Dark Matter phase space distribution under a fully relativistic treatment. Eddington inversion according to general relativity
Abstract
We develop a fully relativistic extension of Eddington inversion to reconstruct the phase-space distribution function of dark matter from its density profile. Treating dark matter as an effective perfect fluid in a static, spherically symmetric spacetime, we use the Tolman-Oppenheimer-Volkoff equations to determine the gravitational background and derive an Abel-type inversion that yields the distribution function in terms of the conserved particle energy measured at infinity. We show that the resulting expression recovers the classical Eddington inversion in the weak-field, nonrelativistic limit. We also investigate the finite-boundary contribution, which can introduce an unphysical divergence, and present a regularization procedure that removes it. We apply our method to a Milky Way model consisting of a generalized Navarro-Frenk-White halo and a Gondolo-Silk dark-matter spike around Sagittarius A*, neglecting baryonic contributions. The relativistic and Newtonian velocity distributions agree at radii far from the Galactic center but begin to differ at distances of order 10−5 pc. Closer to the black hole, the relativistic peak speed can reach approximately twice the Newtonian prediction. Finally, we derive a general expression for dark-matter capture by neutron stars that accommodates arbitrary ambient velocity distributions. Near Sagittarius A*, the relativistic capture factor is nearly an order of magnitude smaller than the Maxwell-Boltzmann prediction, while the Newtonian Eddington result can overestimate it by a factor of approximately two. These results demonstrate the importance of relativistic phase-space reconstruction for dark-matter phenomenology in strong gravitational fields.
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BibTeX
@misc{zink2026dark,
title = {{Dark Matter phase space distribution under a fully relativistic treatment. Eddington inversion according to general relativity}},
author = {Jaime Hoefken Zink and Pellegrino Piantadosi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33150},
eprint = {2610.33150},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33150},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dark Matter phase space distribution under a fully relativistic treatment. Eddington inversion according to general relativity
AU - Jaime Hoefken Zink
AU - Pellegrino Piantadosi
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33150
UR - https://arxiv.org/abs/2610.33150
AB - We develop a fully relativistic extension of Eddington inversion to reconstruct the phase-space distribution function of dark matter from its density profile. Treating dark matter as an effective perfect fluid in a static, spherically symmetric spacetime, we use the Tolman-Oppenheimer-Volkoff equations to determine the gravitational background and derive an Abel-type inversion that yields the distribution function in terms of the conserved particle energy measured at infinity. We show that the resulting expression recovers the classical Eddington inversion in the weak-field, nonrelativistic limit. We also investigate the finite-boundary contribution, which can introduce an unphysical divergence, and present a regularization procedure that removes it. We apply our method to a Milky Way model consisting of a generalized Navarro-Frenk-White halo and a Gondolo-Silk dark-matter spike around Sagittarius A^*, neglecting baryonic contributions. The relativistic and Newtonian velocity distributions agree at radii far from the Galactic center but begin to differ at distances of order 10^−5 pc. Closer to the black hole, the relativistic peak speed can reach approximately twice the Newtonian prediction. Finally, we derive a general expression for dark-matter capture by neutron stars that accommodates arbitrary ambient velocity distributions. Near Sagittarius A^*, the relativistic capture factor is nearly an order of magnitude smaller than the Maxwell-Boltzmann prediction, while the Newtonian Eddington result can overestimate it by a factor of approximately two. These results demonstrate the importance of relativistic phase-space reconstruction for dark-matter phenomenology in strong gravitational fields.
ER -