## Author(s)

Tasinato, Gianmassimo## Abstract

The physics of primordial black holes can be affected by the non-Gaussian statistics of the density fluctuations that generate them. Therefore, it is important to have good theoretical control of the higher-order correlation functions for primordial curvature perturbations. By working at leading order in a $1/|\eta|$ expansion, we analytically determine the bispectrum of curvature fluctuations for single field inflationary scenarios producing primordial black holes. The bispectrum has a rich scale and shape dependence, and its features depend on the dynamics of the would-be decaying mode. We apply our analytical results to study gravitational waves induced at second order by enhanced curvature fluctuations. Their statistical properties are derived in terms of convolution integrals over wide momentum ranges, and they are sensitive on the scale and shape dependence of the curvature bispectrum we analytically computed.

## Figures

\footnotesize The quantity $\Pi(\kappa)$ introduced in eq \eqref{defPa}, evaluated for $\Pi_0=10^3$.

\footnotesize{ {\bf Left}: the squeezed bispectrum satisfies Maldacena consistency relation. Red: $(1-n_\zeta)$, Dashed black the squeezed non-linear parameter $5\,f_{\rm NL}^{\rm sq}/6$ defined in eq \eqref{deffsq}. {\bf Right}: The function $G(\kappa)$ defined in eq \eqref{NLC1}, divided by $\Pi_0^2$ (black line). The quantity $(ns-1)/2$ divided by $\Pi_0^{1/2}$ (dashed blue line). The normalization factors are introduced to obtain comparable variables for the two quantities introduced. In both panels, $\Pi_0=10^3$.}

\footnotesize{ {\bf Left}: the squeezed bispectrum satisfies Maldacena consistency relation. Red: $(1-n_\zeta)$, Dashed black the squeezed non-linear parameter $5\,f_{\rm NL}^{\rm sq}/6$ defined in eq \eqref{deffsq}. {\bf Right}: The function $G(\kappa)$ defined in eq \eqref{NLC1}, divided by $\Pi_0^2$ (black line). The quantity $(ns-1)/2$ divided by $\Pi_0^{1/2}$ (dashed blue line). The normalization factors are introduced to obtain comparable variables for the two quantities introduced. In both panels, $\Pi_0=10^3$.}

\footnotesize{ {\bf Left:} The right hand side of eq \eqref{equaiqu} (black line) versus the left hand side of the same equation (dashed red line) {\bf Right:} The quantity $f_{\rm NL}^{\rm eq}$ defined in eq \eqref{deffeq}. In both panels, $\Pi_0=10^3$. }

\footnotesize{ {\bf Left:} The right hand side of eq \eqref{equaiqu} (black line) versus the left hand side of the same equation (dashed red line) {\bf Right:} The quantity $f_{\rm NL}^{\rm eq}$ defined in eq \eqref{deffeq}. In both panels, $\Pi_0=10^3$. }

\footnotesize Triangular plots of the function $S(\kappa, x_2, x_3)$ of eq \eqref{deffS}, represented with the criteria discussed in the text. {\bf Left} $S(\kappa_{\rm dip}, x_2, x_3)$. {\bf Right} $S(1, x_2, x_3)$.

\footnotesize Triangular plots of the function $S(\kappa, x_2, x_3)$ of eq \eqref{deffS}, represented with the criteria discussed in the text. {\bf Left} $S(\kappa_{\rm dip}, x_2, x_3)$. {\bf Right} $S(1, x_2, x_3)$.

\footnotesize Triangular plots of the function $S(\kappa, x_2, x_3)$ of eq \eqref{deffS}, represented with the criteria discussed in the text. {\bf Left} $S(\kappa_{\rm dip}, x_2, x_3)$. {\bf Right} $S(1, x_2, x_3)$.

\footnotesize {\bf Left:} plot of the gravitational wave energy density induced by a scalar spectrum derived from eq \eqref{defph}, with $\Pi_0=10^3$, under the conditions discussed in the main text. {\bf Right:} plot of the non-linear parameter $f_{\rm NL}^{TTS}$ in eq \eqref{defttsf} computed with the same curvature spectrum of the left figure.

\footnotesize {\bf Left:} plot of the gravitational wave energy density induced by a scalar spectrum derived from eq \eqref{defph}, with $\Pi_0=10^3$, under the conditions discussed in the main text. {\bf Right:} plot of the non-linear parameter $f_{\rm NL}^{TTS}$ in eq \eqref{defttsf} computed with the same curvature spectrum of the left figure.

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