin Gas this is α 0.01. ≤ One can also pass from a Chaplygin Gas to a dynamical Dark Energy-like or Quintessence models in β a direct way. For the first case, one sets ω (a) = κρ− so that the variation of Dark Energy can be x − ch written in terms of the Chaplygin Gas parameters as in (2.21) with

1 B 3β f(a) = ln 1 + a− . (2.26) −β ln a A

On the other hand, we can go from Chaplygin Gas to Quintessence models by assuming that ρch = ρφ: using (2.18) and (2.22), we can solve for the scalar potential and for φ˙ respectively as

1/2 γ 1 γ V (φ) = V (ρ(φ)) = 2ρ(1 ω) + 2κρ− , φ˙ = nρ(t) κρ(t)− . (2.27) − 3 − Hence, if we solve for a(t), we may have from (2.23) ρ = ρ(t) and then we can integrate (2.27) to have φ = φ(t). Furthermore, from (2.18) we may obtain the scalar potential. We can also ﬁnd ρ = ρ(φ) and φ(a) by using equations (2.27) and (2.2) as

1 β 2 3κ 2 β 1 B nβ ρ(φ) = cosh 2nπβ φ , φ(a) = arccosh 1 + a− . n 2nπβ2 "r A # h p i p 18 CHAPTER 2. BACKGROUND COSMOLOGY

2 0.85

α = 1E−2 1E−3 γ = 1.0 1E−4 0.8 1E−5 1.6 0.4 1E−6 0.1 0.8 H

1.2

0.75

Scale Factor 0.8 Age of the Universe / T

0.7 0.4

0 0.65 −1 −0.6 −0.2 0.2 0.6 1 0.0001 0.001 0.01 τ α = H0 t

Figure 2.5: Left: Scale factor for ωch = 0, γ = 1 and diﬀerent values of α3. For α ≤ 0.01 the age becomes undetermined. Right: Age of the universe measured in Hubble times for diﬀerent values of γ, with ωch = 0.

Combining these expressions with (2.27) we can write the induced scalar potential associated to generalized Chaplygin Gas as

β−1 1 2γ β 2 β 1 3κ β 1 3κ − β V (φ) = (1 ω ) cosh 2πnβ2φ + κ cosh 2πnβ2φ − . 2 − ch n 2 n h p i h p i The pure Chaplygin Gas-potential is simpliﬁed to 1 V (φ) = √κ sinh √24πφ tanh √24πφ . 2 From this expression it is simple to show that the slow rolling condition is never satisﬁed, since this potential leads for 1 the equation x2 8 x + 4 0 for x tanh2 √24πφ , which has not real − 3 ≡ solutions. Finally it worths to mention constrains from astronomical data to the set of parameters γ and κ of Chaplygin Gas can be explored in reference [38].

2.3 The Newtonian limit with cosmological constant

In this section we review the Newtonian limit of Einstein’s equations with cosmological constant. In this context, one consider small deviations from a Minkowskian spacetime by writing the line element as g = η + h with h 1, i.e, the eﬀects of gravity are encoded in the metric h . The Newtonian µν µν µν | µν | µν physics is then reached from Einstein’s equation in the limit when c , and the remaining spacetime is → ∞ then treated as a Galilean spacetime. Nevertheless, with cosmological constant the remaining spacetime is not Galilean in the sense that the gravitational potential vanishes at inﬁnite, but one recovers the so called Newton-Hooke spacetime [39]. Let us consider the Lagrangian that generates the geodesic equation for a free particle in the spacetime generated by a region with density ρm. We can write

µ ν dx dx 1 i 2 = gµν 1 + h00 h0iv + (v ). L r− dτ dτ ≈ 2 − O 2.3. THE NEWTONIAN LIMIT WITH COSMOLOGICAL CONSTANT 19

Hence, one then can identify an effective potential Φ = 1 h + h vi. In the the zeroth order, the − 2 00 0i Newtonian potential Φ is related to the metric perturbation hµν as h00 = 2Φ, while in the first order − i k approximation we may write the equation of motion in analogy to Lorentz force as rï = gi+ijkv H , where g = ∂ Φ ∂ h and H = klm∂ h , corresponding to the gravito-electro and the gravito-magnetic i − i N − t 0i k l 0m fields respectively [22, 40]. By working in the zeroth order approximation, one sees that the Newtonian potential is related to the metric via g = (1 + 2Φ). An important consequence of the weak field 00 − approximation is the prediction of gravitational waves through the wave-like equation in the transverse gauge [22] h˜ = 16π , where 1 g with h˜ = h 1 η h and h = ηµν h . Since µν − Sµν Sµν ≡ Tµν − 2 µν T µν µν − 2 µν µν em = 01 then em = em and we have T Tµν Sµν 1 1 = (p + ρ)u u + (ρ p)g + γ g γδ, Sµν µ ν 2 − µν FµγFν − 4 µν FγδF where ρ and p are given as p = pb + δp and ρ = ρb + δρ, corresponding to the contributions of (energy) density and pressure from the background and the configuration respectively. For static configuration the wave equation implies 2h = 16π . Hence from h = 2Φ and using (2.4) we find a modified ∇ 00 − S00 00 − Poisson’s equation for the potential Φ:

δp a¨ 2Φ = 4π δρ + 3 + 2 U 3 , (2.28) ∇ c2 c2 − a where = 1 E2 + B2 is the electromagnetic energy density. Equation (2.28) shows how any contribution U 2 of energy density acts as a source of gravity, although the last two term are strongly suppressed by the 2 1 factor c− , and hence they won’t be included when we work to the first order in c− . Note that an accelerated universe contributes directly in the Newtonian limit through the acceleration equation. The solution for the potential Φ is quite different than in the classical picture, since we now cannot set the Dirichlet boundary conditions at infinite because of the acceleration term, but is must be fixed now at some finite distance R where Φ(r = R) = 0. The solution to the first order of v/c can then be written as [41]

1 a¨ 2 3 Φ = Φgrav r + δρ(r0)G(r, r0) d r0, − 2 a | | 0 ZV where Φgrav encodes the source of gravitation in the Newtonian picture:

δρ(r0) 3 Φgrav = d r0, (2.29) − 0 r r ZV | − 0| 2 satisfying Φ = 4πδρ. The function G(r, r0) is deﬁned as ∇ grav

l l l ∞ 4π r< r> G(r, r0) Y ∗ (θ0, φ0)Y (θ0, φ0), ≡ 2l + 1 R2l+1 lm lm l=0 m= l X X− with r = max r , r0 and r = min r , r0 . This shows that in the weak field limit we not only get an > {| | | |} < {| | | |} external force due the cosmological constant, but also an indirect effect due to the location of boundary conditions at finite distances. Nevertheless, we can neglect the last term if we consider that the value of R lies on cosmological scales and then is large compared to the positions r. It is important however to note that the validity of the Newtonian limit is imposed by the cosmological constant. In this situation, the requirement of the weak field condition yields two strong inequalities of the form

6 2√2 1 R M, Mmax = = 4√2MΛ, (2.30) rΛ 3 √Λ

1 1 This implies prad = 3 ρrad 20 CHAPTER 2. BACKGROUND COSMOLOGY

which are valid up to corrections of the first and second order in M/Mmax. [41, 42]. From (2.28) one determines that the equation of motion for a test particle will be written in the Newtonian limit as d2r a¨(t) i = ∂ Φ + r . (2.31) dt2 − i grav a(t) i Strictly speaking, equation (2.31) implies that energy is no longer conserved in the Newtonian limit because of the time dependence of the scale factor. Nevertheless, the contribution of the acceleration on the universe is often small compared to the contribution from Φgrav. For instance, by assuming the background as a contribution of matter, radiation and Dark Energy, the main effects of the background on an self-gravitating configuration are only due today by the Dark Energy component with equation of state ρ = ω ρ . Note that the contribution from Dark Energy with ω = 1/3 vanishes in this limit. x x x x − On the other hand, the Newton-Hooke space time is defined by (2.31) for ω = 1: the Newtonian limit x − in the Newton-Hooke space time reads as rï = g˜, where the acceleration due to gravitational field is now reduced because of the repulsive force as

H2Ω r r g˜ g 1 0 vac i = g 1 i , ≡ grav − g grav − 3R2 g grav Λ grav where we have used (2.9). In this case, energy conservation holds in the Newtonian limit. Clearly, for small 2 astrophysical scales, the correction term is negligible. For instance, for circular orbits with ggrav = vcirc/R leads

2 2 H0 Ωvacri 1 R 1 δg = v− . ≡ g 3 R circ grav Λ In this simple example the correction factor is strongly suppressed by the squared power. For the solar 20 5 system one gets δg 10− , and for stars orbiting the Milky way one has δg 10− . One then expects ≈ ≈ that the effects associated to the cosmological constant can only be visualized at cosmological scales. Other relevant effects of Λ can be found for instance in gravitational lensing theory [43]. Alternative modifications for the Newtonian gravity have been made through the MOND (Modified New- tonian Dynamics) models where the Poisson’s equation is modified from the l.h.s by introducing an acceleration scale under which the concepts of inertia and/or the law of gravitation are revised. This models seems to fit with high precision with observations done in small astrophysical scales, such as galaxies or star clusters [44], and its relevance lies in the fact that dark matter is not needed to explain the behavior of astrophysical structures.

2.4 Structure formation in expanding universe

Structure formation is based on the considerations of density perturbations (overdensities) on an homogeneous background (Friedmann universe). Such perturbations can be relics of quantum effects that dominates the dynamics of the universe on times of the order of the Planck time after the Big-Bang [45]. The behavior of the density perturbations depends strongly of the medium it traverses, in this case, the expanding universe. Hence, different propagations modes can be found for different models. As long as the perturbation evolves in time, the overdensity increases such that at certain value gravity starts to be more important than the expansion and pressure adjustements, and the overdensity collapses, leaving the Hubble’s flow and forming the virialised structure that we recognize as galaxies and galactic clusters. Structure formation then is one of the best scenarios to test cosmological models. For perturbations with proper wavelength much smaller than the Hubble’s horizon, the evolution of the overdensity can be done 1 with the Newtonian limit. For scales comparable with H0− , general relativistic perturbation theory must be taken into account. The description of a fluid under the effects of a gravitational field can be done through the conservations laws for mass, energy, momentum, the Poisson’s equation and an equation of state. This conservations laws can be derived from an statistical point of view through Vlasov equation or by assuming an ideal gas 2.4. STRUCTURE FORMATION IN EXPANDING UNIVERSE 21 description (see for instance [46, 47]). In this brief explanation, we adopt the latter. In proper coordinates, the conservation of mass (continuity equation) and momentum conservation (Eu- ler’s equation) are written as

∂ρ 1 + ∂ (ρu ) = 0, (∂ + u ∂ ) u + ∂ Φ + ∂ p = 0. (2.32) ∂t i i t k k j i ρ i This, together with Poisson’s equation and an equation of state completes the full picture. Note that these expressions represent the Newtonian limit of the conservation equation (2.2) and Friedmann equation (2.5) for a flat Friedmann universe with the solution ρ = ρcdm, ui = H(t)ri and p = 0, which implies 3 ρ a− . ∼ In order to explore the formation of structures in an expanding universe, we need to cancel the effects of the expansion by writing the fluid equations in comoving coordinates xα through the relation ri(t) = Aiα(t)xα, where the elements Aiα(t) depends on geometry of the volume enclosing the fluid. For spherical symmetry, Aiα = a(t)δiα, where a(t) is the scale factor. The fluid equations for spherical symmetry are written as 1 1 1 1 ∂ ρ + 3ρH + ∂ (ρv ) = 0, ∂ v + Hv + v ∂ v = ∂ p ∂ Φ , (2.33) t a α α t α α a β β α −ρa α − a α grav where Φgrav is defined in (2.29) and vα is the peculiar velocity. As expected, the contributions coming from the expansion cancel out when we transform to comoving coordinates. By applying the operator ∂α∂t in Euler’s equation and using the continuity equation one has 1 1 ρ ∂ (ρv ) + ∂ (ρv v ) + 4ρH(t)v = ∂ p ∂ Φ . (2.34) t α a β α β α −a α − a α grav The formation of structures follows from perturbing (2.33) and (2.34) assuming that the total density ρ(x, t) corresponds to a deviation from the background density characterized by a density contrast δ(x, t) between the overdensity δρ and the background density such that

δρ(x, t) ρ(x, t) ρ¯(t) ρ(x, t) = ρ¯(t) + δρ(x, t) = ρ¯(t)(1 + δ(x, t)), δ(x, t) = − . (2.35) ≡ ρ¯(t) ρ¯(t) The Poisson’s equation in comoving coordinates associated to the overdensity can be then written as 2 2 2 xΦ = 4πa (t)δρ(x, t) = 4πa (t)ρ¯(t)δ(x, t). Taking the divergence of (2.34), using (2.2) for the ∇ grav background density in the Newtonian limit and assuming no peculiar motions for the background, one gets a diﬀerential equation for the perturbation

2 ∂ δ ∂δ 1 1 2 1 + 2H ∂ ∂ [(1 + δ)v v ] = xp + ∂ [(1 + δ)∂ Φ ] , (2.36) ∂t2 ∂t − a2 α β α β a2ρ¯∇ a2 α α grav together with the perturbed form of the continuity equation ∂δ 1 + ∂ v = 0. (2.37) ∂t a α α In the linear regime δ(x, t) = δ (x, t) 1, this diﬀerential equation reduces to a simpler form L 2 ∂ δL ∂δL 1 2 + 2H = xp + 4πρδ˜ . (2.38) ∂t2 ∂t a2ρ˜∇ L

ik x By considering a single Fourier mode δ (x, t) δ˜ (t)e · where k is the comoving wave vector, we have L ∝ L d2δ˜ dδ˜ k2 L + 2H L = 4πρa¯ 2 1 δ˜ , (2.39) dt2 dt − k2 L J where c2 ∂p/∂ρ¯ is the speed of sound squared and k is Jeans wave number, deﬁned as k2 4πρa˜ 2/c2 s ≡ J J ≡ s For an static universe (H = 0) and constant speed of sound, if k > kJ the perturbation will propagate 22 CHAPTER 2. BACKGROUND COSMOLOGY

through the medium as wave of sound. For k < kJ , there can be one growing solution and one decreasing ωt 2 2 2 2 solution, δ˜ (t) e . In this case the dispersion relation reads as k ω = 4πρ¯k k . On the other L ∝ J − J hand, in an expanding universe the Hubble parameter acts as a damping term, i.e, the expansion works against the possible collapsing modes. Hence, the evolution of the perturbation is strongly dependent of the evolution of the background, and we need to solve Friedmann equation in order to solve the evolution equation. By excluding pressure perturbations (and hence a sound-like propagation), the explicit dependence of the (i) comoving coordinate x in (2.38) disappears and we can write the solution as δL(x, t) = i Di(t)∆ (x), where D are two linear independent solutions of 1,2 P d2D dD + 2H 4πρa¯ 2D = 0, (2.40) dt2 dt − referred to a decaying D1 and growing mode D2. It is easy to show from the definition of the Hubble parameter H(t) (equation (2.5)) that it is a solution for the differential equation (2.40). Hence D(t) = H(t) 1 corresponds to the decaying mode (since H(t) t− ), while the function D D, called the growing factor ∝ 2 ≡ gives rise to the formation of structures. Once one has decaying solution H(t), it is possible to write the growing solution as a function of the latter as [48] t dt D(t) H(t)H2 0 . ∝ 0 a(t )2H(t )2 Z0 0 0 In terms of the scale factor, the differential equation (2.40) and its solution, the growing mode D(a) can be written respectively as 2 a 2 2 d D 2 2 dE dD 3 3 3 a E(a) + 3aE(a) + a E(a) = Ω a− D, D(a) = AE(a) [xE(x)]− dx, (2.41) da2 da da 2 cdm Z0 where A is a constant factor and the E(a) is written for DECDM-flat cosmologies as

1 3 f(a) 2 E(a) = Ω a− + (1 Ω )a− , cdm − cdm h i with f(a) given in (2.21). Figure 2.6 shows numerical solutions for the growing factor for different equations of state ωx. The constant A has been chosen so that the growing mode is normalized D(a = 1) = 1. For an EdS universe one has D (t) t2/3 a. For a radiation dominated universe one finds a growing mode + ∝ ∝ as D a2, while for a radiation-matter dominated universe one may find the Mészáros solution2 [46]. For ∼ the DECDM flat universe with ωx constant, the grow factor reads as

1 2 (5+9ωx) E(a)a 3 5 3 5 1 Ωcdm 3ωx D(a) 3 2F1 + , ; 3 + ; a . (2.42) ∼ (1 Ω ) 2 (5 + 9ω ) 2 6ωx 2 6 ωx −1 Ωcdm − cdm x − For the Chaplygin Gas model, the grow factor is reduced to

a β 1 f(a) 3 f(x) 3 1 n 3Ωch 3αn nβ D(a) = Aa− 2 x 2 − dx, f(a) = ln 1 + a− . (2.43) β ln a 3α 8π − n Z0 " n ! # The set of equations (2.36) and (2.37) and the solutions for (2.40) allows one to determine the peculiar velocity ﬁeld in the linear limit without pressure perturbations. The solution for vα is written as [46]

H(a)F (a) 1 δρ(x0, t) 3 v = g , g ∂ Φ (x, t) = a∂ d x0, (2.44) α 4πρ¯ α α ≡ −a α grav α x x ZV | 0 − | where gα is the acceleration field in comoving coordinates due to the potential associated to the inho- mogeneity δ(x, t) and F (a) is the growth index defined as F (a) d ln D/d ln a. The validity of this ≡ solution can be checked by replacing (2.44) in (2.37), which is trivially satisfied, and in (2.36) to obtain (2.40). Equation (2.44) then relates the peculiar velocity at large scales with the large scale densities perturbations.

2 The Mészáros solution D ∼ (a/aeq) + 2/3, where aeq ∼ Ωrad/Ωcdm, describes the Mészáros effect, i.e, δ ∼ constant for the radiation dominated phase, and δ ∼ a for the radiation dominated phase 2.4. STRUCTURE FORMATION IN EXPANDING UNIVERSE 23

0.9

0.8

0.7

0.6

0.5

0.4 EdS ω = −0.4 0.3 x Growing mode D(a) ω x = −0.8 ΛCDM 0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 scale factor a

Figure 2.6: Growing mode normalized at the present time for diﬀerent equations of state and Ωcdm = 0.3, Ωx = 0.3. The solid line represents the EdS universe (Ωcdm = 1) with D ∼ a.

2.4.1 Nonlinear evolution: the spherical collapse model

The spherical collapse model describes the nonlinear evolution of density perturbations (δ 1). In general, the formation of structures at cosmological scales can be divided in three main stages: the expansion of the system along with the universe (Hubble’s flow), the turn-around, when the system leaves at the Hubble flow and starts to collapse, and finally virialisation, corresponding to the final stage where the system reaches equilibrium satisfying the virial theorem. Let us consider a spherical region expanding with the background universe. Following [45], we will consider the overdensity region as composed of thin shells, each with radius r and containing a mass M. If the size of the perturbation is smaller than the Hubble horizon, we can use the Newtonian limit, where the equation of motion for the spherical shell of radius r can be written as in (2.31):

d2r a¨ δM = r . (2.45) dt2 a − r2

We have written the mass enclosed in the shell and associated to the overdensity as

r δM = 4πs2δρ(s) ds. Z0 We now assume contributions from the background in form of a cold dark matter component (cdm) and the x-component (vacuum, Dark Energy, Quintessence). In that case the acceleration equation (2.4) reads

a¨ 4 px(t) = π [ρcdm(t) + ρx(t)ηx(t)] , ηx(t) 1 + 3 . a −3 ≡ ρx(t)

For Dark Energy models or vacuum energy density, the factor ηx is constant. For Quintessence (or Chaplygin Gas), we have ηx = ηx(t). Using the last expression, the equation of motion for each shell is written as

(r, t) 4 4 r¨ = M πρ (t)η (t)r = πr [ρ¯(r, t) + ρ (t)η (t)] , (2.46) − r2 − 3 x x −3 x x 24 CHAPTER 2. BACKGROUND COSMOLOGY where the quantities written above must be deﬁned clearly: ρ¯(r, t) is the mean total density in the collapsing cloud, i.e, ρ¯(r, t) = ρcdm(t) 1 + δ¯(r, t) , while the mean value is understood as

3 s ρ¯(r, t) 4πρ(s, t)s2 ds. (2.47) ≡ 4πr3 Z0 Continuing with equation (2.46), is the total mass enclosed in a shell of radius r, which is given as M 4 (r, t) = πr3ρ(r, t) = M 1 + δ¯(r, t) ), M 3 cdm and Mcdm is the mass enclosed in the collapsing cloud which only belongs to the CDM component, 4 3 ¯ i.e, Mcdm = 3 πρcdm(t)r and δ can be understood as equation (2.47). The mass Mcdm is constant for negligible peculiar motions, since ρ r3 x3. To write the equation (2.46), we have assumed that the cdm ∼ overdensity (source of Φgrav) is written with respect to the cold dark matter in the background, that is, δρ(r, t) = ρcdm(t)δ(r, t). This means that we assume that only the CDM component clusters, i.e, ρ¯ = ρcdm. As usual one assumes that the diﬀerent shells labeled at some initial time t = ti as 1, 2, 3, ... with radius r1 < r2 < r3 < ... does not cross each other during the the expansion or the collapse, which implies that the mass contained within each shell of radius r doesn’t change as long as r changes [45]. That is, for M a shell with radius r, we have δ¯(r, t) = δ¯(r , t ) δ¯ with r = r(t ) and i i ≡ i i i 4 = πr3ρ (t) 1 + δ¯ = constant, (2.48) M 3 cdm i The equation (2.46) resembles the Newtonian equation of motion for a test object under the inﬂuence of gravity and a Hooke-like force with a time dependent spring constant. Hence, in principle certain cases admits analytical solutions, although one could argue that the amount of density associated to the x-component inside the system is very small comparable to ρcdm, and hence we could use energy conservation for certain purposes. If we let the x-component to collapse together with the CDM component, the equation of motion can be written as in (2.46) with the right replacement

Ω (t) = (t) = M 1 + 1 + x δ¯(r, t) . (2.49) M M cdm Ω (t) cdm In this situation, the equation (2.48) is no longer valid and the equation of motion does not allows analytical solutions, since the mass is now time dependent, even for η = 0 or η = 2. M x x − 2.4.2 Collapse with Dark Energy We now explore the spherical collapse in DECDM universe. Clearly, only in certain cases, the equation (2.46) can be solved analytically. For instance, with ρ = ρ and η = 2, the equation of motion x vac x − (2.46) admits a ﬁrst integral of motion yielding a constant when only the cdm component collapses. E The constant is given as

1 4 1 Ω = r˙2 M + πρ r2 = H2r˙2 Ω (1 + δ¯ ) vaci 1 , (2.50) E 2 − r 3 vac −2 i i cdmi i − Ω − cdmi where we have used Hubble’s law without peculiar motions for the kinetic energy term. As usual, this expression imposes a limit on the perturbations in order to obtain a collapsing cloud. That is, if the mean value of the density perturbations at some initial time satisﬁes the inequality

1 + Ωvaci Ωcdmi δ¯i > − , Ωcdmi then the cloud will reach a turn-around at r = rta. This implies that one can treat the the configuration as a closed FRW universe ( < 0 is analogous to k > 0 in Friedmann equation). For flat cosmologies we E have δ¯i > 2Ωvaci/Ωcdmi. At the radius r = rta, the expansion will finish and then collapse comes. The 2.4. STRUCTURE FORMATION IN EXPANDING UNIVERSE 25

0.8

1.0

0.6

ta 0.7 / r i 0.5 r 0.4

Ω 0.2 cdm(t=ti) = 0.3

0 0 1 2 3 4 5 6 7 8 9 10 δ i

Figure 2.7: Ratio between initial radius and radius at virialisation as function of the mean density perturbation for diﬀerent values of the density parameter Ωcdmi for a ﬂat universe

radius at the turn-around can be determined from (2.50) requiring r˙ = 0 at rta. The relation between the radius at turn-around and the initial radius is written as Ω (1 + δ¯ ) Ω r = cdmi i − vaci r . (2.51) ta Ω (1 + δ¯ ) Ω 1 i cdmi i − vaci − This behavior can be seen in figure 2.7. Note that the linear regime δ 1 requires a universe fully i dominated by matter. Once the system reaches the maximum radius rta, it leaves the Hubble’s flow and the gravitational attraction due to the overdensity starts to domain and the collapse takes place. But the configuration does not collapse to a single point. Relaxation processes and dynamical friction transform the potential energy during the collapse to kinetic energy in the form of random motions of the mean components. This give rises to gradients of pressure that work against gravity and lead the system to an equilibrium configuration. Such equilibrium state is represented by the virial theorem. That is, let rvir be the (fixed) radius at virialisation. Then, in equilibrium:

2 (rvir) = (rvir) + x(rvir), (2.52) K WM W where is the kinetic energy. For spherical symmetrym the following expressions for potential energies K are useful: dΦ = 2π ρΦ r2 dr, = 4π ρr3 x dr, (2.53) Wgrav grav Wx − dr Z Z 3 where ρ, defined as in (2.35) is the total energy density inside the configuration . The potentials Φgrav and Φ satisfies Poisson’s equation 2Φ = 4πδρ and 2Φ = 4πη ρ respectively. For spherical x ∇ grav ∇ x x x homogeneous configurations with radius R one has [49]

1 1 Φ (r) = 2πδρ R2 r2 , Φ (r) = 2πρ η R2 r2 . (2.54) grav − − 3 x − x x − 3 The combination of the densities δρ and ρ in the definition of the total mass (equation (2.46)) leads x M 3In the appendix these expressions are derived but the final result is written slightly different by separating the contribution of the background. 26 CHAPTER 2. BACKGROUND COSMOLOGY

Virialisation radius 0.9 Ω Ω vac = 1 − cdm

0.8 vir y 0.7

0.6

0.5 0 0.2 0.4 0.6 0.8 1 Ω cdm i

Figure 2.8: Radius at virialisation normalized to rta for a ﬂat universe with cosmological constant. For a matter dominated universe we reduce to yvir = 1/2. to a new potential Φ whose source is the total density ρ (overdensity plus the CDM component of the M background):

1 Φ (r) = 2πρ R2 r2 . (2.55) M − − 3 The potential energies are then written following (2.53) as

2 3 4 2 = M , x = πρxηx R . (2.56) WM − 5R W −5 M where is given in (2.48) and δ¯ = δ for homogeneous overdensity. Let us assume for a moment M i i that energy conservation holds for the x component. Using the virial theorem and energy conservation − (evaluating the energy at r = rta and at r = rvir) the condition for virial equilibrium is ﬁnally written in terms of the potential energies as 1 1 (rta) + x(rta) = (rvir) + x(rvir). (2.57) WM W 2WM 2W Let us now deﬁne the dimensionless parameters y = r/r , x a/a . Using the formulae (2.56), the ta ≡ ta equilibrium condition (2.57) yields a cubic equation for the radius at virialisation:

y3 + 2y2 ( χ 1) χ = 0. (2.58) vir vir | | − − | | The quantity χ has been deﬁned as

ρ(tta) ξ Ωcdm(tta) f(a) ρ χ = x(t)− , ξ , (2.59) ≡ η (t)ρ (t) η Ω (t ) ≡ ρ x x x x ta cdm ta where in the second equality we have used the function f(a) deﬁned in (2.21) . The solution of the cubic equation can be written as (see appendix C)

1/2 1/2 8 1 27 3/2 y = ( χ 1) sinh arcsinh χ ( χ 1)− . (2.60) vir 3 | | − 3 32 | | | | − " ( )# 2.4. STRUCTURE FORMATION IN EXPANDING UNIVERSE 27

1.5 ta

1 ω x = a / x = − 1/3 ω = − 2/3 Ω x cdm (tta)

ω 0.5 0.1 x = − 1 0.3 0.5 0.7 0.9

0 0 0.6 1.2 1.8 0 0.6 1.2 1.8 0 0.6 1.2 1.8 τ τ τ

Figure 2.9: Evolution of the scale factor normalized to its value at turn around for diﬀerent values of the density parameter of the CDM component for two diﬀerent equations of state of the x-component.

As pointed before, the quantity χ is function of time as can be seen from (2.59) and hence the energy is not strictly conserved inside the conﬁguration. That is, equation (2.60) is only valid for a universe where with Ω Ω or simply by setting ω = 1. The factor ξ has been ﬁtted in reference [50] for arbitrary x cdm x − (but constant) ωx as

2 ρ 3π 0.79+0.35Ω 0.06ω ξ 1 + δ(r ) = = Ω− cdm− x , 1 ω 0 (2.61) ≡ ta ρ 4 cdm |ta − ≤ x ≤ cdm ta This makes yvir = yvir(Ωcdm(tta)) for ρx = ρvac. In the figure 2.8 we plot the function yvir(Ωcdm) for the ΛCDM model. For Ω (t ) 1 we recover the standard value y 1 . As long as the contribution cdm ta → vir → 2 of cosmological constant increases, the radius at virialisation increases and approaches to the radius at the turn-around. For Ω 0.97 y 1, that is, if formation of structures take place at the vacuum vac ≈ vir ≈ dominated era, virialisation then would take place when the cloud leaves the Hubble’s flow. For a Dark Energy component with constant e.o.s, the Friedmann equation and Eq. (2.46) can be written as a set if couple differential equations as [50]

2 dx 1/2 d y 1 ξ 1 Ωcdm(x) 3 = [xΩ (x)]− , = + − η (x)yx− , (2.62) dτ˜ cdm dτ˜2 −2 y2 Ω (x) x cdm where τ˜ H(t ) Ω (x = 1)t. These diﬀerential equations must be solved with the boundary condi- ≡ ta cdm tions dy/dτ˜ = 0 for x = 1 and y = 0 for x = 0 given the function Ω (x). For constant ω we can write p cdm x

1 1 Ωcdm(x = 1) 3ω − Ω (x) = 1 + − x− x . cdm Ω (x = 1) cdm Integration of equation (2.62) for the background gives

2 3/2 1 1 1 1 Ωcdm(x = 1) 3ω τ˜ = x F , , 1 , − x− x . (2.63) 3 2 1 2 −2ω − 2ω Ω (x = 1) x x cdm 28 CHAPTER 2. BACKGROUND COSMOLOGY

In figure 2.9 we have plotted the background evolution as x(τ) for different values or Ωcdm(x = 1) and for two different equations of state for the x-component. The density contrast at different times can be written for a ΛCDM flat universe as

ρ(t) 2 2 3 1 + δ(t) = = 2M3 3 sinh H0t Ωvac , (2.64) ρcdm(t) ΩvacH0 y (t)rta 2 p 3 where we have used ρ a− and equation (2.14). Evaluating this expression at t = t and using cdm ∼ ta (2.61)-(2.64), the time of turn around (when y = 1 and x = 1) can be determined as

ΛCDM 2 ρvacξ(ωx = 1) τta = H0tta = arcsinh − . 3√Ωvac s 2ρ(tta) !

In general, for arbitrary ωx, the time at turn around can be determined from (2.63) as

1 H 1 1 1 1 Ω (x = 1) τ (ω ) = H t = 0 F , , 1 , − cdm . ta x 0 ta H(t ) 2 1 2 −2ω − 2ω Ω (x = 1) Ωcdm(x = 1) ta x x cdm Collpase with scalar ﬁeldsp has been explored in [51]. Numerical results on the density contrast at turn around and virialisation can be found in [33, 52]

2.5 Remarks

In this chapter we have briefly discussed the most relevant aspects of moder cosmology. We have set special interest in the role that a positive cosmological constant plays in the determination of the most relevant features of the universe and the formation of structures. As was pointed in the introduction, the astronomical observations have shown that the cosmological constant rules the dynamics of the universe in a 70%. Modern cosmology has then a very different picture of early times cosmology, where the appearence of a cosmological constant seemed quite unnecessary. Although scientific community agrees with a cosmological constant, problems of standard cosmology has implied the introduction of dynamical models for the vacuum energy density, which, as Quintessence, can supply a mechanism to account for the observed flatness and homogeneity of the universe. Since these models display a cosmological constant behavior at the present epoch, one of the most relevant problems in modern cosmology is to determine which of these models are in best accordance with the universe that we observe today. In the following two tables we summarize the most relevant cosmological parameters and the dark energy models. 2.5. REMARKS 29

Parameter Numerical value

1 1 Hubble’s Parameter H0 = 72 kms− Mpc− 1 Hubble’s time TH = H0− 14 Gyr 1 ≈ 3 Hubble’s distance dH = H0− 6 10 Mpc ≈ 11× 3 Critical density ρc 0.68 10 M Mpc− . ≈ × Curvature parameter Ωk = 0 Dark Energy density parameter Ωvac = 0.7 Matter density parameter Ωmat = 0.3 2 7 2 2 Cosmological constant Λ = 3H0 Ωvac = 1.6837 10− Ωvac h70 Mpc− 1/2 3 × Length scale set by Λ RΛ Λ− 4 10 Mpc ≡ 1/2 ≈ × Time scale set by Λ T Λ− 9.6 Gyr Λ ≡ ≈ 2 √Ωvac+1 Age of the ΛCDM universe 3H √Ω ln √1 Ω 13.5Gyr. 0 vac − vac ≈ Age of globular clusters 13 Gyr ∼ Age of the galaxy 12 Gyr ∼

Table 2.1: Cosmological parameters. Ages of globular clusters and galaxies are included in order to be compared with relevant timescales.

Model Equation of state ρx/ρvac Parameters Induced ρvac

C.C p = ρ 1 Λ Λ vac − vac 8π 3(1+ω ) 1 2 D.E p = ω ρ a− x ω = , x x x x − 3 − 3

0 1 1+ωx(a ) D.D.E p = ω (a)ρ exp 3 0 da0 ω (a), ω (1) = 1 x x x a(t) a x x − h i R 1 1 1+γ γ 3(1+ω)(1+γ) γ+1 κ C.G ωρ(t) κρ(t)− 1 + Aa− ω, κ, γ − 1+ω A = 1+ω ρ(a = 1)1+γ 1 κ − 1 ˙2 Q-S.F ρφ = 2 φ + V (φ) Depends on V (φ) V (φ) V (φ0) (Inﬂation) p = 1 φ˙2 V (φ) φ = φ(a = 1) φ 2 − 0

Table 2.2: Dark energy models: For abbreviation, Cosmological constant C.C, dark energy D.E, dynamical dark energy D.D.E, Chaplygin gas C.G., Quintessence-Scalar ﬁelds Q-S.F 30 CHAPTER 2. BACKGROUND COSMOLOGY CHAPTER 3

Spherical conﬁgurations with cosmological constant

In this chapter we will describe the equilibrium of cosmological and astrophysical structures in an accelerated universe in a stage located far from the time of virialisation. The equilibrium condition will be explored through the virial equation in the Newtonian limit with cosmological constant. This chapter is divided in three sections. In the first one we write down the equations that represent the equilibrium concepts, from Euler’s equation to the virial equation. This sets the theoretical basis for the other two sections. In the second section, we explore spherical configurations in equilibrium, for homogeneous and non homogeneous configurations. We put special attemption in the non homogeneous case, where we study the effects of a nonzero cosmological constant on a polytropic configuration.

3.1 Description of equilibrium

The equations governing the dynamics of a fluid are the conservation of mass, conservation of momentum, conservation of energy together with an equation of state. Conservation of mass is given by equation (2.32) in the Newtonian picture and by equation (2.2) in the cosmological context. Conservation of momentum is given by Euler’s equation, given also in (2.32) (see also appendix A). For a self gravitating fluid influenced by a external magnetic field, Euler’s equation is written as (the statistical approach is given in [47], among others.) 1 (∂ + u ∂ ) u + ∂ Φ + ∂ ( + ) = 0, (3.1) t k k i i ρ j Tij Pij where are the components of the pressure tensor defined in terms of phase-space average values as Pij ρ (v u )(v u ) = δ p + π , u = v , (3.2) Pij ≡ h i − i j − j i ij ik i h ii with p as the pressure and π as its the traceless part associated to viscosity. are the components ik Tij of the Maxwell’s stress tensor. For magnetic component it is given by = 1 δ B2 B B . The full Tij 2 ij − i j description is completed with Poisson’s equation for the potential Φ and an equation of state p = p(ρ, S), where S is the entropy. Euler’s equation requires information of the parameters associated to the fluid as a function of time and the position. We can skip the position dependence and get more useful information by taking exterior products of (3.1) with the vector r to obtain the second order tensor virial equation. For a self gravitating rotating configuration influenced by an external magnetic field, the tensor virial equation with the contribution from the background universe reads as (see appendix A) [53, 54] 1 d2 1 Iik = 2 (T + ) gen + exp + Π r dS + + . (3.3) 2 dt2 ik Rik − |Wik | Wik ik − kPij j 2Lik Qik Z∂V 31 32 CHAPTER 3. SPHERICAL CONFIGURATIONS WITH COSMOLOGICAL CONSTANT where 1 ρr r d3r, and T ρu u d3r (3.4) Iik ≡ i k ik ≡ 2 i k ZV ZV are the moment of inertia tensor and the kinetic energy tensor. The tensor gen contains the contribution Wik from the gravitational potential energy and the magnetic field through in the following way:

gen grav ik(B) ik ik 1 ∆(ik) , ∆(ik) F grav , (3.5) |W | ≡ |W | − ≡ ik |W | with the function and the quantities and Π are deﬁned respectively as F Bik ik 1 (B) δ 2 r δ B2 B B dS , Fik ≡ ikB − Bik − k 2 ij − i j j Z∂V 1 B B d3r, Bik ≡ 2 i k ZV Π d3r. (3.6) ik ≡ Pik ZV The gravitational potential energy tensor gen has been deﬁned by the following expression: Wik

grav ρr ∂ Φd3r. (3.7) Wik ≡ − i k ZV The contribution from the background is proportional to the moment of inertia tensor and it is a function of time. It it can be written from Friedmann equation as 4 exp = exp(t) π [2ρ (t) + ρ (t)η (t)] , (3.8) Wik Wik ≡ −3 cdm x x Iik where ηx = 1 + 3ωx. Here we have assumed that the background is composed of a cold dark matter (CDM) and a component associated to a generalized vacuum energy density with equation of state ωx. Recall that this e.o.s may depend on time, for instance in the quintessence models or Chaplygin gas. On the other hand the tensors and appears when we consider an angular velocity as function of Qik Lik time and positions Ω(r, t) together with internal motions. These tensors are deﬁned in the appendix A, and for simplicity we won’t use them in this chapter. The tensor is deﬁned as Rik 1 ρ [v Ω] r d3r, (v = Ω r), (3.9) Rik ≡ 2 × i k × ZV corresponding to the rotational kinetic energy tensor. The most used version for the equation (3.3) comes by taking the trace to obtain the scalar virial equation. We get

1 d2 I = 2( + ) gen + exp p (r nˆ) dA r π dS . (3.10) 2 dt2 K R − |W | W − · − i ij j Z∂V Z∂V where nˆ is an unitary vector perpendicular to the surface with area A and

gen grav (1 ∆) , |W | ≡ |W | − (B) ∆ F , ≡ grav |W | 1 (B) r δ B2 B B dS . (3.11) F ≡ B − i 2 ij − i j j Z∂V together with 3 1 = T + Π = ρ v2 d3r, Π Tr(Π ) = p d3r. (3.12) K 2 2 h i ≡ ik ZV ZV 3.1. DESCRIPTION OF EQUILIBRIUM 33

Note that in the absence of gravitation and internal motion (i.e, = 0, p = 0), the virial equation becomes K a differential equation for the moment of inertia of the configuration as ¨ , which in principle shows I ∝ I an unstable behavior of the configuration, i.e, any small perturbation will take apart each component of the system. This fact is similar to the behavior of the equation of motion in the Newtonian limit for a test particle where a/a¨ Λ leading to an exponential increasing of the radial distance as r exp[ Λ/3t]. ∼ ∼ Hence the virial equation written as ¨ represents the Newtonian picture of the de Sitter universe I ∝ I p (equation (2.13)).

The steady state is described from Euler’s equation when all time dependence has dropped and the l.h.s of (3.1) vanishes to obtain

ρ∂ Φ = ∂ ( + ) . (3.13) i − j Tij Pij The structure of a given conﬁguration is then determined by the solution of this expression together with Poisson’s equation for a given equation of state, as we will see. The steady state can be described from the virial equation by taking time averages over a large period of time in order to let the system pass through each allowed (micro)state compatible with initial conditions until the conﬁguration reaches the state of minimum energy. If the system is stable or the variations in the moment of inertia are periodic in time, the left hand side of (3.3) and (3.10) vanishes and we obtain the virial theorem 4 1 2 T + + gen π ρ η + Π + + = (Surface) (3.14) h ik Rikiτ hWik iτ − 3 h x xIikiτ h ikiτ h2Lik Aikiτ h ikiτ 4 2 + + gen π ρ η + = Surface , hK Riτ hW iτ − 3 h x xIiτ hLiτ h iτ

Note that the contribution from the background can be only factorized out when we choose ρx = ρvac. In this particular situation, we have to make time averages over parameters associated only to the conﬁgu- ration. The tensor-virial theorem can be written as

gen 8 1 2 Tik + ik τ + ik τ + πρvac ik τ + Πik τ + ik + ik τ = rkπij dSj , (3.15) h R i hW i 3 hI i h i h2L A i ∂V τ D Z E and trace of this expression (written in (3.10)) is what one calls the Λ-virial theorem (ΛVT). Note that 3 the contribution from the matter in the background have been neglected since ρ a− and it is matter ∝ negligible if we make time averages over a period of time large enough. In the general situation when the contribution of the background cannot be factorized out, we can use the virial equation (3.3) and (3.10) by assuming that they describe states of quasi-equilibrium where the variations of the moment of inertia are very small so that the approximation ¨ 0 is valid for some time interval during which Iik ≈ we want to make predictions. Furthermore, when we use those expressions we will also assume that the contribution from the matter in the background is negligible, which implies that we need to be located at times far from the matter dominated era. Indeed, we want to make predictions in the present epoch, when the structures have been yet formed and when the universe in entering in a vacuum dominated era, as discussed in chapter 2. To describe a cosmological and/or astrophysical configuration through the virial theorem, we must take into account the two main parameter describing a system in this context. First we have to determine the geometry, which defines the limits in the integral quantities. Second, we must set or find the density profile. In this chapter we are going to explore structures with spherical geometry, and we will consider homogeneous configurations and some special cases for non-constant densities.

3.1.1 General consequences of ΛVT The ΛVT is a useful tool to describe the equilibrium and stability of cosmological/astrophysical configurations under the effects of an expanding universe. Some relevant consequences can be described from (3.15). In the first place, if we neglect surface integrals and consider an static configuration, the ΛVT can be regarded as an upper bound for the contribution of ρ simply by requiring > 0. We obtain [55] vac K 3 gen ρ |W |. (3.16) vac ≤ 8π I 34 CHAPTER 3. SPHERICAL CONFIGURATIONS WITH COSMOLOGICAL CONSTANT

For constant density it is useful to define pure geometrical quantities as ˜ , ˜ and by W I A grav ˜ ˜ grav 2 ˜ 16π |W 2 |, I , I . (3.17) |W | ≡ ρ I ≡ ρ A ≡ 3 ˜ grav ! |W | Hence, inequality (3.16) can be written for constant density as ρ ρ . (3.18) ≥ A vac Any homogeneous system in equilibrium must satisfies this inequality. The factor will play an important A role through this chapter since it will enhance the the effect from the vacuum energy density, as can be seen from (3.18). A useful generalization can be done for situations in which we use the tensor form of the virial equation (i.e, homogeneous ellipsoidal configurations) [56]

16π ˜ij ij Igen , ρ = constant. (3.19) A ≡ 3 ˜ ! |Wij | Finally, a curious equation can be derived by eliminating Λ from the tensor virial equations: gen 2Tij ij + Πij Iij − |Wgen| = . (3.20) 2T mn + Π I nm − |W | mn mn This expression is only valid if Λ = 0, although the cosmological constant does not enters directly in 6 this equation. This expression will be useful in future sections where we will infer a relation between the geometry and rotational velocity of an ellipsoid.

3.2 Spherical conﬁgurations

The tensor virial equation is trivially satisfied for spherically symmetric configuration without a magnetic field, or other external forces, since grav = δ grav and = δ . Therefore, the study of spherical Wik ikW Iik ikI configurations can be made through the scalar virial theorem which is derived from (3.15) for an external contribution dominated by the cosmological constant. It is written as 8 2 + 2 + gen + πρ + = p (r nˆ) dA. (3.21) K R W 3 vacI L · Z∂V The quantities in this equation are assumed to be time average values (since ρvac = constant). Note that the integral in the r.h.s of (3.21) can be set to zero if the configuration is surrounded by a collisionless background. Hence, in all our applications on equilibrium configurations at the present time, this term is taken as zero.

3.2.1 Homogeneous sphere Explicit expressions can be derived in the spherical case with constant density for the terms involved in the ΛVT. In this case, the potential Φ for the gravitational part and for the background contribution are written as 1 1 Φ (r) = 2πρ R2 r3 , Φ (r) = 2πρ η R2 r2 . grav − − 3 x − x x − 3 where R is the radius of the conﬁguration. Note that these are the same expressions as (2.54) after relabeling the density source of gravity as δρ ρ (i.e, we write M by neglecting ρ ). The → M → cdm gravitational potential energy can be derived from (3.7), and the moment of inertia can be calculated from (3.4). We have 3 M 2 3 grav = , = MR2. (3.22) W −5 R I 5 Combining these expressions and using (3.17) we get = 2. Hence the ratio ρ /ρ does not Aspherical vac get enhanced much by the geometrical factor . Nevertheless, using these values in (3.18) we see that A the cosmological constant allow the existence of low density objects with ρ = 2ρvac satisfying the virial theorem. This would be the lowest value allowed for the density of cosmological conﬁgurations [42]. 3.2. SPHERICAL CONFIGURATIONS 35

3.2.2 Radius at virial equilibrium A relevant consequence of the cosmological constant is the existence of a maximum radius at virial equilibrium for spherical conﬁguration at temperatures1 going to zero. Using the expressions for grav and |W | given in (3.22), the ΛVT (equation (3.21)) can be written as a cubic equation for the radius at virial I equilibrium Rvir for a given mass and temperature:

R3 + 10ηR2 R 3r R2 = 0. (3.23) vir Λ vir − s Λ Here we introduced the dimensionless temperature η as

3kBTv 7 T µ η K = = 4.58 10− 3 , (3.24) ≡ rs 2µ × 10 K M where we wrote = 3 (M/µ)k T with µ as the mass of the average member of the conﬁguration, T the K 2 B v v temperature in virial equilibrium and rs = M (half of the Schwarszchild’s radius). The length scale RΛ set by the the cosmological constant was already deﬁned in (2.10) as

1 3 1 1/2 10 RΛ = 2.4 10 h70− Ωv−ac Mpc 1 10 ly. ≡ rΛ × ≈ × It is also useful to introduce a parameter of mass λ as

rs 23 M 1/2 λ = 1.94 10− h70Ωvac 1. (3.25) ≡ RΛ × M This parameter measures the scale imposed by the cosmological constant with respect to the scale imposed by the Schwarszchild’s radius. The cubic equation (3.23) can be parameterized in terms of the function $(λ, η) R (λ, η)/R (λ, 0) such that ≡ vir vir 3 2/3 $(λ, η) + 10η(3λ)− $(λ, η) 1 = 0, (3.26) − h i where Rvir(λ, 0) is radius for the conﬁguration at Tv = 0, given as

2 1/3 1/3 Rvir(λ, 0) = (3rsrΛ) = (3λ) RΛ. (3.27)

This represents the largest radius that a spherical homogeneous cloud may have in virial equilibrium (i.e, satisfying (3.21)). The function $(λ, η) can be obtained from the solution of the cubic equation (see appendix D). The result is

1/3 1/2 1 3/2 $(λ, η) = 2.53λ− η sinh arcsinh 0.24λη− . (3.28) 3 Figure 3.1 shows the behavior of $(λ, η) for different values of λ. We see that the increase of the temperature implies a decrease of the effects of Λ which can be easily checked if we solve Rvir from the ΛVT by setting Λ = 0 (λ 0) and compare it to the approximation η in (3.28)2: → → ∞ 3 rs Rvir(λ 0, η) = Rvir(λ, η ) Rvir(η?) = , (3.29) → → ∞ ≡ 10 η? We then interpret (3.26) as a radius-temperature relation for a fixed mass applied on astrophysical structures in virial equilibrium in the presence of Λ. That is, given λ (mass) and η (temperature), the radius of the configuration is uniquely determined by (3.28) as a function of Rvir(λ, 0). However we can adopt another point of view for this relation. Imagine a spherical configuration characterized by a constant mass M. In analogy to a thermodynamical reversible process, the configuration may pass from one state of virial equilibrium to another following the curve $ η for λ fixed, that is, satisfying − 1Temperatures in the sense of being the measure of mean velocities 2 which is also clear if we write λη = K/rs 36 CHAPTER 3. SPHERICAL CONFIGURATIONS WITH COSMOLOGICAL CONSTANT

100

−1 10−2 λ = 10

10−4 ) λ = 10−6 η

, −6 λ 10 ( ϖ 10−8 λ = 10−12 10−10

10−12 0 5 10 15 20 25 30 η

Figure 3.1: Ratio between Rvir(λ, η) and Rvir(λ, 0) for different values of λ as function of η = (3/µ)kB T , where µ is the mass of the main average components of the system. the condition ¨ = 0. Such a process leads the system to a final temperature η , which is determined by I ? the process itself. However, since the virial equations are not dynamical, we cannot know which stage, characterized with (η, Rvir) is the final one. If we assume that the effects of Λ are negligible when the length scales are such that η = η?, then using (3.28) and (3.29) we get

1 2/3 Rvir(η?) η? = 0.208$?− λ , $? = . (3.30) Rvir(λ, 0) 3 1 If we write = Mµ− k T , equation (3.30) can then be treated as an equation for the temperature K 2 B vir T? of the an astrophysical configuration where the effects of a cosmological constant have vanished: 2/3 1 2/3 M − kB T? = 0.138µ$?− RΛ− , (3.31) Note that this expression maintains the same dependence of the standard mass-temperature relation derived from the virial theorem, i.e, T M 2/3, which maintains even if Λ = 0. Such relation can be easily ∝ 3 1 determined using (3.21) and again = Mµ− k T as K 2 B vir µ ρ k T = (36πρ)1/3 1 2 vac M 2/3. (3.32) B v 15 − ρ v However the meaning of (3.31) is different from that of (3.32), since equation (3.31) is associated to the temperature that a system acquires in the final stage after going through some reversible processes which took the system through successive states of virial equilibrium with constant mass from a radius Rvir(λ, 0) to a radius Rvir(η?) or vice versa, while (3.32) relates the temperature of any configuration in equilibrium with its mass maintaining constant density, and where the cosmological constant enters as cosmological correction modeled by the factor = 2. Note that if we use equations (3.32) (with Λ = 0) and Aspherical (3.31) assuming that M = Mv and T? = Tv and requiring that M(η = 0) = M(η = η?) we recover ρ(η = 0) 2ρ . To get some idea on the orders of magnitude suggested in (3.31), let us write it as ≈ vac 2/3 3 1 M 2/3 1/3 1 T = 1.3 10− $− µ h Ω K eV− , ? × ? M 70 vac where we have used theexpression of the cosmological constant in terms of cosmological parameters given in (2.9). The relevance of the solution described above solution is that we can understand that a cold system with R = Rmax to be in hydrostatic equilibrium just as a consequence of the contrary effects of gravity and a gradient of (a cosmological) pressure which is identified as (proportional to) the cosmological constant as p Λρr2. ∼ − 3.3. NON-CONSTANT DENSITY: POLYTROPIC CONFIGURATIONS 37

3.3 Non-constant density: Polytropic conﬁgurations

The examination of configurations with non-constant densities can be done in two directions. First, by knowing the density profile ρ(r), we can set up the virial equation and evaluate the equilibrium conditions from equation (3.21). There we can solve for parameters such as radius and total mass and we can therefore investigate their behavior in terms of the external parameter, in this case Λ. In this picture, the effects of Λ are included in the solution for the potential Φ as was shown in chapter 2, with the resulting term acting as an external force. The second option is to assume an specific equation of state (e.o.s) and solve the equilibrium equation (3.13). This way provides us the behavior of the parameters characterizing the configuration as functions of the external parameters and also as a function of internal variables such as the radius. Different e.o.s are used to describe astrophysical configurations. The most used form is the barotropic equation of state where p = p(ρ). One of this e.o.s is the polytropic model, written as 1 p = κργ , γ 1 + , (3.33) ≡ n where κ is a parameter (which depends on the entropy), γ is the polytropic index and n in the characterizing index. Configuration satisfying this equation of state are associated to collisionless systems whose n 3 distribution function can be written in the form f = f(E˜) E˜ − 2 , where E˜ is the relative energy (see ∼ [49]). In the limit when n one is able to make the description of the isothermal sphere where the → ∞ constant κ is proportional to the temperature. Let us consider the equations governing equilibrium for arbitrary geometry: 2Φ = 4πρ Λ, p = ρ Φ. (3.34) ∇ − ∇ − ∇ γ 2 We start combining these expressions by inserting the e.o.s in Euler’s equation; we obtain κγρ − ∂iρ = ∂ Φ. Then we apply ∂ and use Poisson’s equation to obtain after some algebra the following expression: − i j

2 1 1 1 ρ 4πnρ − n ∇ + 2 ln ρ = (1 ζ) , (3.35) n ρ ∇ − κ(n + 1) − where we have defined ζ = ζ(r) 2 (ρ /ρ(r)). Equation (3.35) can written in terms of a variable u ≡ vac defined as u = ρ/ρ0, where ρ0 is a parameter with units of density which will be specified later. Introducing also the associated Jeans’ length a through r = aξ, where

κ(n + 1) a , (3.36) 1 1 ≡ − n s 4πρ0 equation (3.35) is ﬁnally written as

2 − 1 ξu 2 1 2n 1 ρvac ∇ + ln u = ζ u n u n , ζ 2 . (3.37) n u ∇ξ 0 − 0 ≡ ρ 0 This is the Λ-Lane-Emden equation (ΛLE). Note that the initial (or boundary conditions) needed to solve (3.37) are fixed by the parameter ρ0. If we choose for instance ρ0 = ρvac, then ζ is fixed as ζ0 = 2 and the initial conditions will be given by u(0) = ρc/ρvac, where ρc is the central density of the configuration. Note that for constant density, independently of the chosen value for ρ0, we recover ρ = 2ρvac as the first non trivial solution of ΛLE equation. For spherical configurations, equations (3.34) reduces to d2Φ 2 dΦ dp dΦ + = 4πρ Λ, = ρ , (3.38) dr2 r dr − dr − dr

The ΛLE then can be written in a known form by setting ρ0 = ρc and assuming a power-law form for u n n as u = ψ , i.e, ρ = ρcψ (ξ). Following [59], (3.37) is then written as 1 d dψ ρ ξ2 = ζ ψn, ζ 2 vac . (3.39) ξ2 dξ dξ c − c ≡ ρ c 38 CHAPTER 3. SPHERICAL CONFIGURATIONS WITH COSMOLOGICAL CONSTANT

Since one has chosen ρ0 = ρc, this differential equation must be solved with the initial conditions ψ(0) = 1, ψ0(0) = 0 with the requirement that ψ must be a decreasing function of ξ. It is important to notice that in this picture the expected effects of Λ are to be found in the behavior of the density profile since now equation (3.39) implies that its solution is also function of the parameter ζc. One more consequence of the term ζc is that equation ΛLE equation looses its scaling properties according to the homology theorem 2/(1 n) [59], which states that if ψ(ξ) is a solution of (3.39), then ψ¯(ξ¯) = λ − ψ(λξ) is also a solution. The ΛLE under this transformations is written (with ξ¯ = λξ) as

¯ 2n 1 d 2 dψ n − ξ¯ + ψ¯ = λ n 1 ζc, (3.40) ξ¯2 dξ¯ dξ 2n/(1 n) Clearly we recover the same form of (3.39) only if we assume that ρ(ξ = 0) λ − ρ(λξ = 0). If we 2n/(1 n) → assume that this scaling holds for any r = aξ, i.e, ρ(ξ) λ − ρ(λξ), then we may write in terms if n 2n/(1 n) n → ψ ρ(ξ = 0)ψ (ξ) λ − ρ(λξ = 0)ψ (λξ), which leads to ψ(ξ) ψ(λξ). Hence, ΛLE does not satis- → → ﬁes the homology theorem. This fact reduces the possibilities of ﬁnding analytical solutions for the ΛLE once we have one solution, since now it is no possible to build a homologous families of solutions by scaling.

3.3.1 Parameters of the conﬁguration The radius of a polytropic conﬁguration is determined as the value of r when the density of matter acquires some special value. The standard picture suggests a radius located at R = aξ1 such that ψ(ξ1) = 0. In this schema, equation (3.39) yields a new equation to determine ξ1:

1/2 1 d dψ ξ = ξ2 . (3.41) 1 ζ dξ dξ " c ξ1 #

This is a transcenden tal equation for ξ1 once we obtain the solution for ψ(ξ). However, note that this expression is only valid in the presence of Λ. The radius is then given as

1/2 d dψ 1 R = aξ = R κ(n + 1) ξ2 ρ 2n . (3.42) 1 Λ dξ dξ c " ξ1 #

Hence the radius of the configuration is now proportional to RΛ. This is due to the fact that Λ sets a scale for length. However, this does not mean that R will be always of the order rΛ as Λ is also contained in the expression in the square brackets in (3.41) through ξ1 = ξ1(ρvac). Equation (3.41) suggest that not all the solutions for θ(ξ) represents physical solutions of density, that is, solutions with a definite value for ξ1. Hence, in order to get a well definite radius of a given configuration with cosmological constant, the 2 solution must satisfy d ξ ψ0 /dξ > 0. This picture is favored by the fact that the boundary is assumed ξ1 as the region where the density of collapsed matter that built the configuration vanishes, and we assume that only matter and not the matter (energy) associated to the vacuum energy density has collapsed. On the other hand, if we assume that the the boundary is located at the region where the density takes the limiting value ρ = 2ρvac, then (3.41) is no longer valid since the r.h.s. of (3.39) vanishes and then the radius is determined in the standard way, i.e, through the equation for the total mass, as we will see.

The eﬀective mass of the conﬁguration at a given value of ξ can be determined as usual:

aξ ξ 4 3 dψ M(ξ) = 4π(aξ)2ρ(aξ)dr = 4πa3ρ ξ2ψn dξ = πa3ξ3ρ ζ , (3.43) c 3 c c − ξ dξ Z0 Z0 where we used equation (3.39) for the second equality. The total mass is then obtained by evaluating the last expression at ξ = ξ1:

4 3 3 3 dψ M = πa ξ1 ρc ζc + , (3.44) 3 ξ1 dξ " ξ1 #

3.3. NON-CONSTANT DENSITY: POLYTROPIC CONFIGURATIONS 39

As expected, this expression shows that the mass is increased with respect to its value for Λ = 0; this is because Newtonian gravity has to be stronger in order for the conﬁguration to be in equilibrium with Λ = 0. Using (3.43) we can also write the radius in terms of the mass and the central density as 6 1 1 3 3 3 ξ1 − 1/3 1/3 ξ1 2 n R = M ρ− f , f (ζ ; n) ξ ψ (ξ)dξ . (3.45) c 0 0 c ≡ 4π Z0 ! and hence we can write a relation between the mean density evaluated at the boundary and the central density 3 ρ¯b = 3 ρc. (3.46) 4πf0 This expression will be useful when we try to determine the eﬀects of Λ on cosmological structures satisfying (3.33). On the other hand, by combining (3.42) and (3.45) we can write a relation between the total mass and the central density which is only valid with Λ = 0: 6 3/2 3 3 κ(n + 1) d 2 dψ 2n +1 M = R ξ ρc . (3.47) Λ f 2 dξ dξ " 0 ξ1 #

We can also determine the moment of inertia of the conﬁguration by using equations (3.4), (3.39) and (3.43) as

ξ1 ξ1 4 n 5 4 n 2 0 ξ ψ dξ = 4πa ρc ξ ψ (ξ)dξ = MR f1, f1(ζc; n) , (3.48) ξ1 I 0 ≡ ξ2 ξ2ψndξ Z 1R 0 where we have used (3.39) for the second equality. Also the termR proportional to the internal energy Π deﬁned in (3.12) can be written as

ξ ξ1 2 n+1 1 1 1 ξ ψ dξ 1+ n 3 2 n+1 n 0 Π = 4πκρc a ξ ψ (ξ)dξ = κρc Mf2, f2(ζc; n) , (3.49) ξ1 0 ≡ 2 n Z R 0 ξ ψ dξ where we have used again (3.43). These expressions will be useful to writeR the virial equation for polytropic configurations. Analytical solutions of (3.39) can be found for n = 0, 1 and n = 5 if Λ = 0 [59, 58]. As an example, for Λ = 0, we can write the analytical solution in the case n = 1 as 6 sin ξ ψ(ξ) = (1 ζ ) + ζ . (3.50) − c ξ c The radius is R = aξ , where ξ is the solution of the transcendental equation ξ ζ = (1 ζ ) sin ξ . In 1 1 1 c − − c 1 the first order of ζc one finds

1 R = πκ (1 + δRζc) , (3.51) r2 where δR = δR(n; ζc) is a numerical factor of order one. On the other hand, equation (3.50) also implies that there exists some ζcrit such that if ζc > ζcrit, we cannot find a real solution for ξ1. Approximately this gives ζ 0.178, that is, ρ & ρ , which, provided the overall density is not too big, is better that crit ≈ c vac ρ 2ρ which is a result from the general inequality (3.18) for ρ = const and spherical symmetry. ≥ vac Figure 3.2 shows the numerical solutions for different index n ranging from n = 1 to n = 5. We expect that the radius of the configuration is increased by the contribution of ζc and find it confirmed in the figures. However, not always is the radius of the configuration well defined, even if n < 5. For sizable values of ζc (black line) we cannot find physical solutions of (3.39) as the function ψ acquires a positive slope. One might be tempted to claim that the radius of the configuration could be defined in these situations as the position where ψ has its first minimum, but as can be seen for n = 3 such a radius would 40 CHAPTER 3. SPHERICAL CONFIGURATIONS WITH COSMOLOGICAL CONSTANT

1 1 1 ζ c = 0.1 0.005 0.0002 0.8 −5 0.8 0.8 1.2 X 10 6.1 X 10 −7

0.6 0.6 0.6 ψ ψ ψ

0.4 0.4 0.4

0.2 0.2 0.2

n=1 n=1.5 n=2

0 0 0 0 1 2 3 4 0 2 4 6 0 2 4 6 8 ξ ξ ξ

1 1 1

0.8 0.8 0.8

0.6 0.6 0.6 ψ ψ ψ

0.4 0.4 0.4

0.2 0.2 0.2

n=3 n=4 n=5

0 0 0 0 2 4 6 8 10 0 5 10 15 20 0 10 20 30 40 50 ξ ξ ξ

Figure 3.2: Effects of Λ on the behavior of the density of a polytropic configuration for different ratios ζc and different polytropic index. The radius of the configuration is not always defined, even for n < 5. For higher values of ρvac, only the n = 1 case has a definite radius for these values of ζc. For other cases, the configuration is defined only for small ζc.

be smaller than the radius with ζ 0 which contradicts the behavior shown for the other solutions where c → ξ(Λ = 0) > ξ(Λ = 0). As already mentioned above this is the correct hierarchy between the radii because 6 large ζc gives rise to a large external force pulling at the matter. The numerical solutions show that for relatively large values of ζc, only n = 1 has a well defined radius. In this case the effect of Λ is a 13% 1 increase of the matter extension as compared to ζ = 0. As we increase the polytropic index, ζ 10− c c ≈ leads to non-physical solutions while the effect with bigger values of ζc becomes visible only for n = 3. 3 For instance, ζc 10− results in a radius which is 17% bigger than the corresponding value with ζc 0. ≈ 3 → The combination n = 4 and ζc 10− also leads to a non-physical solution, whereas the radius of the 4 ≈ case ζ 10− displays a difference of 13% as compared to ζ 0. Finally, for n = 5, the only physical c ≈ c → solutions are obtained for the lowest values of ζc where ψ0 < 0. This case is particularly interesting as with Λ = 0 it is often used as a viable phenomenological parameterization of densities [49, 61]. The solution a has an asymptotic behavior as r− which has been also found in LSB galaxies [62]. With Λ = 0 the n = 5 6 5 seems less appealing is the matter is diluted. For all values of n, the difference between ζc = 10− and 7 10− is negligible.

At the end of this section we would like to summarize the findings form figure 3.2. In table 3.1 we write the ratio ξ (Λ = 0)/ξ (Λ = 0) for the same ratios ζ and polytropic index as in figure 2. The horizontal line 1 1 6 c represents a non-defined radius. The symbol indicates that the radius is defined only asymptotically ∞ in case of Λ = 0 3.3. NON-CONSTANT DENSITY: POLYTROPIC CONFIGURATIONS 41

ζc n = 1 n = 3/2 n = 2 n = 3 n = 4 n = 5 0.1 0.88 – – – – – ∼ 0.005 1 1 0.98 0.86 – – 4 ∼ ∼ ∼ ∼ 2 10− 1 1 1 1 0.88 – × 5 ∼ ∼ ∼ ∼ ∼ 1.2 10− 1 1 1 1 1 – × 7 ∼ ∼ ∼ ∼ ∼ 6.1 10− 1 1 1 1 1 × ∼ ∼ ∼ ∼ ∼ ∞

Table 3.1: Values of the fraction ξ1(Λ = 0)/ξ1(Λ =6 0) for diﬀerent values of the ratio ζc and the polytropic index. The horizontal lines represents the non well deﬁned radius.

3.3.2 The ΛVT for polytropes Stability of polytropic configurations can be studied from the virial theorem (see [58]), which constitutes a useful tool to determine relation between the parameters of the configuration (mass, radius, temperature) and their behavior in terms of the polytropic index. As was explained in the last section, for Λ = 0 analytical solutions can be only reached on specific cases. For Λ = 0 the spectra of solutions is reduced. Furthermore, numerical solutions are constrained by the 6 value of ζc, and hence, for each index n, there is not a unique solution for θ(ξ) unless ζc acquires a fixed n value. One could be tempted then to set ρ0 = ρvac in (3.37) so that ζc = 2; then ρ(r = aξ) = ρvacψ (ξ), 1/n and we must set the boundary conditions ψ(0) = (ρc/ρvac) and ψ0(0) = 0. We have to fix once again the ratio ρvac/ρc in order to get numerical solutions. This method will be follow later for the isothermal sphere. Numerical values for ξ1(n), θn(ξ1), θn0 (ξ1) for Λ = 0 are found in [59]. Let us consider the ΛVT with for a polytropic configuration in which the contribution to the kinetic energy comes from the pressure term as = 3 Π: K 2 8 grav = πρ + 3Π. (3.52) |W | 3 vacI The gravitational potential energy grav can be obtained following [59]. One uses Euler’s equation and W the definition of grav given in (3.7) (or its equivalent given in (A.26)). With this we can derive the W expression dΦ/dr = (n + 1)(d/dr)(p/ρ), which can be easiliy deduced using (3.38) and the polytropic − equation of state. Integrating and recalling that Φ is the contribution from a Newtonian part Φgrav and the part associated to the background, i.e, cosmological constant, we obtain p 4 Φgrav(r) = Φgrav(R) (n + 1) + πρ r2 R2 . (3.53) − ρ 3 vac − Multiplying by ρ/2 and integrating over all the configuration we finally have

1 M 2 1 2 grav = ρΦd3r = (n + 1)Π + πρ MR2 . (3.54) W 2 − 2R − 2 3 vac I − ZV One relevant quantity is the total energy of the conﬁguration. Eliminating Π from (3.52) and (3.54) we write the gravitational potential energy as

3 M 2 ρ 1 grav = 1 vac (5 + 2n) f 1 , (3.55) W −5 n R − ρ¯ 3 1 − − which reduces to the standard expression for Λ = 0. This shows the standard result for a n = 5 polytrope: it has an infinite potential energy due to the fact that the matter of this polytrope is distributed in a infinite volume (or it is highly concentrated at the center). This can be seen in Figure (3.2). On one side, the cosmological constant only helps the system to enlarge its radius, and hence its effect on a physical n = 5 polytrope is invisible. But also the cosmological constant has a relevant effect on deciding whether a n = 5 polytrope has a physical meaning, because not all values of Λ maintain θ0 < 0, as was explained 42 CHAPTER 3. SPHERICAL CONFIGURATIONS WITH COSMOLOGICAL CONSTANT before. Following the same analysis we can write down the energy of the configuration in terms of the polytropic index as E = grav + exp + nΠ, by using (3.55), (3.54) and (3.52) as W W 3 n 8 3 n ρ M 2 E = − grav πρ = − 1 vac (5f 1) . (3.56) − 3 |W | − 3 vacI − 5 n − ρ¯ 1 − R − b Note that the condition to be fulfilled for a bounded system depends not only on the index n but also on the value of ρvac. For ρvac = 0 we recover the condition γ > 4/3 for bounded and stable configurations. If we maintain this condition in the polytropic index, then polytropic configuration is stable if the term in squared brackets is positive. Using (3.48), a bounded configuration with γ > 4/3 (n < 3) is reached if

ρ¯ > (5f 1) ρ . (3.57) b 1 − vac The transition from bounded to an unbounded configuration then occurs for when the equality holds. The solution for this equation requires the knowledge of the function f1, and hence analytical solutions are not straightforward. Note that equation (3.57) is the generalization to equation ρ > ρ found for A vac homogeneous systems. To see this, we use the definition of ρ¯b together with equations (3.43) and (3.45) to write (3.57) as 4 ρ > πf 3 (5f 1) ρ . (3.58) c 3 0 1 − vac Equation (3.58) is the generalization to (3.18), and it must be clear that the term in parenthesis is also a function of ζc. Any polytropic configuration is in equilibrium as long as it satisfies (3.58). It is clear that the inequality (3.58) can be reversed if the configuration has n > 3, which would imply that equilibriums is reached if the central density of the configuration is smaller than a term proportional to the vacuum energy density. This condition is quite unphysical and hence we assume n < 3.

Mass and Radius Relations between mass, radius and central densities in equilibrium are determined from the virial equation. Using the same expressions, the virial equation is written as

2 1 grav 8 M 1 n 2 2 + 3Π + πρ = + (5 n)κMρc f + πρ R M (5f 1) = 0. (3.59) W 3 vacI − 2R 2 − 2 3 vac 1 − 1 Note that for ρ = 0 and ﬁnite mass, one obtain R (5 n)− , while for ρ = 0 we obtain a cubic vac ∝ − vac 6 equation for the radius (as usual). But we can also solve for the virial mass in terms of the central density by using (3.45). We obtain

3 3−n 2 ( 3n ) M(ρ ) = (ζ )ρc , (3.60) c G c where is constant for Λ = 0, but in this case it is a function of the central density too: G 3 2 κf0f2(5 n) (ζc; n) − . (3.61) G ≡ 1 2 πζ f 3(5f 1) − 3 c 0 1 − The explicitly dependence with the central density has the same form as the usual case with Λ = 0. Nevertheless, the function has a complicated dependence of the central density because of the term ζ . G c Combining equations (3.45) and (3.60) we can write the mass-radius relation

3−n − 2 n − n 3 M = 3 ( n−1 )f n 1 R n−1 . (3.62) G 0 Before end this part, we point again on equation (3.47), which we derived simply from the (3.39) and (3.42). Combining that equation with the virial mass we can write down an equation for the central density as

2 1−n 3 ( n ) 1 α 1 α n + 1 d 2 dψ ρc 2 3 ρc + (5f1 1) = 0, α ξ . (3.63) − R f f2 6 f2 − ≡ n 5 dξ dξ Λ 0 − ξ1 3.3. NON-CONSTANT DENSITY: POLYTROPIC CONFIGURATIONS 43

For Λ = 0 we reduce to the standard form for the central density in terms of the index n. As a matter of completeness, it worths to note that one may obtain the same result by writing the total energy of the conﬁguration and determining its minimum with respect to the central density and maintaining the entropy (per nucleon) constant [63, 57]. That is, equilibrium is represented by the values of M, ρc such that ∂E/∂ρc = 0. Hence, one can write the total energy using (3.41), (3.45), 3.49) and (3.55) together with the ﬁrst equality in (3.48), as

1 5 1 1 n 1 3 1 2 3 E = (n 1)κMρc f M 3 ρc f − 1 πζ f (5f 1) . (3.64) 2 − 2 − 2 0 − 3 c 0 1 − The stability criteria then yields the following expression:

2 1 2 1 1 1 1 5 4 3 ∂fi n − 3 − 3 (n 1)κf2ρc f0− M ρc 1 + πζcf0 (5f1 1) + Gi = 0, (3.65) 2n − − 6 3 − ∂ρc i=0 X where the functions Gi = Gi(ζc; n) are directly defined after taking derivatives. This equation defines a critical mass M which can be written in terms of the central density as (3.60) with = 0, so that if crit G G M > M (M < M ) the configuration is unstable (stable). The function 0 has also a complex dependence c c G of the central density and the mass, and hence Mcrit is written as in (3.60) just in order to be able to compare with the standard case Λ = 0. The function 0: G 3 2 2 3 1 − 2 (n 1)κf2 1 1 n ∂fi 4 3 0(ζc; n) 6f0 − + M − ρc− Gi 1 + πζcf0 (5f1 1) . (3.66) G ≡ 2n ∂ρc 3 − " i=0 !# X The dependence of this function is highly complicated when compared with the one derived from the virial equation, which is the one we will use for the next derivations. The slope in a M ρ diagram then varies − c in terms of the polytropic index as

3 4 ∂M 3 4 1 ∂ 2 (γ 3 ) = γ ρ− + G ρc − . (3.67) ∂ρ 2 − 3 c G ∂ρ c c Following the stability theorem, a radial model of oscillation will change its property of stability if the mass as a function of the central density reaches a maximum (see for instance [24, 63]). That is, stability (instability) stands for ∂M/∂ρc > 0 (∂M/∂ρc < 0). This yields a critical value of the polytropic index γcrit given by 4 2 ∂ ln 4 2 ∂ln γcrit = γcrit(ζc) G = + G , (3.68) ≡ 3 − 3 ∂ ln ρc 3 3 ∂ ln ζc so that polytropic configurations are stable for γ > γc. It is clear that the second term in (3.68) also depends on the polytropic index, and hence expression (3.68) is a transcendental equation for γc. It worths to mention that in the framework of general relativity, the critical value for Γ is also modified as γcrit = (4/3) + Rs/R [24, 58]. On going back to equation (3.60), we can write the mass of the configuration as M = αM M0, where M0 is the mass when Λ = 0 and αM = αM (ζc, n) is the enhancement factor: these quantities are given as

3 3 − 2 2 3 n (n) (n) 2n f0f2 M0 κ(5 n)f f ρc , αM (3.69) 0 2 (n) (n) 2 3 ≡ − ≡ " f f 1 πζcf (5f1 1) # 0 2 − 3 0 − (n) where fi = fi(ζc = 0, n) are numerical factor that can be determined straightforward (see table 3.2 and [59]). Also, by using Eq (3.45), the radius can be written as R = αRR0, where

1 1 − 2 2 1 n (n) (n) (n) 2n f0 f0f2 R0 = κ(5 n)f f f ρc , αR . (3.70) 0 2 0 (n) (n) (n) 2 3 − ≡ f ! " f f 1 πζcf (5f1 1) # 0 0 2 − 3 0 − We now show some examples where the enhancement factors may be relevant for astrophysical or cosmological structures. These examples are still being explored in a in a future article [60], and hence the conclusions around some of them are not written in this work. 44 CHAPTER 3. SPHERICAL CONFIGURATIONS WITH COSMOLOGICAL CONSTANT

(n) (n) (n) n ζc = 0.1 ζc = 0.05 ζc = 0.001 ζc = 0.0001 ζcrit f0 f1 f2 1 1.29, 1.12 1.12, 1.05 1.002, 1.001 1, 1 0.178 0.922 0.392 0.5 ∼ 1.5 1.17, 1.11 1.003, 1.002 1, 1 0.0825 1.126 0.307 0.5305 − ∼ 3 1.01, 1.022 1, 1 0.0065 2.34 0.1123 0.596 − − ∼

(n) Table 3.2: Numerical values for the enhancement factors of mass and radius (αM , αR) and the functions fi . Values for n = 1 have been taken from [56]. The symbol − indicates a non deﬁnite radius.

Example I: newtonian cold white dwarf The polytropic equation of state is not only useful to model the inner regions of mean sequence stars. It is also useful to determine the structure of compact Newtonian stars, but is also useful to explore the structure of compact objects as white dwarfs. In the limit where the thermal energy kBT of the star is small compared to the rest energy of electrons, Newtonian white dwarfs can be treated as n = 3 (in the ultra-relativistic limit, where the mass of electrons are much smaller than Fermi momenta pF) polytrope or as a n = 3/2 (in the non relativistic limit p m ) polytropic conﬁguration. In these cases, the F e parameter κ reads as (see for instance [58, 57])

4 5 ~ 3π2 3 ~2 3π2 3 κ = , κ = , (3.71) 3 12π2 m µ 3/2 15m π2 m µ n e n where mn is the nucleon mass, me is the electron mass and µ is the number of nucleon per electron. From equation (3.68) it is clear that the critical value of the polytropic parameter γcrit = 4/3 is shifted because of Λ and hence Newtonian white dwarfs built with relativistic electrons with are slightly deviated from stability to unstability. Hence the mass of the configuration should be greater in order to be stable, as expected. Using the Newtonian limit with cosmological constant, we can derive the mass and radius of these configurations in equilibrium. In the first case, for n = 3 the mass is written from (3.60) as M = (n = 3), which corresponds approximately to the Chandrasekhar’s limit (strictly speaking a configuration G would have the critical mass, i.e, the Chandrasekhar’s limit, if its polytropic index γ is such that γ = γcrit). 2 1/3 2/3 For this situation one has M0(n = 3) = 5.87µ− M and R0(n = 3) = 6.8(ρ¯ /ρc) µ− R . On the 3 1/2 5/2 other hand, for n = 3/2 one has M0(n = 3/2) = 3.3 10− (ρ¯ /ρc)− µ− M and the radius is 1/6 5/6 × R0(n = 3/2) = 0.27(ρ¯ /ρc) µ− R , where ρ¯ is the mean density of the sun. Since for these 4 configurations the ratio ζc is much smaller than 10− , from table 3.2 we see that the effects of Λ are almost negligible. Indeed, the critical value of the ratio ζc gives for n = 3 the inequality ρc > 307.69ρvac and for n = 3/2 one has ρc > 24.24ρvac.

Example II: Neutrino star

Another interesting possibility is to determine the effects of ρvac on configurations formed by other fermions as massive neutrinos, which was also postulated as candidates for dark matter. Such configurations can be used to model galactic halos. Clearly, these kind of systems will maintain equilibrium also through gravity and the degeneracy pressure as in a white dwarf. For stable configurations, i.e, n = 3/2, one must replace the mass of the electron and nucleon by the mass of the considered fermion and set µ = 1 in (3.71) and (3.60). We then get for the mass and the radius:

1 1 4 2 4 6 3 28 ρc eV 4 ρc − eV M0 = 3.28 10 M , R0 = 1.31 10− Mpc, (3.72) × ρ¯ mf × ρ¯ mf This case represents a possible cosmological configurations when the fermion mass if of the order of eV, for instance, massive neutrinos. As pointed before, such configurations must have a central density greater than 24.4ρvac in order to be in equilibrium. From table 3.2 wee see that the effect on a configuration with ζ 0.05 is represented in an increase in the mass by 17% with respect to M and an increment of 11% c ∼ 0 in the radius. 3.3. NON-CONSTANT DENSITY: POLYTROPIC CONFIGURATIONS 45

100 ρ ρ c / vac = 10 80 20 40 60 80

vac 100 ρ /

ρ 40

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ξ 20

vac 10 ρ / ρ

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ξ

Figure 3.3: Effects of Λ on the behavior of the density for an isothermal sphere for different values of central density. The top figure shows the complete range of densities ranging from 100ρvac to 10ρvac. The lower panel shows a larger interval in ξ, where it is shown how the slope changes its sign. The initial condition on the slope is ψ0(0) = 0.

For n = 3 one as

2 1 2 3 3 18 eV eV ρc − M0 = 5.16 10 M , R0 = 0.14 pc, n = 3, (3.73) × mf mf ρ¯ which corresponds to astrophysical scales for fermion mass of the order of eV.

Example III: Boson stars Boson stars have been also widely discussed as possible candidates for dark matter [66]. These kind of configurations are based on the interaction of a massive scalar field and gravitation which leads to gravitational bounded systems where the gradients of pressure come from the uncertainty principle (contrary to fermion stars where the gradients of pressure are associated to exclusion principle). Since the complete formalism of these systems lies out of this work, one can determine some effects of a positive cosmological constant by considering the energy of such configuration as [65] as a two variable function of the mass and the radius E = E(R, M)

N ~ G m2 N 2 8 E B N B B + πG ρ m N R2 (3.74) ∼ R − R 3 N vac B B The ﬁrst term corresponds to the total kinetic energy written as = N p = N ~c/λ and taking λ R. K B B ∼ The second term is the gravitational potential energy and the third term corresponds to the contribution of the background (see Eq. (3.15). The energy has an extrema at

1/3 1 ~ ~ 3 10 eV 5 eV M 10− M , R 10 R (3.75) ∼ m ∼ m ∼ 8πm ρ ∼ m B B B vac B Such values would lead to a mean density of the order of ρ¯ ρ . The mass given in the last expression ∼ vac is the so called Kaup limit [66]. 46 CHAPTER 3. SPHERICAL CONFIGURATIONS WITH COSMOLOGICAL CONSTANT

Example IV: isothermal sphere In the limit n one obtains the isothermal sphere characterized with the equation of state p = κρ. → ∞ However the results for a finite value of the index n are not well defined in the limit n , and hence → ∞ the analysis for the isothermal sphere must be done in a slightly different way. The alternative treatment consists in use (3.38) with the equation of state for an ideal gas, i.e, p = kB (T/m)ρ, where m is the mass of individual component. The resulting differential equation for the density with cosmological constant can be written as k T d d ln ρ B r2 = 4πρ + Λ, mr2 dr dr − which follows from combining Euler’s equation, Poisson’s equation and the e.o.s , as was done to derive equation (3.39). This differential equation could be treated in the same way as we did for the polytropic equation of state, i.e, by defining a new function ψ ρ/ρ , but here we can also use the fact that the ∼ c cosmological constant imposes certain scales of density, length and time, as commented before. Let us then define the function ψ(ξ) = ln(ρ(aξ)/ρvac) and r = aξ, with a the associated Jeans’ length. Since we are now scaling the density with ρvac, the Jeans’ length a should be scaled by the length scale imposed by Λ. That is

kB T a = RΛ, (3.76) r m where R was deﬁned in equation (2.10). The associated length scale is written with h 0.7 and Λ 70 ≈ Ωvac = 0.7 as σ a = σR = 13.34 Mpc (3.77) Λ 103 km/s where m is the mass of the proton. For an hydrogen cloud with T 2000 K and m = m we have a 40 p ≈ p ≈ kpc (which is approximately the radius of an elliptical galaxy). The resulting diﬀerential equation then reads as 1 d dψ 1 ξ2 = 1 eψ, (3.78) ξ2 dξ dξ − 2

The different behaviors for the solution ψ depends on the initial condition ψ(ξ = 0) = ln(ρc/ρvac). In Fig. 3.4 we show numerical solutions of equation (3.78) for different values of ρc/ρvac. As is common for n > 5, the radius cannot be defined by searching the first zeroth of the density, i.e, the value ξ1 such that eψ(ξ1) = 0. In this case, the behavior of the derivative of the density profile changes with respect to the Λ = 0 case, since as long as ξ increases, the density starts oscillate around the value ρ = 2ρvac such that for ξ one has a solution ρ 2ρ , as can be checked from (3.78), which corresponds to → ∞ → vac the first non trivial solution for ρ. This behavior implies that there exist a value of ξ = ξ11 where the derivative changes sign and hence the validity of the physical condition required for any realistic model, i.e, dρ/dr < 0, should be given up to ξ1, although this does not implies that the physical radius is located at R = aξ1. The second option to define the size of the configuration is to set the radius where the density acquires by the first time its asymptotic value 2ρvac. As seen from Figure 3.4, both definitions would yield to two different values of radius which are of the same order of magnitude. A third option is to fix the radius where ρ = ρvac, but this would implies that the dark energy component has collapsed together with the CDM component, which is not our case. Since the first candidate is based upon a physical condition of the configuration, while the second and the third one they are based on an asymptotic condition imposed by the background, we expect the first definition as the more suitable one. However this definition must be in agreement with the observed values for masses and radius of specific configurations, and the validity of this definition can be proved through the total mass of the configuration, given as

ξ1 3 1 3 2 ψ 15 σ M = RΛσ ξ e dξ = 1.55 10 f(ξ1) M , (3.79) 2 × 103 km/s Z0 3.4. STABILITY IN THE NEWTON-HOOKE SPACETIME 47

ρ ρ c / vac = 10 2.5 20 40 80 100 2 vac

/ 1.5 ρ = ψ e 1

0.5

0 0 5 10 15 20 25 30 35 40 45 50 ξ

Figure 3.4: Isothermal sphere with large values of central density. The horizontal line represents the limit ρ = 2ρvac with f(ξ ) ξ1 ξ2eψdξ. Combinig (3.76) and (3.79) we can write 1 ≡ 0 R 3 R 3 M = 653.55 ξ1− f(ξ1) M . (3.80) × Kpc If we define the radius at the first minimum (see Fig. 3.4), we then may find M 2 109 (R/kpc)3 M . ≈ × Although this might set the right order of magnitud for the mass of a E0 galaxy if we insert the real values for the radius, we must be aware that the radius is fixed by (3.76) as R 5.3 106 σ/103km/s kpc. ≈ × In order to get a radius of the order of kpc with masses of the order of 1010M , we would require 5 σ 10− km/s, which is not in agreement with the measured values Faber-Jackson Law [45] for the ∼ velocity dispersion in elliptical galaxies (σ 300 km/s). ∼ The second way that we take in order to get realistic values is then to fix the value of ξ1. For a configuration 5 with ρc = 10 ρvac, we fix the radius for R 50 kpc in equation (3.76) and using σ 300 km/s we get ∼ 12 ∼ ξ1 0.012, which implies f(ξ1) 0.033. The mass (3.79) is then M 1.5 10 M , while the density at ∼ ∼ ∼ × the boundary is ρ 36000ρ , that is, ρ 2.7ρ(R). The definition of the radius of these configurations R ∼ vac c ∼ is still our interest of research in [60].

3.4 Stability in the Newton-Hooke spacetime

The stability criteria for Newtonian configurations can be determined from the expressions representing equilibrium, that is, the total energy [63], Euler’s equation [78] or the second order virial equation [53, 58]. Here we will explore the stability of the equilibrium state from the second order virial equation. Detailed calculations are found in appendix A. One starts by considering small Lagrangian perturbations on a configuration in whose state of equilibrium is described by equation (3.3). The Lagrangian displacement of an element of the configuration is written as ξ(r, t) = ξ(r)eiωt, and hence the condition ω2 > 0 (ω2 < 0) implies stability (unstability). The variational form of the second order virial equation (3.3) is written as (see appendix A): 1 16 ω2 + πρ ρ (r ξ + ξ r ) d3r = ρξj ∂ Φgravd3r (3.81) − 2 3 vac i k i k j ik ZV ZV + ρ(1 Γ)δ ξj ∂ Φgravd3r − ik j ZV 1 + δ (5 3Γ)ρ (Ω ξ)(Ω r) Ω2(r ξ) d3r, 3 ik − · · − · ZV 48 CHAPTER 3. SPHERICAL CONFIGURATIONS WITH COSMOLOGICAL CONSTANT

1.6

1.2 crit Γ 0.8 ζ = 0.1 0.005 0.0002 0.4

0 0 0.1 0.2 0.3 0.4 β

Figure 3.5: Critical adiabatic index as a function of β for a homogeneous spherical configuration for different values of ζ = 2ρvac/ρ where Γ = (∂ ln p/∂ ln ρ) is the adiabatic index governing the adiabatic perturbations (for polytropic e.o.s, γ = Γ), which we assume constant throughout the configuration. The Lagrangian displacement is constrained with the boundary conditions i), ξ = 0 at r = 0 and ii) ξ must be finite at the surface. The boundary conditions are required to solve the equation of motion for ξ resulting from perturbing Euler’s equation (the Sturm-Liouville eigenvalue equation). The Lagrangian displacement should be in accordance with the geometry of the configuration. Being the simplest geometry the spherical case, the obvious choice for the Lagrangian perturbation is a radial dependence. Hence, let us assume a trial function for the Lagrangian displacement ξ = f(r)ˆr. In order to simplify equation (3.81) and in analogy to (3.17), let us define the following useful quantities:

1 16π I˜ W = %2 W˜ , I = %I˜, A , (3.82) | | 2 | | ≡ 3 W˜ | | where % is a parameter with units of density, so that W˜ and I˜ are only functions of the parametrization of the trial function. These quantities are deﬁned as dΦgrav W ρ(r)f(r) d3r, I ρ(r)f(r)rd3r. ≡ − dr ≡ ZV ZV Other useful quantities are introduced as 1 f(r) R R Ω2 ρ(r) r2 δ r r d3r, B . ≡ 2 rot r − ij i j ≡ W ZV | | By keeping the rotational contribution, we assume that the spherical symmetry is not broken by rotation. Taking the trace in (3.81), we can solve for ω2 in terms of the quantities deﬁned above as

8π% 4 5 2 ρ ω2 = Γ 2 Γ B A vac . (3.83) A − 3 − − 3 − 3 % Hence the critical value for the adiabatic index is written as

4 1 1 ρvac 5 Γ = (1 2B)− 1 + A B , (3.84) crit 3 − 2 % − 2 such that for Γ < Γcrit instability sets in and the system becomes unstable while perturbed. From equation (3.83), we see that through the inclusion of the cosmological constant we (the system) are forced 3.4. STABILITY IN THE NEWTON-HOOKE SPACETIME 49 to choose a bigger adiabatic index. In the simplest case, when f(r) = r, we have W , I , B β. → W → I → Furthermore, for ρ = constant, we get A and % = ρ. The stability condition for f(r) = r then → A becomes

4 1 1 5 Γ > (1 2β)− 1 + ζ β . (3.85) 3 − 4 A − 2 The spherical symmetry which we assumed compels us to write the ratio β for low eccentricities or to parametrize it in terms of the total angular momentum of the conﬁguration. In the ﬁrst approximation, at low eccentricities the ratio β given in (4.10) takes the form

2 1 1 β ζ e2 + (e4). (3.86) ≈ 3 5 − 2 O This equation is useful if we want to calculate a small, but peculiar eﬀect. Insisting that Γ is very close to 4/3 we can convert the stability condition Γ > Γcrit into a condition on eccentricity, namely

5 1/2 2 1/2 e > e 3.35 1 + ζ 3ε ε + ζ , min ≡ 2 − 3 where ε 4 Γ measures a small departure from the critical value 4/3. For the specific case ε = 0, we ≡ 3 − conclude that stability under quasi-radial oscillations is reached if the eccentricity is such that emin < e 1 3 with emin = 2.68√ζ, which is clearly valid only for large values of ρ compared with ρvac, say, ρ 10 ρvac. 2 ≥ For completely spherical configurations with constant density, we can write β = Ωrot/4πρ. Figure 3.5 shows the behavior of the critical adiabatic index as a function of β for different values of ζ. The largest deviation is for low densities as expected. For non rotating configurations the critical adiabatic index is written as 4 2 ρ Γ = Γ (ζ ) + A vac . (3.87) crit crit c ≡ 3 3 %

If we compare this equation with (3.68) for a polytropic configuration (γ = Γ), then we can identify ∂ ln /∂ ln ρ = ρ A/%, where the term A is written from (3.82) for a polytropic configuration as G c − vac 2%κ(n + 1)ρΓ 1f A = c − 4 > 0. ζ Mf 3ρΓκ(n + 1)f c 5 − c 6 with M is the mass (see (3.44)), ζ¯ = 2ρvac/% and the functions f4,5,6 defined as

ξ1 f (ζ ) ξ3f(ξ)ψ(ξ)ndξ, (3.88) 4 c ≡ Z0 ξ1 4 0 ξ f(ξ)dξ f5(ζc) , ≡ ξ1 2 n R0 ξ ψ (ξ)dξ ξ1 dψ(ξ) f (ζ ) R f(ξ)ξψ(ξ)n dξ. 6 c ≡ dξ Z0 Therefore the stability criteria (3.87) becomes

1 4 Mζcf5 2 f4 Mζcf5 − Γcrit = ζc 1 (3.89) 3 − Γcrit − 3 f − Γcrit 3κρc f6 6 3κρc f6 If we work with a constant density proﬁle, the choice of another trial function satisfying the boundary conditions mentioned before will not play a role in the above expressions since A = 2 for any f(r). → A Related results regarding the adiabatic index in cosmologies with non-zero Λ have been obtained in the relativistic frame work in [79]. 50 CHAPTER 3. SPHERICAL CONFIGURATIONS WITH COSMOLOGICAL CONSTANT

3.5 Remarks

We have demonstrated that the effects of a positive cosmological constant on spherical configurations are linked not to the geometry but ti eh behavior of the density. The geometrical factor = 2 for spherical A homogeneous configurations does not represents a great enhacement of the effects of Λ. However, we showed that there exist a maximum radius for a spherical homogeneosu system in equilibrium, which corresponds to a system where gravity is only supported by the expansion, ruled by Λ. For non homogeneous spherical configurations, we have shown with a precise example, namely, polytropic configurations, that the effects of Λ can be relevant, of course, for low density configurations, in stablishing the main parameters of such systems as the mass, the radius and stability conditions. Such stability conditions were explored from equilibria conditions by two different ways, i.e, variatins of the mass with respect to the central density, or small perturbations in the tensor virial equation. CHAPTER 4

Ellipsoidal conﬁgurations

A great number of the structures found in the universe have non spherical symmetry (associated to rotation or internal motions), and three major problems one often find related to such configurations: the origin of rotation, the behavior of rotational curves and the shape of such systems. The origin of such rotational motion is still a line of research [67]. Rotation has been well determined in elliptical and spiral galaxies [68], where the introduction of dark matter must be done in order to account for the observed rotation curves. However, alternative models MOND [44] and analysis derived in the frame work of general relativity [69] have reproduced the observed rotation curves for galaxies without dark matter. On the other hand, astronomical observation have pointed out that galaxy clusters may also have rotation [70], and this subjetc is closely related to the shape of this structures [70, 71]. These aspects motivated us to explore the consequences of a positive cosmological constant on non spherical systems. In this chapter we will show that the parameters of such configurations, spetially in rotating systems, can be considerably affected, as well as the possible shapes allowed in equilibrium.

4.1 Rotation of non spherical conﬁgurations

The effect on spherical configurations is not much enhanced by the geometrical factor , as was seen in the A last chapter. We now want to explore another geometries that could contribute an a more effective way to increase the effect of the cosmological constant. The next step is to explore nonspherical configurations. For this we will consider homogeneous systems with or without rotation. A convenient way to model almost all flat shaped objects is to consider ellipsoids which in the limit of flattened spheroids can be considered as disks. There are three different kinds of elliptical configurations, characterized by three semi-axes a1 = a, a2 = b and a3 = c. Oblate , with a = b < c, prolate with a = b > c and triaxial systems with a > b > c. Here the tensor virial equation provides a tool to determine which of these geometries are compatible with the virial equilibrium. Considering the case Λ = 0 and ρ =constant, or spheroids with confocal density distribution whose isodensity surfaces are similar concentric ellipsoids [49, 72] i.e.

3 x2 ρ = ρ(m2), m2 = a2 i , (4.1) 1 a2 1 i X the oblate ellipsoid emerges as a solution of the virial equations with a bifurcation point to a triaxial ellipsoid. According to the inner kinematics of the ellipsoid, the equilibrium equations has different solutions. Relevant examples are: Maclaurin ellipsoid (oblate ellipsoid because rotation without internal motions), Jacobi ellipsoid (triaxial ellipsoid flat because rotation without internal motions), Dedekind ellipsoids (triaxial ellipsoid flat because of internal streaming without rotation) and Riemann ellipsoid

51 52 CHAPTER 4. ELLIPSOIDAL CONFIGURATIONS

Λ = 0 Elliptical galaxy 0.2 0.2 Galactic cluster

0.15 0.15

πρ β / 2 2

Ω 0.1 0.1

0.05 0.05

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 eccentricity eccentricity

2 Figure 4.1: Eﬀects of Λ on the angular velocity Ωrot and the ratio β for ρvac ≈ 0.7ρcrit

(triaxial ellipsoid, flat because a combination or rotation and internal streaming), among others [73]. We can view the virial equation without Λ as a homogeneous equation. Switching on Λ = 0 this becomes an 6 inhomogeneous equation whose right hand side is proportional to ρvac. It is therefore a priori not clear if in the case of Λ = 0 we can draw the same conclusions as with Λ = 0. Let us consider the tensor virial 6 equation for configurations in equilibrium with rigid-body rotation with angular velocity Ω along the z axis. Writing the kinetic energy tensor as (see appendix A): 1 = Ω2 Ω Ω , Rik 2 rotIik − rotiIkj rotj and using (3.3) with ¨ = 0, we obtain I 8 Ω2 ( δ ) grav + πρ = δ Π. rot Iik − izIzk − |Wik | 3 vacIik − ik By eliminating Π, the equations for diagonal elements yield the following expressions: 8 8 8 Ω2 grav + πρ = Ω2 grav + πρ = grav + πρ . (4.2) rotIxx − |Wxx | 3 vacIxx rotIyy − |Wyy | 3 vacIyy −|Wzz | 3 vacIzz This is a relevant equation, since it gives us information about the angular velocity as a function of the geometry (eccentricities) and also represents the geometrical configurations that, for a given density, can be in equilibrium. Let us determine the angular velocity: clearly we can use any equality in (4.2); however if we take the first equality, the angular velocity would be undetermined for configurations with symmetry in the x y plane. Therefore, we can solve by using the first and the third (or the second and the third) − member of (4.2) for the angular velocity: grav grav 8 Ω2 = |Wxx | − |Wzz | πρ 1 Izz . (4.3) rot − 3 vac − Ixx Ixx The same expression holds if we make the replacement and on the right hand side Wxx → Wyy Ixx → Iyy of (4.2). Also, for a prolate configuration rotating along its major axis locataed along the eˆx direction, the angular velocity is written as in (4.2) by interchanging the x z components. On the other hand we ↔ can eliminate the angular velocity and get grav grav yy xx zz 8π xx yy zz I |Wgrav| − |Wgrav| 1 = ρvac graI v − I grav I . (4.4) yy zz − 3 yy zz Ixx |W | − |W | |W | − |W | Ixx 4.1. ROTATION OF NON SPHERICAL CONFIGURATIONS 53

It worths to mention some properties of this expression for Λ = 0. In that situation, equation (4.4) is a pure geometrical condition and the solution represents the allowed ellipsoidal configurations in virial equilibrium. A first solution is given by the condition grav = grav and = , which corresponds |Wxx | |Wyy | Ixx Iyy to an oblate (prolate) configuration, where a = b < c (a = b > c). Such conditions can be reached for constant densities or for a confocal ellipsoid with the density given in (4.1). The oblate configuration correspond to the Maclaurin ellipsoid. However, this expression admits a second solution, the Jacobi ellipsoids, which corresponds to a triaxial configuration. The bifurcation of these solutions occurs for a fixed value of the ratio q a /a as we will show later [73]. 3 ≡ 3 1 With cosmological constant, note that the oblate (prolate) configuration is still allowed by (4.4). However, the rise of a second solution now depends on the details of the density profile even if we take the latter to be as in (4.1). This is a direct consequence of a ‘pre-existing’ density scale ρvac.

For homogeneous conﬁgurations, the gravitational potential tensor and the moment of inertia tennsor are given for the three ellipsoids respectively as

8 4 grav = π2ρ2a a a a2A δ , = πρa a a a2δ , (4.5) Wik −15 1 2 3 i i ik Iik 15 1 2 3 i ik where the quantities Ai are functions of the eccentricities deﬁned for each case: oblate a1 = a2 > a3, prolate a1 = a2 < a3 and triaxial a1 > a2 > a3 [49]:

√1 e2 arcsin e 2 − √1 e Oblate, e2 e − − 1 e2 1 1 1+e A1 =  −e2 1 e2 2e ln 1 e Prolate, − − −  F (θ,k) E(θ,k)  h − i 2q2q3 k2 sin3 θ Triaxial,   A1 Oblate,

A1 Prolate, A2 =   E(θ,k) (1 k2)F (θ,k) q3 k2 sin θ  − − − q2 2q2q3 (1 k2)k2 sin3 θ Triaxial, −  and 

√1 e2 1 arcsin e 2 − Oblate, e2 √1 e2 e − −  1 e2 h1 1+e i A3 = 2 −e2 2e ln 1 e 1 Prolate,  − −  q sin θ q E(θ,k)  2h −3 i 2q2 (1 k2) sin3 θ Triaxial. −   2 2 2 2 The eccentricities are deﬁned in each case as e = 1 q− , e = 1 q and q a /a , while prolate − 3 oblate − 3 i ≡ i 1 F (θ, k) and E(θ, k) are the incomplete elliptic integrals deﬁned as

θ θ dφ 1 q2 E(θ, k) = 1 k2 sin2 φ dφ, F (θ, k) = , k − 2 , 2 2 0 − 0 1 k2 sin φ ≡ s1 q3 Z q Z − − p 1 with θ arccos (q ). The quantities are then given from (3.19) as = (2/3)A− δ . ≡ 3 Aik Aik i ik

Maclaurin ellipsoid

Maclaurin ellipsoid corresponds to an oblate configuration whose flatness can be associated to its rotation, expressed throught the Maclaurin equation. For a homogeneous oblate configuration, the equations (4.2) and (4.4) are written with the help of (4.5) as

Ω2 2 q2 rot = A (q , q ) q2A (q , q ) ζ 1 3 , (4.6) 2πρ 1 2 3 − 3 3 2 3 − 3 − q2 2 54 CHAPTER 4. ELLIPSOIDAL CONFIGURATIONS

10 10

8 Oblate 8 Prolate

6 6 A

4 Function g (e) 4

2 2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 eccentricity eccentricity

Figure 4.2: Function g(e) and geometrical factor A(e) for prolate and oblate ellipsoids. These functions have their largest values for ﬂat oblates and large prolate.

where ζ = 2ρvac/ρ. From these expressions, we write the constant angular velocity for an oblate ellipsoid from the first line of equation (4.6) as 1 Ω2 = Ω2 1 ζg(e) , (4.7) rot 0 − 2 where Ω0 corresponding to the angular velocity when Λ = 0 is given by the Maclaurin formula: (1 e2)1/2 3 Ω2 = 2πρ − (3 2e2) arcsin e (1 e2) . (4.8) 0 e3 − − e2 − The function g(e) defined from equation (4.6) can be calculated with explicit dependence on the eccentricity as η 4 1 g(e) Axx − xAzz = e5 (1 e2)1/2(3 2e2) arcsin e 3e(1 e2) − . (4.9) ≡ 1 ηx 3 − − − − − h i As is evident from the above equations, Λ has a twofold effect on the angular velocity. Firstly, it reduces the angular velocity with respect to the value Ω0 especially at the local maximum (see Figure 4.1). This is not a small effect and can affect even galaxies. Secondly, we see from (4.8) that Ω 0 for e 1. On 0 → → the other hand we have g(e) 32πq /9, approaching 1 for a very flat oblate configuration and not too → 3 dense matter. Therefore, beyond the local maximum in Ωrot the cosmological constant causes a steeper fall of Ωrot toward 0. Another relevant interesting quantity which can be calculated in this context is the ratio of the rotational over the gravitational energy contributions to the scalar virial equations, i.e., β / ˜ N . In accordance with (4.8) the latter can be written as ≡ R |W | 1 3 e(1 e2)1/2 β = β 1 ζg(e) , β = 1 − 1. (4.10) 0 − 2 0 2e2 − arcsin e − The effects on β are therefore similar to the the ones encountered in Ωrot (see also Figures 4.1). For large values of ζ the function β acquires a maximum value and goes to zero as e goes to 1. Finally, on account of p > 0 we can infer from the virial equations with ρ =const the following inequality 1 0 β (1 ζ ) , (4.11) ≤ ≤ 2 − A 4.2. ALLOWED CONFIGURATIONS IN EQUILIBRIUM 55

1 1

0.995 0.8

0.99

max 0.6

0.985

0.4

eccentricity e 0.98

Oblate 0.2 Prolate 0.975

0 0.97 0 0.1 0.2 0.3 0.4 0.5 0.001 0.01 0.1 ρ ρ ρ ρ vac / min vac / min

Figure 4.3: Critical eccentricity which together with (4.10) results in an inequality for the density of an ellipsoidal conﬁguration:

1 4β g ρ ˜ρ , ˜ A − 0 . (4.12) ≥ A vac A ≡ 2 1 2β − 0 For e 0, we have 2 ˜, while for e 1, ˜, and ˜. Therefore the above inequality is → A → A → A → A A ≥ A slightly weaker than the bound given in (3.18). From (3.12) we see that requiring > 0 is the same than K require that p > 0, since T > 0 by deﬁnition.

4.2 Allowed conﬁgurations in equilibrium

4.2.1 Bifurcation point: From oblate to triaxial As pointed before, the rise of a second solution (Jacobi ellipsoids) for ellipsoidal conﬁgurations in virial equilibrium depends now on the density through the ratio ρvac/ρ. To explore this dependence, we start to write the equation (4.4) as 8 ( ) ( ) = πρ ( ) (4.13) Iyy |Wxx| − |Wzz| − Ixx |Wyy| − |Wzz| 3 vacIzz Ixx − Iyy Using the formulae given in the last section, this expression can be written for a homogeneous triaxial ellipsoid with density ρ and semiaxis a1, a2 and a3 with a1 > a2 > a3 as

2 2 q (A3(q2, q3) ζ) q2 3 − 3 = 0 (4.14) 2 − A (q , q ) A (q , q ) + q2(A (q , q ) 2 ζ) 2 2 3 − 1 2 3 3 3 2 3 − 3 where q a /a (q > q ). Expression (4.13) is a complicated equation which solution is the function i ≡ i 1 2 3 q2 = q2(q3)x which represents the allowed ratios for a configurations in virial equilibrium with cosmological constant. In figure 4.1 we plot these solutions with and without cosmological constant. As usual, the Maclaurin ellipsoid (q2 = 1) is always a solution of (4.13) for any value of q3. Jacobi ellipsoid rises for bif bif q3 < q3 , where the bifurcation point is q3 = 0.583 for Λ = 0. With cosmological constant, we find the following effects. First, as ζ increases, the bifurcation point moves higher values of q3. It means that the 56 CHAPTER 4. ELLIPSOIDAL CONFIGURATIONS

min q bif bif q3 3 q3 1 1 Maclaurin

0.8 0.8

Jacobi 0.6 0.6 2 2 q q

0.4 0.4

0.2 0.2 ρ ρ ρ = 0 vac/ =0.1 vac

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

q3 q3

Figure 4.4: Values of q2 and q3 satisfying the virial condition F (q2, q3) = 0. Left: with ζ = 0.2, Right: ζ = 0

2 2 bif Jacobi solution appears at lower eccentricities (e = 1 q3). For ζ = 0.2, we get q3 0.69. Second, the − min ≈bif min range of allowed q3 also changes. Jacobi ellipsoids are now restricted to q3 < q3 < q3 . Below q3 , the only allowed solution is again the Maclaurin ellipsoid. For ζ = 0.2, the minimum value of q3 is 0.34 bif ≈ The behavior of the bifurcation point q3 can be approximately described as a function of ζ by bif 8.207ζ q3 (ζ) = 0.4826 0.2082e + 1 , (4.15) The equilibrium condition can be ﬁttedas q (q ) = aebq3 c, where the values a, b and c take diﬀerent 2 3 − values as a function of the ratio ζ. In table 4.1 we summarize these values.

bif ζ a b c q3 0 0.34622 2.30358 0.334876 0.582 0.006 0.329658 2.36399 0.314573 0.585 0.06 0.332745 2.332886 0.318074 0.589 0.2 0.348911 1.9253 0.334721 0.657

bq Table 4.1: Fits for the equilibrium condition q2(q3) = ae 3 − c for diﬀerent values of ζ

In Figure 4.6 we plot the angular velocity for different ratios ζ. The black line in that figure corresponds to the modified Maclaurin formula (4.7), while the red line is associated to the Jacobi solution. As pointed before, for high values of ζ, the Jacobi solution is allowed for certain range of the ratio q3. For instance, for ζ = 0.2 one has an allowed triaxial ellipsoid in virial equilibrium for 0.34 < q3 < 0.69.

4.2.2 Minor axis rotation: from prolate to triaxial Minor axis rotation is a rare case, but it has been observed in galaxies [74, 75, 76, 77]. Let us consider a triaxial ellipsoid with an angular velocity along the major axis, which we label as eˆx. The expression for the angular velocity is written in this case as 8 Ω2 = |Wzz| − |Wxx| + πρ Ixx 1 , (4.16) 3 vac − Izz Izz 4.2. ALLOWED CONFIGURATIONS IN EQUILIBRIUM 57

1 1

bif bif q = 0.585 0.8 0.8 3 q3 = 0.582

0.6 0.6 2 2 q q 0.4 0.4

0.2 ζ = 0 0.2 ζ = 0.006

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

q3 q3 1 1

bif bif q3 = 0.589 0.8 0.8 q3 = 0.69

0.6 0.6 2 2 q q 0.4 0.4

0.2 0.2 ζ = 0.2 ζ = 0.06 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

q3 q3

Figure 4.5: Values of q2 and q3 satisfying the virial condition condition F (q2, q3) = 0. Left: with ζ = 0.6, Right: ζ = 0 while the geometrical restriction (4.13) becomes 8 ( ) ( ) = πρ ( ) , (4.17) Iyy |Wzz| − |Wxx| − Izz |Wyy| − |Wxx| 3 vacIxx Izz − Iyy which in terms of the functions Ai is written as

2 2 q (A1(q2, q3) ζ) q2 = 3 − 3 . 2 A (q , q ) + q2(A (q , q ) A (q , q )) 2 ζ 1 2 3 3 2 2 3 − 3 2 3 − 3 Clearly, the prolate solution q = q is trivially satisfied in (4.17) with = and = , 2 3 Izz Iyy Wzz Wyy independent of the value of the ratio ζ. However, with Λ = 0, prolate is the only non trivial allowed configuration in equilibrium. That is, triaxial confiurations with minor axis rotation cannot be in equilibrium with Λ = 0. Surprisingly, for Λ = 0 we find a bifurcation point from which a triaxial solution 6 can be determined. This is shown in figure (4.7), where we have plotted the solution q2(q3) for different ratios ζ. In this figure, the straight line represents the prolate solution q3 = q2. The bifurcation point arises for small values of q , which represents large values in the eccentricity e = 1 q2. For the largest 3 − 3 value of ζ, ζ = 0.1, we obtain the largest value for the bifurcation point, located at q 0.15. Hence p 3 ≈ the triaxial solution arises for e > 0.98. Note that the triaxial solution has also a lower limit of validity for q3. It occurs when q2 approaches 1. For instance, for ζ = 0.1, the triaxial solution is valid in the interval 0.047 < q3 < 0.15 (0.988 < e < 0.998). As long as ζ decreases, the range of validity of the triaxial solution is reduced. The complete set of values of our solutions are given in table 4.2. In figure 5 we bif plotted the angular velocity against q3. The bifurcation point, q3 , is where the prolate curve intersects min the corresponding triaxial case. q3 is represented by the left vertical line beyond which we have no solution. It can be nicely seen that with growing ζ the range of the triaxial solution becomes smaller until it vanishes for ζ = 0. The angular velocity is also not a smooth function as it was the case for the triaxial/oblate case. Indeed, it oscillates (allowing for the moment the physically excluded range of Ω < 0) min when approaching q3 . 58 CHAPTER 4. ELLIPSOIDAL CONFIGURATIONS

W2 W2 2 ΠΡ Ζ = 0 2 ΠΡ Ζ = 0.006 0.25 0.25

0.2 0.2

0.15 0.15

0.1 0.1

0.05 0.05

e e 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 W2 W2 2 ΠΡ Ζ = 0.06 2 ΠΡ Ζ = 0.2 0.2 0.12

0.1 0.15 0.08

0.1 0.06

0.04 0.05 0.02

e e 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Figure 4.6: Maclaurin and Jacobi solutions for diﬀerent ratios ζ. The black line corresponds to the Maclaurin solution. The red line is the Jacobi solution.

bif min ζ q3 q3 5 0.0001 0.0024 4.25 10− × 4 0.001 0.00868 4.25 10− × 0.01 0.0338 0.00423 0.1 0.15 0.0478

Table 4.2: The two bifurcation points for the transition triaxial-prolate for diﬀerent values of ζ.

bif In ﬁgure 4.8 we plotted the angular velocity against q3. The bifurcation point, q3 , is where the prolate min curve intersects the corresponding triaxial case. q3 is represented by the left vertical line beyond which we have no solution. It can be nicely seen that with growing ζ the range of the triaxial solution becomes smaller until it vanishes for ζ = 0. The angular velocity is also not a smooth function as it was the case for the triaxial/oblate case. Indeed, it oscillates (allowing for the moment the physically excluded range min of Ω < 0) when approaching q3 .

4.3 Other eﬀects of Λ for non-rotating conﬁgurations

We can now derive other relevant quantities from the scalar Λ-virial theorem applied to homogeneous ellipsoidal conﬁgurations. In this section we will not consider rotating conﬁgurations, but systems with kinetic energy coming from internal motions. We will focus again on ellipsoidal, oblate and prolate, geometries. As in the preceding section the relevant quantity here is the function . A 4.3. OTHER EFFECTS OF Λ FOR NON-ROTATING CONFIGURATIONS 59

0.04 0.035 ζ = 0.001 0.01 ζ = 0.0001 0.03 0.025

2 2 0.02 q q 0.005 0.015 q2 = q2 (q3) q = q : Prolate 2 3 0.01 0.005 0 0 0 0.001 0.002 0 0.002 0.004 0.006 0.008

q3 q3

0.2 1

ζ = 0.01 0.8 ζ = 0.1 0.15

0.6 2 2

0.1 q q 0.4

0.05 0.2

0 0 0 0.01 0.02 0.03 0 0.04 0.08 0.12

q3 q3

Figure 4.7: Bifurcation point for prolate conﬁguration. The line q2 = q3 represents the prolate solution.

4.3.1 Mean mass-weighted rotational velocity

Not always the deviation from spherical symmetry guarantees that the effect of Λ on observables is sizable. This depends on the densities and the scales we want to compare. If we compare ρvac to densities, the quantities and defined in (3.17) and (3.19) are for flattened objects large enough to enhance the A Aii effect of Λ. If rΛ is combined with the Schwarzschild radius rs to give rmax as in (3.27), the result is still of astrophysical relevance. But if we had to compare one of the axes of an ellipsoid to rΛ (i.e. ai/rΛ) the effect would be negligible unless the extension ai is bigger than Mpc (clusters and superclusters) and the small ratio ai/rΛ is comparable to another small quantity of the same order of magnitude entering the equation under consideration. This happens for instance if we generalize a result ([49], chapter 4) on a mass-weighted mean-square rotation speed v0 of an ellipsoidal object to include Λ. Assume that due to symmetry properties of the object the only relevant components of the tensors in the tensor virial equations are xx and zz. We then obtain

1 grav 2Txx + Πxx + 3 Λ xx xx 1 I = |Wgrav|. (4.18) 2T + Π + Λ zz zz zz 3 Izz |W |

If the only motion is a rotation about the z axis we have Tzz = 0 and we can solve for Txx as

grav xx 1 1 2Txx = |Wgrav| Πzz + Λ zz Πxx + Λ xx . (4.19) zz 3 I − 3 I |W | Using equations (3.2) and (3.12), the quantities entering our equation can be parametrized in the following way:

1 1 2T = ρ v2 d3r = Mv2, Π = Mσ2, Π = (1 δ )Π . (4.20) xx 2 h φi 2 0 xx 0 zz − 0 xx Z 60 CHAPTER 4. ELLIPSOIDAL CONFIGURATIONS

W2 W2 2 ΠΡ Ζ = 0.0001 2 ΠΡ Ζ = 0.01 5 5

4 4

3 3

2 2

1 1

q q 0.001 0.002 0.003 0.004 3 0.002 0.004 0.006 0.008 3 W2 W2 2 ΠΡ Ζ = 0.01 2 ΠΡ Ζ = 0.1 2.5 1

2 0.8

1.5 0.6

1 0.4

0.5 0.2

q q 0.01 0.02 0.03 0.04 3 0.05 0.1 0.15 0.2 3

Figure 4.8: Angular velocity for prolate and triaxial solutions for diﬀerent ratios ζ.

where v0 is the mass weighted mean angular velocity, σ0 is the mass-weighted mean-square random velocity in the x direction 1 σ2 = (v u )2 = ρ (v u )2 d3r, (4.21) 0 h x − x iM M h x − x i ZV and δ measures the anisotropy in Π . If δ is of the order of one, it suffices to compare σ2 with Λ /3M. 0 ii 0 0 Ixx If both are of the same order of magnitude, the effect of Λ is non-negligible. Since v0 and σ0 are of the 2 same order of magnitude and v0 is non-relativistic, we can assume that σ0 < 10− . The quantity ΛIxx/3M 2 can be estimated to be (a1/rΛ) . Hence if ai 1Mpc, v0 (σ0) has to be truly non-relativistic and of the 6 ∼ the order of 10− to gain an appreciable effect of Λ. This improves if ai is one order of magnitude bigger 4 which is possible for large galaxy clusters. The velocities have to be then at most 10− . In these cases we 2 have to keep Λ and while solving (4.20) for v0 one has

Using almost the same set-up as above, we can use equation (3.20) to establish a relation between v0, the mass M and the geometry of the object which we choose below to be oblate. After straightforward algebra we obtain

1 1 ε T = grav (1 ε ) 1 Axx − z (4.23) xx 2|Wxx | − x − 1 ε Azz − x 4.3. OTHER EFFECTS OF Λ FOR NON-ROTATING CONFIGURATIONS 61

For oblate conﬁguration we get

v2 3 G M √1 e2 3 arcsin e 3 e2 1 δ 0 = N − − + − 0 1. (4.24) 2σ2 10 σ2 a3e2 e − √1 e2 1 e2 − 0 0 1 − − Since this relation is derived from (3.20) which in turn is based on the assumption of Λ = 0 it is only valid 6 for non-zero cosmological constant although the latter does not enter the expression.

4.3.2 Critical mass Consider the stability criteria for a homogeneous cloud with mass M and internal mean velocity v in h i mechanical equilibrium with the background with pressure p . The system will collapse under it’s own gravity (p < 0) if its mass is greater than a critical mass Mc. With cosmological constant, this critical mass is increased with respect to its value Mc0 when Λ = 0, which is expected, since now there is an external force associated to Λ that acts against Newtonian gravity and hence the collapse can be postponed. By using the scalar virial theorem (3.21) and setting p = 0 as the criterion for the onset of instability, we can write

1 2 ρvac − 2 v Mc0 Mc = Mc0 1 , Mc0 = h i , ρc0 = . (4.25) − A ρc0 ˜ N V |W | This expression is valid for any geometry. However, we pointed out already that spherical symmetry implies = 2 and the eﬀect is suppressed. Some numerical values can be considered by writing v2 = A h i 3k T/m for a hydrogen cloud with T 500 K and radius R 10pc. One then has B p ≈ ≈

ρvac 8 10− , (4.26) A ρ ≈ c0 which represents a very tiny correction to the critical mass Mc0 for a spherically symmetric object. On the other hand, for ellipsoidal configurations with the same temperature, we have for oblate and prolate configurations respectively: 2 ρvac 5 a δM = 5 10− , (4.27) Aobl ρ ≈ × kpc c0 2 ρvac 1 c δM = 10− , Apro ρ ≈ kpc c0 where these expression are written in the e 1 approximation . We have set c = 10a for the prolate case → and Ω = 0.7. For an oblate ellipsoid with a 50 kpc, one has δM 0.15 while for the prolate ellipsoid vac ≈ ≈ with c 50 kpc, δM 102. ≈ ≈ 4.3.3 Mean velocity and Mass-Temperature relation By virtue of the scalar Λ-virial theorem, we can also write down the mass-temperature relation for an astrophysical structure. Note first that using = 1 M v2 in equation (3.21) we have for Λ = 0 the K 2 h i standard expression for the mean velocity

N ρ2G ˜ N v2 = |W | = N |W |, (4.28) h iΛ=0 M M where the second equality applies to the constant density case. Clearly, with Λ = 0 the mean velocity cannot become zero. Let us contrast it to the case with Λ > 0. One obtains ρ2G ˜ N 1 v2 = N|W | 1 ζ . (4.29) h i 2M − 2 A As can been seen from (4.29) the mean velocity in the ellipsoidal configurations is decreased because of the Λ-external force. A drastic effect of the cosmological constant could be reached for the geometrical 62 CHAPTER 4. ELLIPSOIDAL CONFIGURATIONS factor approaching the critical value = ρ/ρ which is possible for flat objects. In the extreme A Acrit vac the mean velocity can go to zero. Since the mean velocity is proportional to the temperature, ρ = M/V and N and are geometrical parameters, equation (4.29) represents also a mass-temperature relation. W A We can write the equation (4.28) for a cosmological structure, say a galactic cluster, by writing its density as resulting from a perturbation δρ from the background density of the universe ρb(t) as ρgc = ρb(t)(1 + δ(t)), where δ(t) = δρ(t)/ρb(t) (see equation (2.35)). Equation (4.29) is written for the temperature of the cluster as

mp 3 2 Ωvac 1 T = (1 + δ(t))a(t)− Ω H (e) 1 (e) (1 + δ(t))− , (4.30) 10k b 0 F − A 1 Ω B − vac where the geometrical factor (e) is given as F 2 1√ 2 1+e a1e− 1 e ln 1 e Prolate (e) = − − (4.31) 2 1 2 F (a e− √1 e arcsin e Oblate. 1 −

Equation (4.30) assumes Ωmatter + Ωvac = 1. This result is a generalization of a result derived in [50] valid for spherical geometry. In that case one recovers the typical mass temperature relation T M 2/3 ∝ mantaining ρ constant. Although this has the same dependence as in equation (3.31), the meaning is different since (4.30) for = 2 computes the temperature of a certain galactic cluster at some redshift A given its mass while equation (3.31) is associated to a reversible process where a configuration passes from T = 0 to some final T? through states of virial equilibrium maintaining a constant mass.

4.4 Remarks

In this chapter we have shoewn that the effects of the cosmological constant can greatly enhanced as long as the cosmological and astrophysical configurations deviate from spherical geometry, since the enhacement factor has become a function of the eccentricities of ellipsoidal configurations. We have also determined A that the allowed configurations associated to rotating ellipsoids are contrain by the cosmological constant, and we have shown that new configurations are allowed, namely, triaxial homogeneous ellipsoids with rotation along the major axis. Such rare behavior predcted from the virial equation with cosmological constant have been detected in some galaxies as NGC4640. [75], and hence, the main contribution of what we have described in this chapter corresponds to a first theoretical background that can be used in order to understand such configurations and to test some models of dark energy. CHAPTER 5

Scales set by the cosmological constant

As shown in chapter 2, the cosmological constant set scales of distance and length which coincides with the order of magnitude with the size and the age of the universe, in the so called coincidence problem. We also showed how the cosmological constant enters in the Newtonian limit as an harmonic-like potential r2 defining the Newton-Hooke space time. We mentioned that this limit is only valid in certain ranges ∝ of masses and distances given in equation (2.30). In the third chapter we showed that the appearance of Λ implies that an astrophysical configuration can be in equilibrium only if its mass and radius are greater than the expected values for Λ = 0. In this chapter we study the same effects but now we concentrate on the motion of a test particle in the generalized Schwarszchild space time with cosmological constant. Here we will derive another scale defined in terms of the cosmological constant which may have a great relevance at astrophysical scales. We will also show that the introduction of the cosmological constant changes considerably the motion of test particles that traveling cosmological scales [80].

5.1 Dynamics in the Schwarszchild-de Sitter spacetime

Let us consider an spherically symmetric and static configuration with mass M. The solution to the Einstein field equations with cosmological constant for r > R (the radius of the configuration) is given from Birkoff’s theorem as the Swcharszchild-de Sitter metric (see appendix C):

2 ν(r) 2 ν(r) 2 2 2 2 2 2 ds = e dt + e− dr + r dθ + r sin θdφ , (5.1) − where the function ν(r) is deﬁned as

2 ν(r) 2rs r e = 1 2 , (5.2) − r − 3RΛ with rs = M and RΛ defined in (2.10). One would suspect that the inclusion of Λ is irrelevant in this setting. Note, however, that there are two scales involved, the Schwarszchild radius Rs = 2rs and RΛ. We will show that a (justified) combination of the two can lead to a new relevant scale. To start, let us recall that a spherically symmetric configuration has two killing vectors k¯t = ∂t and k¯φ = ∂φ which represents conserved quantities: for the metric (5.1), these quantities are dt dφ g uµkν = eν(r) , L g uµkν = r2 , (5.3) E ≡ µν t dτ ≡ µν φ dτ where uµ is the four-velocity and the affine parameter τ is the proper time. This conserved quantities should not be confused with energy and angular momentum. The equation of motion for a test particle

63 64 CHAPTER 5. SCALES SET BY THE COSMOLOGICAL CONSTANT

L=Lcrit

L>Lcrit

−13 −0.045 0 0.1 0.2 0 5 10 15 20 r (arbitrary units) r (arbitrary units)

Figure 5.1: General form of the eﬀective potential and its shapes for diﬀerent values of rL = L in a spacetime described by the line element (5.1) can then be written as

1 dr 2 C = + U (r), (5.4) 2 dτ eff where the constant C is what really plays the role of energy in the Newtonian sense. Its defined in terms of conserved quantities as 1 L2Λ C = 2 + 1 . 2 E 3 − The function Ueff(r) corresponds to the analog to the effective potential of Newtonian dynamics, defined by

2 2 2 rs 1 r L rsL Ueﬀ(r) = 2 + 2 3 , (5.5) − r − 6 RΛ 2r − r 1 which contains the contribution from the Newtonian potential r− , the harmonic like potential due to the 2 2 3 cosmological constant r , the centrifugal barrier r− and the general relativistic contribution r− . The ∼ shape of the eﬀective potential is shown in Figure 5.1, from where we can see that the introduction of the cosmological constant changes considerably the shape of the potential by displaying a maximum and hence allows unstable circular orbits at cosmological scales.

5.2 Radial motion

We now concentrate on purely radial motion (L = 0) of test particles traveling over astrophysical or/and cosmological scales. The first detail we want to point is that from the definition of C, we have that is E real if 1 C > . (5.6) −2 This simples inequality will play a relevant role in future derivations. For now it is enough to point that in the limiting value C = 1 , we have = 0 which signals an artifact of the Schwarzschild coordinates, − 2 E 5.2. RADIAL MOTION 65 that is, where the function eν(r) vanishes. For Λ = 0, one define a horizon at the Schwarzschild radius Rs = 2M, but with cosmological constant we can introduce a second horizon. These horizons, located at (1) (2) r = r? and r = r? are then solutions of the cubic equation

y3 3y + 6λ = 0, − where y r /r , and where the parameter λ was deﬁned in (3.25). We write it here again: ≡ ? Λ

rs 23 M 1/2 λ = = 1.94 10− h70Ωvac . RΛ × M The cubic equation has two positive solutions which can be found by using machinery described in appendix D. The solutions for y leads approximately to

1 r 2 r(1) = √3R 1 s r , (5.7) ? Λ − 2 R − s " Λ # 4 r 2 r(2) = 2r 1 + s . ? s 3 R " Λ # In other words, the condition C = 1/2 is satisﬁed at the Schwarzschild radius and the distance scale − imposed by the cosmological constant, which is, according to the coincidence problem, of the same order of magnitude than the actual horizon (edge of the observed universe).

5.2.1 Astrophysical scale set by Λ With the same limiting value of C = 1/2, we have U (r ) = 1 . Motion with U (r) 1 becomes − | eﬀ ? | 2 | eﬀ | ≥ 2 unphysical since it corresponds to allow the motion of test particles inside the Schwarzschild radius and beyond the observed universe, which would imply that we may receive superluminous information. Hence, the particles are allowed to be at some r such that

rs < r < √3RΛ, with C < U (r) < 1 (for negative C and U ). The effective potential for radial motion reduces to | | | eff | 2 eff 2 rs 1 r Ueff (r) = 2 . − r − 6 RΛ It is clear that at certain distance, the terms r /r and r2/R2 will become comparable leading to a local − s Λ maximum (ass seen in Figure 5.1) located at

1/3 2 1/3 5 M 2/3 1/3 r = 3r R = 9.5 10− h− Ω− Mpc (5.8) max s Λ × M 70 vac Note that this expression coincides with the largest radius for an spherical configuration in virial equilibrium given in equation (3.27), althought their meanings differ slightly. The effective potential at this local maximum is given by

2/3 16 M 2/3 1/3 U (r ) = 7.51 10− h Ω . (5.9) eﬀ max − × M 70 vac

Beyond rmax, Ueff is a continuously decreasing function. This implies that rmax is the maximum value within we can find a bound solutions for the orbit of a test body around a symmetric and static configuration. The behavior of the effective potential at moderate distances is governed by the scale rs (and L in general). At large distances, rs and RΛ take over till finally at cosmological distances only RΛ is of relevance. Therefore, in a good approximation, (5.8) and (5.9) are also valid for most of the cases of moderate L = 0. 6 66 CHAPTER 5. SCALES SET BY THE COSMOLOGICAL CONSTANT

Mean component M/M rmax/α [pc] Structures

Stars 1 75 Globular cluster Globular cluster 106 7.5 103 Galaxy × Galaxy 1011 3.5 105 Galactic cluster × Galactic cluster 1013 1.6 106 Superclusters ×

Table 5.1: Scales set by the cosmological constant and the Schwarszchild’s radius

We now make an attempt to find the astrophysical meaning of the new scale (5.8). Consider the following chain of matter conglomeration of astrophysical objects: the smaller are star clusters (globular and open) with stars as members (M = M ) and a mass of 106M . We proceed to galaxies and galactic clusters. 2/3 1/3 Within this chain, we find for rmax some numerical values given in table (5.1), with α = h0− Ωv−ac . The value at the first line is of the order of magnitude of the tidal radius of globular clusters [49]. The second line agrees with the extension of an average galaxy. The next two values are about the size of a galaxy cluster. The value 1013M corresponds to a giant elliptic galaxy encountered often at the center of the clusters. Hence, RΛ in combination with rs gives us a surprisingly accurate and natural astrophysical 2 1/3 scales. The combination rmax = (3rsRΛ) from which these scales where calculated is not an arbitrary combination with length dimension, but it is the distance beyond we cannot find bound orbits,as pointed before. Therefore we would expect that rmax sets a relevant astrophysical scale as demonstrated in table 5.1. Of course, we are talking here about scales neglecting dynamical aspects of many body interactions, but no doubt rmax is roughly the scale to be set for bound systems. Indeed, the agreement of the result in table (5.1) with values encountered in nature is striking. The second line in table 5.1 requires some comments. Unlike the other three ones, where the argument M of rmax has been taken to be the mass of the average members of the astrophysical object, it might appear unjustified to take the mass of the globular cluster to obtain the extension of the galaxy. However, in view of the fact that globular clusters are very old objects and are thought to be of importance in the formation of the galaxy, this choice seems justified. Indeed, with Λ > 0, our result strengthens the believe that globular clusters are relics of the formation of the galaxy. For instance, rmax for open star clusters with a mass M = 250M is only 0.5 kpc.

5.2.2 Constrains in the velocities of test particles The above scales still do not complete the full picture. We can also gain some insight into the velocity of particles traveling over Mpc distances. To start, note that particles starting beyond the astrophysical scale rmax with initial values such that

C(r0, v0) < Ueﬀ (rmax) < 0, (5.10) do not reach our galaxy due to the potential barrier beyond rmax. To determine what this exactly means, let us consider a test particle with radial motion and calculate C in terms of the initial (coordinate) velocity v0 and radial coordinate r0 > rmax . From (5.3) and (5.4) we have

2 ν(r0) v0 + 2e Ueff (r0) C = C(v0, r0) = . (5.11) 2 e2ν(r0) v2 − 0 Combining this with inequalit y (5.10)we can solve for the initial velocity. We also use the condition C < 0, which can be achieved in two different situations, namely, the numerator in (5.11) is positive and the denominator is negative (case (i)) and vice versa (case (ii)). In analyzing this situation the power and importance of the inequality (5.6) i.e. C 1/2 together with U (r) 1/2 will become apparent. ≥ − eff ≥ − Let us first establish a hierarchy of some quantities involved in the study. Defining

Ξ(r) < 2e2ν(r) U (r) < e2ν(r). | eﬀ | In the case (i) described by the inequality

v2 > e2ν(r0) > 2 U (r ) e2ν(r0), 0 | eﬀ 0 | we obtain a solution for the initial velocity from (5.10):

Ξ(r ) < 2e2ν(ro) U (r ) < e2ν(r0) < v2. 0 | eﬀ 0 | 0

However, the condition C(v0, r0) 1/2 implies in this case Ueff (r0) 1/2 which would require that (1)| | ≤ (2) | | ≥ our r0 is bigger than r? = √3rΛ or smaller than r? = 2rs. Therefore it might appear that the case (i) is unphysical. To see that this is not so, it is best to consider the problem at hand without the potential (2) (1) barrier posed by Λ i.e. without equation (5.10). If satisfying r? < r < r? (equivalent to Ueff (r) 1/2) ν(r0) | | ≤ we insist on an initial value v0 such that v0 > e for an arbitrary r0, we will violate the fundamental inequality (5.6) i.e. C 1/2. For, choosing C > 0 we automatically get ≥ − v2 < 2e2ν(r0) U (r ) < e2ν(r0), 0 | eff 0 | violating the assumption. If C < 0, then C 1/2 leads to U (r ) 1/2 violating again one of the | | ≤ | eff 0 | ≥ restrictions. Hence for any r0 which we parametrize as r0 = ζrmax, the quantity

1/3 2/3 ν(r ) 8 2/3 15 M 2/3 1/3 v (ζ) = e 0 = 1 λ f(ζ) = 1 1 10− h Ω f(ζ), (5.12) max − 3 − × M 70 vac with the function f(ζ) deﬁned as

2 + ζ3 f(ζ) , ≡ 2ζ represents the maximal value of the initial velocity which we can choose at any r0. This statement is independent of Λ and hinges solely on the conditions C 1/2 and U (r) 1/2. As long as we are ≥ − eff ≥ − above the Schwarzschild radius and below rΛ, this statement is not an artifact of the coordinate system as the condition (5.6) is valid throughout the coordinate system. At least for the simple case of a one-body problem in a central gravitational field, we can then say that there is not only a energy cut-off on high energy cosmic rays traveling over Mpc distances due to the resonant process p+γ ∆ πN [81], but CMB → → also a general relativistic one on the initial energies. This is unusual in the sense that one is accustomed to the freedom of choosing any possible initial condition. It is, however, clear that the restriction (5.6) does not admit all possible velocities v0. As said above, the existence of vmax is independent of Λ. But Λ has an 2 1/2 effect on the functional form of v . It is easy to see from (5.12) that v and γ(ζ) = (1 v (ζ) )− max max − max have a local maximum due to the ζ2 term in f(ζ) which comes from the contribution of the cosmological constant to Ueff . In Figure 5.2 we plot the relativistic factor γ(ζ) versus ζ with and without Λ. Let us 2 now come to the effect of the cosmological constant i.e. to the task of resolving (5.10) with respect to v0 according to the sub-case (ii):

v2 < 2 U (r ) e2ν(r0) < e2ν(r0). 0 | eﬀ 0 | From (5.10) we obtain

v2 < Ξ(r ) < 2e2ν(r0) U (r ) < e2ν(r0). 0 0 | eﬀ 0 | 68 CHAPTER 5. SCALES SET BY THE COSMOLOGICAL CONSTANT

6000

5000 mass ratio = 1012

4000

with Λ Λ γ 3000 without

2000

1000

0 0 1 2 ζ 3 4 5

2 −1/2 Figure 5.2: The relativistic factor γ = (1 − vmax) as a function of ζ = r0/rmax with and without Λ. The mass ratio 12 M/M is 10 .

This deﬁnes a minimum velocity

1 1/6 8 1/3 2f(ζ) 3 v (ζ) Ξ(r ) = λ1/3 1 λ2/3f(ζ) − , (5.13) min ≡ 0 3 − 3 1 (3λ)2/3 " # s p − which can be approximate for λ 1 to 1/3 8 M 1/3 1/6 v (ζ) 2.24 10− 2f(ζ) 3 h Ω . min ≈ × − M 70 vac p This implies that particles whose initial velocity is smaller than vmin do not reach the central object with mass M due to the potential barrier caused by Λ, (where we have used λ 1 for the second line). This means among other things, that cosmic rays have also a lower cut-oﬀ on energy when traveling over Mpc distances in space-times with a positive Λ. Note that the results applies equally to any kind test particle: asteroids, comets, rockets etc. For ζ 1 (again with λ 1), v (ζ) grows with the distance as → min v (ζ) λ1/3(ζ 1). min ∝ − But this is a relative statement since small velocities for cosmic rays are not necessarily called small for macroscopic objects. To see how vmin behaves for large r0 i.e large ζ, we plotted in Figure 5.3 this quantity together with vmax against ζ. Almost for all reasonable values of M, vmin has a local maximum at a ζ value corresponding to rΛ with vmin(rΛ) = 4/27 = 0.385. Clearly this maximum is presently at the edge of the universe in which we live. This is connected to the coincidence problem in the sense that at p later epochs, when the universe is much larger, this will not be the case anymore. The region between vmax and vmin is the region of allowed velocities such that the test particle can reach the central object with mass M. Summarizing the above, there are three distinguished regions of possible initial velocity. Due to the re- 2 2ν(r0) striction C 1/2 the region v0 > e is not accessible. Only, if the initial velocity lies in the range ≥ − ν(r0) Ξ(r0) < v0 < e , they can be detected. If Ξ(r0) > v0 the test particle does not reach the central object. In our estimate, we neglect the fact that i.e. many galaxies is of spiral type and not spherically p p symmetric. This however, does not play a role , since r0 is of the orders of magnitude of Mpc and we consider here the problem of particles transversing Mpc distances. We do not consider here the details of what happens close to the galaxy. It is instructive to compare the above results with ones we would obtain in non-relativistic mechanics 5.3. MOTION WITH ANGULAR MOMENTUM 69

vmax 1 vmin (classical)

vmin

mass ratio = 1012

0.5

r0=rΛ

0 1 1000 2000 ζ 3000 4000

Figure 5.3: vmax and vmin versus ζ for large distances r0 ≥ rmax i.e. ζ ≥ 1. vmin(classical) is the velocity emerging in classical non-relativistic mechanics with Λ. The arrow indicates that the local maximum occurs at rΛ. working in the Newtonian limit with Λ. Let us first point out the similarities between the general relativistic approach and the the one in classical mechanics. For L = 0, the effective potential is the same in both cases which results in the same rmax and Ueff (rmax). In classical mechanics the equation of motion is similar, but not equivalent to (5.4). The constant C has now the meaning of (non-relativistic) energy E over mass and the affine parameter is time. Imposing the condition E/m < Ueff (rmax) corresponding to (5.10) gives us v2 = 2( U (r ) U (r ) ), which grows unlimited unlike the general relativistic min | eff 0 | − | eff max | expression (see Figure 5.3). The difference to general relativity is twofold. First, C is not energy. In addition we have the constraint C 1/2 which is absent in classical mechanics. Of course, by hand we ≥ − could restrict the velocity to account for the fact that in classical mechanics we are in the non-relativistic realm. But this would simply signal an inadequacy of classical mechanics. Secondly, formally the constant C is also included in the derivative with respect to the affine parameter which makes the relation between C and v2 nonlinear (see equation (5.11)) unlike the relation between E and v2 in classical mechanics. In as far, our result on the existence on vmax is a genuine general relativistic result. Similarly the largest value of v 0.4 is due to general relativistic effect. min ∼

5.3 Motion with angular momentum

The full case of non-zero angular momentum i.e. L = 0 requires more complicated calculations since we 6 handle with higher order polynomial equations, and hence only specific situations can be solved analytically. In this section we will concentrate on the maximally possible L, due to Λ, such that bound orbits beyond the Schwarzschild radius are still possible. For Λ = 0, it is well known that there exist a minimum value for the conserved quantity L such that the condition L > L 2√3r must be satisfied to ensure bound min ≡ s orbits. The last circular orbit in a Schwarszchild spacetime is reached when L = Lmin and its located at r = 3r . If r L r this will not change much in the presence of the cosmological constant. s Λ s Qualitatively, for L = 0 and Λ > 0 the functional form of U is as follows. Depending on the values of 6 eff the parameters, there can be three local extrema: first we have a maximum close to rs (unstable circular orbit). Then we find a minimum associated to bound orbits, which is followed by a maximum due to Λ (see Figure 5.1) We will show now that due to Λ there exist also a maximal critical value Lmax such that bound orbits with strictly r > rs are only possible if L < Lmax. The value of L = Lmax correspond to the situation when the local maximum due to Λ and the standard local minimum fall together in a saddle 70 CHAPTER 5. SCALES SET BY THE COSMOLOGICAL CONSTANT point. To find this point, we require that the first and the second derivatives of the effective potential vanish at r = r . From (5.5), these conditions are written as ∗ 5 2 2 2 2 2 2 5 2 2 2 2 2 2 r = 3rsRΛr 3RΛL r + 9RΛL rs, r = 6rsRΛr + 9RΛL r 36RΛL rs, (5.14) ∗ ∗ − ∗ ∗ − ∗ ∗ − These expressions can be combined in a quadratic equation for r . The solutions for the resulting quadratic ∗ equation are for L = Lmax:

2L2 45 r 2 r = max 1 1 s . (5.15) ∗ 3r  − 4 L  s s max   Note that there is not any reference of the cosmological constant term, but it must be clear that (5.15) is only valid if Λ = 0. Both solutions described in (5.13) are positive, but one of them must be discarted. If 6 we assume L r , then we have max s 2 (1) 4Lmax (2) 15 r Rs, r rs. (5.16) ∗ ≈ 3rs ∗ ≈ 2 The smallest root is of the order of the Schwarszchild radius. Hence we consider the largest root r(1), ∗ which represents a cosmological scale. We then can replace this value in the condition dU/dr = 0 (the ﬁrst equation in (5.14)) and solve for a critical value of angular momentum Lmax. The resulting equation is a eighth-order equation for Lmax, which can be converted in a fourth order equation: 3r 4 3r 6 L 2 δ4 s δ 12 s = 0, δ max . (5.17) max − 4R max − 4R max ≡ R Λ Λ Λ Under the assumption L r , the only real and positive root for this fourth order equation gives us max s a simple expression for Lmax (see Appendix D for explicit calculation):

2/3 2/3 3 2 1/3 12 M 1/3 1/6 L = r R = 1.45 10− h− Ω− Mpc. (5.18) max 4 s Λ × M 70 vac Replacing this value in (5.16), we ﬁnd that the last circular orbit allowed by the potential (5.12) when 1 L = Lmax is located at

1 3 1/3 3 1/3 1 M 2/3 2 3 5 1/3 r = rsRΛ = 4− rmax 5.9 10− h70− Ωv−ac Mpc. ∗ 4 ≈ × M 1/3 1/6 The values of Lmax in astronomical units for different masses are given in table 5.2, with α˜ = h70− Ωv−ac . These values are not very large. Indeed, as in the other scales this comes out because we combine a big scale r with a smaller scale r . Note in this context that only on dimensional grounds L r where r Λ s ∝ 0 0 is the initial value of the radial component which can make Lmax relevant for astrophysical objects, even for the solar system. Since L vanishes if the perpendicular velocity v⊥ is zero, we know that L v⊥r . ∝ 0 0 Therefore the relevance of L will grow for relativistic velocities such that L (r ). max ∼ O 0 We can solve for the angular momentum in tertms of the lenght scale RΛ. Consider again the equation (5.3). We will follow now the same analysis done in the first part of the chapter for a more general case, that is, considering angular momentum. In this case, the constant C is written as 1 1 r2 C = 2 + L 1 . 2 E 3 R2 − Λ Hence, by using the first equation of (5.3) together with (5.5), we can solve for this constant as

2 1 L2 2ν(r) 1 r˙ 1 3 R2 2 Ueﬀ(r) e C = − Λ − | | . 2  e2ν(r) r˙2  −   1Note that the solution (5.18) can be simple obtained in if neglect the last term of (5.17), and it is the same if we replace (5.16) in the second expression of (5.14) 5.3. MOTION WITH ANGULAR MOMENTUM 71

M/M Lmax/α˜ (A.U)

1 0.3 106 649 1011 1.3 106 × 1013 3 107 ×

Table 5.2: Maximum angular momentum for stable orbits in astronomical units.

On the other hand, we can proceed in the same way from the second equation of (5.3) to obtain

1 L2e2ν L2 C = + 1 . 2 4 ˙2 3R2 − r φ Λ Combining these two expressions we can solve for the angular momentum as

1/2 eν(r) L = r2φ˙ . e2ν(r) r2φ˙2 r˙2 − − Circular orbits with r = R are allowed as far as the the inequality eν(R) > Rφ˙ is satisﬁed, that is

3 2 1 R RΛ > . 3 R R2φ˙ 2M − − This inequality can be written in terms of time, by writing φ˙ = 2π/T , being T the period of translation; ν(R) we then may write T > 2πe− R. These expressions lead to a precise tool for determining the value of the cosmological constant, once the parameters of a given orbit are measuered. To see this clearer, we pay small attemption on the effects on small scales. Such effects have been widely explored (see [82, 83]). From the numerical values of the scales set by Λ, it is clear that the effects of Λ are almost negligible and no bound can be set from observations at such scales, as the solar system, even with the latest data from precession of perihelia [84]. To see this, let us consider the equation of motion for a test particle orbiting around an static symmetric configuration, which can be easily determined from (5.4) with the help of (5.3) and (5.2). The resulting differential equation whose solution is the trayectory of the test particle r(φ) can be written as

2 2 du 2 1 δ 3 + u = 1 + 3u u− , dφ2 − 3 4 where the variable u and the parameters and δ are deﬁned as usual:

2 L 1 rs L u r− , , δ . ≡ r ≡ L ≡ R s Λ In order to explore the eﬀects on small scales, one can make a Taylor expansion around a circular orbit located at u0 so that u = u0 + x. To the ﬁrst order in x, the equation of motion can be reduced to a harmonic form

2 2 2 d x δ 4 1 δ 3 + ηx = ξ, η = 1 6u u− , ξ 1 u (1 3u ) u− , dφ2 − 0 − 4 0 ≡ − 0 − 0 − 3 2 0

The solution for the radius r can then be written as the equation for an elliptical trayectory with precession 1 given as r(φ) = r0(1 + e cos √ηφ)− , where the parameter of the orbit r0 and the eccentricity are given by

ηL2 A r0 = , e = , (5.19) rs(ηu0 + ξ) ηu0 + ξ 72 CHAPTER 5. SCALES SET BY THE COSMOLOGICAL CONSTANT with the constant A must be ﬁxed by initial conditions. Equation (5.19) then implies that the orbit of a test particle is stable under small perturbations if η > 0. This can be then translated to a bound for the cosmological constant as

1 L2 r L2 Ω < 1 6 s ρ . vac 3 H2R4 − R ⇒ vac R4 0 0 0 0 where the parameters L and R correspond to the angular momentum and the radius of the circular orbit. This expression also allows us to determine the range allowed for the angular momentum in oder to ﬁnd a stable circular orbit with cosmological constant with the help of equation (5.18). We get

2/3 1 2 1 2 1 3 3 3 (8π)− 2 R R− . L Rs R (5.20) 0 Λ ≤ 2 Λ

The values for Lmax are shown in Fig 5.2, while the values of Lmin depend on the size of the orbit, and is very small for local systems as the solar system. Finally, it worths to mention that from equation (5.19) one can also determine the rate of precession for the perihelia Ω˙ for planets, say, in the solar system. With the solution for r(φ), the rate of precession is given by Ω˙ pres = (2π/T ) √η 1 , where T is the period of rotation; in the non-relativistic limit (that − is, neglecting the r 3 term in the eﬀective potential), the precession of perihelia due to Λ takes the form − Ω˙ Λ/φ˙. This, in principle, would lead to a bound on the cosmological constant, once the rates Ω˙ pres ∼ pres are measured. For instance, for mercury one has Ω˙ = 0.0036 0.0050 arcseconds per century [85], pres which lead to a bound for ρ such that 1 1011 < Ω < 2 1010, which clearly is a useless bound vac − × vac × when compared with astrophysical data which yield Ω 0.7. vac ∼ 5.4 Remarks

In this chapter we have shown that the scales set by the cosmological constant an be combined with astrophysical scale in a non arbitrary way in order to generate scales with astrophysical relevance. This eﬀects has been explored from the motion of a test particle in a space time generated by a spherical and static conﬁguration, represented through the Schwarszchild -de Sitter line element, where the cosmological constant enters as a potential barrier which leads to limits in the velocities of such particles traveling over astrophysical distances in order to be detected in th earth, for instance. The upper and lower limits imposed for the velocities open a possibility to consider realistic problems in astrophysics (for example, the gamma ray problem), since these bounds implies also bounds on energies which could range from non relativistic to ultra relativistic particles. The results showed in this chapter implies that the cosmological constant certainly sets the length scales for astrophysical and cosmological systems, which in principle could be interpreted as an extension of the coincidence conjecture. CHAPTER 6

Astrophysical bounds on the cosmological constant

In this chapter we will set some bounds for the cosmological constant derived from astrophysical considerations. In fact, in chapter 3 we wrote a ﬁrst inequality with Λ (equation (3.18)), and we interpreted it as a restriction of the densities. However, such inequality can be also written as a bound for the cosmological constant. These bounds can be derived not only from the Newtonian limit but also from a general relativistic treatment of hydrostatical equilibrium.

6.1 Bounds from the Newtonian limit

As pointed in chapter 2 and 3, the hydrostatic equilibrium is represented by Euler’s equation in absence of ﬂuid motions (see 3.13). For a sphericall symmetric conﬁguration, with the help of equation (2.45), Euler’s equation reduces to the basic equation of Newtonian stellar structure:

dp(r) ρ(r)M(r) 1 r = + rρ(r)Λ, M(r) = 4πs2ρ(s)ds (6.1) dr − r2 3 Z0 where we have used (2.45) to deﬁne the mass inside the radius r. This can be written in a familiar way as function of the mean density ρ¯(r) = 3M(r)/4πr3 (ρ¯ = ρ and ρ¯(r) ρ(r) r) as c c ≥ ∀ dp(r) ρ(r)M(r) ρ = 1 2 vac . (6.2) dr − r2 − ρ¯(r)

Once again we recognize the term ρvac/ρ, which appeared first in the virial equation and then in the ΛLE equation (3.35). We now can determine the same inequality (3.18). In this case, one requires that for a reasonable astrophysical configuration, the pressure is a decreasing function of the radius. This implies ρ¯(r) 2ρ r. This is the same bound imposed from the virial equation by demanding that the kinetic ≥ vac ∀ energy is a definite positive quantity > 0, while (6.3) have been derived from the physical condition K p0 < 0 for realistic configurations. The lower value that the mean density may take is reached at the boundary ρ¯(r = R) = ρ¯b. Hence, the lower bound can be set as

2ρ ρ¯ , Λ 4πρ¯ . (6.3) vac ≤ b ≤ b

For ρ¯ = 2ρvac one sees that the pressure must be constant. The Einstein universe is a good example of this situation. It worths mention that in chapter 3 we derived an inequality for the mena density of a polytropic conﬁg- uration in terms of the cosmological constant. This is no more than an astrophysical bound on Λ, which

73 74 CHAPTER 6. ASTROPHYSICAL BOUNDS ON THE COSMOLOGICAL CONSTANT reads as 1 4 − ρ < (n + 1)f 2 ρ¯ , vac 3 1 − b where n is the polytropic index, f is a function of ζ 2ρ /ρ deﬁned in (3.48) and ρ¯ is the mean 1 c ≡ vac c b density at the boundary of the conﬁguration.

6.2 Bounds from general relativity

Hydrostatic equilibrium for a spherical symmetric configuration can be derived from Einstein field equations by considering the content of matter and energy described as a perfect fluid. Birkoff’s theorem states that the solution for the field equations is given by the Schwarszchilds metric, from which one can derive the Λ-Tolman-Oppenhaimer-Volktoff (ΛTOV) equation (see appendix C)

dp 1 12πpr3 + 3M(r) 8πρ r3 = (ρ + p) − vac . (6.4) 2 M(r) 8 2 dr −3r 1 2 πρvacr ! − r − 3 with M(r) given as in (6.1). This is the general relativistic version of Euler’s equation, which is recovered when p ρ. As in the Newtonian picture, the complete description is reached with an equation of state p(ρ). As pointed before, in realistic configurations the pressure is a monotonically decreasing function of the radius. This is reached if both the numerator and denomiator have the same sign. For a positive numerator one has p(r) + ρ¯(r) > 2ρvac. Hence, the denominator must be positive, i.e, 8 8 1 πρr¯ 2 πρ r2 > 0 (6.5) − 3 − 3 vac It can be shown that for a density which is a monotonically decreasing function of the radius, the following inequality is satisfied [42, 86, 87, 88] 3p(r) + ρ¯(r) 2ρ 3p(r) + ρ¯(r) 2ρ y(r) η(r) − vac − vac r. (6.6) ≥ 3p + ρ 2ρ ≥ 3p + ρ 2ρ ∀ c c − vac c c − vac The quantities y(r) and η(r), defined by

1 4 2 p(r) dp y(r) 1 πρr2 2 + ζ¯ > 0, η(r) exp , (6.7) ≡ − 3 ≡ − p + ρ(p) " Zpc # are the so called Buchdahl variables [88]. In terms of these functions, the Schwarszchild’s solution for r < R reads as ds2 = η(r)2dt2 + y(r)2dr2 + r2dΩ2. Note that the variable y is well deﬁned up to − the boundary (where p = 0) if the remaining terms of the numerator satisfy 2ρvac < ρ¯b. Therefore, the Newtonian upper bound for the cosmological constant given in (6.3) could exactly be reproduced in the general relativistic case.

Solutions with constant density Let us consider the non-physical solution of the ΛTOV equation for an spherical homogeneous configuration. In the first place, let us assume that an equation of state is prescribed, for instance, a barotropic γ equation of state p = κρ . In this case we have p0(r) = 0, that is, we also have a constant pressure (we can take this as an approximation in the sense that p0(r) 0 for a slowly varying density or pressure ≈ profile). Using this in equation (6.4) one has

ρ + 3κργ 2ρ = 0. (6.8) − vac Note that with vanishing cosmological constant Λ = 0, there would not be any acceptable solution for a positive energy density. Thus, we can say that in the presence of ρvac, a system with constant density 6.2. BOUNDS FROM GENERAL RELATIVITY 75 and constant pressure can be in hydrostatical equilibrium. In the case with Λ = 0, such a system could not be in hydrostatical equilibrium: sooner or later the system will collapse under it’s own gravity. From equation (6.8) we then can find the density of the configuration as a function of ρvac and parametrized with the polytropic parameters ρ = ρ(ρvac; κ, γ). To illustrate the consequence of equation (6.8) let us assume that γ = 1. It then follows that an astrophysical object with ρ ρ (the critical density defined ∼ crit in (2.6)) is stable. Such stable objects would have the lowest possible density among stable astrophysical objects and could be e.g. superclusters.

Let us now assume that no equation of state is speciﬁed. For constant density, the equation (6.4) can be written as

1 dp 1 4 − = 4πr(ρ + p(r)) p(r) + ρ(1 ζ) 1 πρr2(2 + ζ) , (6.9) dr − 3 − − 3 where still ζ = 2ρvac/ρ. After inegration we may write the pressure as

(1 ζ) (p + ρ[1 y(r)]) 3p y(r) p(r) = ρ − c − − c , (6.10) 3p (y(r) 1) + ρ(y(r)[1 ζ] 3) c − − − where the function y(r) is given in (6.7). The central pressure can be determined by evaluating (6.10) at the boundary r = R where the pressure vanishes. We have

(1 ζ)(y(R) 1) p = − − ρ (6.11) c 3y(R) (1 ζ) − − Finiteness of the central pressure implies that the denominator must be well deﬁned, hence greater than zero. In the simpliﬁed constant density case this implies the so called Buchdahl inequality y(R) > 1 (1 ζ). 3 − In terms of (6.7) this inequality becomes

M 2ζ ζ2 + 8 < − , (6.12) R 9(2 + ζ) which for ζ = 0 leads to the well known result 9M < 4R, that is, radius of static perfect ﬂuid sphere is larger than the Schwarzschild radius of the corresponding mass. Note that since the r.h.s of (6.12) must be positive, then one obtain a less restrictive condition ρvac < 2ρ. Finally, it is not diﬃcult to se that inequality (6.6) becomes an equality with the solution (6.10) [42].

Generalized Buchdahl inequalities The inequality (6.12) is not only valid for homogeneous conﬁgurations. It can be shown [57, 86] that the same inequality holds for conﬁguration whose density is a decreasing function of r. The generalized Buchdahl inequality is given by

4 2 1 1 πρ¯bR 2 + ζ¯b 1 ζ¯b , (6.13) r − 3 ≥ 3 − 3 with ζ¯b = 2ρvac/ρ¯b. Using again ρ¯b = 3M/(4πR ) explicitely in ζ¯b in (6.13) one has in terms of Λ

R 4 1 R 2 M 2 + (6.14) ≤ 3  9 − 3 R  s Λ   Just as equation (6.12), the the last expression imposes an upper bound to the total mass of the conﬁg- uration. This bound is only valid if the square appearing on the r.h.s of (6.14) is well deﬁned, that is,

4 R RΛ (6.15) ≤ r3 76 CHAPTER 6. ASTROPHYSICAL BOUNDS ON THE COSMOLOGICAL CONSTANT is satisﬁed by the object’s radius R. Inserting the highest possible radius (6.15) in (6.14) we obtain

3 2 4 1 4 2 M = M (6.16) ≤ 9 3 √ 3 Λ r Λ It is now worthwhile to compare the inequalities (6.15) and (6.16) with constraints arising from the Newtonian limit, given in (2.30). Disregarding diﬀerent numerical factors which are of the order of unity and the fact that (2.30) are strong inequalities, we see that both are essentially the same. Inequalities (6.15) and (6.16) can be understood as constraints on Λ to keep the object in hydrostatic equilibrium from which they are derived. On the other hand, (2.30) ensures that the gravitational ﬁelds are not too strong. Hence in both cases one expects a restriction on a positive cosmological constant. What is surprising, however, is the fact that these restrictions are so similar in both cases.

6.2.1 General solutions with equation of state In view of the above results, we can assume a variable density in order to find a more restrictive condition on Λ. First of all, inequality (6.7) is valid for any decreasing density profile. On the other hand, since it must be fulfilled at any r in order that the pressure be a decreasing function of the radius, we can evaluate it at the boundary r = R, defined by the condition p(R) = 0, which leads to ρ¯ (1 ζ¯) y(R) b − . (6.17) ≥ 3p + ρ (1 ζ ) c c − c In this case we can safely assume an polytropic e.o.s, so that the central pressure is only a function of the central density. The function η(p) can be integrated and becomes a function of the central density when evaluated at the boundary γ γ γ(γ 1) γ−1 η(p(R)) = κ ρc − + 1 = η(ρc) . (6.18) Equation (6.17) for the second in bequality then becomes a cubic equation for Λ, namely, f(Λ) = aΛ3 + bΛ2 + cΛ + d 0 , (6.19) ≥ where the coefficients are given explicitly as 1 a = R3, (6.20) −3 8π b = 1 + 8πp R2 R2 (ρ¯ ρ ) 1, c − 3 − c − 64 16 c = 24πp 48π2p2R2 + 64π2ρp¯ R2 + 8πρ¯ 8πρ 32π2p ρ R2 + π2R2ρρ¯ π2R2ρ2, − c − c c − c − c c 3 c − 3 c 128 d = 144π2p2 384π3p2ρR¯ 2 16π2ρ¯2 + 96π2p ρ 256π3p ρρ¯ R2 + 16π2ρ2 π3ρρ¯ 2R2, c − c − c c − c c c − 3 c and where ρ¯ = ρ¯b = ρ¯(R) is the object’s mean density. Thus we have the following parameters R, ρc, pc, ρ¯b, which can be related as follows: the central pressure is connected with the central density through the γ equation of state so that pc = κρc . The radius of the configuration is related with the mean density at the 3 surface as usual R = 3M/4πρ¯b, while the mass M is just the volume integral of the density. We need the density profile and the total mean density (together with an equation of state) to solve the cubic equation (6.19) for Λ. It is interesting to further exploit the inequality (6.17). Although it is quite involved to extract information on the cosmological constant one can solve it for the central energy density. This yields ρ¯ 2ρ ρ ρ = − vac 3p + 2ρ , (6.21) c ≥ c,min y(R) − c vac giving a lower bound on the central energy density. This is not surprising because we have upper bounds on the boundary mean density. Since the energy density for astrophysical models is a decreasing function of the radius, one must find some lower bound, that in particular extends the boundary mean density. A clever choice of new variables could be of help in order to get information on the cosmological constant from (6.17). CHAPTER 7

Conclusions

This work has been devoted to the effects of a positive cosmological constant on scales ranging from cosmological to astrophysical scales. We started with a review of the role of the cosmological constant in determining the dynamics of the universe. Dynamical models as dark energy and Chaplygin Gas that mimic a vacuum energy density at the present time were also presented. Here we also explored the role of a cosmological constant in the formation of large cosmological structures. Since the current dynamics of the universe is dominated by the cosmological constant, it worths to wonder how did this term or its generalization affected the formation of galactic clusters and galaxies. It i found the parameters such as the final radius of a forming structure is enlarge because of a repulsive Hooke-like force associated to Λ. We considered the effects of vacuum dominated universe in cosmological and astrophysical scales. For this purpose, we wrote down the most general expressions describing equilibrium for structures not only located at cosmological scales, but also on astrophysical scales, such as galaxies. As examples, we studied the equilibrium of spherical and non spherical configurations. The exploration of the effects on spherical configurations were divided in two sets: homogeneous and non homogeneous systems. For the first case, we determined the radius of a virialised and pressurless configuration where gravity is only supported by the expansion, mainly, by Λ. Such radius has cosmological scales and hence such configurations may represent the largest virialised structures as galaxy clusters. On the other hand, for non homogeneous structures we took a precise example by assuming an polytropic equation of state; we showed that the parameters of such configuration (as radius and the mass) are affected by Λ and according to the polytropic index, a given value of the cosmological constant can allow a physical configuration. In this context, the effects of Λ are introduced not as an external force but as modification in the Lane-Emden equation and hence the cosmological constant enters as a modification in its solutions, i.e, the density, although it should be noticed that this modifications are due to the ones on mass and radius, and not because the vacuum energy density collapses with conventional matter. We studied examples of configurations found some in astrophysical and cosmological context satisfying the polytropic equation of state, as white dwarfs, neutrino stars and isothermal sphere. Clearly the effects of Λ were negligible for white dwarfs because of their scales of mass and length, but sizable effects were found on neutrino stars. Finally we explore the stability of configurations under small perturbations. It is shown that the cosmological constant modifies the stability condition. We also explored the equilibrium conditions for non-spherical configurations with cosmological constant. Important results we have derived in this context. First of all, we showed that the effects of Λ are enhanced as long as the configuration deviates from spherical symmetry. As some examples, we determined the angular velocity for oblate spheroids, where we found that for the configuration decreases its angular velocity (with respect to its value with Λ = 0) in order to maintain equilibrium. As said before, this

77 78 CHAPTER 7. CONCLUSIONS

Density Geometry % (x) x A

Homogeneous Spherical ρ 2 −

4 3 2 1 Polytropic Spherical ρc 3 πf0 3 (n + 4)f1 1 n = 1 γ − −

4 3 e2 e 2 Homogeneous Oblate ρ − e = 1 q 3 arcsin e 2√1 e2 3 − − p

1 4 e(3 2e2 ) 1+e − 2 − Homogeneous Prolate ρ 3 (1 e2)3/2 ln 1 e e = 1 q3− − − − h i q

Table 7.1: Equilibrium condition % > ρ for different configurations A vac effects is enhanced as long as we consider flattest ellipsoids, applicable to the global structure spiral and elliptical galaxies. Second, we showed that the allowed geometries for homogeneous rotating configurations in equilibrium depends of the ratio ζ 2ρ /ρ and hence, solutions to virial equations for oblate systems, ≡ vac i.e, Maclaurin and Jacobi ellipsoids, bifurcates at different eccentricities as long as the ratio ζ changes. But this is also an important feature of rotating prolates. As known, rotating ellipsoids along the mayor axis has a prolate configuration as solution to the virial equations. However, with cosmological constant a second possible solution arises, corresponding to an allowed triaxial configuration in equilibrium. Our results can be easily generalized to a time dependent vacuum energy density (for instance Chaplygin Gas, Quintessence or Dark Energy), and this analysis can be used as a link between the phenomenology of galaxy and galaxy clusters with generalized dark energy models. In the fourth chapter, we explore the scales of length set by the cosmological constant and it’s effects on the motion of test particles in a space time generated by a spherical symmetric configuration (Schwarzschild metric). It is shown that the cosmological constant allows unstable circular orbits at cosmological scales. As a relevant result, it is shown that the non-arbitrary combination of the scale length imposed by Λ and the Schwarszchild radius associated to the mass-source of the Schwarszchild spacetime leads to a new length scale which is associated to the typical size of virialised configuration. As a possible relic of the coincidence problem, we noted that through this scale, the cosmological constant sets the sizes of gravitational bounded configurations. Finally, we wrote down some bounds on Λ derived from the Newtonian Limit as well from the equation for hydrostatic equilibrium in general relativity. Here we showed that similar bounds can be derived from both contexts. In the table 7.1 we summmarize the equilibrium conditions % > ρ derived from the virial theorem A vac with cosmological constant for different geometries and density profiles. As a global conclusion, we can say that the effects of a positive cosmological constant are sizable and relevant for cosmological structures, furthermore if we realize that the vacuum energy density is the dominating component of the universe. As part of the coincidence conjecture, is was very interesting to note that not only the size of the universe by the relevant scales of astrophysical and cosmological structures are determined by Λ. The results derived here with cosmological constant can be generalized to any model of dark energy, as can be check in the appendix and future publications. In that context framework described in this work becomes a good link between the phenomenological exploration of galaxies and galaxy clusters (where the effects of Λ are stringent) and theoretical aspects of dark energy. This connection could help us to 79 determine which if the current models is the best to describe dark energy. 80 CHAPTER 7. CONCLUSIONS APPENDIX A

The virial equation

Conservations laws (mass, energy, momentum) for a system composed with a large number of components can be derived from statistical analysis starting from the Boltzmann equation (see [47]), or by considering the ideal fluid description. Using the statistical description, Euler’s equation (momentum conservation) for a self gravitating configuration with electromagnetic field is written as

du ρ i = ρ∂ Φ ∂ ( + ) ∂ S , (A.1) dt − i − j Tij Pij − t i with ρ = ρ(r, t) = mn(r, t) is the total density of matter in the fluid and n(r, t) is the mean number of particles in the fluid at certain position r in time t. is the pressure tensor defined by Pij ρ (v u )(v u ) , (A.2) Pij ≡ h i − i j − j i where u = v is the average velocity. This average is defined as a the mean value in the phase-space i h ii 1 u(r, t) = v(r, p, t) f(r, p, t)v(r, p, t)d3p, n(r, t) = f(r, p, t)d3p, (A.3) h i ≡ n(r, t) Z Z where f(r, p, t) is the distribution function, the solution for the collisionless Bolztman equation df/dt = 0. Hence, conservation laws can in principle be applied if we solve for f, which implies that we could write the conservation laws for the different orders of approximation used to find f. In this way, the pressure tensor can be separated into the diagonal part, associated to the thermodynamical pressure p Pik (zeroth order approximation) and a traceless component associated to viscosity (first order approximation): = δ p + π such that Tr(π = 0) and p = 1 Tr( ). On the other hand, the tensor refers to the Pik ik ik ik 3 Pik Tij components of the Maxwell stress energy tensor,

1 = δ E E B B , = E2 + B2 , S = E B, Tij ij Uem − i j − i j Uem 2 × with as the electromagnetic energy density and S the Poynting vector. Euler’s equation together with Uem conservation of mass for the set called Jeans equations when applied to stellar systems. The tensor virial equation of order n is derived by making exterior products of the vector rk and the force acting over the system Fi given through Euler’s equation and integrating over the volume containing the system. With this ones obtains a n + 1 symmetric tensor equation.

du ρ (r r r ) i d3r = (r r r ) [ρ∂ Φ ∂ ( + ) ∂ G ] d3r. (A.4) j k · · · n dt − j k · · · n i − j Tij Pij − t i ZV ZV 81 82 APPENDIX A. THE VIRIAL EQUATION

For instance, the first order t.v.e represents the equation of motion for the center of mass of the system: du d2R ρ d3r = cm = [ρ∂ Φ + ∂ ( + ) + ∂ G ] d3r. dt dt2 − i j Tij Pij t i ZV ZV The meaning of the second order virial equation can only be understood after some steps. From (A.4) we have du r ρ i d3r = r ρ∂ Φd3r r ∂ ( + ) d3r r ∂ G d3r. (A.5) k dt − k i − k j Tij Pij − k t i ZV ZV ZV ZV We now proceed to evaluate each term contained in this expression. For the l.h.s we may write du d d ρr i d3r = ρ (u r )d3r ρu u d3r = ρ (u r )d3r 2T , (A.6) k dt dt i k − i k dt i k − ik ZV ZV ZV ZV where the quantities Tik are defined as 1 T ρu u d3r, (A.7) ik ≡ 2 i k ZV corresponding to the components of the kinetic energy tensor. Note that the l.h.s of (A.6) since being symmetric implies angular momentum conservation, while the first term in the last equality of equation.(A.6) can be modified and be written as d 1 d2 1 d2 ρ (u r )d3r = ρr r d3r = Iik , (A.8) dt i k 2 dt2 i k 2 dt2 ZV ZV where are the components of the moment of inertia tensor defined by Iik ρr r d3r. (A.9) Iik ≡ i k ZV We now take the r.h.s of (A.5) associated to the potential Φ which satisfies the modified Poisson’s equation (2.28) with a solution written as

1 2 (ρ(r, t) ρb(t)) 3 Φ(r, t) = Φ (r, t) f(t) r , Φ (r, t) = − d r0, (A.10) grav − 2 | | grav − r r ZV | − 0| where ρ(r, t) ρ (t) = δρ(r, t) is the overdensity associated to the Newtonian source of gravity (see − b equation (2.35)), ρb(t) is the homogeneous background energy density (which is ρb = ρcdm + ρx when the x-component collapses or simply ρb = ρcdm when only the cdm collapsed to form a virialized structure) and f(t) = a/a¨ is the the acceleration equation for the background. Replacing this expression in (A.5), one realizes that the term associated to Φgrav can be modiﬁed by taking derivatives and interchanging indices, while the second can be written in terms of the moment of inertia tensor such that 1 4 ρ(r)r ∂ Φ d3r = ρΦ d3r + πρ (t) , (A.11) k i grav −2 ik 3 b Iik ZV ZV where Φik is the gravitational potential tensor deﬁned as

(ri ri0)(rk rk0 ) 3 Φik(r) G ρ(r0) − − d r0. (A.12) ≡ − 0 r r 3 ZV | 0 − | Similarly, for the contribution from the acceleration equation for the background we have

1 2 3 ρr ∂ f(t) r d r0 = f(t) . (A.13) k i −2 | | − Iik ZV We then can write the components of a potential energy tensor as the contribution of a pure gravitational part grav 1 and the contribution of the expansion of the universe exp: Wik Wik ρr ∂ Φ(r, t)d3r = grav + exp, (A.14) Wik ≡ − k i Wik Wik ZV 1 known as (the components of) the Chandrasekhar potential energy tensor 83 with 1 4 grav ρΦ d3r, exp f(t) πρ (t) . (A.15) Wik ≡ 2 ik Wik ≡ − 3 b Iik ZV For a spherically symmetric configuration becomes diagonal. Hence, any deviation from non-spherically Wik symmetry can be measured through the non-diagonal elements of . On the other hand. if we assume Wik that the background is filled with a cold dark matter component ρcdm and a x-component (cosmological constant or dark energy with constant e.o.s for instance), we can write 4 exp π [2ρ + ρ (1 + 3ω )] . (A.16) Wik ≡ −3 cdm x x Iik It is important to note that the potential energy grav is not generated by the potential Φgrav but from W the part containing the total density. This makes an small difference in the writing of the potential energies when compared with equations (2.53), where we defined a potential energy grav which is the W one associated only to the overdensity Φgrav δρ, while (A.15) is associated to the total density Φ ρ. → → Collecting these results and applying the divergence theorem in the last term in the r.h.s. of (A.5), we get the second order t.v.e as 1 d2 Iik = 2T + + ( + ) d3r r ∂ G d3r r ( + ) dS . (A.17) 2 dt2 ik Wik Tik Pik − k t i − k Tij Pij j ZV ZV Z∂V Some simplifications can be done in order to explore astrophysical effects. First, most of astrophysical systems are electrically neutral and only a magnetic field prevails through the system; thus, we may write the components of the Maxwell’s stress tensor as = 1 δ B2 B B . Hence we can write (A.17) as Tij 2 ij − i j 1 d2 Iik = 2T + + Π + δ 2 + , (A.18) 2 dt2 ik Wik ik ikB − Bik Sik where 1 B B d3r, Π d3r, (A.19) Bik ≡ 2 i k ik ≡ Pik ZV ZV and the surface terms are encoded in the tensor : Sik 1 r δ B2 B B + dS . (A.20) Sik ≡ − k 2 ij − i j Pij j Z∂V The form of the tensor virial equation given in (A.18) will be useful when we derive the scalar virial equation and to perform Lagrangian perturbations. For the use given in some part of the text, the second order virial equation can be written also as

1 d2 Iik = 2T gen + exp + Π r dS , (A.21) 2 dt2 ik − |Wik | Wik ik − kPij j Z∂V where we have included the volume and surface integrals associated to the magnetic field into the term gen Wik (B) gen grav 1 ∆ , ∆ Fik , (A.22) |Wik | ≡ |Wik | − (ik) (ik) ≡ grav |Wik | where 1 (B) δ 2 r δ B2 B B dS . (A.23) Fik ≡ ikB − Bik − k 2 ij − i j j Z∂V The second simplification comes as follows: since we are not neglecting contributions from the background, one is forced to define the boundary of the configuration as the region where the density and the pressure take the background values. If the matter component of background is pressurless, then the surface 84 APPENDIX A. THE VIRIAL EQUATION integrals with pressure vanishes, but surface integrals with density remains. On the other hand, since the magnetic fields goes beyond the surface of the system in most of the real cases, the surface integrals with magnetic field cannot be taken as zero at less that we consider a system with the surface placed at infinity. We obtain the scalar equation by taking the trace of each tensor involved in (A.21). For the l.h.s we have

Tr( ) = ρ r 2d3r , (A.24) Iik | | ≡ I ZV which corresponds to the moment of inertia about the origin of coordinates. For the r.h.s we find 1 Tr(T ) = ρ u 2d3r T, (A.25) ik 2 | | ≡ ZV which represents the contribution of ordered motions to the kinetic energy of the system. For the tensor we have from (A.14): Wik 1 Tr( ) = ρ(r)Φ(r)d3r = ρ(r)r ∂ Φ(r) , (A.26) Wik 2 − i i ≡ W ZV ZV which represents the potential energy of the system. The terms associated to the magnetic field are 1 Tr( ) = B2d3r , (A.27) Bik 2 ≡ B ZV corresponding to the magnetic energy of the configuration. The scalar virial equation is then written from (A.18) as

1 d2 I = 2T + + + 3Π + , (A.28) 2 dt2 W B S where the surface term and the pressure term Π are deﬁned as 1 (r nˆ)B2 (r B)(B nˆ) + r nˆ dA, Π Tr(Π ) = p d3r, (A.29) S ≡ − 2 · − · · · P · ≡ ik Z∂V ZV with nˆ as an unitary vector perpendicular to the surface wit area A. The scalar virial equation taken from (A.21) can also be written as

1 d2 I = 2 gen + exp r nˆ dA. (A.30) 2 dt2 K − |W | W − · P · Z∂V Following (A.22) and (A.23), the these expressions are defend as (B) 1 gen = grav (1 ∆) , ∆ = F , (B) = (r nˆ)B2 (r B)(B nˆ) dA, (A.31) |W | |W | − grav F B− 2 · − · · |W | Z∂V in accordance with (A.22). The total kinetic energy term is written from (A.2) and (A.21) as K 1 3 = ρ v2 d3r = T + Π, (A.32) K 2 h i 2 ZV and corresponds to the contribution of ordered motions (T ) and the random motion associated to the pressure tensor . Pik Rotational contributions the virial equation Consider a rotating system with angular velocity characterized by a vector Ω(r, t) aligned along an axis passing through the center of mass of the system. The time variations of an arbitrary vector A observed from the inertial system are related to those seen from the rotating system as dA dA = + Ω A. (A.33) dt dt × inertial rotating 85

Hence, the velocities U measured from the rotating system are related to the velocities u referred to the inertial system as u = U + v with v = Ω r. Using (A.33) once again to ﬁnd the time derivative of u, × we write Euler’s equation in the absence of electric ﬁelds and with isotropic pressure tensor as

˙ 1 2 ˙ ρUi = ρ∂iΦ ρ∂iΦext ∂i p + B + ∂j (BiBj ) ρ (Ω v)i 2ρ (Ω U)i + ρ r Ω , (A.34) − − − 2 − × − × × i where the time derivatives are referred to the rotating reference frame so that the Lagrangian operator j reads as (d/dt)rot = ∂t + U ∂j . The last three terms correspond respectively to non-physical forces consequences of the non-inertial reference frame. The total angular momentum of the system is now written as

L = ρ (r U) d3r + (Ω I) d3r, (A.35) × · ZV ZV where I = ρ δ r2 r r . The first term represents the total angular momentum associated to the ik ik − i k motion of elements due to Coriolis force (L ), the second term represents the spin angular momentum. cor For constant angular velocity, it which reduces to Ω ˜ , with ˜ as the moment of inertia (about the iIik Iik axis of rotation) tensor (Tr(˜ ) = 2 ). Iik I To determine the contributions of rotational effects in the virial equation, we again multiply (A.34) by rk and integrate over the volume of the configuration. After this operation, we define the rotational kinetic energy tensor from the first extra-term in (A.34) Rik 1 ρ [v Ω] r d3r. (A.36) Rik ≡ 2 × i k ZV As expected, the trace of this tensor reproduces the rotational kinetic energy of the system: 1 1 1 Tr( ) = ρr [v Ω] d3r = ρv2d3r = (Ω I Ω) d3r . (A.37) Rik 2 · × 2 2 · · ≡ R ZV ZV ZV Now consider the second extra-term in equation (A.34). Defining a local angular velocity $ such that U = $ r we can define a new tensor by × Lik 3 ˜ 3 ik 2 ρrk [Ω U]i d r = 2 Ω d r, (A.38) L ≡ − V × V · M ik Z Z where ˜ is a three-rank-the angular momentum-like density tensor, with components are given by M ˜ r P r P , P = ρr $ , (A.39) Milk ≡ k il − l ki ik i k with P acting as a local momentum density tensor, both and P associated to displacements of mass il Mik ik elements due to Coriolis force. Now we consider the last contribution associated to the time derivative of the angular velocity. Using the continuity equation ∂ ρ + ∂ (ρU ) = ∂ (ρv ), this term can be written t j j − j j after some algebra as 1 ρ r r Ω˙ d3r = , (A.40) ilm k l m Qik − 2Lik ZV where we defined

r r ∂ (ρΩ )d3r v [r ∂ (ρv ) ρU ] d3r ρ v r U dS . (A.41) Qik ≡ ilm k l t m − i k j j − k − b i k j j ZV ZV Z∂V Note that the tensor has a background contribution in a surface integral related to the density. Both Aik and have the form of a kinetic energy tensors. The other terms involved in equation (A.34) are Lik Aik transformed as usual. Nevertheless we must be aware that the velocities involved in the kinetic energy tensor are now Ui. We write the tensor virial equation with rotational eﬀects as

1 d2 1 Iik = 2(T + ) + + Π + δ B 2 + + + . (A.42) 2 dt2 ik Rik Wik ik ik − Bik Sik 2Lik Qik 86 APPENDIX A. THE VIRIAL EQUATION

Now we take the trace in this expression. For is easier to consider the l.h.s of (A.38): Lik Tr( ) = 2 ρr (Ω U) d3r = 2 (Ω l) d3r, (A.43) L ≡ Lik − · × · ZV ZV where l = ρ (r U) is the total angular momentum density, i.e, ld3r = L, associated to motions due × V to Coriolis force. For the last term in (A.42), from (A.41) we see that Tr( ) = 1 (i.e the term R A ≡ Qik 2 L associated to Ω˙ does not contribute in the scalar virial equation). The scalar virial equation (A.30) is written as 1 d2 I = 2 (T + ) + + + 3Π + + . (A.44) 2 dt R W B S L The contributions from vanishes when considering diﬀerential rotation and steady state (ρ = ρ(r), Ω = A Ω(r)) or rigid body rotation. The contribution from only vanishes in the latter case, while the rotational L kinetic energy tensor reduces to 1 = Ω2 Ω Ω . (A.45) Rik 2 Iik − iIkj j Thus, for constant angular velocity and rigid body rotation the tensor and virial equation only acquires the rotational kinetic energy tensor. This shows that the tensor virial equation encodes much more informations, and can be used to explore the equilibrium conditions in a more general situation.

Virial theorem The most used version of the virial equation the is the so-called virial theorem, which states that in a configurations in steady state, the mean value of the potential energy equals twice the mean value of the kinetic energy. The word mean refers to a time average, which according to the ergodic hypothesis, yields to the same results than the phase-space average, defined in (A.3). This means that if we consider a set of initial conditions, the variables describing the dynamics of the system will take every allowed value compatible with the initial conditions such that if we let the system to evolve through a large period time, the configuration would have passed through all the allowed microstates until it reach the state of equilibrium. The ergodic hypothesis then states that the equilibrium reached by this way is the same as the one described through the variables resulting spatial (in the sense of the phase-space) averages with fixed time. For instance, Euler’s equation and the virial equation both describe instantaneously the way the configuration reaches equilibrium (i.e, the state when the l.h.s of both equations vanishes). Let us consider the tensor virial equation and let us take a time average in (A.42) and (A.30) over a time interval large enough so that the equilibrium condition is assumed to be reached. As usual, the time average of the second derivative of the moment of inertia tensor yields

2 τ 2 d ik 1 d ik 1 ˙ ˙ I2 = I2 dt = ik(τ) ik(τ0) 0 as ∆τ , (A.46) dt τ ∆τ τ dt ∆τ I − I −→ → ∞ D E Z 0 which is valid if the variations of the moment of inertia are periodic (i.e, if the system is stable) or if the variations of are only appreciable on time scales much greater than the time interval ∆τ = Iik τ τ . The standard version of the virial theorem then can be written from the scalar virial equation as − 0 2 + = 0 for Λ = 0. With cosmological constant, this can be generalized to hKiτ hWiτ 8 2 T + gen + πρ + Π = (Surface terms) (A.47) h ikiτ hWik iτ 3 vachIikiτ h ikiτ h ikiτ 8 2 + gen + πρ = Surface terms , hKiτ hW iτ 3 vachIiτ h iτ where 2 = 2 T + 3 Π is twice the total mean kinetic energy. In this case the contribution from hKiτ h iτ h iτ the background is separated and we deal with the mean values of the parameters of the conﬁguration. In a more general situation when we consider contribution from dark energy or Quintessence models, this contribution acquires an explicit time dependence, which acts as a non-conservative force which 87 implies that the volume in phase-space is no longer conserved and the system will not reach a deﬁnite equilibrium state. Using (A.16), the average for the contribution of the expanding universe reads as

τ τ exp 8π 4π ik = ρm(t) ik(t)dt ρx(t)(1 + 3ωx(t)) ik(t) dt. (A.48) W τ −3∆τ τ I − 3∆τ τ I D E Z 0 Z 0 3 The matter component of the background varies as ρ (t) a(t)− after the time of decoupling with radi- m ∝ ation. If the scale factor varies as a power law with time a(t) tm, then the ﬁrst term rapidly goes to zero ∝ as ∆τ increases. The second term can be related to the dark energy component (if ωx is constant) or to mn the Quintessence models (if ω = ω (t)). It then varies as t− with n = 3(1 + ω ) and 1 ω 1/3. x x x − ≤ x ≤ − Hence this contribution does not vanishes as fast as the matter contribution, and then we cannot take out this contribution from the time integrals. The virial theorem then has a term ρ which should h xIikiτ be computed in accordance to the evolution of the energy density associated to the x-component. If the the non-conservative terms change slowly compared to typical time scales associated to inner processes in the system, we can assume quasi-equilibrium states described by (A.42) with ¨ = 0, which is Iik the form of virial equation that we have used in this work.

Lagrangian perturbations in the virial equation

Let χ represent some (scalar, vector or tensor) attribute of the fluid. The variation of this quantity due to small perturbations can be treated from two different points of view. First, if we call χ0(x, t) the value of χ at a position x and at a time t in the unperturbed flow, and χ(x, t) the same attribute measured in the same point in the perturbed flow, then one defines the Eulerian variation of a fluid attribute χ as

δ˜χ χ(r, t) χ (r, t). (A.49) ≡ − 0 This corresponds to the macroscopic description. On the other hand, in the microscopic picture the variations of an attribute are written in terms of a displacement vector field ξ(x, t) which connects elements of the perturbed flow with the same elements in the unperturbed configuration. The Lagrangian variations are then defined as

∆χ χ (r + ξ(r, t), t) χ (r, t), (A.50) ≡ − 0 which corresponds to the variation in the attribute χ as measure at the same fluid element which has been displaced by the perturbation from r to r + ξ. From these definitions, we find that Euler and Lagrange variations are related as

j ∆ = δ˜ + ξ ∂j , (A.51) so that ξk = ∆rk. This represents a Lagrangian variation but referred to an Eulerian reference frame, where proper coordinates change with time, while a Lagrangian reference frame is associated to a frame which follows the fluid, i.e, a point within the fluid has always the same (Lagrangian) coordinates as seen from a Lagrangian reference frame. If we want to describe the perturbations from a Lagrangian reference ˜ frame, equation (A.51) must be generalized to ∆ = δ + £ξ,where £ξ is the Lie derivative along the vector field ξ (both descriptions coincide for scalar attributes) [78]. Here we will describe the perturbations from an Eulerian reference frame, and hence we stick to (A.51). It worths mention that comoving coordinates represent a mixture of these types of coordinates, since a comoving reference frame follows the fluid (for instance, the expanding universe) and proper distances change with time according to the scale factor. The next useful identities can be verified from the definitions (A.50) and (A.51):

δ˜∂ = ∂ δ˜ δ∂ = ∂ δ ∆∂ = ∂ ∆ ∂ ξj ∂ t t i i t t − t j (A.52) d d d d d ∆∂ = ∂ ∆ ∂ ξj ∂ ∆ = ∆ δ = ∆ ξj ∂ . i i − i j dt dt dt dt − j dt 88 APPENDIX A. THE VIRIAL EQUATION

This implies that the Lagrangian variation in the velocity is ∆ui = ∆dri/dt = dξi/dt. Once we have deﬁned Lagrangian perturbations, we now consider variations for integrals quantities, which are present in the virial equation. Let the integral

3 I = χ0(x, t)d x, (A.53) ZV denote some attribute of the unperturbed system. The ﬁrst variation of I, δI, produced by a perturbation is deﬁned as

δI = χ(r, t)d3r χ (r, t)d3r. (A.54) − 0 ZV +∆V ZV The first term represents the quantity I defined with respect to the perturbed flow, while the second is the quantity I defined with respect to the unperturbed flow. In order to write this difference in a single integral, we must transform the volume elements with r0 = r ξ, such that to the first order, the Jacobian − of the transformation is = 1 + ∂ ξj , so that the first integral in (A.54) becomes J j 3 j 3 χ(r, t)d r = χ(r0 + ξ, t)(1 + ∂j ξ )d r0. 0 ZV +∆V ZV Changing r0 by r in this integral and inserting it into (A.54), the variation of the integral quantity (A.53) is written with the help of (A.50) as

δI = ∆χ + χ∂ ξj d3r = ∆χ + ∂ (ξj χ) ξj ∂ χ d3r = δ˜χd3r + ∂ (ξj χ)d3r. (A.55) j j − j j ZV ZV ZV ZV If the attribute χ vanishes at the boundary we have

δI = δ˜χd3r. (A.56) ZV Now, if a mass in an arbitrary volume element is conserved, then, setting χ = ρ in (A.55) we have

δ ρd3r = 0, ZV and then equation (A.55) implies that the Lagrangian change in ρ is given by ∆ρ = ρ∂ ξj . (A.57) − j Using (A.51), one can easily derive an expression for the Eulerian change of the density: δ˜ρ = ∂ (ρξj ). (A.58) − j Using these relations it can be easily shown that

δ χρd3r = ρ∆χd3r. ZV ZV The complete description of a system under the influence of gravity and electromagnetic fields is complete with an equation of state p = p(ρ, s), where s is the entropy of the system. If the system undergoes adiabatic perturbations, then ∆s = 0, and the first law of thermodynamics yields ∆p ∆ρ ∂ ln p = Γ , Γ = , p ρ ∂ ln ρ s where the parameter Γ is the adiabatic index that governs the perturbations so that if the equation of state corresponds to that of a polytrope p = κργ , we can identify Γ = γ, the politropic index. We can also write the internal energy of the system as = (ρ, s), so that from the first law we have U U ∂ p = ρ2 U . ∂ρ s Hence, a variation in the internal energy of the system is accompanied by a variation in the density as ρ2∆ = p∆ρ. Finally, integration of the first law of thermodynamics also allows us to write the total U 1 internal energy of the configuration for an adiabatic equation of state as = (γ 1)− Π, where γ is the U − ratio between specific heat capacities. 89

Perturbing the virial equation Given the elements of an integral tensor attribute , its ﬁrst order variation under small perturbation Qik is written as δ eiωt, with ω as the frequency of oscillations about the values at equilibrium. Here we Qik assume that all the parameters associated to the system oscillate under small perturbations with the same frequency. The Lagrangian variation in the coordinates can be written from (A.51)

iωt ∆rk(r, t) = ξk(r, t) = ξk(r)e (A.59)

The tensor virial equation written as (A.18) becomes 1 δ ω2 = 2δ + δ 2δ + δ δ [Π + ] , (A.60) −2 Iik Rik Wik − Bik ik B where we have assumed an isotropic pressure tensor and have neglected any kinetic contribution from internal motions. This equation lead us to derive an expression for the oscillations frequencies about equilibrium, which, in turn will set the stability conditions. We now consider the terms in (A.60):

Moment of inertia: Using (A.59), we write the variations of the moment of inertia and it’s trace as

δ = ρ∆(r r )d3r = ρ (r ξ + ξ r ) d3r, δ = 2 ρ (r ξ) d3r. (A.61) Iik i k i k i k I · ZV ZV ZV Potential energy. We divided the potential energy as a contribution of gravity and the background, as it is in equation (A.14). Here we calculate the variations in the gravitational contribution associated 2 (1) to the potential Φ , which satisfies the Poisson equation Φgrav = 4πGρ (the overscript (1) refers to grav ∇ the first term in Φgrav given in (A.10) which is proportional to the total matter density ρ. In this part we leave it just as Φ). Using the properties (A.52), we may write 2δ˜Φ = 4πGδ˜ρ, which solution can be ∇ written as ˜ δρ(r0) 3 δ˜Φ(r) = d r0. − 0 r r ZV | − 0| It is convenient to define

(ri ri0)(rk rk0 ) 3 δ˜Φik(r) = δ˜ρ(r0) − − d r0. − 0 r r 3 ZV | − 0| so that Tr(δ˜Φ ) = δ˜Φ. With this, we can write the variation of the tensor grav as ik Wik grav 1 3 1 ˜ j 3 δ ik = ρ(r)∆Φikd r = ρ(r) δΦik + ξ ∂j Φik d r. (A.62) W 2 V 2 V Z Z For the ﬁst term, we integrate by parts and use the deﬁnition of Φik

1 3 1 (ri ri0)(rk rk0 ) 3 3 ρ(r)δ˜Φikd r = G ρ(r)δ˜ρ(r0) − − d r0d r (A.63) 2 −2 0 r r 3 ZV ZV ZV | − 0| 1 1 1 = δ˜ρΦ d3r = ∂ (ρξj )Φ d3r = ρξj ∂ Φ d3r. 2 ik −2 j ik 2 j ik ZV ZV ZV where we have used the expression (A.57) and (A.58). Replacing (A.63) in (A.62) and integrating by parts in the second term of (A.62) we ﬁnally ﬁnd the Lagrangian variation of the Chandrasekhar potential energy tensor and its trace as

δ = ρξj ∂ Φ d3r, δ = ρξj ∂ Φd3r. (A.64) Wik j ik W j ZV ZV From this follows that the variation of the gravitational potential energy is generated by a redistribution of mass elements due to perturbations inside the system. 90 APPENDIX A. THE VIRIAL EQUATION

1 Internal energy: We consider the variations in the internal energy = (γ 1)− Π. Assuming adiabatic U − perturbations, we have from equation (A.56)

δ˜pd3r = ∆p ξj ∂ p d3r = (Γ 1) ξj ∂ pd3r Γ pξj dS , (A.65) − j − j − j ZV ZV ZV Z∂V where we have used (A.57), (A.58) and integrated by parts in the last equality. This expression can be rewritten with the help of the the equation of motion (A.34) at equilibrium: 1 ∂ p = ρ∂ Φ ∂ B2 + ∂ (B B ) ρ (Ω v) . j − j − 2 j i i j − × j Replacing this in (A.65) we have 1 δ˜p d3r = (1 Γ) ρξj ∂ Φ ξj ∂ (BiB ) + ξj ∂ (B2) + ρξj (Ω v) d3r, − j − i j 2 j × j ZV Z where we have neglected the contribution coming from the surface integrals of pressure. Integrating by parts we finally find the variation in the internal energy as 1 δΠ = (1 Γ) ρξj ∂ Φ + B B ∂ ξj B2∂ ξj + ρξj (Ω v) d3r. (A.66) − j i j i − 2 j × j Z Magnetic energy: From (A.19) we have 1 1 δ = ∆(B B ) + B B ∂ ξj d3r = B δB + B δB + B B ∂ ξj d3r (A.67) Bik 2 i k i k j 2 i k k i i k j ZV ZV To calculate the Eulerian variations in the magnetic field, consider the Lagrangian displacement of an element of mass. An observer at rest with respect to the perturbed system measures an electric current ˙j k given by Ohm law Ji = σ(δ˜Ei + ijkξ B ). If we assume that the system has infinite electric conductivity σ, then δ˜E = ξ˙j Bk. Applying an Eulerian perturbation to Maxwell’s equation for E, we have i − ijk ∇ × δ˜E = ∂ (δ˜B). Combining these expressions one gets δ˜B = (ξ B). Written by components we ∇ × − t ∇ × × have the Eulerian and Lagrangian variation in the magnetic field respectively as

δ˜B = Bj ∂ ξ B ∂ ξj ξj ∂ B ∆B = Bj ∂ ξ B ∂ ξj . i j i − i j − j i i j i − i j Replacing this expression in (A.67) and rearranging terms we finally write 1 δ = B B ∂ ξ B B ∂ ξj + B B ∂ ξ d3r. (A.68) Bik 2 i j j k − i k j k j j i ZV Taking the trace of this expression we find the Lagrangian variation in the magnetic energy as B 1 δ = B Bj ∂ ξ B2∂ ξj d3r. (A.69) B i j i − 2 j ZV Rotational kinetic energy: The rotational kinetic energy tensor is written as 1 1 = ρ Ω2r r Ω r (Ω r) d3r = L˜ Ω d3r, (A.70) Rik 2 i k − i k · 2 ikj j ZV ZV where L˜ is a three-rank angular momentum defined as L˜ = ρ(r r Ω Ω r r ), with Tr(L˜ ) = L , ikj ikj i k j − i k j ikj j with L given by the second term in (A.35). We now calculate the variation in from (A.70) as j Rik 2δ = ρ Ω2 (r ξ + r ξ ) (Ω ξ) Ω r (Ω r) Ω ξ d3r (A.71) Rik i k k i − · i k − · i k ZV ρ rj ∆Ω Ω r + rj Ω ∆Ω r 2Ωj∆Ω r r d3r. − j i k j i k − j i k ZV The second term is associated to Lagrangian variations in the angular velocity. This can found by requiring the angular momentum to be conserved during the perturbation, i.e, δL˜ikj = 0. This condition implies 91

∆ (r r Ω Ω r r ) = 0. Multiplying this expression by Ω , solving for rj ∆Ω Ω r and replacing in i k j − i k j j j i k (A.71) we ﬁnd

2δ = ρ∆Ωj [r r Ω Ω r r ] d3r. Rik i k j − i k j ZV

The conservation of angular momentum also implies that ∆Lj = 0. From (A.45) we obtain

r2(Ω ∆Ω) (r ∆Ω)(r Ω) = 2(Ω ξ)(Ω r) 2Ω2(r ξ). · − · · · · − · Writing this expression in components and replacing in (A.71) we can ﬁnally write

1 δ = δ δ δ = ρ Ω2(r ξ) (Ω ξ)(Ω r) . (A.72) Rik 3 ik R R − · − · · ZV Variational form for the tensor virial equation We now collect the diﬀerent expressions appearing in (A.60). In order to account for an arbitrary geometry of the conﬁguration, one writes the Lagrangian displacement ξ as a linear combination of r as ξ = r , i i Oik k where is symmetric matrix, so that ∂ ξ = Tr( ) and ∂ ξ = . The variational form of the virial O j j O j i Oij equation then reads as 1 ω2 ρr ( r + r ) d3r = ρ rl∂ Φ d3r + δ (1 Γ) ρ rl∂ Φd3r (A.73) − 2 l Olk i Oil k Ojl j ik ik − Ojl j ZV ZV ZV 1 + δ (Γ 2) B2Tr( ) B B d3r ik − 2 O − i j Oij ZV B Bj + B Bj B B Tr( ) d3r − i Ojk k Oji − i k O ZV 1 + δ (5 3Γ) ρ rl (Ω r)Ω Ω2r d3r + , 3 ik − Ojl · j − j Gik ZV where the tensor δ contains the contribution from surface integrals. Taking the trace of this Gik ≡ Sik expression we have the scalar equation for the oscillation frequency as 1 ω2 = [(4 3Γ) (Tr( )Tr(B) + Tr( W) 2Tr( B)) 2 (5 3Γ) Tr( R) ] , (A.74) Tr( I) − O O − O − − O − G O with Tr( ) = and ˜ = and so on for the other tensors. For spherical symmetry and radial OI Oij Iij W Wij perturbations, one recovers the well known stability criteria

1 4 5 1 ω2 = Γ (1 δ) + 2 Γ β γ , (A.75) ε − 3 − 3 − − 3 where ε = 3 / , δ = / , β = / and γ = / . This has the following implications. In I |W| B |W| R |W| G |W| the first place, a non-rotating configuration is unstable under Lagrangian perturbations for Γ < 4/3, provided that δ < 1. This imposes a lower value for the magnetic field intensity as a function of the gravitational potential energy for the system to be stable (see for instance [53]). On the other hand, rotating configurations with δ 1 are unstable for Γ 4/3. The set of stability occurs for 4/3 < ≤ Γ < 5/3, where the effects of the rotational term is to decrease the oscillation frequency. For Γ = 5/3, the contributions to the oscillation frequency due to rotational effects vanishes, while for Γ > 5/3 the rotational term makes the frequency ω to increase and hence rotation tends make the configuration more stable. 92 APPENDIX A. THE VIRIAL EQUATION APPENDIX B

An eﬀect with generalized vacuum energy density

We assume that the contribution from background comes from a CDM component and a generalized vacuum energy density ρx, i.e, dark energy, Chaplygin Gas or scalar ﬁelds. The cdm component scales as 3 f(a) a− . The vacuum energy density scales as ρ = ρ a− , where ∼ x vac a 3 1 + ωx(a0) f(a) = da0. (B.1) ln a a Z1 0 For the cosmological constant case we get f(a) = 0. For ωx = constant, f(a) = 3(1 + ωx). For Chaplygin Gas one has

1 B nβ f(a) = ln 1 + a− , (B.2) −β ln a A where A and B are constants given in (2.23). We have used the generalized equation of state for Chaplygin β nβ 1/β Gas ωx = p/ρ = ωch κρc−h , with n = 3(1 + ωch). Chaplygin Gas scales as ρch(a) = A + Ba− . − 1 The vacuum energy density derived from Chaplygin Gas reads as ρ A β . With these contributions vac ∼ the potential exp reads W exp 1 2 3 f(a) = H Ω a− + (1 + 3ω (a))Ω a− . (B.3) W −2 0 cdm x vac I For the various eﬀects found in last paper on spherical as well as in non spherical conﬁgurations, we found a correction term proportional to the factor ζ = 2ρvac/ρ, which is the case for a cosmological constant a/a¨ = Λ/3. This appears if one neglects the contribution from the cdm component and take ω = 1. x − For a generalized equation of state and taking into account the CDM component, the factor ζ is replaced by

1 f(a) Ωcdm 3 ζ ζ 1 + 3ω (a) a− a− , (B.4) → 2 | x | − Ω vac where ζ is still ζ = 2ρvac/ρ in the r.h.s of this expression. The quantity ρvac is understood as the value 1/β that each model of vacuum energy density gives at the present time, i.e, V (φ0) for scalar ﬁelds, A for Chaplygin gas e.t.c.

A peculiar eﬀect Let us consider the angular velocity on an oblate ellipsoid. We had Ω2/2πρ = Ω2(1 (1/2)g(e)ζ), where 0 − g(e) is function of the eccentricity e2 = 1 a2/a2 and Ω2 is the angular velocity for ζ = 0. In the − 3 1 0

93 94 APPENDIX B. AN EFFECT WITH GENERALIZED VACUUM ENERGY DENSITY

1.5

1.4

1.3

1.2

1.1

0.9

scale factor 0.8

0.7 Ω vac = 0.7 0.6 0.8 0.9 0.5

0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 eccentricity

Figure B.1: Critical scale factor as a function of the eccentricity for ζ ≈ 0.2 and diﬀerent values of Ωvac for a ﬂat universe generalization we have

2 Ω 2 1 f(a) Ωcdm 3 = Ω 1 g(e)ζ 1 + 3ω (a) a− a− . (B.5) 2πρ 0 − 4 | x | − Ω vac

We now concentrate on the term within the parenthesis and ﬁnd a scale factor ac such that this term vanishes (for a ﬁxed value of e and ρ). Let us consider three cases: ΛCDM, DECDM and Chaplygin Gas.

ΛCDM: We take ω = 1. The value of a is then determine from • x − c 1 3 1 Ωcdm 3 ζg(e)Ωcdm 1 g(e)ζ 1 a− = 0 a = . (B.6) − 2 − 2Ω ⇒ c 2Ω (ζg(e) 2) vac vac −

The last equality implies that the density of the conﬁguration should be as ρ < g(e)ρvac in order to get a > 0.

Dark Energy-CDM: Now consider a dark energy component with ω constant and diﬀerent of • x 1. Deﬁning X = a3, we get the following expression for X: − c

9ζg Ωcdm 9ζg Xωx+1 + Xωx 1 + 3ω = 0. (B.7) 4 Ω − 4 | x| vac Chaplygin Gas. In this case, the critical value for the scale factor is given from the solution of the • equation for the density at ac:

β 4ρvac β 1 3κ ρ + ρ − = 0. (B.8) ch gζ(1 + 3ω ) ch − 1 + 3ω ch ch

For the pure Chaplygin gas with with ωch = 0 (n = 3) and γ = 1 (β = 2), this can be solved as

2 2 1 3g ζ ρch = g− 1 + 1 ρ, (B.9) "r 4 − # 95 which gives an scale factor

6 (B/A) B 9 ac = 2 = 2 1. (B.10) 2 2 2 A 64π α − ζ 1 + 3g ζ 1 1 4g2 4 − − q In the equation for the ratio B/A we have assumed that the parameter κ can be written for the 2β generalized Chaplygin gas as κ = αnH0 and Ωch = 1. H0 is the Hubble’s parameter. It was shoen in chapter 2 that for the pure CG α 0.01 and B/A 0.42 ≈ ≈ 96 APPENDIX B. AN EFFECT WITH GENERALIZED VACUUM ENERGY DENSITY APPENDIX C

Schwarszchild-de Sitter metric

This appendix is devoted to show explicitly the demonstration of Birkoff’s theorem by deriving the Schwarszchild’s solution of Einstein field equation with cosmological constant via the formalism of coordinate basis [89]. In curved spaces, the metric tensor is written as g˜ = g d˜xµ d˜xν , where d˜xµ are the basis forms for µν ⊗ the T ∗ space which are dual to the basis vectors e ∂ . This represents the coordinate basis, in terms p µ ≡ µ of which the usual definitions of Christoffel symbols, curvature tensor are related quantities are written. However, one can assume a coordinate transformation such that the metric can be written simply as the a b Minkowski’s metric tensor, i.e, g˜ = ηabθ˜ θ˜ . One therefore defines a transformation between one-forms ã a ˜ µ ⊗ basis as θ = e µdx such that g˜ = g d˜xµ d˜xν = g eµ eν θã θ˜b = η θã θ˜b, µν ⊗ µν a b ⊗ ab ⊗ µ ν a b which implies ηab = e ae bgµν and gµν = eµ eν ηab. Hence, given a vector V , we can write it’s components µ a in both basis as V = V eµ = V ea, that is, we can relate the components of the vector V in both basis a a µ a a ν µ as V = eµ V , which can also be extended to tensors of higher ranks V b = eµ e bV ν . One can also find the relation between the affin connection in the coordinate basis (Christoffel’s symbols) and the spin connection by writing the covariant derivative of a vector, since it must be independent of the chosen basis. That is, if X is a vector, then X = ( Xν ) d˜xµ e = ∂ Xν + Γν Xλ d˜xµ e . ∇ ∇µ ⊗ ν µ µλ ⊗ ν If we make the same operation ina mixed basis and transforming to the coordinate basis we can find X = ∂ Xν + ω a Xb d˜xµ e = ∂ Xν + (eν ∂ ea) Xλ + eν eb ω a Xλ d˜xµ e . ∇ µ µ b ⊗ a µ a µ λ a λ µ b ⊗ ν Then, the Christoffel’s symbolscan be derived from the spin connection by

ν ν a ν b a Γµλ = e a∂µeλ + e aeλ ωµ b. (C.1) In the absence of Torsion, Cartan’s structure equations for the spin connection and curvature tensor reads respectively as [89] dθã + ωã ω˜c = 0, ã d˜ωã ωã ω˜c = 0. (C.2) c ∧ b R b − b − c ∧ b We now apply this formalism to find the solution to the field equations for an static and spherically symmetric configuration. Let us assume that the metric which is a solution of the field equations for such configuration can be written as ds2 = e2α(r)dt2 + e2β(r)dr2 + r2dΩ2, (C.3) −

97 98 APPENDIX C. SCHWARSZCHILD-DE SITTER METRIC where dΩ2 = dθ2 + sin2 θdφ2 is the metric of a two sphere and where the functions α and β must be such that this line element satisfy the ﬁeld equations with cosmological constant (see equation (1.1)). The metric tensor can be rewritten by introducing the basis one-forms in the orthonormal basis as

ˆ ˆ θ˜tˆ = eα(r)d˜t, θ˜rˆ = eβ(r)d˜r, θ˜θ = rd˜θ, θ˜φ = r sin θd˜φ, (C.4) together with the vector basis

α(r) ∂ β(r) ∂ 1 ∂ 1 ∂ e = e− , e = e− , e = , e = . (C.5) tˆ ∂t rˆ ∂r θˆ r ∂θ φˆ r sin θ ∂φ Using (C.4) in the structure equation (C.2) we obtain for the non zero spin connections:

tˆ dα β tˆ θˆ 1 β θˆ φˆ 1 β φˆ φˆ 1 φˆ ω˜ = e− θ˜ , ω˜ = e− θ˜ , ω˜ = e− θ˜ , ω˜ = cot θθ˜ . (C.6) rˆ dr rˆ r rˆ r θˆ r Using again (C.4) we then may write

tˆ dα α β θˆ β φˆ β φˆ Γ = e − Γ = e− , Γ = sin θe− , Γ = cos θ. trˆ dr θrˆ φrˆ φθˆ Using (C.1) we ﬁnd the nonzero Christoﬀel’s symbols: dα 1 dβ Γt = Γθ = Γφ = Γr = tr dr θr φr r rr dr φ r 2(α+β) dα r 2β 2 Γ = cot θ Γ = e Γ = e− r sin θ . φθ tt dr φφ − r 2β θ 1 φ 1 Γ = re− Γ = sin θ Γ = csc θ θθ − rφ r rθ r From (C.1) and using (C.6) we have for the independent curvature two-form

2 2 tˆ 2β d α dα dβ dα rˆ tˆ ˜ = e− + θ˜ θ˜ , R rˆ dr2 − dr dr dr ∧ " # tˆ 1 2β dα tˆ θˆ ˜ = e− θ˜ θ˜ , R θˆ − r dr ∧ rˆ 1 2β dβ rˆ θˆ ˜ = e− θ˜ θ˜ , R θˆ r dr ∧ rˆ 1 2β dβ rˆ φˆ ˜ = e− θ˜ θ˜ , R φˆ r dr ∧ θˆ 1 2β θˆ φˆ ˜ = 1 e− θ˜ θ˜ . (C.7) R φˆ r2 − ∧ ˜a 1 a ˜c ˜d Hence, we can read the elements of the curvature tensor in this coordinates, since b = 2 bcdθ θ . µ R R ∧ The conversion to the coordinate basis is easily done from = eµ e be ce d a . One ﬁnds R νµλ a ν µ λ R bcd dα dβ d2α dα 2 t = R rtr dr dr − dr2 − dr t 2β dα = re− R θtθ − dr t 2 2β dα = r sin θe− R φtφ − dr 2 2 r 2(α β) d α dα dα dβ = e − + R trt dr2 dr − dr dr " # r 2β dβ = re− R φrφ dr r 2 β dβ = r sin θe− (C.8) R φrφ dr θ 2 2β = sin θ 1 e− . R φθφ − 99

By contracting, we have the diagonal elements of the Ricci tensor as

2 2 2(α β) d α dα dα dβ 2 dα = e − + + Rtt dr2 dr − dr dr r dr " # d2α dα 2 dα dβ 2 dα = + Rrr − dr2 dr − dr dr − r dr " # 2β dβ dα = e− r r 1 + 1 Rθθ dr − dr − = sin2 θ . Rφφ Rθθ Hence, the scalar of curvature is given as

2 2 2 2β 2 dβ 2 dα 1 d α dα dβ dα = + 2e− + . R r2 r dr − r dr − r2 − dr2 dr dr − dr " # The ﬁeld equations for a perfect ﬂuid with cosmological constant are then written as

2β dβ 2β 2 e− 2r + e 1 = 8π (ρ + ρ ) r dr − vac 2β dα 2β 2 e− 1 + 2r e = 8π (p ρ ) r dr − − vac 2 2 2β d α dα dα dβ 1 dα 1 dβ 2 e− + + = 8π (p ρ ) r . (C.9) dr2 dr − dr dr r dr − r dr − vac " # 2β 1 By defining the function m(r) such that e = (1 2m(r)/r)− and using the first equation in (C.9), we − obtain r dm 1 2 2 2 = (8πρ + Λ) r 4πρ0r = m(r) = 4πρ0(r0)r0 dr0. (C.10) dr 2 ≡ ⇒ Z0 The function m(r) can be identified with a enclosed mass associated to the density ρ0 = ρ+ρvac. Replacing this in the second line of equation of (C.9) we find

3 8 3 dα 4πpr + M(r) πρvacr = − 3 , (C.11) dr r(r 2M(r) 8 πρ r3) − − 3 vac where M(r) is the proper mass inclosed at a radius r:

r 2 4 3 M(r) = 4πρ(r0)r0 dr0 = m(r) πρ r . (C.12) − 3 vac Z0 On the other hand, from the conservation of the energy-momentum tensor for a perfect ﬂuid µν = 0 ∇µT we obtain another expression for dα/dr as

dα 1 dp = . (C.13) dr − ρ + p dr By combining these expression one obtains the Tolman-Oppenheimer-Volktoﬀ equation (ΛTOV), which represents hydrostatic equilibrium in general relativity with cosmological constant:

dp 1 p 12πpr3 + 3M(r) 8πρ r3 = ρ 1 + − vac . dr −3r ρ r 2M(r) 8 πρ r3 − − 3 vac Hence, the equation for α(r) is then

r 1 2 2m(r) − dp(r0) α(r) = 1 4π + m(r0) dr0 + α(0). (C.14) r 2 − r dr Z0 0 100 APPENDIX C. SCHWARSZCHILD-DE SITTER METRIC

One can also obtain an equation for α(r) by introducing an equation of state. Our most used example is p = ωργ . Integrating (C.13) one has

ωγ ωργ 1 + 1 ω(γ−1) α(r) = ln c − . (C.15) ωρ(r)γ 1 + 1 − where ρc is the central density. Hence, the line element is written with the help of equations (C.12) and (C.15) as

ωγ − 1 ωργ 1 + 1 ω(γ 1) 2M(r) 8 − ds2 = c − dt2 + 1 πρ r2 dr2 + r2dΩ2. − ωρ(r)γ 1 + 1 − r − 3 vac − The vacuum solution ρ = p = 0 for r > 2M then reads for α and β:

α 2M 8 2 β e = 1 πρ r = e− . − r − 3 vac Then Schwarszchild de Sitter metric for = 0 reads as Tµν 2 2 1 2 2 2 ds = (1 2Φ) dt + (1 2Φ)− dr + r dΩ , − − − where the generalized potential reads as as M 4 Φ = + πρ r2, − R 3 vac which in turns deﬁnes the so called Newton-Hooke space time. APPENDIX D

Roots of third and fourth order equation

Third order equation This solution is based on reference [90]. Consider the third order equation ax3 + bx2 + cx + d = 0, where a, b, c, d are real coefficients and a = 0; this expression is converted to 6 x3 + rx2 + sx + t = 0, (D.1) with r = b/a, s = c/a, t = d/a. We can write (D.1) in a more convenient form as 1 y3 + py + q = 0, with y = x + r, (D.2) 3 where 3s r2 2r3 rs p = − , q = + t. (D.3) 3 27 − 3 Two relevant quantities for the solution of (D.2) are defined: the discriminant D and R 1 1 1 D = p3 + q2, R = sgn(q) p . 27 4 r3| | The roots of (D.2) are given in the next cases as a function of the auxiliary angle φ defined in each case. First, if p < 0 we have i) D 0 : The three roots of (D.2) are given as ≤ φ φ 2π φ 4π y = 2R cos , y = 2R cos + , y = 2R cos + , 1 − 3 2 − 3 3 3 − 3 3 q where cos φ = 2R3 . ii) D > 0: The roots are φ φ φ φ φ y = 2R cosh , y = R cosh +i√3R sinh , y = R cosh i√3R sinh , 1 − 3 2 3 3 3 3 − 3 q where cosh φ = 2R3 On the other hand, if p > 0 we have the solution φ φ φ φ φ y = 2R sinh , y = R sinh + i√3R cosh , y = R sinh i√3R cosh , 1 − 3 2 3 3 3 3 − 3 (D.4) q where sinh φ = 2R3 .

101 102 APPENDIX D. ROOTS OF THIRD AND FOURTH ORDER EQUATION

Fourth order equation Consider the fourth order equation ax4 + bx3 + cx2 + dx + e = 0, where a, b, c, d, e are real coeﬃcients and a = 0. This can be written in a reduced form as 6 b y4 + py2 + qy + r = 0, with y = x + . 4a This expression can be reduced to a third order equation in the form

z3 + 2pz2 + p2 4r z q2 = 0. (D.5) − − Given the roots of(D.5), whic h can be derived from the ﬁrst part, we have [90] 1 y = (√z + √z √z ) , (D.6) 1 2 1 2 − 3 1 y = (√z √z + √z ) , 2 2 1 − 2 3 1 y = ( √z + √z + √z ) , 3 2 − 1 2 3 1 y = ( √z √z √z ) . 4 2 − 1 − 2 − 3 Example We show explicitly the solution of the equation (5.16). Let us write it now as

2 rcrit 3r 2 y4 α2y 12α3 = 0, y L α s . (D.7) − − ≡ R ≡ 4R Λ Λ Hence, we can identify the coeﬃcients as

p = 0, q = α2, r = 12α3 Fourth order. (D.8) − − The reduced third order equation is then

z3 + 48α3z α4 = 0. − We use the method described to ﬁnd the roots of (D.2), with

p = 48α3 > 0 q = α4 Third order. (D.9) − Following (D.4), we can write the three roots of (D.9) as

3/2 1 1 1 1/2 z = 8α sinh sinh− α− (D.10) 1 3 128 3/2 1 1 1 1/2 3/2 1 1 1 1/2 z = 4α sinh sinh− α− 4iα √3 cosh sinh− α− 2 − 3 128 − 3 128 3/2 1 1 1 1/2 3/2 1 1 1 1/2 z = 4α sinh sinh− α− + 4iα √3 cosh sinh− α− . 3 − 3 128 3 128 In the limit α 0, we can use the approximation → 1 1 sinh− x = ln x + x2 + 1 ln(2x) cosh− x, x large, (D.11) ≈ ≈ p so that

1 1 1 1/2 1 1/6 1 1/6 sinh sinh− α− sinh ln α− α− . (D.12) 3 128 ≈ 4 ≈ 8 103

Under these approximations, the three roots are written as

4/3 z1 = α , (D.13) 1 z = α4/3 1 + i√3 = α4/3eiπ/3, 2 −2 − 1 4/3 4/3 iπ/3 z = α 1 i√3 = α e− . 3 −2 − − Then, from the equation (D.6) we have 1 1 y 0, y α2/3, y α2/3 1 i√3 , y α2/3 1 + i√3 . 1 ≈ 2 ≈ 3 ≈ −2 − 4 ≈ −2 Note that the ﬁrst root is not valid without the approximation α 1. The real and positive root for (D.7) under the approximation of α small is then given by

3r 4/3 y = s . 4R Λ

Hence, the solution for the critical value of rL is given as

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