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This result is the same as for the flat Newtonian case (equation 20.21).

For positive curvature (k = +1) and negative curvature (k = —1), the results are mathematically different from the Newtonian case, and are also complicated to express. They do have similar characteristics to their Newtonian counterparts. Namely, the positive curvature case produces a universe that expands, reaches some maximum R, and then contracts. The negative curvature produces an expansion that lasts forever.

Models with non-zero cosmological constant are called Lemaitre models. As we have said, for the sake of simplicity, we consider the empty Lemaitre models. One such model is also shown in Fig. 20.9 these models are useful for any universe dominated by the cosmological constant. As, we said above, these universes will behave roughly

There are different ways in which we could define the distance, since we are dealing with objects whose separation changes between the time a photon is emitted at one galaxy and received in another. A convenient definition of distance in this case is that which we would associate with a distance modulus, m — M. This would tell us how to convert apparent brightnesses (or magnitudes) into absolute brightnesses (or magnitudes).

As light travels from a distant source, the observed brightness decreases as the photons from that source spread out on the surface of a sphere. Let the radius of that sphere be a. If the geometry of space-time is flat, the surface area of that sphere is 4^a2. So, the observed brightness falls as 1/a2, just as for light from nearby stars (Chapter 2). If the geometry is not flat, then

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