An Introduction into the Theory of Cosmological Structure Formation Christian Knobel Institute for Astronomy, ETH Zurich, Zurich 8093, Switzerland January 2013 (Second Edition) This text aims to give a pedagogical introduction into the main concepts of the theory of structure formation in the universe. The text is suited for graduate students of astronomy with a moderate background in general relativity. A special focus is laid on deriv- ing the results formally from first principles. In the first chapter we introduce the homogeneous and isotropic universe defining the framework for the theory of structure formation, which is dis- cussed in the three following chapters. In the second chapter we describe the theory in the Newtonian framework and in the third chapter for the general relativistic case. The final chapter dis- cusses the generation of perturbations in the very early universe for the simplest models of inflation. arXiv:1208.5931v2 [astro-ph.CO] 1 Feb 2013 ii Preface This text aims to give a pedagogical introduction into the main concepts of the theory of structure formation in the universe. The text is suited for graduate students of astronomy with a moderate background in general relativity. A special focus is laid on deriving the results formally from first principles. During my PhD studies on high redshift galaxy groups I had the endeavor to understand the theoretical framework on which my research was based. I did not only want to be able to reproduce statements that were made in the books, but to understand where they came from and what their underlying assumptions were. Therefore, I read several books on cosmology and worked out the basic theory the way I found it the most accessible. In the course of time, I filled a couple of notebooks this way, until I finally started to convert part of them into digital form as an introduction for my PhD thesis. But, as the introduction turned out to be way too long, I eventually did not include it into my thesis. However, since I had already spent much work compiling this text and since I was encouraged by the positive feedback of students who had read it, I decided to assemble it into a self-contained introduction on cosmological structure formation in order to make it also available to other graduate students with the same interest as me. I attempted to make a common theme obvious, which is the question of how structures in the universe were created and grew to the present time large-scale structure of galaxies that is observable today. The selected material is supposed to be self-contained, but nevertheless concise. I put a special focus on clarifying how the results are formally derived from underlying fundamental principles and which assumptions were made. This still does not imply every single calculation to be included in detail. For instance, the derivation of the Robertson-Walker metric is not performed specifically, as \maximally symmetric spaces" are a special part of general relativity rather than astrophysical cosmology. However, I show certain conditions to be satisfied so that I can refer to a common theorem of general relativity in the literature that uniquely leads to the Robertson-Walker metric. Besides it is generally only the simplest possible case worked out in a given context, since these cases often allow elegant, rigorous derivations. This approach is motivated by the fact that the simplest cases already allow a sufficient qualitative understanding and that realistic quantitative results in the context of structure formation usually require large numerical computations. Unfortunately, technical texts with a high aspiration for completeness and rigorousness are prone to become longish, while concise texts have the tendency to be incomplete, inaccurate, or ambiguous. For this reason this introduction contains unusually many footnotes compared to other astronomical texts. In order to keep the central theme as straight and concise as possible, I included many minor comments and sometimes also short derivations in the form of footnotes. That is, the basic text (without footnotes) is self-contained, while the footnotes provide additional comments and assistance. iii It is exactly these footnotes, which can be helpful for students to understand certain subtleties. Those readers who are not interested in these details can just omit them whithout losing the thread. There are still many topics that would fit into this introduction, which were omitted, such as the derivation of the cosmic microwave background fluctuations or a more detailed discussion of the nonlinear regime of the large-scale structure (e.g., the Zel'dovich approximation, numerical N- body simulations). So naturally the material selected represents only a tiny fraction of interesting and important topics in the context of structure formation. Also the literature cited is not comprehensive. I mainly included the material which the text is based on and which I find useful for further reading. If I have accidentally omitted an important reference that should definitely be included, please let me know. Since I aim to maintain this text, all sorts of comments are welcome. If you find typos or if you have suggestions on how to improve the content, I would be grateful, if you sent me a message to [email protected], so that I can update it. Structure of the introduction This introduction is divided into four chapters. The first two are easier to understand than the last two and need only little input from general relativity (only for the derivation of the Robertson-Walker metric and the Friedmann equation). The second chapter is even fully based on Newtonian physics and yet contains most of the results that are presented. On the other hand, the last two chapters and the appendix are entirely based on general relativity and are much more technical than the first two chapters. They require a moderate background in general relativity, although we still derive all results starting with the field equations. Readers who are not interested in the theory of general relativistic structure formation can just omit the latter two chapters and focus on the former ones, which are mostly self-contained. In Chapter1 , we discuss the universe in the homogeneous and isotropic limit. Starting with the Robertson-Walker metric, we derive the Friedmann equation and describe the dynamics of the universe for given energy contents. We introduce many basic concepts that will be needed in further chapters, such as redshift, comoving distance, and horizons. Then we give an overview of the current concordance cosmology (i.e., the ΛCDM model) and briefly summarize the history of the universe. Finally, we provide an introduction into the phenomenology of the simplest models of inflation. In Chapter2 , we present the theory of structure formation based on Newtonian physics which is valid well inside the horizon. Interestingly, using Newtonian physics it is possible to derive basically all of the main results for the formation of structures in the universe (except of the form of the primordial power spectrum) and this is the reason why many books entirely omit a treatment of the general relativistic case. We first derive the equations governing the growth of fluctuations at first (linear) order within an expanding universe from the basic hydrodynamical equations and discuss the different possible fluctuation modes. We afterwards introduce the correlation function and the power spectrum to describe the perturbations in statistical terms. In a further step, we leave the linear regime and describe the formation of nonlinear bound structures (\halos") by means of the (simplistic) \spherical top hat model". We motivate approximative formulas for the number density and spatial correlation of dark matter halos in the universe. Finally, we introduce the \halo model", which is the current scheme for analyzing the clustering of galaxies in the universe. In Chapter3 , we give an introduction into the linear theory of hydrodynamic perturbations in the general relativistic regime, which is needed to understand the evolution of structures outside the horizon. Without this theory, it is impossible to understand how structures that were created during inflation evolved until present time. Compared with the Newtonian linear theory, the general relativistic case is much more complicated and also allows perturbations which have no Newtonian iv counterpart (e.g., gravitational waves). We first introduce the Scalar-Vector-Tensor decomposition to simplify the perturbed field equations, since they decouple into independent scalar, vector, and tensor equations. Next, we further simplify the field equations by introducing \gauge transformations" and choosing particular gauges. We compare our results to the Newtonian ones from the previous chapter for the limiting case well inside the horizon and give a sketch for the general treatment which treats the fluctuations by means of the general relativistic Boltzmann equation. Chapter4 , is the most technical chapter of all and basically consists of one single big calculation. The aim is to compute the form of the primordial dark matter power spectrum that results from the generation of fluctuations by the simplest models of inflation. The chapter is divided up into three parts: First, we quantize the perturbations of a scalar field inside the horizon during inflation and compute the power spectrum of the fluctuations in the ground state, second we show that under certain conditions the perturbations remain constant outside the horizon, and finally we compute the spectrum of the perturbations after they have reentered the horizon during the matter dominated era and compute the deviations from scale invariance that are expected from slow-roll inflation. The appendix gives an introduction into the theory of a classical scalar field in the context of general relativity. We derive the equations of motion for the scalar field in a smooth Friedmann- Robertson-Walker universe and then for the general relativistic linear theory of perturbations.