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Hessian eigenvalue distribution in a random Gaussian landscape

Journal of High Energy Physics volume 2018, Article number: 29 (2018) Cite this article

A preprint version of the article is available at arXiv.

Abstract

The energy landscape of multiverse cosmology is often modeled by a multi-dimensional random Gaussian potential. The physical predictions of such models crucially depend on the eigenvalue distribution of the Hessian matrix at potential minima. In particular, the stability of vacua and the dynamics of slow-roll inflation are sensitive to the magnitude of the smallest eigenvalues. The Hessian eigenvalue distribution has been studied earlier, using the saddle point approximation, in the leading order of 1/N expansion, where N is the dimensionality of the landscape. This approximation, however, is insufficient for the small eigenvalue end of the spectrum, where sub-leading terms play a significant role. We extend the saddle point method to account for the sub-leading contributions. We also develop a new approach, where the eigenvalue distribution is found as an equilibrium distribution at the endpoint of a stochastic process (Dyson Brownian motion). The results of the two approaches are consistent in cases where both methods are applicable. We discuss the implications of our results for vacuum stability and slow-roll inflation in the landscape.

References

  1. R. Bousso and J. Polchinski, Quantization of four form fluxes and dynamical neutralization of the cosmological constant, JHEP 06 (2000) 006 [hep-th/0004134] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. L. Susskind, The Anthropic landscape of string theory, hep-th/0302219 [INSPIRE].

  3. A. Linde, A brief history of the multiverse, Rept. Prog. Phys. 80 (2017) 022001 [arXiv:1512.01203] [INSPIRE].

    ADS  Article  Google Scholar 

  4. M. Tegmark, What does inflation really predict?, JCAP 04 (2005) 001 [astro-ph/0410281] [INSPIRE].

  5. A. Aazami and R. Easther, Cosmology from random multifield potentials, JCAP 03 (2006) 013 [hep-th/0512050] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  6. J. Frazer and A.R. Liddle, Exploring a string-like landscape, JCAP 02 (2011) 026 [arXiv:1101.1619] [INSPIRE].

    ADS  Article  Google Scholar 

  7. D. Battefeld, T. Battefeld and S. Schulz, On the unlikeliness of multi-field inflation: bounded random potentials and our vacuum, JCAP 06 (2012) 034 [arXiv:1203.3941] [INSPIRE].

    ADS  Article  Google Scholar 

  8. T.C. Bachlechner, D. Marsh, L. McAllister and T. Wrase, Supersymmetric vacua in random supergravity, JHEP 01 (2013) 136 [arXiv:1207.2763] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. I.-S. Yang, Probability of slowroll inflation in the multiverse, Phys. Rev. D 86 (2012) 103537 [arXiv:1208.3821] [INSPIRE].

    ADS  Google Scholar 

  10. T.C. Bachlechner, On gaussian random supergravity, JHEP 04 (2014) 054 [arXiv:1401.6187] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. G. Wang and T. Battefeld, Vacuum selection on axionic landscapes, JCAP 04 (2016) 025 [arXiv:1512.04224] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  12. A. Masoumi and A. Vilenkin, Vacuum statistics and stability in axionic landscapes, JCAP 03 (2016) 054 [arXiv:1601.01662] [INSPIRE].

    ADS  Article  Google Scholar 

  13. R. Easther, A.H. Guth and A. Masoumi, Counting vacua in random landscapes, arXiv:1612.05224 [INSPIRE].

  14. A. Masoumi, A. Vilenkin and M. Yamada, Inflation in random Gaussian landscapes, JCAP 05 (2017) 053 [arXiv:1612.03960] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  15. A. Masoumi, A. Vilenkin and M. Yamada, Initial conditions for slow-roll inflation in a random Gaussian landscape, JCAP 07 (2017) 003 [arXiv:1704.06994] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  16. A. Masoumi, A. Vilenkin and M. Yamada, Inflation in multi-field random Gaussian landscapes, JCAP 12 (2017) 035 [arXiv:1707.03520] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. T. Bjorkmo and M.C.D. Marsh, Manyfield inflation in random potentials, JCAP 02 (2018) 037 [arXiv:1709.10076] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  18. J.J. Blanco-Pillado, A. Vilenkin and M. Yamada, Inflation in random landscapes with two energy scales, JHEP 02 (2018) 130 [arXiv:1711.00491] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  19. J.E. Kim, H.P. Nilles and M. Peloso, Completing natural inflation, JCAP 01 (2005) 005 [hep-ph/0409138] [INSPIRE].

  20. S. Dimopoulos, S. Kachru, J. McGreevy and J.G. Wacker, N-flation, JCAP 08 (2008) 003 [hep-th/0507205] [INSPIRE].

    ADS  Article  Google Scholar 

  21. L. McAllister, E. Silverstein and A. Westphal, Gravity waves and linear inflation from axion monodromy, Phys. Rev. D 82 (2010) 046003 [arXiv:0808.0706] [INSPIRE].

    ADS  Google Scholar 

  22. T. Higaki and F. Takahashi, Natural and multi-natural inflation in axion landscape, JHEP 07 (2014) 074 [arXiv:1404.6923] [INSPIRE].

    ADS  Article  Google Scholar 

  23. T.C. Bachlechner, K. Eckerle, O. Janssen and M. Kleban, Systematics of aligned axions, JHEP 11 (2017) 036 [arXiv:1709.01080] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. A.J. Bray and D.S. Dean, Statistics of critical points of Gaussian fields on large-dimensional spaces, Phys. Rev. Lett. 98 (2007) 150201 [INSPIRE].

    ADS  Article  Google Scholar 

  25. Y.V. Fyodorov and C. Nadal, Critical behavior of the number of minima of a random landscape at the glass transition point and the Tracy-Widom distribution, Phys. Rev. Lett. 109 (2012) 167203 [arXiv:1207.6790].

    ADS  Article  Google Scholar 

  26. Y.V. Fyodorov and I. Williams, Replica symmetry breaking condition exposed by random matrix calculation of landscape complexity, J. Stat. Phys. 129 (2007) 1081 [cond-mat/0702601].

  27. F.J. Dyson, A Brownian-motion model for the Eigenvalues of a random matrix, J. Math. Phys. 3 (1962) 1191.

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. Y.V. Fyodorov, Complexity of random energy landscapes, glass transition and absolute value of spectral determinant of random matrices, Phys. Rev. Lett. 92 (2004) 240601.

    ADS  MathSciNet  Article  MATH  Google Scholar 

  29. E.P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math. 62 (1955) 548.

    MathSciNet  Article  MATH  Google Scholar 

  30. 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].

  31. 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].

    ADS  MathSciNet  Google Scholar 

  32. M.L. Mehta and M. Gaudin, On the density of Eigenvalues of a random matrix, Nucl. Phys. 18 (1960) 420.

    MathSciNet  Article  MATH  Google Scholar 

  33. J.S. Chang and G. Cooper, A practical difference scheme for Fokker-Planck equations, J. Comput. Phys. 6 (1970) 1.

    ADS  Article  MATH  Google Scholar 

  34. M. Mohammadi, and A. Borzì, Analysis of the Chang-Cooper discretization scheme for a class of Fokker-Planck equations, J. Num. Math. 23 (2015) 271.

  35. L. Pareschi and M. Zanella, Structure preserving schemes for nonlinear Fokker-Planck equations and applications, arXiv:1702.00088.

  36. C.A. Tracy and H. Widom, Level spacing distributions and the Airy kernel, Commun. Math. Phys. 159 (1994) 151 [hep-th/9211141] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  37. C.A. Tracy and H. Widom, On orthogonal and symplectic matrix ensembles, Commun. Math. Phys. 177 (1996) 727 [solv-int/9509007] [INSPIRE].

  38. V.A. Marcenko and L.A. Pastur, Distribution of eigenvalues for some sets of random matrices, Math. USSR-Sbornik 1 (1967) 457.

    Article  Google Scholar 

  39. B. Greene et al., Tumbling through a landscape: Evidence of instabilities in high-dimensional moduli spaces, Phys. Rev. D 88 (2013) 026005 [arXiv:1303.4428] [INSPIRE].

    ADS  Google Scholar 

  40. M. Dine and S. Paban, Tunneling in theories with many fields, JHEP 10 (2015) 088 [arXiv:1506.06428] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  41. C.W.J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys. 69 (1997) 731 [cond-mat/9612179].

  42. T. Guhr, A. Müller-Groeling and H.A. Weidenmuller, Random matrix theories in quantum physics: Common concepts, Phys. Rept. 299 (1998) 189 [cond-mat/9707301] [INSPIRE].

  43. B. Eynard, T. Kimura and S. Ribault, Random matrices, arXiv:1510.04430 [INSPIRE].

  44. J.E. Kim and G. Carosi, Axions and the strong CP problem, Rev. Mod. Phys. 82 (2010) 557 [arXiv:0807.3125] [INSPIRE].

    ADS  Article  Google Scholar 

  45. P. Svrček and E. Witten, Axions in string theory, JHEP 06 (2006) 051 [hep-th/0605206] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  46. P. Di Vecchia and G. Veneziano, Chiral dynamics in the large-N limit, Nucl. Phys. B 171 (1980) 253 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  47. G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro, The QCD axion, precisely, JHEP 01 (2016) 034 [arXiv:1511.02867] [INSPIRE].

    Article  Google Scholar 

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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.

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Authors and Affiliations

  1. Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA, 02155, U.S.A.

    Masaki Yamada & Alexander Vilenkin

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  1. Masaki Yamada
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  2. Alexander Vilenkin
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Correspondence to Masaki Yamada.

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ArXiv ePrint: 1712.01282

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Yamada, M., Vilenkin, A. Hessian eigenvalue distribution in a random Gaussian landscape. J. High Energ. Phys. 2018, 29 (2018). https://doi.org/10.1007/JHEP03(2018)029

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  • DOI: https://doi.org/10.1007/JHEP03(2018)029

Keywords

  • Cosmology of Theories beyond the SM
  • Random Systems