Toward second-order self-force for eccentric extreme-mass ratio inspirals in Schwarzschild spacetimes

An Extreme Mass Ratio Inspiral (EMRI), which corresponds to a small compact object inspirals around a massive black hole in the center of a galaxy, is one of the most important sources for future space-borne gravitational-wave (GW) detectors such as TianQin and LISA. By analyzing the emitted GW signals, we can probe the theory of gravity and the nature of black holes in the strong field region. To achieve these objectives, the second-order self-force effect should be considered in the waveform modeling. Up to now, the waveform of EMRIs including the second-order self-force effect is only achieved for the circular orbit on Schwarzschild background. In this work, we generalized the calculation of the second-order self-force to the eccentric orbits on Schwarzschild spacetime. We calculated the puncture field, and give the form of two-timescale expansion for the field equations. The corresponding numerical calculation and programming can be performed based on these results.