Inflationary Correlators and Cosmological Perturbation Theory

Inflationary correlators relate a dynamical background to primordial fluctuations. A reliable calculation connects gravitational constraints and canonical variables to the initial state, interaction vertices, and a specified late-time observable.

Enter this chapter

Begin with Inflationary Backgrounds, Broken Time Diffeomorphisms, and Decoupling to identify the clock background and the conditions for decoupling gravitational constraints.

Helpful background. Closed-Time-Path Generating Functionals in Practice and Effective Field Theory as a Controlled Expansion develop the definitions used by the chapter’s opening pages.

Topics

1. Inflationary Backgrounds, Broken Time Diffeomorphisms, and Decoupling — Identifies the clock, constraints, mixing scale, and cutoff.
2. Inflationary Perturbations and Gauge-Invariant Variables — Constructs scalar, vector, and tensor representatives.
3. Mukhanov–Sasaki Scalar Modes — Reduces the scalar sector to its canonical oscillator.
4. Tensor Modes and Primordial Gravitons — Fixes helicity and normalization conventions.
5. Constraint Solving and Gauge-Invariant Observables — Follows a calculation from ADM variables to a late-time quantity.
6. Power Spectra, Horizon Crossing, and Freeze-Out — Distinguishes a crossing estimate from conserved evolution.
7. In-In Cosmological Correlators — Derives normalized expectation values on a closed time path.
8. Contours, iε, and Initial-Boundary Terms — Treats $i\epsilon$, finite initial time, and boundary vertices.
9. Effective Field Theory of Inflation and the Goldstone Mode — Organizes Goldstone interactions and strong-coupling scales.
10. Cubic Interactions and Bispectrum Shapes — Relates operators to shapes and boundary terms.
11. Soft Limits and Inflationary Consistency Relations — States the hypotheses behind consistency relations.
12. Cosmological Loops and Renormalization — Separates ultraviolet renormalization from infrared and secular effects.
13. Secular Growth, Resummation, and Dynamical RG — Identifies observable-specific large-log expansions.
14. Decoherence and the Quantum-to-Classical Claim — Distinguishes squeezing, reduced-state diagonality, and measurement claims.

How the ideas fit together

Solving the constraints removes nondynamical variables before the remaining scalar and tensor modes are quantized. Their interactions generate higher-point correlations on an in/in contour, where the initial-state prescription is part of the answer Maldacena 2003, §§ 2–3.

A squeezed-limit relation additionally assumes an appropriate adiabatic long mode and background evolution; it is not a universal identity for every inflationary model. At loops, ultraviolet subtraction and late-time secular growth require different controls. Squeezing of a mode likewise does not, by itself, establish decoherence into classical statistical alternatives.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

Problem	Required data	Controlled output	Expansion parameter or check	Failure or further method needed
Background and decoupling	Clock solution, constraint equations, EFT coefficients	Goldstone–gravity hierarchy	Compare mixing energy, freeze-out scale, and cutoff	Solve the coupled constrained system when scales overlap
Gauge-invariant variables	Transformation convention and boundary conditions	Gauge-invariant scalar and tensor representatives	Recompute in a second complete gauge	Residual or large transformations require relational data
Scalar modes	$z(\eta)$, sound speed, state	Canonically normalized $v=z\zeta$	Wronskian and long-mode solution	Entropy sources or rapid variation require a larger system
Tensor modes	Effective Planck mass, tensor speed, polarization norm	Two-helicity spectrum	Canonical commutator and both-helicity count	Higher-curvature modes require EFT or new degrees of freedom
Constraint reduction	Lapse, shift, boundary terms, target observable	Reduced action and observable map	Agreement between two gauges	A gauge-coordinate correlator is not automatically observable
Power spectra	State and background evolution through freeze-out	Late-time scalar or tensor spectrum	Compare exact transport with crossing estimate	Nonattractor evolution invalidates freeze-out shorthand
In-in evolution	Density matrix, contour, interaction Hamiltonian	Normalized real-time expectation value	$Z[J,J]=1$, reality, largest-time check	In-out amplitudes do not replace expectation values
Initial boundary	Initial surface, $i\epsilon$, boundary action	State-dependent correlator	Move the surface while running boundary coefficients	Unmatched surface dependence signals an incomplete state EFT
Goldstone EFT	Unitary-gauge operators and cutoff hierarchy	Ordered $\pi$ interactions	Energy and coefficient power counting	Strong coupling before freeze-out ends predictivity
Bispectrum	Cubic action, field map, state	Shape and normalization	Integration-by-parts and field-redefinition invariance	A shape label alone does not identify a model
Soft limit	Single clock, attractor, invariant state, locality	Ward identity for a soft leg	Direct squeezed calculation	Excited states, entropy modes, or nonattractors modify it
Loop correction	Regulator, counterterms, composite observable, state	Renormalized correlator	Independent UV and IR variations	UV poles and infrared logs must not be conflated
Secular resummation	Observable and identified large parameter	Method-specific late-time approximation	Re-expand to fixed order	Resummation of one object need not control another
Decoherence	System split, environment, window, coupling	Reduced density matrix or influence functional	Split variation after matching and positivity	Diagonality neither selects an outcome nor proves classical gravity

Where to go next

Continue to de Sitter Infrared Physics and Stochastic Inflation to identify the field, state, observable, and infrared limits of the calculation.

References

• Maldacena, J., “Non-Gaussian Features of Primordial Fluctuations in Single Field Inflationary Models,” Journal of High Energy Physics 05, 013 (2003), doi:10.1088/1126-6708/2003/05/013.