References

PBHs in a matter era #

• C.-M. Yoo, A. Escrivà, T. Harada, K. Kohri (2026). Simulation of PBH formation in a matter-dominated universe. arXiv:2609.14218 — 3D BSSN with particles and dust; the ζ profile, CMC initial data and the e = 0.2 comparison — our main reference point.
• T. Harada, C.-M. Yoo, K. Kohri, K. Nakao, S. Jhingan (2016). Primordial black hole formation in the matter-dominated phase of the Universe. arXiv:1609.01588 — The fraction β ≈ 0.05556 σ⁵ for non-spherical collapse in the matter era.
• T. Kokubu, K. Kyutoku, K. Kohri, T. Harada (2018). Effect of Inhomogeneity on Primordial Black Hole Formation in the Matter Dominated Era. Phys. Rev. D 98, 123024 (2018). arXiv:1810.03490 — Effect of inhomogeneity on β in the matter era.
• T. Harada, K. Kohri, M. Sasaki, T. Terada, C.-M. Yoo (2022). Threshold of Primordial Black Hole Formation against Velocity Dispersion in Matter-Dominated Era. arXiv:2211.13950 — Threshold against velocity dispersion; the 'Harada' source in our dispersion scan.
• E. Ebrahimian, A. A. Abolhasani, M. Mirbabayi (2025). Primordial black hole formation in matter domination. arXiv:2507.18312 — Newtonian threshold estimate with small-scale kicks; the 'Ebrahimian' source in our scan.
• W. Ye, Y. Gong, T. Harada, Z. Kang, K. Kohri, D. Saito et al. (2025). Primordial Black Hole Formation and Spin in Matter Domination Revisited. arXiv:2508.10070 — β and spin in matter domination revisited.

Scalar fields and numerical relativity #

• E. Milligan, L. E. Padilla, D. J. Mulryne, J. C. Hidalgo (2025). Primordial Black Hole Formation in a Scalar Field Dominated Universe. JCAP 10 (2025) 025. arXiv:2504.02600 — Spherical scalar + fluid (Misner–Sharp); kinetic initial data φ = 0.
• L. E. Padilla, E. Milligan, D. J. Mulryne, J. C. Hidalgo (2025). Primordial Black Hole Formation in a Scalar Field Dominated Universe: Investigation of the Critical nature of the Collapse. arXiv:2509.10431 — Critical collapse for a quartic potential.
• E. de Jong, J. C. Aurrekoetxea, E. A. Lim (2021). Primordial black hole formation with full numerical relativity. JCAP 03 (2022) 029. arXiv:2109.04896 — Full numerical relativity in GRChombo; the difficulty of a direct field perturbation (footnote 1).
• E. de Jong, J. C. Aurrekoetxea, E. A. Lim, T. França (2023). Spinning primordial black holes formed during a matter-dominated era. arXiv:2306.11810 — Spinning PBHs in a matter era, CTTK initial data.
• E. de Jong (2024). Primordial black hole formation processes with full numerical relativity. PhD thesis. arXiv:2403.02878 — Horizon mass at birth ~10⁻² M_H followed by rapid accretion.

Einstein–Vlasov: numerical benchmarks and rigorous results #

• G. Rein, A. D. Rendall, J. Schaeffer (1998). Critical collapse of collisionless matter — a numerical investigation. Phys. Rev. D 58, 044007 (1998). arXiv:gr-qc/9804040 — Benchmark: polar-areal gauge, A* ≈ 0.70.
• I. Olabarrieta, M. W. Choptuik (2001). Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry. Phys. Rev. D 65, 024007 (2002). arXiv:gr-qc/0107076 — Benchmark: maximal-areal gauge, M₀* ≈ 1.3.
• A. Akbarian, M. W. Choptuik (2014). Critical collapse in the spherically-symmetric Einstein-Vlasov model. Phys. Rev. D 90, 104023 (2014). arXiv:1409.5176
• H. Andréasson, G. Rein (2006). A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein-Vlasov system. Class. Quantum Grav. 23, 3659 (2006). arXiv:gr-qc/0601112
• S. Günther, J. Körner, T. Lebeda, B. Pötzl, G. Rein, C. Straub et al. (2020). A numerical stability analysis for the Einstein-Vlasov system. Class. Quantum Grav. 38, 035003 (2021). arXiv:2009.08163
• H. Andréasson, M. Kunze, G. Rein (2007). The formation of black holes in spherically symmetric gravitational collapse. Math. Ann. (2011). arXiv:0706.3787 — Rigorous black-hole formation in collisionless collapse.
• H. Andréasson, G. Rein (2009). Formation of trapped surfaces for the spherically symmetric Einstein-Vlasov system. J. Hyperbolic Differ. Equ. 7, 707 (2010). arXiv:0910.1254
• H. Andréasson, G. Rein (2024). Oppenheimer-Snyder type collapse for a collisionless gas. Commun. Math. Phys. 406, 284 (2025). arXiv:2410.06701 — Oppenheimer–Snyder-type collisionless collapse forms a trapped surface.
• W. E. East (2019). Cosmic Censorship Upheld in Spheroidal Collapse of Collisionless Matter. Phys. Rev. Lett. 122, 231103 (2019). arXiv:1901.04498 — Prolate collisionless collapse forms horizons.
• E. Ames, H. Andréasson, O. Rinne (2023). On the Hoop conjecture and the weak cosmic censorship conjecture for the axisymmetric Einstein-Vlasov system. Phys. Rev. D 108, 064054 (2023). arXiv:2305.04360

Observations and gravitational waves #

• O. Gottlieb, M. Cantiello, C. Norton, K. Van Tilburg, M. Kleban (2026). The Life and Death of Stars That Capture Primordial Black Holes. arXiv:2606.02700 — Weakens limits from PBH capture by stars.
• K. Inomata, K. Kohri, T. Terada (2025). The poltergeist mechanism — Enhancement of scalar-induced gravitational waves with early matter-dominated era. arXiv:2511.07266 — Induced gravitational waves with an early matter era; linear theory fails at P_ζ ~ 10⁻³.

Public codes #

• GRChombo — BSD-3, AMR (Chombo), stage 1
• GRTeclyn — AMReX port of GRChombo, the base of PBHVlasov (AHFinder branch)
• AMReX — adaptive meshes and particles
• GRTresna — CTTK initial data
• CosmoGRaPH — BSSN + particles (MIT), a candidate for V0
• COSMOS — the Yoo et al. code, without the particle module
• OllinSphere-BiB — a spherical scalar on an expanding background