Introduction to Friedmann Cosmology

Introduction to Friedmann Cosmology Andrei Lupu May 29, 2016 Contents Introduction 3 1 Tools of the Trade 4 1.1 Metrics of Surfaces with Constant Curvature . 4 1.2 The Robertson-Walker Metric . 5 1.3 The Friedmann Equation . 9 1.4 The Fluid and Acceleration equations . 10 1.5 The Equations of State . 12 2 Single-Component Universes 14 2.1 Temporal Evolution of the Equations of State . 14 2.2 Cosmological Eras and the Benchmark Model . 16 2.3 Empty Universes . 18 2.4 Spatially Flat Universes . 20 2.5 Radiation Only . 21 2.6 Matter Only . 23 2.7 Lambda Only . 24 3 Conclusion 25 4 References 26 2 Introduction Cosmology is a branch of astrophysics that concerns itself, not with individ- ual celestial bodies, but rather with the universe as a whole, at the largest possible scale. The cornerstone of cosmology is the set of Einstein's ten field equations, which, under its most compact form, can be written as such: 1 8πG R − Rg + Λg = T : (1) µν 2 µν µν c4 µν These equations describe the relation between the matter and the energy permeating space-time at a given point, and the metric used at that point. In other terms, Einstein's equations link the geometry of the space-time, as given by its curvature, to the matter and energy found in it. As a result, the solution set to these equations represents all possible universes that respect general relativity. Although tremendously powerful, Einstien's field equations are extremely complex (the terms in the above equation are tensorial in nature) and thus difficult to solve in their full form. Consequently, decennia of cosmological research have been focused on reducing the equations using restrictive conditions and symmetries. One very successful and highly popular simplification has been achieved by Alexander Friedmann in the 1920's by building on the works of Howard P. Robertson and Arthur Geoffrey Walker. The objective of this paper is therefore, in the first part, to walk through Robertson and Walker's results in a process to understand Friedmann's equation, as well as the acceleration and fluid equations, which are required in Friedmann Cosmology. This process will also allow to define common con- cepts in the field of cosmology, including curvature, metric, homogeneity and isotropy. In the second part, the article will systematically look at specific solutions to the Friedmann's equations that are known as "single-component universes". Note that this article is addressed to an audience assumed to have re- ceived formal education in differential equations, multivariable calculus and introductory astrophysics. 3 1 Tools of the Trade 1.1 Metrics of Surfaces with Constant Curvature Consider a two-dimensional surface; one of its defining property is, quite obviously, its curvature. The curvature could vary from point to point and, locally, the surface can either be flat, negatively curved or positively curved. For the moment, let the curvature be identical at all points on the surface. The flat and positively curved case can then easily be visualized as a plane and the surface of a sphere, respectively. The negatively curved case, however, is effectively impossible to represent accurately in three dimension, but can be approximated by imagining the middlepoint of a horse saddle. Thus, a uniformly flat surface is necessarily infinite, as, by travelling constantly in any given direction, it is impossible to return at the starting point. The same holds in the negatively curved case. A surface with constant positive curvature, on the other hand, will inevitably end up closing upon itself, which means that, by travelling in a constant direction, returning to the starting point becoms inevitable. Therefore, positively curved surfaces are by definition finite. A full awareness of the curvature is necessary when attempting to define distances on the surface. In general spaces, the distance is defined as the length of the geodesic, which is the shortest path between two points. On a plane, the geodesic is simply a straight line of a length determined by Pythagoras' theorem in a cartesian coordinate system, or by its equivalent in polar coordinates. However, when the surface is either positively or negatively curved, the notion of a "straight line" is meaningless and a different approach must be used. Thus, we rely on relations called "metrics" that define the distance ds between two points. On two-dimensional surfaces, the metrics defining the distance ds between the points (r; θ) and (r + dr; θ + dθ) in a polar coordinate system take the following form : κ = 0 ! ds2 = dr2 + r2dθ2 (2) r κ = +1 ! ds2 = dr2 + R2sin2( )dθ2 (3) R r κ = −1 ! ds2 = dr2 + R2sinh2( )dθ2; (4) R where κ is the curvature parameter, with values of 0; +1 or −1 corresponding to a flat, positive or negative curvature respectively, and R is the radius of curvature. 4 The mathematical concept of a metric can be extended to spaces with more than two dimensions. In the three-dimensional case, the metrics of constant curvature have infinitesimal distance ds that can be summarized as follows: 2 2 2 2 2 2 ds = dr + Sκ(r) (dθ + sin (θ)dφ ); (5) where 8 Rsin( r ) ; κ = +1 < R Sκ(r) = r ; κ = 0 (6) : r Rsin( R ) ; κ = −1: Then, by replacing the polar coordinates (r; θ; φ) by (x; θ; φ), such that 2 2 2 2 x ≡ Sκ(r), and if we substitute dΩ ≡ (dθ + sin (θ)dφ ), the metric of a three-dimensional space with constant curvature can be written under the general formula dx2 ds2 = + x2dΩ2: (7) x2 1 − κ R2 1.2 The Robertson-Walker Metric In the same train of thought, it is possible to establish a metric for space-time, which is a four-dimensional manifold composed of three spatial dimensions and one time dimension. This is precisely what Howard P. Robertson and Arthur Geoffrey Walker achieved independently for a spacetime which is homogeneous and isotropic − two properties that we will define shortly. But before studying their result, it is necessary that we clarify the notions of curvature and distance. Firstly, in the surfaces discussed in the previous section, curvature could be understood as a property emerging when considering the additional, higher dimension. The surface of a sphere, for instance, is effectively two-dimensional, as two coordinates are sufficient to describe any point on it. However, the surface could be thought of as going around the interior of the solid ball and thus enclosing a volume, which is a three-dimensional property. However, this need not be the case. Any general n-dimensional manifold can be defined without reference to a (n + 1)-dimensional space. Therefore, a curved four-dimensional space-time doesn't have to be embedded in a space with five or more dimensions. For all intents and purposes, curvature can be, and will be considered as an intrinsic property of space-time itself. The same 5 goes for the expansion of the universe, which will be discussed at length this paper. Secondly, the notion of distance between two points is a purely spatial one. It was appropriate in the previous section, but it becomes limited and incomplete when treating with a universe that evolves in time. Indeed, we need a way to distinguish between two events occuring at exact same point in space, but at different moments in time. Thus, we will substitute distance for separation, and make it account for time as well as three-dimensional space. Just like a two-dimensional surface, space-time can be locally flat, positively curved or negatively curved at any point in space. Then, the curvature of a region can evolve with time, especially if massive bodies such as galax- ies or planets move through it. Einstein's Field Equations can account for the complex interactions between curvature, matter and energy at all points, which is what makes them nearly impossible to solve in the general case and what warrants the efforts of cosmologists to apply restrictive conditions. One set of such restrictions is what made H. P. Robertson and A. G. Walker famous. They focused on the restricted set of universes which satisfy two very powerful conditions of symmetry: homogeneity and isotropy. Homogeneity is the property of a space which is identically the same at all points. In terms of curvature, this means that the curvature parameter κ and the radius of curvature R is constant for every point. An isotropic space is one that looks identically the same, no matter the direction in which you look. Essentially, these are the same curvature conditions as those imposed in section 1.1. But in this case, homogeneity and isotropy must not only hold for the topography of the universe, but also for the matter density and energy density. As a result, the form of a metric describing such a universe becomes much simpler and manageable. Notice that the immediate universe around us is far from uniform: the matter density in a star is obviously different from the matter density of the intergalactic dust, and the same holds for local curvature. However, at very large scales, namely around 100 Mpc and more, the the universe can reasonably be approximated as being homogeneous and isotropic. It is important to remember however, that the equations discussed in this paper cannot be interprated at the human or even the galactic scale. For instance, if a given model describes a universe expanding as a whole, it is fallacious to conclude that Earth is also slowly moving away from the Sun. 6 In addition to imposing those two symmety requirements, Robertson and Walker allowed the universes described to evolve in time, by expanding or contracting.

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