[Paper Review] Non-linear Oscillations of Compact Stars and Gravitational Waves

Andrea Passamonti|OpenGrey (Institut de l'Information Scientifique et Technique)|Jul 31, 2006

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper develops a gauge-invariant, second-order perturbation framework to study non-linear coupling between radial and non-radial oscillations in compact stars, showing that radial pulsations—though silent in linear theory—generate detectable gravitational wave signals via non-linear interactions. Key results include a resonant enhancement of gravitational wave amplitude when radial modes approach the w-mode frequency, with the fourth radial overtone emitting signals ~1000× stronger than the fundamental mode.

ABSTRACT

This thesis investigates in the time domain a particular class of second order perturbations of a perfect fluid non-rotating compact star: those arising from the coupling between first order radial and non-radial perturbations. This problem has been treated by developing a gauge invariant formalism based on the 2-parameter perturbation theory (Sopuerta, Bruni and Gualtieri, 2004) where the radial and non-radial perturbations have been separately parameterized. The non-linear perturbations obey inhomogeneous partial differential equations, where the structure of the differential operator is given by the previous perturbative orders and the source terms are quadratic in the first order perturbations. In the exterior spacetime the sources vanish, thus the gravitational wave properties are completely described by the second order Zerilli and Regge-Wheeler functions. As main initial configuration we have considered a first order differentially rotating and radially pulsating star. Although at first perturbative order this configuration does not exhibit any gravitational radiation, we have found a new interesting gravitational signal at non-linear order, in which the radial normal modes are precisely mirrored. In addition, a resonance effect is present when the frequencies of the radial pulsations are close to the first axial w-mode. Finally, we have roughly estimated the damping times of the radial pulsations due to the non-linear gravitational emission. The coupling near the resonance results to be a very effective mechanism for extracting energy from the radial oscillations.

Motivation & Objective

• To model non-linear gravitational wave emission from compact stars where both radial and non-radial oscillations are excited.
• To develop a gauge-invariant formalism for second-order perturbations in a time-dependent, spherically symmetric spacetime.
• To investigate how radial pulsations couple with non-radial modes to produce gravitational radiation not present in linear theory.
• To quantify the impact of resonances and energy transfer between mode classes on gravitational wave emission.
• To estimate damping times of radial modes due to non-linear gravitational wave emission.

Proposed method

• Adopts a 2-parameter perturbation theory framework with separate parameterization of radial and non-radial perturbations.
• Uses gauge-invariant formalism based on Gerlach & Sengupta (1979) and Gundlach & García (2000), fixing the radial gauge for second-order gauge invariance.
• Expands the time-dependent, spherically symmetric background to derive second-order non-radial perturbations with quadratic source terms in first-order perturbations.
• Solves inhomogeneous partial differential equations for second-order perturbations using finite differencing and explicit numerical schemes.
• Applies the McCormack algorithm for time integration and convergence testing via grid refinement to validate numerical accuracy.
• Monitors numerical stability via L2 norms of perturbations throughout time evolution.

Experimental results

Research questions

• RQ1 Can radial pulsations in compact stars generate gravitational waves through non-linear coupling with non-radial modes, despite being silent in linear theory?
• RQ2 How do the spectral properties of the gravitational wave signal reflect the frequencies of the radial normal modes?
• RQ3 What role do resonances play in enhancing gravitational wave emission when radial mode frequencies approach the w-mode frequency?
• RQ4 How does the amplitude of gravitational waves depend on the radial overtone number, particularly for higher overtones?
• RQ5 What are the damping timescales of radial pulsations due to non-linear gravitational wave emission, and how do they depend on resonance conditions?

Key findings

• Radial pulsations produce a non-linear gravitational wave signal through coupling with non-radial modes, even though radial oscillations alone do not emit gravitational waves in linear theory.
• The gravitational wave spectrum precisely mirrors the frequencies of the radial normal modes, confirming the non-linear origin of the signal.
• A resonance effect significantly amplifies gravitational wave emission when radial mode frequencies are close to the first w-mode frequency.
• For the stellar model studied, the gravitational wave amplitude from the fourth radial overtone is approximately three orders of magnitude stronger than that from the fundamental radial mode.
• The fundamental radial mode damps after about ten billion oscillation periods, while the fourth overtone damps after only ten oscillations, due to strong non-linear emission and resonance.
• Damping timescales are highly sensitive to resonance conditions, indicating that non-linear gravitational wave emission can dominate radial mode energy loss in certain configurations.