arXiv subjects

Jamal Saeed

Publications and source records attributed to Jamal Saeed.

3 recordsLinked to original sources

arXiv2025

Stable initial conditions and analytical investigations of cosmological perturbations in a modified loop quantum cosmology

In this paper, we study cosmological perturbations in a modified theory of loop quantum cosmologies, the so-called mLQC-I model. Our purposes are two-fold: First, using a method developed by Birrell and Davies, we identify an initial state in the remote contracting phase, which turns out to be stable, minimize particle creations and diagonalize the Hamiltonian, despite the fact that at this time some modes may be still outside of the Hubble horizon and not in their adiabatic states. Second, using the uniform asymptotic approximation method, we obtain the first-order approximate solutions of the mode function in terms of either the Airy functions, or the first or second kind of cylindrical functions, depending on the values of the wavenumber. In each case, the mode function contains two integration constants, which are uniquely determined by the initial state.

gr-qc↗

arXiv2024

Universal properties of the evolution of the Universe in modified loop quantum cosmology

In this paper, we systematically study the evolution of the Universe in the framework of a modified loop quantum cosmological model (mLQC-I) with various inflationary potentials, including chaotic, Starobinsky, generalized Starobinsky, polynomials of the first and second kinds, generalized T- models and natural inflation. In all these models, the big bang singularity is replaced by a quantum bounce, and the evolution of the Universe both before and after the bounce is universal and weakly depends on the inflationary potentials, as long as the evolution is dominated by the kinetic energy of the inflaton at the bounce. In particular, the evolution in the pre-bounce region can be universally divided into three different phases: pre-bouncing, pre-transition, and pre-de Sitter. The pre-bouncing phase occurs immediately before the quantum bounce, during which the evolution of the Universe is dominated by the kinetic energy of the inflaton. Thus, the equation of state of the inflaton is about one, w = 1. Soon, the inflation potential takes over, so w rapidly falls from one to negative one. This pre-transition phase is very short and quickly turns into the pre-de Sitter phase, whereby the effective cosmological constant with a Planck size takes over and dominates the rest of the contracting phase. In the entire pre-bounce regime, the evolution of the expansion factor and the inflaton can be approximated by analytical solutions, which are universal and independent of the inflation potentials.

gr-qc↗

arXiv2023

Uniform Asymptotic Approximation Method with P\"oschl-Teller Potential

In this paper, we study analytical approximate solutions of the second-order homogeneous differential equations with the existence of only two turning points (but without poles), by using the uniform asymptotic approximation (UAA) method. To be more concrete, we consider the P\"oschl-Teller (PT) potential, for which analytical solutions are known. Depending on the values of the parameters involved in the PT potential, we find that the upper bounds of the errors of the approximate solutions in general are $\lesssim 0.15\% \sim 10\% $, to the first-order approximation of the UAA method. The approximations can be easily extended to high-order, with which the errors are expected to be much smaller. Such obtained analytical solutions can be used to study cosmological perturbations in the framework of quantum cosmology, as well as quasi-normal modes of black holes.

gr-qc↗