Harvesting primordial black holes from stochastic trees with $\texttt{FOREST}$

Author(s)

Animali, Chiara, Auclair, Pierre, Blachier, Baptiste, Vennin, Vincent

Abstract

We introduce a novel framework to implement stochastic inflation on stochastic trees, modelling the inflationary expansion as a branching process. Combined with the $\delta N$ formalism, this allows us to generate real-space maps of the curvature perturbation that fully capture quantum diffusion and its non-perturbative backreaction during inflation. Unlike lattice methods, trees do not proceed on a fixed background since new spacetime units emerge dynamically as trees unfold, naturally incorporating metric fluctuations. The recursive structure of stochastic trees also offers remarkable numerical efficiency, and we develop the FOrtran Recursive Exploration of Stochastic Trees ($\texttt{FOREST}$) tool and demonstrate its performance. We show how primordial black holes blossom at unbalanced nodes of the trees, and how their mass distribution can be obtained while automatically accounting for the "cloud-in-cloud" effect. In the "quantum-well" toy model, we find broad mass distributions, with mild power laws terminated by exponential tails. We finally compare our results with existing approximations in the literature and discuss several prospects.

Figures

Elementary vertex of a stochastic tree, where one parent Hubble patch $i$ gives rise to two independent Hubble patches $\ell$ and $m$ after expanding for $\Delta N=\ln(2)/3$.

Elementary vertex of a stochastic tree, where one parent Hubble patch $i$ gives rise to two independent Hubble patches $\ell$ and $m$ after expanding for $\Delta N=\ln(2)/3$.


Example of a stochastic tree, made of multiple elementary vertices of the type shown in \cref{fig:elementary:tree}. The leaves are Hubble patches where inflation ends, and the patch labels correspond to their topological coordinates in the tree (see main text).

Example of a stochastic tree, made of multiple elementary vertices of the type shown in \cref{fig:elementary:tree}. The leaves are Hubble patches where inflation ends, and the patch labels correspond to their topological coordinates in the tree (see main text).


: $\Delta N$: we divide the $x$-axis.

: $\Delta N$: we divide the $x$-axis.


: $2 \Delta N$: we divide the $y$-axis.

: $2 \Delta N$: we divide the $y$-axis.


: $3 \Delta N$: we divide the $z$-axis.

: $3 \Delta N$: we divide the $z$-axis.


: Without PBHs.

: Without PBHs.


: With PBHs.

: With PBHs.


Example of a tree realization with five PBHs in the flat-well toy model, with $\mu=0.8$ and starting from $\phi=\Delta\phi_{\mathrm{well}}$. This is the same realization as \cref{fig:cmap}. The black patches are those for which the coarse-shelled curvature perturbation, used as a proxy for the compaction function, exceeds the PBH formation threshold.

Example of a tree realization with five PBHs in the flat-well toy model, with $\mu=0.8$ and starting from $\phi=\Delta\phi_{\mathrm{well}}$. This is the same realization as \cref{fig:cmap}. The black patches are those for which the coarse-shelled curvature perturbation, used as a proxy for the compaction function, exceeds the PBH formation threshold.


: Probability distribution of the final volume.

: Probability distribution of the final volume.


: Probability distribution of the volume-averaged expansion.

: Probability distribution of the volume-averaged expansion.


Volume-weighted probability distribution of the first-passage time $\mathcal{N}_{x_*\to x_\mathrm{end}}$ through the end-of-inflation hypersurface, in the flat well for $\mu = 0.6, 0.7, 0.8, 0.85, 0.89$ and $x_*=1$. These were obtained with, respectively, $10^{11}, 10^{11}, 5\times 10^{10}, 2\times 10^{10}, 10^{10}$ trees. Full coloured lines stand for the results of \texttt{FOREST}, whereas black dashed lines represent the analytical formula~\eqref{eq:FPTVolWeight}.

Volume-weighted probability distribution of the first-passage time $\mathcal{N}_{x_*\to x_\mathrm{end}}$ through the end-of-inflation hypersurface, in the flat well for $\mu = 0.6, 0.7, 0.8, 0.85, 0.89$ and $x_*=1$. These were obtained with, respectively, $10^{11}, 10^{11}, 5\times 10^{10}, 2\times 10^{10}, 10^{10}$ trees. Full coloured lines stand for the results of \texttt{FOREST}, whereas black dashed lines represent the analytical formula~\eqref{eq:FPTVolWeight}.


: Mass fraction of PBHs.

: Mass fraction of PBHs.


: PBH mass distribution.

: PBH mass distribution.


Different prescriptions for the branching time.

Different prescriptions for the branching time.


Different prescriptions for the branching time.

Different prescriptions for the branching time.


Different prescriptions for the branching time.

Different prescriptions for the branching time.


Example tree considered in \cref{sec:discretization:artefact}.

Example tree considered in \cref{sec:discretization:artefact}.


: $\mu=0.6$, $10^7$ trees

: $\mu=0.6$, $10^7$ trees


: $\mu=0.8$, $10^5$ trees

: $\mu=0.8$, $10^5$ trees


: $\mu=0.6$, $10^7$ trees

: $\mu=0.6$, $10^7$ trees


: $\mu=0.8$, $10^7$ trees

: $\mu=0.8$, $10^7$ trees


Demonstration for \cref{eq:alpha} using of the binary representation of nodes $i$ and $j = i+d$. In this example, $i=1101001_2 = 105$ and $j= 1101111_2=111$ are distant by $d=6$. One has $q = 3$ (three green bits) and $q_\mathrm{T} - q = 3$ (three red bits).

Demonstration for \cref{eq:alpha} using of the binary representation of nodes $i$ and $j = i+d$. In this example, $i=1101001_2 = 105$ and $j= 1101111_2=111$ are distant by $d=6$. One has $q = 3$ (three green bits) and $q_\mathrm{T} - q = 3$ (three red bits).


References
  • [1] SDSS collaboration, D. J. Eisenstein et al., Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies, Astrophys. J. 633 (2005) 560–574, [astro-ph/0501171].
  • [2] Planck collaboration, N. Aghanim et al., Planck 2018 results. I. Overview and the cosmological legacy of Planck, Astron. Astrophys. 641 (2020) A1, [1807.06205].
  • [3] M. M. Ivanov, M. Simonović and M. Zaldarriaga, Cosmological Parameters from the BOSS Galaxy Power Spectrum, JCAP 05 (2020) 042, [1909.05277].
  • [4] V. F. Mukhanov and G. V. Chibisov, Quantum Fluctuations and a Nonsingular Universe, JETP Lett. 33 (1981) 532–535.
  • [5] V. F. Mukhanov and G. Chibisov, The Vacuum energy and large scale structure of the universe, Sov. Phys. JETP 56 (1982) 258–265.
  • [6] A. A. Starobinsky, Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations, Phys. Lett. B 117 (1982) 175–178.
  • [7] A. H. Guth and S. Pi, Fluctuations in the New Inflationary Universe, Phys. Rev. Lett. 49 (1982) 1110–1113.
  • [8] S. Hawking, The Development of Irregularities in a Single Bubble Inflationary Universe, Phys. Lett. B 115 (1982) 295.
  • [9] J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Spontaneous Creation of Almost Scale Free Density Perturbations in an Inflationary Universe, Phys. Rev. D 28 (1983) 679.
  • [10] A. A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Adv. Ser. Astrophys. Cosmol. 3 (1987) 130–133.
  • [11] K. Sato, First-order phase transition of a vacuum and the expansion of the Universe, Mon. Not. Roy. Astron. Soc. 195 (1981) 467–479.
  • [12] A. H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Adv. Ser. Astrophys. Cosmol. 3 (1987) 139–148.
  • [13] A. D. Linde, A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Adv. Ser. Astrophys. Cosmol. 3 (1987) 149–153.
  • [14] A. Albrecht and P. J. Steinhardt, Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking, Adv. Ser. Astrophys. Cosmol. 3 (1987) 158–161.
  • [15] A. D. Linde, Chaotic Inflation, Phys. Lett. B 129 (1983) 177–181.
  • [16] E. Asencio, I. Banik and P. Kroupa, A massive blow for ΛCDM − the high redshift, mass, and collision velocity of the interacting galaxy cluster El Gordo contradicts concordance cosmology, Mon. Not. Roy. Astron. Soc. 500 (2020) 5249–5267, [2012.03950].
  • [17] A. H. Gonzalez, S. A. Stanford, M. Brodwin, C. Fedeli, A. Dey, P. R. M. Eisenhardt et al., IDCS J1426.5+3508: Cosmological Implications of a Massive, Strong Lensing Cluster at z = 1.75, ApJ 753 (July, 2012) 163, [1205.3788].
  • [18] S. L. Finkelstein et al., A Rapidly Star-forming Galaxy 700 Million Years After the Big Bang at z=7.51, Nature 502 (2013) 524, [1310.6031].
  • [19] A.-C. Eilers, R. A. Simcoe, M. Yue, R. Mackenzie, J. Matthee, D. Durovcikova et al., EIGER. III. JWST/NIRCam Observations of the Ultraluminous High-redshift Quasar J0100+2802, Astrophys. J. 950 (2023) 68, [2211.16261].
  • [20] C. T. Donnan, D. J. McLeod, J. S. Dunlop, R. J. McLure, A. C. Carnall, R. Begley et al., The evolution of the galaxy UV luminosity function at redshifts z ≈ 8 - 15 from deep JWST and ground-based near-infrared imaging, MNRAS 518 (Feb., 2023) 6011–6040, [2207.12356].
  • [21] Y. Harikane, M. Ouchi, M. Oguri, Y. Ono, K. Nakajima, Y. Isobe et al., A Comprehensive Study of Galaxies at z ∼ 9–16 Found in the Early JWST Data: Ultraviolet Luminosity Functions and Cosmic Star Formation History at the Pre-reionization Epoch, Astrophys. J. Suppl. 265 (2023) , [2208.01612].
  • [22] R. P. Naidu, P. A. Oesch, P. van Dokkum, E. J. Nelson, K. A. Suess, G. Brammer et al., Two Remarkably Luminous Galaxy Candidates at z ≈ 10-12 Revealed by JWST, ApJ 940 (Nov., 2022) L14, [2207.09434].
  • [23] S. L. Finkelstein, M. B. Bagley, P. Arrabal Haro, M. Dickinson, H. C. Ferguson, J. S. Kartaltepe et al., A Long Time Ago in a Galaxy Far, Far Away: A Candidate z ∼ 12 Galaxy in Early JWST CEERS Imaging, ApJ 940 (Dec., 2022) L55, [2207.12474].
  • [24] H. Atek, M. Shuntov, L. J. Furtak, J. Richard, J.-P. Kneib, G. Mahler et al., Revealing galaxy candidates out to z 16 with JWST observations of the lensing cluster SMACS0723, MNRAS 519 (Feb., 2023) 1201–1220, [2207.12338].
  • [25] G. Roberts-Borsani, T. Morishita, T. Treu, G. Brammer, V. Strait, X. Wang et al., Early Results from GLASS-JWST. I: Confirmation of Lensed z ≥ 7 Lyman-break Galaxies behind the Abell 2744 Cluster with NIRISS, ApJ 938 (Oct., 2022) L13, [2207.11387].
  • [26] M. Castellano, A. Fontana, T. Treu, P. Santini, E. Merlin, N. Leethochawalit et al., Early Results from GLASS-JWST. III. Galaxy Candidates at z 9-15, ApJ 938 (Oct., 2022) L15, [2207.09436].
  • [27] P. Santini, A. Fontana, M. Castellano, N. Leethochawalit, M. Trenti, T. Treu et al., Early Results from GLASS-JWST. XI. Stellar Masses and Mass-to-light Ratio of z ¿ 7 Galaxies, ApJ 942 (Jan., 2023) L27, [2207.11379].
  • [28] N. J. Adams, C. J. Conselice, L. Ferreira, D. Austin, J. A. A. Trussler, I. Juodžbalis et al., Discovery and properties of ultra-high redshift galaxies (9 ¡ z ¡ 12) in the JWST ERO SMACS 0723 Field, MNRAS 518 (Jan., 2023) 4755–4766, [2207.11217].
  • [29] A. Escrivà, F. Kuhnel and Y. Tada, Primordial Black Holes, 2211.05767.
  • [30] B. Carr, S. Clesse, J. Garcia-Bellido, M. Hawkins and F. Kuhnel, Observational evidence for primordial black holes: A positivist perspective, Phys. Rept. 1054 (2024) 1–68, [2306.03903].
  • [31] LISA Cosmology Working Group collaboration, E. Bagui et al., Primordial black holes and their gravitational-wave signatures, 2310.19857.
  • [32] K. N. Ananda, C. Clarkson and D. Wands, The Cosmological gravitational wave background from primordial density perturbations, Phys. Rev. D 75 (2007) 123518, [gr-qc/0612013].
  • [33] D. Baumann, P. J. Steinhardt, K. Takahashi and K. Ichiki, Gravitational Wave Spectrum Induced by Primordial Scalar Perturbations, Phys. Rev. D 76 (2007) 084019, [hep-th/0703290].
  • [34] EPTA, InPTA collaboration, J. Antoniadis et al., The second data release from the European Pulsar Timing Array - IV. Implications for massive black holes, dark matter, and the early Universe, Astron. Astrophys. 685 (2024) A94, [2306.16227].
  • [35] NANOGrav collaboration collaboration, A. Afzal et al., The NANOGrav 15 yr Data Set: Search for Signals from New Physics, The Astrophysical Journal Letters 951 (2023) L11, [2306.16219].
  • [36] D. S. Salopek and J. R. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models, Phys. Rev. D 42 (1990) 3936–3962.
  • [37] M. Sasaki and E. D. Stewart, A General analytic formula for the spectral index of the density perturbations produced during inflation, Prog. Theor. Phys. 95 (1996) 71–78, [astro-ph/9507001].
  • [38] D. Wands, K. A. Malik, D. H. Lyth and A. R. Liddle, A New approach to the evolution of cosmological perturbations on large scales, Phys. Rev. D 62 (2000) 043527, [astro-ph/0003278].
  • [39] D. H. Lyth and D. Wands, Conserved cosmological perturbations, Phys. Rev. D 68 (2003) 103515, [astro-ph/0306498].
  • [40] G. I. Rigopoulos and E. P. S. Shellard, The separate universe approach and the evolution of nonlinear superhorizon cosmological perturbations, Phys. Rev. D 68 (2003) 123518, [astro-ph/0306620].
  • [41] D. H. Lyth and Y. Rodriguez, The Inflationary prediction for primordial non-Gaussianity, Phys. Rev. Lett. 95 (2005) 121302, [astro-ph/0504045].
  • [42] D. Artigas, J. Grain and V. Vennin, Hamiltonian formalism for cosmological perturbations: the separate-universe approach, JCAP 02 (2022) 001, [2110.11720].
  • [43] J. H. P. Jackson, H. Assadullahi, A. D. Gow, K. Koyama, V. Vennin and D. Wands, The separate-universe approach and sudden transitions during inflation, JCAP 05 (2024) 053, [2311.03281].
  • [44] A. A. Starobinsky, Multicomponent de Sitter (Inflationary) Stages and the Generation of Perturbations, JETP Lett. 42 (1985) 152–155.
  • [45] M. Sasaki and T. Tanaka, Superhorizon scale dynamics of multiscalar inflation, Prog. Theor. Phys. 99 (1998) 763–782, [gr-qc/9801017].
  • [46] D. H. Lyth, K. A. Malik and M. Sasaki, A General proof of the conservation of the curvature perturbation, JCAP 05 (2005) 004, [astro-ph/0411220].
  • [47] A. A. Starobinsky, Stochastic De Sitter (inflationary) stage in the early universe, Lect. Notes Phys. 246 (1986) 107–126.
  • [48] K. Enqvist, S. Nurmi, D. Podolsky and G. Rigopoulos, On the divergences of inflationary superhorizon perturbations, JCAP 04 (2008) 025, [0802.0395].
  • [49] T. Fujita, M. Kawasaki, Y. Tada and T. Takesako, A new algorithm for calculating the curvature perturbations in stochastic inflation, JCAP 12 (2013) 036, [1308.4754].
  • [50] V. Vennin and A. A. Starobinsky, Correlation Functions in Stochastic Inflation, Eur. Phys. J. C 75 (2015) 413, [1506.04732].
  • [51] C. Pattison, V. Vennin, H. Assadullahi and D. Wands, Quantum diffusion during inflation and primordial black holes, JCAP 10 (2017) 046, [1707.00537].
  • [52] J. M. Ezquiaga, J. Garcı́a-Bellido and V. Vennin, The exponential tail of inflationary fluctuations: consequences for primordial black holes, JCAP 03 (2020) 029, [1912.05399].
  • [53] G. Panagopoulos and E. Silverstein, Primordial Black Holes from non-Gaussian tails, 1906.02827.
  • [54] C. Pattison, V. Vennin, D. Wands and H. Assadullahi, Ultra-slow-roll inflation with quantum diffusion, JCAP 04 (2021) 080, [2101.05741].
  • [55] E. Tomberg, A numerical approach to stochastic inflation and primordial black holes, J. Phys. Conf. Ser. 2156 (2021) 012010, [2110.10684].
  • [56] G. Rigopoulos and A. Wilkins, Inflation is always semi-classical: diffusion domination overproduces Primordial Black Holes, JCAP 12 (2021) 027, [2107.05317].
  • [57] A. Achucarro, S. Cespedes, A.-C. Davis and G. A. Palma, The hand-made tail: non-perturbative tails from multifield inflation, JHEP 05 (2022) 052, [2112.14712].
  • [58] J. M. Ezquiaga, J. Garcı́a-Bellido and V. Vennin, Massive Galaxy Clusters Like El Gordo Hint at Primordial Quantum Diffusion, Phys. Rev. Lett. 130 (2023) 121003, [2207.06317].
  • [59] C. Animali and V. Vennin, Primordial black holes from stochastic tunnelling, JCAP 02 (2023) 043, [2210.03812].
  • [60] Y.-F. Cai, X.-H. Ma, M. Sasaki, D.-G. Wang and Z. Zhou, Highly non-Gaussian tails and primordial black holes from single-field inflation, JCAP 12 (2022) 034, [2207.11910].
  • [61] A. D. Gow, H. Assadullahi, J. H. P. Jackson, K. Koyama, V. Vennin and D. Wands, Non-perturbative non-Gaussianity and primordial black holes, EPL 142 (2023) 49001, [2211.08348].
  • [62] V. Briaud and V. Vennin, Uphill inflation, JCAP 06 (2023) 029, [2301.09336].
  • [63] V. Vennin and D. Wands, Quantum diffusion and large primordial perturbations from inflation, 2402.12672.
  • [64] R. Inui, H. Motohashi, S. Pi, Y. Tada and S. Yokoyama, Constant roll and non-Gaussian tail in light of logarithmic duality, 2409.13500.
  • [65] D. Sharma, Stochastic inflation and non-perturbative power spectrum beyond slow roll, 2411.08854.
  • [66] K. Ando and V. Vennin, Power spectrum in stochastic inflation, JCAP 04 (2021) 057, [2012.02031].
  • [67] Y. Tada and V. Vennin, Statistics of coarse-grained cosmological fields in stochastic inflation, JCAP 02 (2022) 021, [2111.15280].
  • [68] C. Animali and V. Vennin, Clustering of primordial black holes from quantum diffusion during inflation, JCAP 08 (2024) 026, [2402.08642].
  • [69] A. D. Linde, D. A. Linde and A. Mezhlumian, From the Big Bang theory to the theory of a stationary universe, Phys. Rev. D 49 (1994) 1783–1826, [gr-qc/9306035].
  • [70] M. Jain and M. P. Hertzberg, Statistics of Inflating Regions in Eternal Inflation, Phys. Rev. D 100 (2019) 023513, [1904.04262].
  • [71] J. Felsenstein, Inferring Phylogenies. Sinauer Associates, Sunderland, Massachusetts, 2004.
  • [72] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, 05, 1991, 10.1093/oso/9780198545996.001.0001.
  • [73] L. Breiman, Random Forests, Machine Learning 45 (2001) 5–32.
  • [74] P. Glasserman, Monte Carlo Methods in Financial Engineering. Springer, 2003, 10.1007/978-0-387-21617-1.
  • [75] I. Hanski, Metapopulation Ecology. Oxford University Press, 1999.
  • [76] W. E. East, M. Kleban, A. Linde and L. Senatore, Beginning inflation in an inhomogeneous universe, JCAP 09 (2016) 010, [1511.05143].
  • [77] K. Clough, E. A. Lim, B. S. DiNunno, W. Fischler, R. Flauger and S. Paban, Robustness of Inflation to Inhomogeneous Initial Conditions, JCAP 09 (2017) 025, [1608.04408].
  • [78] J. K. Bloomfield, P. Fitzpatrick, K. Hilbert and D. I. Kaiser, Onset of inflation amid backreaction from inhomogeneities, Phys. Rev. D 100 (2019) 063512, [1906.08651].
  • [79] C. Joana and S. Clesse, Inhomogeneous preinflation across Hubble scales in full general relativity, Phys. Rev. D 103 (2021) 083501, [2011.12190].
  • [80] M. Corman and W. E. East, Starting inflation from inhomogeneous initial conditions with momentum, JCAP 10 (2023) 046, [2212.04479].
  • [81] G. N. Felder and I. Tkachev, LATTICEEASY: A Program for lattice simulations of scalar fields in an expanding universe, Comput. Phys. Commun. 178 (2008) 929–932, [hep-ph/0011159].
  • [82] A. V. Frolov, DEFROST: A New Code for Simulating Preheating after Inflation, JCAP 11 (2008) 009, [0809.4904].
  • [83] R. Easther, H. Finkel and N. Roth, PSpectRe: A Pseudo-Spectral Code for (P)reheating, JCAP 10 (2010) 025, [1005.1921].
  • [84] D. G. Figueroa, A. Florio, F. Torrenti and W. Valkenburg, CosmoLattice: A modern code for lattice simulations of scalar and gauge field dynamics in an expanding universe, Comput. Phys. Commun. 283 (2023) 108586, [2102.01031].
  • [85] A. Caravano, E. Komatsu, K. D. Lozanov and J. Weller, Lattice simulations of inflation, JCAP 12 (2021) 010, [2102.06378].
  • [86] Y. Mizuguchi, T. Murata and Y. Tada, STOLAS: STOchastic LAttice Simulation of cosmic inflation, 2405.10692.
  • [87] M. Shibata and M. Sasaki, Black hole formation in the Friedmann universe: Formulation and computation in numerical relativity, Phys. Rev. D 60 (1999) 084002, [gr-qc/9905064].
  • [88] T. Harada, C.-M. Yoo, T. Nakama and Y. Koga, Cosmological long-wavelength solutions and primordial black hole formation, Phys. Rev. D 91 (2015) 084057, [1503.03934].
  • [89] I. Musco, Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations, Phys. Rev. D 100 (2019) 123524, [1809.02127].
  • [90] L. Pinol, S. Renaux-Petel and Y. Tada, A manifestly covariant theory of multifield stochastic inflation in phase space: solving the discretisation ambiguity in stochastic inflation, JCAP 04 (2021) 048, [2008.07497].
  • [91] H. Firouzjahi, A. Nassiri-Rad and M. Noorbala, Stochastic Ultra Slow Roll Inflation, JCAP 01 (2019) 040, [1811.02175].
  • [92] C. Pattison, V. Vennin, H. Assadullahi and D. Wands, Stochastic inflation beyond slow roll, JCAP 07 (2019) 031, [1905.06300].
  • [93] G. Ballesteros, J. Rey, M. Taoso and A. Urbano, Stochastic inflationary dynamics beyond slow-roll and consequences for primordial black hole formation, JCAP 08 (2020) 043, [2006.14597].
  • [94] S. Raatikainen, S. Räsänen and E. Tomberg, Primordial Black Hole Compaction Function from Stochastic Fluctuations in Ultraslow-Roll Inflation, Phys. Rev. Lett. 133 (2024) 121403, [2312.12911].
  • [95] J. H. P. Jackson, H. Assadullahi, A. D. Gow, K. Koyama, V. Vennin and D. Wands, Stochastic inflation beyond slow roll: noise modelling and importance sampling, 2410.13683.
  • [96] J. Grain and V. Vennin, Stochastic inflation in phase space: Is slow roll a stochastic attractor?, JCAP 05 (2017) 045, [1703.00447].
  • [97] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations. Springer Berlino, Heidelberg, 1992, 10.1007/978-3-662-12616-5.
  • [98] S. Young, C. T. Byrnes and M. Sasaki, Calculating the mass fraction of primordial black holes, JCAP 07 (2014) 045, [1405.7023].
  • [99] I. Musco, V. De Luca, G. Franciolini and A. Riotto, Threshold for primordial black holes. II. A simple analytic prescription, Phys. Rev. D 103 (2021) 063538, [2011.03014].
  • [100] A. Kehagias, D. Perrone and A. Riotto, Why the Universal Threshold for Primordial Black Hole Formation is Universal, 2405.05208.
  • [101] S. Young, The primordial black hole formation criterion re-examined: Parametrisation, timing and the choice of window function, Int. J. Mod. Phys. D 29 (2019) 2030002, [1905.01230].
  • [102] K. Tokeshi, K. Inomata and J. Yokoyama, Window function dependence of the novel mass function of primordial black holes, JCAP 12 (2020) 038, [2005.07153].
  • [103] A. Kalaja, N. Bellomo, N. Bartolo, D. Bertacca, S. Matarrese, I. Musco et al., From Primordial Black Holes Abundance to Primordial Curvature Power Spectrum (and back), JCAP 10 (2019) 031, [1908.03596].
  • [104] C. Germani and I. Musco, Abundance of Primordial Black Holes Depends on the Shape of the Inflationary Power Spectrum, Phys. Rev. Lett. 122 (2019) 141302, [1805.04087].
  • [105] A. Escrivà, C. Germani and R. K. Sheth, Universal threshold for primordial black hole formation, Phys. Rev. D 101 (2020) 044022, [1907.13311].
  • [106] J. C. Niemeyer and K. Jedamzik, Near-critical gravitational collapse and the initial mass function of primordial black holes, Phys. Rev. Lett. 80 (1998) 5481–5484, [astro-ph/9709072].
  • [107] K. Jedamzik, The Cloud in cloud problem in the Press-Schechter formalism of hierarchical structure formation, Astrophys. J. 448 (1995) 1–17, [astro-ph/9408080].
  • [108] S. Winitzki, Reheating-volume measure for random-walk inflation, Phys. Rev. D 78 (2008) 063517, [0805.3940].
  • [109] S. Dubovsky, L. Senatore and G. Villadoro, The Volume of the Universe after Inflation and de Sitter Entropy, JHEP 04 (2009) 118, [0812.2246].
  • [110] V. Vennin, Stochastic inflation and primordial black holes, Habilitation thesis (2020) , [2009.08715].
  • [111] W. H. Press and P. Schechter, Formation of galaxies and clusters of galaxies by selfsimilar gravitational condensation, Astrophys. J. 187 (1974) 425–438.
  • [112] A. A. Starobinsky and J. Yokoyama, Equilibrium state of a selfinteracting scalar field in the de Sitter background, Phys. Rev. D 50 (1994) 6357–6368, [astro-ph/9407016].
  • [113] N. A. Chernikov and E. A. Tagirov, Quantum theory of scalar fields in de Sitter space-time, Ann. Inst. H. Poincare A Phys. Theor. 9 (1968) 109.
  • [114] E. A. Tagirov, Consequences of field quantization in de Sitter type cosmological models, Annals Phys. 76 (1973) 561–579.
  • [115] T. S. Bunch and P. C. W. Davies, Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting, Proc. Roy. Soc. Lond. A 360 (1978) 117–134.
  • [116] J. H. P. Jackson, H. Assadullahi, K. Koyama, V. Vennin and D. Wands, Numerical simulations of stochastic inflation using importance sampling, JCAP 10 (2022) 067, [2206.11234].
  • [117] P. Auclair and B. Blachier, Small-scale clustering of primordial black holes: Cloud-in-cloud and exclusion effects, Phys. Rev. D 109 (2024) 123538, [2402.00600].
  • [118] J. F. N. n. Siles and J. Garcı́a-Bellido Capdevila, Primordial Black Hole clusters, phenomenology & implications, 2405.06391.
  • [119] W. R. Coulton, O. H. E. Philcox and F. Villaescusa-Navarro, The Impact of Non-Gaussian Primordial Tails on Cosmological Observables, 2406.15546.