Dark energy effects in the Schr\"odinger-Newton approach

TL;DR

This paper extends the Schr"odinger-Newton model by incorporating dark energy effects via a cosmological constant, revealing regimes where dark energy dominates quantum and gravitational dynamics, with potential laboratory and cosmological implications.

Contribution

It introduces dark energy into the Schr"odinger-Newton framework and analyzes its impact on quantum wave packet dynamics, identifying a minimal delocalization width and novel collapse-expansion phenomena.

Findings

Dark energy dominates for sufficiently delocalized objects.

A minimal delocalization width of about 67 meters is identified.

Wave packet dynamics show collapse near the center and expansion outward.

Abstract

The Schr\"odinger-Newton equation is a proposed model to explain the localization 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 a cosmological constant $\Lambda$, that drives the late-time acceleration of the Universe, into the Schr\"odinger-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 delocalized objects with an arbitrary mass and that there exists a minimal delocalization width of about $67$ m. While extremely macroscopic from a quantum perspective, the value is in principle accessible to laboratories on Earth. Hence, we analyze, numerically, how the dynamics of an initially spherical Gaussian wave packet is modified in the presence of $\Lambda > 0$. A notable feature is the gravitational collapse of part of the wave packet, in the core region close to the center 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.