Inflation with a graceful exit in a random landscape

  • F. G. PedroEmail author
  • A. Westphal
Open Access
Regular Article - Theoretical Physics


We develop a stochastic description of small-field inflationary histories with a graceful exit in a random potential whose Hessian is a Gaussian random matrix as a model of the unstructured part of the string landscape. The dynamical evolution in such a random potential from a small-field inflation region towards a viable late-time de Sitter (dS) minimum maps to the dynamics of Dyson Brownian motion describing the relaxation of non-equilibrium eigenvalue spectra in random matrix theory. We analytically compute the relaxation probability in a saddle point approximation of the partition function of the eigenvalue distribution of the Wigner ensemble describing the mass matrices of the critical points. When applied to small-field inflation in the landscape, this leads to an exponentially strong bias against small-field ranges and an upper bound N ≪ 10 on the number of light fields N participating during inflation from the non-observation of negative spatial curvature.


Flux compactifications Superstring Vacua 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    D. Baumann and L. McAllister, Inflation and String Theory, arXiv:1404.2601.
  2. [2]
    C. Brodie and M.C.D. Marsh, The Spectra of Type IIB Flux Compactifications at Large Complex Structure, JHEP 01 (2016) 037 [arXiv:1509.06761] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    D. Marsh, L. McAllister and T. Wrase, The Wasteland of Random Supergravities, JHEP 03 (2012) 102 [arXiv:1112.3034] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    B. Freivogel, R. Gobbetti, E. Pajer and I.-S. Yang, Inflation on a Slippery Slope, arXiv:1608.00041 [INSPIRE].
  5. [5]
    M.C.D. Marsh, L. McAllister, E. Pajer and T. Wrase, Charting an Inflationary Landscape with Random Matrix Theory, JCAP 11 (2013) 040 [arXiv:1307.3559] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    G.E. Uhlenbeck and L.S. Ornstein, On the Theory of the Brownian Motion, Phys. Rev. 36 (1930) 823 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  7. [7]
    F.J. Dyson A Brownian-Motion Model for the Eigenvalues of a Random Matrix, J. Math. Phys. 3 (1962) 1191.Google Scholar
  8. [8]
    D.S. Dean and S.N. Majumdar, Large deviations of extreme eigenvalues of random matrices, Phys. Rev. Lett. 97 (2006) 160201 [cond-mat/0609651] [INSPIRE].
  9. [9]
    A. Edelman and N.R. Rao, Random matrix theory, Acta Numerica 14 (2005) 233.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    J.-P. Bouchard and M. Potters, Financial Applications of Random Matrix Theory: a short review, to appear in the Handbook on Random Matrix Theory, Oxford University Press, arXiv:0910.1205.
  11. [11]
    B. Freivogel, M. Kleban, M. Rodriguez Martinez and L. Susskind, Observational consequences of a landscape, JHEP 03 (2006) 039 [hep-th/0505232] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    F. Denef and M.R. Douglas, Distributions of flux vacua, JHEP 05 (2004) 072 [hep-th/0404116] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    F. Denef and M.R. Douglas, Distributions of nonsupersymmetric flux vacua, JHEP 03 (2005) 061 [hep-th/0411183] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    M. Rummel and Y. Sumitomo, Probability of vacuum stability in type IIB multi-Kähler moduli models, JHEP 12 (2013) 003 [arXiv:1310.4202] [INSPIRE].zbMATHGoogle Scholar
  15. [15]
    C. Long, L. McAllister and P. McGuirk, Heavy Tails in Calabi-Yau Moduli Spaces, JHEP 10 (2014) 187 [arXiv:1407.0709] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    K. Sousa and P. Ortiz, Perturbative Stability along the Supersymmetric Directions of the Landscape, JCAP 02 (2015) 017 [arXiv:1408.6521] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    M.C.D. Marsh and K. Sousa, Universal Properties of Type IIB and F-theory Flux Compactifications at Large Complex Structure, JHEP 03 (2016) 064 [arXiv:1512.08549] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    A. Aazami and R. Easther, Cosmology from random multifield potentials, JCAP 03 (2006) 013 [hep-th/0512050] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    R. Easther and L. McAllister, Random matrices and the spectrum of N-flation, JCAP 05 (2006) 018 [hep-th/0512102] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    F.G. Pedro and A. Westphal, The Scale of Inflation in the Landscape, Phys. Lett. B 739 (2014) 439 [arXiv:1303.3224] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M.L. Mehta, Random Matrices, third edition, Academic Press, 2004.Google Scholar
  22. [22]
    F.J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 (1962) 140 [INSPIRE].
  23. [23]
    D.S. Dean and S.N. Majumdar, Extreme value statistics of eigenvalues of Gaussian random matrices, Phys. Rev. E 77 (2008) 041108 [arXiv:0801.1730].ADSMathSciNetGoogle Scholar
  24. [24]
    F.G. Tricomi, Integral Equations, Pure Appl. Math. V, Interscience, London U.K. (1957).Google Scholar
  25. [25]
    S.L. Paveri-Fontana and P.F. Zweifel, The half-Hartley and the half-Hilbert transform, J. Math. Phys. 35 (1994) 2648.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    K. Dutta and A. Maharana, Inflationary constraints on modulus dominated cosmology, Phys. Rev. D 91 (2015) 043503 [arXiv:1409.7037] [INSPIRE].ADSGoogle Scholar
  27. [27]
    M. Cicoli, K. Dutta, A. Maharana and F. Quevedo, Moduli Vacuum Misalignment and Precise Predictions in String Inflation, JCAP 08 (2016) 006 [arXiv:1604.08512] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    G. Wang and T. Battefeld, Vacuum Selection on Axionic Landscapes, JCAP 04 (2016) 025 [arXiv:1512.04224] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    A. Westphal, Tensor modes on the string theory landscape, JHEP 04 (2013) 054 [arXiv:1206.4034] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    E. Silverstein, Les Houches lectures on inflationary observables and string theory, arXiv:1311.2312 [INSPIRE].
  31. [31]
    T.C. Bachlechner, Inflation Expels Runaways, JHEP 12 (2016) 155 [arXiv:1608.07576] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Departamento de Física Teórica and Instituto de Física Teórica UAM/CSICUniversidad Autónoma de MadridMadridSpain
  2. 2.Deutsches Elektronen-Synchrotron DESY, Theory GroupHamburgGermany

Personalised recommendations