Dark energy effects in the Schrödinger-Newton approach

Abstract

The Schrödinger-Newton equation is a proposed model to explain the localisation of macroscopic particles by suppressing quantum dispersion with the particle's own gravitational attraction. On cosmic scales, however, dark energy also acts repulsively, as witnessed by the accelerating rate of universal expansion. Here, we introduce the effects of dark energy in the form of cosmological constant Λ, that drives the late-time acceleration of the universe, into the Schrödinger-Newton approach. We then ask in which regime dark energy dominates both canonical quantum diffusion and gravitational self-attraction. It turns out that this happens for sufficiently delocalised objects with arbitrary mass and that there exists a minimal delocalisation width of about 67 meters. While extremely macroscopic from a quantum perspective, the value is in principle accessible to laboratories on Earth. Hence, we analyse, numerically, how the dynamics of an initially spherical Gaussian wave packet is modified in the presence of Λ > 0. A notable feature is the gravitational collapse of part of the wave packet, in the core region close to the centre of mass, accompanied by the accelerated expansion of the more distant shell surrounding it. The order of magnitude of the distance separating collapse from expansion matches analytical estimates of the classical turnaround radius for a spherically symmetric body in the presence of dark energy. However, the time required to observe these modifications is astronomical. They can potentially be measured only in physical systems simulating a high effective cosmological constant, or, possibly, via their effects on the inflationary universe.

Problems with modified commutators

“…It is then claimed that the link between GURs and nonlocal gravity is provided by the semi-classical approach (Nicolini and Niedner, 2011;Lake, 2022a) in which the classical curvature of space-time is sourced by the expectation value of the energy-momentum operator for the quantum matter fields, ⟨ T μν ⟩ ψ . In the weak field limit, the semiclassical field equations reduce to Poisson's equation, with the square of the wave function as a source term, neglecting the subdominant dark energy contribution (Møller, 1962;Rosenfeld, 1963;Kelvin et al, 2020),…”

The effects of the dark energy on the static Schrödinger-Newton system -- an Adomian Decomposition Method and Padé approximants based approach

“…In order to apply these mathematical methods we will consider the equivalent mathematical formulations of the Schrödinger-Newton-Λ system as a system of integral equations. In the following we will look for a series solution of the system (19) and (20), by assuming that S(r) = ∞ n=0 S n (r), and V (r) = ∞ n=0 V n (r), respectively. As for the nonlinear terms S(x)V (x) and S 2 (x), we will decompose them in terms of the Adomian polynomials according to…”

“…Recently, an extension of the standard Schrödinger-Newton system was proposed and investigated in, 20 by including in the mathematical formalism the effects of the dark energy, represented by a cosmological constant Λ. Presently, it is assumed that dark energy drives the late-time acceleration of the Universe, and plays a determining role in the late cosmological evolution; 21 for alternative models of dark energy as modified gravity see, 22 and references therein.…”

Novel exact traveling wave solutions of Newton-Schrödinger system using Nucci reduction and Sardar sub-equation methods

“…Thus, it is feasible to utilize specialized numerical techniques designed for optical systems to analyze the behavior of matter in gravitational fields. This similarity has facilitated the exploration of various phenomena, including dark matter [43], dark energy [44], structure formation [45], and alternative theories of gravity [46]. However, the similarity goes beyond surface-level comparisons and also facilitates the replication of astronomical phenomena, such as boson stars, through analogous optical experiments conducted in a laboratory [42].…”