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Observational Constraints on Models with an Interaction between Dark Energy and Dark Matter

In this thesis we go beyond the standard cosmological ΛCDM model and study the effect of an interaction between dark matter and dark energy. Although the ΛCDM model provides good agreement with observations, it faces severe challenges from a theoretical point of view. In order to solve such problems, we first consider an alternative model where both dark matter and dark energy are described by fluids with a phenomenological interaction given by a combination of their energy densities. In addition to this model, we propose a more realistic one based on a Lagrangian density with a Yukawa-type interaction. To constrain the cosmological parameters we use recent cosmological data, the CMB measurements made by the Planck satellite, as well as BAO, SNIa, H0 and Lookback time measurements.


Introduction
The large amount of precise astronomical data released in the past few years provided opportunities to answer questions in cosmology and astrophysics. Such a precision allows us to test cosmological models and determine cosmological parameters with high accuracy.
The simplest cosmological model one can build that reasonably explains the current data is the ΛCDM model. This model consists in a cosmological constant Λ to account for the observed acceleration of the Universe, plus cold dark matter (CDM) necessary to produce the gravitational potential wells inferred on galactic to cosmological scales.
However, theoretically the ΛCDM model itself faces challenges, the cosmological constant problem [1] and the coincidence problem [2]. The first one refers to the small observed value of the cosmological constant, incompatible with the vacuum energy description in field theory. The second one refers to the fact that we have no natural explanation for why the energy densities of dark matter and vacuum energy are of the same order today.
These problems open the avenue for alternative models of dark energy to substitute the cosmological constant description.
One way to alleviate the coincidence problem, which embarrasses the standard ΛCDM cosmology, is to consider an interaction between dark energy and dark matter. Considering that dark energy and dark matter contribute with significant fractions of the contents of the Universe, it is natural, in the framework of field theory, to consider an interaction between them. The interaction between dark energy and dark matter will affect significantly the expansion history of the Universe and the evolution of density perturbations, changing their growth. The possibility of the interaction between dark sectors has been widely discussed in the literature . Determining the existence of dark matter and dark energy interactions is an observational endeavor that could provide an interesting insight into the nature of the dark sectors.
Since the physical properties of dark matter and dark energy at the present moment INTRODUCTION are unknown, we cannot derive the precise form of the interaction from first principles. For simplicity, most considerations of the interaction in the literature are from phenomenology.
Attempts to describe the interaction from field theory have been proposed in [37][38][39]. One possibility is a phenomenological model of the interaction, Q, between dark matter and dark energy, which is in a linear combination of energy densities of the dark sectors Q = 3H(ξ 1 ρ c + ξ 2 ρ d ) [13,29,40]. In this interaction, H is the Hubble parameter, ξ 1 and ξ 2 are dimensionless parameters, assumed to be time independent, for simplicity, and ρ c and ρ d are the energy densities of dark matter and dark energy, respectively.
Such a model was widely studied in [13,18,34,[41][42][43][44]. It was disclosed by the late integrated Sachs-Wolf (ISW) effect [25,27] that the interaction between dark matter and dark energy influences the cosmic microwave background (CMB) at low multipoles and at high multipoles through gravitational lensing [44,45]. With the WMAP data [25,27] together with galaxy cluster observations [34,35] and also recent kinetic Sunyaev-Zel'dovich effect observations [46], it was found that this phenomenological interaction between dark energy and dark matter is viable and the coupling constant is positive, indicating that there is energy flow from dark energy to dark matter, which is required to alleviate the coincidence problem and to satisfy the second law of thermodynamics [16].
It is of great interest to build alternative models of the Universe and employ the latest high-precision data to further constrain them. This is the main motivation of the present work. We will combine the CMB data from Planck [47][48][49] with other cosmological probes such as the baryonic acoustic oscillations (BAO) [50][51][52], supernovas [53], the latest constraint on the Hubble constant [54] and lookback time [55,56]. We want to see how these different probes will influence the cosmological parameters and put tight constraints on the interaction between dark sectors.
This thesis is organized as follows: in the first chapter we introduce some fundamental aspects of cosmology and present the contents of the Universe. Chapter two goes beyond the homogeneous and isotropic universe and describe the linear perturbations. Our models of interactions between dark energy and dark matter are presented in chapter three. The data analysis and our fitting results appear in chapter four. Finally, we conclude and discuss some perspectives for future works.

Homogeneous and Isotropic Universe
General relativity establishes that the geometry of the spacetime is determined by the energy content of the Universe and this geometry governs the motion of free particles. In general, the geometry of the spacetime is described by the line element where g µν are the components of the metric tensor. The indices µ, ν are defined such that x 0 represents the time coordinate and x i , with i = 1, 2, 3, represent the spatial coordinates.
We are using the Einstein's convention where repeated indices are summed.
To determine the form of the line element of a given cosmological model we use the underlying symmetries. The simplest cosmological model can be built assuming that the constituents of the Universe present the properties of statistical homogeneity and isotropy, known as the cosmological principle. In fact, observations of the cosmic microwave background have shown isotropy of one part in 100.000 [58]. Also, evidences of galaxy surveys suggest that the Universe is homogeneous on large scales [59].
Using the cosmological principle, the metric of the spacetime must be the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric [57], which is given by ds 2 = −dt 2 + a(t) 2 dr 2 1 − Kr 2 + r 2 dθ 2 + sin 2 θdφ 2 . (1.2) Here a(t) is a scale factor accounting for the expansion or contraction of the Universe and K is a constant that establishes the geometry of the spatial section. If K > 0 the spatial section is closed, while for K = 0 it is flat and K < 0 means that the spatial section is open. Throughout this work we are using natural units such thath = c = k B = 1.
The motions of free particles follow the geodesic equations where λ is a monotonically increasing parameter that parameterizes the particle's path and Γ µ αβ are the Christoffel symbols. The Christoffel symbols are related to the metric tensor by the expression where g µν is the inverse of the metric tensor such that g µα g αν = δ µ ν with δ µ ν , the Kronecker delta, defined as zero unless µ = ν in which case it is equal to one.
To obtain a(t) and K we need the dynamical equations governing the Universe. In the context of general relativity it is determined by the Einstein equations [60] R µν − 1 2 g µν R = 8πGT µν , (1.5) where R µν is the Ricci tensor, which can be written in terms of the Christoffel symbol as (1.6) R = g µν R µν is the Ricci scalar, or curvature scalar, G is the Newton gravitational constant and T µν is the energy-momentum tensor of the content in the Universe. All the quantities in the left hand side of (1.5) are geometrical quantities, while the right hand side presents the energy content. Thus, the Einstein equations relate the geometry of the Universe to the energy content.
On large scales the constituents of the Universe can be treated as a fluid. The most general energy-momentum tensor for a fluid component "A", T A µν , is given by where ρ A is the energy density, P A is the pressure, u A µ is the four-velocity vector, π A µν is the anisotropic stress and q A µ is the heat flux vector relative to u A µ , all these quantities with respect to the A-fluid. The total energy-momentum tensor is T µν = A T A µν . However, if the fluid has at each point a velocity v, such that an observer with this velocity sees the fluid around him as isotropic, this is known as a perfect fluid [57] and in this case the anisotropic stress and the heat flux are null.  (1.8) and the space-space components result iṅ , (1.9) where H(t) ≡ȧ(t)/a(t) is called Hubble parameter, ρ, P denote the total energy density and pressure and a dot means differentiation with respect to the cosmic time t.
We can define a critical density ρ crit by , (1.10) such that the abundance of a substance in the Universe can be expressed with respect to it. Constructing the density parameter Ω as (1.11) the Friedmann equation (1.8) can be rewritten as . (1.12) From this equation we can see that the curvature of the spacial section K is determined by the energy content of the Universe. In fact, There is another relation that can be obtained from equations (1.8) and (1.9). Eliminating K/a 2 from those equations, we obtain a a = − 4πG 3 (ρ + 3P ). (1.15) This equation tells us that an accelerated expansion only occurs if ρ + 3P < 0. As the energy density must be a positive quantity, this means that in order to realize an accel-erated expansion, the Universe must contain some component with a negative pressure.
Considering a fluid with a linear barotropic equation of state 1 , P = ωρ, the accelerated expansion can occur if w < −1/3.

Cosmic Distances
In the previous section we established a theoretical model for a homogeneous and isotropic universe. Solving the system of equations (1.8) and (1.14) allows us to determine the evolution of the scale factor a(t) and consequently the history of the Universe. However, to solve that system of equations we need to determine the values of some parameters as K, the initial value of H(t), the initial energy densities of all the constituents of the Universe and their equation of state, usually assumed to be of the form ω = P/ρ. To describe the real Universe these parameters must be in agreement with observations, which means we need observables that allow us to compare theory with observations.
A fundamental step to compare theory with observations is the measurement of distances on cosmological scales. These measurements enable us to relate physical observables with the parameters in our model such that we can constrain it and make predictions.
Actually, there are several ways to define distances in cosmology as we show below.
From the FLRW line element (1.2), a light ray traveling along the radial direction satisfies the geodesic equation ds 2 = −dt 2 + a 2 (t)dχ 2 = 0, (1.16) where we defined dχ ≡ dr/(1 − Kr 2 ) 1/2 . Therefore, considering that a light ray have traveled from the time t = 0, we can find the total comoving distance that it could travel until the time t as η ≡ , (1.18) where the subscript "0" represents quantities at the present time.
The comoving coordinates are constant over the expansion history of the Universe.
Thus, using Eq. (1.18), a light ray emitted at time t + δt and observed at time t 0 + δt 0 Manipulating the limits of integration, we can write (1.20) At first order in δt we have δt If δt is the period of emission of the light ray and δt 0 the period of detection, as the wave frequency ν is the inverse of the period and the wave length is defined as λ = c/ν, we This expression defines a cosmological Doppler effect associated with the expansion or contraction of the Universe. To account with this effect we defined the redshift z. The above equation allows us to relate the redshift of a distant object to the scale factor when the light ray was emitted.
Basically, there are two ways of inferring distances in astronomy: using a standard ruler or a standard candle. With the knowledge of trigonometry astronomers have inferred lengths for a long time. Measuring the angle θ subtended by an object of known physical size l (a standard ruler), the distance to that object is where we used the definitions of χ and Ω k ≡ −K/H 2 (t)a 2 (t), and we are using a normalization such that a 0 = 1.
Another important technique to determine distances is to find an object of known intrinsic brightness, a standard candle, such that any difference between the apparent brightness of two of these objects is a result of their different distances from us. Given an object of known luminosity L, the observed flux F a distance d L from the source is (1.26) On an expanding universe we can write a similar equation considering a comoving grid as , (1.27) where L(χ) is the luminosity of the source through a comoving spherical shell with radius r(χ). Assuming that the photons are emitted with the same energy, the luminosity L(χ) is the energy multiplied by the number of photons crossing the shell per unit time. As the Universe expands, the number of photons passing through the spherical shell per unit time becomes smaller by a factor of a. On the other hand, Eq. (1.22) tells us that the wave lengths of the photons are stretched by a factor of 1/a. Thus, as the energies of the photons are inversely proportional to the wave length, they will decrease accordingly.
Therefore, the energy per unit time on the spherical shell at r(χ) will be a factor of a 2 smaller than the luminosity at the source . (1.28) If we define the luminosity distance d L as (1.29) we observe that which is valid in general since the flux is conserved [61].

The Standard Model of Particle Physics
The standard model of particle physics contains our present knowledge of the fundamental particles that compose all the material content in the Universe and the interactions between them. The standard model consists in a gauge group where Q denotes the electric charge generators.
The standard model of particle physics agrees pretty well with the observed particles and the corresponding interactions. However, from a theoretical point of view, there are some remarkable difficulties. Therefore, it is a consensus that a more fundamental theory must exist coinciding with the standard model in the low-energy limit. It should also be noted that the SM does not include gravitation.
As we said before, on large scales the behavior of the particles are governed by the gravitational interaction. In fact the strong and weak interactions act only in the nuclear range. On the other hand, the atoms that build the matter content are neutral and have spin oriented randomly so that on large scale matter do not interact with each other electromagnetically. Actually, these interactions are important in the early Universe when it was hotter and denser, but can be neglected at more recent epochs.

Photons
For a dilute weakly-interacting gas with g * internal degrees of freedom, the number density n, energy density ρ and pressure P are given by [60,62] where E 2 = | p| 2 + m 2 and f ( x, p) is the phase space distribution function (or occupation number) which counts the number of particles around position x and momentum p in phase space. If some component is in kinetic equilibrium, i.e. is in equilibrium at temperature T , the distribution function is we can see that their energy density evolves as ρ γ ∝ a −4 .
Photons can be described as a gas with a temperature given by the COBE satellite and a chemical potential µ = 0, since they can be freely created or destroyed. In fact, observationally, the limits on a chemical potential are |µ|/T < 9 × 10 −5 [64], thus µ can be safely neglected. With these assumptions, and knowing that photons have two degenerate states given by their polarizations, we obtain from (1.38) Since ρ γ ∝ a −4 this tells us that the temperature of the CMB must vary as T ∝ a −1 .
With respect to the critical density today, the photon energy density is where h parameterizes the Hubble constant H 0 = 100h kmsec −1 M pc −1 and we used that 1eV = 11605K. Substituting the observational value of h above, h = 0.72 [65], and using a normalization of a given by a 0 = 1, we have Ω γ0 ≈ 5 × 10 −5 .

Baryons
Generally in cosmology, we call the protons, neutrons and electrons that together build the atoms, as baryons. Although electrons are not baryons, but leptons, because their masses are so small in comparison with that of the protons and neutrons, we can consider that atoms are made of baryons. In this way, the baryons form all the known matter content in the Universe.
Combining these equations we can construct an equation of state P (ρ) = ωρ ≈ 0 for nonrelativistic particles. In an ideal case we consider that ω = const. = 0. Thus, as baryons are non-relativistic particles, we consider that they obey an equation of state with ω = 0.
With this equation of state, the continuity equation (1.14) gives us that ρ b ∝ a −3 . Now, we know how the energy density of baryons evolves with the scale factor. In this way, if we obtain the value of the energy density at some epoch, all the history will be established. However, unlike the CMB photons which can be described by a gas with a temperature T and zero chemical potential, the above equations show that the energy density for non-relativistic particles does not depend on the temperature T only.
Therefore, the energy density for non-relativistic particles must be measured directly from observations.
There are four methods to measure the density of baryons and all of them are in good agreement [66]. The first method consists in observing baryons in galaxies today, the baryon density can be obtained estimating the mass of stars and mainly the mass of gas in the groups of galaxies. The second way is obtained by observing the spectra of distant quasars and the amount of light absorbed by the intervening hydrogen [67].
The anisotropies in the Universe also depend on the baryon density and studying them constitutes another form to infer the baryon density [68]. At last, the light element abundances are able to pin down the baryon density [69]. All of these observations restrict the baryon density in the Universe to 2 − 5% of the critical density.

Neutrinos
Neutrinos were in equilibrium with the initial cosmic plasma, but lost contact with it slightly before the annihilation of electrons and positrons when the temperature was of the order of the electron mass. Therefore, neutrinos did not receive any energy contribution from this annihilation while the photons did. Then, photons are hotter than the neutrinos.
From the second law of thermodynamics The neutrino temperature varies proportionally to a −1 , i.e. a 1 T 1 = a 2 T ν , thus the above Actually, neutrinos seem to be massive as observed from oscillations of solar [70] and atmospheric neutrinos [71]. Nevertheless, at epochs where the temperature is much larger than the predicted mass of the neutrinos, we can consider them as massless. Just when k B T ∼ m ν or less, we have to consider the mass of the neutrinos. For a massive neutrino, the relative energy density will be [60] Ω ν0 = m ν 94h 2 eV . (1.55) Finally, we emphasize that, unlike baryons and photons, cosmic neutrinos have not been observed. Their contributions come from theoretical arguments.

Dark Matter
There is a large number of evidences in ranges from the galactic to cosmological scales indicating the presence of a new component in the Universe or some deviation from the known laws of gravitation. If it actually is a new component, it cannot interact electromagnetically, since its presence can only be detected via gravitational effects. Thus, it is dubbed dark matter.
On galactic scales, the most convincing and direct evidence of dark matter comes from observations of the rotation curves. The rotation curves of galaxies are a measure of circular velocities of stars and gas as a function of their distance from the galactic center.
Theoretically, using Newtonian dynamics, we expect the circular velocities to be v(r) = GM (r) r , (1.56) where M (r) ≡ 4π ρ(r)r 2 dr is the mass interior to the radius r and ρ(r) is the mass density profile. The above equation tells us that beyond the optical disc the circular velocity should scale as v(r) ∝ 1/ √ r. However, observationally, rotation curves usually exhibit a flat behavior at large distances from the galactic center as can be seen in Fig. 1.1. Actually, the first indication of dark matter was obtained by Zwicky in 1933 [74].
Studying the Coma cluster he inferred a mass-to-light ratio of 400 solar masses per solar luminosity, two orders of magnitude greater than observed in the solar neighborhood. The mass of a galaxy cluster can be determined in several ways: applying the virial theorem to the observed distribution of radial velocities, by weak gravitational lensing, and from the X-ray emitted by the hot gas in the cluster. All of these measurements are consistent with Ω m ∼ 0.2 − 0.3 [75][76][77].
Finally, on cosmological scales, the anisotropies of the cosmic microwave background

Dark Energy
Observations of anisotropies in the CMB have shown that the geometry of the spatial section is very close to a flat one [79][80][81][82]. Actually, we also expect this theoretically from inflationary scenarios in the early Universe [60]. This means that the total energy density should be equal to the critical density. However, summing the contributions of all the components described so far, we obtain that they contribute with around 30% of the critical density. Thus, there is a lack of 70% in the energy content of the Universe.
Using the luminosity distance (1.29) we can find the apparent magnitude m of an object with intrinsic magnitude M . Conventionally we have where K is a shifting correction factor of the spectrum into the wavelength range measured.
In 1998, two groups measured the apparent magnitude of various supernovae Type Ia and established that a universe dominated by a vacuum energy density, i.e. a cosmological constant, with equation of state ω = −1 is favored by the data [83,84], as shown in Figure   1.2. These two sets of observational evidences show that beyond dark matter there must be another unknown component in the Universe, which is called dark energy. Then, we conclude that our present Universe is composed of around 5% of baryons, 25% of dark matter and 70% of dark energy, with insignificant contributions of photons and neutrinos.
Nowadays, the best model explaining the data is the ΛCDM , with cold dark matter (CDM ) and where dark energy is the cosmological constant Λ.
The ΛCDM model is the simplest one that fits the data. However, it suffers of two theoretical problems. The first occurs when we try to associate the cosmological constant with the vacuum energy density. In quantum field theory, we generally introduce a cut off on the energy beyond which our theory cannot describe the physics. If we introduce this cut off at the Planck reduced mass M P l = (8πG) −1/2 = 2.436 × 10 18 GeV [85], the vacuum energy density will be given by ρ vac ∼ M 4 P l ∼ 10 73 (GeV ) 4 . On the other hand, the energy density of dark energy today is ρ Λ = 3M 2 P l H 2 0 Ω Λ ∼ 10 −47 (GeV ) 4 . Thus, there is a difference of 120 orders of magnitude, which is known as the cosmological constant problem.
The second problem arises when we consider the evolution of the components in the Universe. As we already saw, the non-relativistic matter (baryons and dark matter) scales as ρ m ∝ a −3 , radiation scales as ρ r ∝ a −4 and the cosmological constant has a density ρ Λ = constant. Therefore, each component scales in a very different way. But, as mentioned above, matter and dark energy have the same order of magnitude today.
Thus, the question arise: "why matter and dark energy have the same order of magnitude exactly now in the whole history of the Universe?". This is called the coincidence problem.
These difficulties in the ΛCDM model have motivated the search for new models of dark energy. In conclusion we observe that both dark matter and dark energy have no explanation in the standard model of particle physics.

Chapter 2 Cosmological Perturbations
The description of the Universe was, up to now, performed considering it as homogeneous and isotropic, as described by the cosmological principle. However, the structures around us imply that at some point we have to break these assumptions and introduce inhomogeneities and anisotropies in our model. Because on large scales the cosmological principle leads to correct results, we shall do that using perturbation theory.

Perturbed Metric
The metric of a flat FLRW universe with small perturbations can be written as where (0) g µν corresponds to the unperturbed part and |δg µν | | (0) g µν |. Using the conformal time (1.17), the most general components of the metric tensor are given by [86] g 00 = −a 2 (1 + 2ψ), From the three types of perturbations, the scalars are the most important in cosmology because they present gravitational instability and can lead to structure formation in the Universe. The vector perturbations are responsible for rotational motions of the fluid and decay very quickly. Finally, tensor perturbations describe gravitational waves, but in the linear approximation they do not induce perturbations in the perfect fluid. Moreover, the decomposition theorem [60,86] states that these three types of perturbations evolve independently. Thus, in this work, we will only be interested in scalar perturbations. In this case the metric becomes Given a coordinate transformation where ξ µ ≡ (ξ 0 , ξ i ⊥ + ζ i ) are infinitesimally small functions of space and time, the metric tensor has its components changed bỹ at first order in δg and ξ. The new coordinate system also allows to split the metric into a background and a perturbed part where (0) g µν (x ρ ) is the Friedmann metric in the new coordinates. On the other hand, expanding the background part of (2.1) in a Taylor series around the coordinatesx, we Therefore, comparing Eqs. (2.8) and (2.9) and also using Eq. (2.10), we obtain the gauge transformation law Using the gauge transformation law (2.11), the metric (2.6) has its components changed Instead of working with gauge-invariant functions, we can impose two conditions on the coordinate transformation, which is equivalent to a gauge choice. In particular, the conformal-Newtonian gauge is obtained with coordinates ξ 0 and ζ such that B = E = 0.
Another gauge, widely used in the literature, is the synchronous gauge. It corresponds to the gauge choice ψ = B = 0. However, unlike the conformal-Newtonian gauge, the synchronous gauge does not fix the coordinates uniquely. If the conditions ψ = B = 0 are satisfied in a coordinate system x µ ≡ (η, x), they will also be satisfied in any coordinate systemx µ given bỹ where C 1 (x j ) and C 2 (x j ) are arbitrary functions of the spatial coordinates.

Einstein Equations
Now we know the components of the perturbed metric tensor. We can thus obtain their evolution using the Einstein equations (1.5). Assuming the perturbations are small we can expand them in a Taylor series. Thus, as we did to the metric tensor, it is possible to split the Einstein tensor G µ ν and the energy-momentum tensor T µ ν into background and perturbed parts: The zeroth order terms correspond to the homogeneous and isotropic background and the others give us the perturbed Einstein equations The geometric part of the Einstein equations (2.19) can be solved following the same procedure adopted in the last chapter. Thus, using the perturbed metric (2.6) in the conformal-Newtonian gauge and keeping only first-order terms, we obtain where θ denotes a generic perturbation and the subscript k represents a Fourier mode for each wavenumber k, the resulting Fourier amplitudes obey ordinary differential equations.
Thus, working in the Fourier space makes things easier. Furthermore, the Fourier modes θ k evolve independently in the linear regime. Therefore, instead of solving an infinite number of coupled equations, we can solve for one k-mode at a time. In practice, each perturbation quantity θ and its derivatives can be substituted as where we are omitting the subscript k to simplify the notation.
and the longitudinal traceless projection of the space-space components These equations were written in the Fourier space and we defined the unit direction wavevectork i =k i , which satisfies δ ijk ikj = 1.

Boltzmann Equation
To solve the Einstein equations (2.27) and (2.28) we need the first-order components of the energy-momentum tensor. This could be obtained using the hydrodynamic equations as in the homogeneous and isotropic case. However, those equations can be obtained by the first moments of the Boltzmann equation, which means that it is more general.
Furthermore, the radiation description is made using fluctuations of the temperature, and the Boltzmann formalism is more natural in this case.
The Boltzmann equation formalizes the statement that the variation in the distribution of some species is equal to the difference between the rates of ingoing and outgoing particles of that species. In its differential form, the Boltzmann equation is where f is the distribution function and C[f ] is a functional of the distribution function, which establishes all possible collision terms of the particle of interest. In general, f is a function of the spacetime point x µ = (t, x) and also of the four-dimensional momentum vector in comoving frame where λ parametrizes the particle's path. The four-vector P µ is related to the physical momentum four-vector p µ by P µ = ∂x µ ∂x ν p ν , withx ν in the physical frame.
In order to calculate the perturbed energy-momentum tensor, we develop below the Boltzmann equation for each component in the Universe.

Photons
Photons satisfy the energy-momentum relation Let us begin to solve equation (2.32). The last term does not contribute at first order in perturbation theory. In fact, the zeroth order part of f depends only on p, which means ∂f /∂p i is nonzero only in a perturbed level. In the same way, a photon moves in a straight line in the absence of the potentials Ψ and Φ, therefore, dp i /dt is also a perturbed quantity. Thus, the last term of (2.32) must be of second order.
Using the definition of the comoving energy-momentum vector (2.30), we obtain From (2.31) the time-component of P µ is given by where the last equality is valid at first order. On the other hand, the spatial component is proportional to the unit direction vectorp i Plugging (2.35) in the definition of the spatial magnitude, we find which gives |P | = p(1 − Φ)/a at first order. Therefore, combining equations (2.34) and The next term we need to evaluate is dp/dt. This factor can be calculated from the time component of the geodesic equation dP 0 /dλ = −Γ 0 µν P µ P ν in a simple, although tedious, way [60]. Here we just show the final result dp dt Combining the terms obtained so far, the left hand side of the Boltzmann equation for where we neglected the product of ∂f /∂x i and either Ψ or Φ because they are second-order terms.

Photons in a homogeneous and isotropic distribution with an equilibrium temperature
T obey the Bose-Einstein statistics given by equation (1.40). There, T is a function of time only. To describe perturbations about this distribution, we have to introduce inhomogeneities, so that it must have an x dependence, and anisotropies, which means a dependence with the direction of propagationp. Thus, for photons, the distribution function is given by where T (t) is the zero-order temperature and Θ ≡ δT /T characterizes the perturbation to the distribution function. Here we have assumed that Θ does not depend on the magnitude p, which is a valid assumption since in a Compton scattering p is approximately conserved.
Expanding up to first order, The last step necessary to calculate the Boltzmann equation for photons is the collision term. Photons interact with electrons through the Compton scattering where the momentum of each particle is indicated. To obtain the effect of the Compton scattering in the distribution function of photons with momentum p, we must sum the contributions of all the other momenta to the collision term [60] At the epochs of interest the energies are the relativistic limit for photons E(p) = p and the non-relativistic limit for electrons E e (q) = m e + q 2 /(2m e ) ≈ m e . Thus, we can use the three-dimensional delta function to eliminate the integral over q For non-relativistic Compton scattering very little energy is transferred , which holds since q is much larger than p and p . In this limit, the scattering is nearly elastic p ≈ p. As the change in the electron energy is small, it makes sense to expand the final electron kinetic energy ( q + p − p ) 2 /(2m e ) around its zeroth order value q 2 /(2m e ). Therefore, we can make the formal expansion in the delta function Using this expansion and , we obtain The amplitude for Compton scattering can be found using the Feynman rules [87] where σ T is the Thomson cross section,ˆ andˆ are the polarization vectors of the initial and final photons, respectively. For simplicity, we average over all polarizations and angular dependence which results in a constant amplitude Thus, using this amplitude and the expansion of the distribution function (2.41), the collision term can be written as where we did the integrals over momentum q using the number density definition (1.37) and we define the velocity as The spin degeneracy g * can be incorporated into the phase space distribution f .
To solve the integral over p in Eq. (2.50), we split the radial and angular parts of the differential d 3 p . Then, keeping only first-order terms in perturbations, the integral over the solid angle yields where we define the multipoles where n e is the electron density and v b is the velocity of the electrons, which is associated with the baryons.
At last the first-order Boltzmann equation ( As usual we assume the fluid to be irrotational, which means we can write v i b = v b k i /k. Thus, passing to the conformal time and Fourier space we obtaiñ where we defined the optical depth When deriving Eq. (2.57) we use a constant amplitude given by (2.49). Actually, the Compton scattering has an angular dependence and it also couples the temperature field to the strength of the polarization field Θ P . The general expression can be obtained following the same procedure we did, but using the complete amplitude (2.48). The where Π =Θ 2 +Θ P 2 +Θ P 0 . Θ P 0 and Θ P 2 are the monopole and quadrupole of the polarization field, which satisfies The above equations show that even if we are interested only in the temperature field, it is influenced by the polarization field.

Baryons
The formalism presented in the last subsection can be used to obtain the Boltzmann equation for any constituent in the Universe. We now move to consider the behavior of massive particles, for which we have Defining the energy where p 2 is the same as in equation (2.36), the four-momentum of a massive particle is given by Instead of the momentum p, massive particles must be described by the energy E.
Therefore, the left hand side of the Boltzmann equation (2.29) is expanded as Following the same steps as for the case of photons, we can obtain the coefficients dx i /dt and dE/dt such that Eq. (2.64) can be rewritten as Once again we neglected the last term in Eq. (2.64) as it does not contribute at first order.
The main difference of this equation to the photon case is the presence of factors p/E, which arise from the energy-momentum constraint. The massless case can be recovered When we considered photons, to complete the left hand side of the Boltzmann equation, we needed the knowledge of the distribution function. For massive particles, the treatment can be simplified if they are nonrelativistic. In this case, we do not need a detailed information about the distribution function, all we need is to take moments of the Boltzmann equation taking into account that terms second-order in v = p/E must be neglected because of the nonrelativistic behavior.
The zeroth moment is obtained integrating the Boltzmann equation as where in the third step we integrated by parts. The last term of (2.66) does not contribute at first order since the integral over the direction vector is null at zeroth order and that integral multiplies a metric perturbation, thus this term is at least of second order. Finally, The above result for the zeroth moment of the Boltzmann equation has introduced two unknown variables: n and v i . Thus, we need one more equation to close the system.
This additional equation can be obtained taking the first moment The first term on the r.h.s. can be identified with the partial time derivative of nv j .
The second one is neglected at first order since it depends on (p/E) 2 . To calculate the last terms, we follow the same procedure used in Eq. (2.67). Therefore, keeping only first-order terms, the first moment of the Boltzmann equation is Electrons and protons are coupled by Coulomb scattering whose rate is much larger than the expansion rate at all epochs of interest. Because of this tight coupling, electrons and protons have the same overdensities Besides the Coulomb scattering, baryons interact with photons through the Compton scattering. Thus, the unintegrated Boltzmann equation for baryons is given by where c eγ pp q represents the Compton collision term, as in Eq. (2.44), and the subscripts represent which momenta are being integrated. In principle, the collision term should have an additional term account for the proton-photon Compton scattering, however, the cross section for this process is much smaller than the electron-photon scattering and can be ignored. We are also considering a simplified model with all electrons ionized.
To solve Eq. (2.73) we first take the zeroth moment integrating in the electron momentum q. Thus, using Eq. (2.68) we obtain The collision term in the right hand side vanishes since the integration is symmetric under the interchange of p ↔ p and q ↔ q while it is antisymmetric in the distribution function factors. The number density can be split into a background part and a perturbed part as Thus, collecting the zeroth order and first-order terms, and passing to conformal time and Fourier space, the perturbed part gives us Using the conservation of the total momentum q + p, we have that c eγ q pp q q = − c eγ p pp q q . Passing to Fourier space and multiplying Eq. (2.76) byk j , the right-hand side becomes − c eγ pµ pp q q . In equation (2.55), we already computed c eγ p q q , thus we just need to multiply that result by pµ and integrate over all p, In the second line we split the integration over p into a radial part and an angular part.
The integral over the radial part p can be made integrating by parts similar to Eq.(2.67), the result is −4ρ γ . The first term in the µ-integration vanishes, the second term is the dipole component of Θ and the last term reduces to v b /3. Therefore, collecting these results in Eq. (2.76) and switching to conformal time, we obtaiñ

Energy-Momentum Tensor
Now that we know the equations governing the evolution of perturbations in the components of the Universe, we are in position to complete the Einstein equations (2.27) and (2.28) with the perturbed energy-momentum tensor. In general, the energy-momentum tensor of a fluid with a distribution function f (t, x, p) is given by Remember that the Einstein equations take into account the contributions of all species in the Universe, thus from the time-time component of (2.82), we obtain The distribution functions above consist in a background part and a perturbed part.
For nonrelativistic particles, the integral produces −ρ i (1 + δ i ), while for relativistic ones, whereÑ 0 is the monopole term of the neutrino temperature perturbations.
The second Einstein equation (2.28) needs the longitudinal traceless projection of the space-space energy-momentum tensor We observe that the factor µ 2 −1/3 is equal to 2/3P 2 (µ), where P 2 (µ) is the second Legendre polynomial. Therefore, the integral picks out the quadrupole part of the distributions.
We already saw that nonrelativistic particles do not contribute to the second moment and higher, thus only photons and neutrinos must be taken into account. For example, let us calculate the integral in the case of photons: Finally, combining (2.28) and (2.85) we obtain the second Einstein equation We emphasize that, in principle, we should have a term for perturbations in the dark energy in the Einstein equations above. However, until now, we consider it to be smooth and important only very recently as described by the cosmological constant model.

Initial Conditions
Equations ( system of nine first-order differential equations. To solve them we need a set of nine initial conditions. However, in the early Universe, we can relate all perturbation variables to the gravitational potential Φ, such that we will actually need just one initial condition.
Considering times so early that for all k-modes of interest kη 1, we can neglect all terms in the Boltzmann equations multiplied by k. This condition means that on early times all perturbations of interest had wavelengths much larger than the distance of causal contact. Thus, an observer within the causal horizon sees a uniform sky, which means that higher multipoles, e.g. Θ 1 , Θ 2 , ..., are much smaller than the monopole term.
Therefore, using these considerations, equations (2.59), (2.79), (2.75) and (2.80) can be written asΘ respectively. The velocities v b and v c are of the same order of magnitude of the dipole moment of the temperature distribution, thus they can be neglected at first. We are also setting the polarization equal to zero.
Using the Einstein equation (2.84) at early times, and knowing that radiation dominates the energy content at those epochs, we obtaiñ where η i means that this is valid for some initial conformal time. Combining (2.88) with (2.75) and (2.80), we obtain that their initial conditions are given bỹ If the constants above are zero we have adiabatic initial conditions, but if they are nonzero the initial conditions are called isocurvature ones.
Although we have neglected the initial conditions for velocities, there are situations where they must be taken into account. Using (2.59), (2.78), (2.79) and (2.81), we can show they are given byΘ The above equations relate all variables of the initial perturbations to the initial gravitational potentialΦ(η i , k). Then, it remains to be known how these primordial perturbations were generated and which initial value they should have. The theory of inflation, which was created to account with the horizon problem, also provides a mechanism responsible for the origin of perturbations in the Universe. In this theory, quantum mechanical fluctuations during inflation are responsible for the variations around the smooth background. At any given time, there are regions where the fields are slightly larger and regions where they are smaller, so that the average value is zero However, the variance is nonzero where P Φ (k) is the power spectrum of the primordial perturbations to the gravitational potential Φ.
To account with an inflationary scenario we need a field with a negative equation of state, the inflaton, which is responsible for the exponential acceleration of the Universe at early times. Quantum mechanical fluctuations on this field feed perturbations in the metric. Assuming a canonical scalar field is responsible for inflation, we can obtain the post-inflation power spectrum of Φ from the horizon-crossing spectrum of the scalar field as [60] where = −H /aH 2 is the slow roll parameter of the scalar field. In the second equality, we rewrote the primordial power spectrum defining the scalar spectral index n s and the scalar amplitude δ H 4 . This convention means that a scale-invariant scalar spectrum corresponds to n s = 1.

Inhomogeneities: Matter Power Spectrum
The perturbation equations developed in sections 2.2 and 2.3, with the initial conditions obtained in the last section, allow us to calculate the inhomogeneities and anisotropies in the Universe. Inhomogeneities in the matter density at the early Universe will grow up due to gravitational instabilities and they will build the structures we observe. Actually, to describe correctly the structures observed, e.g. galaxy distributions, we have to take into account nonlinearities and gas dynamics, but even on small scales the linear regime is a starting point.
Solving the cosmological perturbations it is verified that their evolution can be divided in three stages. Thus, the density contrast of matter at some conformal time η, or equivalently with scale factor a,δ m (a, k), is related to the primordial density contrast where the transfer function T (k) describes the evolution of perturbations through the epochs of horizon crossing and transition from radiation to matter. On the other hand, the 4 Camb code parameterizes the primordial super-horizon power spectrum P Φ (k) of curvature perturbations as P Φ (k) = A s ( k ks ) (ns−1) . The input parametes are the pivot scale k s , the spectral index n s and the amplitude at the pivot scale A s . As we are interested in using the Camb code for numerical calculations, we shall use such convention for the primordial power spectrum in the next chapters. growth function D δ (a) describes the late time growth which is independent of wavelength.
To compare with observations we note that, as for the gravitational potentialΦ( k), the density contrast of matter also has a vanishing average, δ m (a, k) = 0. (2.102) Its variance equals Therefore, combining (2.101) with the above variance, we see that the matter power spectrum is given by

Anisotropies: CMB Power Spectrum
One way to solve the perturbation equation for photons (2.59) is expanding the Θ function in Legendre polynomials such that (2.59) is split in a hierarchical system of differential equations as In fact this was done in the first numerical programs to calculate the anisotropies in the CMB. However, this method requires to solve thousands coupled differential equations for small angular scales θ ≈ 1/l, besides the need of a small time step to obtain good results.
Therefore, nowadays, another method is used, the line of sight approach. Following this strategy we can obtain an approximate analytic solution given by [60] where η * is the conformal time at recombination and j l is the spherical Bessel function.
The solutions of (2.59) are given by Θ l . We need to know how to make comparisons with the observations. We thus first expand the temperature perturbation in spherical harmonics, Using the orthogonality property of the spherical harmonics, we can invert the expansion above to obtain However, we cannot make predictions for a particular a lm , only their distributions are known which are originated in quantum mechanical fluctuations at the inflationary epoch.
Therefore, in the linear regime, they have a gaussian distribution, where The matter power spectrum obtained in the last section and the CMB power spectrum in the above equation give us a prediction for the variance in the matter and CMB temperature, respectively. Both depend on the whole system of Einstein-Boltzmann equations developed in this chapter. Thus, their predictions take into account all the components in the Universe. In the next chapter, we will extend the standard scenario presented so far, considering models with an interaction in the dark sector. Using these power spectra we will be able to make predictions about the behavior of an interaction and constraint the models comparing with some observational data.

Chapter 3 Interacting Dark Energy
In the last years several works on the possibility of an interaction between dark energy and dark matter have appeared [3,10,18,21,22,36,42,[89][90][91][92][93][94]. One of the motivations is that a model with an interaction in the dark sector can provide a mechanism to solve, or at least to alleviate, the coincidence problem. Moreover, because both dark energy and dark matter are, until now, two unknown components, when considering that they are originated by physical fields from quantum field theory it is natural to assume that they interact.

Phenomenological Model
The first chapters dealt with general aspects of cosmology. We started with a homogeneous and isotropic model of the Universe and treated its components. Then, in order to explain the structures around us, we introduced inhomogeneities and anisotropies about this background through a perturbation theory. We pass now to discuss a specific model which consists in a universe filled with baryons, photons, neutrinos, dark matter and dark energy where we introduce an interaction between the components in the dark sector.
If dark matter and dark energy are coupled with each other, the energy-momentum where ∇ µ is the covariant derivative, Q ν (λ) is the four-vector governing the energy-momentum transfer between dark components and the subscript (λ) can refer to dark matter (c) or dark energy (d), respectively. Including an interaction in the dark sector, dark matter and dark energy components are not separately conserved, while the energy-momentum tensor of the whole dark sector is still conserved: We assume a spatially flat Friedmann-Lemaitre-Robertson-Walker background given by Eq. (1.2) and an energy-momentum tensor of a perfect fluid (1.7). From the energy conservation of the full energy-momentum tensor, we can derive the equations of evolution of the background dark matter and dark energy densities, where the derivatives and the Hubble parameter H are in conformal time, ρ c and ρ d are the energy densities for dark matter and dark energy, respectively, ω = p d /ρ d is the equation of state of dark energy and Q was chosen to be the energy transfer in cosmic time coordinates. We emphasize that the homogeneity and isotropy of the background require the spatial components of Q ν (λ) to be zero.
In this section we concentrate on the phenomenological interaction as a linear combination of energy densities of dark sectors with the form of Q = 3H(ξ 1 ρ c +ξ 2 ρ d ), describing the energy transfer. With the above expression for the continuity equations, if Q > 0, we have that the dark energy transfers energy to the dark matter. In studying the curvature perturbation, it has been made clear that when the interaction is proportional to the energy density of dark energy (Q = 3Hξ 2 ρ d ), we get a stable curvature perturbation except for ω = −1; however, when the interaction is proportional to the dark matter density (Q = 3Hξ 1 ρ c ) or total dark sectors (Q = 3Hξ(ρ c + ρ d )), the curvature perturbation can only be stable when the constant dark energy equation of state satisfies ω < −1 [13].
For the case of a time-dependent dark energy equation of state, the stability of curvature perturbations was discussed in [14,15]. With the interaction, the effective background equations of state for the dark matter and dark energy change to We summarize different forms of the interaction with the effective background equation of state in Table 3.1; we label our models with Roman numerals [44].
In order to solve the coincidence problem, we require the ratio of the energy densities of dark matter and dark energy, = ρ c /ρ d , to be a constant in the expansion history of . This leads to a quadratic equation, The solutions of this equation can lead to unphysical results, such as negative energy density of cold dark matter in the past or complex roots. For different phenomenological models of the interaction between dark sectors, the conditions to obtain physical results, positive energy densities and real roots were obtained in [44] as shown in Table 3.1. Figure   3.1 illustrates the behavior of for the four interacting models. We observe that, for the interaction proportional to the energy density of dark energy, a positive interaction can help to alleviate the coincidence problem as there is a longer period for the energy densities of dark matter and dark energy to be comparable. In contrast, a negative interaction can not alleviate the coincidence problem. For the interaction proportional to the energy density of dark matter or to the sum of both energies, the ratio presents a scaling behavior.
From the background dynamics we see that when we introduce the phenomenological interaction between dark sectors, it is possible to have the scaling solution of the ratio between dark matter and dark energy, which can help to alleviate the coincidence problem.
However, in the background dynamics there appears an inevitable degeneracy between the   Table 4.4), and the blue dotted line, to Planck best-fit Model IV (see Table 4.5).
coupling in dark sectors and the dark energy equation of state. In general this degeneracy cannot be broken by just investigating the dynamics of the background spacetime, except in the case when the coupling is proportional to the dark matter density (Model III) [44]. It is expected that the degeneracy between the coupling and other cosmological parameters can be solved in the perturbed spacetime by considering the evolution of the perturbations of dark energy and dark matter.
Although we introduced perturbations in the components of the Universe through the Boltzmann equation in the last chapter, the dark matter could be completely described using the energy-momentum tensor as we observed earlier. Thus, for the interacting model, we will consider perturbations through the conservation equation (3.1). To calculate the covariant derivative we use the perturbed metric (2.6) and we assume that the energy-momentum tensor is given by a perfect fluid even in the perturbed case: where δ(η, x) is the density contrast δ ≡ δρ/ρ and the four-velocity reads v (λ) can be written as minus the gradient of a peculiar velocity potential v (λ) plus a zero divergence vector. Only the first one contributes to scalar perturbations. Thus, taking the perturbed part of the energy-momentum conservation results and To solve equations (3.8) and (3.9) we need a relation for δP λ . In order to find such a relation, we first observe that the sound speed c s of a fluid or scalar field is the propagation speed of pressure fluctuations in its rest frame [12]: For a scalar field ϕ, the sound speed is equal to the speed of light, c 2 sϕ = 1. On the other hand, we can define the "adiabatic sound speed" for any medium as If a fluid is barotropic, the sound speed is equal to the adiabatic sound speed, c 2 s = c 2 a , and if its equation of state is constant, then c 2 a = ω. At first sight, the dark-energy fluid with ω = const. is a barotropic adiabatic model. However, this results in an imaginary sound speed which leads to instabilities in the dark energy. Thus, we impose that c 2 sd > 0 by hand and it is natural to adopt the scalar field value.
The rest frame is defined as the comoving (v| rf = 0) and orthogonal (B| rf = 0) frame, from the rest frame to a general gauge, we obtain 1 Using the definition of the rest frame, the first equality above results δη = −v + B. Thus, following [12], the perturbed pressure of dark energy is given by where δP (nad) is the intrinsic non-adiabatic pressure perturbation in the dark energy fluid.
For the coupled case, we have where δ d = δρ d /ρ d is the density contrast, c 2 sd is the effective sound speed at the rest frame, which we set to one, and c 2 ad is the adiabatic sound speed, all with respect to dark energy.
The perturbed four-vector δQ ν (λ) can be decomposed into Here the ± sign refers to dark matter or dark energy, respectively, and δQ p(λ) is the potential of the perturbed energy-momentum transfer δQ i (λ) . Q I p(λ) t is the external nongravitational force density and v t is the average velocity of the energy transfer. In this section we consider that there is no nongravitational interaction between dark energy and dark matter; only an inertial drag effect appears due to stationary energy transfer. Thus and v t vanish, which implies that δQ i (λ) = 0.
Finally we are in position to write the equations governing the evolution of linear perturbations for dark matter and dark energy. As we are interested to implement the interacting model in the public code CAMB [95], it will be more appropriate to obtain the perturbed equations in the synchronous gauge. Thus, collecting equations (3.14) and (3.15) in (3.8) and (3.9), and passing to the Fourier space 2 , we obtaiṅ where h = 6φ is the synchronous gauge metric perturbation. In practice, we fix the remaining freedom in the synchronous gauge setting a comoving frame with respect to the dark matter fluid, such that the peculiar velocity of dark matter v c vanishes.
To solve equations (3.16)-(3.19) we set initial conditions according to [13]. Using the gauge-invariant quantity [96] ζ = φ − H δρ ρ (3.20) and the gauge-invariant relative entropy perturbation we get adiabatic initial conditions In the linear perturbation formalism, the influence of the interaction between dark energy and dark matter on the CMB and the matter power spectrum can be calculated by modifying the CAMB code [95].  was observed that there are possible ways to break the degeneracy between the interaction, dark energy equation of state and the dark matter abundance in the perturbation theory [44]. This can help to get tight constraint on the interaction between dark energy and dark matter.

Lagrangian Model
It would be desirable to construct a more fundamental model for dark matter, dark energy and a possible interaction between them. As we know, the fundamental particles and the corresponding interactions in the standard model are described by quantum field theories.
In the same way, we would like to account with dark matter and dark energy in the context of a quantum field theory. However, as we already saw, the standard model of particle physics have no explanation for the dark sector. Thus, in this section, we postulate a model for dark matter interacting with dark energy.

The Tetrad Formalism
We are interested in the behavior of the fields on cosmological scale which means we need a field theory on a curved spacetime. We could be tempted to follow the general approach to transform the special-relativistic equations into general-relativistic equations changing the tensors T α··· β··· , the derivatives ∂/∂ α and the metric η αβ to tensors T µ··· ν··· , covariant derivatives ∇ µ and metric g µν in the general coordinate frame. However, there are no representations of the general-coordinate transformation group which behave like spinors under the Lorentz subgroup. Thus, to deal with a fermionic field in the context of general relativity we need the tetrad formalism [57].
Using the Principle of Equivalence, we can establish, at every point X, a set of coordinates ξ α X that are locally inertial. In the locally inertial frame, the metric is the Minkowski one, η αβ . Thus, we can obtain the metric in a general noninertial coordinate system x µ Fixing the locally inertial coordinates ξ α X at each physical point X, the partial derivatives e α µ change under a general coordinate transformation x µ → x µ as Thus, we can identify e α µ with four covariant vector fields, and this set of four vectors is called a tetrad, or vierbien. On the other hand, the tetrad behaves like a Lorentz contravariant vector under a Lorentz transformation Λ α β (x) of the locally inertial frame: Given a contravariant vector A µ , we can contract it with the tetrad e α µ such that the resulting object A α = e α µ A µ transforms as a set of four scalars under a general coordinate transformation, while it transforms as a Lorentz contravariant vector under a Lorentz transformation of the locally inertial coordinate system. In general, a tensor n times contravariant and m times covariant T µ 1 ···µn ν 1 ···νm , having its contravariant indices contracted with e α µ and its covariant indices contracted with the inverse tetrad e ν β = η αβ g µν e α µ , will transform as a set of n × m scalars under a general transformation, and as a Lorentz tensor n times contravariant and m times covariant under a Lorentz transformation of the local frame.
A matter action physically acceptable must be both a scalar under a general coordinate transformation and under a locally inertial Lorentz transformation. If an action could be obtained solely from fields, this condition would be automatically achieved. However, any physically sensible action must also involve derivatives of the fields. Let us consider a field Ψ (where Ψ can be a field of multiple components), which transforms under a Lorentz transformation as Ψ (x) = D(Λ(x))Ψ(x), where D(Λ) is a matrix representation of the Lorentz group. What we need is to define a covariant derivative that for a local Lorentz transformation behaves as∇ β Ψ = Λ α β D(Λ)∇ α Ψ. We can verify that the following covariant derivative obeys such a condition [57]: where ∇ µ e δν (x) ≡ ∂ µ e δν (x) − Γ λ µν e δλ (x) and σ βδ are the generators of the Lorentz group. For example, scalars have σ βδ = 0 and∇ α = ∂ α , while spinors have σ βδ = 1 4 [γ β , γ δ ], where γ β are Dirac matrices.
Finally we have a prescription to generalize a field theory to curved spacetimes: given a Lagrangian which is a function of the fields and their derivatives, we contract all vectors, tensors, etc., with the tetrad (e.g. A α → e α µ A µ ) and substitute the derivatives ∂ α for covariant derivatives∇ α . Using this prescription, the action will be a scalar under a general coordinate transformation and under a local Lorentz transformation.
The energy-momentum tensor can be defined as [57] T where S M denotes the action of the matter fields and det[e(x)] = √ −g, where g is the determinant of the metric tensor. It is possible to show that the energy-momentum tensor defined in this way is symmetric and obeys the conservation equation ∇ µ T µν = 0. Using the definition above, we can obtain the energy-momentum tensor for fields with spin in a curved spacetime.

Yukawa-Type Interaction
Let us consider now a model of interaction between dark energy and dark matter from a Lagrangian density. In this model, the dark energy will be described by a scalar field φ and the dark matter will be described by a massive fermionic field of spin-1 2 Ψ 4 . We will suppose that the interaction between them is realized by an Yukawa type interaction, which is the only interaction between a scalar field and a fermionic field of spin-1 2 that is renormalizable. Thus, the action for this model will be given by where β is a dimensionless interaction constant and V (φ) is the scalar potential. γ µ are matrices in the curved spacetime identified with the Dirac matrices γ α multiplied by the tetrad as γ µ ≡ e µ α γ α and they obey the anti-comutation relation whereM ≡ M − βφ. Using these equations, we can show that the current defined as On the other hand, varying the action with respect to the scalar field φ, we obtain its equation of motion where ∇ µ is the covariant derivative in the curved spacetime.
The energy-momentum tensor of the system can be calculated using Eq. (3.29). For our scalar and fermionic fields described by the action (3.30), we can show that [97] T We have split the energy-momentum tensor by defining a scalar part and a fermionic part such that (3.38) and the interaction was included in the fermionic part.
On large scales, we assume that the components in the Universe can be described by a fluid with energy-momentum tensor given by Eq. (1.7). Thus, from the energy-momentum tensor of a perfect fluid, we find that Therefore, using the energy-momentum tensor of the scalar field (3.36), we can identify an energy density and a pressure given by and On the other hand, from the energy-momentum tensor of the fermionic field (3.37), we and We observe that in the relativistic limitM = 0, we recover the equation of state for an ideal relativistic gas, P Ψ = ρ Ψ /3.
Let us assume that the Universe obeys the FLRW metric and the fields are homogeneous such that ∂ i φ = 0, ∂ i Ψ = ∂ iΨ = 0 and ∇ iΨ γ i Ψ −Ψγ i∇ i Ψ = 0. Thus, the energy density and pressure of the fields result From the above equations we observe that the scalar field possesses an equation of state given by From Eq. (3.46), we see that the interaction Q ≡ βφΨΨ can be rewritten as Thus, we observe that the fermionic field can be completely described by a fluid, since the energy balance equations depend on the fermionic field only through the energy density ρ Ψ and a constant mass M . On the other hand, the scalar field cannot be completely described by the energy balance equations because the equation of state ω φ and the interaction term Q depend on the scalar field and its derivative. Therefore, the background equations describing the evolution of the fermionic and scalar fields arė Let us introduce now linear perturbations in our model. To facilitate the study of linear perturbations, we assume that the fermionic field can be completely described by the fluid equations even in the perturbed level. The interaction between the dark energy and dark matter can be generalized into a covariant form as Thus, the balance equations can be generalized to and where T µν Ψ assumes the form of a perfect fluid as in Eq. (3.6). The scalar field can be decomposed into a homogeneous part and a perturbed part, φ(t) + ϕ(t, x). Thus, using the metric (2.6) in the synchronous gauge, the balance equations (3.57) and (3.58) yield the perturbed equations in the Fourier space and conformal time.
To complete the discussion about the linear perturbations we need the initial conditions of the perturbed quantities. We assume that the dark matter and the dark energy satisfy adiabatic initial conditions Thus, the adiabatic initial condition (3.62) can be rewritten as As the scalar field obeys a second-order differential equation, we need two initial conditions. The second initial condition can be obtained considering that the intrinsic entropy Finally, solving equations (3.64) and (3.66) for ϕ and ϕ , we obtain the perturbed initial conditions for the scalar field We modified the CAMB code to include the Lagrangian model above. We considered that the scalar potential is given by where A is a normalization constant, λ is a dimensionless parameter and M pl is the reduced Planck mass. We set A to the value of the cosmological constant energy density A = ρ Λ .
Thus, λ = 0 and β = 0 is a measurement of how our model differ from the cosmological constant model. We also see from Eq. (3.53) that the interaction β and the fermion mass M are degenerate, we can only know the ratio r ≡ β M M pl . Therefore, we use r instead of β as our interaction parameter. This has the advantage of decreasing one degree of freedom in the analysis, at the cost that we are unable to know the individual values of β or M .
The appendix B shows the most import steps to introduce the Lagrangian model in the CAMB code. Below we present some graphs for the CMB and matter power spectrum obtained from the Lagrangian model for different values of the parameters. Figure 3.6 shows that the scalar potential parameter λ has a small effect on the CMB and matter power spectrum, affecting mainly the low-l CMB power spectrum. On the other hand, as observed in the phenomenological case, we see from Fig. 3.7 that in addition to modifying the CMB spectrum at low l, the coupling between dark sectors can shift the acoustic peaks at large multipoles. However, comparing with the phenomenological models, the Yukawa interaction has a more dramatic effect on low multipoles and less effect on the acoustic peaks. We also observe that the power spectra present an almost symmetric behavior around the zero value for the interaction. However, such a symmetry is broken when we look at background quantities, e.g. the age of the Universe.

Chapter 4 Analysis
In the previous chapter we presented some models of interacting dark energy. We showed some aspects of their behavior, especially on the power spectra. We now wish to test such models with current observational data in order to constrain the cosmological parameters.
The results of this chapter are matter of publication in [98][99][100].

Methods for Data Analysis
Let us suppose we want to estimate a random physical quantity x with a probability density function (PDF) f (x|θ) that depends on an unknown parameter θ. Naturally, we cannot know the exact value of x, since there are several uncertainties in the measurement, which we denote as σ. Then, the probability of obtaining x in the interval ∆x around x is Such a probability is called a conditional probability of having the data x given the theoretical parameter θ. If the interval of integration above is small, we can approximate the probability as P ≈ f (x|θ)∆x.
The law of joint probability tells us that for several measurements x i , i = 1, ..., N , the probability of having x 1 in the interval ∆x 1 around x 1 , x 2 in the interval ∆x 2 around x 2 and so forth is This procedure is correct given the fact that the measurements are mutually independent and considering equally spaced small intervals ∆x i . In the above expression we defined the Except for a constant value, the likelihood function gives us the probability of having the outcomes x i given the parameter θ. In general, the data can be correlated and depend on several parameters θ j . For correlated data, L(x i |θ j ) will not be the simple product of the PDFs, but its statistical interpretation remains the same.
For every set of parameters θ j , the likelihood function L(x i |θ j ) will assume a different value. It is logic to define the best θ j s as the parameters that maximize the likelihood.
Thus, the maximum likelihood method of parameter estimation consists in finding the parameters that maximize the likelihood solving the system The solutions of these equations,θ j s, are functions of the random data x i s, therefore they are random too. In the frequentist approach, one tries to find the distribution of theθ j s given the distribution of the data x i s; if this is possible, one can associate probabilities to intervals ofθ j s, for instance determine the interval ofθ j that has 95% probability that a set of data were obtained from the theoretical distribution. However, it is often too difficult to derive theθ j 's distributions analytically and very demanding to derive them numerically using simulated datasets. Moreover, this approach does not take into account our previous knowledge about the theoretical parameters, e.g. the results of preceding experiments.
These issues can be worked more appropriately using the Bayesian approach.
In the Bayesian approach, instead of looking for the probability of having the data given the model, L(x i |θ j ), we look for the probability of the model given the data, P (θ j |x i ). This is possible thanks to the Bayes's theorem, which relates the conditional probability of having an event B given the event A occurred, P (B|A, I), with the conditional probability P (A|B, I) of A given B and the probabilities P (B|I) and P (A|I) of the events B and A, respectively. The letter "I" denotes that these probabilities depend on some information I we assume to be true.
In our case we have the data A ≡ x i and the parameters B ≡ θ j for some theoretical The function p(θ j |x i , M ) is the posterior probability distribution, p(x i |M ) is the PDF of the data x i , sometimes called evidence, and p(θ j |M ) is the prior probability for the parameters θ j , it represents our previous knowledge for the parameters before we make the experiment. Therefore, the posterior contains the information we are looking for: the probability of having the parameters θ j given the data x i and some previous knowledge about the parameters.
The posterior is a probability distribution and consequently it must be normalized to Thus, we can think the evidence as a normalization factor. On the other hand, the prior is often unknown. Usually, if we know nothing about the parameters we want to estimate, we choose a uniform prior in some interval and zero outside it. However, if we have some estimate of the parameters from previous experiments, we can adopt a Gaussian centered at the estimated value as our prior. In general, the choice of priors affect the posterior distribution, but if we have enough data they will dominate the posterior.
Given the posterior p(θ j |x i , M ) we can find the maximum likelihood estimatorsθ j as We can also derive the regions of confidence for the parameters, which are defined as where the posterior must be normalized, which means 0 < α < 1. To find these regions, the limits of integration must be the lowest ones such that the integral is valid. Typical choices of α are α = 0.683, 0.954, 0.997 which denote the 1σ, 2σ and 3σ confidence levels.
Frequently, we are interested in a subset of the parameter space and we consider the others as "nuisance", thus we integrate (marginalize) over the uninterested parameters.
In general, the distribution function depends on the characteristics of the quantity it describes and its uncertainties. In cosmology we usually assume a Gaussian distribution for the data even when each data was not extracted from a Gaussian distribution. This is possible thanks to the central limit theorem [101,102], which establishes that for a set of N variables with some distribution each one with a finite variance, in the limit N → ∞, the average distribution tends to a Gaussian one. Thus, if the data have a Gaussian distribution and are independent of each other, then (4.10) Therefore, maximizing the posterior distribution (4.5) is equivalent to minimizing the above exponent (4.11) If the data are correlated, the above likelihood must be generalized as where X is the data vector and C is the covariance matrix.

Data
To constrain the cosmological parameters in our interacting models, we use several data sets, the measurements of CMB anisotropies, BAO, SNIa, the direct measurement of the Hubble constant H 0 and Lookback Time. Below we describe the likelihood for these measurements.

CMB Measurements
The Planck data set we use is a combination of the low-l TT likelihood, which includes measurements for l = 2 − 49, combined with the high-l TT likelihood, which includes measurements from l = 50 up to a maximum multipole number of l max = 2500 [47][48][49].
Together with the Planck data, we include the polarization measurements from the nine year Wilkinson Microwave Anisotropy Probe (WMAP) [103], the low-l (l < 32) TE, EE, BB likelihoods.
The CMB power spectrum likelihood for low-l multipoles can be written as where n is the number of observed pixels, M(C l ) = C(C l ) + N is the data covariance matrix, which is split into the CMB, C, and noise, N, covariance matrices, and m = s + n is the observed map. Actually, the estimated C l distribution is not well approximate by a Gaussian at low l (l < 50), because of the limited degrees of freedom per l. However, if foregrounds and instrumental systematics are negligible, then the CMB signal s and instrumental noise n are nearly Gaussian and the above likelihood is valid.
In the general case, the data vector m includes both temperature and linear polarization. For the temperature, the signal covariance matrix gives where P l are Legendre polynomials calculated at angle θ i 1 i 2 between the centers of pixels i 1 and i 2 , b l is the effect of instrumental beam and W l is the window function. For polarization correlations, we obtain similar expressions.

BAO Measurements
In addition to the CMB data sets, we also consider measurements of Baryon Acoustic Oscillations (BAO) in the matter power spectrum. We combine the results from three redshift surveys: the 6dF Galaxy Survey measurement at redshift z = 0.106 [50], the SDSS DR7 BAO measurement at redshift z = 0.35 as analysed by Padmanabhan et al. [51] and the BOSS DR9 measurement at z = 0.57 [52].
These redshift surveys measure the distance ratio where r s (z drag ) is the comoving sound horizon at the baryon drag epoch, the epoch when baryons became dynamically decoupled from photons, and D V (z) combines the angular diameter distance d A (z) and the Hubble parameter H(z), in a way appropriate for the analysis of spherically-averaged two-point statistics, (4.16) The comparison with BAO measurements is made using χ 2 statistics

SNIa Measurements
We use the SNIa data from the Supernova Cosmology Project (SCP) Union 2.1 compilation [53], which has 580 samples. The Union 2.1 uses SALT2 [104] to fit supernova lightcurves. The SALT2 model fits three parameters to each supernova: an overall normalization, x 0 , to the time dependent spectral energy distribution of a SNIa, the deviation from the average lightcurve shape x 1 and the deviation from the mean SNIa B -V color c. Combining these parameters, the distance modulus is given by The best-fit cosmology is determined by minimizing the χ 2 , We use this measurement of the Hubble constant as an additional data.

Lookback Time Measurements
The background observables treated so far, the measurements of H 0 and SNIa, are firmed on distance measurements. To pin down the cosmological parameters we will consider another indicator, the lookback time, which is based on ages instead of distances. The lookback time t L (z) is the difference between the present age of the Universe, t 0 , and its age at redshift z, t(z), . (4.21) For an object at redshift z i , we can calculate its age from the difference between the age of the Universe at redshift z i and the age of the Universe when the object was formed at where we used the lookback time definition (4.21). Thus, the above equation tells us that the observed lookback time t obs L (z i ) to an object i at redshift z i is where df ≡ t obs 0 − t L (z F ) is the delay factor which encodes our ignorance of the formation redshift z F .
In order to constrain the cosmological parameters, we implement a CosmoMC module to calculate the likelihood for lookback time measurements (Appendix C shows the details of this module). We use the data of 32 passively evolving galaxies in the redshift interval 0.117 ≤ z ≤ 1.845 with an uncertainty in the age measurements of 12% at one standard deviation [56]. Additionally, we use the ages of 6 galaxy clusters in the redshift range 0.10 ≤ z ≤ 1.27 with an uncertainty of 1Gyr at one standard deviation [55]. The likelihood is a Gaussian with where p denotes the theoretical cosmological parameters. The delay factor is a nuisance parameter that is fitted simultaneously with the cosmological parameters.

Phenomenological Model
We would like to put constraints on the four phenomenological coupled dark energy models listed in Table 3.1. For this purpose, we will use the recent measurements of the cosmic microwave background anisotropies as measured by the Planck satellite mission. We will also consider the combined constraints from the Planck data plus BAO, SNIa and H 0 measurements. In our analysis, we will choose our priors of different cosmological parameters as listed in Table 4.1. The results were published in [98].
We will allow the equation of state of dark energy to vary. We also choose the helium abundance Y p from a big bang nucleosynthesis (BBN) consistent scenario. Thus, the primordial helium abundance Y p is predicted as a function of the baryon density Ω b h 2 and number of extra radiation degrees of freedom ∆N . We will use interpolated results from the PArthENoPE code [107] to set Y p , following [108]. We will take the relativistic number of degrees of freedom N ef f = 3.046, the total neutrino mass m ν = 0.06eV and the spectrum lensing normalization A L = 1. To compare theory with observations, we employ the Markov chain Monte Carlo (MCMC) methodology through a modified version    Now we present the fitting results for the coupling Model II in Table 4    is proportional to the energy density of dark matter. In this model, to ensure stability of the curvature perturbation, if the equation of state of dark energy is constant, it has to be smaller than −1 [13]. Looking at the new constraints on this coupled dark energy model from the recent measurements of CMB from the Planck satellite mission alone, in Table   4.4, we find that the Hubble constant value is consistent with low redshift observations, but it is much higher than that of the ΛCDM result. The coupling constant is more tightly constrained in this coupled dark energy model than those in Models I and II, which is in agreement with the findings in the WMAP constraints [25,44]. The value of the coupling parameter ξ 1 is small positive, which meets the requirement to alleviate the coincidence problem. The evolution of the ratio between energy densities of dark matter and dark energy with this small positive coupling was shown in the Fig.3 Age/Gyr when ξ is positive and has the attractor solution with the ratio between dark energy and dark matter energy densities ∼ constant in the past. We also consider the combined constraints from the Planck data plus other measurements. The results are listed in Table   4.4, which shows stronger evidence for this coupled dark energy model with small positive coupling. We plot the 1-D posteriors for the parameters in  we consider the interaction between dark energy and dark matter is proportional to the energy density of the total dark sectors. In order to ensure the stability of the curvature perturbation, the constant equation of state of dark energy has to be in the phantom range. This was disclosed in [13]. As observed in the WMAP fitting results, this type of interaction has very similar constraints to the Model III [25,44]. Confronting the model to the Planck data alone and the combined observational data, we list the constraints in Table 4.5. We show the 1-D posteriors for the parameters in Fig.4.7 and plot the main parameter degeneracies in Fig.4  sities to be comparable in the expansion of the Universe as shown in Fig.3.1

. The Model
IV has an attractor solution with ∼ constant in the future. In the joint constraints, by including other observational data, we find that the coupled dark energy model IV is fully compatible with astronomical observations. It is a viable model.
We constrained, up to now, the four phenomenological models based on Planck measurements and additional data from BAO, SNIa and H0. These additional data sets are based on measurements of distances. As we pointed out in the previous section, another  kind of data can be obtained based on measurements of ages instead of distances. One such technique is the lookback time. Thus, to take this kind of data into account we implemented the likelihood for lookback time in the CosmoMC code (see Appendix C).
We choose our priors for the different cosmological parameters as listed in Table 4 Table 4.7 present the best fit and 68% C.L. for the prior given in Table 4.6. The columns in the middle redo the calculations with a restrict prior for ξ 2 as that in    at 68% C. L., while Planck is consistent with a cosmological constant. The lookback time measurements favor a positive interaction parameter, which is in tension with more than 2σ as compared with the result from Planck. Such difference in the interaction has an effect in the time when the structures were formed, as described by the delay factor df , and also in the amount of dark matter and dark energy at present.
We pass now to consider the behaviour of Model II with respect to LBT data. Table   4.8 presents the best fit values and marginalizations for the parameters.

Lagrangian Model
Let us consider now our Lagrangian model described in the previous chapter. We want to put constraints on the cosmological parameters and verify if the Yukawa interaction is favored by the observational data. The priors that we use are listed in Table 4.11. At first we allow the parameter of the scalar potential λ to vary freely. We fixed the helium    abundance as Y p = 0.24. The number of relativistic degrees of freedom is adjusted to N ef f = 3.046 and the total neutrino mass is set to m ν = 0.06eV . At last the spectrum lensing normalization is A L = 1. To finish the MCMC we set the Gelman and Rubin criterion to R − 1 = 0.03. The results obtained for this model will be published soon [99].
To constrain the Yukawa-type interacting dark energy, we use the measurements of the CMB anisotropies made by Planck together with BAO, SNIa and H 0 measurements. Using the priors listed in Table 4.11 we run the MCMC. The results are shown in Table 4.12, the 1-D posteriors for the parameters are given in Fig. 4.18 and some parameter degeneracies are in Fig. 4.17. We observe that the Planck data alone is not able to constrain the scalar potential λ and it constrains the interaction parameter r symmetrically around the zero value. This is what we expected from the discussion about the power spectra of the We then consider the case when we fix the scalar potential parameter λ. We have learned that as we increase the value for λ, the interaction becomes more favored. For instance, λ = 3/2 produces the results in Table 4.13 and the 1-D posterior distributions

Conclusions
The aim of this thesis was to propose models of interaction between dark energy and dark matter and discover from observational data if the interaction is favored.
Chapter one dealt with basic aspects of cosmology. We introduced the simple homogeneous and isotropic model for the Universe and the equations governing its evolution.
Then, in order to put the cosmological models on experimental basis, we described definitions of cosmic distances which allow us to constrain the cosmological parameters of a model from observations and make predictions that can be falsifiable. For instance, observations of surpernova type Ia put constraints from luminosity distances and measurements of baryon acustic oscillations are based on angular diameter distances. Finally, we discussed how current observations lead to the necessity of two unknown components: dark matter and dark energy.
The structures around us imply that we must go beyond the homogeneous and isotropic universe if our model intends to explain the real Universe. On large scales, we can improve our model through small perturbations in the background. Chapter two introduced such perturbations and developed their dynamical equations. The linear perturbations over the smooth background lead to predictions about the cosmic microwave background anisotropies and matter inhomogeneities which can be tested by anisotropy probes, e.g.
Planck satellite, and galaxy surveys, respectively.
The first two chapters were concerned to establish the cosmological basis we needed.
In chapter three we explore the possibility of an interaction in the dark sector. The simplest cosmological model, the ΛCDM, is in good agreement with current observations.
However, it suffers from two theoretical problems: the cosmological constant problem and the coincidence problem. Thus, we consider alternative models: the first is a phenomenological model with an interaction given by a linear combination of the dark fluids Q = 3H(ξ 1 ρ c + ξ 2 ρ d ); and, as a tentative to build a more fundamental framework, the second model comes from a Lagrangian density of a scalar field, describing dark energy, and a fermionic field, for dark matter, with an Yukawa interaction between them.
The last chapter compares the theoretical predictions of the interacting models with several observational data. The data show that an interaction is allowed and in some cases it is favored. We also observe some tension between the results from Planck with the low redshift measurements from lookback time. From this thesis we see at least two possible directions for future works: first, there is still much space for alternative models of dark energy and dark matter, and different kinds of interaction; second, there are other data sets that were not considered here and new techniques and probes should be taken into account. These studies will certainly augment our knowledge of the nature of dark matter and dark energy.