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Unformatted text preview: Chapter 11 History of the Universe Knowing how the scale factor a ( t ) grew in the past, and predicting how it will behave in the future, is an important goal of cosmologists. The Friedmann equation tells us that the growth of the scale factor is related to the energy density of the universe. It is useful to divide the energy content into radiation (relativistic particles), matter (non-relativistic particles), and a cosmological constant. This is because each of these components has an energy density with a different dependence on the scale factor. A cosmological constant has an energy density u Λ that is constant with time. To see how the energy density of radiation and matter behaves as the universe expands, consider a volume V that expands with the universe, so that V ( t ) ∝ a ( t ) 3 . If particles are neither created nor destroyed, then the number density of particles, n , is diluted by the expansion of the universe at the rate n ( t ) ∝ V ( t )- 1 ∝ a ( t )- 3 , as illustrated in Figure 11.1. The energy of the non-relativistic particles is contributed entirely by their rest mass, ε = mc 2 , which remains constant as the universe expands. Thus, for non- relativistic particles, alias “matter”, the energy density has the dependence u m ( t ) = n ( t ) ε = n ( t ) mc 2 ∝ a ( t )- 3 . (11.1) The energy of relativistic particles, such as photons, has the dependence ε ( t ) = hc/λ ( t ) ∝ a ( t )- 1 . Thus, for relativistic particles, alias “radiation”, the energy density has the dependence u r ( t ) = n ( t ) ε ( t ) = n ( t ) hc/λ ( t ) ∝ a ( t )- 4 . (11.2) Given the different rates of decrease for the energy density, we find that the total energy density u was contributed mainly by radiation at early times, 254 11.1. THE CONSENSUS MODEL 255 Figure 11.1: Dilution of non-relativistic particles (“matter”) and relativistic particles (“radiation”). when a ¿ 1 (Figure 11.2). In the language of cosmologists, the early uni- verse was “radiation dominated”. If the universe has a positive cosmological constant Λ, then it becomes “lambda dominated” if it reaches a sufficiently large scale factor. 11.1 The Consensus Model In recent years, cosmologists (ordinary a contentious bunch) have found themselves approaching an approximate consensus on the curvature, con- tents, and age of the universe. The curvature is flat (or nearly so), implying that the energy density today is close to the critical density u ≈ u c, ≈ 5200 MeV m- 3 . To see how this energy density is allocated among the differ- ent components, let’s do a census of the universe. Most of the energy density of photons is provided by the Cosmic Mi- crowave Background; although stars have been shining away for ∼ 13 Gyr, starlight still provides less than 10% of the total photon energy of the uni- verse....
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- Winter '06
- Astronomy, Big Bang, Energy density, Consensus model