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Description
The adiabatic growth of a black hole embedded in a collisionless dark-matter halo can induce a substantial rearrangement of the surrounding phase-space distribution, leading to the formation of a steep central overdensity commonly referred to as a dark-matter spike. Such structures may enhance the dark-matter annihilation rate by several orders of magnitude and therefore constitute potentially relevant targets for indirect searches with gamma-ray observatories. In this work, we present an independent numerical reconstruction of the formalisms introduced by Gondolo and Silk and subsequently extended to intermediate-mass black holes by Bertone, Zentner, and Silk. For the Gondolo–Silk scenario, the final dark-matter distribution was reconstructed from first principles by numerically evaluating the phase-space integrals associated with the adiabatic transformation of the initial halo. The implementation explicitly accounts for the transition to a black-hole-dominated Keplerian potential and for the conservation of the relevant adiabatic invariants, including angular momentum and radial action. We recover the isothermal solution and construct the corresponding families of cusped profiles for different values of the initial inner logarithmic slope, γ. The analysis includes both the asymptotic power-law approximation and the full radial correction gγ(r), as well as the dependence of the spike radius, Rsp = αγr0 MBH ρ0r3 0 1/(3−γ) , on the normalization coefficient αγ. A detailed comparison between the direct numerical reconstruction and the published parametrization reveals significant normalization discrepancies associated with αγ and with the resulting spike density ρDM(r) for non-isothermal initial conditions. For the configurations investigated here, a literal implementation of the reported expressions produces relative deviations reaching approximately two orders of magnitude. These differences are absent, or substantially reduced, in the isothermal limit, suggesting that they originate in the normalization of the generic cusped solutions rather than in the numerical treatment of the adiabatic invariants. We subsequently implemented the Bertone–Zentner–Silk framework using an initial Navarro–Frenk–White halo and explicitly incorporated its methodological differences with respect to the Gondolo–Silk construction, including the mini-spike normalization, the finite annihilation plateau, and the definition of the inner cutoff radius. The resulting density profile was coupled to a numerical evaluation of the annihilation J-factor and to a modular pipeline for computing the differential gamma-ray flux, dΦγ dE = ⟨σv⟩ 8πm2 χ dNγ dE J. Finally, a parametric toy detector-response model was developed to map the predicted physical flux into reconstructed event distributions. This end-to-end framework provides a reproducible connection between black-hole-induced dark-matter dynamics, annihilation phenomenology, and detector-level gamma-ray observables, and constitutes a basis for its future integration into sensitivity studies for next-generation gamma-ray instruments.