Cosmology in .NET framework

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Cosmology
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A at universe is described by Euclidean geometry on large scales and will expand forever. In this case, k = 0 (see Fig. 12-4).
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Whether or not the universe is open or closed is determined by the density of stuff in the universe. In other words, is there enough matter, and therefore enough gravity, to slow down the expansion enough so that it will stop and reverse If so, we would live in a closed universe. The density required to have a closed universe is called the critical density. It can be de ned in terms of the Hubble constant, Newton s gravitational constant, and the speed of light as c =
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(12.8)
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At the present time, the ratio of the observed density to the critical density is very close to unity, indicating that the universe is not closed. However, keep in mind the uncertainty in the Hubble constant.
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This is the ratio of the observed density to the critical density. = 8 G = 2 c 3H0 (12.9)
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The density used here is obtained by adding contributions from all possible sources (matter, radiation, vacuum). If < 1, this corresponds to k < 0 and the universe is open. If = 1 then k = 0 and the universe is at. Finally, if > 1 then we have k > 0 and a closed universe. As we indicated above, it appears that 1. =
The Robertson-Walker Metric and the Friedmann Equations
To model the large-scale behavior of the universe such that Einstein s equations are satis ed, we begin by modeling the matter and energy in the universe by
Cosmology
a perfect uid. The particles in the uid are galaxy clusters and the uid is described by an average density and pressure P. Moreover, in co-moving coordinates, t = 1 and xi = 0 giving u a = (1, 0, 0, 0). Therefore, we set 0 = 0 0 0 0 0 P 0 0 0 P 0 0 0 P
T ab
(12.10)
Using the metric a 2 (t) dr 2 a 2 (t)r 2 d 2 a 2 (t)r 2 sin2 d 2 ds = dt 2 1 kr
we can use the metric to derive the components of the curvature tensor in the usual way. This was done in Example 5-3. From there we can work out the components of the Einstein tensor and use Einstein s equation to equate its components to the stress-energy tensor. We remind ourselves how Einstein s equation relates the curvature to the stress-energy tensor: G ab gab = 8 Tab
The details were worked out in Example 7-3. Note we used a different signature of the metric in that example. For the signature we are using in this case, the result is found to be 3 k + a 2 = 8 2 a a 1 2 + 2 k + a 2 = 8 P a a
(12.11)
We can augment these equations by writing down the conservation of energy equation using the stress-energy tensor (see 7): a T a t = a T a t +
a b ab T t
a at T b
Since the stress-energy tensor is diagonal, this simpli es to a T a t +
a b ab T t
a at T b
= t T t t + t tt T t t + t tt T t t
t rt T t + r r rt T r r
t t t T t + t T t t T t T
Cosmology
In the chapter Quiz you will derive the Christoffel symbols for the RobertsonWalker metric. The terms showing up in this equation are given by
t r tt rt
=0 =
t
a a
Using this together with T t t = and T r r = T = T = P, we have
r t rt T t
and
r r rt T r
a a a r t t t rt T t + t T t + t T t = 3 a =
t t T t
t t T t
t T
t T
a a a a = ( P) ( P) ( P) = 3 P a a a a
Therefore, the conservation equation becomes a = 3 ( + P) t a (12.12)
This is nothing more than a statement of the rst law of thermodynamics. The volume of a slice of space is given by V a 3 (t) and the mass energy enclosed in the volume is E = V . Then (12.12) is nothing more than the statement dE + P dV = 0. In the present matter-dominated universe, we can model the matter content of the universe as dust and set the pressure P = 0. In this case, the second equation of (12.11) can be written as a 1 2 + 2 k + a2 a a =0 (12.13)
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