arXiv subjects
Publications and source records attributed to Qin Sheng.
We develop a technique to construct analytical solutions of the linear perturbations of inflation with a nonlinear dispersion relation, due to quantum effects of the early universe. Error bounds are given and studied in detail. The analytical solutions describe the exact evolution of the perturbations extremely well even when only the first-order approximations is used.
We present a technique, {\em the uniform asymptotic approximation}, to construct accurate analytical solutions of the linear perturbations of inflation after quantum effects of the early universe are taken into account, for which the dispersion relations generically become nonlinear. We construct explicitly the error bounds associated with the approximations and then study them in detail. With the understanding of the errors and the proper choice of the Liouville transformations of the differential equations of the perturbations, we show that the analytical solutions describe the exact evolution of the linear perturbations extremely well even only in the first-order approximations. As an application of the approximate analytical solutions, we calculate the power spectra and indices of scalar and tensor perturbations in the slow-roll inflation, and find that the amplitudes of the power spectra get modified due to the quantum effects, while the power spectrum indices remain the same as in the linear case.
A new $z$-stretching finite difference method is established for simulating the paraxial light beam propagation through a lens in a cylindrically symmetric domain. By introducing proper domain transformations, we solve corresponding difference approximations on a uniform grid in the computational space for great efficiency. A specialized matrix analysis method is constructed to study the numerical stability. Interesting computational results are presented.