Wave effect of gravitational waves intersected with a microlens field II: An adaptive hierarchical tree algorithm and population study

The gravitational lensing wave effect generated by a microlensing field embedded in a lens galaxy is an inevitable phenomenon in strong lensed gravitational waves (SLGWs). This effect presents both challenges and opportunities for the detection and application of SLGWs. However, investigating this wave effect requires computing a complete diffraction integral over each microlens in the field. This is extremely time-consuming due to the large number of microlenses (10³–10⁶). Therefore, simply adding all the microlenses is impractical. Additionally, the complexity of the time delay surface makes the lens plane resolution a crucial factor in controlling numerical errors. In this paper, we propose a trapezoid approximation-based adaptive hierarchical tree algorithm to meet the challenges of calculation speed and precision. We find that this algorithm accelerates the calculation by four orders of magnitude compared to the simple adding method and is one order of magnitude faster than the fixed hierarchical tree algorithm proposed for electromagnetic microlensing. More importantly, our algorithm ensures controllable numerical errors, increasing confidence in the results. Together with our previous work (Sci. China-Phys. Mech. Astron. 66, 239511, 2023), this paper addresses all numerical issues, including integral convergence, precision, and computational time¹). Finally, we conducted a population study on the microlensing wave effect of SLGWs using this algorithm and found that the microlensing wave effect cannot be ignored, especially for Type II SLGWs (from saddle position of the time delay surface) due to their intrinsic geometric structures and their typical intersection with a denser microlensing field. Statistically, more than 33% (11%) of SLGWs have a mismatch larger than 1% (3%) compared to the unlensed waveform. Additionally, we found that the mismatch between signal pairs in a doubly imaged GW is generally larger than 10⁻³, and 61% (25%) of signal pairs have a mismatch larger than 1% (3%). Therefore, the microlensing-induced mismatch can reduce the SLGW identification ability using the overlapping method.