Rick's Cosmology Tutorial: Chapter 5C Abstract

Cosmic Geometry: Part 2

In this second part of 'Cosmic Geometry' we continue to derive the basic geometrical consequences of general relativistic cosmologies. Various different distance measures are explained. In particular it is explained why the observable universe is larger than the velocity of light times the age of the universe. It is shown how the factor R/ct can be calculated. The size of the observable universe is deduced using current consensus values for the cosmological parameters.

The boundary of the observable universe is called the horizon. It is shown that, looking backwards in time, the horizon encompasses an ever decreasing amount of matter as the Big Bang is approached. In the limit, all particles of matter were causally isolated at sufficiently early times. This is paradoxical since the high degree of isotropy of the cosmic microwave background implies that the universe was in thermal equilibrium at early times. This is known as the Horizon Problem.

The red shift is defined and its relationship to the various distance measures is derived. The red shift is divergent at the horizon.

The Hubble relationship between distance and recessional speed is derived. The distance measure, R, to employ in the Hubble relation is discussed. It is explained in what sense the recessional 'speed' can exceed the speed of light.

The contribution of pressure in causing gravitation within relativity theory is derived. Hence the relativistic formulation of radiation as a source of gravitation is deduced. The relevance of this in the early, radiation dominated, universe is discussed. The interpretation of the cosmological constant as a positive energy density plus a negative pressure is explained. Hence it is shown why a positive cosmological constant produces a net repulsive gravtitation.

Finally, the possibility that the universe has a non-trivial topology is discussed briefly. Thus, the universe could appear to be larger than it actually is - like a room full of mirrors.

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Carina Nebula NGC3372: Emission Nebula in the Milky Way (N. Smith, 2007 / Hubble, NASA)