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Scalar correlation functions for a double-well potential in de Sitter space
General Relativity and Quantum Cosmology · Submitted 13 Jan 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The long-distance asymptotic behaviour is given by the lowest state in the spectrum, but we demonstrate that the magnitude of the coeffients of different terms can be very different, and the correlator can be dominated by different terms at different distances.
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The abstract, sentence by sentence
We use the spectral representation of the stochastic Starobinsky-Yokoyama approach to compute correlation functions in de Sitter space for a scalar field with a symmetric or asymmetric double-well potential. The terms in the spectral expansion are determined by the eigenvalues and eigenfunctions of the time-independent Fokker-Planck differential operator, and we solve them numerically. The long-distance asymptotic behaviour is given by the lowest state in the spectrum, but we demonstrate that the magnitude of the coeffients of different terms can be very different, and the correlator can be dominated by different terms at different distances. This can give rise to potentially observable cosmological signatures. In many cases the dominant states in the expansion do not correspond to small fluctuations around a minimum of the potential and are therefore not visible in perturbation theory. We discuss the physical interpretation these states, which can be present even when the potential has only one minimum.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We use the spectral representation of the stochastic Starobinsky-Yokoyama approach to compute correlation functions in de Sitter space for a scalar field with a symmetric or asymmetric double-well potential. | contribution 0.55 | 0.94 | 0.07 | asserted 0.99 |
| s1 | The terms in the spectral expansion are determined by the eigenvalues and eigenfunctions of the time-independent Fokker-Planck differential operator, and we solve them numerically. | method 1.00 | 0.93 | 0.04 | asserted 1.00 |
| s2 | The long-distance asymptotic behaviour is given by the lowest state in the spectrum, but we demonstrate that the magnitude of the coeffients of different terms can be very different, and the correlator can be dominated by different terms at different distances. | result 0.99 | 0.93 | 0.43 | asserted 1.00 |
| s3 | This can give rise to potentially observable cosmological signatures. | interpretation 0.98 | 0.40 | 0.23 | hedged possibility 0.98 |
| s4 | In many cases the dominant states in the expansion do not correspond to small fluctuations around a minimum of the potential and are therefore not visible in perturbation theory. | result 0.80 | 0.78 | 0.20 | asserted 0.81 |
| s5 | We discuss the physical interpretation these states, which can be present even when the potential has only one minimum. | interpretation 0.96 | 0.69 | 0.13 | asserted 0.71 |
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The paper
Original abstract
We use the spectral representation of the stochastic Starobinsky-Yokoyama approach to compute correlation functions in de Sitter space for a scalar field with a symmetric or asymmetric double-well potential. The terms in the spectral expansion are determined by the eigenvalues and eigenfunctions of the time-independent Fokker-Planck differential operator, and we solve them numerically. The long-distance asymptotic behaviour is given by the lowest state in the spectrum, but we demonstrate that the magnitude of the coeffients of different terms can be very different, and the correlator can be dominated by different terms at different distances. This can give rise to potentially observable cosmological signatures. In many cases the dominant states in the expansion do not correspond to small fluctuations around a minimum of the potential and are therefore not visible in perturbation theory. We discuss the physical interpretation these states, which can be present even when the potential has only one minimum.
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References
How the literature uses it
19 arXiv papers cite this in 31 sentences
Papers citing it · 15 sentences without a confident label
Uses its method or tool 6 papers
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In order to determine the power spectrum of the spectator sourced part $ζ_χ$, we apply the stochastic formalism [ref] and the spectral expansion method [ref] .
Stochastic Gravitational Waves from Modulated Reheating · Curvature perturbation · 1.00
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We use the stochastic formalism [ref] and the spectral expansion method [ref] to compute the infrared spectrum of $ζ_χ$ in our setup where $⟨χ⟩= 0$ and Eq. [ref] cannot be expanded in small perturbations around a mean field.
Primordial black holes from a curvaton scenario with strongly non-Gaussian perturbations · The spectrum of ζχ · 1.00
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In this paper, we go beyond that by using the spectral expansion method [ref] , which allows the numerical determination of the exact asymptotic long-distance behaviour of any correlation function in de Sitter spacetime within the stochastic approximation.
Stochastic isocurvature constraints for axion dark matter with high-scale inflation · Introduction · 0.83
Cites it as background 6 papers
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For more details we refer the reader to the Refs. [ref] .
Novel mechanism for CMB modulation in the Standard Model · Spectator Fields · 0.99
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See also e.g. [ref] and references therein for recent developments on stochastic cosmic inflation.
Massless minimal quantum scalar field with an asymmetric self interaction in de Sitter spacetime · Introduction · 0.95
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We further refer our reader to [ref] and references therein for recent developments.
Non-perturbative analysis for a massless minimal quantum scalar with $V(\phi)=\lambda \phi^4/4!+\beta \phi^3/3!$ in the inflationary de Sitter spacetime · Introduction · 0.94
What it cites · 14 sentences without a confident label
Builds on 1 paper
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This work is a continuation of Ref. [ref] where only potentials with manifest $ℤ_2$ symmetry and a single minimum were considered.
Scalar correlation functions in de Sitter space from the stochastic spectral expansion · Conclusions · 1.00
Uses the method or tool of 2 papers
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Following Ref. [ref] , we solve the eigenfunctions and eigenvalues of Eq. ([ref]) numerically with the 'overshoot/undershoot' or otherwise known as 'wag the dog' method.
Scalar correlation functions in de Sitter space from the stochastic spectral expansion · A numerical approach · 0.99
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In the stochastic formalism one may express any correlator involving noncoincident spacetime points as a spectral expansion [ref] , by solving the eigenvalues and eigenfunctions, $Λ_n$ and $ψ_n$, from the eigenvalue equation [equation] with [equation] Since the input in the eigenvalue equation ([ref]) is $W(ϕ)$ (and not $V(ϕ)$) it is the $W(ϕ)$ that will turn out to be the fundamental quantity providing a qualitative understanding of the behaviour of the eigenfunctions and -values.
Equilibrium State of a Massless Self-Interacting Scalar Field in the De Sitter Background · The eigenvalue equation · 0.91
Cites as background 45 papers
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The spatial correlators have been addressed for example in Refs. [ref] .
Feynman Diagrams for Stochastic Inflation and Quantum Field Theory in de Sitter Space · Introduction · 0.96
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The spatial correlators have been addressed for example in Refs. [ref] .
On Resumming Inflationary Perturbations beyond One-loop · Introduction · 0.96
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The spatial correlators have been addressed for example in Refs. [ref] .
Higgs condensation as an unwanted curvaton · Introduction · 0.96
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