×

Amplification of primordial perturbations from the rise or fall of the inflaton. (English) Zbl 1487.83129


MSC:

83E05 Geometrodynamics and the holographic principle
83C57 Black holes
83F05 Relativistic cosmology
35B20 Perturbations in context of PDEs
83C35 Gravitational waves
47A10 Spectrum, resolvent
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Starobinsky, Alexei A.Khalatnikov, I. M.Mineev, V. P.A new type of isotropic cosmological models without singularityPhys. Lett. B19809199102 · Zbl 1371.83222
[2] Sato, K.First order phase transition of a vacuum and expansion of the universeMon. Not. Roy. Astron. Soc.1981195467479
[3] Guth, Alan H.Fang, Li-ZhiRuffini, R.The inflationary universe: a possible solution to the horizon and flatness problemsPhys. Rev. D198123347356
[4] Linde, Andrei D.Fang, Li-ZhiRuffini, R.A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problemsPhys. Lett. B1982108389393
[5] Albrecht, AndreasSteinhardt, Paul J.Fang, Li-ZhiRuffini, R.Cosmology for grand unified theories with radiatively induced symmetry breakingPhys. Rev. Lett.19824812201223
[6] Linde, Andrei D.Chaotic inflationPhys. Lett. B1983129177181
[7] CMBPol Study Team CollaborationBaumann, Danielet al.Dodelson, ScottBaumann, DanielCooray, AsanthaDunkley, JoannaFraisse, AurelienJackson, Mark G.Kogut, AlanKrauss, LawrenceSmith, KendrickZaldarriaga, MatiasCMBPol mission concept study: probing inflation with CMB polarizationAIP Conf. Proc.2009114110120
[8] CMB-S4 CollaborationAbazajian, Kevork N.et al.CMB-S4 science book, first edition2016
[9] Bartolo, Nicolaet al.Science with the space-based interferometer LISA. Part IV. Probing inflation with gravitational wavesJCAP201612026
[10] Meerburg, P. Danielet al.Primordial non-Gaussianity2019
[11] Y.B Zel’dovich and I.D. Novikov, The hypothesis of cores retarded during expansion and the hot cosmological model, Sov. Astron.10 (1967) 602.
[12] Hawking, StephenGravitationally collapsed objects of very low massMon. Not. Roy. Astron. Soc.197115275
[13] Carr, Bernard J.Hawking, S. W.Black holes in the early UniverseMon. Not. Roy. Astron. Soc.1974168399415
[14] Carr, Bernard J.The Primordial black hole mass spectrumAstrophys. J.1975201119
[15] Bird, Simeonet al.Did LIGO detect dark matter?Phys. Rev. Lett.2016116201301
[16] Clesse, SebastienGarcía-Bellido, JuanThe clustering of massive primordial black holes as dark matter: measuring their mass distribution with Advanced LIGOPhys. Dark Univ.201715142147
[17] Sasaki, MisaoSuyama, TeruakiTanaka, TakahiroYokoyama, ShuichiroPrimordial black hole scenario for the gravitational-wave event GW150914Phys. Rev. Lett.2016117061101[Erratum: Phys.Rev.Lett. 121, 059901 (2018)]
[18] Kashlinsky, A.LIGO gravitational wave detection, primordial black holes and the near-IR cosmic infrared background anisotropiesAstrophys. J. Lett.2016823L25
[19] Kashlinsky, A.Arendt, R. G.Atrio-Barandela, F.Cappelluti, N.Ferrara, A.Hasinger, G.Looking at cosmic near-infrared background radiation anisotropiesRev. Mod. Phys.201890025006
[20] García-Bellido, JuanNuño Siles, José FranciscoRuiz Morales, EsterBayesian analysis of the spin distribution of LIGO/Virgo black holesPhys. Dark Univ.202131100791
[21] Sasaki, MisaoSuyama, TeruakiTanaka, TakahiroYokoyama, ShuichiroPrimordial black holes—perspectives in gravitational wave astronomyClass. Quant. Grav.201835063001 · Zbl 1386.83001
[22] Carr, BernardKohri, KazunoriSendouda, YuuitiYokoyama, Jun’ichiConstraints on primordial black holesRept. Prog. Phys.202184116902
[23] Green, Anne M.Kavanagh, Bradley J.Primordial black holes as a dark matter candidateJ. Phys. G202148043001
[24] Scott, PatSivertsson, SofiaGamma-rays from ultracompact primordial dark matter minihalosPhys. Rev. Lett.2009103211301[Erratum: Phys.Rev.Lett. 105, 119902 (2010)]
[25] Bringmann, TorstenScott, PatAkrami, YasharImproved constraints on the primordial power spectrum at small scales from ultracompact minihalosPhys. Rev. D201285125027
[26] Nakama, TomohiroSuyama, TeruakiKohri, KazunoriHiroshima, NagisaConstraints on small-scale primordial power by annihilation signals from extragalactic dark matter minihalosPhys. Rev. D201897023539
[27] Kawasaki, MasahiroNakatsuka, HiromasaNakayama, KazunoriConstraints on small-scale primordial density fluctuation from cosmic microwave background through dark matter annihilation2021
[28] R.A. Sunyaev and Y.B. Zeldovich, Small-scale fluctuations of relic radiation, Astrophys. Space Sci.7 (1970) 3.
[29] R.A. Sunyaev and Y.B. Zeldovich, The interaction of matter and radiation in the hot model of the Universe, II, Astrophys. Space Sci.7 (1970) 20.
[30] K. Tomita, Non-linear theory of gravitational instability in the expanding universe, Prog. Theor. Phys.37 (1967) 831. · doi:10.1143/PTP.37.831
[31] Matarrese, SabinoPantano, OrnellaSaez, DiegoGeneral relativistic dynamics of irrotational dust: Cosmological implicationsPhys. Rev. Lett.199472320323 · Zbl 0973.83529
[32] Matarrese, SabinoMollerach, SilviaBruni, MarcoSecond order perturbations of the Einstein-de Sitter universePhys. Rev. D199858043504
[33] Ananda, Kishore N.Clarkson, ChrisWands, DavidThe cosmological gravitational wave background from primordial density perturbationsPhys. Rev. D200775123518
[34] Baumann, DanielSteinhardt, Paul J.Takahashi, KeitaroIchiki, KiyotomoGravitational wave spectrum induced by primordial scalar perturbationsPhys. Rev. D200776084019
[35] Saito, RyoYokoyama, Jun’ichiGravitational wave background as a probe of the primordial black hole abundancePhys. Rev. Lett.2009102161101[Erratum: Phys.Rev.Lett. 107, 069901 (2011)]
[36] Saito, RyoYokoyama, Jun’ichiGravitational-Wave Constraints on the Abundance of Primordial Black HolesProg. Theor. Phys.2010123867886[Erratum: Prog.Theor.Phys. 126, 351-352 (2011)]
[37] Kohri, KazunoriTerada, TakahiroSemianalytic calculation of gravitational wave spectrum nonlinearly induced from primordial curvature perturbationsPhys. Rev. D201897123532
[38] Inomata, KeisukeNakama, TomohiroGravitational waves induced by scalar perturbations as probes of the small-scale primordial spectrumPhys. Rev. D201999043511
[39] Adshead, PeterLozanov, Kaloian D.Weiner, Zachary J.Non-Gaussianity and the induced gravitational wave backgroundJCAP202110080 · Zbl 1486.83017
[40] Domènech, GuillemScalar induced gravitational waves reviewUniverse20217398
[41] Chluba, J.et al.New horizons in cosmology with spectral distortions of the cosmic microwave backgroundExper. Astron.20215115151554
[42] Kogut, A.et al.The Primordial Inflation Explorer (PIXIE): a nulling polarimeter for cosmic microwave background observationsJCAP201107025
[43] Weltman, A.et al.Fundamental physics with the Square Kilometre ArrayPubl. Astron. Soc. Austral.202037e002
[44] LISA CollaborationAmaro-Seoane, Pauet al.Laser Interferometer Space Antenna2017
[45] Kawamura, Seijiet al.Current status of space gravitational wave antenna DECIGO and B-DECIGOPTEP2021202105A105
[46] Ivanov, P.Naselsky, P.Novikov, I.Inflation and primordial black holes as dark matterPhys. Rev. D19945071737178
[47] Leach, Samuel M.Liddle, Andrew R.Inflationary perturbations near horizon crossingPhys. Rev. D200163043508
[48] Leach, Samuel MSasaki, MisaoWands, DavidLiddle, Andrew REnhancement of superhorizon scale inflationary curvature perturbationsPhys. Rev. D200164023512
[49] Inoue, ShogoYokoyama, Jun’ichiCurvature perturbation at the local extremum of the inflaton’s potentialPhys. Lett. B20025241520 · Zbl 0981.83062
[50] Tsamis, N. C.Woodard, Richard P.Improved estimates of cosmological perturbationsPhys. Rev. D200469084005
[51] Kinney, William H.Horizon crossing and inflation with large etaPhys. Rev. D200572023515
[52] Garcia-Bellido, JuanRuiz Morales, EsterPrimordial black holes from single field models of inflationPhys. Dark Univ.2017184754
[53] Ezquiaga, Jose MariaGarcia-Bellido, JuanRuiz Morales, EsterPrimordial black hole production in critical Higgs inflationPhys. Lett. B2018776345349
[54] Kannike, KristjanMarzola, LucaRaidal, MarttiVeermäe, HardiSingle field double inflation and primordial black holesJCAP201709020 · Zbl 1515.83156
[55] Germani, CristianoProkopec, TomislavOn primordial black holes from an inflection pointPhys. Dark Univ.201718610
[56] Ballesteros, GuillermoTaoso, MarcoPrimordial black hole dark matter from single field inflationPhys. Rev. D201897023501
[57] Hertzberg, Mark P.Yamada, MasakiPrimordial black holes from polynomial potentials in single field inflationPhys. Rev. D201897083509
[58] Cheng, Shu-LinLee, WolungNg, Kin-WangSuperhorizon curvature perturbation in ultraslow-roll inflationPhys. Rev. D201999063524
[59] Byrnes, Christian T.Cole, Philippa S.Patil, Subodh P.Steepest growth of the power spectrum and primordial black holesJCAP201906028
[60] Passaglia, SamuelHu, WayneMotohashi, HayatoPrimordial black holes and local non-Gaussianity in canonical inflationPhys. Rev. D201999043536
[61] Drees, ManuelXu, YongOvershooting, critical Higgs inflation and second order gravitational wave signaturesEur. Phys. J. C202181182
[62] Carrilho, PedroMalik, Karim A.Mulryne, David J.Dissecting the growth of the power spectrum for primordial black holesPhys. Rev. D2019100103529
[63] Ng, Kin-WangWu, Yi-PengConstant-rate inflation: primordial black holes from conformal weight transitionsJHEP202111076 · Zbl 1521.83144
[64] Romano, Antonio EneaSound speed induced production of primordial black holes2020
[65] Chen, ChaoCai, Yi-FuPrimordial black holes from sound speed resonance in the inflaton-curvaton mixed scenarioJCAP201910068 · Zbl 1515.83132
[66] Fu, ChengjieWu, PuxunYu, HongweiPrimordial black holes from inflation with nonminimal derivative couplingPhys. Rev. D2019100063532
[67] Kawai, ShinsukeKim, JinsuCMB from a Gauss-Bonnet-induced de Sitter fixed pointPhys. Rev. D2021104043525
[68] Kawai, ShinsukeKim, JinsuPrimordial black holes from Gauss-Bonnet-corrected single field inflationPhys. Rev. D2021104083545
[69] Cai, Rong-GenGuo, Zong-KuanLiu, JingLiu, LangYang, Xing-YuPrimordial black holes and gravitational waves from parametric amplification of curvature perturbationsJCAP202006013 · Zbl 1491.83009
[70] Inomata, KeisukeKawasaki, MasahiroMukaida, KyoheiYanagida, Tsutomu T.NANOGrav results and LIGO-Virgo primordial black holes in axionlike curvaton modelsPhys. Rev. Lett.2021126131301
[71] Ando, KentaInomata, KeisukeKawasaki, MasahiroMukaida, KyoheiYanagida, Tsutomu T.Primordial black holes for the LIGO events in the axionlike curvaton modelPhys. Rev. D201897123512
[72] Inomata, KeisukeKawasaki, MasahiroMukaida, KyoheiYanagida, Tsutomu T.Double inflation as a single origin of primordial black holes for all dark matter and LIGO observationsPhys. Rev. D201897043514
[73] Palma, Gonzalo A.Sypsas, SpyrosZenteno, CristobalSeeding primordial black holes in multifield inflationPhys. Rev. Lett.2020125121301
[74] Fumagalli, JacopoRenaux-Petel, SébastienRonayne, John W.Witkowski, Lukas T.Turning in the landscape: a new mechanism for generating primordial black holes2020
[75] Fumagalli, JacopoRenaux-Petel, SébastienWitkowski, Lukas T.Oscillations in the stochastic gravitational wave background from sharp features and particle production during inflationJCAP202108030 · Zbl 1495.83019
[76] Covi, LauraHamann, JanMelchiorri, AlessandroSlosar, AnzeSorbera, IreneInflation and WMAP three year data: features have a future!Phys. Rev. D200674083509
[77] Hamann, JanCovi, LauraMelchiorri, AlessandroSlosar, AnzeNew constraints on oscillations in the primordial spectrum of inflationary perturbationsPhys. Rev. D200776023503
[78] Mortonson, Michael J.Dvorkin, CoraPeiris, Hiranya V.Hu, WayneCMB polarization features from inflation versus reionizationPhys. Rev. D200979103519
[79] Adshead, PeterDvorkin, CoraHu, WayneLim, Eugene A.Non-Gaussianity from step features in the inflationary potentialPhys. Rev. D201285023531
[80] Miranda, ViníciusHu, WayneInflationary steps in the Planck dataPhys. Rev. D201489083529
[81] Adshead, PeterHu, WayneBounds on nonadiabatic evolution in single-field inflationPhys. Rev. D201489083531
[82] Cai, Yi-FuFerreira, Elisa G. M.Hu, BinQuintin, JeromeSearching for features of a string-inspired inflationary model with cosmological observationsPhys. Rev. D201592121303
[83] Miranda, ViniciusHu, WayneHe, ChenMotohashi, HayatoNonlinear excitations in inflationary power spectraPhys. Rev. D201693023504
[84] Kefala, K.Kodaxis, G. P.Stamou, I. D.Tetradis, N.Features of the inflaton potential and the power spectrum of cosmological perturbationsPhys. Rev. D2021104023506
[85] Wolfson, IraAnalytic correlation of inflationary potential to power spectrum shape: limits of validity, and “no-go” for small field model analyticsJCAP202201036 · Zbl 1486.83158
[86] Mishra, Swagat S.Sahni, VarunPrimordial Black Holes from a tiny bump/dip in the Inflaton potentialJCAP202004007 · Zbl 1491.83035
[87] Özsoy, OganParameswaran, SushaTasinato, GianmassimoZavala, IvonneMechanisms for primordial black hole production in string theoryJCAP201807005 · Zbl 1527.83061
[88] Inomata, KeisukeMcDonough, EvanHu, WaynePrimordial black holes arise when the inflaton fallsPhys. Rev. D2021104123553
[89] Dalianis, I.Kodaxis, G. P.Stamou, I. D.Tetradis, N.Tsigkas-Kouvelis, A.Spectrum oscillations from features in the potential of single-field inflationPhys. Rev. D2021104103510
[90] Birrell, N. D.Davies, P. C. W.Quantum Fields in Curved Space1984 · Zbl 0972.81605
[91] Deng, HelingVilenkin, AlexanderPrimordial black hole formation by vacuum bubblesJCAP201712044 · Zbl 1515.83139
[92] Garriga, JaumeVilenkin, AlexanderZhang, JunBlack holes and the multiverseJCAP201602064
[93] Atal, VicenteGarriga, JaumeMarcos-Caballero, AiramPrimordial black hole formation with non-Gaussian curvature perturbationsJCAP201909073 · Zbl 07483222
[94] Atal, VicenteCid, JudithEscrivà, AlbertGarriga, JaumePBH in single field inflation: the effect of shape dispersion and non-GaussianitiesJCAP202005022 · Zbl 1491.83044
[95] Pollack, JasonSpergel, David N.Steinhardt, Paul J.Supermassive black holes from ultra-strongly self-interacting dark matterAstrophys. J.2015804131
[96] Blinov, NikitaDolan, Matthew J.Draper, PatrickShelton, JessieDark matter microhalos from simplified modelsPhys. Rev. D2021103103514
[97] Sasaki, MisaoLarge scale quantum fluctuations in the inflationary universeProg. Theor. Phys.1986761036
[98] Mukhanov, Viatcheslav F.Quantum theory of gauge invariant cosmological perturbationsSov. Phys. JETP19886712971302
[99] Planck CollaborationAghanim, N.et al.Planck 2018 results. VI. Cosmological parametersAstron. Astrophys.2020641A6[Erratum: Astron.Astrophys. 652, C4 (2021)]
[100] Chung, Daniel J. H.Kolb, Edward W.Long, Andrew J.Gravitational production of super-Hubble-mass particles: an analytic approachJHEP201901189
[101] S. Weinberg, Cosmology, Oxford University Press, Oxford U.K. (2008). · Zbl 1147.83002
[102] S. Dodelson, Modern cosmology, Academic Press, San Diego U.S.A. (2003).
[103] Cheung, CliffordCreminelli, PaoloFitzpatrick, A. LiamKaplan, JaredSenatore, LeonardoThe effective field theory of inflationJHEP200803014
[104] Behbahani, Siavosh R.Dymarsky, AnatolyMirbabayi, MehrdadSenatore, Leonardo(Small) resonant non-Gaussianities: signatures of a discrete shift symmetry in the effective field theory of inflationJCAP201212036
[105] Baumann, DanielGreen, DanielEquilateral non-Gaussianity and new physics on the horizonJCAP201109014
[106] Goodhew, HarryJazayeri, SadraPajer, EnricoThe cosmological optical theoremJCAP202104021 · Zbl 1485.83139
[107] Pattison, ChrisVennin, VincentAssadullahi, HooshyarWands, DavidQuantum diffusion during inflation and primordial black holesJCAP201710046 · Zbl 1515.83170
[108] Biagetti, MatteoFranciolini, GabrieleKehagias, AlexRiotto, AntonioPrimordial Black Holes from Inflation and Quantum DiffusionJCAP201807032 · Zbl 1527.83047
[109] Ezquiaga, Jose MaríaGarcía-Bellido, JuanQuantum diffusion beyond slow-roll: implications for primordial black-hole productionJCAP201808018 · Zbl 07461347
[110] Ezquiaga, Jose MaríaGarcía-Bellido, JuanVennin, VincentThe exponential tail of inflationary fluctuations: consequences for primordial black holesJCAP202003029 · Zbl 1490.83076
[111] Figueroa, Daniel G.Raatikainen, SamiRasanen, SyksyTomberg, EemeliNon-Gaussian tail of the curvature perturbation in stochastic ultraslow-roll inflation: implications for primordial black hole productionPhys. Rev. Lett.2021127101302
[112] Pattison, ChrisVennin, VincentWands, DavidAssadullahi, HooshyarUltra-slow-roll inflation with quantum diffusionJCAP202104080 · Zbl 1486.83151
[113] Cheng, Shu-LinLee, Da-ShinNg, Kin-WangPower spectrum of primordial perturbations during ultra-slow-roll inflation with back reaction effects2021
[114] Caravano, AngeloKomatsu, EiichiroLozanov, Kaloian D.Weller, JochenLattice simulations of inflationJCAP202112010 · Zbl 1487.83116
[115] Green, Anne M.Liddle, Andrew R.Malik, Karim A.Sasaki, MisaoA new calculation of the mass fraction of primordial black holesPhys. Rev. D200470041502
[116] Ando, KentaInomata, KeisukeKawasaki, MasahiroPrimordial black holes and uncertainties in the choice of the window functionPhys. Rev. D201897103528
[117] Young, SamThe primordial black hole formation criterion re-examined: parametrisation, timing and the choice of window functionInt. J. Mod. Phys. D2019292030002
[118] Tokeshi, KokiInomata, KeisukeYokoyama, Jun’ichiWindow function dependence of the novel mass function of primordial black holesJCAP202012038 · Zbl 1484.83073
[119] Kawasaki, MasahiroNakatsuka, HiromasaEffect of nonlinearity between density and curvature perturbations on the primordial black hole formationPhys. Rev. D201999123501
[120] De Luca, V.Franciolini, G.Kehagias, A.Peloso, M.Riotto, A.Ünal, C.The ineludible non-Gaussianity of the primordial black hole abundanceJCAP201907048 · Zbl 1515.83161
[121] Young, SamMusco, IliaByrnes, Christian T.Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbationJCAP201911012 · Zbl 07502067
[122] Byrnes, Christian T.Copeland, Edmund J.Green, Anne M.Primordial black holes as a tool for constraining non-GaussianityPhys. Rev. D201286043512
[123] Young, SamByrnes, Christian T.Primordial black holes in non-Gaussian regimesJCAP201308052
[124] Davies, Matthew W.Carrilho, PedroMulryne, David J.Non-Gaussianity in inflationary scenarios for primordial black holes2021
[125] Fixsen, D. J.Cheng, E. S.Gales, J. M.Mather, John C.Shafer, R. A.Wright, E. L.The Cosmic Microwave Background spectrum from the full COBE FIRAS data setAstrophys. J.1996473576
[126] Chluba, JensErickcek, Adrienne L.Ben-Dayan, IdoProbing the inflaton: small-scale power spectrum constraints from measurements of the CMB energy spectrumAstrophys. J.201275876
[127] Kohri, KazunoriNakama, TomohiroSuyama, TeruakiTesting scenarios of primordial black holes being the seeds of supermassive black holes by ultracompact minihalos and CMB μ-distortionsPhys. Rev. D201490083514
[128] Cheng, Shu-LinLee, WolungNg, Kin-WangPrimordial black holes and associated gravitational waves in axion monodromy inflationJCAP201807001 · Zbl 1527.83049
[129] Özsoy, OganSynthetic gravitational waves from a rolling axion monodromyJCAP202104040 · Zbl 1485.83016
[130] Özsoy, OganLalak, ZygmuntPrimordial black holes as dark matter and gravitational waves from bumpy axion inflationJCAP202101040 · Zbl 1484.83017
[131] Anber, Mohamed M.Sorbo, LorenzoNaturally inflating on steep potentials through electromagnetic dissipationPhys. Rev. D201081043534
[132] Barnaby, NeilPajer, EnricoPeloso, MarcoGauge field production in axion inflation: consequences for monodromy, non-Gaussianity in the CMB, and gravitational waves at interferometersPhys. Rev. D201285023525
[133] Adshead, PeterGiblin, John T.Scully, Timothy R.Sfakianakis, Evangelos I.Gauge-preheating and the end of axion inflationJCAP201512034
[134] Enqvist, KariSloth, Martin S.Adiabatic CMB perturbations in pre-Big Bang string cosmologyNucl. Phys. B2002626395409
[135] Lyth, David H.Wands, DavidGenerating the curvature perturbation without an inflatonPhys. Lett. B2002524514 · Zbl 0981.83063
[136] Moroi, TakeoTakahashi, TomoEffects of cosmological moduli fields on cosmic microwave backgroundPhys. Lett. B2001522215221[Erratum: Phys.Lett.B 539, 303-303 (2002)]
[137] Kolb, Edward W.Long, Andrew J.Completely dark photons from gravitational particle production during the inflationary eraJHEP202103283
[138] McDonough, EvanGuth, Alan H.Kaiser, David I.Nonminimal couplings and the forgotten field of axion inflation2020
[139] Cai, Yi-FuJiang, JieSasaki, MisaoVardanyan, ValeriZhou, ZihanBeating the Lyth bound by parametric resonance during inflationPhys. Rev. Lett.2021127251301
[140] Turner, Michael S.Weinberg, Erick J.Widrow, Lawrence M.Bubble nucleation in first order inflation and other cosmological phase transitionsPhys. Rev. D19924623842403
[141] Coleman, Sidney R.The Fate of the False Vacuum. 1. Semiclassical TheoryPhys. Rev. D19771529292936[Erratum: Phys.Rev.D 16, 1248 (1977)]
[142] Coleman, Sidney R.De Luccia, FrankGravitational effects on and of vacuum decayPhys. Rev. D1980213305
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.