Tunneling in theories with many fields

The energy landscape of multiverse cosmology is often modeled by a multidimensional 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.

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“…This is √ N times larger than the estimate of Ref. [40], so the vacuum stability is significantly enhanced. (We note that if we used Eq.…”

Section: Vacuum Stabilitymentioning

confidence: 58%

“…A simple analytic treatment was given by Dine and Paban in Ref. [40]. They assume (i) that the most probable decay channels are typically in the directions of the smallest Hessian eigenvalues and (ii) that the vacuum decay rate is controlled mainly by the quadratic and cubic terms in the expansion of U (φ) about the potential minimum.…”

Section: Vacuum Stabilitymentioning

confidence: 99%

Hessian eigenvalue distribution in a random Gaussian landscape

Yamada

Vilenkin

2018

J. High Energ. Phys.

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“…We calculate the ratio R by taking µ's, γ's, and λ's as random variables as in Ref. [4]. The ranges of the parameters are taken as…”

Section: B Multi Scalar Fieldsmentioning

confidence: 99%

Absolute Lower Bound on the Bounce Action

Sato

Takimoto

2018

Phys. Rev. Lett.

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The decay rate of a false vacuum is determined by the minimal action solution of the tunneling field: bounce. In this Letter, we focus on models with scalar fields which have a canonical kinetic term in N(>2) dimensional Euclidean space, and derive an absolute lower bound on the bounce action. In the case of four-dimensional space, we show the bounce action is generically larger than 24/λ_{cr}, where λ_{cr}≡max[-4V(ϕ)/|ϕ|^{4}] with the false vacuum being at ϕ=0 and V(0)=0. We derive this bound on the bounce action without solving the equation of motion explicitly. Our bound is derived by a quite simple discussion, and it provides useful information even if it is difficult to obtain the explicit form of the bounce solution. Our bound offers a sufficient condition for the stability of a false vacuum, and it is useful as a quick check on the vacuum stability for given models. Our bound can be applied to a broad class of scalar potential with any number of scalar fields. We also discuss a necessary condition for the bounce action taking a value close to this lower bound.

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“…[5][6][7][8][9][10]. The statistics of vacuum decay rates in random landscapes has been investigated in [9,11,12]. Another crucial set of problems is to determine the likelihood of slow-roll inflation in the landscape, the expected number of e-folds, and the expected spectrum of density perturbations.…”

Section: Introductionmentioning

confidence: 99%

Inflation in random Gaussian landscapes

Masoumi

Vilenkin

Yamada³

2017

J. Cosmol. Astropart. Phys.

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We develop analytic and numerical techniques for studying the statistics of slow-roll inflation in random Gaussian landscapes. As an illustration of these techniques, we analyze small-field inflation in a one-dimensional landscape. We calculate the probability distributions for the maximal number of e-folds and for the spectral index of density fluctuations n s and its running α s . These distributions have a universal form, insensitive to the correlation function of the Gaussian ensemble. We outline possible extensions of our methods to a large number of fields and to models of large-field inflation. These methods do not suffer from potential inconsistencies inherent in the Brownian motion technique, which has been used in most of the earlier treatments.

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Tunneling in theories with many fields

Cited by 22 publications

References 16 publications

Hessian eigenvalue distribution in a random Gaussian landscape

Hessian eigenvalue distribution in a random Gaussian landscape

Absolute Lower Bound on the Bounce Action

Inflation in random Gaussian landscapes

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