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Register a new userCosmological Perturbation Theory Using the Schrodinger Equation
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We introduce the theory of non-linear cosmological perturbations using the correspondence limit of the Schrodinger equation. The resulting formalism is equivalent to using the collisionless Boltzman (or Vlasov) equations which remain valid during the whole evolution, even after shell crossing. Other formulations of perturbation theory explicitly break down at shell crossing, e.g. Eulerean perturbation theory, which describes gravitational collapse in the fluid limit. This paper lays the groundwork by introducing the new formalism, calculating the perturbation theory kernels which form the basis of all subsequent calculations. We also establish the connection with conventional perturbation theories, by showing that third order tree level results, such as bispectrum, skewness, cumulant correlators, three-point function are exactly reproduced in the appropriate expansion of our results. We explicitly show that cumulants up to N=5 predicted by Eulerian perturbation theory for the dark matter field $delta$ are exactly recovered in the corresponding limit. A logarithmic mapping of the field naturally arises in the Schrodinger context, which means that tree level perturbation theory translates into (possibly incomplete) loop corrections for the conventional perturbation theory. We show that the first loop correction for the variance is $sigma^2 = sigma_L^2+ (-1.14+n)sigma_L^4$ for a field with spectral index $n$. This yields 1.86 and 0.86 for $n=-3,-2$ respectively, and to be compared with the exact loop order corrections 1.82, and 0.88. Thus our tree-level theory recovers the dominant part of first order loop corrections of the conventional theory, while including (partial) loop corrections to infinite order in terms of $delta$.
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Cosmological perturbation theory is crucial for our understanding of the universe. The linear theory has been well understood for some time, however developing and applying the theory beyond linear order is currently at the forefront of research in theoretical cosmology. This thesis studies the applications of perturbation theory to cosmology and, specifically, to the early universe. Starting with some background material introducing the well-tested standard model of cosmology, we move on to develop the formalism for perturbation theory up to second order giving evolution equations for all types of scalar, vector and tensor perturbations, both in gauge dependent and gauge invariant form. We then move on to the main result of the thesis, showing that, at second order in perturbation theory, vorticity is sourced by a coupling term quadratic in energy density and entropy perturbations. This source term implies a qualitative difference to linear order. Thus, while at linear order vorticity decays with the expansion of the universe, the same is not true at higher orders. This will have important implications on future measurements of the polarisation of the Cosmic Microwave Background, and could give rise to the generation of a primordial seed magnetic field. Having derived this qualitative result, we then estimate the scale dependence and magnitude of the vorticity power spectrum, finding, for simple power law inputs a small, blue spectrum. The final part of this thesis concerns higher order perturbation theory, deriving, for the first time, the metric tensor, gauge transformation rules and governing equations for fully general third order perturbations. We close with a discussion of natural extensions to this work and other possible ideas for off-shooting projects in this continually growing field.
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