Application of factorisation and boost transformations Procedures on single field inflationary cosmological Perturbations

Abstract

Inflation is a period of an accelerated expansion of the universe. Cosmological perturbations are created by the amplification of quantum vacuum fluctuations of matter and metric perturbations during inflation. The equation of dynamics governing the evolution of cosmological perturbations within the cosmological model in a single field inflationary scenario takes the form of a standard non-linear second-order differential equation, whose exact analytical solution has not been obtained to date. The various methods of approximation that have been used in solving this equation of dynamics have varied limitations that include: inadequate error control; difficulty in improving the accuracy beyond the leading order (are not systematically extendable); complicated/tedious mathematical formulations; and series expansions that may be also divergent at some order. This study provides a systematically extendable method of approximation for the study of single field cosmological perturbations during inflation, which removes the divergence in the Wentzel-Kramers-Brillouin (WKB) approximation, based on a factorization and boost transformation procedures up to zeroth-order. The equation of dynamic is factorized and then converted into a matrix equation with its corresponding Hamiltonian. By using appropriately defined boost transformation operator, the resultant matrix equation undergoes successive boost transformations along suitable axes to new dynamical frames of high accuracy levels, characterized by an approximation parameter that becomes smaller with increasing number of boost transformations and is safely neglected at the highest level of approximation (accuracy). Diagonalization of the boost frame Hamiltonian leads to a simple analytical solution of the boost frame matrix equation through direct integration, and once the time evolution operator in the form obtained through the diagonalization procedure, is used into the analytical solution obtained, the desired general solution of the equation of dynamics governing the evolution of cosmological perturbations in single field inflationary scenario follow easily up to order.The zeroth-order derivative of the approximation parameter produces an expression that is more exact and similar to the standard WKB approximation parameter , though with different co-efficient. Each order of approximation provides an amplitudemodulated “plane” mode function specified by a renormalized time-dependent frequency, that facilitates exact evaluation of the phase accumulation integral for various forms of the potential . The zeroth-order solution is exactly the leading order/first- order standard WKB solution, and takes exactly the same form of the assumed solution (ansatz) in the standard WKB approximation. Furthermore, it does not require any matching condition about any particular point, that is to say, a turning point as outlined in the results for the WKB mode function and therefore the issue of divergence at a turning point as encountered in the WKB approximation does not arise in the approximation procedure developed in this study. This goes a long way to improving the understanding of inflationary perturbations. The general expressions for the zeroth-order power spectrum constitute the main results of this study.