Cosmological Optical Theorem

• Cosmological Optical Theorem is a framework that extends unitarity from flat-space scattering to cosmological boundary observables, defining relations among wavefunction coefficients.
• It employs diagrammatic cutting rules and geometric cosmological polytopes to capture analytic structures and enforce phase constraints in expanding universe scenarios.
• The theorem underpins practical methods for extracting non-Gaussian features in cosmic data and supports rigorous consistency checks in quantum gravity models.

The Cosmological Optical Theorem is a rigorous extension of the principle of unitarity—probability conservation in quantum mechanics and field theory—into cosmological observables, with particular relevance for analyses of the wavefunction of the universe in expanding backgrounds such as de Sitter (dS) and flat Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes. Unlike the familiar optical theorem in flat-space scattering theory (which relates the imaginary part of a forward amplitude to the total cross section), the cosmological optical theorem encodes unitarity constraints as functional and analytic relations among wavefunction coefficients or correlators measured at the cosmological boundary. Recent developments have mapped these relations into both diagrammatic (cosmological cutting rules) and geometric (cosmological polytopes) frameworks, revealing deep connections between symmetry, analytic continuation, and the bootstrap of cosmological correlators.

1. Unitarity in Cosmology and the Emergence of the Cosmological Optical Theorem

In cosmology, observables are not S-matrix elements but rather the coefficients of late-time wavefunctionals (often denoted $\psi_n$ for $n$-point functions) or correlators obtained via in-in (Schwinger–Keldysh) formalisms. The time-evolution operator remains unitary ($U^\dagger U = 1$), but due to the absence of spatial infinity and the time-dependent background, the consequences for boundary observables are nontrivial.

The cosmological optical theorem (COT) is formulated as a set of exact relations among the $\psi_n$ coefficients, generated by unitarity constraints (Goodhew et al., 2020). For contact interactions, these relations fix the analytic continuation and discontinuity structure: $$\psi'_n(k_a,\ldots) + \Bigl[\psi'_n(-k_a-i\epsilon,\ldots)\Bigr]^* = 0$$ where primed coefficients denote the stripping of the overall momentum-conserving delta function. For exchange diagrams (such as four-point functions), unitarity relates the higher-point coefficients to products of lower-point coefficients (e.g., $\psi_4$ in terms of $\psi_3$), with the full relation incorporating propagators for exchanged fields: $$\psi'^s_4(\{k_i\},s) + \Bigl[\psi'^s_4(-\{k_i\},s)\Bigr]^* = P_\sigma(s)\Bigl[\psi_3^{\phi\phi\sigma}(k_1,k_2,s)-\psi_3^{\phi\phi\sigma}(k_1,k_2,-s)\Bigr]\Bigl[\psi_3^{\phi\phi\sigma}(k_3,k_4,s)-\psi_3^{\phi\phi\sigma}(k_3,k_4,-s)\Bigr]$$ This structure enforces factorization and selection rules (e.g., correlators with an odd number of conformally-coupled scalar fields vanish).

2. Analytic Structure, Cutting Rules, and Diagrammatic Bootstrap

The analytic structure imposed by the COT manifests in the singularities of cosmological correlators, particularly the locations and orders of energy poles. The total-energy pole encodes the flat-space limit, while subdiagram singularities reflect factorization on on-shell intermediate states (Goodhew et al., 2020).

The development of cosmological cutting rules (Melville et al., 2021, Goodhew et al., 2021) generalizes the COT to arbitrary diagrams and loop orders. The rules prescribe that the discontinuity of a cosmological diagram (with respect to the internal energies/momenta) can be systematically constructed by summing over all ways of "cutting" internal lines and replacing bulk-to-bulk propagators with pairs of bulk-to-boundary propagators and power-spectrum insertions. Schematically: $$i\, \text{Disc}_{\text{internal}}[i\,\psi^{(D)}] = \sum_{\text{cuts}} \prod_{\text{cut lines}} \left(\int P \right) \prod_{\text{subdiagrams}} \left(-i\, \text{Disc}[i\,\psi^{(\text{subdiagram})}]\right)$$ These rules fix loop-induced discontinuities in terms of tree-level data and have extensive applicability across inflationary EFTs and general FLRW backgrounds.

3. Geometric Formulation: Cosmological Polytopes and Non-Convexity

Recent work has embedded these unitarity relations in a geometric framework using "cosmological polytopes" (Albayrak et al., 2023). The canonical form of the polytope associated to a diagram encodes the universal integrand for the Bunch-Davies wavefunction. Unitarity is mapped to a special non-convex subregion—the "optical polytope"—whose subdivisions correspond to different cutting rules.

$$\Delta\psi_{\mathcal{G}} = \psi_{\mathcal{G}}(|p|, \dots) + \psi_{\mathcal{G}}^\dagger(-|p|, \dots) = - \sum_{\text{cuts}}$$

Two invariant definitions for non-convex geometry of the optical polytope are given: one as a degenerate limit of convex polytopes and another via compatibility conditions on the polytope facets. This combinatorial-geometric perspective facilitates the identification of cutting rules and reveals connections between analytic continuation, pole structure, and the geometry of cosmological amplitudes.

4. Discrete Symmetry Constraints and Phase Formulae

In flat-space QFT, unitarity and Lorentz invariance imply CPT symmetry, which for cosmological settings generalizes to combinations of Reflection Reality (RR), discrete dilatations (D$_{-1}$), and CRT (Goodhew et al., 30 Aug 2024). In flat FLRW and de Sitter backgrounds, the imposition of any two of these symmetries automatically implies the third. Nonperturbative reality constraints can be translated to exact relations among the phases of wavefunction coefficients at the future boundary:

$$\arg(\overline{\psi}_n^{(L)}) = -\frac{\pi}{2} \left[ (d+1)(L-1) + dn - \sum_{\alpha=1}^n \Delta_\alpha \right] + \pi \mathbb{N}$$

This constraint fixes (up to signs) the phases of $n$-point functions, with direct consequences for the structure of central charges and correlation functions in candidate dual CFTs in holographic models of de Sitter space.

5. Relation to Flat-Space Optical Theorem and S-Matrix Unitarity

In the flat-space limit, the cosmological optical theorem reduces to the familiar Cutkosky rules, establishing the equivalence between the analyticity and unitarity in cosmological contexts and those in standard quantum field theory. When the total energy singularity of the boundary correlator is reached, the canonical forms and discontinuities of the cosmological polytope degenerate to the flat-space S-matrix and its optical theorem (Albayrak et al., 2023). This correspondence is crucial for externally connecting cosmological measurements to scattering amplitudes and constraining model-building in quantum gravity and inflation.

6. Operator Content, Positivity, and Bootstrap in Celestial and Cosmological Contexts

The celestial optical theorem (Liu et al., 29 Apr 2024) generalizes the concept to celestial conformal field theories (CCFT), yielding nonperturbative bootstrap equations for conformal partial wave (CPW) coefficients. S-matrix unitarity translates to positivity constraints on the imaginary part of CPW coefficients, with the pole structure (simple poles for massless and double-trace poles for massive exchanges) encoding the allowed operator content of the theory. In contrast to AdS/CFT where double-trace dimensions acquire anomalous corrections, in CCFT they remain unrenormalized. These constraints are essential for any viable cosmological bootstrap program and for distinguishing unitary holographic duals (Liu et al., 29 Apr 2024).

7. Practical Implications and Observational Significance

Direct implementation of the cosmological optical theorem informs the computation and interpretation of cosmological correlators (bispectrum, trispectrum, etc.) relevant for cosmic microwave background and large scale structure analyses. It sharpens the extraction of physical non-Gaussianity signals and helps in identifying selection rules and null-test correlators. Diagrammatic and geometric approaches streamline both analytic and numerical treatments, from perturbative inflationary EFTs to nonperturbative bootstrap programs. The phase constraints and positivity theorems serve as diagnostic tools in model selection for quantum gravity and early universe cosmology.

Table: Key Conceptual Features

Feature	Description	Reference
Unitarity constraint	Relations among wavefunction coefficients at boundary	(Goodhew et al., 2020, Albayrak et al., 2023)
Cutting rules	Diagrammatic prescription for wavefunction discontinuities	(Melville et al., 2021, Goodhew et al., 2021)
Cosmological polytopes	Geometric encoding of the integrand and cuts	(Albayrak et al., 2023)
Phase formula at boundary	Symmetry-fixed phase of n-point wavefunction coefficients	(Goodhew et al., 30 Aug 2024)
Positivity of CPW coefficients	Imaginary part non-negative, operator spectrum constraints	(Liu et al., 29 Apr 2024)

Concluding Perspective

The cosmological optical theorem encapsulates the far-reaching consequences of bulk unitarity for cosmological boundary correlators, imposing powerful analytic, diagrammatic, and geometric constraints on the structure, phase, and positivity of observables in expanding spacetimes. These constraints play a central role in contemporary cosmological bootstrap approaches and holographic dualities, and provide an authoritative basis for both the interpretation of precision cosmological data and the foundational consistency checks of quantum gravity models.