qr code reader c# .net Meanwhile, the conservation equation can be written as a = 3 t a in .NET

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Meanwhile, the conservation equation can be written as a = 3 t a
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This can be rearranged to give
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da d = 3 a ln = 3 ln a = ln a 3 where we are ignoring constants of integration to get a qualitative answer. The result is that for a matter-dominated universe a 3 we set = gives
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in the rst of the Friedmann equations listed in (12.11). This 3 k + a2 a2 8 a3
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Now multiply through by a 3 and divide by 3 to obtain a k + a2 1 3 8 a = 3 3
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We can rearrange this equation to obtain a relation giving the time variation of the scale factor with zero pressure: a2 = 1 2 8 a k+ 3 3a (12.14)
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The solutions of this equation give rise to different Friedmann models of the universe. Before we move on to consider these models in the next section, we brie y return to the conservation equation (12.12). We have just worked out a reasonable approximation to the present universe by considering the modeling of a matter-dominated universe by dust. Now let s consider the early and possible future states of the universe by considering a radiation-dominated universe and a vacuum-dominated universe, respectively. As the universe expands, photons get redshifted and therefore loose energy. Using the electromagnetic eld tensor, it can be shown that the equation of state (which relates the pressure to the density for a uid) for radiation is given by = 3P
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We can use this to replace P in (12.12) to obtain 1 a a = 3 ( + P) = 3 + t a a 3 Rearranging terms, we obtain da d = 4 a Following the procedure used for matter density, we integrate and ignore any constants of integration, which gives ln = 4 ln a = ln a 4 Therefore, we conclude that the energy density a 4 in the case of radiation. The density falls off faster than matter precisely because of the redshift we mentioned earlier. To close this section, we consider the vacuum energy density. The vacuum energy density is a constant, and therefore remains the same at all times. Since matter density and radiation energy density are decreasing as the universe expands but the vacuum energy density remains the same, eventually the universe will become dominated by vacuum energy. a = 4 a
Different Models of the Universe
We now turn to the problem of considering the evolution of the universe in time, which amounts to solving for the scale factor a (t). In this section we will be a bit sloppy from time to time, because we are interested only in the qualitative behavior of the solutions. Therefore, we may ignore integration constants etc. First we consider the very early universe which was dominated by radiation. In that case we use = 3P. For simplicity, we set the cosmological constant to zero and the Friedmann equations can be written as 3 k + a 2 = 8 2 a 1 8 a 2 + 2 k + a2 = a a 3
Using the rst equation to replace in the second, we obtain a 1 1 2 + 2 k + a2 = 2 k + a2 a a a Rearranging terms, we have a+ 1 k + a2 = 0 a
Cosmology
In the very early universe we can neglect the k/a term, which gives a+ a2 =0 a
This can be rewritten as a a + a 2 = 0. Notice that d (a a) = a a + a 2 dt and so we can conclude that a a = C, where C is a constant. Writing this out, we nd a da =C dt a da = C dt
Integrating both sides (and ignoring the second integration constant), we nd a2 = Ct 2 a (t) t As we will see later, this expansion is more rapid than the later one driven by matter (as dust). This is due to the radiation pressure that dominates early in the universe. For a complete examination of the behavior of the universe from start to nish, we begin by considering a simple case, the de Sitter model. This is a at model devoid of any content (i.e., it contains no matter or radiation). Therefore, we set k = 0 and the line element can be written as ds 2 = dt 2 a 2 (t) dr 2 a 2 (t)r 2 d 2 a 2 (t)r 2 sin2 d 2
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