Radial Stress and the Innermost Stable Circular Orbit of a Bardeen Black Hole in a Dark Matter Halo: A First-Order Response Criterion
A dark matter density profile alone does not determine the spacetime around a black hole; the radial stress must be prescribed separately and governs the strong-field orbital response. We study a Bardeen black hole in a Hernquist halo through three spherical models sharing the same density, mass function, and cutoff but differing in radial stress: a halo with $p_r^{\mathrm{DM}}=-ρ_{\mathrm{DM}}$, its truncated form, and a truncated Einstein cluster with $p_r^{\mathrm{DM}}=0$. Treating the halo as a small perturbation, we derive a first-order criterion for the leading innermost stable circular orbit shift, $C_λ=C_ρ+λC_p$. The two truncated closures shift this orbit in opposite directions for the Hernquist profile, and unexpanded calculations for five density profiles confirm the predicted signs. Checks using a Hayward background and a published Dehnen-halo result show that the criterion is not restricted to the Bardeen-Hernquist system. A continuous stress interpolation identifies a critical closure at which the leading shift vanishes. For a representative four-year extreme-mass-ratio inspiral, changing the radial stress at fixed density and cutoff produces a phase difference of several radians in a leading-order adiabatic treatment.