Unitarity, Recursion and Soft Limits in (EA)dS through Dressing

Abstract: Using the recently developed framework in which cosmological correlators in (E)AdS are represented as flat-space amplitudes dressed by auxiliary propagators, we show that several structural properties of the cosmological observables have a direct flat-space origin. We derive cosmological cutting rules for spinning correlators from the flat-space optical theorem, obtain the cosmological tree theorem from the Feynman tree theorem, and uplift BCFW recursion relations to (E)AdS via dressing. We also show that flat-space soft theorems reproduce the soft limits of (E)AdS correlators, and find indications of an emergent universal structure in subleading soft limits. These results provide evidence that key features of cosmological correlators can be systematically understood as dressed manifestations of flat-space physics.

• The paper presents a dressing protocol that maps flat-space amplitudes to (EA)dS correlators, enabling systematic computation and interpretation of cosmological observables.
• It rigorously uplifts unitarity and cutting rules from flat-space to cosmological settings, maintaining essential analytic and discontinuity structures.
• The study extends on-shell recursion (BCFW) and soft theorems to (EA)dS correlators, providing a unified framework applicable to scalar and spinning fields at both tree and loop levels.

Unitarity, Recursion, and Soft Limits in (EA)dS via Dressing: A Comprehensive Technical Analysis

Motivation and Technical Framework

The theoretical complexity of cosmological correlators in de Sitter (dS) and Euclidean Anti-de Sitter ((EA)dS) spaces arises from the absence of global time-translation symmetry and a well-defined asymptotic S-matrix. Instead, the observables are encoded in boundary correlators (wavefunction coefficients) or in-in correlators, whose analytic structure and physical interpretation remain challenging. The paper leverages a recently developed paradigm in which (EA)dS correlators are formulated as flat-space scattering amplitudes "dressed" by auxiliary propagators that encode the effects of the curved background [Chowdhury:2025ohm, Chowdhury:2025nnk]. This dressing mechanism systematically uplifts analytic and structural properties from flat-space to cosmological correlators, facilitating both computation and interpretation.

Dressing Protocol and Wavefunction Representation

The dressing operation constitutes a mapping between flat-space amplitudes and boundary correlators in cosmological settings. For a given theory (e.g., conformally coupled scalars, scalar QED, non-Abelian gauge fields, gravity), the relevant Feynman diagram is supplemented by kernel integrals that reflect the curved spacetime dynamics. The precise form of these dressing factors depends on the bulk theory and boundary conditions; for instance, in $\phi^4$ theory, the dressing factor $$\hat{\Delta}(k_{\text{ext}}, p) = p/(p^2 + k_{\text{ext}}^2)$$ enters at each vertex, as detailed in the explicit worked example (cf. Eq. (1) of the paper).

The resulting correlator decomposes into a product of flat-space amplitude structure and (EA)dS-specific factors:

$$\psi_{n,s}(\{k_i\}, \vec{s}) = \int dp\, [\text{dressing}_L][\text{dressing}_R][\text{flat-space amplitude}]$$

This structural splitting persists for spinning fields, with longitudinal and transverse components handled via their respective dressing rules.

Unitarity and Cosmological Cutting Rules

The paper rigorously uplifts the flat-space optical theorem, which encodes unitarity via discontinuities across branch cuts, to cosmological correlators in (EA)dS. Flat-space discontinuity relations,

$- i (T - T^\dagger) = TT^\dagger\,,$

are precisely mapped, via dressing, to discontinuity relations among cosmological wavefunction coefficients. Both for tree-level and loop-level diagrams, the dressed discontinuity (Disc) operation yields cosmological cutting rules. The explicit transformation maintains the distinction between transverse and longitudinal contributions, reconciling polarization completeness, and demonstrates that the action of cutting rules on longitudinal modes is trivial (cuts vanish) while transverse parts retain nontrivial discontinuities.

This approach generalizes previous results for scalar fields to spinning sectors, including scalar QED, Yang-Mills, and gravity. Loop-level unitarity (Cutkosky rules) in flat-space translates into cosmological cutting relations for loop correlators via dressing, with integrand-level correspondence maintained [Melville:2021lst].

Cosmological Tree Theorem via Dressed Feynman Tree Theorem

The cosmological tree theorem (CTT), which relates loop-level cosmological correlators to sums over tree-level diagrams (i.e., cuts of internal lines), is shown to arise from a dressed version of the Feynman Tree Theorem (FTT) in flat-space. The decomposition of Feynman propagators into retarded and on-shell (delta-function) contributions (cf. Eq. (2) of the paper),

$\Pi_F(p, s) = \Pi_R + \frac{\pi}{s}\delta(p + s)$

when dressed, leads to the correct analytic structure for cosmological loop correlators. Both explicit diagrammatic calculations and analytic kernel manipulations verify this uplift, including intricate cancellation of terms necessary for the proper (EA)dS retarded structure. The procedure is validated for $\phi^4$ and $\phi^3$ theory at both tree and loop orders, and the resulting wavefunction coefficients precisely match those computed directly in (EA)dS using the CTT [AguiSalcedo:2023nds].

Recursion Relations: BCFW Uplift in (EA)dS

On-shell recursion, particularly BCFW recursion, is a cornerstone in modern amplitude theory. The paper demonstrates that (EA)dS recursion relations for boundary correlators can be systematically obtained by dressing the flat-space BCFW recursion:

$$\mathcal{A}_n = \sum_{\text{factorizations}} \frac{\mathcal{A}_L \mathcal{A}_R}{\text{Propagator}}$$

The dressing protocol incorporates auxiliary propagators for each threshold channel, yielding an energy-dependent kernel that replaces the propagator denominators. The analytic structure of BCFW (complex momentum shifts, factorization channels, on-shell residues) is preserved under dressing, with only minor modifications due to the lack of global energy conservation in (EA)dS.

For Yang-Mills and gravity, the recursion relations yield integrals over dressed three-point correlators in the appropriate channel. The universal forms for energy denominators and pole behavior are shown to persist, indicating a robust uplift from S-matrix to cosmological correlators [Raju:2010by].

Soft Theorems: Dressing and Universal Limits

Soft theorems articulate the behavior of amplitudes as one external momentum vanishes ($k_n \to 0$), revealing universal factors determined by gauge symmetry or diffeomorphism invariance. The paper details the uplift of flat-space (leading and subleading) soft theorems to (EA)dS correlators via dressing:

• Leading Soft Limit: The dressed correlator factorizes onto a lower-point correlator, with the energy derivative (w.r.t. the hard leg) replacing the conventional soft pole. The structure is verified for scalar QED, Yang-Mills, and gravity correlators [Chowdhury:2024wwe].
• Subleading Soft Limit: Although (EA)dS lacks a simple universal form for the subleading soft factor, the paper shows that, after appropriate dressing, a quasi-universal subleading structure emerges. The leading dressing of the subleading amplitude, subleading dressing of the leading amplitude, and off-shell longitudinal contributions are explicitly classified, demonstrating a nuanced but systematic mapping from flat-space soft physics to cosmological correlators.

Implications and Speculative Directions

The technical results have several implications:

• Unified Bootstrap and Analyticity: The dressing paradigm provides a systematic bridge between flat-space bootstrap techniques and cosmological correlators, facilitating the imposition of unitarity, analyticity, and locality constraints in (EA)dS.
• Positivity Constraints: The authors suggest exploring whether positivity bounds (widely studied in flat-space) can be imported and adapted to cosmology via the dressing prescription.
• Nonperturbative Extensions: There is potential for extending the dressing framework beyond perturbation theory, possibly via resummation or functional integral techniques, to obtain nonperturbative cosmological observables.
• Holography and Celestial Amplitude Correspondence: The dressing structure might expose connections between flat-space holography (celestial amplitudes, Carrollian symmetry) and cosmological boundary correlators.

Conclusion

This paper establishes a technically rigorous and comprehensive dictionary between flat-space amplitude theory and cosmological correlators in (EA)dS via dressing. The protocol preserves unitarity, analytic properties, recursive structure, and soft behavior, enabling generalization to spinning fields and loop orders. While certain aspects (subleading soft universality, longitudinal mode dynamics) require nuanced handling, the overall framework unifies disparate analytic features of cosmological observables under the umbrella of dressed flat-space physics. This approach creates avenues for future exploration of analytic, bootstrap, and holographic constraints in quantum cosmology (2606.05289).