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Cosmology in .NET framework
Cosmology QR Code Reader In Visual Studio .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Print QR Code ISO/IEC18004 In VS .NET Using Barcode encoder for .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications. FLAT UNIVERSE
Scanning Quick Response Code In VS .NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET applications. Bar Code Encoder In .NET Using Barcode printer for .NET Control to generate, create bar code image in Visual Studio .NET applications. A at universe is described by Euclidean geometry on large scales and will expand forever. In this case, k = 0 (see Fig. 124). Barcode Reader In .NET Framework Using Barcode scanner for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Painting QR Code In Visual C# Using Barcode creator for VS .NET Control to generate, create Quick Response Code image in .NET framework applications. CRITICAL DENSITY
Creating QR Code In .NET Using Barcode drawer for ASP.NET Control to generate, create QR image in ASP.NET applications. QR Code 2d Barcode Encoder In VB.NET Using Barcode creator for .NET framework Control to generate, create QR Code 2d barcode image in .NET applications. 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 = Creating UCC  12 In Visual Studio .NET Using Barcode generation for .NET framework Control to generate, create Universal Product Code version A image in VS .NET applications. Paint Bar Code In .NET Using Barcode creator for .NET Control to generate, create barcode image in .NET framework applications. 2 3H0 8 G
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EAN 13 Creator In Java Using Barcode creator for Java Control to generate, create EAN / UCC  13 image in Java applications. Code128 Drawer In Java Using Barcode generator for Java Control to generate, create Code 128B image in Java applications. This is the ratio of the observed density to the critical density. = 8 G = 2 c 3H0 (12.9) EAN13 Encoder In None Using Barcode maker for Online Control to generate, create EAN13 image in Online applications. Recognizing USS Code 128 In VB.NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET applications. 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 RobertsonWalker Metric and the Friedmann Equations
To model the largescale 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 comoving 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 53. From there we can work out the components of the Einstein tensor and use Einstein s equation to equate its components to the stressenergy tensor. We remind ourselves how Einstein s equation relates the curvature to the stressenergy tensor: G ab gab = 8 Tab The details were worked out in Example 73. 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 stressenergy tensor (see 7): a T a t = a T a t + a b ab T t
a at T b
Since the stressenergy 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 matterdominated 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)

