[Paper Review] Non-Gaussian inflationary shapes beyond Horndeski

Matteo Fasiello, Sébastien Renaux‐Petel|arXiv (Cornell University)|Jul 27, 2014

Cosmology and Gravitation Theories1 citations

TL;DR

This paper investigates non-Gaussian signatures in higher-order correlators of inflationary curvature fluctuations within a class of healthy, non-Horndeski theories—specifically Generalized Horndeski (G³) models. Despite complex cubic interactions, the leading-order bispectrum reduces to a linear combination of two standard k-inflationary shapes, suggesting a deep connection to stability, covariance, and dispersion relations that may extend beyond cubic order.

ABSTRACT

We consider the possible signatures of a recently introduced class of healthy theories beyond Horndeski models on higher-order correlators of the inflationary curvature fluctuation. Despite the apparent large number and complexity of the cubic interactions, we show that the leading-order bispectrum generated by the Generalized Horndeski (also called $G^3$) interactions can be reduced to a linear combination of two well known $k$-inflationary shapes. We conjecture that said behavior is not an accident of the cubic order but a consequence dictated by the requirements on the absence of Ostrogradski instability, the general covariance and the linear dispersion relation in these theories.

Motivation & Objective

• To analyze higher-order correlators in a recently proposed class of healthy, non-Horndeski inflationary models.
• To determine whether the complexity of cubic interactions in Generalized Horndeski (G³) theories leads to novel non-Gaussian shapes.
• To investigate whether the simplification of the bispectrum into known shapes is a coincidence or a consequence of fundamental physical principles.
• To explore the role of Ostrogradski stability, general covariance, and linear dispersion relations in constraining the structure of inflationary correlators.

Proposed method

• Analyzes the cubic-order curvature fluctuation interactions in Generalized Horndeski (G³) theories using effective field theory techniques.
• Identifies the full set of cubic interaction terms in the G³ Lagrangian and computes their contributions to the inflationary bispectrum.
• Reduces the resulting bispectrum to a linear combination of two well-known k-inflationary shape templates via mathematical decomposition.
• Applies symmetry and stability constraints—specifically absence of Ostrogradski instability, general covariance, and linear dispersion relation—as guiding principles to explain the observed simplification.
• Compares the derived bispectrum with standard non-Gaussian templates to confirm its equivalence to known shapes.
• Proposes a conjecture that the reduction to two shapes is not accidental but a consequence of fundamental physical requirements beyond cubic order.

Experimental results

Research questions

• RQ1 Can the complex cubic interactions in G³ theories produce novel non-Gaussian shapes in the inflationary bispectrum?
• RQ2 Why does the leading-order bispectrum in G³ models reduce to only two known k-inflationary shapes despite the apparent complexity of the interactions?
• RQ3 Is the simplification of the bispectrum into two templates a coincidence or a consequence of deeper physical principles such as stability and covariance?
• RQ4 To what extent do the absence of Ostrogradski modes, general covariance, and linear dispersion relations constrain the structure of higher-order correlators in non-Horndeski models?
• RQ5 Does this behavior persist beyond cubic order, suggesting a universal feature of healthy beyond-Horndeski theories?

Key findings

• The leading-order bispectrum in Generalized Horndeski (G³) models reduces to a linear combination of two standard k-inflationary shapes, despite the large number of cubic interaction terms.
• This simplification is not accidental but likely arises from fundamental constraints such as the absence of Ostrogradski instability and linear dispersion relations.
• The reduction suggests that the structure of higher-order correlators in these theories is highly constrained by general covariance and stability conditions.
• The authors conjecture that this behavior extends beyond cubic order, implying a universal feature of healthy beyond-Horndeski models.
• The results indicate that non-Gaussian signatures in such models remain within a limited class of shapes, limiting their observational distinguishability from standard k-inflation.