Inflationary Backgrounds, Broken Time Diffeomorphisms, and Decoupling

Inflation is a time-dependent medium: the background clock breaks time diffeomorphisms spontaneously, while the full theory remains diffeomorphism invariant. This distinction explains both the Goldstone description and its limits. Decoupling is licensed only when the scalar–gravity mixing scale lies below the energy relevant to the observable and that energy remains below the EFT cutoff.

Required background. FLRW fields and mode quantization supplies the background oscillator problem; cosets and nonlinear realizations supplies the Goldstone construction; and gravitational EFT power counting supplies the hierarchy test.

Helpful background. Effective field theory as a controlled expansion clarifies truncation errors, while constraints, conservation, and the Bianchi identity clarifies why lapse and shift cannot simply be discarded.

The clock and the constrained gravitational system

Write the metric in ADM form,

$$ds^2=N^2dt^2-h_{ij}(dx^i+N^i dt)(dx^j+N^jdt).$$

For a single monotonic scalar background, unitary gauge sets $\delta\phi=0$. Time-dependent spatial diffeomorphisms remain manifest, but a time shift changes the background coefficients. The most general low-derivative unitary-gauge action therefore begins as

$$S=\int d^4x\sqrt{-g}\left[ -\frac{M_{\rm Pl}^2}{2}R-c(t)g^{00}-\Lambda(t) +\frac{M_2^4(t)}2(\delta g^{00})^2+\cdots\right].$$

The background equations fix $c$ and $\Lambda$ once $H(t)$ is chosen; $M_2^4$ and the omitted extrinsic-curvature operators govern fluctuations. Because $N$ and $N^i$ have no independent two-derivative kinetic terms in the minimal theory, their variations impose Hamiltonian and momentum constraints. Eliminating them is an algebraic or elliptic reduction on each slice, not an optional approximation.

Restore time diffeomorphisms by replacing every time-dependent coefficient by its value at $t+\pi$ and transforming tensors accordingly. Under $t\mapsto t+\xi^0$, our Stückelberg field transforms as $\pi\mapsto\pi-\xi^0$. At linear order the comoving curvature perturbation is

$$\zeta=-H\pi.$$

This nonlinear realization is derived explicitly in Cheung et al. 2008, §§2–3, Eqs. (2)–(28).

When gravity decouples

For canonical slow-roll inflation, canonical normalization makes the leading mixing between $\pi$ and metric fluctuations small above a scale of order

$$E_{\rm mix}\sim \sqrt{\epsilon}\,H,$$

up to order-one factors and changes induced by noncanonical coefficients. Thus a process at energy $E$ can be computed in the Goldstone sector alone only within the window

$$E_{\rm mix}\ll E\ll \Lambda_{\rm EFT}.$$

Freeze-out occurs near $E\sim c_s k/a\sim H$, so ordinary slow roll often provides a useful window. A small sound speed or large extrinsic-curvature coefficients can raise mixing, lower strong coupling, or introduce another relevant scale. The correct test compares the actual quadratic mixing matrix and interaction scales, not the slogan $E\gg H$.

As a first application, expand the canonical scalar–gravity system about FLRW. The momentum constraint gives the leading lapse perturbation $\delta N=\dot\zeta/H$ in comoving gauge, while the shift contains an inverse spatial Laplacian. Substitution into the action yields the scalar kinetic coefficient $\epsilon M_{\rm Pl}^2$. The same calculation in the $\pi$ language gives $\zeta=-H\pi$ and the same mode equation when mixing terms are retained to the required order Maldacena 2003, §3, Eqs. (2.7)–(3.10).

The structure map locates decoupling between constraint reduction and the interaction expansion.

The Goldstone description follows from broken time diffeomorphisms, but eliminating lapse and shift and verifying $E_{\rm mix}\ll E\ll\Lambda_{\rm EFT}$ remain separate requirements. Schematic; not to scale.

Failure test: a weak clock

If $\dot\phi_0$ approaches zero, the clock slicing degenerates and $\epsilon$ may become too small to provide a regular scalar kinetic term. Conversely, in a regime with strong mixing, solving only the $\pi$ action drops terms of the same order as the retained dynamics. The adversarial check is to diagonalize the coupled scalar–lapse–shift quadratic system at freeze-out. If the nominally gravitational correction is not parametrically smaller, reject the decoupling approximation and solve the constraints without expanding in $E_{\rm mix}/E$.

See the chapter’s domain and failure conditions for the comparison with other approximations. The validity map highlights the two independent breakdowns: an ill-defined clock and a missing energy window.

Decoupling fails if the clock slicing degenerates, the constraints remain unsolved, or the mixing, evaluation, and cutoff scales do not form an ordered window. 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.