tokens: T / T c = 10 −2 . ( e ) GC3 WC, T / T c = 10 −4 . ( f ) GC3 WC, T / T c = 10 −6 . ( g ) GC4 LC, T / T c = 10 −2 . ( h ) GC4 LC, T / T c = 10 −4 . ( i ) GC4 LC, T / T c = 10 −6 . ( j ) GC4 WC, T / T c = 10 −2 . ( k ) GC4 WC, T / T c = 10 −4 . ( l ) GC4 WC, T / T c = 10 −6 . The orbital decay time for a wide range of parameters is shown in Fig. 8 for the two GCs inside the core radius of the Fornax dSph, GC3 and GC4, in the two density profiles considered. τ DF generally either attains a minimum value, τ DF, min , or approaches infinity. The minimum values in the region V > c − and δ ρ / ρ 0 1 are summarized in Table 3 , with τ DF in the range 67 Myr−197 Myr. These timescales are considerably smaller than the CDM result assuming the same density profiles, 515 Myr−1327 Myr, with the dynamical friction given by Binney Tremaine (2008) Table 3. Minimum orbital decay time τ DF, min of GC3 and GC4 in the LC and WC models for the Fornax dSph, found in the region V > c − and δ ρ / ρ 0 1 (for which perturbation theory is properly valid), with the CDM result given by Eq. ( 87 ) for comparison. Notes. The listed values are essentially constant for all temperatures for which the assumptions made are valid, T / T c ≲ 0.1. For V c − , perturbation theory instead predicts the dynamical friction to quickly vanish, causing τ DF to become infinite. The criterion on g / m 2 due to the hydrostatic core radius, Eq. ( 82 ), is also listed. The values for g / m 2 are given in units of eV −4 . where Λ ≈ r δ v 2 / G M , , δ v is the velocity dispersion of CDM particles, taken to be δ v ≈ V , and erf is the error function. The decay time remains small even if parameters used to model the Fornax dSph, the GCs, and the dynamical friction are varied, as illustrated in Fig. 6 . A notable exception is the position of the GC, for which τ DF is considerably shorter when closer to the halo center, and likewise longer when further away. This implies that the value for τ DF obtained from Eq. ( 79 ) overestimates the time it takes the GC to fully decay from its current position, but it also implies that the migration towards the halo center was slower in the past when the GCs were at larger radial distances. Indeed, estimates of τ DF, min for GC1, GC2, and GC5, all of which are located at r ≳ 1 kpc, give decay times in excess of 4 Gyr. In the CDM case, the decay times for these GCs are even longer: 17 Gyr and more. These estimates do not suggest a timing problem for the outer GCs, even if their decay times are considerably shorter for SIBEC-DM compared to CDM. However, we note that these GCs are near or outside the radius r s , where the density profile of the dSph falls sharply, and therefore we do not expect the result for the dynamical friction, nor Eq. ( 79 ), to necessarily provide a reasonable estimate of τ DF . Nonetheless, the present results show that for a large region of the relevant parameter space of the SIBEC-DM model considered here, GC3 and GC4 are currently
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