Total contribution to the SGWB from weakly hyperbolic encounters (orange area). In this plot, the virial velocity varies within 5 km s⁻¹ ≤ v_i ≤ 18  km s⁻¹. The black lines around the orange area show the fitted curves for this SGWB; the bottom line represents the pessimistic scenario, while the top line is the optimistic scenario (see Sect. 4.2 for more details). Moreover, BNS, LIGO-A⁺, and Voyager curves correspond to the energy density of GW background of the binary neutron star (Abbott et al. 2023), and the sensitivity curves of the Advanced Laser Interferometer Gravitational-wave Observatory (Mishra et al. 2010), and the Voyager (Abbott et al. 2017; Sathyaprakash et al. 2019; see also (LIGO Scientific Collaboration et al. 2021)) detectors respectively for one year of observations.