Primordial Non-Gaussianity and the Maldacena Consistency Relation: An EFT-of-Inflation Test

The Effective Field Theory of Inflation

1. Symmetry Breaking and the Goldstone Boson

   Inflation is fundamentally characterized by the spontaneous breaking of time-translation invariance. The evolving background energy density, driven by a scalar field or alternative mechanism, establishes a preferred physical clock across the universe. In the Effective Field Theory of Inflation, this symmetry breaking is non-linearly realized through a Goldstone boson, denoted as π. The formalism allows us to construct a general action by writing down all operators that respect spatial diffeomorphisms while explicitly breaking time diffeomorphisms, parameterized by functions of time such as the Hubble parameter H and its derivative Ḣ.

   By defining surfaces of constant inflaton field (the unitary gauge), the perturbations of the scalar degree of freedom are temporarily hidden in the metric. However, using the Stueckelberg trick, we can explicitly reintroduce the Goldstone boson π to restore full spacetime diffeomorphism invariance. At high energies, the mixing with gravity decouples, and the dynamics of π dominate the generation of primordial fluctuations. The lowest-order terms in the EFT action, governed by the background evolution, fix the speed of sound of the perturbations.

   c_s⁻² = 1 - (2 M₂⁴) / (M_pl² Ḣ)

   This relation demonstrates how the coefficient M₂⁴ of the lowest-order symmetry-breaking operator dictates the departure of the sound speed from unity. A reduced sound speed naturally enhances the interactions of the Goldstone boson, leading to observable non-Gaussian signatures in the primordial plasma.

2. The Goldstone Action and Higher-Order Operators

   To capture the full phenomenology of single-field inflation, we expand the action to cubic order in the Goldstone boson π. The leading non-trivial operators in the EFT expansion are parameterized by mass scales M₂ and M₃. While the quadratic term proportional to M₂⁴ modifies the sound speed and introduces a specific cubic interaction, the M₃⁴ term represents an independent self-interaction. When transformed from the unitary gauge, the relevant cubic Lagrangian for π emerges directly from the combination of these operators.

   ℒ_π ⊃ ( M₂⁴ - (4/3) M₃⁴ ) π̇³ - M₂⁴ π̇ (∂_iπ)² / a²

   The coefficients of the π̇³ and π̇(∂_iπ)²/a² operators dictate the amplitude and shape of the resulting primordial bispectrum. Because these operators contain spatial and time derivatives, their Fourier counterparts scale with distinct momentum dependencies. The intricate interplay between these two interaction terms allows the EFT framework to naturally span the parameter space of both equilateral and orthogonal non-Gaussian configurations, providing a direct map between high-energy Lagrangian parameters and late-time cosmological observables.

Primordial Non-Gaussianity and Bispectrum Shapes

1. Equilateral and Orthogonal Configurations

   The three-point correlation function of primordial curvature perturbations, evaluated in Fourier space, defines the bispectrum. The geometry of the momentum triangle formed by the three wavevectors k₁, k₂, and k₃ determines the "shape" of the non-Gaussianity. The derivative interactions inherent in the EFT operators—specifically π̇³ and π̇(∂_iπ)²/a²—peak when the interacting modes have comparable wavelengths. This naturally generates an equilateral bispectrum, where the signal is maximized for configurations where k₁ ≈ k₂ ≈ k₃.

   However, by carefully tuning the ratio of the M₂⁴ and M₃⁴ coefficients, one can construct an orthogonal shape. The orthogonal bispectrum peaks both at equilateral configurations and at folded configurations (where k₁ ≈ k₂ ≈ k₃/2), but with opposite signs. This orthogonal template represents an independent basis vector in the space of bispectrum shapes. Current constraints aim to measure the amplitudes of these shapes, denoted f_NL^equil and f_NL^ortho, which serve as direct diagnostic tools for determining the sound speed of the inflaton and the mass scale of its self-interactions.

2. The Squeezed Limit and Local Non-Gaussianity

   While derivative interactions predominantly source equilateral and orthogonal shapes, models involving multiple fields or non-standard initial states often generate local-type non-Gaussianity. The local shape peaks in the squeezed limit, where one wavevector is much smaller than the other two (e.g., k₁ ≪ k₂ ≈ k₃). In real space, this corresponds to a modulation of short-wavelength power by a long-wavelength background perturbation.

   In standard single-field inflation, the generation of local non-Gaussianity is tightly constrained. Because a long-wavelength mode has already exited the Hubble horizon, it effectively acts as a background coordinate transformation for the short-wavelength modes still undergoing quantum fluctuations. This fundamental physical insight strictly bounds the amplitude of f_NL^local in all single-field models, paving the way for one of the most robust null-tests in theoretical cosmology.

The Maldacena Consistency Relation

1. Derivation from Spatial Diffeomorphisms

   The Maldacena consistency relation provides an exact, non-perturbative link between the two-point function (power spectrum) and the squeezed limit of the three-point function (bispectrum). The derivation relies on the behavior of the curvature perturbation ζ. In the squeezed limit, the long-wavelength mode k₁ acts as a constant background field from the perspective of the highly oscillatory short modes k₂ and k₃. By absorbing the long mode into a rescaling of the spatial coordinates, x' = x exp(ζ_long), we can evaluate the impact of this background on the short-scale power spectrum.

   ⟨ ζ(k₁) ζ(k₂) ζ(k₃) ⟩ ≈ - (n_s - 1) P_ζ(k₁) P_ζ(k₃) (2π)³ δ³(k₁ + k₂ + k₃)

   This fundamental relation reveals that the squeezed bispectrum is purely a kinematic effect of the spatial diffeomorphism induced by ζ_long. Because the variation of the power spectrum with scale is governed by the scalar spectral index n_s, the resulting bispectrum in the squeezed limit is directly proportional to the deviation from exact scale invariance, (n_s - 1). This mathematical proof holds for any single-field model regardless of the specific Lagrangian, provided the perturbations originate from a single clock and initially reside in the Bunch-Davies vacuum.

2. Single-Field Predictions

   Translating the Maldacena consistency relation into the standard phenomenological parameterization involves matching the theoretical bispectrum to the local shape template. The local non-Gaussianity parameter, f_NL^local, is defined relative to the amplitude of the bispectrum in this squeezed configuration. When we map the spatial rescaling effect to the f_NL framework, the prefactor yields a remarkably simple predictive formula.

   f_NL^local = 5 (1 - n_s) / 12

   Given that cosmic microwave background observations consistently measure a spectral index slightly less than unity, this relation predicts a non-zero but infinitesimally small local non-Gaussianity for all standard single-field models. A robust detection of f_NL^local significantly greater than this value would immediately falsify the entire class of single-field inflation, forcefully indicating the presence of multiple active fields, non-trivial vacuum states, or alternative universe-generating mechanisms.

Observational Constraints and Future Prospects

1. Current CMB Bounds from Planck PR4 and ACT DR6

   Testing the EFT of Inflation and the Maldacena relation requires highly sensitive observations of the Cosmic Microwave Background. Recent analysis of the Planck PR4 (NPIPE) datasets by Jung et al. (2025) provides the most stringent constraints on the primordial bispectrum to date. The study reports f_NL^local = -0.1 ± 5.0, f_NL^equil = 6 ± 46, and f_NL^ortho = -8 ± 21. These values are highly consistent with Gaussian primordial perturbations and place tight bounds on the EFT parameters M₂ and M₃, confirming that the inflaton sound speed cannot be drastically smaller than the speed of light.

   Concurrently, the Atacama Cosmology Telescope Data Release 6 (ACT DR6), as detailed by Louis et al. (2025), has refined our measurement of the scalar spectral index, yielding n_s = 0.9743 ± 0.0034. Inserting this precise value into the Maldacena formula predicts a single-field squeezed limit of f_NL^local ≈ 0.011. This theoretical target remains well below the current detection threshold of Planck PR4, reaffirming that standard single-field models are perfectly compatible with the latest high-resolution CMB maps.

2. Large-Scale Structure and the Path to SPHEREx

   While the primary CMB anisotropies have historically dominated non-Gaussianity constraints, Large-Scale Structure (LSS) is rapidly emerging as a competitive frontier. Local primordial non-Gaussianity leaves a distinct imprint on LSS by inducing a scale-dependent bias in the clustering of dark matter halos at very large scales. A recent cross-correlation analysis between DESI Luminous Red Galaxies (LRGs) and Planck CMB lensing maps by Bermejo-Climent et al. (2025) yielded f_NL^local = 24⁺²⁰₋₂₁. Although statistically consistent with zero at the 1σ level, this result highlights the growing power of LSS surveys to probe inflationary physics.

   To definitively test the Maldacena consistency relation and cross the critical f_NL^local ~ 1 threshold, observational cosmology relies on next-generation missions. The NASA SPHEREx mission, an all-sky near-infrared spectral survey, is specifically optimized to measure the 3D distribution of galaxies over an unprecedented volume. By mapping the scale-dependent bias across multiple cosmic epochs, SPHEREx targets a precision of σ(f_NL) ≲ 1. Reaching this milestone will either confirm the single-field paradigm or usher in a new era of multi-field inflationary physics.

Conclusion