Physical Consistency Condition in Inflation

• Physical consistency condition is a squeezed-limit relation that connects the three-point function of curvature perturbations with the scale-dependent two-point function in inflationary models.
• It has been generalized to dissipative single-clock inflation, where additional degrees of freedom and controlled slow-roll dynamics preserve the standard single-field result.
• The condition originates from a soft theorem, showing that long-wavelength adiabatic modes can be reabsorbed as coordinate transformations, ensuring consistent squeezed-limit behavior.

The physical consistency condition, in inflationary cosmology, denotes the squeezed-limit consistency relation that ties the three-point function of the curvature perturbation to the scale dependence of its two-point function when the long-wavelength perturbation is an adiabatic mode equivalent, at leading order in gradients, to a coordinate transformation. In "The consistency condition for the three-point function in dissipative single-clock inflation," Lopez Nacir, Porto, and Zaldarriaga generalized this statement from standard single-field inflation to dissipative, multi-field, single-clock models and showed that, under a preferred-clock assumption and controlled dissipative dynamics, the standard single-clock relation survives unchanged at leading non-trivial order (Nacir et al., 2012).

1. Definition, observables, and squeezed kinematics

The central variable is the curvature perturbation $\zeta$, defined on uniform density slices. In the EFT of Inflation it is related to the Goldstone boson $\pi$ that nonlinearly realizes time diffeomorphisms. To leading order, $\zeta \simeq -H\pi$, while the second-order relation is

$$\zeta = -H \pi + H \pi \dot{\pi} + \frac{1}{2}\dot{H}\pi^2 + \alpha,$$

with

$$\alpha = \frac{1}{a^2}\left[-\partial_i\pi \partial_i\pi + \partial^{-2}\partial_i\partial_j(\partial_i\pi \partial_j\pi)\right].$$

The two-point function defines the power spectrum,

$$\langle \zeta_{\mathbf{k}} \zeta_{\mathbf{k}'} \rangle = (2\pi)^3 \delta^3(\mathbf{k}+\mathbf{k}') P_\zeta(k),$$

and the dimensionless spectrum is $\Delta_\zeta^2(k) \equiv k^3 P_\zeta(k)$. The bispectrum is defined by

$$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\rangle = (2\pi)^3 \delta^3(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3) B_\zeta(k_1,k_2,k_3).$$

The scalar tilt is

$$n_s-1 \equiv \frac{d\ln(k^3 P_\zeta)}{d\ln k} = \frac{d\ln \Delta_\zeta^2}{d\ln k}.$$

The relevant limit is the squeezed configuration, in which one momentum is soft, $q \equiv k_L \to 0$, while the other two are hard and nearly back-to-back, $\pi$0 with $\pi$1. In this regime one studies $\pi$2 and asks whether the long mode changes the local short-scale physics or merely rescales coordinates.

2. Generalized relation in dissipative single-clock inflation

The generalized consistency relation proved in the dissipative single-clock EFT is

$\pi$3

Equivalently, in the exact squeezed limit,

$\pi$4

This is the same form as in standard single-clock attractor inflation, despite the presence of additional dissipative degrees of freedom (Nacir et al., 2012).

The nontrivial point is that dissipative response and stochastic noise introduce new contributions that could have modified the squeezed limit. In the decoupling limit, the squeezed bispectrum takes the form

$\pi$5

After including mixing with gravity and the nonlinear $\pi$6--$\pi$7 relation, an extra $\pi$8 contribution appears, and all terms recombine into the tilt,

$\pi$9

so that the final squeezed result is again

$\zeta \simeq -H\pi$0

The generalized slow-roll parameters are

$\zeta \simeq -H\pi$1

$\zeta \simeq -H\pi$2

The function $\zeta \simeq -H\pi$3 is known explicitly, and for $\zeta \simeq -H\pi$4 it obeys

$\zeta \simeq -H\pi$5

A central structural result is that the first correction to the squeezed expansion scales as $\zeta \simeq -H\pi$6. No $\zeta \simeq -H\pi$7 term survives once the constraints and symmetries are implemented.

3. Symmetry origin and persistence of the adiabatic mode

The physical origin of the consistency condition is a soft theorem. In single-clock EFT, broken time diffeomorphisms are nonlinearly realized by $\zeta \simeq -H\pi$8, and a long $\zeta \simeq -H\pi$9 mode acts as a background dilation or time reparameterization. At zeroth order in gradients,

$$\zeta = -H \pi + H \pi \dot{\pi} + \frac{1}{2}\dot{H}\pi^2 + \alpha,$$0

The long mode can therefore be reabsorbed by a coordinate transformation, and the leading squeezed limit is fixed by a Ward identity relating the soft bispectrum to the tilt of the short-scale power spectrum (Nacir et al., 2012).

The dissipative sector does not invalidate this argument provided the long mode remains adiabatic outside the horizon. In the models analyzed, dissipative couplings generate friction terms of the form $$\zeta = -H \pi + H \pi \dot{\pi} + \frac{1}{2}\dot{H}\pi^2 + \alpha,$$1 and stochastic sources, but the constraint equations imply on superhorizon scales

$$\zeta = -H \pi + H \pi \dot{\pi} + \frac{1}{2}\dot{H}\pi^2 + \alpha,$$2

As a result, the dissipative response obeys

$$\zeta = -H \pi + H \pi \dot{\pi} + \frac{1}{2}\dot{H}\pi^2 + \alpha,$$3

so any dissipative backreaction vanishes when $$\zeta = -H \pi + H \pi \dot{\pi} + \frac{1}{2}\dot{H}\pi^2 + \alpha,$$4 is constant. The conserved long mode remains a pure adiabatic perturbation, and this is the key reason the standard squeezed-limit relation survives dissipation.

This mechanism clarifies a frequent misconception. The relevant requirement is not the literal absence of extra fields, but the existence of a preferred clock. A broad class of multi-field models with additional degrees of freedom can still satisfy the standard squeezed-limit theorem if the long mode is adiabatic and no long-lived isocurvature perturbation sources $$\zeta = -H \pi + H \pi \dot{\pi} + \frac{1}{2}\dot{H}\pi^2 + \alpha,$$5 on superhorizon scales.

4. EFT structure, dissipation, and stochastic forcing

The dissipative extension of the EFT of Inflation is formulated in unitary gauge with action

$$\zeta = -H \pi + H \pi \dot{\pi} + \frac{1}{2}\dot{H}\pi^2 + \alpha,$$6

Here $$\zeta = -H \pi + H \pi \dot{\pi} + \frac{1}{2}\dot{H}\pi^2 + \alpha,$$7 is a composite operator of the additional degrees of freedom, and $$\zeta = -H \pi + H \pi \dot{\pi} + \frac{1}{2}\dot{H}\pi^2 + \alpha,$$8 is their own action. The EFT parameters define

$$\zeta = -H \pi + H \pi \dot{\pi} + \frac{1}{2}\dot{H}\pi^2 + \alpha,$$9

After the Stückelberg restoration $$\alpha = \frac{1}{a^2}\left[-\partial_i\pi \partial_i\pi + \partial^{-2}\partial_i\partial_j(\partial_i\pi \partial_j\pi)\right].$$0, the dissipative response is expanded locally in derivatives:

$$\alpha = \frac{1}{a^2}\left[-\partial_i\pi \partial_i\pi + \partial^{-2}\partial_i\partial_j(\partial_i\pi \partial_j\pi)\right].$$1

with $$\alpha = \frac{1}{a^2}\left[-\partial_i\pi \partial_i\pi + \partial^{-2}\partial_i\partial_j(\partial_i\pi \partial_j\pi)\right].$$2. At linear order, the $$\alpha = \frac{1}{a^2}\left[-\partial_i\pi \partial_i\pi + \partial^{-2}\partial_i\partial_j(\partial_i\pi \partial_j\pi)\right].$$3 equation of motion becomes

$$\alpha = \frac{1}{a^2}\left[-\partial_i\pi \partial_i\pi + \partial^{-2}\partial_i\partial_j(\partial_i\pi \partial_j\pi)\right].$$4

The noise is taken to be local in time and Gaussian to leading order,

$$\alpha = \frac{1}{a^2}\left[-\partial_i\pi \partial_i\pi + \partial^{-2}\partial_i\partial_j(\partial_i\pi \partial_j\pi)\right].$$5

Nonlinear consistency requires the stochastic kernel to transform as a bi-scalar under the $$\alpha = \frac{1}{a^2}\left[-\partial_i\pi \partial_i\pi + \partial^{-2}\partial_i\partial_j(\partial_i\pi \partial_j\pi)\right].$$6-dependent time slicing. This is implemented by replacing $$\alpha = \frac{1}{a^2}\left[-\partial_i\pi \partial_i\pi + \partial^{-2}\partial_i\partial_j(\partial_i\pi \partial_j\pi)\right].$$7, giving

$$\alpha = \frac{1}{a^2}\left[-\partial_i\pi \partial_i\pi + \partial^{-2}\partial_i\partial_j(\partial_i\pi \partial_j\pi)\right].$$8

and, to first order,

$$\alpha = \frac{1}{a^2}\left[-\partial_i\pi \partial_i\pi + \partial^{-2}\partial_i\partial_j(\partial_i\pi \partial_j\pi)\right].$$9

This produces the additional squeezed-limit contribution proportional to $$\langle \zeta_{\mathbf{k}} \zeta_{\mathbf{k}'} \rangle = (2\pi)^3 \delta^3(\mathbf{k}+\mathbf{k}') P_\zeta(k),$$0 that is needed for the full recombination into $$\langle \zeta_{\mathbf{k}} \zeta_{\mathbf{k}'} \rangle = (2\pi)^3 \delta^3(\mathbf{k}+\mathbf{k}') P_\zeta(k),$$1.

In the strongly dissipative regime $$\langle \zeta_{\mathbf{k}} \zeta_{\mathbf{k}'} \rangle = (2\pi)^3 \delta^3(\mathbf{k}+\mathbf{k}') P_\zeta(k),$$2, with freezeout defined by $$\langle \zeta_{\mathbf{k}} \zeta_{\mathbf{k}'} \rangle = (2\pi)^3 \delta^3(\mathbf{k}+\mathbf{k}') P_\zeta(k),$$3, the power spectrum satisfies

$$\langle \zeta_{\mathbf{k}} \zeta_{\mathbf{k}'} \rangle = (2\pi)^3 \delta^3(\mathbf{k}+\mathbf{k}') P_\zeta(k),$$4

5. Conditions of validity and mechanisms of violation

The validity of the physical consistency condition rests on a specific set of structural assumptions (Nacir et al., 2012). There must be a single clock: a preferred time variable whose fluctuations nonlinearly realize time reparameterizations, with the long-wavelength adiabatic mode conserved outside the horizon up to slow-roll corrections. Long-lived isocurvature modes must be absent, or at least must not source $$\langle \zeta_{\mathbf{k}} \zeta_{\mathbf{k}'} \rangle = (2\pi)^3 \delta^3(\mathbf{k}+\mathbf{k}') P_\zeta(k),$$5 on superhorizon scales.

The dissipative sector must admit a local derivative expansion and approximately local-in-time retarded response. The composite operators of the additional degrees of freedom must have response kernels that are analytic in soft frequency and momentum. An emergent shift symmetry in $$\langle \zeta_{\mathbf{k}} \zeta_{\mathbf{k}'} \rangle = (2\pi)^3 \delta^3(\mathbf{k}+\mathbf{k}') P_\zeta(k),$$6 is also required, so that the response depends on derivatives of $$\langle \zeta_{\mathbf{k}} \zeta_{\mathbf{k}'} \rangle = (2\pi)^3 \delta^3(\mathbf{k}+\mathbf{k}') P_\zeta(k),$$7 rather than on a constant $$\langle \zeta_{\mathbf{k}} \zeta_{\mathbf{k}'} \rangle = (2\pi)^3 \delta^3(\mathbf{k}+\mathbf{k}') P_\zeta(k),$$8 itself. The leading stochastic noise is assumed Gaussian with a local correlator. The analysis is performed to leading non-trivial order in generalized slow-roll parameters and in a perturbative expansion in mixing with gravity scales; the friction $$\langle \zeta_{\mathbf{k}} \zeta_{\mathbf{k}'} \rangle = (2\pi)^3 \delta^3(\mathbf{k}+\mathbf{k}') P_\zeta(k),$$9 may be large compared to $\Delta_\zeta^2(k) \equiv k^3 P_\zeta(k)$0, but the local approximation must remain valid.

Several mechanisms can violate or modify the relation. If extra light fields source $\Delta_\zeta^2(k) \equiv k^3 P_\zeta(k)$1 outside the horizon, as in curvaton-like or genuinely multifield scenarios with nontrivial turns, the long mode is no longer purely adiabatic. If the background is non-attractor, so that $\Delta_\zeta^2(k) \equiv k^3 P_\zeta(k)$2 evolves on superhorizon scales, the soft theorem can fail. Strongly nonlocal-in-time dissipation can obstruct the separate-universe argument. Non-Bunch--Davies initial states can add independent squeezed contributions. Strong mixing with gravity beyond the controlled expansion, or explicit breaking of the emergent $\Delta_\zeta^2(k) \equiv k^3 P_\zeta(k)$3 shift symmetry, can also generate terms not tied to $\Delta_\zeta^2(k) \equiv k^3 P_\zeta(k)$4.

Operationally, such failures appear as

$\Delta_\zeta^2(k) \equiv k^3 P_\zeta(k)$5

A measured $\Delta_\zeta^2(k) \equiv k^3 P_\zeta(k)$6 significantly different from this value would therefore rule out not only standard single-field inflation but also a large class of dissipative multi-field models that still possess a single effective clock.

6. Related soft theorems, higher-order effects, and observational significance

The generalized dissipative result sits within a broader hierarchy of inflationary soft theorems. "A Note on the Consistency Condition of Primordial Fluctuations" (Senatore et al., 2012) emphasized that in single-clock inflation a long mode can be reabsorbed into the background cosmology, so the squeezed limit is controlled by the scale dependence of the short-mode correlator, and that the first genuine physical correction begins at order $\Delta_\zeta^2(k) \equiv k^3 P_\zeta(k)$7. "The Physical Squeezed Limit: Consistency Relations at Order $\Delta_\zeta^2(k) \equiv k^3 P_\zeta(k)$8" (Creminelli et al., 2013) developed this point further by showing that the order-$\Delta_\zeta^2(k) \equiv k^3 P_\zeta(k)$9 effect of the long mode is equivalent to placing the short modes in a locally curved FRW universe.

A later refinement distinguished the ordinary adiabatic-mode argument from a sharper physical-mode criterion. "Inflationary soft theorems revisited: A generalized consistency relation" (Hui et al., 2018) formulated a generalized early--late-time Ward identity and argued that the nonlinear part of the symmetry transformation must match the time dependence of the dominant, long wavelength physical mode for the standard late-time consistency relation to follow. This clarifies why the standard theorem can fail in non-attractor regimes even when the residual symmetry still exists.

The same logic also illuminates noninflationary backgrounds. "Can non-minimal coupling restore the consistency condition in bouncing universes?" (Nandi et al., 2019) argued that minimally coupled bouncing models often violate the squeezed-limit theorem because scalar or tensor perturbations grow near the bounce rather than freezing on super-Hubble scales, but that suitable non-minimal couplings can restore freeze-out and the associated consistency relation in the tensor sector.

Observationally, dissipative single-clock models predict

$$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\rangle = (2\pi)^3 \delta^3(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3) B_\zeta(k_1,k_2,k_3).$$0

numerically of order a percent for $$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\rangle = (2\pi)^3 \delta^3(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3) B_\zeta(k_1,k_2,k_3).$$1--$$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\rangle = (2\pi)^3 \delta^3(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3) B_\zeta(k_1,k_2,k_3).$$2 (Nacir et al., 2012). The relevant signal is the local squeezed shape, with subleading $$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\rangle = (2\pi)^3 \delta^3(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3) B_\zeta(k_1,k_2,k_3).$$3 corrections. CMB bispectrum measurements, large-scale structure through scale-dependent halo bias, and CMB $$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\rangle = (2\pi)^3 \delta^3(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3) B_\zeta(k_1,k_2,k_3).$$4-distortion are all sensitive to such squeezed couplings. Within the assumptions of the dissipative single-clock EFT, departures from the relation would therefore exclude not only ordinary single-field inflation but also a broad class of multi-field models with dissipative additional degrees of freedom, local retarded response, and a preferred clock.