---
title: Physical Consistency Condition in Inflation
url: https://www.emergentmind.com/topics/physical-consistency-condition
type: topic
---

# Physical Consistency Condition in Inflation

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 [1206.7083].

## 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, $|k_2| \simeq |k_3| \equiv k_S$ with $q \ll k_S$. In this regime one studies $B_\zeta(q,k,-k)$ 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
$$
\lim_{q\to 0} B_\zeta(q,k,-k)
=
-(n_s(k)-1)\,P_\zeta(q)\,P_\zeta(k)
+
O\!\left((q/k)^2\right).
$$
Equivalently, in the exact squeezed limit,
$$
f_{NL}^{sq} = -\frac{5}{12}(n_s-1).
$$
This is the same form as in standard single-clock attractor inflation, despite the presence of additional dissipative degrees of freedom [1206.7083].

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
$$
f_{NL}^{sq} \simeq -\frac{5}{12}\left[(\epsilon_{\nu_O}+2\epsilon_f)-3s-2\epsilon_{N_c}+R(\gamma/H)\epsilon_\gamma\right].
$$
After including mixing with gravity and the nonlinear $\zeta$--$\pi$ relation, an extra $-(5/12)(2\epsilon)$ contribution appears, and all terms recombine into the tilt,
$$
n_s - 1 \simeq \epsilon_{\nu_O} + 2 \epsilon_f - 3 s - 2 \epsilon_{N_c} + R(\gamma/H)(\epsilon + \epsilon_\gamma),
$$
so that the final squeezed result is again
$$
f_{NL}^{sq} = -\frac{5}{12}(n_s-1).
$$

The generalized slow-roll parameters are
$$
\epsilon \equiv -\frac{\dot H}{H^2},\quad
\epsilon_{\nu_O} \equiv \frac{\dot\nu_O}{H\nu_O},\quad
\epsilon_f \equiv \frac{\ddot f}{H\dot f},
$$
$$
\epsilon_\gamma \equiv \frac{\dot\gamma}{2H\gamma},\quad
s \equiv \frac{\dot c_s}{H c_s},\quad
\epsilon_{N_c} \equiv \frac{\dot N_c}{H N_c}.
$$
The function $R(x)$ is known explicitly, and for $\gamma/H \gg 1$ it obeys
$$
R(x) \simeq -\frac{1}{2} + \frac{3}{8}\frac{H}{\gamma} + O\!\left((H/\gamma)^2\right).
$$

A central structural result is that the first correction to the squeezed expansion scales as $(q/k)^2$. No $O(q/k)$ 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 $\pi$, and a long $\zeta_L$ mode acts as a background dilation or time reparameterization. At zeroth order in gradients,
$$
x^i_{\mathrm{phys}} = e^{\zeta_L} x^i,
\qquad
k_{S,\mathrm{phys}} = e^{-\zeta_L} k_S.
$$
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 [1206.7083].

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 $\gamma(\dot\pi-\delta N)$ and stochastic sources, but the constraint equations imply on superhorizon scales
$$
\delta N_L \simeq \epsilon H \pi_L,
\qquad
\dot\pi_L \simeq \epsilon H \pi_L,
\qquad
\Rightarrow\qquad
\dot\zeta_L \simeq 0.
$$
As a result, the dissipative response obeys
$$
\delta R^s \propto (\dot\pi-\delta N) \propto \dot\zeta,
$$
so any dissipative backreaction vanishes when $\zeta$ 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 $\zeta$ 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
$$
S = \int d^4x \sqrt{-g}\left[
\frac{M_{\mathrm{pl}}^2}{2}R
-
M_{\mathrm{pl}}^2\dot H(t)(g^{00}+1)
-
M_2^4(t)(\delta g^{00})^2
-
\frac{1}{2}\bar M_1^3(t)\delta K^\mu_\mu(1+g^{00})
-
f(t) O
+
S_O
+\ldots
\right].
$$
Here $O$ is a composite operator of the additional degrees of freedom, and $S_O$ is their own action. The EFT parameters define
$$
c_s^2 = \frac{\bar p + \bar\rho + H\bar M_1^3}{\bar p + \bar\rho + 4M_2^4},
\qquad
N_c \equiv \bar p + \bar\rho + 4M_2^4.
$$

After the Stückelberg restoration $t\to t+\pi$, the dissipative response is expanded locally in derivatives:
$$
\delta_R^s(\pi) \simeq V_O(t)(\dot\pi-\delta N)+\ldots,
$$
with $V_O \propto \gamma N_c/\dot f$. At linear order, the $\pi$ equation of motion becomes
$$
\ddot\pi_k + (3H+\gamma)\dot\pi_k + \frac{c_s^2 k^2}{a^2}\pi_k
=
-\frac{\dot f}{N_c}\,\delta_S(t,k).
$$
The noise is taken to be local in time and Gaussian to leading order,
$$
\langle \delta_S(t,k)\delta_S(t',q)\rangle
=
\nu_O(t)\,\delta(t-t')\,(2\pi)^3\delta^3(k+q)/a^3.
$$

Nonlinear consistency requires the stochastic kernel to transform as a bi-scalar under the $\pi$-dependent time slicing. This is implemented by replacing $\nu_O(t)\to \nu_O(t+\pi)$, giving
$$
\langle \delta_S(t,k)\delta_S(t',q)\rangle
=
\nu_O(t+\pi)\,\delta(t-t')\,(2\pi)^3\delta^3(k+q)/[a^3(1+\delta N)]
$$
and, to first order,
$$
\simeq
\nu_O(t)\left[1+(\epsilon_{\nu_O}-\epsilon)H\pi\right]
\delta(t-t')(2\pi)^3\delta^3(k+q)/a^3.
$$
This produces the additional squeezed-limit contribution proportional to $\epsilon_{\nu_O}$ that is needed for the full recombination into $n_s-1$.

In the strongly dissipative regime $\gamma \gg H$, with freezeout defined by $c_s k/a \simeq \sqrt{\gamma H}$, the power spectrum satisfies
$$
\Delta_\zeta \equiv k^3 P_\zeta(k)
\simeq
\frac{\dot f_*^2 \nu_{O,*} \sqrt{\pi H_*/\gamma_*}\, H_*^2}
{2 c_{s,*} (c_{s,*}N_{c,*})^2}.
$$

## 5. Conditions of validity and mechanisms of violation

The validity of the physical consistency condition rests on a specific set of structural assumptions [1206.7083]. 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 $\zeta$ 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 $\pi$ is also required, so that the response depends on derivatives of $\pi$ rather than on a constant $\pi$ 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 $\gamma$ may be large compared to $H$, but the local approximation must remain valid.

Several mechanisms can violate or modify the relation. If extra light fields source $\zeta$ 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 $\zeta$ 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 $\pi$ shift symmetry, can also generate terms not tied to $n_s-1$.

Operationally, such failures appear as
$$
f_{NL}^{local} \neq -\frac{5}{12}(n_s-1).
$$
A measured $f_{NL}^{local}$ 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" [1203.6884] 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 $q^2$. "The Physical Squeezed Limit: Consistency Relations at Order $q^2$" [1307.0503] developed this point further by showing that the order-$q^2$ 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" [1811.05951] 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?" [1904.13254] 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
$$
f_{NL}^{local} = -\frac{5}{12}(n_s-1) \simeq O(\text{tilt}),
$$
numerically of order a percent for $n_s \simeq 0.96$--$0.97$ [1206.7083]. The relevant signal is the local squeezed shape, with subleading $O((q/k)^2)$ corrections. CMB bispectrum measurements, large-scale structure through scale-dependent halo bias, and CMB $\mu$-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.

Source: https://www.emergentmind.com/topics/physical-consistency-condition