Rick's Cosmology Tutorial: Chapter 5B Abstract

Cosmic Geometry: Part 1

Cosmological models are usually based on the assumption that the universe in-the-large is homogeneous and isotropic. The formulation of the general relativistic gravitational field equations is presented. The general form of homogeneous, isotropic cosmological solution is stated (the Friedmann-LeMaitre-Robertson-Walker, FLRW, spacetimes). The generic behaviour of such a universe is not static. In most cases the solutions have a singular origin with divergent initial 'velocity', dR/dt, i.e. a Big Bang occurs.

In the case when pressure is negligible, and there is no cosmological constant, it is shown that the expansion of the universe in a FLRW cosmology, i.e. the behaviour of R as a function of time, can be interpreted in a Newtonian manner. In these cases the closed form solutions for R(t) are given explicitly. There are two types of solution: those that are spatially infinite and will expand forever, and those that are spatially finite and will ultimately recontract. In the relativistic interpretation, the latter correspond to 3D spaces of positive intrinsic curvature (k>0) which can be embedded as a hypershere in a hypothetical 4D Euclidean space. The spatially infinite solutions may either be flat (k=0) or have negative intrinsic curvature (k<0). The latter cannot be embedded in a higher dimensional Euclidean space. The solutions with k>0, k=0, k<0 relate to the density being respectively greater than, equal to, or less than the critical density.

In the case of non-zero cosmological constant, as now seems appropriate for our universe, the solutions for R(t) in FLRW cosmologies are described in qualitative terms. A negative cosmological constant would manifest itself as an additional attractive gravitational force, becoming larger at large distances. Such universes would always be subject to recontraction, whatever their spatial curvature. Thus, the identification of recontraction with positive curvature ceases to hold for non-zero cosmological constant. It is possible to envisage spatially infinite universes (with k<0 or k=0) which recontract after a finite time.

However, a positive cosmological constant currently appears more likely to describe our universe. This corresponds to a repulsive form of gravity which increases in magnitude with increasing distance. Such a universe will expand forever if it is spatially flat or negatively curved, ultimately at an exponentially increasing rate. However, even a space of positive curvature (k>0) may expand forever, provided that the cosmological constant is sufficiently large compared with k.

More complex behaviours are possible for k>0 and a positive, but sufficiently small, cosmological constant. Amongst these are the earliest solutions discovered, and associated with the names Einstein, de Sitter and LeMaitre. The static Einstein universe is one of them, but is pathological and unstable. These solutions are unlikely to be appropriate for our universe.

Our universe appears to have a positive cosmological constant and a curvature parameter, k, consistent with zero. Frustratingly, it seems that if k is exactly zero, then measurements will always be consistent with negative and positive curvature as well. Measurements will never reveal whether a perfectly flat universe might not be slightly curved one way or the other. Consequently, we may never know if the universe is spatially finite or infinite.

In a universe with non-zero cosmological constant, the correspondence between k>0, k=0, k<0 and whether the matter density is greater than, equal to, or less than the critical density no longer holds. This correspondence persists if an effective contribution of the cosmological constant to the density is included. Thus, in our universe it appears that the total matter density, including dark matter, is not more than ~30% critical. Consistency with a flat geometry (k~0) prevails because the cosmological constant accounts for the remaining ~70% of the critical density.

For all cosmological solutions, the curvature parameter (k) is a constant of the motion. Hence, the universe is either born finite and remains finite for all finite times, or is born infinite and remains infinite forever.

Perhaps the most important contribution of relativity theory to cosmology is that it provides an elegant solution to every child's question, "what lies beyond the edge of the universe?" The concept of a space of positive intrinsic curvature allows space to be both finite and yet without an edge. But this solution, elegant though it is, may not be needed. The universe may actually be infinite.

Finally, in this Chapter we show how to calculate the age of the universe in terms of the Hubble parameter. In Chapter 2 we noted that the age of the early, radiation dominated, universe was just half the reciprocal of the Hubble parameter. In the present epoch the age of the universe turns out to be very close to just the reciprocal of the Hubble parameter. We show why, and how this would vary with different cosmological models.

Read Chapter 5B (pdf): Cosmic Geometry Part 1

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Pandora, one of the smaller of Saturn's currently known 19 moons (taken by the Cassini probe)