Spherical Collapse Model and Cosmic Acceleration

UNIVERSIDADE FEDERAL DO RIO DE JANEIRO INSTITUTO DE FISICA

Duvan Ricardo Herrera Herrera

Ph.D. Thesis presented to the Graduate Program in Physics of the Institute of Physics of the Federal Uni- versity of Rio de Janeiro - UFRJ, in partial fulﬁllment of the requirements for the degree of Doctor of Philoso- phy(Physics).

Advisor: Ioav Waga Co-Advisor: Sergio E. Jor´as

Rio de Janeiro

April, 2019 ii iii

CIP - Catalogação na Publicação

Herrera Herrera, Duvan Ricardo H 565s Spherical Collapse Model and Cosmic Acceleration / Duvan Ricardo Herrera Herrera. -- Rio de Janeiro, 2019. 106 f.

Orientador: Ioav Waga. Coorientador: Sergio E Jorás. Tese (doutorado) - Universidade Federal do Rio de Janeiro, Instituto de Física, Programa de Pós Graduação em Física, 2019.

1. Dark energy. 2. Spherical collapse. 3. f(R) theories. 4. Structure Formation. 5. Cosmology. I. Waga, Ioav, orient. II. Jorás, Sergio E, coorient. III. Título.

Elaborado pelo Sistema de Geração Automática da UFRJ com os dados fornecidos pelo(a) autor(a), sob a responsabilidade de Miguel Romeu Amorim Neto - CRB-7/6283. iv

Resumo

Modelo de Colapso Esféricoe Acelera¸cãoCósmica

Duvan Ricardo Herrera Herrera

Orientador: Ioav Waga

Coorientador: Sergio E. Jor´as

Resumo da Tese de Doutorado apresentada ao Programa de PósGradua¸cão em F´ısicado Instituto de F´ısicada Universidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necessáriosàobten¸cãodo t´ıtulode Doutor em Ciências(F´ısica).

Compreender a influênciada acelera¸cãocósmica na forma¸cãode estruturas éatual- mente um dos grandes desafios em Cosmologia, jáque ela pode ajudar distinguir os diferentes modelos cosmológicosviáveis. Neste trabalho estudamos o modelo de colapso esféricoe determinamos seus principais resultados (como o contraste de densidade cr´ıtico, a evolu¸cãodo raio e densidade do númerode halos) quando a acelera¸cão cósmicaémod- elada de duas formas. Na primeira, consideramos os modelos de gravidade modificada f(R) na abordagem métrica;em particular, usamos os chamados limite de campo forte (F = 1/3, onde a gravidade émodificada com uma constante de Newton maior) e limite de campo fraco (F = 0, onde a gravidade nãoémodificada devido ao efeito camaleão). No últimocaso, contemplamos modelos de energia escura “phantom” e “non-phantom”, onde ela pode se aglomerar parcialmente (ou totalmente), de acordo com um parâmetro livre γ.

Palavras-chave: Energia escura. Teorias f(R). Colapso esf´erico. L´ımitede campo fraco. L´ımitede campo “forte”. Cosmologia. Forma¸c˜aode estruturas. v

Abstract

Spherical Collapse Model and Cosmic Acceleration

Duvan Ricardo Herrera Herrera

Advisor: Ioav Waga Co-Advisor: Sergio E. Jor´as

Abstract da Tese de Doutorado apresentada ao Programa de PósGradua¸cão em F´ısicado Instituto de F´ısicada Universidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necessáriosàobten¸cãodo t´ıtulode Doutor em Ciências(F´ısica)

Understanding the influence of cosmic acceleration on the formation of structures is currently a major challenge in Cosmology, since it can distinguish otherwise degenerated viable models. In this work we study the Spherical collapse model and determinate its outcomes (such as critical density contrast, radius evolution and number density of halos) when the cosmic acceleration is modeled of two ways. In the former we consider f(R) modified-gravity models in the metric approach; in particular, we use the so-called large (F = 1/3, where the gravity is modified with a larger Newton’s constant) and small-field (F = 0, where the gravity is not modified due chameleon effect) limits. In the latter we contemplate phantom and non-phantom dark energy models, which can partially (or totally) cluster, according to a free parameter γ.

Keywords: Dark energy. f(R) theories . Spherical collapse. Small ﬁeld limit. Large

ﬁeld limit. Cosmology. Structure Formation. vi

Acknowledgments

My deepest gratitude to Professors Ioav Waga and Sergio Jorásfor the great help, dedication, and time spent for me could conclude this work. Their knowledge in both cosmology and program were fundamental to solve the theoretical and computational problems presented throughout this thesis. Also thank to my friends Dalma,Daniel, Leonardo, Guarumo, Omar, Matu and Rhonald for the discussions on physics, politics, philoso- phy, among others, and my mother for her infinite and unconditional support. Finally, I thank to CAPES for the financial support that made possible the accomplishment of this Ph.D, and to officials (especially them of the Postgraduate) and professors of the Physics Institute. vii

Contents

Contents vii

List of Figuresx

1 Introduction1

2 Relativistic Cosmology and Standard ΛCDM Model5

2.1 The Cosmological Principle...... 5

2.2 Background Evolution...... 7

2.3 Energy content of the Universe...... 11

2.3.1 Baryonic matter...... 11

2.3.2 Dark matter...... 11

2.3.3 Relativistic components (rc)...... 15

2.3.4 Dark energy...... 19

2.4 ΛCDM model...... 24

3 Dark Energy and cosmic acceleration 26

3.1 f(R) gravity theories...... 26

3.1.1 Hu-Sawicki model...... 33

3.1.2 γ gravity model...... 36

4 Spherical collapse model 39

4.1 Spherical Model in a Universe dominated by matter...... 39 viii

4.1.1 Virialization...... 42

4.2 Spherical collapse in the ΛCDM model...... 43

4.2.1 Virialization...... 44

5 Calculation of the critical overdensity in the spherical-collapse approximation 46

5.1 Spherical collapse in f(R) theories...... 48

5.2 Calculating the critical density contrast δc ...... 50 5.2.1 The diﬀerential-radius method...... 51

5.2.2 The constant-inﬁnity method and its mending...... 52

5.3 Comparing the results from diﬀerent approaches...... 54

6 Number density of halos in f(R) theories 60

6.1 Comparing the f(R) models with ΛCDM...... 60

7 Top-Hat Spherical Collapse with Clustering Dark Energy: Radius Evo- lution and Critical Density Contrast 67

7.1 Spherical collapse with dark energy perturbations...... 68

7.2 Bubble Evolution...... 73

7.2.1 Radius...... 74

7.2.2 The critical contrast density...... 76

8 Conclusions and future works 84

Bibliography 86

A SC model equations in f(R) theories 97

B Solution of the diﬀerential equation for the density contrast in the linear

regime 100 ix

C Converting from Mv to Mh 103

D Solution of linear equations for δm and δde 105 x

List of Figures

2.1 Full sky map of the CMB temperature anisotropies observed by Planck

satellite. Figure taken from Ref. [1]...... 6

2.2 Hubble diagram for a compilation of data. Type Ia supernovae (squares),

Tully-Fisher clusters (solid circles), fundamental plane clusters (triangles), surface brightness ﬂuctuation galaxies (diamonds) and Type II supernovae (open squares). Figure taken from [2]...... 7

2.3 Representation the spiral galaxy inside a dark matter halo...... 12

2.4 Angular Temperature spectrum. Figure taken from Ref[1]...... 17

3.1 Eﬀective Dark Energy Equation of State for the Hu-Sawicki f(R) model

0 as a function of redshift z. Figure taken from [3] where fR ≡ fR0...... 35

3.2 Eﬀective equation-of-state parameter wde as a function of z for n = 1, 2 and 3 and diﬀerent values of α. Figure taken from [4]...... 38

5.1 (a,left panel) Critical density contrast δc as a function of Ωm0 for F = 0 (solid blue line) and F = 1/3 (dotted red line) when the collapse occurs

at redshift zc = 0, following our approach. (b,right panel) Critical density

contrast δc as function of redshift collapse zc for F = 0 (solid blue line) and

F = 1/3 (dotted red line) with Ωm0 = 0.3, following our approach...... 54 xi

5.2 Relative errors for δc (see text for definition) between our method and (a) the differential-radius one (dotted black line), the constant-infinity one,

with (b) Inff = 105 (dashed red line) and (c) Inff = 108 (solid blue line) for F = 0 (left panel) and F = 1/3 (right panel). For all models, we assumed

Ωm0 = 0.3...... 55

5.3 Relative errors for nln M (see text for deﬁnition) between our method and

the constant-inﬁnity one, with (a) Inff = 108 (solid blue line) and (b) Inff = 105 (dashed red line) for redshift z = 0 in the cases F = 0 (left

panel) and F = 1/3 (right panel). For all models, we assumed Ωm0 = 0.3 and h = 0.6774...... 57

5.4 Relative errors for nln M (see text for deﬁnition) between our method and

the constant-inﬁnity one, with (a) Inff = 108 (solid blue line) and (b) Inff = 105 (dashed red line) as a function of redshift for F = 0, F = 1/3

13 −1 and virial mass of 10 h M (upper panels). The same plots, are shown

15 −1 for 10 h M (lower panels). For all models, we assumed Ωm0 = 0.3 and h = 0.6774...... 58

5.5 Nbin (see text for deﬁnition) as a function of redshift for F = 0 and virial

13 14 −1 14 15 −1 masses between 10 and 10 h M (left panel) and 10 and 10 h M

(right panel). As always, we assumed Ωm0 = 0.3 and h = 0.6774...... 58

5.6 Relative errors for Nbin (see text for deﬁnition) between our method and

the constant-inﬁnity one, with (a) Inff = 105 (dashed red line) and (b) Inff = 108 (solid blue line) as a function of redshift for virial masses between

13 14 −1 10 and 10 h M and F = 0 (left panel) and F = 1/3 (right panel).

For all models, we assumed Ωm0 = 0.3 and h = 0.6774...... 59 xii

HS Λ −4 −5 −6 6.1 ∆n300/n300, for fR0 = 10 , 10 and 10 (from top to bottom). In all panels, the case F = 0 (F = 1/3) is given by the upper red dashed (lower

solid blue) curve...... 64

γ Λ 6.2 ∆n300/n300 for n = 2 and α = 1.05, 1.18 and 1.5 (from top to bottom). In all panels, the case F = 0 (F = 1/3) is given by the upper red dashed (lower solid blue) curve...... 64

6.3 Eﬀective equation-of-state parameter wDE of γG as a function of the red-

shift z with α = 0.86, n = 3 e Ωm0 = 0.267. The amplitudes of the peaks decrease with larger n, while their redshifts slowly grow with n...... 65

γG Λ 6.4 ∆n300/n300 for α = 0.86, n = 3 e Ωm0 = 0.267...... 65

7.1 Behavior of the equation-of-state parameter inside the bubble (wc) with

respect to time, for the labeled parameters...... 72

2 7.2 Evolution of the scale radius of the collapsing sphere for cs = w = {−0.9, −1.1} and diﬀerent values of γ. The solid blue line corresponds to the ΛCDM

model...... 76

2 7.3 Evolution of the scale radius of the collapsing sphere for cs = 0, w = {−0.9, −1.1} and diﬀerent values of γ. The solid blue line corresponds to the ΛCDM model...... 77

2 7.4 Evolution of the scale radius of the collapsing sphere for cs = 1, w = {−0.9, −1.1} and diﬀerent values of γ. The solid blue line corresponds to the ΛCDM model. Note the non-collapsing curve (γ = 0, w = −1.1). We

2 also point out that the curve given by γ = 0, cs = 1 and w = −0.9, that crosses ΛCDM close to the collapse, is also dissonant in Fig. 7.7...... 77

2 7.5 Evolution of the scale radius of the collapsing sphere for cs = γ, w = {−0.9, −1.1} and diﬀerent values of γ = {0, 0.8}. The solid blue line cor-

responds to the ΛCDM model...... 78 xiii

2 7.6 Evolution of the critical contrast density for cs = 0 and diﬀerent values of w and γ. The solid black line corresponds to ΛCDM model...... 80

2 7.7 Evolution of the critical contrast density for cs = 1 and diﬀerent values of w and γ. The solid black line corresponds to ΛCDM model...... 81

2 7.8 Evolution of the critical contrast density for cs = w and diﬀerent values of w and γ. The solid black line corresponds to ΛCDM model in all panels. As before, here we ﬁnd the largest deviations from ΛCDM...... 82

2 7.9 Evolution of the critical contrast density for : cs = γ and diﬀerent values of w and γ. The solid black line corresponds to ΛCDM model in all panels. 83 xiv Notation and Conventions

1. Speed of light: c = 1

2. Gravitational constant: G = 1

3. Planck constant: ~ = 1

4. Boltzmann constant: kB = 1

5. κ2 = 8πG

6. Metric tensor: gµν

7. Metric signature: (− + ++)

8. Greek indices changes run 0 to 3

9. Latin indices changes run 1 a 3

10. Coordinates: xµ = {x0, x1, x2, x3}

∂f 11. Partial derivative: ∂xµ = f,µ

˙ df 12. Time derivative: f = dt

α α α δ δ α 13. Covariant derivative: ∇µGβ = Gβ,µ + ΓµδGβ − ΓµβGδ

α αβ 14. Christofell connections: Γµν = g (gµβ,ν + gνβ,µ − gµν,β)

λ λ λ η λ η λ 15. Riemann Tensor: Rµνκ = Γµν,κ − Γµκ,ν + ΓµνΓκη − ΓµκΓνη

α 16. Ricci Tensor: Rµν = Rµαν

µν 17. Ricci Scalar: R = g Rµν

18. Momentum-energy tensor: Tµν xv

df(R) 19. fR(R) = dR

d2f(R) 20. fRR(R) = dR2

21. Scale factor: a

1 22. Redshift: z = a − 1

a˙ 23. Hubble parameter H = a xvi 1

Chapter 1

Introduction

According to the ΛCDM standard cosmological model, the process of structures formation at large scales (such as galaxies, clusters and superclusters of galaxies, etc.) has its origin at high redshifts. In this picture, through gravitational instability matter density perturbations grew, especially in the matter era forming gravitationally binding objects that we observe today [5]. While deviations are small, the growth of perturbations can be studied by linear theory. When deviations become large, the linear theory fails, nonlinear effects become important, and then it is necessary to study their evolution in the nonlinear regime. The most accepted scenario for the structure formation at large scales is the so-called botton-up, in which structures with small masses were formed first and later, by means of fusion of them, the large and massive structures were formed. This seems not possible without a fundamental characteristic for dark matter, it must be cold, that is, non-relativistic at the moment of its decoupling [6]. In 1972 James E. Gunn et al. proposed the so-called spherical collapse (SC) model for a fluid without pressure in his famous article “On the infall of matter into clusters of galaxies and some effects on their evolution”[7]. According to this model it is possible to determine the non -linear evolution of matter perturbations in a spherical region that is immersed in an expanding universe. Such a region is considered homogeneous and isotropic in relation to its center and therefore its pressure and density (which is infinitesimally greater than the 2 density of the background fluid) are functions only on time. To determine the evolution of perturbations in the mentioned region it is common to analyze the behavior of density contrast δm, that for SC depends only of time. According to this analysis the spherical region expands to a maximum radius rta and then collapses until it reaches a radius that is theoretically zero. The moment at which the region reaches the maximum radius is called ”turn-around” and the one at which the radius reaches zero is called the collapse moment.

It is clear that the spherical collapse is an idealization since a real density perturbation is neither spherical nor homogeneous. Besides, in general (with baryons) it is necessary to consider the pressure gradient in the evolution of perturbations. In this way the system will be stable, that is, the region does not contract or expands, in a certain radius greater than zero, called the viral radius. Even so, it is possible to calculate the virial radius in the spherical collapse model through an energy analysis assuming that the system satisﬁes the virial theorem.

The SC model provides a fundamental quantity for estimating the number count of halos at diﬀerent redshift and halo-mass bins, which is called the critical density contrast

δc. It indicates if, in a region of the Universe with a given value of δm, a gravitationally bound object is formed (speciﬁcally, the formation happens if δm > δc). For the SC model,

δc only depends on time due to the conditions imposed on the sphere.

On other hand, recent results [1,8] from independent cosmological observations — such as anisotropies in the Cosmic Microwave Background (CMB), Baryon Acoustic Oscilla- tions (BAO), type-Ia Supernovae (SNe Ia) and the Large-Scale Structure of the Universe (LSS) — imply that the expansion of the Universe is speeding up. [9, 10]. The responsi- ble for this effect is dubbed “dark energy” (DE), whose physical nature is still unknown. If we model dark energy as a fluid, according to General Relativity, it needs to have negative pressure. In particular, the cosmological model that better fits observations is the cold-dark-matter with Cosmological-Constant model (ΛCDM). However, this model 3 presents difficulties at theoretical level [11, 12], motivating the search for alternatives such as quintessence [13, 14, 15, 16], phantom dark energy [17], k-essence [18], decaying vacuum models [19, 20] or even modifications of General Relativity, such as f(R) theories [21], among others. A great difficulty is that many of these models behave very similarly to ΛCDM at the background level, making it difficult to distinguish them through cosmological kinematical tests (those that depend essentially only on distance). Therefore, it is crucially important to study the evolution of perturbations and the structure formation in those models, where they are expected to have different (and measurable) consequences from those obtained by ΛCDM.

The simplest way to study the structure formation with dark energy is through the

Top-Hat Spherical-Collapse (SC) approach, which was initially used in Einstein-de Sitter (EdS) background (as an useful benchmark since it yields an exact analytical result for the critical density), in the standard cold-dark-matter scenario [7], and later in ΛCDM

[22]. The SC model has also been extended to quintessence ﬁelds [23, 24, 25], decaying vacuum models [26], f(R) theories [27, 28, 29, 30] and DE with constant equation-of-state (EoS) models [31, 32, 33, 34].

In this thesis we study the SC model and determinate its outcomes (such as critical density contrast, radius evolution and number density halos) when the Universe acceleration is modeled in two ways. In the first we consider f(R) modified-gravity models in the metric approach; in particular, we use the so-called large (F = 1/3, where the gravity is modified with a larger Newton’s constant) and small-field (F = 0, where the gravity is not modified due chameleon effect) limits. In the second we contemplate phantom and non-phantom dark energy models, which can partially (or totally) cluster, according to a free parameter γ.

The structure of this thesis is as follows: In chapter 2 we show the set of basic equations that describes the dynamics of the Universe (without perturbations) that satisfies the Cosmological principle. In chapter 3 we show the characteristics of the modified grav- 4 ity theories f(R) and we obtain the modified field equations necessary to describe the spherical collapse in this formalism. In chapter 4 we describe the standard SC model and determinate its main outcomes. In chapter 5 we study the SC model in f(R) theories for the large and small-field, where we discussed the different methods to calculate the critical density contrast and their consequences in the comoving number density of halos.

The results of this work were published in [30]. In chapter 6 we determine the comoving number density of halos for theses models in the small-ﬁeld and large-ﬁeld limits using the collapse spherical approximation when the background is given by two models of f(R) gravity (γ gravity model and the Hu-Sawicki one). In chapter 7 we consider the SC model with dark energy, which can partially (or totally) cluster, according to a free parameter γ. Besides, we determine characteristic quantities for the SC model, such as the critical contrast density and radius evolution, with particular emphasis on their dependence on the clustering parameter γ. The results of this work are shown in the article “Top-Hat Spherical Collapse with Clustering Dark Energy I:Radius Evolution and Critical Contrast

Density” [35]. Finally, we conclude in chapter 8. 5

Chapter 2

Relativistic Cosmology and Standard ΛCDM Model

2.1 The Cosmological Principle

The Cosmological Principle is the basic hypothesis for the cosmological standard model. According to this principle, we are typical observers that, on very large scales (about 100 Mpc or greater) the Universe is spatially homogeneous and isotropic. Many cosmological observations support this hypothesis; for example the anisotropies in the Cosmic Microwave Background (CMB) of the order of ∆T/T ∼ 10−5 is a proof of the isotropy in the early phases of the Universe. In ﬁgure (2.1), the full sky map of the CMB temperature anisotropies is shown with an angular resolution of 5 arcmin, obtained thanks to the measurements of the Planck satellite [1,8]. In this map, the temperature variations with respect to mean the value go from −0.0003K for the coldest sky regions (blue) to 0.0003K for the hottest (red) ones. 6

Figure 2.1: Full sky map of the CMB temperature anisotropies observed by Planck satellite. Figure taken from Ref. [1]

Likewise, different statistical analyses of galaxies distribution confirm the spatial homogeneity in more recent times. For example in refs. [36, 37, 38] is estimated the homogeneity of different Gamma-Ray Bursts samples. On large scales those galaxy samples become homogeneous, which is in accordance with the Cosmological Principle.

Another main characteristic of the Universe is that it is expanding. In 1927 Georges Lemaˆıtreproposed and observationally demonstrated [39] it when he analyzed the measurements of the distance of galaxies and their velocities via Doppler eﬀect at small redshift z . He found that the farther a galaxy is, the faster it recedes from us. He then formulated a linear relation between the distance r and the recession velocity v,

v = H0r, (2.1)

where H0 is constant (now known as the Hubble constant). Two years latter, the American astronomer Edwin Hubble conﬁrmed this relation. Recently, analysis of diﬀerent measurements corroborates this law. For example, in

ﬁgure (2.2) is shown a diagram (known as Hubble diagram) for a compilation of data which includes Type-Ia supernovae (squares), Tully-Fisher clusters (solid circles), fundamental- plane clusters (triangles), surface-brightness-ﬂuctuation galaxies (diamonds) and Type-II supernovae (open squares), where we can observe a linear relation between the distance 7

Figure 2.2: Hubble diagram for a compilation of data. Type Ia supernovae (squares), Tully-Fisher clusters (solid circles), fundamental plane clusters (triangles), surface brightness ﬂuctuation galaxies (diamonds) and Type II supernovae (open squares). Figure taken from [2]

−1 −1 and recession velocity with H0 = 72 ± 8 km s Mpc [2].

2.2 Background Evolution

With those geometrical properties, the Universe on large scales is described by the Friedmann- Lemaˆıtre-Robertson-Walker (FLRW) metric, which in spherical coordinates (r, θ, ψ) can be written as:

dr2 ds2 = −dt2 + a(t)2 + r2(dθ2 + sin2θdϕ2) , (2.2) 1 − Kr2 8 where a(t) is known as the scale factor, which determines the physical—distance evolution

1 — its current value a0 is conventionally set to one , and t is called the cosmic time. The constant K is associated with the spatial curvature and it is positive, negative or zero. For an observer at the origin of the coordinate system, the physical position of an object, in a FLRW spacetime is R = a(t)x, where x is its comoving coordinate. Deriving the physical position R with respect to time t, we get

a˙ R˙ = R + ax.˙ (2.3) a

. d where ≡ dt . The second term on the right-hand side is known as the peculiar velocity of the object; for a comoving object it is zero and therefore (2.3) takes the form:

R˙ = H(t)R, (2.4) where R˙ in this case is known as the recession velocity and H(t) ≡ a/a˙ is the Hubble parameter, which indicates the expansion rate of Universe. This expression is known as the Hubble law. Measurements done by the Planck satellite [8] indicate that the Hubble

−1 −1 constant’s current value is H0 = 67.74 ± 0.40km s Mpc at 68% conﬁdence level. Note that for small redshifts we can consider H(t) ≈ H0 in equation (2.4) and so equation (2.1) is recovered. Usually H0 is expressed in 100km/s/Mpc units, that is:

H0 = 100h km/s/Mpc. (2.5)

So, from the value of H0 we get that h = 0.6774 ± 0.0046. Due to the cosmological principle, the components of the Universe can be approximated as perfect fluids (that is, without anisotropic stress or heat flow) being both their pressure p and their energy density ρ functions of time t, so the stress-energy tensor for each fluid ”i” is given by

i i i Tµν = (ρi + pi)UµUν + pigµν, (2.6)

1which can always be done if K = 0. 9

i where gµν is the metric and Uµ is the ﬂuid four-velocity. In the ﬂuid rest frame Uµ =

µ (1, 0, 0, 0) and Tν = diag(−ρ, p, p, p), therefore, using the covariant conservation of the stress-energy tensor,

(i)µ ∇µTν = 0, (2.7) with a metric (2.2) we get the continuity equation:

ρ˙i + 3H(ρi + pi) = 0. (2.8)

Note that we assume that each ﬂuid satisﬁes equation (2.8), which indicates that there is no energy transfer between them.

To determine the evolution of the energy density through the continuity equation, it is necessary to establish a relation between it and the pressure. This relation is known as the equation of state (EoS), taking the form

pi = wiρi, (2.9) for the diﬀerent cosmic components, since they are considered both dilute and barotropic

ﬂuids according to the cosmological principle. The equation-of-state parameter, wi, in general, is a function of time. Integrating equation (2.8) and using the EoS (with wi constant), a relation between the energy density and the scale factor is found and it is given by

−3(1+wi) ρi = ρi0a , (2.10) where ρi0 is the current energy density.

We can associate an energy density ρK to the term related to the spatial curvature K, given by: 3K ρ ≡ − , (2.11) K 8πGa2 and a pressure equal to K p ≡ , (2.12) K 8πGa2 10

so, its EoS parameter is wK = −1/3. The evolution of Universe is determined by the scale factor alone, whose dynamics is deﬁned by the Einstein ﬁeld equations (without Cosmological constant):

1 R − Rg = 8πGT , (2.13) µν 2 µν µν

P i where Rµν is the Ricci tensor, R is the Ricci scalar and Tµν = i Tµν is the total stress- energy tensor. The Einstein ﬁeld equations relate the space-time geometry (left-hand side) with the energy content (right-hand side). The curvature of the space-time is due both to the energy density and the pressure. Using the equations (2.2), (2.6), (2.9) and (2.13), the Friedman and acceleration equations, respectively, are obtained:

2 a˙ 8πG X H2 ≡ = ρ , (2.14) a 3 i i a¨ 4πG X = − ρ (1 + 3w ). (2.15) a 3 i i i Thus, the Friedman and acceleration equations together with Eq. (2.10) describe the unperturbed Universe dynamics or background evolution. In order to quantify how much a ﬂuid component ”i” contributes to the total energy density of the Universe, it is usually introduced the concept of density parameter Ωi, which is deﬁned as

ρi Ωi ≡ , (2.16) ρc where ρc is know as the critical density, which is given by

3H2 ρ = , (2.17) c 8πG and it represents the energy density of a ﬂat universe (K = 0) expanding at a rate H. Therefore the Friedman equation (2.14) can be written as:

2 H X Ωi0 = (2.18) H a3(1+wi) 0 i 11

where Ωi0 is the current value of density parameter of ﬂuid component ”i”.Necessarily, the density parameters satisfy the following normalization condition:

X Ωi = 1. (2.19) i 2.3 Energy content of the Universe

The evolution of the scale factor depends clearly on the diﬀerent ﬂuids contained in the

Universe, which are characterized trough their EoS parameter wi at the background level (since at the perturbative level the effective sound speed can also be used to identify them). In the Standard model (ΛCDM) the energy content of the Universe is given by the following fluids: Baryonic matter (bm), Dark matter (dm), relativistic components (radiation (r) and neutrinos) and Dark energy (de), being the interaction between them only gravitational. The main characteristics of those fluids will be described below.

2.3.1 Baryonic matter

All non-relativistic matter composed of protons, neutrons and electrons is considered as baryonic matter. Hydrogen constitutes nearly 75% of all of this component while Helium- 4 corresponds to about 25%. The other light elements and metals have only very small abundances. The baryonic matter is approximatedly a low-density gas of non-relativistic massive particles, so its pressure is much smaller than the energy density: pbm ρbm.

Therefore the EoS parameter for baryonic matter is wbm = 0.

2.3.2 Dark matter

In 1933 the astronomer Fritz Zwicky observed that the dispersion of the relative velocities of the galaxies in the Coma cluster were much larger than the escape velocity calculated from the luminous mass of the cluster [40]. In order to keep the galaxies in the cluster, Zwicky concluded that there must be a large amount of “Dark matter” (DM). Later, the rotation-curve measurements of individual galaxies provided more evidences for the 12

Figure 2.3: Representation the spiral galaxy inside a dark matter halo existence of dark matter [41]. According to Kepler’s third law, the velocity v of a star √ in a galaxy would fall as v ∝ 1/ R at large radii, where R is the distance from the galactic centre2. However, astronomers found [41] that at large radii the velocity was approximately constant. This seems to indicate the presence of another mass component in the galaxies which should have a different density profile than the visible mass in the galaxy, so that it could be subdominant in the inner parts of the galaxy, but would become dominant in the outer parts. This dark component appears to extend well beyond the visible parts of galaxies, forming a dark halo surrounding the galaxy. This is illustrated in the figure 2.3 where it is shown a spiral galaxy inside a dark matter halo.

On other hand, it is possible to indirectly measure the dark matter in a cluster via gravitational lensing. Indeed, with the help of this technique and X-ray detection, it was

2in a simple spherically-symmetric galaxy. 13 observed the existence of dark matter in a pair of colliding galaxy clusters (Bullet Cluster [42]). Nowadays, this observation is considered as the major proof of the existence of dark matter in clusters.

At cosmological scales, dark matter is also necessary: the proof of its existence are the CMB temperature anisotropies, baryon acoustic oscillations (BAO) [43] and structure formation [44]. As a matter of fact, since baryons can only start clustering once they decouple from the CMB photons, they would not have enough time to grow from such small initial fluctuations and form the observed structures. On the other hand, dark matter only interacts gravitationally with photons and baryons because it has a very low cross- section for electroweak interaction (according to some models [45, 46]). Therefore, dark matter decoupled from the photon-baryon fluid earlier, so that their fluctuations grow via gravitational instability until they form non-linear structures or dark halos that then created large gravitational potential wells where later the baryonic matter fell into. Pho- tons from the CMB were also affected, being gravitationally redshifted via Sachs-Wolfe effect [5] and then contributed to the CMB temperature anisotropies. This dark matter perturbations come from quantum fluctuations of the scalar field which drives inflation (called inflaton) [47, 48]. On scales close to the Planckian length, these fluctuations may have substantial amplitudes; however, during the inflationary stage, they are stretched to galactic scales with nearly unchanged amplitudes (so they are scale independent). This process occurs at end of inflation, where the inflaton reaches and oscillates coherently around the minimum of its potential decaying into a thermal mix of elementary particles

[49] (a process known as reheating), so leading to a radiation-dominated universe. Gen- erally the study of dark-matter perturbations δρdm(~r, t) is determined by the evolution of the density matter contrast field δdm(~r, t) ≡ δρdm(~r, t)/ρ¯dm in Fourier space. Within a large comoving box of comoving volume V, the density matter contrast field δdm(~r, t) can 14 be expressed as V Z δ (~r, t) = δ (t)e−i~k.~rd3~k, (2.20) dm (2π)3 k where δk(t) is the amplitude of the component of δdm(~r, t) with wavelength 2π/k and wave-vector ~k. The mean-square amplitudes of the Fourier components defines the power spectrum:

2 P (k, t) = |δk(t)| (2.21)

Given that quantum fluctuations are scale-independent, most inflationary scenarios pre- dict that the density fluctuation created by inflaton will be an isotropic, homogeneous and

Gaussian ﬁeld, so the expected power spectrum for primordial dark matter perturbations is given by the following power law:

P (k) ∝ kns−1 (2.22)

where ns is called spectral index, whose expected value (in a naive approximation) is ns = 1. In fact, according to current observation ns = 0.965 ± 0.004 at 68% conﬁdence level [1].

Moreover all observational evidences of DM and observations of cosmological large- scale structure suggest that it must be cold (with wdm = 0) [50], i.e, with negligible pressure and with non-relativistic velocity distribution at decoupling moment, so the structures in the universe were formed through bottom-up mechanism (structure grows hierarchically, with small objects collapsing under their self-gravity ﬁrst and merging in a continuous hierarchy to form larger and more massive objects).

Thus, given that for both dark matter and baryonic matter the EoS parameter are same (wdm = wbm = 0), according to equation (2.10) the energy density for both decreases as ρ ∝ a−3 due to increase in volume resulting from the expansion. On other hand, we can write the energy density as ρ = nm, where n is the number density of particles of mass m. So, in a comoving volume, the number of particles is conserved. Using the equation 15

(2.14) for a Universe with zero spatial curvature (K = 0) in which non-relativistic matter is dominant, the scale factor evolves as

2 a ∝ t 3 . (2.23)

The cosmological model with this behavior is known in the literature as Einstein-de-Sitter

Universe.

2.3.3 Relativistic components (rc)

The radiation and neutrinos are considered as a ﬂuid composed by ultra-relativistic particles.The greatest contribution to its energy density is given by the kinetic energy and therefore its EoS parameter is given by wrc = 1/3. According to equation (2.10) its energy density decreases as ρ ∝ a−4. Although the numerical density of the relativistic components decreases in the same way as the numerical density of the non-relativistic particles, in this case there is an energy loss due to the redshift (the frequency ν decreases as ν ∝ a−1).

In a Universe with negligible spatial curvature and dominated by relativistic components, the scale factor is given by

1 a ∝ t 2 . (2.24)

Note that both matter and relativistic components satisfy the strong energy condition

ρ + 3p > 0 and therefore generate gravitational attraction, sincea ¨ ≤ 0 according to the equation (2.15). The main radiation components in the whole Universe comes from the photons from the CMB.

CMB

After recombination, a plasma of photon-baryon was in thermal equilibrium because the number of photons was large enough to ionize the baryons [51]. Then, CMB photons are described by Bose-Einstein distribution and therefore their energy density ργ is given by 16

[5]: Z d3p p π2 ρ = 2 = T 4, (2.25) γ (2π)3 ep/T − 1 15 where the factor of “2” comes from the two states of polarization, p is photon momentum and T its temperature. Note that according to the above equation and (2.15), the temperature decreases as T ∝ 1/a.

However, as a result of the Universe expansion the interaction rate between photons and baryons dropped quickly until it was comparable with the expansion rate and then the radiation decouples from the baryonic matter. After the decoupling the CMB photons started to free-stream through the Universe without any further scattering but being redshifted by dark matter overdensities, generating ﬂuctuations in its temperature spectrum. Since perturbations δT are deﬁned on a celestial sphere, it is useful to expand the temperature contrast δT/T in terms of spherical harmonics:

∞ l δT (θ, φ) T (θ, φ) − hT i X X ≡ = a Y (θ, φ), (2.26) T hT i lm lm l=0 m=−l where θ and φ are the angular position of the line of sight, Ylm(θ, φ) is the spherical harmonic, alm is the multipole coeﬃcient and hT i is the mean temperature, whose current value is 2.7255 ± 0.0006K at 95% conﬁdence level according to the FIRAS experiment [52]. Thus l introduces an angular scale in the sky θ ∼ π/l (in the same way as the

2π wavenumber k introduces a scale λ = k in a Fourier expansion), while m is just a phase. Using the closure relation of spherical harmonics, the mean magnitude of the temperature ﬂuctuations squared is given by

* 2+ δT (θ, φ) X 2l + 1 = C , (2.27) T 4π l l where Cl is called the angular power spectrum, which is deﬁned as

2 Cl ≡ |alm| . (2.28) 17

Figure 2.4: Angular Temperature spectrum. Figure taken from Ref[1]

So, Cl represent the squared amplitude of temperature ﬂuctuations on angular scale θ ∼

π/l. Moreover, since CMB fluctuations comes from random process, Cl is considered as theoretical power spectrum, since it is calculated for a given ensemble of realizations: * l + 1 X C = |a |2 = |a |2 (2.29) l lm 2l + 1 lm m=−l Ensemble ˆ However, the observed power spectrum Cl is computed for a given realization and it is defined as l 1 X Cˆ = |a |2 (2.30) l 2l + 1 lm m=−l ˆ The expected squared difference between Cl and Cl is called the cosmic variance and it is given by D E 2 (Cˆ − C )2 = C2 (2.31) l l 2l + 1 l ˆ We see that the expected relative difference between Cl and Cl is smaller for higher l. Nonetheless, it is large for small l and therefore the cosmic variance limits the accuracy of comparison of CMB observations with theory. Figure (2.4) shows the observed angular spectrum according to Planck measurements

[1]. The observational results are the data points with error bars. The dark green curve 18

is the theoretical Cl from a best-fit model (ΛCDM model), and the cyan band around it represents the cosmic variance corresponding to this Cl. An important feature of the figure (2.4) is the position of first peak around θ ≈ 1◦. Note that the Hubble distance at the

−1 redshift of the last scattering surface — d = H (zls with zls ≈ 1100 — is approximately 0.2Mpc and the angular diameter distance to the last scattering is close to 13 Mpc.

Therefore, an object on the last scattering surface with physical length d has an angular size θH , as seen from Earth, of

0.2Mpc θ = ≈ 1◦ (2.32) H 13Mpc

This suggests that physical mechanism which origins the temperature perturbations is different for θ > θH (large scales) and θ < θH (small scales) due to causality. Indeed, note that in the former case, the temperature fluctuations are causally disconnected. On large scales the fluctuations are generated through the Sachs-Wolfe effect. On other hand, the fluctuations on small scales are produced by the baryon acoustic oscillations

(that generate the peaks in figure (2.4) and gravitational lensing. In fact, before the decoupling, the dynamics of the photon-baryon fluid was determined by dark matter. If this fluid happens to finds itself in a potential well of dark matter, it will fall to the center of the well. The compression of the fluid heats it, and by response enhances the internal pressure leading to an expansion. The gravitational attraction will eventually slow down the expansion and further return the system to the initial condition and start to go through all this process again while the decoupling is not reached. The inwards and outwards oscillations of photon-baryon fluid are called baryon acoustic oscillations (BAO). Thus, if the photon-baryon fluid within a potential well is at maximum compression at the decoupling moment, its energy density will be higher than average and since T ∝ ρ1/4, the liberated photons will be hotter that average generating the peaks in the observed angular spectrum. On the contrary, if the photon-baryon fluid within a potential well is at maximum expansion at the decoupling time, the liberated photons will be slightly 19 cooler than the average.

Neutrinos

At early times, the main contribution to the energy density comes from relativistic particles (mainly photons), the three neutrino species, barions and their antiparticles. Since all particles are in thermal equilibrium, then neutrinos are described by Fermi-Dirac distribution and therefore their energy density ρν is given by [5]:

Z d3p p π2 7 ρ = N = N T 4, (2.33) ν eff (2π)3 ep/T + 1 eff 30 8 where Neff is the number of degrees of freedom compatible with 3 neutrino species . As a result of the Universe expansion, all three neutrino species thermally decouple before the electron-positron pairs begin to annihilate. After decoupling, the neutrinos propagate without further scatterings, preserving their thermal spectrum. Their temperature decreases in inverse proportion to the scalar factor and is not inﬂuenced by the subsequent electron-positron pairs annihilation. The energy released in this process is thermalized and as a result the radiation is heated. Therefore, the temperature of the radiation must be larger than the neutrino temperature. In fact, after decoupling, the neutrino’s and the other particles’s entropies are conserved and, therefore, the ratio between the neutrino and photon temperatures is [5]: T 111/3 γ = . (2.34) Tν 4

So, using equations (2.25) and (2.33), the energy density of the cosmic neutrinos ρν is approximately 68% of the CMB one if we assume Neff = 3. Besides, according to equation

(2.34), the massless primordial neutrinos should have a temperature today of Tν ≈ 1.95K.

2.3.4 Dark energy

In the 90s, the estimations of the age of the oldest stars in the globular clusters [53] (around

13.5 ± 2 Gyrs and therefore older than the CDM Universe) and later measurements of 20 the luminosity distance of type Ia supernovae (SN Ia) [9, 10] show evidences that the Universe has begun a process of accelerated expansion (ä > 0) in the recent past. The cause of such cosmic acceleration is still unknown. Recently, observations of the CMB anisotropies, BAO, gravitational lensing and the distribution of large-scale clustering of galaxies in the sky confirm the existence of accelerated expansion. For example, the accelerated expansion leads to a shift of the position of acoustic peaks in CMB anisotropies (due that angular distance is altered) as well as a modification of the large-scale CMB spectrum through the so-called integrated Sachs-Wolfe effect [12]. The detection of a peak of BAO reported in [54] at the average redshift z = 0.35 from the observations of luminous red galaxies in the Sloan Digital Sky Survey is another independent test of the accelerated expansion. On other hand, the power spectrum of matter distributions also favors a Universe with accelerated expansion rather than the CDM Universe [55].

Usually the accelerated expansion is attributed to a new component, called generally dark energy [56], which would be a exotic fluid with negative pressure, such that its equation-of-state parameter satisfies the condition wde < −1/3 (you can see it from the acceleration equation (2.15) for a single fluid). This negative pressure leads to the accelerated expansion of the Universe by counteracting the standard classical gravitational force.

The simplest candidate for dark energy is the cosmological constant Λ, which is so called because its energy density is constant in time and space, and, accordingly its equation-of-state parameter is wΛ = −1— see equation (2.10). Initially, the cosmological constant was introduced by Albert Einstein at the beginning of the last century to make compatible his ﬁeld equations with the idea of a static Universe as it was believed at that epoch. As the gravitation is attractive, he was forced to introduce Λ as a repulsive force to get a static solution. Thus, the Einstein ﬁeld equations with cosmological constant are given by: 21

1 R − Rg + g Λ = 8πGT , (2.35) µν 2 µν µν µν So, the Friedman and acceleration equations for an ﬂat Universe that contains both matter and cosmological constant take the form:

8πG Λ H2 = ρ + , (2.36) 3 m 3

a¨ 4πGρ Λ = − m + . (2.37) a 3 3 Introducing the cosmological constant in the Einstein equations is equivalent to adding one new component in the universe with energy density:

Λ ρ ≡ , (2.38) Λ 8πG and pressure Λ p ≡ − . (2.39) Λ 8πG

Note that if Λ > 4πGρm, from the equation (2.37) we obtaina ¨ > 0, in other words, the cosmological constant does generate cosmic acceleration. From the combined analysis of

SN Ia, CMB anisotropies, BAO and gravitational lensing was found that wde = −1.006 ± 0.045 at the 95% conﬁdence level [1] , which is consistent with the expected value for a cosmological constant.

Despite its simplicity and concordance with the observational data, the introduction of Λ presents some difficulties that have motivated scientists to look for alternatives capable of explaining the cosmic acceleration. The first difficulty encountered is the so-called problem of the cosmological constant [57]. According to the equation (2.36), to get cosmic acceleration today, it is necessary that the cosmological constant is of the order of the square of the present Hubble parameter H0 :

2 −42 2 Λ ≈ H0 = (2.1332h x 10 GeV) . (2.40) 22

So, the energy density observed of the cosmological constant is given, according to equation (2.38), by:

−47 4 −123 4 ρΛ ≈ 10 GeV ≈ 10 mpl, (2.41)

19 where mpl ≈ 10 GeV is the Planck mass and h ≈ 0.7. On other hand, if the energy density of cosmological constant comes from the vacuum energy of the empty space, the energy density ρvac associated to some ﬁeld of mass m with momentum k , frequency ω √ and zero-point energy E = ω/2 = k2 + m2 is [12]:

Z kmax 3 √ dk 1 2 2 ρvac = 3 k + m , (2.42) 0 (2π) 2 where kmax is a cut-oﬀ scale. Assuming the cut-oﬀ scale as the Plank mass, the vacuum energy density can be estimated as: m4 ρ ≈ pl ≈ 1074GeV4. (2.43) vac 16π2

121 Therefore, ρvac is about 10 times larger than the observed value ρΛ. Note that all particles (bosons or fermions) contribute for the vacuum energy density. One can show that the contributions from bosons and fermions would cancel out and the vacuum energy density would be zero if supersymmetry were an exact symmetry of nature. However, this symmetry has been not yet observed in the nature and therefore it could have been broken in some energy range, smaller than the Planck energy and which we still do not have access to. A rather conservative possibility would be to assume as cut-off scale the energy value of the electroweak phase transition (≈ 102 GeV). Even assuming such low cut-off energy, the discrepancy between the theoretical value and the observed value would be very large, requiring a strong fine adjustment. Another problem related to the cosmological constant is the so-called problem of cosmic coincidence [58] which consists in the fact that there is an approximate coincidence between the value of the energy density of cosmological constant and the value of the energy density of matter in the current universe. This is particularly strange given that the rela-

3 tive balance between these energies varies rapidly as the universe expands (ΩΛ/Ωm ∝ a ). 23

In fact, in the primordial universe the energy density of the cosmological constant was negligible in comparison to matter, whereas recently the situation has reversed and the energy density of the cosmological constant has begun to dominate. There is then a relatively short period in the history of the Universe where those energy densities are comparable and it seems a strange coincidence that this period is precisely around the present. In function of redshift z = 1/a − 1 and neglecting radiation, the energy density

ρm coincides with ρΛ at 1/3 ΩΛ0 zcoinc = − 1 (2.44) 1 − ΩΛ0

For ΩΛ = 0.7, we get zcoinc ' 0.3. With those problems for the cosmological constant, the emergence of alternative theories with purpose of describing the cosmic acceleration and explaining the nature of dark energy seems inevitable. Basically there are three approaches to construct models to explain the cosmic acceleration. The first category accepts the Relativity General and proposes the existence of new fields in nature modifying the right-hand side of the Ein- stein equations by considering specific forms of the energy-momentum tensor Tνµ with a negative pressure. The representative models that belong to this class are the so-called quintessence [13, 16], k-essence [18], and perfect fluid models. The quintessence makes use of scalar fields with slowly varying potentials, whereas in k-essence it is the non-trivial scalar-field kinetic energy that drives the acceleration. The perfect fluid models are based on a perfect fluid with a specific equation of state such as the Chaplygin gas model [59], phantom dark energy [17] and decaying vacuum models [20]. The second group of cosmological models are based on the modification of the GR, where the cosmic acceleration is caused by geometric effects due to the fact that gravity is weak in cosmological scales. The representative models that belong to this class (denoted as “modified gravity”) are the so-called f(R) theories [60, 61], scalar-tensor theories [62], and braneworld models [63]. However, this models can also be seen as dark-energy ones when one defines an effective

ﬂuid, whose energy density and pressure depend on the Ricci scalar R. Finally, the last 24 category considers cosmological models where the Cosmological Principle is violated [64].

2.4 ΛCDM model

The model of the Universe that better fits current observations is known as ΛCDM model or concordance cosmic model. In this model, the Universe is spatially flat (K = 0) and contains relativistic particles (photons and neutrinos), matter (dark matter and barions) and a Cosmological constant. The fact that current observations indicate that the spatial curvature of the universe is very close to zero is in agreement with the theoretical predictions of the inflationary scenario.

Thus, assuming K = 0 the Friedman’s metric (2.2) in Cartesian coordinates is given by:

ds2 = −dt2 + a2(t)dx2 + dy2 + dz2. (2.45)

On other hand, for ΛCDM model the Friedman equation (2.18) can be written as:

2 H Ωr0 Ωm0 = 4 + 3 + ΩΛ0. (2.46) H0 a a

According to combined analysis of CMB anisotropies, BAO, SN Ia and gravitational lensing observations, the best ﬁt values for the cosmological parameters reported by the Planck experiment are[1,8]:

H0 = 67.74 ± 0.40km/s/Mpc,

ΩΛ0 = 0.6911 ± 0.0062, (2.47)

Ωm0 = 0.3089 ± 0.0062,

at 68% conﬁdence level. Here Ωm0 is the current value of baryon density parameter (Ωb0) plus the cold dark matter one (Ωc0). Indeed, they are reported as[1,8]:

2 2 Ωb0h = 0.02230 ± 0.00014, Ωc0h = 0.1188 ± 0.0010. (2.48) 25

From the value of H0 in equation (2.47) we get h = 0.6774 ± 0.0046. From (2.48), the current value of the density parameters for baryonic matter and cold dark matter are respectively:

Ωb0 = 0.0486 ± 0.0010, Ωc0 = 0.2589 ± 0.0057. (2.49)

The current density parameter for radiation is given by CMB radiation density parameter (Ωγ0) plus neutrino’s (Ων0):ΩR0 = Ωγ0 + Ων0. Using equation (2.25) and the deﬁnition of the density parameter, the best ﬁt values are:

2 −5 2 −5 Ωγ0h = 2.47282 × 10 , Ων0h = 1.71062 × 10 , (2.50)

7 4/3 where we have used that for massless neutrinos Ων = 8 (4/11) Neff Ωγ with Neff = 3.046 as the cosmological ﬁt for neutrino number of species [65]. Therefore, ΩR0 = 9.23640 × 10−5.

As we can see in equation (2.46), at small values of scale factor the universe expansion is driven by radiation since its energy density decreases as a−4. The radiation energy density is equal to matter energy density at the scale factor:

ΩR0 −4 arm = ≈ 2.96 × 10 (2.51) Ωm0 which corresponds to zrm ≈ 3371. Then, at scale factors a > arm, the Universe dynamics is driven by matter. At this stage, the factor scale evolution change from a ∝ t1/2 to a ∝ t2/3. Since the matter energy density decreases as a−3 and the cosmological constant’s remains constant, the equality between them happens at scale factor:

Ωm0 aΛ = ≈ 0.447 (2.52) ΩΛ0 which correspond to zΛm ≈ 1.237. So, at scalar factors a > aΛ, the Universe is dominated by a cosmological constant experimenting an accelerated expansion. 26

Chapter 3

Dark Energy and cosmic acceleration

As it was mentioned in the previous chapter, the models that can explain the current acceleration are divided in three classes. The first category accepts the General Relativity and proposes the existence of new fields in nature. Such models are known as Dark-energy models. The second group of cosmological models are based on the modification of the GR and are known as modified gravity models. The third class considers cosmological models where the Universe is assumed inhomogeneous. In this work we will focus on f(R) gravity — which belongs to the second category — and phantom (w < −1) and non-phantom (w > −1) dark energy models. In f(R) theories, the Einstein equations are modified so that the late-time accelerated expansion of the Universe is realized without recourse to an explicit unknown component. This means that the space-time geometry interacts in a different way with its matter-energy content than in GR, so the curvature

— namely, the Ricci scalar — is no more proportional to the energy density and pressure. Indeed, an accelerated-expansion solution is obtained without requiring any kind of exotic unknown ﬂuid.

3.1 f(R) gravity theories

The Einstein equations are obtained through the action, 27

S = SEH + Sm(gµν, ψa), (3.1) where Sm is the matter action with matter ﬁelds ψa and SEH is the Einstein-Hilbert action, given by: Z √ 1 S = d4x −g R, (3.2) EH 2κ2 whose lagrangian density L ≡ (1/2κ2)R is a linear function in R. Thus, the simplest form of modifying the Einstein equations is considering the lagrangian density L as a non-linear function of Ricci scalar R.

Therefore, the action for f(R) theories is given by [60, 61]:

Z √ 1 S = d4x −g [R + f(R)] + S (g , ψ ), (3.3) 2κ2 m µν a where f(R) is a nonlinear function of R. Note that if f(R) = −2Λ, the Einstein equation with cosmological constant are obtained. There are two approaches to derive ﬁeld equations from the action (3.3). The ﬁrst approach is the so-called metric formalism in

α which the connections Γβγ are the usual connections deﬁned in terms of the metric gµν.

α The second approach is the so-called Palatini formalism in which Γβγ and gµν are treated as independent variables. In this work will use the ﬁrst approach; for more information about Palatini formalism you can see [12, 66]. Variation of action (3.3) with respect to the metric gµν yields the modiﬁed Einstein equations:

1 G + f R − f g − [∇ ∇ − g ]f = κT , (3.4) µν R µν 2 µν µ ν µν R µν

df where Gµν is the Einstein tensor and fR ≡ dR . Note now that the ﬁeld equations (3.4) are fourth-order (i.e. contain fourth derivatives of the metric), which is expected for a modiﬁed gravity theory according to Lovelock’s Theorem [67], with some notable exceptions [68].

Taking the trace of Eq. (3.4) one gets: 28

2 (fR − 1) R − 2f + 3 fR = κ T, (3.5) where T is the trace of the energy-momentum tensor Tµν. This equation is a generalization of the trace of Einstein equations (without cosmological constant):

R = −κ2T (3.6)

In the general case, R is related to T through a diﬀerential equation, which allows a greater variety of solutions than in the case of General Relativity [60]. For a FRLW Universe, we have:

R = 12H2 + 6HH0, (3.7) where 0 ≡ d/dy and y ≡ ln a.

Taking the covariant derivative in both sides of eq. (3.4), using the Bianchi equations,

µ ∇ Gµν = 0, (3.8) and using the non-cummutativity of the covariant derivatives,

µ (∇ν − ∇ν) = Rµν∇ , (3.9)

µν one gets that ∇νT = 0. Therefore, the energy-momentum in f(R) theories is also conserved [69]. In the metric formalism, a new degree of freedom arises in this theories: the scalar field fR, dubbed “scalaron” [56, 70], which generates a fifth force that may drive the cosmic acceleration in low curvatures, i.e, in the present era. To make this field explicit, consider the action [60, 61]

Z 1 √ h i S = d4x −g (φ˜ + f(φ˜)) + (1 + f )(R − φ˜) + S , (3.10) 2κ2 φ˜ m ˜ ˜ ˜ where φ is a new scalar ﬁeld and fφ˜ ≡ df/dφ. Variation with respect to the ﬁeld φ yields

˜ ˜ fφ˜φ˜(φ)(R − φ) = 0. (3.11) 29

˜ ˜ Therefore, if fφ˜φ˜ 6= 0 then R = φ so the action (3.3) is reobtained. Redeﬁning the ﬁeld φ ˜ by χ ≡ fφ˜(φ) + 1 and setting

φ˜(χ)χ − (φ˜(χ) + f[φ˜(χ)]) V (χ) ≡ (3.12) 2κ2 the action (3.10) takes the form (in the so-called Jordan-frame)

Z √ 1 S = d4x −g χR − V (χ) + S . (3.13) 2κ2 m

Meanwhile the action in Brans-Dicke (BD) theory [61] with a potential V (χ) is given by:

Z √ 1 ω S = d4x −g χR − BD gµν∂ χ∂ χ − V (χ) + S , (3.14) 2 2χ µ ν m where ωBD is the BD parameter. Comparing eq. (3.13) with eq. (3.14), it follows that f(R) theory in the metric formalism is equivalent to BD theory with the parameter

2 ωBD = 0 (in the unit κ = 1). The ﬁeld equations corresponding to the action (3.13) are

κ2 1 1 G = T − g V (χ) − (∇ ∇ χ − g χ), (3.15) µν χ µν 2χ µν χ µ ν µν

R = V 0(χ), (3.16) where V 0(χ) = ∂V/∂χ. By taking the trace of eq. (3.15) in order to replace R in eq. (3.16), one gets

0 2 3χ + 2V (χ) − χV (χ) = κ T (3.17)

This last equation determines the dynamics of χ for a given matter source T . Deﬁning the eﬀective potential Veff by ∂Veff /∂fR ≡ χ, eq.(3.17) can be rewritten as:

∂Veff 1 2 = R − fRR + 2f + κ T (3.18) ∂fR 3

Thus, the eﬀective mass of that scalar ﬁeld is given by

2 2 ∂ Veff 1 1 + fR µ = 2 = − R . (3.19) ∂ fR 3 fRR 30

1 The inverse of this mass scale defines the comoving Compton wavelength λc ≡ µ that sets the range of the scalar field interactions. When the scalar mass is small, the reach of the fifth force mediated by the scalar field fR is longer, and conversely, if the mass is large (e.g., in regions of high density), λc is small and the range of the fifth force is more limited. The mechanism whereby the scalar field acquires a mass in dense regions such as stars and planets is known as the chameleon mechanism [71], which is necessary for f(R) models to pass local tests of gravity. One can write the action (3.10) in the so-called Einstein frame by making the conformal transformation

g˜µν ≡ χgµν (3.20)

hq 2 i and deﬁningχ ˜ ≡ fR = exp 3 x˜ . Therefore, Z h i 4 1 p ˜ µν 1 S = d x 2 −g˜ (R − g˜ ∂µχ∂˜ νχ˜ − 2V (˜χ) + Sm , ψa . (3.21) 2κ χ˜g˜µν

In this frame, f(R) theories are equivalent to GR with the scalar field fR and , consequently, the field equations in this frame are second order. As f(R) models are proposed to be an alternative to ΛCDM , they should be in excellent agreement with current observations. Thus, a given f(R) model is said to be viable if it predicts an evolution of both cosmological and local parameters according to the observational data. In fact, there are very strong restrictions on the possible f(R) functions. They are based on local test of gravitation, stability and viability conditions of the model. On other hand, it is desirable that f(R) model is distinguishable from the standard one, so that it can be falsified.

In a nutshell, the viability conditions that a f(R) model must satisfy are [72, 73]:

1. fRR > 0,

2.1+ fR > 0,

f 3. lim = 0 and lim fR = 0, R→∞ R R→∞ 31

4. kfR| 1 at recent epochs.

The ﬁrst condition follows from requiring the existence of a stable high-curvature regime [74, 75], such as the matter-dominated universe. Such condition can be obtained from the following argument: Let us consider local ﬂuctuations on a background characterized by a

2 curvature R0 = −κ T . Under a weak-ﬁeld approximation, eq. (3.5) can expand in powers of ﬂuctuations. One can decompose the quantities R and gµν into the background part and the perturbed part: R = R0 + R1 and gµν = ηµν + hµν, where we have assumed that

0 the approximation that the zero-order term gµν corresponds to the Minkowski space-time metric ηµν. Therefore, eq. (3.5) takes the form [76, 77]:

2 ¨ 2 2κ fRRR ˙ ˙ ~ ~ 1 − fR R1 − ∇ R1 − (T R1 − ∇T · ∇R1) + R1 = (3.22) fRR 3fRR 2 2 ¨ 2 2 κ fRRR ˙ 2 ~ ~ 1 2 = κ T − κ ∇ T − (T − ∇T · ∇T ) − (κ T fR + 2f). fRR 3fRR

Assuming a homogeneous perturbation and empty space ( T = 0), the eq. (3.22) is reduced to: ¨ 1 − fR 2fR R1 + R1 = . (3.23) 3fRR 3fRR

For the models where the deviation from the ΛCDM model is small, fR 1, so if fRR < 0 the perturbation R1 exhibits an exponential instability. Then the condition fRR > 0 is needed for the stability of cosmological perturbations. The second condition comes from the study of the growth of matter perturbations in f(R) theories, where the Newton constant G is modiﬁed for an eﬀective constant Geff .

On sub-horizon scales (i.e. k/a H), the matter density perturbation δm approximately satisﬁes equation [70] ¨ ˙ δm + 2Hδm − 4πGeff ρ¯mδm = 0, (3.24) where G 1 + 4mk2/(a2R) Geff = 2 2 , (3.25) 1 + fR 1 + 3mk /(a R) 32

with m ≡ RfRR/(1 + fR). To avoid anti-gravity, Geff must be always positive, so the second condition is necessary. On other hand, given tight constraints from Big Bang nucleosynthesis and from the Cosmic Microwave Background, GR must be recovered at early times in any f(R) model; this is guaranteed by the third condition.

For the consistency with local gravity constraints in the solar system [61, 73, 78], the function f(R) needs to be close to the ΛCDM model in high-density regions (where the Ricci scalar R is much larger than the cosmological Ricci scalar today), so the last condition is necessary.

For a given f(R) model, it is not enough to satisfy the stability constraints. It should also reproduce the known cosmological history, namely a matter-dominated phase followed by an (desirable) accelerated attractor (late-time de Sitter point). The conditions on f(R) functions to present such features are [79]:

dm 1 + f m(r ≈ −1) ≈ 0+ and (r ≈ −1) > −1, with r ≡ −R R , (3.26) dr f + R for a matter-dominated phase and

0 < m(r = −2) ≤ 1 (3.27) for a ﬁnal accelerating attractor.

As it was previously mentioned, f(R) theories are equivalent to GR with a scalar field in the Einstein frame. However, in cosmology,such theories are also analogous to GR with a new component dubbed “curvature fluid”, whose pressure and energy density depend on the scalar curvature R and its derivatives. This can be easily observed considering the modified Einstein equations for a FLRW universe filled with matter energy density ρm, given by:

1 3H2 = κ2ρ + (Rf − f) − 3Hf˙ − 3Hf (3.28) m 2 R R R ˙ 2 ¨ ˙ ˙ −2H = κ ρm + fR − HfR + 2HfR (3.29) 33

From the above equations , one can deﬁne ρx, px and wx ≡ px/ρx, respectively the energy density, pressure and the equation-of-state parameter of the curvature ﬂuid [72]:

1 κ2ρ ≡ (f R − f) − 3Hf˙ − 3H2f (3.30) x 2 R R R 1 κ2p ≡ f¨ + 2Hf˙ − (2H˙ + 3H2)f + (f − f R) (3.31) x R R R 2 R These definitions are such as to guarantee that the curvature fluid is conserved [80]. They are not unique, but the wx we choose is finite along the evolution of the Universe because

ρx 6= 0.

3.1.1 Hu-Sawicki model

One of the few known viable f(R) model with the interesting feature of being able to satisfy solar system tests of gravity is the Hu-Sawicki model [3, 78], where the f(R) function is given by: 2 n hs 2 c1(R/m ) f (R) = −m 2 n (3.32) c2(R/m ) + 1 with

2 2 m ≡ Ωm0H0 , (3.33)

2 c1, c2 positive constants and n positive integer. For high curvatures, R m , eq. (3.32) takes the form: 2 n hs c1 2 c1 2 m f (R) ≈ − m + 2 m . (3.34) c2 c2 R From the above equation one can identify the ﬁrst term on the right side as a cosmological

2 constant, i.e. 2Λ ≡ c1/c2m . Thus, to mimic a ΛCDM evolution with a cosmological constant ΩΛ0 and matter density Ωm0 is requested that:

ΩΛ0 c1 = 6 c2 (3.35) Ωm0

The last term on the right side of equation (3.34) corresponds to a deviation from the cosmological constant Λ, which becomes more important at low curvatures. 34

The scalar ﬁeld fR in this case is given by: −(n+1) hs c1 R fR (R) = −n 2 2 (3.36) c2 m

Consequently, if it is assumed that the current value of curvature R0 is given by the ΛCDM model (neglecting radiation), according to equation (3.7) one obtains: 2 12 R0 ≈ m − 9 . (3.37) Ωm0

hs Therefore the current value for fR is: −(n+1) hs c1 12 fR0 = −n 2 − 9 , (3.38) c2 Ωm0 and the constant c2 is given by: −(n+1) ΩΛ0 n 12 c2 = −6 − 9 (3.39) Ωm0 fR0 Ωm0

Then the Hu-Sawicki model presents two new parameters, fR0 and n. In term of those parameters the Hu-Sawicki function (3.34) takes the form :

f R R n f hs(R) = −2Λ − R0 0 0 (3.40) n R

So, when fR0 → 0, the Hu-Sawicki model behaves like ΛCDM. Combining equations (3.28) and (3.29) one obtains:

1 ρ¯ H2 − f (HH0 + H2) + f + H2f R0 = κ2 m . (3.41) R 6 RR 3

To solve these equations, one can re-express them in terms of the new variables

H2 R y ≡ − a−3, y ≡ − 3a−3 . (3.42) H m2 R m2

Equations (3.7) and (3.41) then become a set of coupled ordinary diﬀerential equations

1 y0 = y − 4y , (3.43) H 3 R H 0 −3 1 1 1 1 −3 1 f yR = 9a + −3 2 yH − fR yR − yR − a + 2 . (3.44) yH + a m fRR 6 2 6 m 35

Figure 3.1: Eﬀective Dark Energy Equation of State for the Hu-Sawicki f(R) model as a 0 function of redshift z. Figure taken from [3] where fR ≡ fR0.

The initial conditions are taken at high redshifts, where the model behaves like ΛCDM.

Finally, the equation-of-state parameter for eﬀective ﬂuid is given by:

0 1 yH wDE = − − 1 (3.45) 3 yH

In ﬁgure (3.1) it is shown wDE for the Hu-Sawicki f(R) model as a function of redshift z for diﬀerent values of fR0 and n. Note that in all cases the ΛCDM model is recovered at high redshifts. On other hand, wDE oscillates around −1 at small redshifts where cosmic acceleration is important. Note that the redshift value for the minimum (or maximum) of wDE is determined by n and its amplitude by fR0. An appreciable discrepancy between the standard model and the Hu-Sawicki one occurs for large values of |fR0| and small n. However, in order to satisfy local gravity tests, the values of fR0 are restricted to

−6 |fR0| < 10 [3, 78], making it diﬃcult to distinguish between this model and the standard one. 36

3.1.2 γ gravity model

Another f(R) model, which obeys the stability and viability conditions described previously, is the “γ gravity” [4] where the f(R) function is given by:

αR 1 R n f(R) = − ∗ γ , , (3.46) n n R∗ where α, n and R∗ are positive constants, and

Z x γ(a, x) = e−tta−1dt (3.47) 0 is the “lower incomplete gamma function” [81]. From the deﬁnition (3.46), the ﬁrst derivative of f(R) with respect to R is given by

n (−R/R∗) fR = αe (3.48)

Therefore, at high curvatures (R/R∗ 1), f(R) is approximately constant (fR 1) and the model behaves like ΛCDM. Besides, when n increases, the steepness of f(R) (i.e, how fast the curve tends to GR with an eﬀective cosmological constant) also does. So, if n > 1 the steepness is larger than in the exponential case which implies that, cosmologically, when we go back in time from the present (increasing R), the ΛCDM regime is achieved faster. Imposing that f(R) → −2Λ in this regime we get the following relation

R 6nd ∗ = (3.49) m2 αΓ(1/n)

Above, Γ(x) is the (standard) gamma function.

The γ gravity model presents two new parameters: α and n. To solve equations (3.41) and (3.7) in this case (including now radiation), one can re-write them in terms of the new variables

H2 x (y) ≡ − e−3y − d − a e−4y, (3.50) 1 m2 eq R x (y) ≡ − e−3y − 12(d + x (y)), (3.51) 2 m2 1 37

2 −4 where d = αR∗Γ(1/n)/6nm and aeq =ρ ¯r0/ρ¯m0 ≈ 2.9 x 10 . With the above deﬁnitions, one obtains:

x (y) x0 (y) = 2 (3.52) 1 3 R0 x0 (y) = + 9e−3y − 4x (y), (3.53) 2 m2 2 where

0 −3y −4y R e + aeqe 1 f fR R 2 = 2 − 2 1 + 2 + 2 2 − 1 (3.54) m H fRR m fRR 6H m fRR 6H

From the equations above, the evolution of FRLW background is given by:

−4y aeqe Ωr(y) = −3y −4y , (3.55) d + x1 + e + aeqe x1 + d Ωde(y) = −3y −4y , (3.56) d + x1 + e + aeqe 1 x2 wde = −1 − , (3.57) 9 x1 + d and Ωm(y) = 1 − Ωr(y) − Ωde(y). Figure (3.2) show the eﬀective dark-energy equation-of-state parameter as a function of the redshift for models with n = 1, 2 and 3 and diﬀerent values of the parameter

α. For higher redshift values ( R R∗), wde gets closer to −1. Besides, all models behave like ΛCDM at early times. For z → −1, one also has wde → −1 indicating that, asymptotically (t → ∞), the models have a de Sitter ﬁnal attractor. Note that, for

ﬁxed n, as the parameter α increases, the models approach the ΛCDM expansion history behavior. The redshift of phantom crossing decreases with increasing α for ﬁxed n. Note also that models with larger values of n start the phantom phase later in the Universe evolution. For instance, if n = 1 this deviation occurs at z ∼ 2.5, while for n = 2 it occurs at z ∼ 1.5, and for n = 3 at z ∼ 1.0. 38

Figure 3.2: Eﬀective equation-of-state parameter wde as a function of z for n = 1, 2 and 3 and diﬀerent values of α. Figure taken from [4] 39

Chapter 4

Spherical collapse model

In the spherical collapse model, one considers a spherical overdense region immersed in an expanding Universe [7, 82, 83]. This region is homogeneous and isotropic around its center, so its density ρ (with a top-hat profile) and pressure p only depend on time. The gravitational attraction in this region is strong enough so it can decouple from the background flow, reaching a maximum radius (the so-called turn-around) and then collapses. Initially, the density ρ(t) is infinitesimally greater than backgroundρ ¯(t), i.e,

δρ(ti) ≡ ρ(ti) − ρ¯(ti) ρ¯(ti) at an initial time ti.

4.1 Spherical Model in a Universe dominated by matter

If the Universe and the spherical region only contain non-relativistic matter (both the

2 pressure pm and the eﬀective sound speed cs are negligible), its energy and mass are conserved. Therefore, this sphere can be considered as an (isolated) closed Universe,

(l) with local scale factor r, local density parameter of dark matter Ωm and local Hubble parameter h =r/r ˙ . The evolution of r is given by the Friedmann equation [84]

1 dr = H [Ω(l) (r/r )−3 + (1 − Ω(l) )(r/r )−2]1/2, (4.1) r dt i mi i mi i 40

where quantities with subscript “i” are evaluated at initial time ti and hi = Hi,assuming, initially, the spherical region expands together with the background ﬂuid.

The equation (4.1) has the following parametric solution:

r(θ) = A(1 − cos θ),

t(θ) = B(θ − sin θ), (4.2) where A and B are constants given by

(l) riΩmi A = (l) , 2(Ωmi − 1) H−1Ω(l) B = i mi , (4.3) (l) 3/2 2(Ωmi − 1) and the parameter θ is deﬁned as:

(l) 1/2 θ = Hiη(Ωmi − 1) . (4.4)

Here η represents the conformal time, which is given by:

Z t dt η = . (4.5) 0 r(t)

According to equation (4.2), the maximum radius occurs when θ = π, so its value and turn-around time tta are:

rmax = 2A,

tta = Bπ. (4.6)

The collapse (where r = 0) happens at θ = 2π, which corresponds to

tcol = 2πB = 2tta. (4.7)

The mass inside the sphere, which is conserved, is given by:

4π 3H2 M = r3Ω(l) i , (4.8) 3 i mi 8πG 41 which relates the constants A and B through the expression

A3 = GMB2. (4.9)

In the linear regime, i.e at small θ, equation (4.2) takes the form:

1 1 lim r(θ) = A θ2 − θ4 , θ→0 2 24 1 1 lim t(θ) = B θ3 − θ5 . (4.10) θ→0 6 120

From equations (4.9) and (4.10), one obtains the following expression for the radius in the linear regime: " # 1 1 6t2/3 r (θ) = (6t)2/3(GM)1/3 1 − . (4.11) L 2 20 B

Therefore, the density in the linear regime ρL for spherical region is given by: " # 3M 1 3 6t2/3 ρL = 3 = 2 1 + . (4.12) 4πrL 6πt G 20 B Using the background density, 1 ρ¯ (θ) = , (4.13) m 6πGt2(θ) which is given for an Einstein-de Sitter Universe, the linear density contrast takes the form 2/3 L ρL − ρ¯m 3 6t(θ) δm(θ) = = . (4.14) ρ¯m 20 B L 2/3 From the above expression, one can note that δm ∝ t ∝ a, which it is in agreement with the solution obtained through the linear perturbation theory for non-relativistic matter in an Einstein-de Sitter Universe. At the collapse, the value of the linear density contrast is:

3 3π 2/3 δL (θ = 2π) ≡ δ = ' 1.68647. (4.15) m c 5 2

The linear density contrast calculated at the collapse time is called the “critical overdensity

L contrast” (δc). As it will be discussed later, in regions of Universe where δm > δc, one 42

can aﬃrm that a halo of dark matter was formed. So the critical overdensity δc is a key concept in estimating the number count of halos for diﬀerent redshift and halo-mass bins, and therefore, it is a powerful tool to compare cosmological models to observations.

4.1.1 Virialization

A real density perturbation is neither spherical nor homogeneous, so collapse does not occur at a point of inﬁnite density because a pressure gradient appears to avoid it. In general the particles present velocities that are not exactly radial and have a small angular momentum. At the moment of collapse, there is a redistribution of energy and angular momentum in the system, so that, although the individual parts still orbit, the system as a whole neither contract nor expand and reaches the virial equilibrium at a radius rvir, known as the virialization radius[83, 84]. In virial equilibrium, the system satisﬁes the virial theorem: its potential energy

n W ∝ r is related to its kinetic energy Tk by[85] n T = W. (4.16) k 2 For a spherical and homogeneous region, the gravitational potential energy is: 3 GM 2 W = − . (4.17) 5 r To determinate the virialization radius, one considers the energy conservation between the turn-around and virialization . The energy at the turn-around Eta is given by 3 GM 2 Eta = Wta = − , (4.18) 5 rta while at virialization moment it takes the form: 1 E = T + W = W (4.19) vir vir vir 2 vir where it was used equation (4.16). Using the energy conservation between the turn-around and virialization, we get that: 1 r = r . (4.20) vir 2 ta 43

From equations (4.2) and (4.6), one obtains that θvir = 3π/2 and

tvir ≈ 1.81tta. (4.21)

On the other hand, given that tvir is close to tcol, the overdensity contrast at virialization moment ∆vir is approximated to

ρ(θvir) ρ(3π/2) 2 ∆vir ≈ = = 18π ' 177.65, (4.22) ρ¯m(θcol) ρ¯m(2π)

3 whereρ ¯m is given by the equation (4.13) and ρ(θvir) = 3M/4πr (θvir).

4.2 Spherical collapse in the ΛCDM model

The evolution of the spherical overdensity region depends, of course, on the background dynamics. So, the characteristic quantities such as the critical contrast density, radius evolution and virial density will be aﬀected by the dark energy. If the background is driven by the cosmological constant Λ, then the acceleration equation for the spherical region is given by [83, 84, 86]:

r¨ 4πG = − (ρ − 2¯ρ ) . (4.23) r 3 m Λ

Assuming that both dark matter and cosmological constant interact only gravitationally and are separately conserved, we get for the background:

ρ¯˙m + 3Hρ¯m = 0, (4.24) and

ρ¯˙Λ = 0. (4.25)

On other hand, inside of the spherical region, due to its standard attractive character, dark matter always tends to cluster (its mass is conserved), so the local continuity equation takes a similar form as the continuity equation for the background ﬂuid, that is: 44

r˙ ρ˙ + 3 ρ = 0, (4.26) m r m

Of course, it is clear that dark matter will actually cluster only if the initial δρm is large enough to overcome the eﬀects from both the background expansion and DE.

Diﬀerentiating twice the deﬁnition of the dark matter density contrast δm ≡ ρm/ρ¯m −1 and using the equations above we obtain the following nonlinear evolution equation:

˙2 2 ¨ ˙ 4δm 3H δm + 2Hδm − = (1 + δm)Ωmδm. (4.27) 3(1 + δm) 2 whose linear solution is given by [87]:

L 1 11 (1 − Ωm0) 3 δm ∝ a 2F1 1, ; ; − a (4.28) 3 6 Ωm0 where 2F1 is the Hypergeometric function. Of course, if Ωm0 = 1, one recovers the linear

L L 1 relation between δm and the scale factor (δm ∝ a), as expected . In the spherical collapse model with dark energy, the critical density contrast will depend of the collapse redshift zc, i.e., δc = δc(zc). However, at high redshifts , when

Ωm → 1, one hopes to recover the standard value δc = 1.68647. Likewise, since it does not exist an analytical solution for non linear equation (4.27), it is necessary to implement numerical methods to compute δc. These methods and their consequences in the number count of halos will be the subject of the next chapter.

4.2.1 Virialization

In this case, given that the cosmological constant does not cluster or virialize [86], its sole eﬀect is to contribute to the potential energy of the dark-matter component. Therefore, the potential energy for the system is given now by:

W = Wg + WΛ, (4.29)

1 Because 2F1(a, b; c; 0) = 1 ∀{a, b, c}. 45

where Wg is the gravitational potential energy (4.17) and WΛ is the potential energy associated with the cosmological constant, that for a spherical region is given by[86]:

4πGρ¯ W = − Λ Mr2. (4.30) Λ 5

Again, using the virial theorem and the energy conservation between the turn-around and virialization, one obtains a relation between the virial radius and the turn-around one, which is given by [27]: 2s − 1 η = , (4.31) 2s3 − s where s ≡ rvir/rta, 2ΩΛ0 η ≡ −3 , (4.32) Ωm0ata (1 + δm(ata)) and ata is scale factor at turn-around moment. In particular, note that if ΩΛ0 = 0 one obtains s = 1/2, as expected. Finally, in this model the overdensity contrast at virialization moment ∆vir also depends on zc. Besides, at high redshifts one hopes to

2 recover the standard value ∆vir = 18π . 46

Chapter 5

Calculation of the critical overdensity in the spherical-collapse approximation

In GR, Birkhoff’s theorem holds, and in the spherical collapse (SC) approximation, an initial top-hat density profile keeps being a top hat. Such straightforwardness enables analytical results in the Einstein-de Sitter (EdS) background (Ωm = 1) — and it is therefore the standard benchmark for all more realistic initial conditions. In modified gravity theories, however, the fifth force mediated by the new scalar degree of freedom

— the scalaron [88] — and the so-called chameleon mechanism [71] play a crucial role. Indeed, the chameleon mechanism is a key ingredient to hide the ﬁfth-force eﬀects in high-density environments such as the Solar System and at Galactic scales.

Evidently, the validity of the SC approximation itself has been the subject of a large dispute in the current literature. Borisov et al. [28] numerically solved the full modified gravity equations for the model proposed by Hu and Sawicki [89] and found that an initial top-hat profile develops shell crossing during its evolution, and therefore, its shape changes. An improvement of the SC numerical calculation for again the same f(R) model is found in Ref. [29], using as an initial condition the average density profile around a density peak. Using the results for the SC found in Ref. [71], Lombriser et al. [90] and subsequently Cataneo et al. [91] have taken into account the chameleon suppression of 47 modifications in high-density regions.

Precisely in order to circumvent such problems and to gain some insight on the role played by the chameleon mechanism, we work in the so-called large- and small-ﬁeld limits

(see Sec 5.1). Indeed, such an approach has been proven eﬀective before: In Refs. [27, 92], the density proﬁles and the linear bias of the cluster halos were determined in such limits, showing good agreement with N-body simulations.

As mentioned before, the key quantity in the SC is the critical overdensity δc(zc) at a given collapse redshift zc. It is defined as the final value (i.e, at redshift zc) of the linear evolution of a given spherical top-hat initial perturbation that actually collapses at zc according to the full nonlinear equations. Theoretically, the latter value should be infinite, but in practice one has to deal with numerical infinities, and here is where it lies a potential problem.

There are currently two different prescriptions in the literature for its calculation, namely the “differential-radius” and the “constant-infinity” methods. In this work we show that the latter yields precise results only if we are careful in the definition of the so- called “numerical infinities” (from now on, this will be referred to as “our method”). As we shall see later on, different “numerical infinities” give rise to different results for δc(z). Although the subtleties we point out are crucial ingredients for an accurate determination of δc both in GR and in any other gravity theory, we focus on f(R) modified gravity models in the metric approach; in particular, we use the aforementioned large- (F = 1/3) and small-field (F = 0) limits1. For both of them, we calculate the relative errors (between our method and the others) in the critical density δc, in the comoving number density of halos per logarithmic mass interval nln M and in the number of clusters at a given redshift in a given mass bin Nbin, as functions of the redshift. We have also derived an analytical expression for the density contrast in the linear regime as a function of the collapse redshift zc and Ωm0 for any F .

1The parameter F will be deﬁned in equation (5.10). 48

5.1 Spherical collapse in f(R) theories

In f(R) theories the Einstein-Hilbert action is modiﬁed to

Z √ R + f(R) S = d4x −g + L , (5.1) 2κ2 m

2 where Lm is the Lagrangian of the ordinary matter, κ ≡ 4πG, and throughout this thesis, we use c = ~ = 1. Variation of Eq. (5.1) with respect to the metric gµν yields the modiﬁed Einstein equations:

1 G + f R − f g − [∇ ∇ − g ]f = κT , (5.2) µν R µν 2 µν µ ν µν R µν

df where Gµν is the Einstein tensor and fR ≡ dR . Taking the trace of Eq. (5.2), one gets

fR R − 2f + 3 fR − R = κT, (5.3)

where T is the trace of the energy-momentum tensor Tµν. The SC model considers a homogeneous and isotropic region (a top-hat profile) with density ρ(t) =ρ ¯(t) + δρ(t), whereρ ¯ is the background fluid density. We suppose that this region contains only non-relativistic matter (both the pressure pm and the effective sound

2 speed cs are negligible). Such a region can be described as a perturbation in an otherwise homogeneous universe with densityρ ¯(t), scale factor a(t), and Hubble parameter H ≡ a/a˙ , whose metric is given by

2 2 2 i j ds = −(1 + 2φ)dt + a (t)(1 + 2ψ)δijdx dx . (5.4)

In GR, in the considered case (since the anisotropic stress vanishes), the gravitational potential φ would be equal to the negative of the second potential (φ + ψ = 0). In modiﬁed theories, however, the extra scalar ﬁeld acts as a source of the deviation between them.

δρ The nonlinear continuity and Euler equations [93, 94] for the density contrast δ(t) ≡ ρ¯ 49 and velocity-ﬁeld perturbation ~v are, respectively,

1 δ˙ + (1 + δ)∇~ · ~v = 0 and (5.5) a 1 1 ~v˙ + (~v · ∇~ )~v + H~v = − ∇~ φ, (5.6) a a in comoving spatial coordinates. Combining Eqs. (5.5),(5.6) and considering ∇~ · ~v as time function (spherical-collapse approximation), one obtains a second-order diﬀerential equation for δ: 4 δ˙2 1 δ¨ + 2Hδ˙ − = ∇2φ (1 + δ). (5.7) 3 (1 + δ) a2 When gravity is modeled by f(R) theories, the potential φ is modiﬁed accordingly, as follows. A perturbation in the matter density produces a perturbation in the metric gµν, which can be translated into a perturbation in the Ricci scalar R. Using Eqs. (5.2) and

(5.4), one gets the equation for the modiﬁed potential φ (see appendix A):

16πG a2 ∇2φ = a2δρ − δR(f ), (5.8) 3 m 6 R where δR ≡ R − R¯ and R¯ is the background Ricci scalar. Accordingly, the function ¯ df ¯ ¯ fR(R) ≡ dR (R) is perturbed by δfR ≡ fR(R) − fR(R). Using (5.3), we get a2 ∇2δf = [δR(f ) − 8πGδρ ]. (5.9) R 3 R m ¯ To obtain Eqs. (5.8) and (5.9), we have considered |fR(R)| 1 and the quasi-static ˙ ¯ ~ ¯ approximation fR(R) |∇fR(R)|. The former condition indicates that the background is similar to ΛCDM, while the latter assumes that the time scale of the collapse is much smaller than the time scale of the expansion of the Universe. Therefore, any time variation of the (background) scalar ﬁeld fR is negligible in a typical time scale of collapse. Such an approximation is equivalent to focus on the evolution of the perturbations inside the

Hubble radius when the background evolution is close to ΛCDM (fR 1), as it has been shown in Ref. [95]. However, in Ref. [96], it was shown that the deviation in the global matter power spectrum between static and nonstatic simulations is only 0.2%. 50

Therefore, although in principle the static approximation is not supposed to be accurate, the corrections are actually small.

The large- and small-field limits, opposite but also inherent to any f(R), can be represented by a single factor F [27]. When the curvature in the spherical region is large, the fluctuations of the field δfR are very small, so that the Laplacian in Eq. (5.9) can be neglected, yielding δR ≈ 8πGδρm and the usual Poisson equation is recovered; this defines the so-called small-field limit. On the other hand, if the curvature R in the spherical region is similar to the background curvature R¯, its fluctuation δR is small.

Thus, it can be neglected in Eq. (5.8), which increases the gravitational potential by a global 4/3 factor on the right-hand side of that equation. This regime is the so-called large-field limit, i.e., |δfR| ∼ |φ|. In the former case, gravity is not modified due to the chameleon effect; the opposite situation occurs in the large-field limit where gravity is strengthened, becoming more attractive. For both the limits above, Eq. (5.7) can be cast as

4 δ˙2 3 δ¨ + 2Hδ˙ − = (1 + δ)H2Ω (t)(1 + F )δ (5.10) 3 (1 + δ) 2 m where F = 0 reproduces the small-ﬁeld case and F = 1/3 corresponds to the large-ﬁeld limit. It is convenient to write the above equation in terms of y ≡ ln a (we take the present value of the scale factor a0 = 1):

d2δ dH(y) dδ H2(y) + 2H2(y) + H(y) − (5.11) dy2 dy dy 4H2(y) dδ 2 3 − = (1 + δ)H2(y)Ω (y)(1 + F )δ. 3(1 + δ(y)) dy 2 m

5.2 Calculating the critical density contrast δc

The critical density can be calculated following two different procedures: one directly from the time evolution of δ(y) given by Eq. (5.11) and another from the difference in evolution between the bubble radius and the background scale factor, as we see below. In 51 the former, one has to deal with numerical infinities. We start with the latter, where one can circumvent this problem.

5.2.1 The diﬀerential-radius method

We deﬁne (following Ref. [27]) the diﬀerential radius

r a q(y ≡ ln a) ≡ − , (5.12) ri ai where ri is the initial bubble radius when the scale factor is ai . The full nonlinear evolution equation for q(y) obtained from mass conservation and Eq. (5.11) is

0 −3 00 H 0 1 Ωm0a − 2ΩΛ0 q + q = − −3 q − (5.13) H 2 Ωm0a + ΩΛ0 −3 1 Ωm0a a − −3 (1 + F ) + q σ, 2 Ωm0a + ΩΛ0 ai where 1 3 σ ≡ (1 + δi) − 1. (5.14) qai/a + 1

The initial conditions — usually set at a high redshift, when matter dominates (Ωm ∼ 1)

0 — are q(yi) ≡ qi = 0 and qi = −δi(1 + p)/(3(1 + δi)), with ! 5 r 24 p ≡ p(F ) ≡ −1 + 1 + F . (5.15) 4 25

One then solves Eq. (5.13) requiring that the collapse, deﬁned by q(zc) = −ac/ai, takes

0 place at a given redshift zc ≡ 1/ac −1. Such requirement constraints qi and, consequently, the value of δi ≡ δm(ai) which we parametrize as

1+p δi = C ai , (5.16)

inspired by the linear growth of δc at such high redshifts — see Eq.(5.17).

The critical density δc is the linear evolution — determined by Eq. (B.3) — of such initial density perturbation δi, given by Eq. (B.11) at the collapse (a = ac): 1+p (1 − Ωm0) 3 δc = C ac 2F1 (p), b(p); c(p); − ac Ωm0 52

for any Ωm0 and F — see Eqs. (B.12) for deﬁnitions of (p), b(p) and c(p).

The explicit dependence on Ωm0 is usually taken for granted because the full evolution (i.e, until the collapse) is supposed to happen while matter still dominates. In that case there is no dependence of the hypergeometric function on the collapse scale factor ac for any value of F , since 2F1(, b; c; 0) = 1 ∀{, b, c} and one recovers the standard power-law dependence

1+p δc(a) = C ac . (5.17)

Of course, for p(F = 0) = 0 and Ωm0 = 1, we recover the standard value δc = 1.68647, as expected. For F = 1/3 and Ωm0 = 1, our results yield a constant δc = 1.70605 for any zc, which agrees with the result given in Ref. [27] for zc = 0.

5.2.2 The constant-inﬁnity method and its mending

As mentioned before, the full solution of Eq. (5.11) should be infinite at the collapse. However, since this equation can be solved only numerically, it is necessary to establish the infinite as a very large number for a given collapse redshift zc. As we see below, this numerical infinity depends both on the collapse redshift zc and on Ωm0.

Here, we consider an initial value for the redshift zi = 1000, where the Universe is completely dominated by nonrelativistic matter (Ωm = 1). We then have to deal with the linear version of Eq. (5.11) when the background behaves like an EdS universe:

d2δ 1 dδ 3 + = (1 + F )δ , (5.18) dy2 2 dy 2 m whose linear growing solution is given by

y(1+p) δl(y) = Ce , (5.19) where C is again a constant which clearly depends on the collapse redshift. The initial conditions are then given by

yi(1+p) 0 δi = Ce , δi = (1 + p)δi . (5.20) 53

The constant C is obtained requiring the collapse occurs at y = yc, i.e, δ(yc) → ∞. To

ﬁx the numerical value of inﬁnity, we use the known values of δc for an EdS universe:

 3 3π 2/3  5 2 , for F = 0 δc = (5.21)  1.70605 , for F = 1/3, where the latter value was obtained in the previous section. The constant C is then given by 2/3  3 3π −yc  5 2 e , for F = 0 C(yc) = (5.22)  1.70605 e−yc(1+p) , for F = 1/3. Evolving Eq. (5.11) with the initial conditions (5.20) and (5.22), the numerical inﬁnity is deﬁned by

Inf(yc) ≡ δm(Ωm0 = 1, yc), (5.23) which clearly depends on zc. Some papers [31, 97] do not take this dependence into

5 8 account and assume a constant value Inff ≡ Inf(yc = 0) = 10 or 10 . While both Inff

5 and Inf have roughly the same order of magnitude (∼ 10 ) at zc = 0, Inf(zc) decreases

4 monotonically with zc and can be as low as 10 at z ∼ 3 (for both F = 0 and F = 1/3). Later on (see Figs. 5.2, 5.3, and 5.6), we point out the numerical diﬀerences in the ﬁnal outcome from this approximation.

Once the so-called infinity Inf(yc) is established, we then use Eq. (5.11) for different values of collapse redshifts zc and for different values of Ωm0 in order to calculate the constants Cj ≡ C(yc, Ωm0) that satisfy the condition for the collapse δm(Ωm0, yc) = Inf(yc).

The corresponding value of δc is given by Eq. (B.11). If F = 0, we get yc 1 11 (1 − Ωm0) 3yc δc(Ωm0, yc) = Cj e 2F1 , 1; ; − e . (5.24) 3 6 Ωm0

If F = 1/3, we get

1 √ (−1+ 33)yc δc(Ωm0, yc) = Cj e 4 × (5.25) " √ √ √ # 7 + 33 −1 + 33 6 + 33 (1 − Ωm0) 3yc × 2F1 , ; ; − e . 12 12 6 Ωm0 54

1.705 F=1/3 F=1/3 1.705 1.700 1.700 1.695 1.695 1.690 c c δ F=0 δ 1.690 1.685 F=0 1.680 1.685

1.675 1.680

zc =0 Ωm0 = 0.3 1.670 1.675

0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 6 Ωm0 zc

Figure 5.1: (a,left panel) Critical density contrast δc as a function of Ωm0 for F = 0 (solid blue line) and F = 1/3 (dotted red line) when the collapse occurs at redshift zc = 0, following our approach. (b,right panel) Critical density contrast δc as function of redshift collapse zc for F = 0 (solid blue line) and F = 1/3 (dotted red line) with Ωm0 = 0.3, following our approach. 5.3 Comparing the results from diﬀerent approaches

In this section we plot the relative errors between our method and the previously mentioned ones, for both values F = 0 and F = 1/3, when one calculates the critical density

δc (as a function of the collapse redshift), the comoving number density of halos per logarithmic mass interval nln M — see Eq. (5.27) for its deﬁnition — and the number of clusters at a given redshift in mass bins Nbin, as a function of either the redshift or the mass interval.

If the value of zc is ﬁxed, our method describes the evolution of δc in terms of Ωm0

(Fig. 5.1, left panel). On the other hand, when we fix the value of Ωm0, the method describes the evolution of δc as function of redshift zc (Fig. 5.1, right panel). In the following figures we compare the results of our method with the differential- radius equation and with the method where the numerical infinity assumes large constant values (we pick Inff = 105 and 108 for the sake of comparison with previous results in the literature [31, 97]) for both values F = 0 and F = 1/3. We define the ratio between the other method and ours as i i δc ∆ ≡ pw (5.26) δc where the superscript i stands for the different methods in the literature, specified in the 55

Radius 0.002 0.000 0.001 -0.002 Radius 0.000 -0.004 105 -0.001 105 -0.006 -0.002 1 - Δ 1 - Δ 108 -0.008 8 -0.003 10 -0.010 -0.004 F=1/3 Ω = 0.3 -0.012 F=0 m0 Ωm0 = 0.3 -0.005 -0.014 0 1 2 3 4 5 6 0 1 2 3 4 5 6

zc zc

Figure 5.2: Relative errors for δc (see text for definition) between our method and (a) the differential-radius one (dotted black line), the constant-infinity one, with (b) Inff = 105 (dashed red line) and (c) Inff = 108 (solid blue line) for F = 0 (left panel) and F = 1/3 (right panel). For all models, we assumed Ωm0 = 0.3.

ﬁgures, and the superscript “ pw” stands for the approach introduced in the present work.

For the ﬁgures we always assume Ωm0 = 0.3.

i In Fig 5.2 we show 1 − ∆ as a function of the collapse redshift zc. Note that the differential-radius method and ours are equivalent. On the other hand, the relative differences between our method (or the differential-radius method) and the constant-infinity one increase (in magnitude) with the collapse redshift because the critical density contrast calculated with our method stabilizes at the EdS value for large zc (as it is supposed to), while it grows unbounded in the latter approaches.

Next, we calculate and compare the mass function using the results for δc obtained by our method and by the method where the infinite is fixed (we pick 105 and 108 again). We use the Sheth-Tormen mass function [98], which analytically determines the distribution of these objects as a function of their virial masses and redshifts. The virial mass M is defined such that the average density inside the virial radius rv is ∆v times the critical density. We should mention that in order to relate SC to virialized halos, the virial theorem has to be modified in f(R) theories. See, for instance, Ref. [99] for more on this topic. In this formalism, the comoving number density of halos per logarithmic interval 56

(in the virial mass M) is

dn ρ dν n = (M, z) = m0 f(ν) (5.27) ln M d ln M M d ln M where ρm0 is the present matter density of the Universe, the peak threshold ν = δc(z)/σ(M, z),

r 2 f(ν) = A aν[1 + (aν2)−q] exp[−aν2/2)] (5.28) π and σ is the variance of the linear contrast density ﬁeld in spheres of a comoving radius r containing the mass M. The normalization constant A is deﬁned such that R f(ν)dν = 1.

Here, we use the following approximation [100] for σ:

M −γ(M)/3 σ(M, z) = σ80 D(z) , (5.29) M8 where 1 M γ(M) ≡ (0.3Γ + 0.2) 2.92 + log , (5.30) 3 M8 and shape parameter Γ is given by [100] Ωb Γ ≡ Ωm0h exp −Ωb − . (5.31) Ωm0

14 −1 −1 Above, M8 = 5.95 x 10 Ωm0h M is the mass inside a sphere of radius 8h Mpc, h

−1 −1 2 is the current Hubble constant in units of 100 km s Mpc ,Ωb = 0.02230/h is the baryonic density parameter, and D(z) is the growing solution of Eq. (B.1) normalized in z = 0. We assume that σ80 = 0.8159 [101], q = 0.3, and a = 0.707 [98]. We have checked that if, instead of the ﬁt given by Eq. (5.29) — which is numerically simpler to deal with

— we had used a more accurate approach, our results would not change signiﬁcantly. In Fig. 5.3, we show the relative error

i pw i pw 1 − ∆n ≡ nln M − nln M /nln M (5.32)

for z = 0 between our method and the others when Ωm0 = 0.3 for both values F = 0 and F = 1/3. 57

0.01 0.05 F=0 108

0.04 Ωm0 = 0.3 0.00 105 h=0.6774 -0.01 0.03 108 n z = 0 n F=1/3 0.02 -0.02 1 - Δ 1 - Δ Ωm0=0.3 -0.03 0.01 h=0.6774 -0.04 z = 0 0.00 105 -0.05 -0.01 13.0 13.5 14.0 14.5 15.0 15.5 16.0 13.0 13.5 14.0 14.5 15.0 15.5 16.0 M M Log10[M*h/ ⊙] Log10[M*h/ ⊙]

Figure 5.3: Relative errors for nln M (see text for deﬁnition) between our method and the constant-inﬁnity one, with (a) Inff = 108 (solid blue line) and (b) Inff = 105 (dashed red line) for redshift z = 0 in the cases F = 0 (left panel) and F = 1/3 (right panel). For all models, we assumed Ωm0 = 0.3 and h = 0.6774.

In Fig. 5.4 we show relative errors as a function of the redshift when the virial mass

13 −1 15 −1 takes the value 10 h M (upper panels) and 10 h M (lower panels). From these figures, it is clear that the relative errors are more significant for larger virial masses and higher redshift. The observed quantity, however, is the number of clusters at a given redshift and in a given mass bin. It is defined as

Z Z Msup dnln M dV Nbin ≡ dΩ dM, (5.33) 4π Minf dV dz dΩ

dV r2 where dV is the comoving volume at redshift z, dz dΩ = H(z) , and the comoving distance r(z) is given by Z z r(z) = H−1(z0) dz0. (5.34) 0

Figure 5.5, obtained with our method, shows Nbin for F = 0 and virial masses in the

13 14 −1 14 15 −1 ranges 10 − 10 h M (left panel) and 10 − 10 h M (right panel). Similar results are obtained by using F = 1/3. Note that for higher-mass bins, the peak in the

Nbin is lower and located at a smaller z. From this piece of information and from our

i i 13 14 −1 results above — both ∆ and ∆n increase with z — we expect the bin 10 − 10 h M to be the most sensitive one to the small diﬀerences in δc and nln M we have been pointing out. 58

0.30 0.25 F=0 F=1/3 108 0.25 0.20 8 Ωm0 = 0.3 10 Ωm0 = 0.3 0.20 h=0.6774 0.15 h=0.6774 n n

13 -1 0.15 13 -1 5 M= 10 h M M= 10 h M⊙ 10 ⊙ 1 - Δ 1 - Δ 0.10 0.10 5 0.05 10 0.05

0.00 0.00 0 1 2 3 4 5 6 0 1 2 3 4 5 6 z z 1.0 F=1/3 F=0 108 0.8 108

0.8 Ωm0 = 0.3 Ωm0=0.3 0.6 h=0.6774 h=0.6774 0.6 105 n n M 15 -1 M M= 10 15 h-1 M = 10 h ⊙ 0.4 ⊙ 1 - Δ 1 - Δ 0.4 105 0.2 0.2

0.0 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 z z

Figure 5.4: Relative errors for nln M (see text for deﬁnition) between our method and the constant-inﬁnity one, with (a) Inff = 108 (solid blue line) and (b) Inff = 105 (dashed red 13 −1 line) as a function of redshift for F = 0, F = 1/3 and virial mass of 10 h M (upper 15 −1 panels). The same plots, are shown for 10 h M (lower panels). For all models, we assumed Ωm0 = 0.3 and h = 0.6774.

2.5×10 7 F=0 600 000 F=0

Ωm0 = 0.3 Ω = 0.3 7 m0 2.0×10 13 -1 14 -1 500 000 10 h M⊙

N 300 000 1.0×10 7 200 000 5.0 10 6 × 100 000

0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 z z

Figure 5.5: Nbin (see text for deﬁnition) as a function of redshift for F = 0 and virial 13 14 −1 14 15 −1 masses between 10 and 10 h M (left panel) and 10 and 10 h M (right panel). As always, we assumed Ωm0 = 0.3 and h = 0.6774. 59

0.10 0.06 F=1/3 F=0 0.05 Ωm0 = 0.3 0.08 Ωm0 = 0.3

13 -1 14 -1 13 -1 14 -1 0.04 10 h M⊙

bin 0.03 N bin

N 8

1 - Δ 10 0.04 108 0.02

0.02 0.01 105 105 0.00 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 z z

Figure 5.6: Relative errors for Nbin (see text for deﬁnition) between our method and the constant-inﬁnity one, with (a) Inff = 105 (dashed red line) and (b) Inff = 108 (solid blue 13 14 −1 line) as a function of redshift for virial masses between 10 and 10 h M and F = 0 (left panel) and F = 1/3 (right panel). For all models, we assumed Ωm0 = 0.3 and h = 0.6774.

We then deﬁne one last relative error, namely,

1 − ∆i ≡ N pw − N i /N pw (5.35) Nbin bin bin bin

13 14 −1 and plot it, for virial masses between 10 and 10 h M , in Fig. 5.6 for F = 0 (left panel) and F = 1/3 (right panel). Notice that, in the redshift range of interest, it is at most of the order of 0.01. 60

Chapter 6

Number density of halos in f(R) theories

In this chapter we investigate the discrimination power of the cosmological evolution of density perturbations and their outcomes (such as the number density of halos per mass bin) between GR with cosmological constant and two models of f(R) gravity: γ gravity(γG) and Hu-Sawicki(HS).

In particular, we determine the comoving number density of halos for these models in the small-ﬁeld and large-ﬁeld limits using the spherical-collapse approximation. Dif- ferently from chapter 5, where it is assumed a ΛCDM background, here we consider a f(R) background. In Refs. [27, 92], the number density of halos was determined in the aforementioned limits for the Hu-Sawicki model, showing good agreement with N-body simulations. We extend those works to γ gravity and study the role of its free parameters on the number density of halos. At the end of the current chapter, we compare our results to the outcome from N-body simulations of γ gravity [102].

6.1 Comparing the f(R) models with ΛCDM

Any viable f(R) theory can be made to agree with the observed evolution of the back- ¯ ground by a suitable choice of its free parameters. Indeed, for HS, if fR0 ≡ fR(R0) 1, the background expansion history is indistinguishable from that of ΛCDM (at the current 61 observational level). Therefore, no strong constraints will ever come from the background cosmology itself, because most of the proposed modifications behave almost exactly the same — i.e, as a cosmological constant — for large values of R¯ (or equivalently, z ≥ 1). Tests at solar-system scales also yield potentially strong constraints, but in f(R) theories they can always be satisfied via the so-called chameleon effect [71].

On the other hand, the cosmological evolution of density perturbations probe a large range of R both in the background and in the perturbed patch itself. Therefore, it is expected that diﬀerences between f(R) models and the standard model can be observed at perturbative level. Besides, one way to understand the role both the chameleon and the scalar ﬁeld fR mechanics in the matter perturbations is trough their observable outcomes (such as the number density of halos per mass bin).

Equation (5.27) determines the comoving number density of halos per logarithmic interval virial mass Mv. If one assumes a particular density proﬁle, it is possible (see below) to rescale the virial mass to M300 in such a way that the comoving number density of halos per logarithmic mass M300 is given by [27]:

dn dlnMv nlnM300 ≡ = nlnMv . (6.1) dlnM300 dlnM300

The parameter M300 is deﬁned as the mass of a halo whose density is 300 times the background density — its radius is denoted r300 [103]: 4 M ≡ πr3 300¯ρ. (6.2) 300 3 300

The mass Mh of a spherical top-hat halo with radius rh can also be defined as [103] 4 4 M = π r3 ρ = π r3 ∆ ρ,¯ (6.3) h 3 h h 3 h h where ρh is the halo density,ρ ¯ is the background density and ∆h ≡ ρh/ρ¯. In this work we used an the Navarro-Frenk-White density profile (NFW) [104]. The correspondence between the virial mass Mv and M300 by considering this density profile is shown in the appendix B. 62

Λ We deﬁne n300 as the comoving number density of halos per logarithmic mass M300 f in the ΛCDM model. Accordingly, n300 is the corresponding quantity in an universe described by an alternative theory f(R), where the label f stands for either HS or γG. We now will explore the behavior of the relative diﬀerences between such quantities

Λ and n300: f f ∆n300 n300 Λ ≡ 1 − Λ . (6.4) n300 n300 Diﬀerently from chapter 5, in this section we consider the full expression for the variance of the linear contrast density ﬁeld σ [84]: Z inf 2 1 2 2 dk σ(R, z) = 2 k PL(k, z)W (kR) , (6.5) 2π 0 k

where PL(k) is the linear power spectrum and sin(kR) cos(kR) W (kR) ≡ 3 − (6.6) (kR)3 (kR)2

3 R is a top-hat window function with mass M = 8h−1Mpc M8, where M8 = 5.96 × 14 −1 −1 10 Ωm0h M is the mass inside a comoving sphere of radius 8h Mpc. We write the linear power spectrum as

3+ns 2 2 PL(z, k) = P0k D (z, k) T (k), (6.7) where ns is the spectral index, D(z, k) is the growth function (normalized at z = 0) and T (k) is the BBKS Transfer function [105], which can be written as

− 1 ln(1 + 0.171x) 4 T (x ≡ k/k ) = 1 + 0.284x + (1.18x)2 + (0.399x)3 + (0.490x)4 (6.8) eq 0.171x

−1 2 2 −1 for Ωb0 Ωm0, where keq = (0.73h Ωm0) is the comoving scale at the radiation-dust energy-density equality. In order to calculate the quantity (6.1) we consider the value of

δc given by the spherical collapse model in the quasi-static approximation (for large-field and small-field limits). Besides, the relative difference between the linear power spectrum in f(R) and ΛCDM is ∆P f δf (k, z)2 L = − 1, (6.9) ˜ PL δ(z) 63 where δ(z) is the growing mode of the density contrast in ΛCDM [87]: y 1 11 (1 − Ωm0) 3y δ(y) = e 2F1 , 1, , − e , (6.10) 3 6 Ωm0

f with y ≡ ln a and a0 = 1, and δ is the growing mode of the density contrast in f(R) gravity that obeys the equation (assuming |fR| 1) [73]:

0 2 00f 0f H f −3y 1 − 2Q 3H0 Ωm0 δ + δ 2 + − δ e 2 = 0, (6.11) H 2 − 3Q H (1 + fR) where 2 2fRRk Q(k, y) ≡ − 2y , (6.12) (1 + fR)e and 0 ≡ d/dy. The evolution of the Hubble parameter H(y) depends, as already mentioned, on the particular f(R) adopted. For HS model, this evolution is given by equations (3.43-3.44) while for γG , it is given by equations (3.52-3.53).

Λ The dependence of ∆n300/n300 with the halo mass is shown in Fig. 6.1 for HS and in Fig. 6.2 for γG — see the corresponding insets for the values of the free parameters. In all the panels of both ﬁgures, the upper red dashed curves represent the case F = 0, while the lower blue solid bound is given by F = 1/3.

For HS, the lower fR0 is, the closer the upper curve (F = 0) is to the standard ΛCDM case. This is in agreement with the behavior of the background in this limit for HS. The values of the free parameters for γG in Fig. 6.2 were chosen so that the eﬀective equation- of-state parameter would disagree from a plain cosmological constant (wDE = −1) by no

γG more than 3% — see Fig. (3.2). When α increases, n300 gets closer to its corresponding ΛCDM value for both F = 0 and F = 1/3, except for larger masses.

For both models, the lower curve (F = 1/3) displays a large discrepancy at higher mass values. Although our results do agree with the conclusions from Ref. [27], we point out that the static approximation used through out the present work might not be very accurate for large-mass clusters since their collapse time is much larger than for smaller 64

2.0 0.8 Hu & Sawicki Hu & Sawicki n=2 n=2 1.5 FR0=10^-4 0.6 FR0=10^-5 Ωm0=0.24 Ωm0=0.24 = ns 0.96314 ns=0.96314 1.0 0.4 lnM lnM = lnM

h 0.6774 lnM

n = n

Δ n h 0.6774 Δ n Sheth & Tormen Sheth & Tormen BBKS 0.2 0.5 BBKS

0.0 0.0

12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 Log [M *h/M ] 10 300 ⊙ Log10[M300*h/M⊙] 0.4 Hu & Sawicki n=2 0.3 FR0=10^-6 Ωm0=0.24 ns=0.96314 0.2 h=0.6774 Sheth & Tormen

0.1 BBKS lnM lnM n Δ n

0.0

-0.1

-0.2

-0.3 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5

Log10[M300*h/M⊙] HS Λ −4 −5 −6 Figure 6.1: ∆n300/n300, for fR0 = 10 , 10 and 10 (from top to bottom). In all panels, the case F = 0 (F = 1/3) is given by the upper red dashed (lower solid blue) curve.

0.7 γ-Gravity 0.6 0.6 γ-Gravity n=2 n=2 α=1.05 0.5 α=1.18 Ωm0=0.267 0.4 Ω = ns=0.96314 0.4 m0 0.267 h=0.71 ns=0.96314 lnM lnM 0.3 lnM lnM = n n h 0.71 Δ n Δ n 0.2 0.2

0.1 0.0

0.0

-0.2 -0.1 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 Log M h M Log10[M300*h/M⊙] 10[ 300* / ⊙] 0.3 γ-Gravity n=2 0.2 α=1.5 Ωm0=0.267 ns=0.96314 0.1 h=0.71 lnM lnM n Δ n

0.0

-0.1

-0.2 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5

Log10[M300*h/M⊙] γ Λ Figure 6.2: ∆n300/n300 for n = 2 and α = 1.05, 1.18 and 1.5 (from top to bottom). In all panels, the case F = 0 (F = 1/3) is given by the upper red dashed (lower solid blue) curve. 65

-0.98

-0.99

-1.00 DE w

-1.01

-1.02

-1.0 -0.5 0.0 0.5 1.0 z

Figure 6.3: Effective equation-of-state parameter wDE of γG as a function of the redshift z with α = 0.86, n = 3 e Ωm0 = 0.267. The amplitudes of the peaks decrease with larger n, while their redshifts slowly grow with n. masses (and might be of the same order of magnitude than the Hubble time) — that would actually affect both limits. Nevertheless, is was shown in Ref. [96] that the deviation in the global matter power spectrum between static and non-static simulations is only about 0.2%. This question remains to be fully clarified.

The role played by the steepness in γG can be seen in Figure 6.4, which shows

γG Λ ∆n300/n300 gravity with n = 3 (where the eﬀective equation-of-state parameter would disagree from cosmological constant by no more than 2.5% – see Fig.6.3). Notice that the disagreement between γG and ΛCDM for small masses is even smaller for both F = 0 and F = 1/3 (when compared to n = 2), as a direct consequence of the behavior of the eﬀective cosmological constant: although the peak divergence from w = −1 of both cases are approximately the same same order, in the latter case (n = 3) such displacement happens at a smaller z.

0.4

0.2 lnM

lnM 0.0 n Δ n

-0.2

-0.4

12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 M M Log10[ 300*h/ ⊙] γG Λ Figure 6.4: ∆n300/n300 for α = 0.86, n = 3 e Ωm0 = 0.267. 66

As mentioned before, Ref. [102] uses γG in N-body simulations. As such, they do not consider the spherical collapse approximation, but it is expected that their outcomes (eg, the number counts) are between the large- and small-field limits presented in the current work [27, 92, 28]. Although they do not present error bars (see their fig. 5) and their mass normalization is different (in comparison to our results), their discrepancy with the ΛCDM model is very large compared to our results. Besides, the authors of Ref. [102] conclude that γG can be falsified at perturbative level, which is not possible according with our results — except, maybe, for large-mass clusters, where the statistics is low because the number count of halos in this mass range is small. 67

Chapter 7

Top-Hat Spherical Collapse with Clustering Dark Energy: Radius Evolution and Critical Density Contrast

In this chapter we consider the Top-Hat Spherical-Collapse (SC) model with dark energy, which can partially (or totally) cluster, according to a free parameter γ. The lack of energy conservation has to be taken into account accordingly, as we will show. In particular, Ref. [31] investigated constant phantom, constant non-phantom and varying DE EoS parameter. The authors have focused only in the limiting cases, namely, fully clustered and completely homogeneous DE. In Ref. [34] the SC model with fully clustered DE is considered assuming a linear relation between the matter contrast density and the DE one, according to a free parameter r. In Refs. [31, 32], as well as in [34], it is also assumed that the DE EoS is the same inside and outside the collapsed region 1.

We relax the aforementioned hypotheses and generalize those results. Following the Ansatz suggested in Ref. [86] (see also [23]), we investigate the SC model with DE, assuming that it can cluster partially or totally, according to a normalized parameter: if

1As we will show further down, this assumption is equivalent to requiring that the DE EoS parameter 2 is equal to its speed of sound squared: w = cs. 68

γ = 0, DE is fully clustered; if γ = 1, DE is completely homogeneous. Besides, we determine characteristic quantities for the SC model, such as the critical density contrast and the radius evolution, with particular emphasis on their dependence on the clustering parameter γ.

7.1 Spherical collapse with dark energy perturbations

For a ﬂat, homogeneous and isotropic universe with dark matter and dark energy, the

Einstein equations are given by:

a˙ 2 8πG ≡ H2 = (¯ρ +ρ ¯ ) , (7.1) a 3 m de a¨ 4πG = − [¯ρ + (1 + 3w)¯ρ ] . (7.2) a 3 m de

In the equations above, a is the scale factor, H is the Hubble parameter, w ≡ p¯de/ρ¯de is the

EoS parameter of DE (assumed to be constant), andρ ¯m,ρ ¯de andp ¯de are the (background) energy densities of matter and DE and the DE pressure, respectively. A dot over a given quantity denotes its time derivative.

Assuming that both dark matter and DE interact only gravitationally and are separately conserved, we get

ρ¯˙m + 3Hρ¯m = 0, (7.3)

ρ¯˙de + 3H(1 + w)¯ρde = 0. (7.4)

Here we investigate the nonlinear evolution of the gravitational collapse and, to this aim, we consider the Top-Hat Spherical-Collapse (SC) model. The SC model considers a spherical region with a top-hat proﬁle and uniform density ρ(t) =ρ ¯(t) + δρ(t), immersed in a homogeneous universe with energy densityρ ¯(t). Here δρ initially is a small perturbation of the background ﬂuid energy density. We suppose that this region also contains nonrelativistic matter (pm =p ¯m = 0) and DE. Such a spherical region can be described as 69 a separated universe with (local) scale factor r. The acceleration equation for this region is given by: r¨ 4πG = − (ρ + ρ + 3p ) , (7.5) r 3 m de de where pde(t) =p ¯de(t) + δp(t) is the DE pressure inside the spherical region and δp(t) a small pressure perturbation. The DE EoS parameter in the spherical region is given by [? ] p (c2 − w)δ wc ≡ de = w + s de , (7.6) ρde 1 + δde

c 2 where the superscript “ ” stands for “clustered”, cs ≡ δpde/δρde is the DE sound speed squared (assumed to be constant) and δde is the DE density contrast (see its deﬁnition

2 below). Note that only if cs = w (or homogeneous DE, i.e., δde = 0) the DE EoS parameter in the collapsing region is equal to that of the background (wc = w).

Due to its standard attractive character, dark matter always tends to cluster, so the local continuity equation takes a similar form as the continuity equation for the background fluid, that is: r˙ ρ˙ + 3 ρ = 0, (7.7) m r m where r is the local scale factor. Of course, it is clear that dark matter will actually cluster only if the initial δρm is large enough to overcome the effects from both the background expansion and DE. In the present work we assume that DE can also collapse — although not necessarily together with the matter content, since it can flow away from the collapsing sphere. This is precisely the reason for the lack of energy conservation in the perturbed region. Therefore, we parameterize such physical phenomenon writing the local continuity equation for DE as [86] (see also [23]) :

r˙ ρ˙ + 3(1 + wc) ρ = γΓ , with 0 ≤ γ ≤ 1 , (7.8) de r de where r˙ a˙ Γ ≡ 3(1 + wc) − ρ . (7.9) r a de 70

Here, Γ describes the leaking of DE away from the spherical collapsing region and 0 ≤ γ ≤ 1 is the aforementioned clustering parameter. The non-clustering, i.e, homogeneous

R c DE corresponds to γ = 1. Notice that in this case, we have ρde ∝ exp[−3 (1 + w )da/a]

−3(1+w) whileρ ¯de scales asρ ¯de ∝ a . So, in principle, even if the DE energy densities were initially equal, they would evolve diﬀerently. However, as we will show further down, when γ = 1, in the linear regime, there is no growing mode and δρde rapidly tends to zero. Therefore, it is not possible to distinguish in this case the behavior of the DE inside and

c outside the spherical region: ρde =ρ ¯de and consequently w = w. In this case (and also for γ > 0) the total energy of the system is not conserved [86]. In contrast, the case of full clustering, i.e, when γ = 0, ensures that ρde 6=ρ ¯de, such that the spherical region is completely segregated from the background and it is considered an isolated system, which conserves energy. We shall also consider intermediate values of γ in our analysis. Notice that, diﬀerently from Ref. [86], we are not assuming that the DE EoS is the same inside and outside the collapsing spherical region. As remarked above, this is only the case when

2 dark energy is homogeneous (γ = 1) or cs = w.

Diﬀerentiating twice the density contrast δj ≡ ρj/ρ¯j − 1 for both dark matter (δm) and dark energy (δde) and using the equations above we obtain the following nonlinear evolution equations :

˙2 2 ¨ ˙ 4δm 3H 2 δm + 2Hδm − = (1 + δm) Ωmδm + (1 − Ωm)δde(1 + 3cs) , (7.10) 3(1 + δm) 2

¨ ˙ ˙ δde = −3(h − H)(1 + w)(1 − γ)δde − 3(h(1 − γ) + γH)δwδde − ˙ ˙ − 3(1 + w)(1 − γ) h − H (1 + δde) − ˙ ˙ − 3δw(1 − γ) h(1 − γ) + γH (1 + δde) − ˙ − 3 (h(1 − γ) + γH) δw (1 + δde) . (7.11) 71

In the expression above, δw ≡ wc − w (see eq. (7.6)),

˙ 2 r˙ δde + 3H ((1 + w)(−1 + γ ) + (−1 − w + γ + γcs) δde) h ≡ = 2 , (7.12) r 3 (−1 + γ) (1 + w + (1 + cs) δde)

r¨ h˙ = − h2, (7.13) r r¨ H2 = − Ω (1 + δ ) + (1 − Ω )(1 + 3c2)δ + 1 + 3w (7.14) r 2 m m m s de and 3 H˙ = − H2 (1 + w (1 − Ω )) . (7.15) 2 m

Here Ωm = Ωm(t) is the background nonrelativistic matter energy-density parameter at the instant t.

In the expressions above we assume, obviously, that γ 6= 1, since, as mentioned before,

2 if γ = 1 DE does not cluster. We note that in the particular case in which cs = w [31, 32, 34], such that δw = 0, eq. (7.11) reduces to

˙2 ¨ ˙ 4 + 3w − 3γ(1 + w) δde δde + 2Hδde − = 3(1 + w)(1 − γ) (1 + δde) 3H2 = (1 + δ )(1 − γ)(1 + w)[Ω δ + (1 − Ω )δ (1 + 3w)]. (7.16) 2 de m m m de

If we further impose γ = 0, we then recover Eq. (7) of Ref. [31] for the case in which w is constant.

To determine the initial conditions for δm and δde, we consider the linear approximation of Eqs. (7.10) and (7.11) in a matter-dominated universe (Ωm ∼ 1 and Ωde ∼ 0):

3 δ0 3 δ00 + m − δ = 0 (7.17) m 2 a 2a2 m 3 δ0 δ00 + − 3(w − c2) de − de 2 s a 3 − (1 + w)(1 − γ)δ + (w − c2)δ ) = 0, (7.18) 2a2 m s de where 0 ≡ d/da . Since we are interested in the formation of structures, the decreasing 72

2 γ = cs =0,w=-1.1

c 0 w 2 γ = cs =0,w=-0.9 2 γ =0.5, cs =1,w=-0.9

-2 2 γ = 0.5, cs =1,w=-1.1

2 γ =0, cs =1,w=-1.1

-4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 t/tcolΛCDM Figure 7.1: Behavior of the equation-of-state parameter inside the bubble (wc) with respect to time, for the labeled parameters. mode of the above equations will not be considered. The growing mode solutions are:

δm(a) = C a and (7.19) (1 + w)(1 − γ) δde(a) = 2 δm(a). (7.20) 1 − 3(w − cs)

As remarked above, if γ = 1 we obtain δde = 0. We assume in our analysis that 1 − 3(w −

2 cs) > 0 which implies that for phantom models (w < −1) δde < 0 (i.e, there is less dark energy inside the bubble than in the background). Note that if δde < −1, then ρde < 0. Although such case is exotic, in principle, it is allowed in some modiﬁed gravity models

c [106]. Whenever δde crosses −1, which happens only if w < −1, then w goes from −∞ to +∞ (see Eq. (7.6) and Fig. 7.1). Note, however, that such divergence does not aﬀect the evolution of the bubble, since wc does not appear explicitly in the equations of motion for the radius r (or, actually, for the variable y), as we will show next.

Given the contrast density for each ﬂuid, the evolution of the local scale factor is given 73 by Eq. (7.5), which in terms of y ≡ r − a can be written as: ri ai

1 H0 1 y00 + y0 + + = (7.21) ai H a 1 H 2 a = − 0 y + × 2 Ha ai −3 × Ωm0a 1 + δm + 2 −3(1+w) + 1 + 3w + (1 + 3cs)δde 1 − Ωm0 a 1 + δde ,

where Ωm0 is the present value of the matter density parameter. An initial condition for

0 Eq. (7.21) is naturally y(ai) = 0. To obtain y (ai), we consider that, initially, the mass of the spherical region is given only by the contribution from dark matter:

4 M = πR3(1 + δ )¯ρ . (7.22) i 3 i mi mi

The (possible) contribution from DE is negligible since ρde ρm when the initial conditions are set, in a matter-dominated universe.

In the above equations, Ri ≡ r(ti)X is the physical radius of the collapsing sphere at instant ti, X is its coordinate radius and δmi is the initial matter density contrast. Since dark matter always collapses (depending, of course, on the initial conditions of the matter perturbations), the mass M inside the spherical region will always be a constant. Thus,

0 0 −5 we have y (ai) = −δmi/[3(1 + δmi)]. We adopt in our numerical calculations ai = 10 .

7.2 Bubble Evolution

In this section we investigate the bubble evolution, namely its radius as a function of time, and one of the main results from the SC model: the critical density contrast — a crucial quantity to determine the number of collapsed objects. Throughout the paper we assume that Ωm0 = 0.3. We also keep the same initial conditions for dark-matter perturbations, such that the collapse in ΛCDM always occurs at the present time. 74

We pay special attention to the dependence of the outcomes in the free parameters

2 of our model: γ, cs and w. Some situations are particularly interesting and express the richness of the present parametrization:

2 • cs = 0, in which there is no DE pressure perturbation. It is interesting to point

out that, in this case, in the ﬁnal stages of the collapse (δde → ∞), the local dark energy does behave as dark matter, since wc → 0 — see eq. (7.6). Note also, from

eq. (7.20), that for phantom dark energy one will always get δde < 0: there is less dark energy inside the bubble than in the background.

2 c • cs = w, which indicates that the clustered DE EoS parameter (w ) and the background one (w) are equal.

2 • cs = γ. We intend to model a continuous “turning on” of the clustering in scalar ﬁeld models [86]. In quintessence and k-essence models, usually, two choices are

made:

2 a) cs = 1, in which case the standard quintessence scalar ﬁeld (i.e, a minimally coupled scalar ﬁeld with a canonical kinetic term) does not cluster, remaining

homogeneous on subhorizon scales [16], and

2 b) cs = 0 (or more generally sub-luminal behaviour) are considered in k-essence scalar ﬁelds [24].

The new parameter γ models the lack of energy conservation, which happens whenever

2 a fraction of DE does not cluster. Note that when γ = 1, results from diﬀerent cs should coincide, since the latter does not play a role if DE is homogeneous.

7.2.1 Radius

We now investigate the evolution of the spherical-region radius, as given by eq. (7.21).

As mentioned before, the initial conditions for dark-matter perturbations in all models 75 are ﬁxed such that the collapse in ΛCDM model always occurs at the present time. The initial conditions for dark-energy perturbations are given by Eq. (7.20). We point out some noteworthy features in a few particular cases:

2 c • cs = w (w = w)

The collapsing time tcol is earlier than ΛCDM tcol,ΛCDM only for phantom DE. This

is a reasonable outcome, since δde < 0 if w < −1 (as mentioned above): the lack of DE in the clustered region accelerates the collapse.

For non-phantom, DE starts to dominate earlier when compared to ΛCDM for any

γ. On the other hand, a smaller γ corresponds to a larger δde, which will delay the

collapse, since in this case δde > 0.

There is no strong dependence on γ, except for a small drift towards ΛCDM when γ → 1 (homogeneous DE), as expected. Besides, the term that inhibits the collapse

in Eq. (7.5), namely δρde + 3δpde, although always present, will be less important in this limit. See Fig. 7.2.

2 • cs = 0

We also get tcol < tcol,ΛCDM only for phantom DE, as anticipated. The dependence on γ is very weak. There is a slight drift away from ΛCDM as γ → 1. Such opposite

2 behavior (as compared to the previous case) happens because here δpde ≡ csδρde = 0. Without any pressure support, the collapse is expedited if γ → 0 and w > −1.

Nevertheless, with phantom DE (w < −1), one has δρde < 0 and the clustering of

DE (slightly) delays the collapse — one can (barely) see the tiny shift to larger tcol when γ decreases from 0.8 to 0 in Fig. 7.3.

2 • cs = 1 (standard quintessence-like DE)

As before, tcol < tcol,ΛCDM for phantom DE but one can also expedite the collapse if w > −1. See Fig. 7.4. The most striking feature is the possibility to entirely 76

1.0 Ωm0 = 0.3 c2 w s = γ=0 0.8

γ=0.5 γ=1 0.6 w=-0.9 ta

r γ=1 r γ=0.5 0.4 w=-1.1

γ=0 0.2

ΛCDM

0.0 0.0 0.2 0.4 0.6 0.8 1.0

t/tcol,ΛCDM 2 Figure 7.2: Evolution of the scale radius of the collapsing sphere for cs = w = {−0.9, −1.1} and diﬀerent values of γ. The solid blue line corresponds to the ΛCDM model.

prevent the collapse. This is not completely unexpected if there is enough stiﬀ DE

in the initial perturbation.The other ingredients for the bounce are phantom dark energy and no energy leaking. The full consequences of such behavior will be the subject of a future work.

Here, the collapse time tcol is deﬁned as:

Z ac da tcol(w) = , (7.23) 0 H(w, a)a where ac is the scale factor at collapse and H(w, a) is the Hubble parameter of the wCDM model. Of course, tcol(w = −1) represents the collapse time of ΛCMD model, tcol,ΛCDM .

2 The curves γ = 1 (homogeneous DE) from all the panels coincide, regardless of cs, as expected.

7.2.2 The critical contrast density

As can be seen in Eq. (7.5) and from the discussions in the previous section, the DE perturbations do contribute to the collapse. Therefore, the deﬁnition of the critical density 77

1.0 Ωm0 = 0.3 2 c s =0

0.8

w=-0.9 0.6 γ= 0.8 ta r r ΛCDM γ=0 0.4

w=-1.1

0.2 γ=0

γ= 0.8

0.0

0.0 0.2 0.4 0.6 0.8 1.0

t/tcol,ΛCDM 2 Figure 7.3: Evolution of the scale radius of the collapsing sphere for cs = 0, w = {−0.9, −1.1} and diﬀerent values of γ. The solid blue line corresponds to the ΛCDM model.

1.0 Ωm0 = 0.3 2 c s =1

0.8 γ= 0.8

w=-0.9

0.6 γ=0 ta r r γ=0

0.4 w=-1.1

γ= 0.8 0.2

ΛCDM 0.0 0.0 0.2 0.4 0.6 0.8 1.0

t/tcol,ΛCDM 2 Figure 7.4: Evolution of the scale radius of the collapsing sphere for cs = 1, w = {−0.9, −1.1} and diﬀerent values of γ. The solid blue line corresponds to the ΛCDM model. Note the non-collapsing curve (γ = 0, w = −1.1). We also point out that the 2 curve given by γ = 0, cs = 1 and w = −0.9, that crosses ΛCDM close to the collapse, is also dissonant in Fig. 7.7. 78

1.0 Ωm0 = 0.3 2 c s =γ 0.8

0.6 w=-0.9 ta r r w=-1.1 0.4

0.2 ΛCDM

0.0 0.0 0.2 0.4 0.6 0.8 1.0

t/tcol,ΛCDM 2 Figure 7.5: Evolution of the scale radius of the collapsing sphere for cs = γ, w = {−0.9, −1.1} and diﬀerent values of γ = {0, 0.8}. The solid blue line corresponds to the ΛCDM model. contrast must be modiﬁed in order to take this contribution into account. So, let us consider the expression [33, 107]

Ωde δtot = δm + δde, (7.24) Ωm as the total perturbation. Note that, when Ωde → 0, the conventional deﬁnition for the critical contrast is recovered. As usual, the critical contrast δc is determined by its linear evolution — given by Eqs. (7.17) and (7.18) — at the collapse redshift zc (obtained from requiring that r(z = zc) → 0):

lin δc = δtot (zc). (7.25)

Using the diﬀerential-radius method [30], the dependence of δc with zc is shown in Fig. 7.6,

2 7.7, 7.8, and 7.9 for diﬀerent values of the free parameters cs, w and γ, and ﬁxed Ωm0 = 0.3.

Dark-energy overdensities (δde > 0) inhibits the growth of dark-matter perturbations

(δm) due to its repulsive nature. On the other hand, dark-energy underdensities (δde < 0) enhance the growth of δm. The former case occurs in non-phantom models (w > −1), while the latter generally happens when w < −1. Indeed, as one can see in Figs. 7.6 to 79

7.9, the critical overdensity for a collapsing structure (δc) is smaller in phantom cases. Therefore, one should expect an enhancement on the number of collapsed objects in this case. The choice γ = 0 yields extreme variations of δc with respect to ΛCDM, because, in this case, there is no leakage of DE away from the collapsing regions, which maximizes its eﬀects.

In all the presented cases, δc tends to the expected EdS value at high zc. Note also that δc is always larger (smaller) than the standard ΛCDM value for w < −1 (w > −1) and γ 6= 1 (i.e, in the presence of DE perturbations). When γ = 1 (homogeneous DE), this behavior is inverted.

2 The most striking feature in Fig. 7.6( cs = 0) is the strong dependence of δc(zc = 0) on w alone. That piece of information by itself reassures the importance of studying the critical density for breaking the degeneracy among diﬀerent DE models. The dependence on γ alone is not so strong (∼ 2%). Changing both parameters at a time yields larger modiﬁcations on the curves, of course. The possibility of constraints on this parameters from observational data is beyond the scope of this paper.

2 In Fig. 7.7, where we keep cs = 1, we note once again the dependence on w, although

2 about half as strong as in the previous case. One can notice a dissonant curve (γ = 0, cs = 1, w = −0.9), which corresponds to the one that crosses over ΛCDM in Fig. 7.4. It might be a sign of incompatibility of such parameters, since γ = 0 means that there is no DE

2 leaking away from the collapsing matter bubble, but at the same time cs = 1 corresponds to a stiﬀ behavior of the former, which should (at least) delay the DE collapsing process.

The strongest dependence of δc(zc = 0) on the parameters is observed in Fig. 7.8,

2 where we keep cs = w. Observe also that, for larger w, δc(zc = 0) rapidly increases. For

(non)phantom DE, a (larger) smaller γ decreases δc(zc = 0). On the other hand, a larger failure on energy conservation (i.e, larger γ) in the collapsing region does move any of the curves towards ΛCDM.

2 The cases cs = γ are depicted in Fig. 7.9. As expected, the curves tend to ΛCDM 80

1.74 2 Ω = 0.3 γ=0 , cs =0,w=-0.9 m0 1.72 2 γ=0.5 , cs =0,w=-0.9 1.70

ΛCDM

c 1.68 δ 1.66 2 γ=0.5 , cs =0,w=-1.1 1.64

2 1.62 γ=0 , cs =0,w=-1.1

1.60 0 1 2 3 4

zc 2 Figure 7.6: Evolution of the critical contrast density for cs = 0 and diﬀerent values of w and γ. The solid black line corresponds to ΛCDM model.

2 whenever γ → 1, regardless of the values of cs. We also notice that if γ = 1 (without DE perturbation), the phantom-DE curve is slightly above ΛCDM, as opposed to all the other cases presented here. The non-phantom is also inverted (below ΛCDM in this case alone). 81

1.74 Ωm0 = 0.3

1.72

2 1.70 γ=0.5 , cs =1,w=-0.9

ΛCDM

c 1.68 2 δ γ=0 , cs =1,w=-0.9 2 1.66 γ=0.5 , cs =1,w=-1.1

1.64

1.62

1.60 0 1 2 3 4

zc 2 Figure 7.7: Evolution of the critical contrast density for cs = 1 and diﬀerent values of w and γ. The solid black line corresponds to ΛCDM model. 82

1.9 Ωm0 = 0.3 2 γ=0 , cs =w=-0.9 1.8 2 γ=0.5 , cs =w=-0.9

1.7 ΛCDM c δ

2 1.6 γ=0.5 , cs =w=-1.1

2 γ=0 , cs =w=-1.1 1.5

1.4 0 1 2 3 4

zc 2 Figure 7.8: Evolution of the critical contrast density for cs = w and diﬀerent values of w and γ. The solid black line corresponds to ΛCDM model in all panels. As before, here we ﬁnd the largest deviations from ΛCDM. 83

1.74 2 Ωm0 = 0.3 γ = cs =0,w=-0.9 1.72

2 1.70 γ = cs = 0.5 ,w=-0.9

γ = 1, w = -1.1

c 1.68

δ ΛCDM γ = 1 , w = -0.9 1.66 2 γ =cs = 0.5,w=-1.1 1.64

2 1.62 γ = cs =0,w=-1.1

1.60 0 1 2 3 4

zc 2 Figure 7.9: Evolution of the critical contrast density for : cs = γ and diﬀerent values of w and γ. The solid black line corresponds to ΛCDM model in all panels. 84

Chapter 8

Conclusions and future works

In summary, we have shown that our approach matches the results from the differential- radius method and pointed out that the so-called constant-infinity method does need corrections in the calculation of a key quantity, namely, the critical density δc. We have also derived an analytical expression for the critical density δc as a function of Ωm0, zc, and F . In spite of being more rigorous and more accurate than the constant-infinity method used in the literature, we should mention that, for the the current stage of the observations, the procedure presented here does not yield observable differences in the cluster number count Nbin. The small discrepancies pointed out (less than 1%) may be of use in the future, when more precise data become available.

We remark that our results are based on the validity of the SC approximation and of the Sheth and Tormen prescription. It would be interesting to compare the diﬀerences we found, due to distinct ways of calculating δc, using a semianalytic approach, with those from N-body simulations.

Besides, as shown in Ref. [108], departures from spherical symmetry aﬀect chameleon screening and a detailed comparison of semianalytical methods and simulations are re- quired to determine the correct functional form of the mass function. This is an important task to have in mind in the upcoming large-scale surveys.

We have also calculated the comoving number density of halos for γ gravity model 85 and Hu-Sawicki one in the small-field and large-field limits using the collapse spherical approximation studying the role of their free parameters. Besides, we compare our results with those obtained from N-body simulations [102] and determine that the γ gravity model can not be falsified at perturbative level in the SC approximation, except, maybe, at high-mass range.

Additionally, we have shown the non-linear equations that describe the evolution of the perturbations for both the dark matter and dark energy in the SC model when the clustering fraction of the latter is deﬁned by a parameter γ, which consequently also models the lack of energy conservation in the collapsing region.

We have determined the critical contrast density δc for diﬀerent values of γ, obtaining larger values for stronger DE clustering. The largest discrepancies from ΛCDM happen

2 when cs = w (both clustered and smooth DE have the same EoS) and γ = 0 (fully clustered DE). In a next paper, we will explore the consequences of the results presented here, namely deviations on the number density of collapsed objects, and the possibility of constraining the free parameters with current and future observational data. 86

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Appendix A

SC model equations in f(R) theories

Let us calculate equations for ﬁelds φ e ψ. Consider the metric

2 2 2 i j ds = −(1 + 2φ)dt + a(t) (1 + 2ψ)δijdx dx . (A.1)

The modiﬁed Einstein equations are given by:

f G + f R − − f g − ∇ ∇ f = κ2T . (A.2) αβ R αβ 2 R αβ α β R αβ

Trace of (A.2) is:

2 fR − R + fRR − 2f = −κ ρm. (A.3)

In the quasic-static limit and considering |fR| << 1, (A.3) takes the form:

1 ∇2f = (−κ2ρ + R + 2f). (A.4) r R 3 m

Making a variation in (A.4), we get:

1 ∇2δf = (−κ2δρ + δR), (A.5) r R 3 m

¯ where δf = fRδR was negligible. On other hand, we have that δfR = fR(R) − fR(R) ≡ ¯ ¯ fR − fR , δR = R − R e δρm = ρm − ρ¯m. In comoving spatial coordinates (A.5) is given by: 1 ∇2δf = a2 (−κ2δρ + δR). (A.6) R 3 m 98

Equation (A.2) can be written as:

f + R (1 + f )R − g − f − ∇ ∇ f = κ2T . (A.7) R αβ αβ 2 R α β R αβ

In the quasic-static limit and considering |fR << 1| in (A.7), we get:

f ∇2 G − − f g − ∇ ∇ f = κ2T . (A.8) αβ 2 a2 R αβ α β R αβ

The (0, 0) component of the Einstein tensor to the ﬁrst order (A.1) is:

−2∇2ψ G = + 3H2 + 6Hψ,˙ (A.9) 00 a2 and the (0, 0) component of the momentum-energy tensor is given by:

T00 =ρ ¯m + 2φρ¯ + δρm. (A.10)

On other hand, the (0, 0) component of equation (A.8) is:

f G − − ∇2f g − ∇ ∇ f = κ2T , (A.11) 00 2 r R 00 0 0 R 00

f G + ∇2f − g = κ2T . (A.12) 00 r R 2 00 00

Using equations (A.4,A.9,A.10) e T = −(¯ρm + δρm), we get:

2 h 2 ¯ ¯ i −2∇ ψ 2 ˙ −κ [¯ρm+δρm]+R+δR+2[f+δf] ¯ a2 + 3H + 6Hψ − (1 + 2φ) 3 − 2(f + δf) (A.13)

2 = κ (¯ρm + 2φρ¯m + δρm).

Note that ﬁrst-order terms are:

−2∇2ψ 2 κδρ 2 δR 4f¯ + 6Hψ˙ + κ2φρ¯ + m − Rφ¯ − − φ + 4φf¯ = κ2(2φρ¯ + δρ ), (A.14) a2 3 m 3 3 3 3 m m and zero order terms are:

κ2ρ¯ R¯ 2 3H2 + m − − f¯+ 2f¯ = κ2ρ¯ , (A.15) 3 3 3 m κ2ρ¯ R¯ 2 3H2 = − m + + f¯− 2f¯+ κ2ρ¯ . (A.16) 3 3 3 m 99

In addition, we have that:

0 ˙ Gi = 2∂i(ψ − Hφ),

0 2 Ti = a ρ¯mvi, (A.17)

where vi is a peculiar velocity. Therefore, the (0, i) component of (A.8) is

˙ 2∂i(ψ − Hφ) = 0, (A.18)

where we choose gauge in which vi = 0. From (A.18) is obtained the relation:

ψ˙ = Hφ. (A.19)

Substituting equations (A.19,A.16) in (A.14), the expression for the potential ψ is:

κ2δρ δR ∇2ψ = −a2 m + . (A.20) 3 6

The (i, j) component of Einstein Tensor is given by:

Gij = −∂i∂j(ψ + φ). (A.21)

So, the (i, j) component (A.8) is:

∂i∂j(φ + ψ) + ∂i∂j(fR) = 0. (A.22)

Therefore, we get:

2 2 ∇ (φ + ψ) = −∇ δfR. (A.23)

Using (A.6,A.21) in (A.23), the expression for the potential φ is:

16πGδρ δR ∇2φ = a2 m − , (A.24) 3 6 100

Appendix B

Solution of the diﬀerential equation for the density contrast in the linear regime

Consider the linear equation for the perturbations

3 δ¨ + 2Hδ˙ − (1 + F )Ω (t)H2(t)δ = 0, (B.1) 2 m where ˙ ≡ d/dt. Considering a change of variable t → a(t) and using equation for the background in the standard ΛCDM scenarium (neglecting radiation), namely

2 2 −3 H = H0 (Ωm0a + (1 − Ωm0)), (B.2) then Eq. (B.1) is written as 2 00 3/2Ωm0 0 a δ + 3 − 3 aδ − (B.3) Ωm0 + (1 − Ωm0a ) 3 − (1 + F )Ω (a)δ = 0, 2 m where 0 ≡ d/da. In an EdS universe, the above expression becomes

3 3 a2δ00 + aδ0 − (1 + F )δ = 0. (B.4) 2 2

Suppose a solution to (B.4) of the form

δ = Ca1+p, (B.5) 101 where C is a constant. Replacing this solution in equation (B.4) yields

3 3 ap+1 p(1 + p) + (1 + p) − (1 + F ) = 0, (B.6) 2 2 whose non trivial solution is given by

5 5r 24 p = − ± 1 + F. (B.7) 4 4 25

Now consider a solution to Eq. (B.3) of the form

1+p δm ∝ a G(a), (B.8)

Substituting this solution in Eq. (B.3) we get

3 a2G00 + 5 + 2p − Ω (a) aG0 + (B.9) 2 m 3 1 + F +(1 + p) (p + 3) − Ω (a) 1 + G = 0. 2 m 1 + p

Making the change of variable

(1 − Ω ) u(a) ≡ − m0 a3, (B.10) Ωm0 one can recognize Eq. (B.9) as an Hypergeometric diﬀerential equation. Thus, the growing solution of Eq. (B.3) is

1+p δm ∝ a 2F1(, b; c; u), (B.11) where 2F1(, b; c; u) is the Hypergeometric function, and

1 h i ≡ (p) ≡ (2 + p) + p(2 + p)2 − (p + 3)(p + 1) , 3 (p + 3)(1 + p) b ≡ b(p) ≡ h i, (B.12) 3 (2 + p) + p(2 + p)2 − (p + 3)(p + 1) 1 3 c ≡ c(p) ≡ 7 + 2p − . 3 2

For F = 0, one obtains the solution [87] 1 11 (1 − Ωm0) 3 δm ∝ a 2F1 1, ; ; − a , (B.13) 3 6 Ωm0 102 and for F = 1/3, we get

√ (−1+ 33)/4 δm ∝ a × (B.14) " √ √ √ # 7 + 33 33 − 1 6 + 33 (1 − Ωm0) 3 ×2F1 , ; ; − a . 12 12 6 Ωm0 103

Appendix C

Converting from Mv to Mh

In this appendix we follow Ref. [103]. The mass Mh of a spherical top-hat halo with radius rh is 4 4 M = π r3 ρ = π r3 ∆ ρ,¯ (C.1) h 3 h h 3 h h

One can obtain a correspondence between the virial mass Mv and Mh by considering, for instance, the NFW density proﬁle [104]:

ρs ρ(r) = 2 (C.2) (r/rs)(1 + r/rs) for which the mass inside a radius rh is

3 Mh = 4πρs rh f(rs/rh), (C.3) where f(x) = x3 ln(1 + x−1) − (1 + x)−1 . (C.4)

Using Eqs. (C.1) and (C.3) one can write the virial mass as

4 M = πr3∆ ρ¯ = 4πρ r3f(1/c) (C.5) v 3 v v s v where rv is the virial radius and c ≡ rv/rs. From Eqs. (C.1), (C.3) and (C.5), one gets

∆h f(rs/rh) = f(1/c), (C.6) ∆v 104 which yields rs ∆h = x fh = f(1/c) , (C.7) rh ∆v where

2p 2 −1/2 x(f) = [a1f + (3/4) ] + 2f, (C.8)

2 −3 with p ≡ a2 + a3 ln f + a4(ln f) , a1 ≡ 0.5116, a2 ≡ −0.4283, a3 ≡ −3.13 × 10 ,

−5 a4 ≡ −3.52 × 10 and 9 M¯ 0.13 c(Mv, z) ≡ . (C.9) z + 1 Mv ¯ ¯ Here, M is such that σ(M, z = 0) = δc and σ is the variance. Finally, from Eqs. (C.1) and (C.5) one arrives at

3 ∆h rh Mh(z, Mv) = Mv (C.10) ∆v crs 3 ∆h(z) 1 = Mv ∆v(z) c(Mv, z)x(Mv, z) 105

Appendix D

Solution of linear equations for δm and δde

The linear equations for perturbations δm and δde in a Universe dominated by matter are: 3 δ¨ + 2Hδ˙ − H2δ = 0 (D.1) m m 2 m 3 δ¨ + Hδ˙ (2 − 3(w − c2)) − H2[(1 + w)(1 − γ)δ + (w − c2)δ ] = 0 (D.2) de de s 2 m s de

Making a change of variable of t to a, we get:

3 3 δ00 + δ0 − δ = 0, (D.3) m 2a m 2a2 m and, δ0 3 3 δ00 + de − 3(w − c2) − ((1 + w)(1 − γ)δ + (w − c2)δ ), (D.4) de a 2 s 2a2 m s de

2 −3 where we used H = Ωm0a The solution of (D.3) is:

δm = Ca (D.5)

Therefore we can write equation (D.4) as:

δ0 3 3 δ00 + de − 3(w − c2) − ((1 + w)(1 − γ)Ca + (w − c2)δ ). (D.6) de a 2 s 2a2 s de 106

If we suppose a solution for δde as (growing mode):

δde = βa. (D.7)

According with (D.6), we have:

(1 + w)(1 − γ)C β = 2 (D.8) 1 − 3(w − cs)

Therefore, (1 + w)(1 − γ) δde = 2 δm (D.9) 1 − 3(w − cs)