[Paper Review] Gravitational wave propagation beyond general relativity: waveform distortions and echoes

José María Ezquiaga, Wayne Hu+2 more|arXiv (Cornell University)|Aug 24, 2021

Pulsars and Gravitational Waves Research94 references50 citations

TL;DR

This paper investigates gravitational wave (GW) propagation beyond general relativity (GR) in cosmological backgrounds, focusing on interactions with an additional tensor field. Using a parametrized phenomenological framework, it analytically and numerically shows that such interactions lead to GW waveform distortions, dispersion shifts, and coherent/decoupled echoes due to eigenstate mixing. The key result is that modified dispersion relations cause multiple signal copies to arrive at different times, potentially mimicking independent GW events, with observable signatures including phase distortions, amplitude modulations, and birefringence—offering new probes for modified gravity and dark energy with current and future GW detectors.

ABSTRACT

We study the cosmological propagation of gravitational waves (GWs) beyond general relativity (GR) across homogeneous and isotropic backgrounds. We consider scenarios in which GWs interact with an additional tensor field and use a parametrized phenomenological approach that generically describes their coupled equations of motion. We analyze four distinct classes of derivative and non-derivative interactions: mass, friction, velocity, and chiral. We apply the WKB formalism to account for the cosmological evolution and obtain analytical solutions to these equations. We corroborate these results by analyzing numerically the propagation of a toy GW signal. We then proceed to use the analytical results to study the modified propagation of realistic GWs from merging compact binaries, assuming that the GW signal emitted is the same as in GR. We generically find that tensor interactions lead to copies of the originally emitted GW signal, each one with its own possibly modified dispersion relation. These copies can travel coherently and interfere with each other leading to a scrambled GW signal, or propagate decoherently and lead to echoes arriving at different times at the observer that could be misidentified as independent GW events. Depending on the type of tensor interaction, the detected GW signal may exhibit amplitude and phase distortions with respect to a GW waveform in GR, as well as birefringence effects. We discuss observational probes of these tensor interactions with both individual GW events, as well as population studies for both ground- and space-based detectors.

Motivation & Objective

• To investigate how gravitational waves (GWs) propagate beyond general relativity (GR) in a cosmological, homogeneous, and isotropic universe when coupled to an additional tensor field.
• To model and analyze the effects of four distinct interaction types—mass, friction, velocity, and chiral—on GW propagation and waveform evolution.
• To determine whether modified dispersion relations lead to coherent interference or decoherent echoes in the detected GW signal, and how these could be misidentified as independent events.
• To provide analytical and numerical tools to predict observable deviations from GR, including amplitude, phase, and polarization distortions, for use in testing modified gravity and dark energy.
• To enable observational constraints using both individual GW events and population studies with ground- and space-based detectors.

Proposed method

• Uses a parametrized phenomenological approach to describe coupled equations of motion between GWs and an additional tensor field, covering derivative and non-derivative interactions.
• Applies the Wentzel–Kramers–Brillouin (WKB) approximation to analytically solve the time-dependent, coupled equations of motion under cosmological evolution, valid when GW periods are short compared to Hubble timescales.
• Employs a toy Gaussian wavepacket model with numerical integration to validate analytical predictions for amplitude, shape, and polarization evolution under each interaction type.
• Transforms initial wavepackets between frequency and momentum space using the stationary phase approximation (SPA), ensuring consistency between analytical and numerical solutions.
• Derives eigenfrequencies and eigenvectors of the coupled system to identify propagating eigenstates with distinct group velocities and dispersion relations.
• Uses polarization parameters {A, φ, β, χ} to quantify changes in amplitude, phase, circular polarization, and orientation due to propagation effects, including phase birefringence and polarization rotation.

Experimental results

Research questions

• RQ1 How do mass, friction, velocity, and chiral interactions between GWs and an additional tensor field alter the dispersion relation and propagation speed of GWs in cosmological backgrounds?
• RQ2 Under what conditions do GW eigenstates remain coherent or decohere into time-separated echoes, and how do these echoes affect the detectability and interpretation of GW signals?
• RQ3 To what extent do modified dispersion relations induce phase distortions, amplitude modulations, or birefringence (amplitude and phase) in the observed GW waveform compared to GR predictions?
• RQ4 Can the observed GW signal from compact binary coalescences be scrambled by interference between eigenstates, and how can this be distinguished from decoherent echoes?
• RQ5 What are the observable signatures of these tensor interactions in both individual GW events and population-level studies across ground- and space-based GW detectors?

Key findings

• Tensor interactions lead to multiple eigenstates of GWs with distinct dispersion relations, each propagating at different group velocities, resulting in time-delayed copies of the original signal.
• When eigenstates propagate coherently, they interfere, causing frequency- and time-dependent amplitude oscillations and phase distortions in the net GW signal, even in the absence of echoes.
• When eigenstates decohere due to differing group velocities, they arrive at different times, producing detectable echoes that could be misclassified as independent GW events.
• Phase birefringence (velocity birefringence) and amplitude birefringence are observed, particularly in chiral interactions, where the polarization state of the GW rotates and changes ellipticity during propagation.
• The analytical WKB solutions agree well with numerical simulations of Gaussian wavepackets, confirming that amplitude, shape, and polarization evolve significantly under all four interaction types.
• The condition for coherent mixing is satisfied when the difference in wavevector or group velocity between eigenstates is small, and it is most easily met at high frequencies, such as during the chirping phase of binary coalescence.