Cosmology in .NET

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Cosmology
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The fact that this models a universe that contains no matter or radiation means that we set P = = 0. However, we leave the cosmological constant in the equations. So while this is a toy model, it can give an idea of behavior in the very late history of the universe. Since the expansion of the universe appears to be expanding and the matter and radiation density will eventually drop to negligible levels, in the distant future the universe may be a de Sitter universe. With these considerations, (12.11) becomes 3 2 a =0 a2 a 1 2 + 2 a2 = 0 a a Obtaining a solution using the rst equation is easy. We move the cosmological constant to the other side and divide by 3 to get a2 = a2 3 Now we take the square root of both sides 1 da = a dt This can be integrated immediately to give a (t) = C e
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(12.15)
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where C is a constant of integration that we won t worry about. We are interested only in the qualitative features of the solution, which can be seen by plotting. For the de Sitter universe, (see Fig. 12-5) the line element becomes ds 2 = dt 2 a 2 (t) dr 2 e
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r d 2 a 2 (t)r 2 sin2 d 2
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Now let s move on to a universe that contains matter. Direct solution of the Friedmann equations is in general dif cult, and basically requires numerical analysis. Since recent observations indicate that the universe is at (therefore k = 0), we don t lose anything by dropping k. With this in mind, let s consider a universe that contains matter, but we set k = 0 = . We can obtain a solution
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Cosmology
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Fig. 12-5. The de Sitter solution represents a universe without matter. As time increases
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the universe expands exponentially.
by using (12.14). Setting k = 0 =
, we have a2 = 8 3a
Rearranging terms and taking the square root of both sides, we have da 8 = a dt 3 8 dt a da = 3 Integrating on the left side, we have 2 a da = a 3/2 3 while on the right, ignoring the integration constant (we are interested only in the qualitative features), we have 8 /3 t. This leads to a (t) t 2/3 (12.16)
A plot of this case is shown in Fig. 12-6. This describes a universe that evolves with a continuous expansion and a deceleration parameter given by q = 1/2. Remember we found that the radiation-dominated early universe had a (t) t. We can also consider cases with positive and negative curvature. Adding more nonzero terms makes things dif cult, so we proceed with the positive curvature
Cosmology
Fig. 12-6. A at universe that contains matter but with zero cosmological constant.
case and leave the cosmological constant zero. This means we set k = +1 this describes a universe which collapses in on itself, as shown in Fig. 12-7. In this case, we have a2 = 8 C a 1= 3a a (12.17)
where we have called 8 /3 = C for convenience. This equation can be solved parametrically. To obtain a solution, we de ne a = C sin2 where = (t). Then we have d da = 2C sin cos d dt Squaring, we nd that a 2 = 4C 2 sin2 cos2 Using these results in (12.17), we obtain 4C 2 sin2 cos2 d dt
(12.18)
d dt
cos2 C C sin2 = C sin2 sin2
Canceling terms on both sides, we arrive at the following: 2C sin2 d = dt
Cosmology
Fig. 12-7. Dust- lled universe with zero cosmological constant, and positive curvature.
After expanding to a maximum size, the universe reverses and collapses in on itself.
Integrating, the right side becomes 2C sin2 d = 2C = 1 cos 2 1 d = C sin 2 2 2
C (2 sin 2 ) 2
We can use the same trig identity used in the integral to write (12.18) as a= C (1 cos 2 ) 2
These equations allow us to obtain a (t) parametrically. At t = = 0, the universe begins with size a (0) = 0, i.e., at the big bang with zero size. The radius then increases to a maximum, and then contracts again to a (0) = 0. The maximum radius is the Schwarzschild radius determined by the constant C. We leave the case k = 1 as an exercise. Unfortunately, because of limited space our coverage of cosmology is very limited. We are leaving out a discussion of the big bang and in ation, for example. For a detailed overview of the different cosmological models, consider D Inverno (1992). To review a discussion of recent observational evidence and an elementary but thorough description of cosmology, examine Hartle (2002). For a complete discussion of cosmology, consult Peebles (1993).
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