Cosmology in .NET framework

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Cosmology
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this corresponds to positive curvature, while k = 1 corresponds to negative curvature, and k = 0 is at. We consider each of these cases in turn.
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Spaces of Positive, Negative, and Zero Curvature
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With the normalized curvature k there are three possibilities to consider: positive, negative, and zero curvature. To describe these three surfaces, we write (12.5) in the more general form d 2 = d 2 + r 2 ( ) d 2 + r 2 ( ) sin2 d 2 (12.6)
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A space with positive curvature is speci ed by setting k= 1 in (12.5) or by setting r ( ) = sin in (12.6). Doing so we obtain the metric d 2 = d 2 + sin2 ( ) d 2 + sin2 ( ) sin2 d 2 In order to understand this space, we examine the surface we obtain if we set to some constant value: we take = /2. The surface then turns out to be a two sphere (see Fig. 12-2). The line element with = /2 is d 2 = d 2 + sin2 ( ) d 2
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Fig. 12-2. An embedding diagram for a space of positive curvature. We take = /2,
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for which the two surface is a sphere.
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Cosmology
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A three-dimensional space that has constant curvature has two analogies with the surface of a sphere. If we start at some point and travel in a straight line on the sphere, we end up at the same point eventually. This would also be true moving in the three-dimensional space of a universe with positive curvature. Second, if we add up the angles of a triangle drawn on the surface, we would nd that the sum was greater than 180 . Next we consider a space of negative curvature, which means that we take k = 1. In this case, we set r = sinh and the line element becomes d 2 = d 2 + sinh2 ( ) d 2 + sinh2 ( ) sin2 d 2 When we use this as the spatial line element for the universe dt 2 a 2 (t) d 2 , spatial slices have the remarkable property that they have in nite volume. In this case, the sum of the angles of a triangle add up to less than 180 . Once again considering = /2 in order to obtain an embedding diagram, we obtain d 2 = d 2 + sinh2 ( ) d 2 The embedding diagram for a surface with negative curvature is a saddle (see Fig. 12-3). Finally, we consider the case of zero curvature for which k = 0. It turns out that current observations indicate that this is the closest approximation to the real universe we live in. In this nal case, we set r = and we can write the line element as d 2 = d 2 + 2 d 2 + 2 sin2 d 2
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Fig. 12-3. A surface of negative curvature is a saddle.
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Cosmology
Fig. 12-4. When k = 0, space is perfectly at.
which is nothing but good old at Euclidean space. Setting = /2, we obtain a at plane for the embedding diagram (see Fig. 12-4).
Useful De nitions
We now list several de nitions that you will come across when reading about cosmology.
THE SCALE FACTOR
The universe is expanding and therefore its size changes with time. The spatial size of the universe at a given time t is called the scale factor. This quantity is variously labeled R (t) and a (t) by different authors. In this chapter we will denote the scale factor by a (t). Observation indicates that the universe is expanding as time moves forward and therefore a (t) > 0.
THE GENERAL ROBERTSON-WALKER METRIC
The Robertson-Walker metric is de ned by ds 2 = dt 2 a 2 (t) d 2 where d 2 is given by one of the constant curvature metrics described by (12.6) in spaces of positive, negative, and zero curvature.
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