Cosmology, the branch of science dealing with the origin, evolution and structure of the Universe on large scales, is closely related to the study of galaxy formation and evolution. Cosmology provides not only the space-time frame within which galaxy formation and evolution ought to be described, but also the initial conditions for the formation of galaxies. Modern cosmology is founded upon Einstein’s theory of general relativity (GR), according to which the space-time structure of the Universe is determined by the matter distribution within it. This perspective on space-time is very different from that in classical physics, where space-time is considered eternal and absolute, independent of the existence of matter.
A complete description of GR is beyond the scope of this book. As a remedy, we provide a brief summary of the basics of GR in Appendix A and we refer the reader to the references cited there for details. It should be emphasized, however, that modern cosmology is a very simple application of GR, so simple that even a reader with little knowledge of GR can still learn it. This simplicity is owing to the simple form of the matter distribution in the Universe, which, as we have seen in the last chapter, is observed to be approximately homogeneous and isotropic on large scales. We do not yet have sufficient evidence to rule out inhomogeneity or anisotropy on very large scales, but the assumption of homogeneity and isotropy is no doubt a good basis for studying the observable Universe. If indeed the matter distribution in the Universe is completely homogeneous and isotropic, as is the ansatz on which modern cosmology is based, GR would imply that space itself must also be homogeneous and isotropic. Such a space is the simplest among all possibilities. To see this more clearly, let us consider a two-dimensional space, i.e. a surface. We all know that the properties of a general two-dimensional surface can be very complicated. But if the surface is homogeneous and isotropic, we are immediately reminded of an infinite plane and a sphere. These two surfaces differ in their overall curvature. The plane is flat, while the sphere is said to have a positive curvature. In both cases the distance between any two infinitesimally close points on the surface can be written as
where 
for a plane and 
for a sphere. In the case of a plane, 
are just the polar coordinates and a is a length scale (scale factor) relating the coordinate radius 
to distance. To see that corresponds to a sphere, we make the coordinate transformation 
In terms of 
, the distance measure becomes 
which is clearly that of a sphere in terms of the spherical coordinates, with a being the radius of the sphere. In this case, is a spherical coordinate in the three-dimensional space in which the two-dimensional surface is embedded; r is not a distance measure on the surface, but rather a coordinate used to label positions on the surface. Actual distances have to be computed from the metric (1). Only in the case with is r both coordinate and distance measure.
Mathematically it can be shown that there is another two-dimensional homogeneous and isotropic surface for which 
. Changing 
, we can write the distance measure on such a surface as 
where the factor a is again a length scale relating coordinates to distance. This negatively curved, hyperbolic surface, which is locally similar to the surface near a saddle point, is not very familiar to us because it cannot be embedded in a three-dimensional Euclidean space. The existence of a low-dimensional 'space' which cannot be embedded in a space of higher dimensionality is, however, not as strange as it might seem; for example it is easy to envision that it is impossible to embed a hairspring (an intrinsically one-dimensional object) into a plane.
These examples show that the description of a homogeneous and isotropic two-dimensional surface is extremely simple. What we need to do is just to determine the value of 
, which specifies the global geometry of the surface, and the scale factor a, which relates coordinates to distances. In general, the scale factor a can change with time without violating the requirement of homogeneity and isotropy, corresponding to a surface that is uniformly expanding or contracting.
The above discussion can be extended to three-dimensional spaces..., a homogeneous and isotropic space is also completely determined by the curvature signature 
(again equal to 
), which determines the global geometry of the space, and the scale factor a(t) as a function of time. Thus, as far as the space-time geometry is concerned, the task of modern cosmology is simply to determine the value of K and the functional form of a(t) from the matter content of the Universe .
According to GR, the relationships among cosmological events are assumed to be governed by the physical laws that we are familiar with, while the effects of gravity are included in the properties of the space-time (i.e. in the transformations of reference frames). This equivalence principle (that a local gravitational field can be transformed away by choosing an appropriate frame of reference) allows one to derive physical equations in GR from their ordinary forms by general coordinate transformations (see Appendix A). Hence, once the value of K and the functional form of 
are known, the relationships among cosmological events can be described in terms of physical laws. Similarly, if we believe that physical laws are applicable on cosmological scales, the predictions for these relationships will depend only on the space-time geometry and so observations of such relationships can be used to test cosmological models.
One of the most important observations in cosmology is that the Universe is expanding [i.e. a(t) increases with time], which implies that it must have been smaller in the past. Together with the observational fact that our Universe is filled with microwave photons, this time evolution of the scale factor determines the thermal history of the Universe. Because the Universe was denser in the past, it must also have been hotter. Since high density and temperature imply high probabilities for particles to collide with each other with high energy, the early Universe is an ideal place for the creation and transmutation of matter..., the applications of particle, nuclear and atomic physics to the thermal history of the early Universe lead to important predictions for the current matter content of the Universe. Although many of these predictions are still uncertain, they provide the basis for calculating relations between the dominant mass components of the Universe...
