Next Contents Previous


2.1. Cosmology for beginners

All of modern cosmology stems essentially from an application of the Copernican principle: we are not at the center of the universe. In fact, today we take Copernicus' idea one step further and assert the "cosmological principle": nobody is at the center of the universe. The cosmos, viewed from any point, looks the same as when viewed from any other point. This, like other symmetry principles more directly familiar to particle physicists, turns out to be an immensely powerful idea. In particular, it leads to the apparently inescapable conclusion that the universe has a finite age. There was a beginning of time.

We wish to express the cosmological principle mathematically, as a symmetry. To do this, and to understand the rest of these lectures, we need to talk about metric tensors and General Relativity, at least briefly. A metric on a space is simply a generalization of Pythagoras' theorem for the distance ds between two points separated by distances dx = (dx, dy, dz),

Equation 1 (1)

We can write this as a matrix equation,

Equation 2 (2)

where etai j is just the unit matrix,

Equation 3 (3)

The matrix etai j is referred to as the metric of the space, in this case a three-dimensional Euclidean space. One can define other, non-Euclidean spaces by specifying a different metric. A familiar one is the four-dimensional "Minkowski" space of special relativity, where the proper distance between two points in spacetime is given by

Equation 4 (4)

corresponding to a metric tensor with indices µ, nu = 0,..., 3:

Equation 5 (5)

In a Minkowski space, photons travel on null paths, or geodesics, ds2 = 0, and massive particles travel on timelike geodesics, ds2 > 0. Note that in both of the examples given above, the metric is time-independent, describing a static space. In General Relativity, the metric becomes a dynamic object, and can in general depend on time and space. The fundamental equation of general relativity is the Einstein field equation,

Equation 6 (6)

where Tµnu is a stress energy tensor describing the distribution of mass in space, G is Newton's gravitational constant and the Einstein Tensor Gµnu is a complicated function of the metric and its first and second derivatives. This should be familiar to anyone who has taken a course in electromagnetism, since we can write Maxwell's equations in matrix form as

Equation 7 (7)

where Fµnu is the field tensor and Jµ is the current. Here we use the standard convention that we sum over the repeated indices of four-dimensional spacetime nu = 0, 3. Note the similarity between Eq. (6) and Eq. (7). The similarity is more than formal: both have a charge on the right hand side acting as a source for a field on the left hand side. In the case of Maxwell's equations, the source is electric charge and the field is the electromagnetic field. In the case of Einstein's equations, the source is mass/energy, and the field is the shape of the spacetime, or the metric. An additional feature of the Einstein field equation is that it is much more complicated than Maxwell's equations: Eq. (6) represents six independent nonlinear partial differential equations of ten functions, the components of the (symmetric) metric tensor gµnu(t, x). (The other four degrees of freedom are accounted for by invariance under transformations among the four coordinates.)

Clearly, finding a general solution to a set of equations as complex as the Einstein field equations is a hopeless task. Therefore, we do what any good physicist does when faced with an impossible problem: we introduce a symmetry to make the problem simpler. The three simplest symmetries we can apply to the Einstein field equations are: (1) vacuum, (2) spherical symmetry, and (3) homogeneity and isotropy. Each of these symmetries is useful (and should be familiar). The assumption of vacuum is just the case where there's no matter at all:

Equation 8 (8)

In this case, the Einstein field equation reduces to a wave equation, and the solution is gravitational radiation. If we assume that the matter distribution Tµnu has spherical symmetry, the solution to the Einstein field equations is the Schwarzschild solution describing a black hole. The third case, homogeneity and isotropy, is the one we will concern ourselves with in more detail here [1]. By homogeneity, we mean that the universe is invariant under spatial translations, and by isotropy we mean that the universe is invariant under rotations. (A universe that is isotropic everywhere is necessarily homogeneous, but a homogeneous universe need not be isotropic: imagine a homogeneous space filled with a uniform electric field!) We will model the contents of the universe as a perfect fluid with density rho and pressure p, for which the stress-energy tensor is

Equation 9 (9)

While this is certainly a poor description of the contents of the universe on small scales, such as the size of people or planets or even galaxies, it is an excellent approximation if we average over extremely large scales in the universe, for which the matter is known observationally to be very smoothly distributed. If the matter in the universe is homogeneous and isotropic, then the metric tensor must also obey the symmetry. The most general line element consistent with homogeneity and isotropy is

Equation 10 (10)

where the scale factor a(t) contains all the dynamics of the universe, and the vector product dx2 contains the geometry of the space, which can be either Euclidian (dx2 = dx2 + dy2 + dz2) or positively or negatively curved. The metric tensor for the Euclidean case is particularly simple,

Equation 11 (11)

which can be compared to the Minkowski metric (5). In this Friedmann-Robertson-Walker (FRW) space, spatial distances are multiplied by a dynamical factor a(t) that describes the expansion (or contraction) of the spacetime. With the general metric (10), the Einstein field equations take on a particularly simple form,

Equation 12 (12)

where k is a constant that describes the curvature of the space: k = 0 (flat), or k = ±1 (positive or negative curvature). This is known as the Friedmann equation. In addition, we have a second-order equation

Equation 13 (13)

Note that the second derivative of the scale factor depends on the equation of state of the fluid. The equation of state is frequently given by a parameter w, or p = wrho. Note that for any fluid with positive pressure, w > 0, the expansion of the universe is gradually decelerating, addot < 0: the mutual gravitational attraction of the matter in the universe slows the expansion. This characteristic will be central to the discussion that follows.

Next Contents Previous