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Numerical simulations of stochastic inflation using importance sampling. (English) Zbl 1515.83150


MSC:

83C57 Black holes
81V22 Unified quantum theories
62D05 Sampling theory, sample surveys
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics

Software:

DLMF

References:

[1] Starobinsky, Alexei A.; Khalatnikov, I. M.; Mineev, V. P., A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B, 91, 99-102 (1980) · Zbl 1371.83222 · doi:10.1016/0370-2693(80)90670-X
[2] Sato, K., First Order Phase Transition of a Vacuum and Expansion of the Universe, Mon. Not. Roy. Astron. Soc., 195, 467-479 (1981)
[3] Guth, Alan H.; Fang, Li-Zhi; Ruffini, R., The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Phys. Rev. D, 23, 347-356 (1981) · Zbl 1371.83202 · doi:10.1103/PhysRevD.23.347
[4] Linde, Andrei D.; Fang, Li-Zhi; Ruffini, R., A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Phys. Lett. B, 108, 389-393 (1982) · doi:10.1016/0370-2693(82)91219-9
[5] Albrecht, Andreas; Steinhardt, Paul J.; Fang, Li-Zhi; Ruffini, R., Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking, Phys. Rev. Lett., 48, 1220-1223 (1982) · doi:10.1103/PhysRevLett.48.1220
[6] Linde, Andrei D., Chaotic Inflation, Phys. Lett. B, 129, 177-181 (1983) · doi:10.1016/0370-2693(83)90837-7
[7] SDSS Collaboration; Eisenstein, Daniel J., SDSS-III: Massive Spectroscopic Surveys of the Distant Universe, the Milky Way Galaxy, and Extra-Solar Planetary Systems, Astron. J., 142, 72 (2011) · doi:10.1088/0004-6256/142/3/72
[8] Blake, Chris, The WiggleZ Dark Energy Survey: mapping the distance-redshift relation with baryon acoustic oscillations, Mon. Not. Roy. Astron. Soc., 418, 1707-1724 (2011) · doi:10.1111/j.1365-2966.2011.19592.x
[9] BOSS Collaboration; Dawson, Kyle S., The Baryon Oscillation Spectroscopic Survey of SDSS-III, Astron. J., 145, 10 (2013) · doi:10.1088/0004-6256/145/1/10
[10] BOSS Collaboration; Alam, Shadab, The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample, Mon. Not. Roy. Astron. Soc., 470, 2617-2652 (2017) · doi:10.1093/mnras/stx721
[11] Planck Collaboration; Ade, P. A. R., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys., 594, A13 (2016) · doi:10.1051/0004-6361/201525830
[12] Saadeh, Daniela; Feeney, Stephen M.; Pontzen, Andrew; Peiris, Hiranya V.; McEwen, Jason D., How isotropic is the Universe?, Phys. Rev. Lett., 117 (2016) · doi:10.1103/PhysRevLett.117.131302
[13] Planck Collaboration; Aghanim, N., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys., 641, A6 (2020) · doi:10.1051/0004-6361/201833910
[14] Mukhanov, Viatcheslav F.; Chibisov, G. V., Quantum Fluctuations and a Nonsingular Universe, JETP Lett., 33, 532-535 (1981)
[15] Mukhanov, Viatcheslav F.; Chibisov, G. V., The Vacuum energy and large scale structure of the universe, Sov. Phys. JETP, 56, 258-265 (1982)
[16] Starobinsky, Alexei A., Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations, Phys. Lett. B, 117, 175-178 (1982) · doi:10.1016/0370-2693(82)90541-X
[17] Guth, Alan H.; Pi, S. Y., Fluctuations in the New Inflationary Universe, Phys. Rev. Lett., 49, 1110-1113 (1982) · doi:10.1103/PhysRevLett.49.1110
[18] Hawking, S. W., The Development of Irregularities in a Single Bubble Inflationary Universe, Phys. Lett. B, 115, 295 (1982) · doi:10.1016/0370-2693(82)90373-2
[19] Bardeen, James M.; Steinhardt, Paul J.; Turner, Michael S., Spontaneous Creation of Almost Scale - Free Density Perturbations in an Inflationary Universe, Phys. Rev. D, 28, 679 (1983) · doi:10.1103/PhysRevD.28.679
[20] Planck Collaboration; Ade, P. A. R., Planck 2015 results. XX. Constraints on inflation, Astron. Astrophys., 594, A20 (2016) · doi:10.1051/0004-6361/201525898
[21] D.H. Lyth and A.R. Liddle, The primordial density perturbation: Cosmology, inflation and the origin of structure, Cambridge University Press (2009). · Zbl 1167.83303
[22] Liddle, Andrew R.; Leach, Samuel M., How long before the end of inflation were observable perturbations produced?, Phys. Rev. D, 68 (2003) · doi:10.1103/PhysRevD.68.103503
[23] Chluba, Jens; Hamann, Jan; Patil, Subodh P., Features and New Physical Scales in Primordial Observables: Theory and Observation, Int. J. Mod. Phys. D, 24 (2015) · Zbl 1337.83001 · doi:10.1142/S0218271815300232
[24] Christensen, Nelson, Stochastic Gravitational Wave Backgrounds, Rept. Prog. Phys., 82 (2019) · doi:10.1088/1361-6633/aae6b5
[25] Hawking, Stephen, Gravitationally collapsed objects of very low mass, Mon. Not. Roy. Astron. Soc., 152, 75 (1971)
[26] Carr, Bernard J.; Hawking, S. W., Black holes in the early Universe, Mon. Not. Roy. Astron. Soc., 168, 399-415 (1974)
[27] Carr, Bernard J., The Primordial black hole mass spectrum, Astrophys. J., 201, 1-19 (1975) · doi:10.1086/153853
[28] García-Bellido, Juan; Giardini, Domencio; Jetzer, Philippe, Massive Primordial Black Holes as Dark Matter and their detection with Gravitational Waves, J. Phys. Conf. Ser., 840 (2017) · doi:10.1088/1742-6596/840/1/012032
[29] LIGO Scientific, Virgo Collaboration; Abbott, B. P., GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs, Phys. Rev. X, 9 (2019) · doi:10.1103/PhysRevX.9.031040
[30] LIGO Scientific, VIRGO Collaboration; Abbott, R., GWTC-2.1: Deep Extended Catalog of Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run (2021)
[31] LIGO Scientific, Virgo Collaboration; Abbott, R., Properties and Astrophysical Implications of the 150 M_⊙ Binary Black Hole Merger GW190521, Astrophys. J. Lett., 900, L13 (2020) · doi:10.3847/2041-8213/aba493
[32] Clesse, Sebastien; Garcia-Bellido, Juan, GW190425, GW190521 and GW190814: Three candidate mergers of primordial black holes from the QCD epoch (2020)
[33] KAGRA, VIRGO, LIGO Scientific Collaboration; Abbott, R., First joint observation by the underground gravitational-wave detector KAGRA with GEO 600, PTEP, 2022 (2022) · doi:10.1093/ptep/ptac073
[34] Carr, Bernard; Kuhnel, Florian; Sandstad, Marit, Primordial Black Holes as Dark Matter, Phys. Rev. D, 94 (2016) · doi:10.1103/PhysRevD.94.083504
[35] Carr, Bernard; Kohri, Kazunori; Sendouda, Yuuiti; Yokoyama, Jun’ichi, Constraints on primordial black holes, Rept. Prog. Phys., 84 (2021) · doi:10.1088/1361-6633/ac1e31
[36] Carr, Bernard; Kuhnel, Florian, Primordial Black Holes as Dark Matter: Recent Developments, Ann. Rev. Nucl. Part. Sci., 70, 355-394 (2020) · doi:10.1146/annurev-nucl-050520-125911
[37] Green, Anne M.; Kavanagh, Bradley J., Primordial Black Holes as a dark matter candidate, J. Phys. G, 48 (2021) · doi:10.1088/1361-6471/abc534
[38] Shibata, Masaru; Sasaki, Misao, Black hole formation in the Friedmann universe: Formulation and computation in numerical relativity, Phys. Rev. D, 60 (1999) · doi:10.1103/PhysRevD.60.084002
[39] Harada, Tomohiro; Yoo, Chul-Moon; Nakama, Tomohiro; Koga, Yasutaka, Cosmological long-wavelength solutions and primordial black hole formation, Phys. Rev. D, 91 (2015) · doi:10.1103/PhysRevD.91.084057
[40] Escrivà, Albert; Germani, Cristiano; Sheth, Ravi K., Universal threshold for primordial black hole formation, Phys. Rev. D, 101 (2020) · doi:10.1103/PhysRevD.101.044022
[41] Musco, Ilia, Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations, Phys. Rev. D, 100 (2019) · doi:10.1103/PhysRevD.100.123524
[42] Young, Sam; Musco, Ilia; Byrnes, Christian T., Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation, JCAP, 11 (2019) · Zbl 07502067 · doi:10.1088/1475-7516/2019/11/012
[43] Musco, Ilia; Miller, John C.; Polnarev, Alexander G., Primordial black hole formation in the radiative era: Investigation of the critical nature of the collapse, Class. Quant. Grav., 26 (2009) · Zbl 1181.83131 · doi:10.1088/0264-9381/26/23/235001
[44] Garcia-Bellido, Juan; Ruiz Morales, Ester, Primordial black holes from single field models of inflation, Phys. Dark Univ., 18, 47-54 (2017) · doi:10.1016/j.dark.2017.09.007
[45] Ezquiaga, Jose Maria; Garcia-Bellido, Juan; Ruiz Morales, Ester, Primordial Black Hole production in Critical Higgs Inflation, Phys. Lett. B, 776, 345-349 (2018) · doi:10.1016/j.physletb.2017.11.039
[46] Germani, Cristiano; Prokopec, Tomislav, On primordial black holes from an inflection point, Phys. Dark Univ., 18, 6-10 (2017) · doi:10.1016/j.dark.2017.09.001
[47] Motohashi, Hayato; Hu, Wayne, Primordial Black Holes and Slow-Roll Violation, Phys. Rev. D, 96 (2017) · doi:10.1103/PhysRevD.96.063503
[48] Ballesteros, Guillermo; Taoso, Marco, Primordial black hole dark matter from single field inflation, Phys. Rev. D, 97 (2018) · doi:10.1103/PhysRevD.97.023501
[49] Rasanen, Syksy; Tomberg, Eemeli, Planck scale black hole dark matter from Higgs inflation, JCAP, 01 (2019) · doi:10.1088/1475-7516/2019/01/038
[50] Geller, Sarah R.; Qin, Wenzer; McDonough, Evan; Kaiser, David I., Primordial black holes from multifield inflation with nonminimal couplings, Phys. Rev. D, 106 (2022) · doi:10.1103/PhysRevD.106.063535
[51] Pattison, Chris; Vennin, Vincent; Assadullahi, Hooshyar; Wands, David, Quantum diffusion during inflation and primordial black holes, JCAP, 10 (2017) · Zbl 1515.83170 · doi:10.1088/1475-7516/2017/10/046
[52] Biagetti, Matteo; Franciolini, Gabriele; Kehagias, Alex; Riotto, Antonio, Primordial Black Holes from Inflation and Quantum Diffusion, JCAP, 07 (2018) · Zbl 1527.83047 · doi:10.1088/1475-7516/2018/07/032
[53] Ezquiaga, Jose María; García-Bellido, Juan, Quantum diffusion beyond slow-roll: implications for primordial black-hole production, JCAP, 08 (2018) · Zbl 1536.83169 · doi:10.1088/1475-7516/2018/08/018
[54] Firouzjahi, Hassan; Nassiri-Rad, Amin; Noorbala, Mahdiyar, Stochastic Ultra Slow Roll Inflation, JCAP, 01 (2019) · Zbl 1542.83052 · doi:10.1088/1475-7516/2019/01/040
[55] Ballesteros, Guillermo; Rey, Julián; Taoso, Marco; Urbano, Alfredo, Stochastic inflationary dynamics beyond slow-roll and consequences for primordial black hole formation, JCAP, 08 (2020) · Zbl 1492.83077 · doi:10.1088/1475-7516/2020/08/043
[56] Pattison, Chris; Vennin, Vincent; Wands, David; Assadullahi, Hooshyar, Ultra-slow-roll inflation with quantum diffusion, JCAP, 04 (2021) · Zbl 1486.83151 · doi:10.1088/1475-7516/2021/04/080
[57] Prokopec, Tomislav; Rigopoulos, Gerasimos, ΔN and the stochastic conveyor belt of ultra slow-roll inflation, Phys. Rev. D, 104 (2021) · doi:10.1103/PhysRevD.104.083505
[58] Rigopoulos, Gerasimos; Wilkins, Ashley, Inflation is always semi-classical: diffusion domination overproduces Primordial Black Holes, JCAP, 12 (2021) · Zbl 1487.83133 · doi:10.1088/1475-7516/2021/12/027
[59] Starobinsky, Alexei A., STOCHASTIC DE SITTER (INFLATIONARY) STAGE IN THE EARLY UNIVERSE, Lect. Notes Phys., 246, 107-126 (1986) · doi:10.1007/3-540-16452-9_6
[60] Nambu, Yasusada; Sasaki, Misao, Stochastic Stage of an Inflationary Universe Model, Phys. Lett. B, 205, 441-446 (1988) · doi:10.1016/0370-2693(88)90974-4
[61] Nambu, Yasusada; Sasaki, Misao, Stochastic Approach to Chaotic Inflation and the Distribution of Universes, Phys. Lett. B, 219, 240-246 (1989) · doi:10.1016/0370-2693(89)90385-7
[62] Kandrup, Henry E., STOCHASTIC INFLATION AS A TIME DEPENDENT RANDOM WALK, Phys. Rev. D, 39, 2245 (1989) · doi:10.1103/PhysRevD.39.2245
[63] Nakao, Ken-ichi; Nambu, Yasusada; Sasaki, Misao, Stochastic Dynamics of New Inflation, Prog. Theor. Phys., 80, 1041 (1988) · doi:10.1143/PTP.80.1041
[64] Nambu, Yasusada, Stochastic Dynamics of an Inflationary Model and Initial Distribution of Universes, Prog. Theor. Phys., 81, 1037 (1989) · doi:10.1143/PTP.81.1037
[65] Mollerach, Silvia; Matarrese, Sabino; Ortolan, Antonello; Lucchin, Francesco, Stochastic inflation in a simple two field model, Phys. Rev. D, 44, 1670-1679 (1991) · doi:10.1103/PhysRevD.44.1670
[66] Linde, Andrei D.; Linde, Dmitri A.; Mezhlumian, Arthur, From the Big Bang theory to the theory of a stationary universe, Phys. Rev. D, 49, 1783-1826 (1994) · doi:10.1103/PhysRevD.49.1783
[67] Starobinsky, Alexei A.; Yokoyama, Junichi, Equilibrium state of a selfinteracting scalar field in the De Sitter background, Phys. Rev. D, 50, 6357-6368 (1994) · doi:10.1103/PhysRevD.50.6357
[68] Vennin, Vincent; Starobinsky, Alexei A., Correlation Functions in Stochastic Inflation, Eur. Phys. J. C, 75, 413 (2015) · doi:10.1140/epjc/s10052-015-3643-y
[69] Pattison, Chris; Vennin, Vincent; Assadullahi, Hooshyar; Wands, David, Stochastic inflation beyond slow roll, JCAP, 07 (2019) · Zbl 1515.83424 · doi:10.1088/1475-7516/2019/07/031
[70] Starobinsky, Alexei A., Multicomponent de Sitter (Inflationary) Stages and the Generation of Perturbations, JETP Lett., 42, 152-155 (1985)
[71] Sasaki, Misao; Stewart, Ewan D., A General analytic formula for the spectral index of the density perturbations produced during inflation, Prog. Theor. Phys., 95, 71-78 (1996) · doi:10.1143/PTP.95.71
[72] Sasaki, Misao; Tanaka, Takahiro, Superhorizon scale dynamics of multiscalar inflation, Prog. Theor. Phys., 99, 763-782 (1998) · doi:10.1143/PTP.99.763
[73] Lyth, David H.; Malik, Karim A.; Sasaki, Misao, A General proof of the conservation of the curvature perturbation, JCAP, 05 (2005) · Zbl 1236.83043 · doi:10.1088/1475-7516/2005/05/004
[74] Salopek, D. S.; Bond, J. R., Nonlinear evolution of long wavelength metric fluctuations in inflationary models, Phys. Rev. D, 42, 3936-3962 (1990) · doi:10.1103/PhysRevD.42.3936
[75] Wands, David; Malik, Karim A.; Lyth, David H.; Liddle, Andrew R., A New approach to the evolution of cosmological perturbations on large scales, Phys. Rev. D, 62 (2000) · doi:10.1103/PhysRevD.62.043527
[76] Lyth, David H.; Wands, David, Conserved cosmological perturbations, Phys. Rev. D, 68 (2003) · doi:10.1103/PhysRevD.68.103515
[77] Rigopoulos, G. I.; Shellard, E. P. S., The separate universe approach and the evolution of nonlinear superhorizon cosmological perturbations, Phys. Rev. D, 68 (2003) · doi:10.1103/PhysRevD.68.123518
[78] Lyth, David H.; Rodriguez, Yeinzon, The Inflationary prediction for primordial non-Gaussianity, Phys. Rev. Lett., 95 (2005) · doi:10.1103/PhysRevLett.95.121302
[79] Enqvist, K.; Nurmi, S.; Podolsky, D.; Rigopoulos, G. I., On the divergences of inflationary superhorizon perturbations, JCAP, 04 (2008) · doi:10.1088/1475-7516/2008/04/025
[80] Fujita, Tomohiro; Kawasaki, Masahiro; Tada, Yuichiro; Takesako, Tomohiro, A new algorithm for calculating the curvature perturbations in stochastic inflation, JCAP, 12 (2013) · doi:10.1088/1475-7516/2013/12/036
[81] Ando, Kenta; Vennin, Vincent, Power spectrum in stochastic inflation, JCAP, 04 (2021) · Zbl 1485.83111 · doi:10.1088/1475-7516/2021/04/057
[82] Tada, Yuichiro; Vennin, Vincent, Statistics of coarse-grained cosmological fields in stochastic inflation, JCAP, 02 (2022) · Zbl 1487.83137 · doi:10.1088/1475-7516/2022/02/021
[83] Ezquiaga, Jose María; García-Bellido, Juan; Vennin, Vincent, The exponential tail of inflationary fluctuations: consequences for primordial black holes, JCAP, 03 (2020) · Zbl 1490.83076 · doi:10.1088/1475-7516/2020/03/029
[84] Martin, Jerome; Vennin, Vincent, Stochastic Effects in Hybrid Inflation, Phys. Rev. D, 85 (2012) · doi:10.1103/PhysRevD.85.043525
[85] Kawasaki, Masahiro; Tada, Yuichiro, Can massive primordial black holes be produced in mild waterfall hybrid inflation?, JCAP, 08 (2016) · doi:10.1088/1475-7516/2016/08/041
[86] De, Aritra; Mahbub, Rafid, Numerically modeling stochastic inflation in slow-roll and beyond, Phys. Rev. D, 102 (2020) · doi:10.1103/PhysRevD.102.123509
[87] Figueroa, Daniel G.; Raatikainen, Sami; Rasanen, Syksy; Tomberg, Eemeli, Non-Gaussian Tail of the Curvature Perturbation in Stochastic Ultraslow-Roll Inflation: Implications for Primordial Black Hole Production, Phys. Rev. Lett., 127 (2021) · doi:10.1103/PhysRevLett.127.101302
[88] Figueroa, Daniel G.; Raatikainen, Sami; Rasanen, Syksy; Tomberg, Eemeli, Implications of stochastic effects for primordial black hole production in ultra-slow-roll inflation, JCAP, 05 (2022) · Zbl 1505.83017 · doi:10.1088/1475-7516/2022/05/027
[89] Mahbub, Rafid; De, Aritra, Smooth coarse-graining and colored noise dynamics in stochastic inflation, JCAP, 09 (2022) · Zbl 1511.83046 · doi:10.1088/1475-7516/2022/09/045
[90] T. Kloek and H.K. van Dijk, Bayesian estimates of equation system parameters: An application of integration by monte carlo, Econometrica46 (1978) 1. · Zbl 0376.62014 · doi:10.2307/1913641
[91] Mazonka, O.; Jarzynski, C.; Blocki, J., Computing probabilities of very rare events for Langevin processes: A New method based on importance sampling, Nucl. Phys. A, 641, 335-354 (1998) · doi:10.1016/S0375-9474(98)00478-3
[92] P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer Berlin Heidelberg (1992), [DOI]. · Zbl 0752.60043 · doi:10.1007/978-3-662-12616-5
[93] Pinol, Lucas; Renaux-Petel, Sébastien; Tada, Yuichiro, A manifestly covariant theory of multifield stochastic inflation in phase space: solving the discretisation ambiguity in stochastic inflation, JCAP, 04 (2021) · Zbl 1485.83173 · doi:10.1088/1475-7516/2021/04/048
[94] A. Araújo, The Central Limit Theorem for Real and Banach Valued Random Variables, Wiley, New York, U.S.A. (1980). · Zbl 0457.60001
[95] S. Redner, A Guide to First-Passage Processes, Cambridge University Press (2001), [DOI]. · Zbl 0980.60006 · doi:10.1017/CBO9780511606014
[96] Assadullahi, Hooshyar; Firouzjahi, Hassan; Noorbala, Mahdiyar; Vennin, Vincent; Wands, David, Multiple Fields in Stochastic Inflation, JCAP, 06 (2016) · doi:10.1088/1475-7516/2016/06/043
[97] Vennin, Vincent; Assadullahi, Hooshyar; Firouzjahi, Hassan; Noorbala, Mahdiyar; Wands, David, Critical Number of Fields in Stochastic Inflation, Phys. Rev. Lett., 118 (2017) · doi:10.1103/PhysRevLett.118.031301
[98] H. Shen, L.D. Brown and H. Zhi, Efficient estimation of log-normal means with application to pharmacokinetic data, Statist. Med.25 (2006) 3023. · doi:10.1002/sim.2456
[99] R. D’Agostino and E.S. Pearson, Tests for departure from normality. empirical results for the distributions of b_2 and √(b_1), Biometrika60 (1973) 613. · Zbl 0271.62025 · doi:10.2307/2335012
[100] X.-H. Zhou and S. Gao, Confidence intervals for the log-normal mean, Statist. Med.16 (1997) 783. · doi:10.1002/(sici)1097-0258(19970415)16:7<783::aid-sim488>3.0.co;2-2
[101] D. Freedman, R. Pisani and R. Purves, Statistics, W.W. Norton, New York, U.S.A. (1998). · Zbl 1351.62002
[102] Garcia-Bellido, Juan; Linde, Andrei D.; Wands, David, Density perturbations and black hole formation in hybrid inflation, Phys. Rev. D, 54, 6040-6058 (1996) · doi:10.1103/PhysRevD.54.6040
[103] Clesse, Sébastien; García-Bellido, Juan, Massive Primordial Black Holes from Hybrid Inflation as Dark Matter and the seeds of Galaxies, Phys. Rev. D, 92 (2015) · doi:10.1103/PhysRevD.92.023524
[104] Noorbala, Mahdiyar; Firouzjahi, Hassan, Boundary crossing in stochastic inflation with a critical number of fields, Phys. Rev. D, 100 (2019) · doi:10.1103/PhysRevD.100.083510
[105] Hooshangi, Sina; Talebian, Alireza; Namjoo, Mohammad Hossein; Firouzjahi, Hassan, Multiple field ultraslow-roll inflation: Primordial black holes from straight bulk and distorted boundary, Phys. Rev. D, 105 (2022) · doi:10.1103/PhysRevD.105.083525
[106] Perreault Levasseur, Laurence; Vennin, Vincent; Brandenberger, Robert, Recursive Stochastic Effects in Valley Hybrid Inflation, Phys. Rev. D, 88 (2013) · doi:10.1103/PhysRevD.88.083538
[107] Cohen, Timothy; Green, Daniel; Premkumar, Akhil; Ridgway, Alexander, Stochastic Inflation at NNLO, JHEP, 09, 159 (2021) · doi:10.1007/JHEP09(2021)159
[108] Biagetti, Matteo; De Luca, Valerio; Franciolini, Gabriele; Kehagias, Alex; Riotto, Antonio, The formation probability of primordial black holes, Phys. Lett. B, 820 (2021) · doi:10.1016/j.physletb.2021.136602
[109] Kitajima, Naoya; Tada, Yuichiro; Yokoyama, Shuichiro; Yoo, Chul-Moon, Primordial black holes in peak theory with a non-Gaussian tail, JCAP, 10 (2021) · Zbl 1486.83110 · doi:10.1088/1475-7516/2021/10/053
[110] NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/20.7E30, Release 1.0.14 of 2016-12-21.
[111] F. Edgeworth, The law of error I, Proc. Cambridge Philos. Soc.20 (1905) 3665.
[112] D.L. Wallace, Asymptotic approximations to distributions, Annals Math. Statist.29 (1958) 635. · Zbl 0086.34004 · doi:10.1214/aoms/1177706528
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.