Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inflationary non-Gaussianities in the most general second-order scalar-tensor theories

Antonio De Felice, Shinji Tsujikawa

TL;DR

This work provides a comprehensive analysis of primordial non-Gaussianities in the most general second-order single-field scalar-tensor theories (Horndeski-type). By deriving the full second- and third-order actions, it shows that the bispectrum is well described by an equilateral template and presents a compact slow-variation expression for the equilateral non-linearity parameter $f_{ m NL}^{\rm equil}$, highlighting the central role of the sound speed $c_s$ and the ratio $\lambda/\Sigma$. The results are applied to a broad range of models—$k$-inflation, Galileon and its extensions, nonminimal couplings, and Gauss-Bonnet couplings—demonstrating conditions under which large equilateral non-Gaussianities can arise or remain small. This framework provides a practical tool for interpreting upcoming observations (e.g., Planck) and distinguishing among single-field inflationary scenarios with generalized kinetic and gravitational couplings.

Abstract

For very general scalar-field theories in which the equations of motion are at second-order, we evaluate the three-point correlation function of primordial scalar perturbations generated during inflation. We show that the shape of non-Gaussianities is well approximated by the equilateral type. The equilateral non-linear parameter f_NL^equil is derived on the quasi de Sitter background where the slow-variation parameters are much smaller than unity. We apply our formula for f_NL^equil to a number of single-field models of inflation--such as k-inflation, k-inflation with Galileon terms, potential-driven Galileon inflation, nonminimal coupling models (including field-derivative coupling models), and Gauss-Bonnet gravity.

Inflationary non-Gaussianities in the most general second-order scalar-tensor theories

TL;DR

This work provides a comprehensive analysis of primordial non-Gaussianities in the most general second-order single-field scalar-tensor theories (Horndeski-type). By deriving the full second- and third-order actions, it shows that the bispectrum is well described by an equilateral template and presents a compact slow-variation expression for the equilateral non-linearity parameter , highlighting the central role of the sound speed and the ratio . The results are applied to a broad range of models—-inflation, Galileon and its extensions, nonminimal couplings, and Gauss-Bonnet couplings—demonstrating conditions under which large equilateral non-Gaussianities can arise or remain small. This framework provides a practical tool for interpreting upcoming observations (e.g., Planck) and distinguishing among single-field inflationary scenarios with generalized kinetic and gravitational couplings.

Abstract

For very general scalar-field theories in which the equations of motion are at second-order, we evaluate the three-point correlation function of primordial scalar perturbations generated during inflation. We show that the shape of non-Gaussianities is well approximated by the equilateral type. The equilateral non-linear parameter f_NL^equil is derived on the quasi de Sitter background where the slow-variation parameters are much smaller than unity. We apply our formula for f_NL^equil to a number of single-field models of inflation--such as k-inflation, k-inflation with Galileon terms, potential-driven Galileon inflation, nonminimal coupling models (including field-derivative coupling models), and Gauss-Bonnet gravity.

Paper Structure

This paper contains 14 sections, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background equations
  3. Second-order action
  4. Three-point correlation function
  5. The shapes of non-Gaussianities
  6. The non-linear parameter under the slow-variation approximation
  7. Applications to concrete models of inflation
  8. k-inflation
  9. k-inflation with the terms $G_i(X)$ ($i=3,4,5$)
  10. Potential-driven Galileon inflation
  11. Nonminimal coupling models
  12. Potential-driven inflation with a Gauss-Bonnet term
  13. Conclusions
  14. The second-order term in $f_{\rm NL}^{\rm equil\,(3)}$

Figures (1)

  • Figure 1: The shape functions ${\cal B}_{{\cal R}}^{(7)}(1,k_{2}/k_{1},k_{3}/k_{1}) (k_{2}/k_{1})^{2}(k_{3}/k_{1})^{2}$ (left) and ${\cal B}_{{\cal R}}^{(8)}(1,k_{2}/k_{1},k_{3}/k_{1}) (k_{2}/k_{1})^{2}(k_{3}/k_{1})^{2}$ (right). The two functions are plotted in the domain $1-k_{2}/k_{1}\leq k_{3}/k_{1}\leq k_{2}/k_{1}$. The lower boundary is given by the triangular inequality, whereas the upper boundary is chosen in order not to repeat twice the same physical configuration. Finally, the functions ${\cal B}_{{\cal R}}^{(i)}$ are multiplied by the measure $(k_{2}/k_{1})^{2}(k_{3}/k_{1})^{2}$ following Ref. Babich. The plots are normalized to have a unit value at the point $k_{2}/k_{1}=k_{3}/k_{1}=1$.