Hawking Radiation in de Sitter Space: Calculation of the Reflection Coefficient for Quantum Particles

TL;DR

This paper clarifies the calculation of the reflection coefficient for quantum particles in de Sitter space, concluding it is zero under certain conditions and arguing that the reflection coefficient is unnecessary due to the absence of a potential barrier.

Contribution

It demonstrates that the reflection coefficient in de Sitter space is zero when specific quantum number constraints are met, and shows that higher-order corrections do not alter this result.

Findings

The reflection coefficient $R_{\epsilon j}$ is exactly zero under certain conditions.

Higher-order contributions do not change the zero value of the reflection coefficient.

No potential barrier exists in the effective potential in de Sitter space, making the reflection coefficient calculation unnecessary.

Abstract

Though the problem of Hawking radiation in de Sitter space-time, in particular details of penetration of a quantum mechanical particle through the de Sitter horizon, has been examined intensively there is still some vagueness in this subject. The present paper aims to clarify the situation. A known algorithm for calculation of the reflection coefficient $R_{\epsilon j}$ on the background of the de Sitter space-time model is analyzed. It is shown that the determination of $R_{\epsilon j}$ requires an additional constrain on quantum numbers $\epsilon R / \hbar c >> j$, where $R$ is a curvature radius. When taking into account this condition, the value of $R_{\epsilon j}$ turns out to be precisely zero. It is shown that the basic instructive definition for the calculation of the reflection coefficient in de Sitter model is grounded exclusively on the use of zero order approximation in the expansion of a particle wave function in a series on small parameter $1/R^{2}$, and it demonstrated that this recipe cannot be extended on accounting for contributions of higher order terms. So the result $R_{\epsilon j}=0$ which has been obtained from examining zero-order term persists and cannot be improved. It is claimed that the alculation of the reflection coefficient $R_{\epsilon j}$ is not required at all because there is no barrier in the effective potential curve on the background of the de Sitter space-time, the later correlate with the fact that the problem in de Sitter space reduces to a second order differential equation with only three singular points. However all known quantum mechanical problems with potentials containing one barrier reduce to a second order differential equation with four singular points, the equation of Heun class.