SGWB as detected using weakly hyperbolic encounters. The shaded regions show the SGWB and are produced from Eq. (46), where we set r_i in two sets of ranges. The small range varies from {0.03 pc, 0.53 pc}, and the larger range varies from {0.03 pc, 9.03 pc}. In both cases, the eccentricity covers the range of }e_imin,e_imin + 4 × 10⁻⁹{. For the top-left plot, we set N_tot = 10⁶, R_gc = 10 pc (see case 1 in the text for more details.); for the top-right plot, we set N_tot = 5 × 10⁵, R_gc = 10 pc (see case 2 for more details); for the bottom-left plot, we set N_tot = 10⁶, R_gc = 15 pc (see case 3 for more details); for the bottom- right plot, we set N_tot = 5 × 10⁵, R_gc = 15 pc (see case 4 for more details). In all four cases, we integrate up to redshift z = 5. The gray region shows the case for the small r_i and the orange region shows the case for the larger r_i. Moreover, ET, CE, and DECIGO curves correspond to the sensitivity of the Einstein Telescope (Punturo et al. 2010), the Cosmic Explorer (Evans et al. 2021), and the Deci-hertz Interferometer Gravitational-wave Observatory (Hou et al. 2021) detectors, respectively, for one year of observation (Thrane & Romano 2013).