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Restrictions on initial conditions in cosmological scenarios and implications for simulations of primordial black holes and inflation

Thomas W. Baumgarte1,*, Katy Clough2,†, and John T. Giblin, Jr.3,4,5,‡

  • *Contact author: tbaumgar@bowdoin.edu
  • †Contact author: k.clough@qmul.ac.uk
  • ‡Contact author: giblinj@kenyon.edu
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Phys. Rev. D 112, 123528 – Published 17 December, 2025

DOI: https://doi.org/10.1103/9nsk-jy7f

Abstract

Numerical relativity simulations provide a means by which to study the evolution and end point of strong overdensities in cosmological spacetimes. Specific applications include studies of primordial black hole formation and the robustness of inflation. Here we adopt a toy model previously used in asymptotically flat spacetimes to show that, for given values of the overdensity and the mean curvature, solutions to the Hamiltonian constraint need not exist, and if they do exist they are not unique. Specifically, pairs of solutions exist on two branches, corresponding to strong-field and weak-field solutions, that join at a maximum beyond which solutions cease to exist. As a result, there is a limit to the extent to which an overdensity can be balanced by intrinsic rather than extrinsic curvature on the initial slice. Even below this limit, iterative methods to construct initial data may converge to solutions on either one of the two branches, depending on the starting guess, leading to potentially inconsistent physical results in the evolution.

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References (84)

  1. Bernard J. Carr and S. W. Hawking, Black holes in the early Universe, Mon. Not. R. Astron. Soc. 168, 399 (1974).
  2. Bernard J. Carr, The primordial black hole mass spectrum, Astrophys. J. 201, 1 (1975).
  3. Charles W. Misner and David H. Sharp, Relativistic equations for adiabatic, spherically symmetric gravitational collapse, Phys. Rev. 136, B571 (1964).
  4. Michael M. May and Richard H. White, Hydrodynamic calculations of general-relativistic collapse, Phys. Rev. 141, 1232 (1966).
  5. I. D. Novikov and A. G. Polnarev, The hydrodynamics of primordial black hole formation—Dependence on the equation of state, Astrpn. Zh. 57, 250 (1980).
  6. G. Bicknell and Richard Henriksen, Formation of primordial black holes, Astrophys. J. 232, 670 (1979).
  7. Walter C. Hernandez and Charles W. Misner, Observer time as a coordinate in relativistic spherical hydrodynamics, Astrophys. J. 143, 452 (1966).
  8. Thomas W. Baumgarte, Stuart L. Shapiro, and Saul A. Teukolsky, Computing supernova collapse to neutron stars and black holes, Astrophys. J. 443, 717 (1995).
  9. Jens C. Niemeyer and K. Jedamzik, Dynamics of primordial black hole formation, Phys. Rev. D 59, 124013 (1999).
  10. Masaru Shibata and Misao Sasaki, Black hole formation in the Friedmann universe: Formulation and computation in numerical relativity, Phys. Rev. D 60, 084002 (1999).
  11. Anne M. Green, Andrew R. Liddle, Karim A. Malik, and Misao Sasaki, A new calculation of the mass fraction of primordial black holes, Phys. Rev. D 70, 041502 (2004).
  12. I. Hawke and J. M. Stewart, The dynamics of primordial black hole formation, Classical Quantum Gravity 19, 3687 (2002).
  13. Ilia Musco, John C. Miller, and Luciano Rezzolla, Computations of primordial black hole formation, Classical Quantum Gravity 22, 1405 (2005).
  14. J. C. Miller and S. Motta, Computations of spherical gravitational collapse using null slicing, Classical Quantum Gravity 6, 185 (1989).
  15. John C. Miller and Luciano Rezzolla, Hydrodynamics of the cosmological quark—hadron transition in the presence of long range energy and momentum transfer, Phys. Rev. D 51, 4017 (1995).
  16. Alexander G. Polnarev and Ilia Musco, Curvature profiles as initial conditions for primordial black hole formation, Classical Quantum Gravity 24, 1405 (2007).
  17. Ilia Musco, John C. Miller, and Alexander G. Polnarev, Primordial black hole formation in the radiative era: Investigation of the critical nature of the collapse, Classical Quantum Gravity 26, 235001 (2009).
  18. Ilia Musco and John C. Miller, Primordial black hole formation in the early universe: Critical behaviour and self-similarity, Classical Quantum Gravity 30, 145009 (2013).
  19. Anshul Saini and Dejan Stojkovic, Modified hoop conjecture in expanding spacetimes and primordial black hole production in FRW universe, J. Cosmol. Astropart. Phys. 05 (2018) 071.
  20. Ilia Musco, Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations, Phys. Rev. D 100, 123524 (2019).
  21. Ilia Musco and Theodoros Papanikolaou, Primordial black hole formation for an anisotropic perfect fluid: Initial conditions and estimation of the threshold, Phys. Rev. D 106, 083017 (2022).
  22. Jolyon Bloomfield, Daniel Bulhosa, and Stephen Face, Formalism for primordial black hole formation in spherical symmetry, arXiv:1504.02071.
  23. Heling Deng, Jaume Garriga, and Alexander Vilenkin, Primordial black hole and wormhole formation by domain walls, J. Cosmol. Astropart. Phys. 04 (2017) 050.
  24. Heling Deng and Alexander Vilenkin, Primordial black hole formation by vacuum bubbles, J. Cosmol. Astropart. Phys. 12 (2017) 044.
  25. Vicente Atal, Judith Cid, Albert Escrivà, and Jaume Garriga, PBH in single field inflation: The effect of shape dispersion and non-Gaussianities, J. Cosmol. Astropart. Phys. 05 (2020) 022.
  26. Albert Escrivà, Simulation of primordial black hole formation using pseudo-spectral methods, Phys. Dark Universe 27, 100466 (2020).
  27. Albert Escrivà, Cristiano Germani, and Ravi K. Sheth, Universal threshold for primordial black hole formation, Phys. Rev. D 101, 044022 (2020).
  28. Albert Escrivà and Antonio Enea Romano, Effects of the shape of curvature peaks on the size of primordial black holes, J. Cosmol. Astropart. Phys. 05 (2021) 066.
  29. Albert Escrivà, Eleni Bagui, and Sebastien Clesse, Simulations of PBH formation at the QCD epoch and comparison with the GWTC-3 catalog, J. Cosmol. Astropart. Phys. 05 (2023) 004.
  30. Albert Escrivà, Yuichiro Tada, Shuichiro Yokoyama, and Chul-Moon Yoo, Simulation of primordial black holes with large negative non-Gaussianity, J. Cosmol. Astropart. Phys. 05 (2022) 012.
  31. Albert Escrivà and Javier G. Subils, Primordial black hole formation during a strongly coupled crossover, Phys. Rev. D 107, L041301 (2023).
  32. Albert Escrivà and Chul-Moon Yoo, Primordial black hole formation from overlapping cosmological fluctuations, J. Cosmol. Astropart. Phys. 04 (2024) 048.
  33. Tomohiro Harada, Chul-Moon Yoo, and Yasutaka Koga, Revisiting compaction functions for primordial black hole formation, Phys. Rev. D 108, 043515 (2023).
  34. Chul-Moon Yoo, Tomohiro Harada, and Hirotada Okawa, Threshold of primordial black hole formation in nonspherical collapse, Phys. Rev. D 102, 043526 (2020); 107, 049901(E) (2023).
  35. Albert Escrivà and Chul-Moon Yoo, Simulations of ellipsoidal primordial black hole formation, Phys. Rev. D 112, 083518 (2025).
  36. David J. E. Marsh, Axion cosmology, Phys. Rep. 643, 1 (2016).
  37. Juan Garcia-Bellido and Ester Ruiz Morales, Primordial black holes from single field models of inflation, Phys. Dark Universe 18, 47 (2017).
  38. Guillermo Ballesteros and Marco Taoso, Primordial black hole dark matter from single field inflation, Phys. Rev. D 97, 023501 (2018).
  39. Cristiano Germani and Ilia Musco, Abundance of primordial black holes depends on the shape of the inflationary power spectrum, Phys. Rev. Lett. 122, 141302 (2019).
  40. P. Ivanov, P. Naselsky, and I. Novikov, Inflation and primordial black holes as dark matter, Phys. Rev. D 50, 7173 (1994).
  41. Bernard Carr, Florian Kuhnel, and Marit Sandstad, Primordial black holes as dark matter, Phys. Rev. D 94, 083504 (2016).
  42. Bernard Carr, Kazunori Kohri, Yuuiti Sendouda, and Jun’ichi Yokoyama, Constraints on primordial black holes, Rep. Prog. Phys. 84, 116902 (2021).
  43. Bernard Carr and Florian Kuhnel, Primordial black holes as dark matter: Recent developments, Annu. Rev. Nucl. Part. Sci. 70, 355 (2020).
  44. Albert Escrivà, Florian Kuhnel, and Yuichiro Tada, Primordial black holes, in Black Holes in the Era of Gravitational-Wave Astronomy (Elsvier, New York, 2024), 11.
  45. Norbert Duechting, Supermassive black holes from primordial black hole seeds, Phys. Rev. D 70, 064015 (2004).
  46. Sébastien Clesse and Juan García-Bellido, Massive primordial black holes from hybrid inflation as dark matter and the seeds of galaxies, Phys. Rev. D 92, 023524 (2015).
  47. Robert Brandenberger, Initial conditions for inflation—A short review, Int. J. Mod. Phys. D 26, 1740002 (2016).
  48. Andrei Linde, On the problem of initial conditions for inflation, Found. Phys. 48, 1246 (2018).
  49. Dalia S. Goldwirth and Tsvi Piran, Inhomogeneity and the onset of inflation, Phys. Rev. Lett. 64, 2852 (1990).
  50. William E. East, Matthew Kleban, Andrei Linde, and Leonardo Senatore, Beginning inflation in an inhomogeneous universe, J. Cosmol. Astropart. Phys. 09 (2016) 010.
  51. Katy Clough, Raphael Flauger, and Eugene A. Lim, Robustness of inflation to large tensor perturbations, J. Cosmol. Astropart. Phys. 05 (2018) 065.
  52. Josu C. Aurrekoetxea, Katy Clough, Raphael Flauger, and Eugene A. Lim, The effects of potential shape on inhomogeneous inflation, J. Cosmol. Astropart. Phys. 05 (2020) 030.
  53. Cristian Joana and Sébastien Clesse, Inhomogeneous preinflation across Hubble scales in full general relativity, Phys. Rev. D 103, 083501 (2021).
  54. Maxence Corman and William E. East, Starting inflation from inhomogeneous initial conditions with momentum, J. Cosmol. Astropart. Phys. 10 (2023) 046.
  55. David Garfinkle, Anna Ijjas, and Paul J. Steinhardt, Initial conditions problem in cosmological inflation revisited, Phys. Lett. B 843, 138028 (2023).
  56. Cristian Joana, Beginning inflation in conformally curved spacetimes, Phys. Rev. D 110, 063534 (2024).
  57. Matthew Elley, Josu C. Aurrekoetxea, Katy Clough, Raphael Flauger, Panagiotis Giannadakis, and Eugene A. Lim, Robustness of inflation to kinetic inhomogeneities, J. Cosmol. Astropart. Phys. 01 (2025) 050.
  58. Josu C. Aurrekoetxea, Katy Clough, and Eugene A. Lim, Cosmology using numerical relativity, Living Rev. Relativity 28, 5 (2025).
  59. Thomas W. Baumgarte and Stuart L. Shapiro, Numerical Relativity: Solving Einstein’s Equations on the Computer (Cambridge University Press, Cambridge, England, 2010).
  60. David Garfinkle and Lawrence Mead, Cosmological initial data for numerical relativity, Phys. Rev. D 102, 044022 (2020).
  61. Eloisa Bentivegna, Solving the Einstein constraints in periodic spaces with a multigrid approach, Classical Quantum Gravity 31, 035004 (2014).
  62. Josu C. Aurrekoetxea, Katy Clough, and Eugene A. Lim, CTTK: A new method to solve the initial data constraints in numerical relativity, Classical Quantum Gravity 40, 075003 (2023).
  63. Thomas W. Baumgarte, Niall O Murchadha, and Harald P. Pfeiffer, The Einstein constraints: Uniqueness and non-uniqueness in the conformal thin sandwich approach, Phys. Rev. D 75, 044009 (2007).
  64. Harald P. Pfeiffer and James W. York, Jr, Uniqueness and non-uniqueness in the Einstein constraints, Phys. Rev. Lett. 95, 091101 (2005).
  65. J. W. York, Jr., Kinematics and dynamics of general relativity, Sources of Gravitational Radiation, edited by L. L. Smarr (1979), pp. 83–126.
  66. Jonathan Thornburg, Initial data for dynamic black hole space-times in (3+1) numerical relativity, Phys. Rev. D 59, 104007 (1999).
  67. Charles W. Misner and Kip S. Thorne, Gravitation (W. H. Freeman, San Francisco, 1973).
  68. Koichiro Uehara, Albert Escrivà, Tomohiro Harada, Daiki Saito, and Chul-Moon Yoo, Primordial black hole formation from a type II perturbation in the absence and presence of pressure, J. Cosmol. Astropart. Phys. 08 (2025) 042.
  69. J. W. York, Jr., Initial data for collisions of black holes and other gravitational miscellany, in Frontiers in Numerical Relativity, edited by Charles R. Evans, Lee S. Finn, and David W. Hobill (Cambridge University Press, Cambridge, 1989), pp. 89–109.
  70. Valerio Faraoni, Cosmological and Black Hole Apparent Horizons (Springer, Cham, 2015), Vol. 907, ISBN [Amazon][WorldCat], [Amazon][WorldCat].
  71. G. W. Gibbons and S. W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15, 2738 (1977).
  72. Robert M. Wald, The entropy of black holes, Gen. Relativ. Gravit. 57, 87 (2025).
  73. Abhay Ashtekar and Badri Krishnan, Dynamical horizons and their properties, Phys. Rev. D 68, 104030 (2003).
  74. Abhay Ashtekar and Gregory J. Galloway, Some uniqueness results for dynamical horizons, Adv. Theor. Math. Phys. 9, 1 (2005).
  75. Raphael Bousso and Netta Engelhardt, New area law in general relativity, Phys. Rev. Lett. 115, 081301 (2015).
  76. William H. Press, Saul A. Teukolsky, William T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Third Edition) (Cambridge University Press, Cambridge, England, 2007).
  77. Viatcheslav F. Mukhanov, H. A. Feldman, and Robert H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions, Phys. Rep. 215, 203 (1992).
  78. David H. Lyth, Karim A. Malik, and Misao Sasaki, A general proof of the conservation of the curvature perturbation, J. Cosmol. Astropart. Phys. 05 (2005) 004.
  79. Karim A. Malik and David Wands, Cosmological perturbations, Phys. Rep. 475, 1 (2009).
  80. James M. Bardeen, Paul J. Steinhardt, and Michael S. Turner, Spontaneous creation of almost scale—free density perturbations in an inflationary universe, Phys. Rev. D 28, 679 (1983).
  81. Robert H. Brandenberger, Ronald Kahn, and William H. Press, Cosmological perturbations in the early universe, Phys. Rev. D 28, 1809 (1983).
  82. R. Penrose, Difficulties with inflationary cosmology, Ann. N.Y. Acad. Sci. 571, 249 (1989).
  83. Sean M. Carroll and Jennifer Chen, Spontaneous inflation and the origin of the arrow of time, arXiv:hep-th/0410270.
  84. Andreas Albrecht and Lorenzo Sorbo, Can the universe afford inflation?, Phys. Rev. D 70, 063528 (2004).

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Phys. Rev. D 112, 123528– Published 17 December, 2025
  • Received 6 October 2025
  • Accepted 2 December 2025
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