On the Issue of the ζSeries Convergence and Loop Corrections in the Generation of Observable Primordial Non-Gaussianity in Slow-Roll Inflation. Part II: the Trispectrum

TL;DR

<3-5 sentence high-level summary>This paper investigates whether the ζ‑series in slow‑roll inflation with canonical kinetic terms can be reliably truncated and how loop corrections influence primordial non‑Gaussianity beyond the bispectrum. Focusing on a two‑field quadratic model with a σ=0 trajectory, it demonstrates that one‑loop contributions can dominate the trispectrum T_ζ, producing observable values of τ_NL even when the spectrum is set by tree‑level terms, and it revisits the classicality and diffusion issues that govern perturbativity. The analysis characterises parameter regions in which τ_NL can be sizable while f_NL remains small or undetectable, and it discusses the probability of observing a non‑Gaussian distribution in a realistic ensemble of universes. The results emphasize that loop effects are not universally suppressed and can significantly alter inflationary phenomenology, suggesting that trispectrum measurements may be especially informative for these models.

Abstract

We calculate the trispectrum T_ζof the primordial curvature perturbation ζ, generated during a {\it slow-roll} inflationary epoch by considering a two-field quadratic model of inflation with {\it canonical} kinetic terms. We consider loop contributions as well as tree level terms, and show that it is possible to attain very high, {\it including observable}, values for the level of non-gaussianity τ_{NL} if T_ζis dominated by the one-loop contribution. Special attention is paid to the claim in JCAP {\bf 0902}, 017 (2009) [arXiv:0812.0807 [astro-ph]] that, in the model studied in this paper and for the specific inflationary trajectory we choose, the quantum fluctuations of the fields overwhelm the classical evolution. We argue that such a claim actually does not apply to our model, although more research is needed in order to understand the role of quantum diffusion. We also consider the probability that an observer in an ensemble of realizations of the density field sees a non-gaussian distribution. In that respect, we show that the probability associated to the chosen inflationary trajectory is non-negligible. Finally, the levels of non-gaussianity f_{NL} and τ_{NL} in the bispectrum B_ζand trispectrum T_ζof ζ, respectively, are also studied for the case in which ζis not generated during inflation.

Figures (7)

• Figure 1: (a). Tree-level Feynman-like diagram for $T_\zeta$. (b). One-loop Feynman-like diagram for $T_\zeta$. The internal dashed lines correspond to two-point correlators of field perturbations.
• Figure 2: Contours of $\tau_{NL}$ in the $r$ vs $|\eta_\sigma|$ plot. The intermediate (high) $\phi_\star$$T$-region corresponds to the shaded (white) region. The observationally expected $2\sigma$ range of values, for WMAP, PLANCK, and even the 21 cm background anisotropies, and for positive $\tau_{NL}$, $\tau_{NL} > 20$ are completely inside the intermediate $\phi_\star$$T$-region. Notice that the boundary line between the high (see Subsubsection \ref{['hiphit']}) and the intermediate $\phi_\star$$T$-regions matches almost exactly the $\tau_{NL} = 0.04$ line.
• Figure 3: Contours of $f_{NL}$ in the $r$ vs $|\eta_\sigma|$ plot. The intermediate (high) $\phi_\star$ region corresponds to the shaded (white) region. The WMAP (and also PLANCK) observationally allowed $2\sigma$ range of values for negative $f_{NL}$, $-9 < f_{NL}$, is completely inside the intermediate $\phi_\star$ region. Notice that the boundary line between the high and the intermediate $\phi_\star$ regions matches almost exactly the $f_{NL} = -1.667$ line. (This figure has been taken from Ref. cogollo).
• Figure 4: Contours of both $f_{NL}$ and $\tau_{NL}$ in the $r$ vs $|\eta_\sigma|$ plot. The intermediate (high) $\phi_\star$ region corresponds to the shaded (white) region. Lines for constant $\tau_{NL}$ almost exactly match lines for constant $f_{NL}$. According to this figure, and to the present observational status, non-gaussianity is more likely to be detected through the trispectrum than through the bispectrum, for the inflationary model studied in this paper with concave downward potential, and from the WMAP, PLANCK, and even the 21 cm background anisotropies observations. These lines also show some consistency relations between the values of $f_{NL}$ and $\tau_{NL}$ that will be useful at testing the inflationary model considered with concave downward potential against observations.
• Figure 5: Contours of $\tau_{NL}$ in the $r$ vs $n$ plot, for $2.58 \leq n \leq 200$, when $\zeta$ is not generated during inflation. The allowed parameter space corresponds to the white region. The constraint in Eq. (\ref{['plnon1']}) almost matches (visually) the horizontal axis. The largest possible value $\tau_{NL}$ may take in this range is 15.