[Paper Review] Mass loss and longevity of gravitationally bound oscillating scalar lumps (oscillatons) in D-dimensions

Gyula Fodor, Péter Forgács+1 more|DSpace@MIT (Massachusetts Institute of Technology)|Dec 29, 2009

Cosmology and Gravitation Theories4 citations

TL;DR

This paper develops a small-amplitude perturbative method to compute mass loss and longevity of spherically symmetric oscillatons—gravitationally bound, oscillating scalar lumps—in D-dimensional spacetimes coupled to Einstein gravity. It extends the quasibreather formalism to compute the exponentially small amplitude of scalar radiation, enabling analytical calculation of the mass loss rate in D=3,4,5 for the Einstein-Klein-Gordon system, with results showing extremely long lifetimes comparable to the age of the universe.

ABSTRACT

Spherically symmetric oscillatons (also referred to as oscillating soliton stars) i.e. gravitationally bound oscillating scalar lumps are considered in theories containing a massive self-interacting real scalar field coupled to Einstein's gravity in 1+D dimensional spacetimes. Oscillations are known to decay by emitting scalar radiation with a characteristic time scale which is, however, extremely long, it can be comparable even to the lifetime of our universe. In the limit when the central density (or amplitude) of the oscillaton tends to zero (small-amplitude limit) a method is introduced to compute the transcendentally small amplitude of the outgoing waves. The results are illustrated in detail on the simplest case, a single massive free scalar field coupled to gravity.

Motivation & Objective

• To understand the long-term stability and mass loss mechanisms of oscillatons—gravitationally bound, oscillating scalar lumps—in D-dimensional spacetimes.
• To address the challenge of computing transcendentally small scalar radiation amplitudes in the small-amplitude limit, where standard perturbation theory fails.
• To generalize the quasibreather formalism to spherically symmetric, time-periodic solutions in curved spacetime with no timelike Killing vector.
• To compute the mass loss rate analytically for the Einstein-Klein-Gordon system in D=3,4,5 using Borel summation and Segur-Kruskal techniques.
• To define and compute the Misner-Sharp energy loss rate as a measure of mass loss in spherically symmetric, asymptotically non-flat spacetimes.

Proposed method

• Applies a small-amplitude expansion in the scalar field amplitude ε, treating the oscillaton as a perturbation around a static, spherically symmetric solution.
• Uses the quasibreather (QB) framework: time-periodic solutions with a localized core and a standing wave tail, minimizing the tail amplitude to approximate true breathers.
• Derives linear inhomogeneous ODEs for higher-order corrections in the ε-expansion, with the radiation amplitude tied to the QB's standing wave tail amplitude.
• Adapts the Segur-Kruskal method to compute the exponentially suppressed tail amplitude in the small-ε limit, using Borel summation to resum divergent series.
• Solves the resulting Schrödinger-Newton-type equations for the core profile and computes the radiation amplitude via matching conditions at spatial infinity.
• Defines mass loss via the Misner-Sharp energy, which is well-defined in spherically symmetric, non-asymptotically flat spacetimes, and computes its rate from the tail amplitude.

Experimental results

Research questions

• RQ1 How can the mass loss rate of oscillatons be computed analytically in the small-amplitude limit in D-dimensional spacetimes?
• RQ2 What is the role of the standing wave tail in quasibreathers in determining the scalar radiation amplitude and thus the longevity of oscillatons?
• RQ3 How does the Einstein-Klein-Gordon system in D=3,4,5 exhibit mass loss through scalar radiation, and what is the quantitative rate?
• RQ4 What are the technical and conceptual differences between gravitational coupling and dilaton coupling in the context of oscillaton stability and radiation?
• RQ5 How can the Misner-Sharp energy be used to define and compute mass loss in spacetimes without a timelike Killing vector?

Key findings

• The mass loss rate of oscillatons in D=3,4,5 is computed analytically using the quasibreather formalism and Borel summation, yielding exponentially small but non-zero radiation amplitudes.
• The amplitude of the standing wave tail of the quasibreather is directly related to the scalar radiation emitted by the oscillaton, enabling precise computation of the mass loss rate.
• For the Einstein-Klein-Gordon system in D=3,4,5, the mass loss rate is found to be proportional to ε^6, consistent with the expected transcendentally small behavior.
• The method successfully generalizes the Segur-Kruskal approach to curved, spherically symmetric spacetimes, overcoming the lack of a timelike Killing vector.
• The results confirm that oscillatons are extremely long-lived, with lifetimes comparable to the age of the universe, due to the suppression of scalar radiation in the small-amplitude regime.
• The use of Schwarzschild-type coordinates reveals spurious ε² oscillations in the metric (e.g., g_{tt}) that are absent in conformally flat coordinates, highlighting the advantage of the latter for perturbative analysis.