Modulation of the gravitational waveform by the effect of radiation reaction

When we calculate gravitational waveforms from extreme-mass-ratio inspirals by metric perturbation, it is a common strategy to use the adiabatic approximation. Under that approximation, we first calculate the linear metric perturbation induced by geodesics orbiting a black hole, then we calculate the adiabatic evolution of the parameters of geodesics due to the radiation reaction effect through the calculation of the self-force. This procedure is considered to be reasonable, however, there is no direct proof that it can actually produce the correct waveform we would observe. In this paper, we study the formal expression of the second order metric perturbation and show that it can be expressed as the linear metric perturbation modulated by the adiabatic evolution of the geodesic. This evidence supports the assumption that the adiabatic approximation can produce the correct waveform, and that the adiabatic expansion we propose in Ref. [Y. Mino, Prog. Theor. Phys. 115, 43 (2006); Y. MinoProg. Theor. Phys.113, 733 (2005); Y. Mino and R. Price (unpublished).] is an appropriate perturbation expansion for studying the radiation reaction effect on the gravitational waveform.