The mean-free path of the subcomponent is negligible at the scale of 10 kpc if $f{\sigma }_m\gtrsim 100{\mathrm{cm}}^2/\mathrm{g}$. This is satisfied in nearly all the parameter space we consider, except for the $f{\sigma }_m\sim 1–100{\mathrm{cm}}^2/\mathrm{g}$ range. Since the efficiency of heat conduction peaks when the mean-free path matches the system size, the subcomponent in that regime may behave very differently from perfect fluid. Nevertheless, this will not affect our results as we will express them in terms of the galactic fraction $f$ rather than the cosmic fraction $\overline{f}$.

The results of these simulations depend on the initial cosmic temperature of the subcomponent prior to structure formation; higher temperatures tend to suppress its fraction in the central region of the CDM halo. For initial temperatures comparable to or less than that of realistic baryonic gas ($\sim 10\mathrm{meV}$) the suppression is not significant [41, 43]. Also, these simulations are not to be confused with those that studied the fluid limit of SIDM as the dominant DM, which found that the resulting halos resemble cuspy isothermal spheres, with density profiles ${\rho }_{\chi }(r)\propto r^{-2}$ [44, 45].

When ${\lambda }_{\mathrm{MFP}}\gtrsim r$, the local-thermalization condition ${\tau }_{\mathrm{col}}\lesssim {\tau }_{\mathrm{cond}}$ holds only marginally. Nevertheless, the results of gravothermal (with proper calibration of the conductivity coefficients) and N-body simulations have been shown to agree well even in those marginal cases [51, 52, 53, 54].

Although the temperature profile of Eq. (3) is consistent with the boundary condition $T_m(r\to \infty )=T_m^{\infty }$, it behaves as $T_m^{\mathrm{ad}}-T_m^{\infty }\propto r^{-1}$ beyond the radius of gravitational influence of the compact object ${\lambda }_{\mathrm{grav}}\equiv G{M}_{\star }/T_m^{\infty }$. Thus, such a profile leads to an unphysical heat-conduction luminosity that is growing as $L\equiv 4\pi r^2\mathcal{F}\propto r^2$ at $r\gtrsim {\lambda }_{\mathrm{grav}}$. The details of the small deviation $T_m-T_m^{\infty }$ are important because $L$ depends on ${\partial }_r{T}_m$ and not just $T_m$. In reality, collisions of the $\chi$ gas bound to the compact object with the ambient, unbound gas (which are important only at $r\gtrsim {\lambda }_{\mathrm{grav}}$) would bring $T_m$ exponentially close to $T_m^{\infty }$ after several ${\tau }_{\mathrm{col}}$. To account for this effect, in practice, we introduce an exponential factor to the $T_m^{\mathrm{ad}}$ profile to make it decay faster: $T_m^{\mathrm{ad}}=T_m^{\infty }-(2/5)\mathrm{\Phi }{e}^{-r^2/{\lambda }_{\mathrm{grav}}^2}$.

As a check, we run simulations with hypothetical compact objects with mass $M_{\star }=1M_{\odot }$ and radii $R_{\star }=600\mathrm{km}$ (0.1 times WD radius) and $R_{\star }=6\times {10}^4\mathrm{km}$ (10 times WD radius) and found that all the results are in agreement with our understanding of the WD case.

Subcomponent particles with sufficiently small mean-free paths can be shielded from ram stripping in dwarf galaxies. The fact that gas-rich dwarf galaxies exist suggests that this outcome is possible [84, 85].

Black holes have $v_{\mathrm{esc}}=1$ but their absorptive inner boundary conditions act as a central sink that hinders pileup.