Perturbation Theory

Einstein's Theory of General Relativity is highly non-linear. It is therefore very difficult to deal with all but the simplest situations using the full theory. Fortunately the universe appears to he homogeneous and isotropic to a remarkable degree, so the Friedmann-LeMaitre-Robertson-Walker is adequate for many purposes. For instance, once known local features are removed, the Cosmic Microwave Background is isotropic to an accuracy of 1 in 100000.

However if we want greater resolution or more detail than the Friedmann-Robertson-Walker solution, the approximation has to take into account anisotropy and inhomogeneity. At present this cannot be done in full generality since we do not have the appropriate exact solutions to Einstein's equations. This is not surprising, given their highly non-linear nature. To deal with this problem cosmologists have resorted to perturbation methods, which have also proved effective in other areas of physics, such as fluid dynamics.

The essential idea behind perturbation theory is very simple, and best explained by an example for which we choose the metric tensor. We assume that we can approximate the full metric tensor of the universe by a power series expansion. The background is the Friedmann-Robertson-Walker metric with appropriate spatial curvature, according to the assumptions made about the universe. The remaining terms are the perturbations of the background.

Having set up the expansion, we have to substitute it into the Einstein equations to obtain approximate solutions at the required order of approximation for the application we have in mind. This is more difficult than one might imagine. Firstly, perturbations of the metric imply perturbations of the energy momentum tensor, but more importantly, calculation of the connection coefficients and the Einstein tensor involves raising and lowering indices and involves terms of different orders. At zero and first order this is not a problem, but at higher orders it makes the calculations much more complicated and so the choice of coordinates or form of the metric can be important. Already at second order we have "proper" second order terms and terms quadratic in the first order quantities.

Recently second order perturbation theory has become the focus of research. This has happened not so much to attain higher accuracy, but to be able to deal with quantities quadratic in the first order quantities, which are by definition of similar magnitude as the second order quantities themselves. Second order perturbation theory is essential to accurately calculate higher order observables, such as non-gaussianity.

For two particularly lucid reviews on cosmological perturbation theory see Malik and Matravers and Malik and Wands.

Topic revision: r4 - 2010-02-11 - KarimMalik
This site is powered by the TWiki collaboration platformCopyright © 2008-2020 by the contributing authors. All material on this collaboration platform is the property of the contributing authors.
Ideas, requests, problems regarding TWiki? Send feedback