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By David H. Lyth, Andrew R. Liddle
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Additional info for The Primordial Density Perturbation: Cosmology, Inflation and the Origin of Structure
Example text
It is believed that these structures have formed over time through gravitational attraction, from a distribution that was more homogeneous in the past. It is convenient then to break up the dynamics of the observable Universe into two parts. The large-scale behaviour of the Universe can be described by assuming a homogeneous and isotropic background. On this background, we can superimpose the short-scale irregularities. For much of the evolution of the observable Universe, these irregularities can be considered to be small perturbations on the evolution of the background (unperturbed) Universe.
1 Scale factor To arrive at the form of the Robertson–Walker metric, we first choose the time coordinate so that spacetime slices of fixed t are homogeneous and isotropic. In other words, physical conditions on each slice are the same at every position, and in every direction. We choose the threading to be orthogonal to the slicing, corresponding to g0i = 0. Isotropy requires that an observer moving with the threading measures zero velocity for the cosmic fluid, or in other words zero momentum density.
4 Fluid dynamics For the rest of this chapter we look at the relativistic description of fluids. A fluid is a medium whose properties can be regarded as smooth functions of spacetime position. 1 Velocity field The 4-velocity uµ (x, t) of a fluid element defines the fluid flow. The 4-acceleration can be written a µ ≡ u ν ∂ν u µ . 4 Fluid dynamics 15 To see this one can use the chain rule. More elegantly, one can just notice that it is valid in the local rest frame of each fluid element and hence (being a covariant expression) is valid in any inertial frame for the whole fluid.