Low-frequency gravitational wave memory from gamma-ray burst afterglows with energy injection

Ultrarelativistic gamma-ray burst (GRB) jets are strong gravitational wave (GW) sources with memory-type signals. The plateau (or shallow decay) phases driven by the energy injection might appear in the early X-ray afterglows of GRBs. In this paper, we investigate the GW signal as well as X-ray afterglow emission in the framework of GRB jets with energy injection, and both short- and long-duration GRBs are considered. We find that, regardless of the case, because of the antibeaming and time delay effects, a rising slope emerging in the waveform of GW signal due to the energy injection lags far behind the energy ejection, and the typical frequency of the characteristic amplitudes falls within a low-frequency region of $\sim10^{-4}-10^{-6} \,{\rm Hz}$; and we consider that the GW memory triggered by GRB jets with energy injection are previously unaware and the nearby GRBs with strong energy injection might disturb the measurement of the stochastic GW background. Such GW memory detection would provide a direct test for models of energy injection in the scenario of GRB jets.