In the expanding Universe, a comoving volume element expands along with the cosmological flow, getting physically larger over time.

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1 Cosmological models In the expanding Universe, a comoving volume element expands along with the cosmological flow, getting physically larger over time. The expansion is described by the scale factor R(t). If the of the volume element is r, then its physical size at time t is R(t)r. Cosmological models describe the way R varies with t. It is one of the big successes of cosmology that the scale factor calculated from general relativity assuming space that is isotropic and homogenous is approximately consistent with that observed.

2 Steady-state theory One of the first theories describing the expanding Universe was the steady-state theory. The density of the Universe was kept constant in this theory by continuous creation of matter. The expansion rate would then be related to the matter creation rate. The matter is created at a very small rate, one atom per cc per 10 billion years. This would be far too small to measure. This theory does not require a hot big bang. Energy conservation is being violated continuously in a very small way in the steady-state theory but is violated in a large way at one instant.

3 Problems with the steady-state model The steady-state theory was popular in the 1950s and 1960s, but is generally discounted because it cannot predict the relative abundances of the different elements. Predicting the observed initial helium abundance is one of the major successes of the Hot Big Bang Theory. The discovery of quasars in 1963 was also a problem for the steady-state model. The model could not explain why they are preferentially seen at high redshift.

4 Coordinate systems A coordinate system uses one or more numbers to uniquely determine the position of a point.

5 Scalars and Vectors Vectors are expressed differently in different coordinate systems. (1,1,1) means a completely different thing in Cartesian coordinates to in spherical polar coordinates

6 Tensors

7 Tensor Contraction Tensors contain a great deal of information, and often much if it is redundant. The important information can be extracted by contracting the tensor to a smaller number of dimensions. A two-dimensional vector can be contracted to a one-dimensional number by taking a scalar product. An example is pressure which is in reality a three-dimensional tensor to give the familiar scaler quantity pressure, as in atmospheric pressure or tyre pressure.

8 Metric The metric is a tensor relating a physical entity to the coordinate system. Here is the metric for Cartesian coordinates in three dimensions. In special relativity, the metric is used to describe events in space-time. The negative sign in the diagonal time component follows because time operates in a different way to the spatial coordinates and can be represented using complex numbers. A powerful feature of the metric is that the components can depend on the coordinates themselves. This is important for general relativity where space-time is curved. An example is the Schwarzchild metric which describes space near a black hole.

9 Schwarzchild metric The quantity r = 2GM/c 2 results in a singularity is called the event horizon or Schwarzchild radius of a black hole. The Schwarzchild radius of the Earth is 8.7 mm.

10 Riemannian geometry

11 Calculus

12 Robertson-Walker metric The metric describing the Universe must be isotropic and homogeneous. Isotropy requires the diagonal elements of the Riemannian curvature tensor are equal. Homogeneity requires that the fully contracted Riemannian curvature tensor is a constant. The metric satisfying these constraints is the Robertson-Walker tensor.

13 General Relativity Left side of equation depends on geometry Right side of equation depends on mass

14 Robertson-Walker metric ( ) dr ds 2 = R(t) kr c 2 dt 2 General Relativity 16 equations R 1 Rg = 8πGT 2 Equation of State Relates pressure experienced due to space In a matter dominated Universe P = 0 For dark energy P = w ρ Putting the metric and GR together leads to the cosmological field equations Ṙ 2 + k = 8πG 3 ρr2

15 Combining with an equation of state leads to an expansion law. A matter dominated Universe with k = 0 (Einstein de Sitter) has a simple expansion law R t 2 3 What we actually measure is not R but the luminosity distance which is an interal over time depending on R(t) The cosmological constant can be introduced either as a term in the GR equations or as a component with the equation of state w = 1.

16 Scale Factor in Cosmological Models The curvature k depends on the density of the Universe in a matter dominated Universe. If the density is greater than the critical density, the curvature is positive. If the density is lower than the critical density, the curvature is negative.

17 de Sitter Universe In a de Sitter Universe, the scale factor increases exponentially with time. R ~ exp(ht) During inflation, the scale factor goes through about 60 e-foldings.

18 Measurements We cannot directly measure the scale factor. Instead we measure the Hubble constant, the rate of change of the scale factor divided by the scale factor. The Hubble constant only gives information about the scale factor at the previous time. Information about the time variation of the scale factor comes from measurements of the luminosity distance. The luminosity distance is normally expressed as a distance modulus. m - M = 5 log(d/10 pc)

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