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Cosmology and general relativity

Oscillating universe. In this picture t0 is the current time.The universe is the single cycle that contains t0. If the universe is oscillating, then, after all the material comes back together, the expansion starts again.

~ 1 X 10—29 g/cm3. In the final section of this chapter, we will discuss observations that can determine the actual value of O.

There is a final point to consider if the universe turns out to be closed. After the expansion stops, a collapse will start. Eventually, all of the matter will come together into a dense, hot state for the first time since the big bang. Some people have taken to calling this event the big crunch. It is natural to ask what will happen after the big crunch. It has been suggested that the universe might reach a high density and then bounce back, starting a new expansion phase. If this can happen, then it might happen forever into the future, and might have happened for all of the past, as indicated in Fig. 20.5. Such a universe is called an oscillating universe.

If our universe turns out to be closed, can we tell if it is oscillating? Some theoreticians have argued that the big crunch/big bang in an oscillating universe strips everything down to elementary particles, and therefore destroys all information on what has come before. Others have argued that there are certain thermody-namic properties of the universe that might tell us if it is oscillating. Others have taken a wait-and-see attitude, pointing out that we will need a quantum theory of gravity to understand the densest state that is reached.

### 20.4.1 Geometry of the universe

When Einstein developed the general theory of relativity, he realized that it should provide a correct description of the universe as a whole. Einstein was immediately confronted with a result equivalent to equation (20.17), which says that if the density is not zero, the universe must be expanding or collapsing. This was before Hubble's work, and most believed in a static (steady-state) universe.

To get around this problem, Einstein introduced a constant, called the cosmological constant, A, into general relativity. It had no measurable effect on small scales, but altered results on cos-mological scales. For example, in equation (20.17) the effect of the cosmological constant would be to replace the density p by p — A/8-^G. This makes it possible # to have a non-zero density, but a zero value for R. Einstein withdrew the cosmological constant when he heard of Hubble's work, declaring the cosmological constant to be his biggest mistake. However, theoreticians have tended to keep it in the theory, and then formally set it to zero, or consider models with a non-zero A. The best determinations of various cosmological parameters, discussed below and in the next chapter, suggest that A may have a non-zero value.

Following Einstein's work, a number of people worked out cosmological theories, using different simplifying assumptions. The models are generally named after the people who developed them. The de Sitter models are characterized by k = 0 and a non-zero (positive) cosmological constant; the Friedmann models have a zero cosmo-logical constant and also zero pressure (a good approximation at low density); Lemaitre models have non-zero density and a cosmological constant. As a result of his work, Lemaitre noted that (independent of the value of A) there must have been a phase in its early history when the universe was very hot and dense. This phase is called the big bang.

Many of the general relativistic results are similar to those we obtained in the previous section. This is because both depend on the fact that, in a spherically symmetric mass distribution, matter outside a sphere has no effect on the evolution of matter inside the sphere. One modification is the replacement of p by p — A/8vG if you want a non-zero cosmological constant.

However, the general relativistic approach gives us a deeper insight by providing a geometric interpretation of the results. For example, the space-time interval, in spherical coordinates, becomes

(This is sometimes called the Robertson-Walker metric.) In this equation, R(t) has the same meaning as before, and r, 0 and p are the usual spherical coordinates of objects such as galaxies. In cosmology, it is important to use what is known as a co-moving coordinate system. This system expands with the universe. The form of this metric ensures that the models are homogeneous and isotropic (the cosmological principle). (Note that in books on general relativity, authors sometimes use different symbols: the scale factor is written as a(t), and k becomes 1/R2, where R is the "radius of curvature" of the universe. These books also often take a system of units in which c = 1, so it does not appear explicitly in equations.)

In general relativity, whether the universe is open, closed or on the boundary tells us something about the geometry of space-time. That information is contained in the constant, k, which has the value zero for the boundary, +1 for a closed universe, and —1 for an open universe. However, k now tells us something about the curvature of space-time (Fig. 20.6). If k = 0, spacetime is flat (Euclidean). The sums of the angles of

Schematic representations showing the relationship between cosmological model and the geometry of space-time.The top graph shows an open model, with negative curvature (like a saddle).The middle graph shows a universe that will expand forever, but is on the boundary, and the geometry is flat The bottom graph shows a closed model, and the geometry has a positive curvature. [© Edward L.Wright, used with permission]

Schematic representations showing the relationship between cosmological model and the geometry of space-time.The top graph shows an open model, with negative curvature (like a saddle).The middle graph shows a universe that will expand forever, but is on the boundary, and the geometry is flat The bottom graph shows a closed model, and the geometry has a positive curvature. [© Edward L.Wright, used with permission]

triangles are always 180°. In this case space is infinite. If k = +1, space-time has a positive curvature, like the surface of a sphere. The sums of the angles of triangles are always greater than 180°. Space must be finite, just as the surface of a sphere is finite. Finally, if k = —1, we say that space-time has a negative curvature. The sums of the angles of triangles are always less than 180°. In this case, space is infinite. The relationship between geometry and the type of cosmological model is summarized in Table 20.1.

We can get a feel for the geometry of the universe by considering a two-dimensional analogy.

Table 20.1. |
Parameters of various cosmological models. |

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