{"arxiv_id":"2001.04494","incoming":{"papers":19,"sentences":31,"groups":[{"kind":"method_or_tool","label":"Uses its method or tool","papers":6,"sentences":9,"quotes":[{"arxiv_id":"2510.05967","title":"Stochastic Gravitational Waves from Modulated Reheating","url":"/paper/2510.05967","section":"Curvature perturbation","sentence":"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] .","intent_p":1.0,"contests_p":null},{"arxiv_id":"2307.03078","title":"Primordial black holes from a curvaton scenario with strongly non-Gaussian perturbations","url":"/paper/2307.03078","section":"The spectrum of ζχ","sentence":"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.","intent_p":1.0,"contests_p":null},{"arxiv_id":"2107.07948","title":"Stochastic isocurvature constraints for axion dark matter with high-scale inflation","url":"/paper/2107.07948","section":"Introduction","sentence":"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.","intent_p":0.83,"contests_p":null},{"arxiv_id":"2112.01505","title":"Primordial black holes from spectator field bubbles","url":"/paper/2112.01505","section":"Evolution of the probability distribution","sentence":"The spatial profile of a spectator field can then be derived from its temporal evolution by using de Sitter invariance, which maps time intervals to equivalent spatial intervals [ref] .","intent_p":0.82,"contests_p":null},{"arxiv_id":"2008.07497","title":"A manifestly covariant theory of multifield stochastic inflation in phase space: solving the discretisation ambiguity in stochastic inflation","url":"/paper/2008.07497","section":"Stochastic equations","sentence":"Using this time-scale correspondence, Starobinsky and Yokoyama have shown in Ref. [ref] how, once the Fokker-Planck operator for the one-point probability density function (PDF) is known, one can determine the evolution equation for the two-point PDF, or even any $n$-point PDF (at different spatial and temporal locations), and thus, at least in principle, retrieve all the statistical information (see, e.g., [ref] for recent applications).[Naturally, given the hard cutoff in the spatial correlations of the stochastic noises, spatial correlations can be reliably computed only when the relevant length scales are well above $H^-1$, but that is overwhelmingly the case for observationally relevant scales.] This logic is also put to good use in the stochastic-$δN$ approach, with which one can compute Fourier space correlation functions of the observable large-scale curvature perturbation $ζ$ (a","intent_p":0.8,"contests_p":null}]},{"kind":"background","label":"Cites it as background","papers":6,"sentences":7,"quotes":[{"arxiv_id":"2006.14404","title":"Novel mechanism for CMB modulation in the Standard Model","url":"/paper/2006.14404","section":"Spectator Fields","sentence":"For more details we refer the reader to the Refs. [ref] .","intent_p":0.99,"contests_p":null},{"arxiv_id":"2202.01593","title":"Massless minimal quantum scalar field with an asymmetric self interaction in de Sitter spacetime","url":"/paper/2202.01593","section":"Introduction","sentence":"See also e.g. [ref] and references therein for recent developments on stochastic cosmic inflation.","intent_p":0.95,"contests_p":null},{"arxiv_id":"2211.12027","title":"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","url":"/paper/2211.12027","section":"Introduction","sentence":"We further refer our reader to [ref] and references therein for recent developments.","intent_p":0.94,"contests_p":null},{"arxiv_id":"2002.03402","title":"Feynman Rules for Stochastic Inflationary Correlators","url":"/paper/2002.03402","section":"Introduction","sentence":"Recently, the use of stochastic methods to compute IR 2-point functions for a spectator scalar in inflation has further been developed in [ref] .","intent_p":0.82,"contests_p":null},{"arxiv_id":"2310.19436","title":"Resummation of local and non-local scalar self energies via the Schwinger-Dyson equation in de Sitter spacetime","url":"/paper/2310.19436","section":"Introduction","sentence":"The stochastic formalism is very useful and popular for computing scalar correlation functions, see e.g. [ref] for recent progresses on it.","intent_p":0.95,"contests_p":null}]}],"unlabelled_sentences":15},"outgoing":{"papers":50,"sentences":69,"groups":[{"kind":"extends","label":"Builds on","papers":1,"sentences":1,"quotes":[{"arxiv_id":"1904.11917","title":"Scalar correlation functions in de Sitter space from the stochastic spectral expansion","url":"/paper/1904.11917","section":"Conclusions","sentence":"This work is a continuation of Ref. [ref] where only potentials with manifest $ℤ_2$ symmetry and a single minimum were considered.","intent_p":1.0,"contests_p":null}]},{"kind":"method_or_tool","label":"Uses the method or tool of","papers":2,"sentences":8,"quotes":[{"arxiv_id":"1904.11917","title":"Scalar correlation functions in de Sitter space from the stochastic spectral expansion","url":"/paper/1904.11917","section":"A numerical approach","sentence":"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.","intent_p":0.99,"contests_p":null},{"arxiv_id":"astro-ph/9407016","title":"Equilibrium State of a Massless Self-Interacting Scalar Field in the De Sitter Background","url":"/paper/astro-ph/9407016","section":"The eigenvalue equation","sentence":"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.","intent_p":0.91,"contests_p":null}]},{"kind":"background","label":"Cites as background","papers":45,"sentences":46,"quotes":[{"arxiv_id":"1412.4893","title":"Feynman Diagrams for Stochastic Inflation and Quantum Field Theory in de Sitter Space","url":"/paper/1412.4893","section":"Introduction","sentence":"The spatial correlators have been addressed for example in Refs. [ref] .","intent_p":0.96,"contests_p":null},{"arxiv_id":"0801.1845","title":"On Resumming Inflationary Perturbations beyond One-loop","url":"/paper/0801.1845","section":"Introduction","sentence":"The spatial correlators have been addressed for example in Refs. [ref] .","intent_p":0.96,"contests_p":null},{"arxiv_id":"1208.2316","title":"Higgs condensation as an unwanted curvaton","url":"/paper/1208.2316","section":"Introduction","sentence":"The spatial correlators have been addressed for example in Refs. [ref] .","intent_p":0.96,"contests_p":null},{"arxiv_id":"1210.2497","title":"Consequences of a stochastic approach to the conformal invariance of inflationary correlators","url":"/paper/1210.2497","section":"Introduction","sentence":"The spatial correlators have been addressed for example in Refs. [ref] .","intent_p":0.96,"contests_p":null},{"arxiv_id":"1310.0367","title":"Infrared Correlations in de Sitter Space: Field Theoretic vs. Stochastic Approach","url":"/paper/1310.0367","section":"Introduction","sentence":"The spatial correlators have been addressed for example in Refs. [ref] .","intent_p":0.96,"contests_p":null}]}],"unlabelled_sentences":14},"claims":[],"thresholds":{"intent_p":0.8,"supports_p":0.95,"contests_p":0.7},"method":"Citing sentences come from each paper's LaTeX source; every \\cite marks the sentence and the works it cites. Pith's classifier typed each sentence. A label is shown at intent probability 0.8 or more (0.95 for support), a contest at 0.7 or more. Measured accuracy against two blind model families is at https://pith.science/accuracy. The quoted sentences are the citing authors' own.","held_up":null,"_links":{"html":"https://pith.science/paper/2001.04494"}}