---
title: Factorisation of 1-Loop Cosmological Correlators
url: https://www.emergentmind.com/topics/factorisation-property-of-one-loop-cosmological-correlators
type: topic
---

# Factorisation of 1-Loop Cosmological Correlators

The factorisation property of one-loop cosmological correlators refers to the precise manner in which cosmological wavefunction coefficients or in-in correlators at one loop admit a decomposition into products of amplitudes, typically lower-point tree amplitudes or on-shell flat-space loop amplitudes, at certain kinematic singularities. This property establishes a deep connection between the analytic structure of cosmological observables and the unitarity cuts of flat-space quantum field theory, underlining the role of residue calculus, canonical forms, and cohomological methods in the analysis of correlators in Friedmann–Robertson–Walker (FRW), de Sitter (dS), or Anti-de Sitter (AdS) backgrounds.

## 1. Cohomological Foundations and Dual Structure

In the FRW wavefunction formalism, the wavefunction coefficients are defined via twisted integrals over a complex domain with hyperplane singularities, with the integrand encoding both cosmological and flat-space dynamics:
$$
\psi_n^{(\ell)} = \int_0^\infty u(x)\,\Psi_n^{(\ell)}(x;X,Y)
$$
where $u(x) = \prod_i x_i^\epsilon$ is the universal twist, and $\Psi_n^{(\ell)}$ involves a shifted canonical flat-space form $\Omega_n^{(\ell)}(x+X,Y)\,d^n x$ [2411.09695].

A crucial insight is the distinction between the FRW cohomology $H^n(M\setminus B;\nabla)$ and the dual (relative) twisted cohomology $H^n(M,B;\nabla^\dagger)$. The dual cohomology, organized in terms of cuts (sequential residues), selects a closed subspace whose forms are in one-to-one correspondence with physical factorization channels of the wavefunction into flat-space amplitudes. The intersection pairing computes weighted residues and ensures only compatible residue sequences contribute.

## 2. Graphical and Polytope-Based Classification

For a general one-loop $n$-site graph, physical channels are classified using the graphical tubing approach and the geometry of weighted cosmological polytopes. Each hyperplane $S_k(x+X,Y)=0$ represents a possible cut, and allowed cut sequences (tubings) are subject to three universal rules [2411.09695, 2401.05207]:
- Good-cut condition: Only compatible nesting of tubings (respecting minimal linear hyperplane relations) contributes.
- Degenerate-cut condition: Degenerate codimension boundaries are counted exactly once.
- One-dimensional cohomology: Only those cuts preserving a bounded chamber in the remaining coordinates survive.

In weighted cosmological polytopes, ordinary facets $E_\mathfrak{g}=0$ correspond to total energy going to zero (soft total energy), and internal facets $y_e=0$ coincide with cuts putting loop-edges on-shell. The canonical form of the polytope encodes the universal loop integrand structure, and its residues precisely yield the factorisation into tree-level correlators associated with the facets [2401.05207].

## 3. Factorisation Theorems: Residues and Cutting Rules

At singular kinematics, the correlator exhibits a logarithmic singularity whose leading residue factorises:
$$
\lim_{\mathcal E_B + k_e \to 0} [(\mathcal E_{\rm tot})\,\mathcal C_n^{\rm 1\text{-}loop}] = (-1)^V \sum_{\pm} \psi(\{X_v: v\in B\}, \pm k_e) \times \Res_{P_e^2 \to 0}\, \mathcal A_{\rm loop}(P_e,\{P_{\bar B}\})
$$
where $\psi$ is the shifted tree correlator and $\mathcal A_{\rm loop}$ is the flat-space loop amplitude continued on-shell [2601.00952].

Explicit graphical rules for cuts directly enumerate all factorisation channels, with each sequence of residues corresponding to a unique factorisation into subamplitudes (e.g., a one-loop amplitude splits into products of tree and loop amplitudes upon sequential cutting of the total energy and internal edge hyperplanes) [2411.09695]. This procedure is underpinned by intersection theory and closes under kinetic differentiation.

In Mellin-Barnes and in-in approaches, the one-loop correlator decomposes as:
$$
G^{\rm 1\text{-}loop}(k_i) = G^{\rm fact}(k_i) + G^{\rm bkg}(k_i)
$$
where the entire nonlocal signal content (oscillatory features stemming from massive exchanges) resides in $G^{\rm fact}$, a sum over products of tree-level pieces and a universal spectral measure, while $G^{\rm bkg}$ contains only local and UV-divergent contributions, removable by counterterms [2411.13636, 2304.13295, 2308.14802].

## 4. Explicit Example: Four-Site One-Loop Graph

A canonical case is the four-site loop (square), whose hyperplane structure encodes all allowed cuts:
- 1-cut (total energy): Residue yields the one-loop flat-space box amplitude.
- 2-cuts (single-edge or loop-facet): Further residues yield factorisation into tree × one-loop or triangle × tree channels.
- 3-cuts: Complete factorisation into three tree amplitudes.

Intermediate cuts are regulated by compatibility conditions and Steinmann-like relations, with degenerate channels counted via unique $d\log$ combinations. The cohomological closure guarantees the dimension of the physical subspace matches the count of physical factorisation channels [2411.09695].

## 5. Analytic Structure, OPE, and Factorisation in Various Contexts

Factorisation extends beyond FRW and flat space:
- In AdS/dS, the worldsheet (scattering equations) formulation shows one-loop correlators inherit the factorisation structure of tree-level puncture collisions, with residues corresponding to products of lower-point correlators and differential operators encoding on-shell exchanges [2112.12695].
- The OPE (operator product expansion) interpretation applies in the squeezed or soft limit, with nonanalytic terms in the correlator expansion mapping precisely to partial-wave blocks and anomalous dimensions extracted from loop-induced corrections [2204.07217, 2304.13295].
- For arbitrary 1PI one-loop inflation diagrams, the general factorisation theorem yields nonanalytic "clock" signals as products of tree-like components and bubble signals, both in leading and subleading (higher-momentum-ratio) asymptotics [2304.13295, 2308.14802].

## 6. Geometric and Combinatorial Constraints

The analytic integrity of the factorisation structure is maintained by Steinmann-like vanishing conditions: no two incompatible (overlapping) facets can simultaneously produce singularities, as encoded in the polytope boundary structure [2401.05207]. This ensures a consistent causal and combinatorial organisation of channels analogous to flat-space unitarity but adapted to cosmological settings.

Canonical forms, polytope adjoints, and the mutual closure of physical residues establish a mathematical foundation for the correspondence between cosmological and flat-space factorisation, enabling the enumeration and calculation of all physical channels without explicit computation of the entire integrand.

## 7. Generalisations, Limitations, and UV Renormalisation

The factorisation property holds for general one-loop ($n$-site, arbitrary topology) graphs, with UV divergences restricted to bubble topologies and renormalised by local counterterms mimicking flat-space procedure [2411.13636, 2601.00952]. For higher-point loops and mixed cosmological backgrounds, the structure generalises via the dual cohomology and polytope approaches.

Mass ranges and background symmetries influence the detailed implementation; for instance, the worldsheet formula is strictly valid within the complementary series of dS (or BF-bound in AdS), ensuring convergence and manifest conformal invariance [2112.12695].

The factorisation property thus provides a unified theoretical and computational framework, merging residue calculus, cohomological structure, and geometric representation, for understanding the analytic and physical bases of one-loop cosmological correlators across quantum cosmology, inflationary, and holographic regimes.

Source: https://www.emergentmind.com/topics/factorisation-property-of-one-loop-cosmological-correlators