Inflationary Correlators and Cosmological Perturbation Theory

Inflationary correlation functions are predictions only after the dynamical clock, gravitational constraints, canonical variables, initial density matrix, time contour, effective-field-theory hierarchy, and late-time observable have been specified. This chapter builds that chain from the background solution to power spectra, bispectra, loops, resummation, and reduced quantum states. It also marks where familiar shortcuts—horizon-crossing evaluation, decoupling, a squeezed-limit identity, or a claim of classicality—cease to be controlled.

Helpful background. Closed-time-path generating functionals supplies normalized real-time evolution; effective field theory as a controlled expansion supplies power counting; vacuum choice and initial-state effects supplies state diagnostics; and momentum-space conformal Ward identities supplies the symmetry language used in soft limits.

From a clock background to a correlator

For a canonical single field, a useful reference theory in the site’s conventions is

$$S=\int d^4x\sqrt{-g}\left[-\frac{M_{\rm Pl}^2}{2}R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi-V(\phi)\right].$$

On a spatially flat FLRW solution, $ds^2=dt^2-a^2(t)d\mathbf x^2$, define

$$H=\frac{\dot a}{a},\qquad \epsilon=-\frac{\dot H}{H^2},\qquad \dot H=-\frac{\dot\phi_0^2}{2M_{\rm Pl}^2}.$$

The evolving scalar background selects a clock and spontaneously breaks time reparameterizations while preserving time-dependent spatial diffeomorphisms. Lapse and shift remain constraints. Solving them produces one scalar adiabatic mode and two tensor helicities in the minimal model; additional fields, higher derivatives, or modified gravity change that inventory.

Our scalar sign is fixed by writing the spatial metric in comoving gauge as

$$h_{ij}=a^2e^{2\zeta}(e^\gamma)_{ij}, \qquad \partial_i\gamma_{ij}=0, \qquad \gamma_{ii}=0.$$

Thus the curvature perturbation used throughout this chapter is $\zeta$; when $\mathcal R$ appears, it means $\mathcal R=\zeta$. In the Goldstone description below, the Stückelberg field is defined by restoring the clock as $t\mapsto t+\pi$, so $\zeta=-H\pi$ at linear order. These declarations prevent the two common sign translations from being hidden in otherwise identical formulas.

The structure map shows that a correlator is assembled through a sequence of reductions, each with a separate check.

Background evolution, constraint solving, state preparation, in-in evolution, EFT power counting, and observable extraction are successive inputs to an inflationary correlator. Schematic; not to scale.

Routes through the chapter

1. Background symmetry breaking and decoupling identifies the clock, constraints, mixing scale, and cutoff.
2. Gauge-invariant perturbations constructs scalar, vector, and tensor representatives.
3. Mukhanov–Sasaki modes reduces the scalar sector to its canonical oscillator.
4. Tensor modes fixes helicity and normalization conventions.
5. Constraint solving and the observable dictionary follows a calculation from ADM variables to a late-time quantity.
6. Power spectra and freeze-out distinguishes a crossing estimate from conserved evolution.
7. In-in correlators derives normalized expectation values on a closed time path.
8. Contours and initial boundaries treats $i\epsilon$, finite initial time, and boundary vertices.
9. The EFT of inflation organizes Goldstone interactions and strong-coupling scales.
10. Cubic interactions and bispectra relates operators to shapes and boundary terms.
11. Soft limits states the hypotheses behind consistency relations.
12. Cosmological loops separates ultraviolet renormalization from infrared and secular effects.
13. Secular resummation identifies observable-specific large-log expansions.
14. Decoherence distinguishes squeezing, reduced-state diagonality, and measurement claims.

Domain and failure conditions

This table is the chapter’s canonical comparison of controlled claims and their failure tests.

Problem	Required data	Controlled output	Expansion parameter or check	Failure or handoff
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

The validity map collects the most consequential ways this chain can fail. Inspect where the proposed approximation is compared with the physical scale at which the correlator is evaluated.

An inflationary correlator is controlled only when constraints, gauge completion, initial data, contour normalization, EFT hierarchy, renormalization, and observable scope all pass their own checks. Schematic; not to scale.

References

• Cheung, C., P. Creminelli, A. L. Fitzpatrick, J. Kaplan, and L. Senatore, “The Effective Field Theory of Inflation,” Journal of High Energy Physics 03, 014 (2008), doi:10.1088/1126-6708/2008/03/014.
• 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.
• Mukhanov, V. F., H. A. Feldman, and R. H. Brandenberger, “Theory of Cosmological Perturbations,” Physics Reports 215, 203–333 (1992), doi:10.1016/0370-1573(92)90044-Z.
• Weinberg, S., “Quantum Contributions to Cosmological Correlations,” Physical Review D 72, 043514 (2005), doi:10.1103/PhysRevD.72.043514.