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Trispectrum versus Bispectrum in Single-Field Inflation

Kevin T. Engel, Keith S. M. Lee, Mark B. Wise

TL;DR

This work examines whether a single-field inflation model can yield a detectable trispectrum without a detectable bispectrum. By employing general $P(X,\\phi)$ Lagrangians and analyzing the non-Gaussianity parameters $f_{NL}$ and $\\tau_{NL}$, the authors show that a trispectrum-dominated signal is possible but requires non-canonical kinetic terms and substantial fine-tuning of the potential. They construct two explicit models (the $\\alpha$-model and the log-model) that realize large $\\tau_{NL}$ with small $f_{NL}$, and they perform numerical studies illustrating the sensitivity of $\\tau_{NL}$ to perturbations in the potential, implying tuning at the level of $10^{-7}$ to $10^{-4}$. The results suggest that a Planck detection of a trispectrum without a bispectrum would favor multi-field scenarios over canonical single-field inflation. Overall, the paper provides a meticulous exploration of the trade-offs and limitations of single-field models in generating trispectrum-dominant non-Gaussianities, highlighting the tension with naturalness and guiding future observational and theoretical work.

Abstract

In the standard slow-roll inflationary cosmology, quantum fluctuations in a single field, the inflaton, generate approximately Gaussian primordial density perturbations. At present, the bispectrum and trispectrum of the density perturbations have not been observed and the probability distribution for these perturbations is consistent with Gaussianity. However, Planck satellite data will bring a new level of precision to bear on this issue, and it is possible that evidence for non-Gaussian effects in the primordial distribution will be discovered. One possibility is that a trispectrum will be observed without evidence for a non-zero bispectrum. It is not difficult for this to occur in inflationary models where quantum fluctuations in a field other than the inflaton contribute to the density perturbations. A natural question to ask is whether such an observation would rule out the standard scenarios. We explore this issue and find that it is possible to construct single-field models in which inflaton-generated primordial density perturbations have an observable trispectrum, but a bispectrum that is too small to be observed by the Planck satellite. However, an awkward fine tuning seems to be unavoidable.

Trispectrum versus Bispectrum in Single-Field Inflation

TL;DR

This work examines whether a single-field inflation model can yield a detectable trispectrum without a detectable bispectrum. By employing general Lagrangians and analyzing the non-Gaussianity parameters and , the authors show that a trispectrum-dominated signal is possible but requires non-canonical kinetic terms and substantial fine-tuning of the potential. They construct two explicit models (the -model and the log-model) that realize large with small , and they perform numerical studies illustrating the sensitivity of to perturbations in the potential, implying tuning at the level of to . The results suggest that a Planck detection of a trispectrum without a bispectrum would favor multi-field scenarios over canonical single-field inflation. Overall, the paper provides a meticulous exploration of the trade-offs and limitations of single-field models in generating trispectrum-dominant non-Gaussianities, highlighting the tension with naturalness and guiding future observational and theoretical work.

Abstract

In the standard slow-roll inflationary cosmology, quantum fluctuations in a single field, the inflaton, generate approximately Gaussian primordial density perturbations. At present, the bispectrum and trispectrum of the density perturbations have not been observed and the probability distribution for these perturbations is consistent with Gaussianity. However, Planck satellite data will bring a new level of precision to bear on this issue, and it is possible that evidence for non-Gaussian effects in the primordial distribution will be discovered. One possibility is that a trispectrum will be observed without evidence for a non-zero bispectrum. It is not difficult for this to occur in inflationary models where quantum fluctuations in a field other than the inflaton contribute to the density perturbations. A natural question to ask is whether such an observation would rule out the standard scenarios. We explore this issue and find that it is possible to construct single-field models in which inflaton-generated primordial density perturbations have an observable trispectrum, but a bispectrum that is too small to be observed by the Planck satellite. However, an awkward fine tuning seems to be unavoidable.

Paper Structure

This paper contains 8 sections, 51 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Formalism
  3. Non-Gaussianities
  4. Model building
  5. Numerical studies
  6. $\alpha$ model
  7. Log model
  8. Conclusion

Figures (3)

  • Figure 1: \ref{['vel1']}$\dot{\phi}$ (solid) plotted with speed limit of $-\sqrt{\frac{2}{f}}$ (dashed). \ref{['ns1']} Spectral index $n_s$. \ref{['taunl1']}$\tau_{\rm NL}$.
  • Figure 2: $\tau_{\rm NL}$ plotted for various values of $\delta$. From top to bottom, $\delta = 0\, , \, 10^{-15} \, , \, -10^{-12} \, , \, 10^{-9} \, , \, -10^{-7}$. For positive $\delta$, $\dot{\phi}$ crosses the speed limit before leveling off, creating the spikes shown in $\tau_{\rm NL}$.
  • Figure 3: \ref{['vel1']}$\dot{\phi}$ (solid) plotted with speed limit of $-\sqrt{\frac{2}{f}}$ (dashed). \ref{['ns1']} Spectral index $n_s$. \ref{['taunl1']}$\tau_{\rm NL}$. \ref{['fnl1']}$f_{\rm NL}$.