Fireballs' Whispers of Their Central Engine: Relativistic Filtering of Afterglow QPOs

TL;DR

This work introduces a linear-response framework that connects intrinsic central-engine variability in gamma-ray bursts to observed afterglow signals through an angular-delay transfer kernel defined on equal-arrival-time surfaces. By deriving the convolution form $F(T) = \int K(\tau) S(T- T_0-\tau) d\tau$ and its Fourier counterpart $H(\omega)$, the authors show that relativistic propagation acts as a frequency-dependent filter, often suppressing or distorting high-frequency QPOs. In the coasting phase the kernel is stationary with $K(\tau) \propto (1+\tau/\tau_0)^{-3}$ and $H(\omega) \approx 1/(1+i\omega\tau_0)$, implying observed periods largely track engine periods while amplitudes fade. When Blandford–McKee deceleration is included, the kernel becomes time-dependent, causing a secular drift in the observed frequency and amplitude; this can explain observed QPO evolution such as in GRB 220711B without invoking changes in the central engine.

Abstract

Quasi-periodic oscillations (QPOs) in gamma-ray bursts (GRBs) afterglows have been suggested as probes of the central engine. Such interpretations generally assume that the observed modulation frequency directly corresponds to an intrinsic oscillation frequency of the source. We show that this assumption is not generally valid and that interpreting such features without accounting for relativistic propagation may lead to misleading inferences about the engine nature. We show that relativistic propagation effects - most importantly integration over equal-arrival-time surfaces - act as a frequency-dependent filter that can significantly modify or suppress intrinsic variability. In the constant- $Γ$ case, the angular kernel acts as a stationary low-pass filter that suppresses high-frequency variability without altering its frequency, whereas Blandford-McKee deceleration renders the filter time-dependent and manifests observationally as an apparent frequency drift.

Figures (1)

• Figure 1: Magnitude of the transfer function $|H(\omega)|$ for the angular-delay kernel. Multiplying a central-engine signal by $|H(\omega)|$ reduces the amplitude of each frequency component, while the frequency remains unchanged in the constant-$\Gamma$ coasting phase. The solid black line shows the exact kernel, the dashed red line is the low-frequency approximation $1/\sqrt{1+(\omega \tau_0)^2}$, and the vertical dotted grey line marks $\omega \tau_0 = 1$, the approximate cutoff where high-frequency variability is suppressed.