Related papers: A Tree Theorem for Inflation
An algebraic formalism, developped with V. Glaser and R. Stora for the study of the generalized retarded functions of quantum field theory, is used to prove a factorization theorem which provides a complete description of the generalized…
The Matrix-Tree Theorem states that the number of spanning trees of a graph is given by the absolute value of any cofactor of the Laplacian matrix of the graph. We propose a very short proof of this result which amounts to comparing Taylor…
We exploit a new numerical technique for evaluating the tree order contributions to the primordial scalar and tensor power spectra for scalar potential models of inflation. Among other things we use the formalism to develop a good analytic…
By using an approach of the invariant theory we obtain a new formula for the ordinary generating function of the numbers of the simple graphs with $n$ nodes.
Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the…
We consider the number of spanning trees in circulant graphs of $\beta n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of $\beta n$ factors, while we derive a formula of $\beta-1$…
Classical solutions generating tree form-factors are defined and constructed in various models.
We propose a new method to numerically calculate higher-order correlation functions of primordial fluctuations generated from any early-universe scenario. Our key-starting point is the realization that the tree-level In-In formalism is…
The early Universe inflation is well known as a promising theory to explain the origin of large scale structure of Universe and to solve the early universe pressing problems. For a resonable inflation model, the potential during inflation…
The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have…
One can often make inferences about a growing network from its current state alone. For example, it is generally possible to determine how a network changed over time or pick among plausible mechanisms explaining its growth. In practice,…
Any one-field inflation is actually realized in a multifield configuration because the inflaton must have couplings with other fields to reheat the universe and is coupled to all other fields at least gravitationally. In all single inflaton…
The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial…
The ``in-in'' formalism is reviewed and extended, and applied to the calculation of higher-order Gaussian and non-Gaussian correlations in cosmology. Previous calculations of these correlations amounted to the evaluation of tree graphs in…
The quantum theory of cosmological perturbations in single field inflation is formulated in terms of a path integral. Starting from a canonical formulation, we show how the free propagators can be obtained from the well known…
Motivated by machine learning, we introduce a novel method for randomly generating inflationary potentials. Namely, we treat the Taylor coefficients of the potential as weights in a single-layer neural network and use gradient ascent to…
Let $F(x)=\sum\limits_{n=1}^\infty\tau(n)x^n$ be the generating function for the number $\tau(n)$ of spanning trees in the circulant graphs $C_{n}(s_1,s_2,\ldots,s_k).$ We show that $F(x)$ is a rational function with integer coefficients…
Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given graph with a fixed 2-core. We use this generalization to obtain an analog of the matrix-tree theorem for the root system $D_n$ (the classical…
Let $c_n$ denote the number of nodes at a distance $n$ from the root of a rooted tree. A criterion for proving the rationality and computing the rational generating function of the sequence $\{c_n\}$ is described. This criterion is applied…
We argue, using a phenomenological holographic approach, that walking, strongly coupled gauge theories generate a suitable potential for a small field inflation model. We show that the effective description is a model of a single inflaton.…