Cosmological Dressing Rules

• Cosmological dressing rules are diagrammatic and geometric prescriptions that lift flat-space amplitudes to gauge-invariant in-in correlators in curved spacetimes.
• They employ auxiliary vertex dressings and a global energy conservation constraint to systematically account for time translation symmetry breaking and gravitational redundancies.
• This framework unifies flat-space QFT techniques with gravitational observables, addressing issues like infrared divergences, UV regularization, and higher-form symmetry effects.

Cosmological dressing rules constitute a set of diagrammatic and geometric prescriptions that uplift flat-space amplitudes or unitarity-based techniques to cosmological observables, such as in-in correlators in de Sitter (dS) or inflationary backgrounds. They systematically encode the breaking of time translation symmetry and the diffeomorphism/gauge redundancy intrinsic to gravitational and cosmological settings. These rules enable the computation of physical, gauge-invariant observables in a manifestly covariant way, bridging the gap between traditional flat-space S-matrix amplitudes and cosmological correlation functions, including in scenarios involving infrared divergences, UV regularization, and higher-form symmetries (Chowdhury et al., 13 Mar 2025, Chowdhury et al., 3 Feb 2026).

1. Physical Motivation and Problem Setting

In flat Minkowski space, scattering amplitudes are constructed from Feynman diagrams subject to strict local energy-momentum conservation at every interaction vertex. In cosmological settings, such as dS or expanding universes, time translation symmetry is broken and so energy is not conserved at individual vertices. Moreover, the observables of interest are typically in-in correlators as opposed to in-out amplitudes. Standard calculations in the Schwinger–Keldysh or wavefunctional formalisms lead to complicated nested time integrals, but final results often display unexpected simplicity and structural similarity to flat-space amplitudes. Cosmological dressing rules make this hidden structure explicit by showing how to systematically lift flat-space diagrams to the corresponding cosmological observable via vertex-associated dressings or coordinate/geometric redefinitions (Chowdhury et al., 13 Mar 2025, Chowdhury et al., 3 Feb 2026).

An analogous structure appears in the context of defining gauge-invariant observables in general relativity and AdS/dS gauge theories, where physical observables must be constructed from diffeomorphism or gauge-variant fields by means of dressing transformations (François et al., 21 Oct 2025, Ankur et al., 7 Jan 2026).

2. Diagrammatic Dressing Rules for Cosmological Correlators

The principal prescription for cosmological dressing consists of the following:

1. Flat-space starting point: Begin with the conventional flat-space Feynman diagrams of a given theory, but do not impose energy conservation at vertices.
2. Auxiliary vertex dressing: For each interaction vertex $v$ (carrying a set of external spatial momenta $k_{\rm ext}^{(v)}$ and an auxiliary energy variable $p_v$), attach a model-specific auxiliary propagator $G_{\rm aux}^{(v)}(k_{\rm ext}^{(v)},p_v)$ that encodes the failure of local energy conservation and the effect of the curved spacetime background. These auxiliary lines can be classified by type (e.g., “dashed,” “dotted”), with explicit formulas provided for various theories such as $\phi^4$ or $\phi^3$ (Chowdhury et al., 13 Mar 2025).
3. Global energy-conservation node: Enforce a single auxiliary energy conservation at the diagrammatic level via a constraint $\sum_v p_v = 0$.
4. Integration: Integrate over the flat-space loop momenta and over the auxiliary energies $\{p_v\}$ at each vertex.
5. Model-dependent kernels: The explicit dressing kernels are fixed by the polynomial structure of the interaction and the field content, appearing as simple rational or integral expressions (see Table for auxiliary propagators in (Chowdhury et al., 13 Mar 2025)).

This prescription holds to all orders in perturbation theory for scalar theories (including those with IR divergences) and applies at tree level and in multi-loop contexts. The summation over shadow diagrams and auxiliary lines collapses to a single dressed diagram at each loop order due to trigonometric identities in the shadow formalism (Chowdhury et al., 13 Mar 2025, Chowdhury et al., 3 Feb 2026).

Theory	Propagator type	Auxiliary dressing $G_{\rm aux}(k,p)$
cc $\phi^4$	dashed	$-2k/(p^2+k^2)$
cc $\phi^3$	dashed	$-2i\int_0^\infty ds\, p/(p^2+(s+k)^2)$
massless $\phi^4$	dashed	$$-2\cos(\frac{\pi\varepsilon}{2})\int_0^\infty ds\, \mathcal{N}_4(s;\{k,p\},\varepsilon)/(p^2+(s+k)^2)$$
massless $\phi^4$	dotted	$$+2i\sin(\frac{\pi\varepsilon}{2})\int_0^\infty ds\, \widetilde{\mathcal{N}}_4(s;\{k,p\},\varepsilon)/(p^2+(s+k)^2)$$
massless $\phi^3$	dashed/dotted	similar expressions with $\mathcal{N}_3,\widetilde{\mathcal{N}}_3$

(Here cc denotes conformally coupled; $\varepsilon$ is the IR regularization parameter.)

3. Analytic, Geometric, and Relational Formulations

The diagrammatic rules correspond to a broad principle: cosmological correlators can be represented as flat-space amplitudes “dressed” by one-dimensional propagators per vertex (or, equivalently, by bulk-to-boundary/bulk-to-bulk kernels), enforcing global rather than local energy conservation. This can be interpreted geometrically or relationally in the context of diffeomorphism-invariant theories.

In the gravitational context, the dressing field method (DFM) provides a construction in which the full set of dynamical (including reference) fields is used to define a “physical” coordinate system. Here, a matter sector (e.g., a dust fluid, represented by four scalar fields $\varphi^a$) is used to extract a field-dependent map $\upsilon[\varphi]:\mathbb{R}^4\to M$ whose transformation law neutralizes the action of the Diff$(M)$ gauge symmetry. This map serves as the dressing field; all tensorial fields $X$ are pulled back to new coordinates $x^a = \varphi^a(x^\mu)$, ensuring that the resulting observables $X^\upsilon$ are diffeomorphism invariant (François et al., 21 Oct 2025). The perturbative expansion in the deformation $\chi^a$ yields gauge-invariant corrections to observables such as galaxy rotation curves.

4. Unitarity, Cutting Rules, and Dispersion Relations

Cosmological dressing rules enable powerful mapping from flat-space unitarity, cutting, and discontinuity machinery to cosmological correlators in (E)dS. In this formalism:

• Flat-space unitarity cuts (Cutkosky rules, expressed via delta-functions or imaginary parts) become discontinuity ($\mathrm{Disc}$) operations in the energy variables of the cosmological correlator after dressing.
• Auxiliary kernel propagators are attached to each vertex. For a generic $\phi^4$ theory, the dressed $n$-point correlator takes the schematic form:

$$C_n(\{x\},\{y\}) = \sum_{\text{flat graphs }A_n} \prod_{v=1}^V \int dp_v \frac{1}{p_v^2 - x_v^2 + i0}\, \delta\Bigl(\sum_v p_v\Bigr) A_n(\{p_v\};\{y\})\, x_1\cdots x_V$$

where $x_v$ are external energy variables and $y_j$ are internal.

• Discontinuities with respect to internal energies $y_j^2$ correspond to unitarity cuts of flat-space amplitudes, then dressed by the auxiliary kernels (Chowdhury et al., 3 Feb 2026). Discontinuities with respect to external energies $x_v^2$ correspond to cutting auxiliary lines, yielding sum rules constraining the full correlator.
• Sequential application of these cutting rules constrains the full cosmological correlator such that, when all auxiliaries are set on shell, the correlator reduces to the flat-space amplitude with global energy conservation imposed.

This framework allows for the use of powerful analytic and algebraic techniques (dispersion relations, cutting rules, double discontinuities) for cosmological correlators—techniques previously considered restricted to S-matrix or boundary CFT settings (Ansari et al., 13 Jan 2026, Chowdhury et al., 3 Feb 2026).

5. Algorithmic Summary and Practical Calculation

A concise algorithmic statement of the cosmological dressing rules (see (François et al., 21 Oct 2025, Chowdhury et al., 13 Mar 2025, Chowdhury et al., 3 Feb 2026)):

1. Identify the gauge symmetry and field content (e.g., Diff$(M)$ and $g_{\mu\nu}$ plus matter).
2. Extract a dressing field from the matter sector, yielding a field-dependent, invertible map that neutralizes the gauge symmetry.
3. Construct dressed fields by pull-back, ensuring gauge/diffeomorphism invariance.
4. Express all quantities in the dressed (physical) coordinates.
5. Evaluate physical observables, such as correlators or rotation curves, in these coordinates.
6. In diagrammatic rules: Dress each flat-space amplitude by appropriate vertex auxiliary propagators, integrate over auxiliary energies, and impose a single conservation law.

When calculating cosmological correlators, one draws all contributing flat-space diagrams, attaches auxiliary lines at every vertex, integrates over auxiliary energies with global conservation, and contracts with the appropriate external kinematic factors.

6. Geometric and Higher-Form Symmetry Interpretations

In AdS/dS gauge theories and for gravitational systems, dressing rules are not merely diagrammatic recipes but embody a geometric principle for constructing physical, gauge-invariant observables. For example, in AdS or its dS continuation:

• Local charged operators are not gauge invariant and must be dressed using geodesic Wilson lines extending from the bulk insertion to the boundary (Ankur et al., 7 Jan 2026).
• The presence or absence of boundary tilt operators (from higher-form symmetry breaking) determines whether a bulk line can end at the boundary—encoding the relation between bulk gauge group breaking and boundary observable structure.
• Dressed operators transform covariantly under the AdS/dS conformal group, with the Wilson line ensuring covariance and decoupling of unphysical modes in the presence of gauge redundancy.
• One-loop computations confirm that dressing decouples longitudinal (unphysical) photon modes, with explicit cancellation verified by Ward identities.

This geometric viewpoint is closely related to the relational formulation provided by the DFM in GR contexts, where observables are rendered gauge invariant by coordinatization with respect to the matter sector (François et al., 21 Oct 2025).

7. Phenomenological Consequences and Assumptions

The cosmological dressing framework recovers observed phenomenology—such as rotation curves of spiral galaxies—by yielding corrections to observables entirely dictated by the chosen dressing profiles. For instance, the DFM applied to Schwarzschild backgrounds with a quadratic profile $\chi^r(r) \propto r^2$ yields a velocity profile combining a Keplerian term and a constant offset, matching the empirical flatness of galaxy rotation curves and mimicking dark matter effects (François et al., 21 Oct 2025).

There are explicit technical limitations:

• The analysis is perturbative in the dressing parameter $\chi^a$ and neglects higher-order backreaction.
• Phenomenology is sensitive to the ansatz for the dressing profile.
• Validity is restricted to regimes where the perturbative approximation is justified (weak field, large radius).
• For cosmological correlators, additional assumptions may include the mass/interaction structure and boundary conditions (Dirichlet/Neumann vs. mixed).
• The empirical connection to other dark-matter-sensitive probes (e.g., Tully–Fisher, gravitational lensing, CMB) is not yet fully formulated within the dressing framework (François et al., 21 Oct 2025).

8. Outlook and Implications

Cosmological dressing rules unify previously distinct perspectives—relational observables in general relativity, analytic continuation/amplitude bootstrapping in field theory, and gauge-invariant operator construction in AdS/dS gauge theory. They form a technical foundation for the systematic computation and analytic understanding of cosmological correlation functions, their symmetries, sum rules, and their connections to flat-space S-matrix theory (Chowdhury et al., 13 Mar 2025, Chowdhury et al., 3 Feb 2026, Ankur et al., 7 Jan 2026).

A plausible implication is that these rules may streamline perturbative and non-perturbative computations in quantum gravity and cosmology, including the generalization to spinning fields, gravitational waves, slow-roll inflationary backgrounds, and higher-form symmetry observables. They also provide a direct pathway for importing unitarity, dispersion, and double-copy machinery from flat-space QFT into cosmological settings (Ansari et al., 13 Jan 2026, Chowdhury et al., 3 Feb 2026).