Non-Gaussian Trispectrum in Ghost Inflation

• The paper rigorously examines the non-Gaussian trispectrum term as the connected four-point correlation function, providing insights beyond power spectra and bispectra.
• It employs systematic calculational techniques, including in‐in formalism to disentangle contact and scalar exchange contributions in ghost inflation.
• The study demonstrates that a large amplitude (O(10^4)) and distinct shape dependence of the trispectrum offer a robust observational test to differentiate between inflationary models.

The non-Gaussian trispectrum term refers to the connected four-point correlation function of cosmological perturbations, notably in scenarios where the primordial fluctuations produced during inflation deviate from Gaussian statistics. This term provides information that is inaccessible to the power spectrum (two-point function) and the bispectrum (three-point function), enabling discrimination among inflationary models that produce identical power spectra and bispectra but predict distinct four-point statistics. The trispectrum encapsulates the highest-order non-Gaussian signatures that are currently within theoretical and observational reach and is therefore a central quantity in the refinement of early-universe physics, inflation model selection, and primordial feature searches.

1. Definition and Structure of the Non-Gaussian Trispectrum

The primordial curvature perturbation ζ has a nontrivial four-point function in non-Gaussian inflationary scenarios:

$$\langle \zeta_{\mathbf{k}_1} \zeta_{\mathbf{k}_2} \zeta_{\mathbf{k}_3} \zeta_{\mathbf{k}_4} \rangle_c = (2\pi)^3 \delta^3(\mathbf{k}_1 + \mathbf{k}_2 + \mathbf{k}_3 + \mathbf{k}_4) T_\zeta(\mathbf{k}_1, \mathbf{k}_2, \mathbf{k}_3, \mathbf{k}_4)$$

where $T_\zeta$ is the trispectrum, the subscript $c$ denotes the connected part, and momentum conservation applies. For isotropic statistics, the trispectrum depends on six invariants (four magnitudes and two independent angles or diagonals) characterizing the quadrilateral formed by the momenta.

In ghost inflation, two dominant contributions to $T_\zeta$ are identified (Izumi et al., 2010):

• A contact term from a quartic interaction in the effective action of the Goldstone mode π.
• A scalar-exchange term from two cubic interactions, corresponding to a tree-level diagram with an internal scalar propagator.

In the equilateral configuration ($|\mathbf{k}_i| = k$), angular dependence is efficiently parameterized by inner products $C_i = (\mathbf{k}_1 \cdot \mathbf{k}_i)/k^2$ for $i = 2,3,4$, subject to $1+C_2+C_3+C_4=0$.

An associated nonlinearity parameter $\tau_{NL}$ is defined in analogy with $f_{NL}$ for the bispectrum:

$$\tau_{NL} \propto \frac{T_\zeta(\{\mathbf{k}_i\})}{(2\pi^2 P_\zeta)^3} \prod_{i=1}^4 k_i^3$$

where $P_\zeta$ is the observed curvature perturbation power spectrum amplitude.

2. Calculational Techniques and Operator Structure

The evaluation of trispectra necessitates a systematic identification of all higher-order interactions in the inflationary action. In ghost inflation, the relevant operators respect a residual shift symmetry and a $\mathbb{Z}_2$ symmetry ($\pi \rightarrow -\pi$), restricting the quartic term to

$$S_4 = -\frac{\gamma}{8 M^4} \int dt\, d^3x\, a^3 \left( \frac{(\nabla \pi)^4}{a^4} \right)$$

with Hamiltonian density

$$\mathcal{H}_{int,4} = \frac{\tilde{\gamma}}{8 M^4} a^{-1} (\nabla \pi)^4, \quad \tilde{\gamma} = \gamma + 2\beta^2$$

where $\beta$ is the coefficient of the leading cubic term.

The scalar exchange contribution involves vertices from the cubic interaction

$$S_3 = -\frac{\beta}{2 M^2} \int dt\, d^3x\, a^3 (\partial_t \pi) \frac{(\nabla \pi)^2}{a^2}$$

Each term is evaluated using the in-in (Schwinger–Keldysh) formalism for expectation values in time-dependent backgrounds.

Generic $n$-point calculations in the effective field theory (EFT) formalism enumerate all operators up to the desired order, segregated by imposed symmetries (e.g., $S_1: \pi \rightarrow -\pi$, $S_2: \pi \rightarrow -\pi$, $t \rightarrow -t$), and classify contributions as exchange or contact, with their coefficients (e.g., $M_n$, $\bar{M}_n$) parameterizing theory space (Bartolo et al., 2010).

In the equilateral configuration, the trispectrum is frequently normalized by the cubic power of $P_\zeta$, and its amplitude is compared across models by extracting $\tau_{NL}$ or $g_{NL}^{equil}$ coefficients.

3. Magnitude, Shape Dependence, and Distinctive Signatures

The trispectrum amplitude in ghost inflation is characteristically large:

• Contact contribution: $\tau_{NL}^{(cc)} \simeq -1\times10^4$ at $C_2 = C_3 = C_4 = -1/3$
• Scalar exchange contribution: $\tau_{NL}^{(se)} \simeq 3.5\times10^4$ in the symmetric configuration

Thus, the typical overall $\tau_{NL}$ in ghost inflation is $O(10^4)$, well above the sensitivity of Planck-class experiments.

Shape dependence is pronounced: the contact term's absolute value is minimized at the symmetric point while the scalar exchange term is maximized, leading to smooth variation over the allowed $(C_2, C_3)$ domain. Near extremal values ($C_i \rightarrow -1$), the trispectrum decreases smoothly, in contrast to the near-constant ("plateau-like") behavior seen in DBI inflation.

A critical discriminant is the sign of the trispectrum in the equilateral configuration. DBI inflation, where the amplitude is set by the (positive) sound speed, always yields a positive $\tau_{NL}$, whereas in ghost inflation, the quartic coupling is a free parameter and $\tau_{NL}$ can have either sign.

4. Model Differentiation and Experimental Implications

The non-Gaussian trispectrum term in ghost inflation provides a robust and distinctive observational template:

• The large amplitude, $\tau_{NL} \sim O(10^4)$, is an order of magnitude above local-type predictions and is, in principle, within reach of Planck and future surveys.
• The specific angular (shape) dependence in the equilateral configuration, with a smooth decrease toward boundary configurations and negative values at the symmetric point, directly contrasts with DBI and other single-field models.
• The freedom in the sign of $\tau_{NL}$ allows a clear experimental test: observation of a negative equilateral trispectrum would strongly favor ghost inflation over DBI-like scenarios.

Consequently, joint measurements of the bispectrum and trispectrum—including amplitude, angular dependence, and sign—have the potential to uniquely identify ghost inflation and rule out large classes of alternative models.

5. Relation to Broader Non-Gaussianity Program

Trispectrum estimation serves multiple purposes in early-universe cosmology:

• Extension beyond the bispectrum: While the bispectrum ($f_{NL}$) distinguishes among models, corresponding trispectrum parameters ($\tau_{NL}$, $g_{NL}$) provide independent and often sharper constraints.
• Independence and complementarity: In the high-statistics regime, bispectrum and trispectrum estimators become statistically independent, thus failures of the Cramer–Rao bound argument do not apply; the trispectrum indeed supplies additional, non-redundant information (Kamionkowski et al., 2010).
• Consistency checks and discriminants: Models such as ghost inflation, which permit large and sign-variable trispectra, challenge the typical expectation that $\tau_{NL} \geq (6 f_{NL}/5)^2$; they may even allow negative $\tau_{NL}$ in squeezed configurations, demanding careful joint analyses and requiring that loop corrections are not neglected (Bramante et al., 2011).
• Template matching and estimators: Separable estimators for the trispectrum facilitate efficient comparison to data; a high template overlap ($\sim$87% for single-field DBI, lower for multi-field scenarios) ensures robust extraction of $g_{NL}^{equil}$ even amid model variations (Mizuno et al., 2010, Izumi et al., 2011).
• Propagation to observables: The four-point function imprints on the CMB, galaxy clustering, and large-scale structure, with unique configurations (tetrahedral, squeezed, flattened) revealing the non-Gaussian origin—primordial or late-time (Lewis, 2011).

6. Summary Table: Key Results for Ghost Inflation Trispectrum

Contribution	Formula (Equil., Symm.)	Numerical Value	Shape/Sign Features
Contact term	$\tau_{NL}^{(cc)} \approx -1.0 \times 10^{4}$	$-1\times10^4$	Minimized at symm., sign free
Scalar exchange	$\tau_{NL}^{(se)} \approx 3.5 \times 10^{4}$	$3.5\times10^4$	Maximized at symm.
Combined trispectrum	$\tau_{NL} \sim O(10^4)$	$O(10^4)$	Smooth shape variation, possible negative sign

The large amplitude, nontrivial angular dependence, and freedom in sign expected for $\tau_{NL}$ in ghost inflation define a unique template for non-Gaussian trispectrum searches.

7. Outlook and Theoretical Significance

The non-Gaussian trispectrum term is a critical marker for inflationary model selection and discrimination. Its theoretical calculation provides sensitive tests for symmetries and interaction structures present during inflation, and its observational determination, particularly for its amplitude, sign, and momentum dependence, offers the possibility of distinguishing between models with identical lower-order signatures. Ghost inflation, with an unusually large and shape-distinctive trispectrum that permits a negative equilateral value, epitomizes the discriminative power of four-point correlation analysis for reconstructing the physics of the primordial universe (Izumi et al., 2010).