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Anisotropies in the Cosmic Microwave Background Anisotropies in the Cosmic Microwave Background Author: Francesc Redondo Fontrodona Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain. Advisor: Eduard Salvador Sol´e Abstract: The anisotropies in the CMB conform nowadays the most precious dataset in cosmol- ogy and because of that its detection in 1992 supposed a crucial event. The analysis of this data provides evidences of dark matter and the inflation process. It also exists a relation between these anisotropies and the density perturbations, which evolved making the universe structures. In this work, we will explain what density fluctuations are, how they evolved and how can they be math- ematically represented to understand the shape of the anisotropies spectrum. We also explain how this spectrum can be used to give the cosmological parameters of a model and how the anisotropies can be represented to make a mathematical analysis. I. INTRODUCTION lution since their appearance after inflation until time of recombination trec where we want to compute the result- In 1965 the Cosmic Microwave Background was de- ing temperature anisotropies in the CMB. tected causing a great impact in cosmology. In 1948, In the homogeneous and isotropic cosmic fluid, we mea- Gamow and collaborators noted that the Big Bang model sure a density fluctuation at a point x with respect to predicts a background of cosmic radiation which should the mean densityρ ¯ through the dimensionless density be a black body radiation. Its temperature had been contrast field estimated to be between 5K and 50K [1]. δρ(x) ρ(x) − ρ¯ 17 years later, two radio astronomers of the Bell Lab- δ(x) = = (1) oratories, Penzias and Wilson, discovered an isotropic ρ¯ ρ¯ radiation with a temperature of 3K. They had detected Any perturbation can be represented as a superposi- the CMB. This discovery meant the establishment of the tion of plane waves (by the Fourier representation theo- Big Bang model of cosmology instead of the steady state rem) model. Immediately after the detection of CMB cosmolo- gists started to work with it, measuring the blackbody ra- X diation in all the wavelengths and searching anisotropies δ(x) = δk exp(ik · x) (2) in its radiation. k The research for anisotropies was motivated by the where δk are the Fourier coefficients. As long as pertur- models of structure's formation. These models worked bations are in the linear regime, the waves of different k with the assumption that all structure in the Universe evolve independently of each other. formed from small initial density fluctuations, which Although the mean value of the perturbation δ(x) ≡ evolve with gravitational instabilities. As we shall see, δ¯ across the statistical ensemble is identically zero by these density fluctuations are related to anisotropies in definition, its mean square value, i.e. its variance σ2, is the CMB spectrum. not. In the limit where the volume of the perturbation In the 80s cosmologists started to work with inflation- is Vu ! 1, the variance is ary models and the CMB became important again. Infla- tionary models predict the kind of initial density fluctu- 1 Z 1 σ2 = P (k)k2dk (3) ations and its shape, so looking for anisotropies in CMB 2π it is also an observational test of inflationary models. 0 In 1989, the NASA satellite COBE was launched to 2 where P (k) = δk is the power spectrum of the pertur- measure the CMB spectrum with extreme precision and bation δ. The power spectrum will characterize the per- find some fluctuations. These fluctuations were detected turbations. The variance does not depend on spatial po- in 1992 and the next years WMAP and Planck satellites sition but on time, because the perturbation amplitudes have been launched in order to obtain more and more 2 δk evolve. Therefore, the quantity σ inform us about precision. the amplitude of perturbations, but does not carry infor- mation about their spatial structure [4]. For power spectra of the power law P (k) = Akn, where II. STATISTICAL TREATMENT OF DENSITY n is called the spectral index, the convergence of the vari- FLUCTUATIONS ance in Eq. (3) requires n > 3 for k ! 0 and n < 3 for k ! 1. The inflation process predicts a primordial The aim of studying cosmological inhomogeneities is power spectrum P (k; tp) of this form with np ≈ 1 which to characterize them statistically and monitor their evo- is called the Harrison{Zel'dovich spectrum. Fluctuations Anisotropies in the Cosmic Microwave Background Francesc Redondo Fontrodona like that enter the cosmological horizon with a constant hydrodynamically like nonrelativistic matter [3], includ- value of the variance [4]. ing a radiation pressure term in the motion equations. As we see in Eq. (2), δ(x) is simply a sum over a As we shall see, our models need Dark Matter (DM) large number of Fourier modes. If the phases of each to make reasonable predictions about structure formation of these modes are random, then the central limit theo- and anisotropies in CMB. DM decouples from baryonic rem guarantees that the resulting superposition will be matter and radiation at early time. We work with Cold close to a Gaussian distribution if the number of modes Dark Matter (CDM) which is nonrelativistic when it de- is large. This class of density perturbation field, generi- couples. The individual CDM particles are assumed to cally predicted by inflationary models, is statistically de- move too slowly for them to produce an appreciable pres- termined by the covariance or autocorrelation function sure so we can describe them like baryonic nonrelativistic hδ(xi)δ(xj)i = ξ(rij). As the covariance function is re- matter without pressure term in the equations of motion lated with the power spectrum (ξ(0) = σ2), P (k) is par- [3]. ticularly important because it provides a complete sta- We will work in comoving coordinates, so it is like fol- tistical characterisation of the density field as long as it lowing the Hubble expansion in an unperturbed universe. is Gaussian. Comoving coordinates are get dividing the Eulerian co- The net effect of dissipation processes due to pho- ordinates by the scale factor a(t). This is very useful ton and matter diffusion is, in Gaussian fluctuations, because displacements respect these coordinates mean the change of the shape of the original power spectrum, gravity acting which is the most direct signature of inho- 2 P (k) = P (k0) × T (k) , where T (k) is the transfer func- mogeneity. tion. The study of the dissipation of a perturbation can We look for solutions in the form of plane waves be achieved by studying its transfer function T (k). δ = D(t) exp(ik · r) (4) where D(t) is the amplitude. The solutions for the ve- III. EVOLUTION OF DENSITY locity and gravitational field perturbations are similar. FLUCTUATIONS There are two advantages in expressing the solution in plane waves: first, all the time dependence is carried by The density content can be divided into nonrelativis- the amplitude and this eliminates the spatial dependence tic matter and radiation. There are two different rele- in our equations; and second, we can work in the Fourier vant kinds of fluctuations which have different relations space easily. The wave vector which appears in Eq. (4) between the density components: the adiabatic fluctua- is the comoving wave vector, or comoving wavelength λ, tions maintain the entropy invariant and it has the re- so in our coordinates it does not evolve, but the physi- 4 lation δr = 3 δm; the isocurvature fluctuations keep the 2π cal wavelength of the perturbation λ0 = , evolves in total density homogeneous so that there is no perturba- k0 tion in the spatial curvature and it satisfies the relation time. The solution modes in which we are interested in ρrδr = −ρmδm [2]. are the increasing mode where the perturbation grows Primordial fluctuations are created during the infla- due to gravitational instability, and the oscillating mode tion era. After this process only the adiabatic fluctua- where the perturbation oscillates like an acoustic wave tions survive, so we will work only with these fluctua- due to pressure and expansion effects. tions. However, we will see later other reasons for not From the wavelength scale λ we can define the mass considering isocurvature fluctuations. scale of the corresponding perturbation as The evolution of nonrelativistic matter can be de- π scribed by the Euler and energy equations of a fluid, and M = ρλ3 (5) 6 by the Poisson equation of a gravitational field. Indeed, since small perturbations imply weak gravitational fields, Similarly, we define the mass of the horizon as we can work in the Newtonian approach instead of gen- π 3 eral relativity as long as we have causality. To have the MH = ρRH (6) evolution equations for the perturbations, we must lin- 6 earize those equations, that is, we must write the per- where RH is the radius of the particle horizon. turbation of the homogeneous background to first order, If M > MH there is not casual contact inside the per- for example, ρ = ρ0 + δρ = ρ0(1 + δ), and the same for turbation and we have to work with general relativity the velocity, gravitational field and pressure, and then which is a local theory. Perturbations outside the hori- simplify the equations for the unperturbed field. zon always have increasing modes and never oscillating The radiation must be worked out in a relativistic way. modes because we don't have any pressure effect. These It is too far complicated to allow an analytic solution. perturbations can be treated with the method of autoso- Until near the time of recombination, trec, the rate of lution. This method is based on the property that these collisions of photons with free electrons was so great that type of spherical perturbations evolve in exactly the same photons were in local thermal equilibrium with the bary- manner as a universe model, which is a consequence of onic plasma.
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