## Author(s)

Caravano, Angelo, Franciolini, Gabriele, Renaux-Petel, Sébastien## Abstract

Violating the slow-roll regime during the final stages of inflation can significantly enhance curvature perturbations, a scenario often invoked in models producing primordial black holes and small-scale scalar induced gravitational waves. When perturbations are enhanced, one approaches the regime in which tree-level computations are insufficient, and nonlinear corrections may become relevant. In this work, we conduct lattice simulations of ultra-slow-roll (USR) dynamics to investigate the significance of nonlinear effects, both in terms of backreaction on the background and in the evolution of perturbations. Our systematic study of various USR potentials reveals that nonlinear corrections are significant when the tree-level curvature power spectrum peaks at $\mathcal{P}^{\rm max}_{\zeta} = {\cal O}(10^{-3})-{\cal O}(10^{-2})$, with 5%$-$10% corrections. Larger enhancements yield even greater differences. We establish a universal relation between simulation and tree-level quantities $\dot\phi = \dot\phi_{\rm tree}\left(1+\sqrt{\mathcal{P}^{\rm max}_{\zeta,\rm tree}}\right)$ at the end of the USR phase, which is valid in all cases we consider. Additionally, we explore how nonlinear interactions during the USR phase affect the clustering and non-Gaussianity of scalar fluctuations, crucial for understanding the phenomenological consequences of USR, such as scalar-induced gravitational waves and primordial black holes. Our findings demonstrate the necessity of going beyond leading order perturbation theory results, through higher-order or non-perturbative computations, to make robust predictions for inflation models exhibiting a USR phase.

## Figures

Evolution of $\eta (N)$ (top panel) and $\epsilon(N)$ (bottom panel) as a function of number of $e$-folds considered in this work. Solid, dashed and dotted line indicate different cases, corresponding to the approximate Wands duality (WD), repulsive and attractive case, respectively. See the main text for more details. Different colors (from purple to yellow) correspond to different USR durations $N_{\rm end} - N_{\rm in}$, leading to a tree-level curvature power spectrum at its maximum ranging from ${\cal P}_\zeta^{\rm max} =10^{-4}$ to ${\cal P}^{\rm max}_\zeta =1$. In all cases, we fixed $N_{\rm in } = 0$.

Reconstructed inflationary potentials built with the reverse engineering approach and choosing the parameters reported in Tab.~\ref{tab:usr_cases}. From top to bottom, we show the potential and its first and second derivatives with respect to the inflaton field $\phi$, normalised to their quantities at $N_{\rm in}$, i.e. $\phi_{\rm ref}$ and $V_{\rm ref}$. {\it Left panel:} case I (WD); {\it Center panel:} case II (repulsive); {\it Right panel:} case III (attractive).

Reconstructed inflationary potentials built with the reverse engineering approach and choosing the parameters reported in Tab.~\ref{tab:usr_cases}. From top to bottom, we show the potential and its first and second derivatives with respect to the inflaton field $\phi$, normalised to their quantities at $N_{\rm in}$, i.e. $\phi_{\rm ref}$ and $V_{\rm ref}$. {\it Left panel:} case I (WD); {\it Center panel:} case II (repulsive); {\it Right panel:} case III (attractive).

Reconstructed inflationary potentials built with the reverse engineering approach and choosing the parameters reported in Tab.~\ref{tab:usr_cases}. From top to bottom, we show the potential and its first and second derivatives with respect to the inflaton field $\phi$, normalised to their quantities at $N_{\rm in}$, i.e. $\phi_{\rm ref}$ and $V_{\rm ref}$. {\it Left panel:} case I (WD); {\it Center panel:} case II (repulsive); {\it Right panel:} case III (attractive).

Illustration of the tree-level predictions of Standard Perturbation Theory, considering for definiteness case I and $\Delta N = 2$ (corresponding to a very good approximation with $\mathcal{P}^{\rm max}_{\zeta,\rm tree}=10^{-2}$). {\it Left panel:} Curvature perturbation mode evolution as a function of number of $e$-folds. At early times, modes evolve from the Bunch-Davies vacuum. The orange line indicates the evolution of a mode crossing the Hubble sphere before the USR phase. Indeed, we see it freezing out, and then experiencing a change of behavior for the decaying mode, until the USR phase ends again. Brown and red lines correspond to modes crossing the Hubble scale at the onset and end of USR, respectively. The yellow line corresponds to a mode crossing the Hubble sphere after USR has ended, but sufficiently close to it to be affected. {\it Right panel:} Curvature power spectrum near the peak. Vertical dashed lines indicate the scales with the same color code adopted in the left panel. For reference, the lattice simulations stop at around $N=5$, after the relevant modes have frozen.

Illustration of the tree-level predictions of Standard Perturbation Theory, considering for definiteness case I and $\Delta N = 2$ (corresponding to a very good approximation with $\mathcal{P}^{\rm max}_{\zeta,\rm tree}=10^{-2}$). {\it Left panel:} Curvature perturbation mode evolution as a function of number of $e$-folds. At early times, modes evolve from the Bunch-Davies vacuum. The orange line indicates the evolution of a mode crossing the Hubble sphere before the USR phase. Indeed, we see it freezing out, and then experiencing a change of behavior for the decaying mode, until the USR phase ends again. Brown and red lines correspond to modes crossing the Hubble scale at the onset and end of USR, respectively. The yellow line corresponds to a mode crossing the Hubble sphere after USR has ended, but sufficiently close to it to be affected. {\it Right panel:} Curvature power spectrum near the peak. Vertical dashed lines indicate the scales with the same color code adopted in the left panel. For reference, the lattice simulations stop at around $N=5$, after the relevant modes have frozen.

{\it Top panel:} Evolution of $\nu^2$ as a function of number of $e$-folds across SR-USR-SR transitions in the three cases considered in this work, with $\Delta N = 2$ in each case. While all cases presents a sizeable evolution of $\nu^2$ at the onset of USR, in case I, $\nu^2 $ is nearly constant from deep in the USR phase. {\it Bottom panel:} first derivative of $\nu^2$ with respect to number of efolds. By construction, at the end of USR around $N=2$, it is ${\cal O}(1)$ and positive (resp. negative) in case II (resp. case III), which makes the former a ``repulsive'' scenario while the latter can be consider as ``attractive''. See the main text for an explanation of the terminology adopted here. By contrast, in case I with approximate Wands duality, $(\nu^2)'$ is two orders of magnitude smaller than the other cases at $N=2$, corresponding to an approximately free theory.

{\it Top panel:} Evolution of $\nu^2$ as a function of number of $e$-folds across SR-USR-SR transitions in the three cases considered in this work, with $\Delta N = 2$ in each case. While all cases presents a sizeable evolution of $\nu^2$ at the onset of USR, in case I, $\nu^2 $ is nearly constant from deep in the USR phase. {\it Bottom panel:} first derivative of $\nu^2$ with respect to number of efolds. By construction, at the end of USR around $N=2$, it is ${\cal O}(1)$ and positive (resp. negative) in case II (resp. case III), which makes the former a ``repulsive'' scenario while the latter can be consider as ``attractive''. See the main text for an explanation of the terminology adopted here. By contrast, in case I with approximate Wands duality, $(\nu^2)'$ is two orders of magnitude smaller than the other cases at $N=2$, corresponding to an approximately free theory.

Simulation results for case I (Wands duality) of Table \ref{tab:usr_cases}. {\it Top panels:} Plot of background quantities $\langle\phi\rangle$ (left) and $\langle\dot\phi\rangle$ (right) computed as averages over the $N_{\rm pts}^3$ lattice points. The relative difference with respect to the purely homogeneous result is shown in the lower panel. {\it Middle panels:} Power spectrum of $\zeta$ (left) and of $\phi$ (right) at the end of the simulation ($N=4.96$). Colors range from $\mathcal{P}^{\rm max}_{\zeta,\rm tree}=10^{-4}$ (blue) to $\mathcal{P}^{\rm max}_{\zeta,\rm tree}=1$ (yellow), similarly to Figs.~\ref{fig:dyn} and \ref{fig:dyn_pot}. Tree-level results from SPT are shown as dashed lines, with relative differences shown in the lower panel. {\it Bottom panels:} Nonlinear correction to $\dot\phi$ (left panel) and to the power spectrum peak (right panel) as a function of the linear theory prediction. Each point of this plot is obtained from one lattice simulation.

Simulation results for case I (Wands duality) of Table \ref{tab:usr_cases}. {\it Top panels:} Plot of background quantities $\langle\phi\rangle$ (left) and $\langle\dot\phi\rangle$ (right) computed as averages over the $N_{\rm pts}^3$ lattice points. The relative difference with respect to the purely homogeneous result is shown in the lower panel. {\it Middle panels:} Power spectrum of $\zeta$ (left) and of $\phi$ (right) at the end of the simulation ($N=4.96$). Colors range from $\mathcal{P}^{\rm max}_{\zeta,\rm tree}=10^{-4}$ (blue) to $\mathcal{P}^{\rm max}_{\zeta,\rm tree}=1$ (yellow), similarly to Figs.~\ref{fig:dyn} and \ref{fig:dyn_pot}. Tree-level results from SPT are shown as dashed lines, with relative differences shown in the lower panel. {\it Bottom panels:} Nonlinear correction to $\dot\phi$ (left panel) and to the power spectrum peak (right panel) as a function of the linear theory prediction. Each point of this plot is obtained from one lattice simulation.

Simulation results for case I (Wands duality) of Table \ref{tab:usr_cases}. {\it Top panels:} Plot of background quantities $\langle\phi\rangle$ (left) and $\langle\dot\phi\rangle$ (right) computed as averages over the $N_{\rm pts}^3$ lattice points. The relative difference with respect to the purely homogeneous result is shown in the lower panel. {\it Middle panels:} Power spectrum of $\zeta$ (left) and of $\phi$ (right) at the end of the simulation ($N=4.96$). Colors range from $\mathcal{P}^{\rm max}_{\zeta,\rm tree}=10^{-4}$ (blue) to $\mathcal{P}^{\rm max}_{\zeta,\rm tree}=1$ (yellow), similarly to Figs.~\ref{fig:dyn} and \ref{fig:dyn_pot}. Tree-level results from SPT are shown as dashed lines, with relative differences shown in the lower panel. {\it Bottom panels:} Nonlinear correction to $\dot\phi$ (left panel) and to the power spectrum peak (right panel) as a function of the linear theory prediction. Each point of this plot is obtained from one lattice simulation.

Same as Fig. \ref{fig:caseI}, but for case II (repulsive).

Same as Fig. \ref{fig:caseI}, but for case II (repulsive).

Same as Fig. \ref{fig:caseI}, but for case II (repulsive).

Same as Fig. \ref{fig:caseI}, but for case II (repulsive).

Same as Fig. \ref{fig:caseI}, but for case II (repulsive).

Same as Fig. \ref{fig:caseI}, but for case II (repulsive).

Same as Fig. \ref{fig:caseI}, but for case III (attractive).

Same as Fig. \ref{fig:caseI}, but for case III (attractive).

Same as Fig. \ref{fig:caseI}, but for case III (attractive).

Same as Fig. \ref{fig:caseI}, but for case III (attractive).

Same as Fig. \ref{fig:caseI}, but for case III (attractive).

Same as Fig. \ref{fig:caseI}, but for case III (attractive).

2D snapshots of the inflaton field at the end of the simulation. Different columns correspond to different $\mathcal{P}^{\rm max}_{\zeta,\rm tree}$, while the rows refer to the three different cases. These simulations are run with the same initial conditions to highlight the differences induced by the USR evolution between the three cases.

1-point PDF of $\delta\phi=\phi-\langle\phi\rangle$ at the final time $N=4.96$. Colors correspond to different peak values of $\mathcal{P}_{\zeta,\rm tree}^{\rm max}$ in the range $10^{-4 \div 0}$. As we normalize by the standard deviation $\sqrt{\langle\delta\phi^2\rangle}$, any deviation from the nearly Gaussian case $\mathcal{P}_{\zeta,\rm tree}^{\rm max} = 10^{-4}$ (blue) represents a deviation from Gaussianity.

1-point PDF of $\delta\phi=\phi-\langle\phi\rangle$ at the final time $N=4.96$. Colors correspond to different peak values of $\mathcal{P}_{\zeta,\rm tree}^{\rm max}$ in the range $10^{-4 \div 0}$. As we normalize by the standard deviation $\sqrt{\langle\delta\phi^2\rangle}$, any deviation from the nearly Gaussian case $\mathcal{P}_{\zeta,\rm tree}^{\rm max} = 10^{-4}$ (blue) represents a deviation from Gaussianity.

1-point PDF of $\delta\phi=\phi-\langle\phi\rangle$ at the final time $N=4.96$. Colors correspond to different peak values of $\mathcal{P}_{\zeta,\rm tree}^{\rm max}$ in the range $10^{-4 \div 0}$. As we normalize by the standard deviation $\sqrt{\langle\delta\phi^2\rangle}$, any deviation from the nearly Gaussian case $\mathcal{P}_{\zeta,\rm tree}^{\rm max} = 10^{-4}$ (blue) represents a deviation from Gaussianity.

Phase space trajectory $\dot\phi(\phi)$ from the simulation, compared with the purely homogeneous prediction (dashed lines). Different colors correspond to different $\mathcal{P}_{\zeta,\rm tree}^{\rm max}$ in the range $10^{-4 \div 0}$, like in all other plots.

Phase space trajectory $\dot\phi(\phi)$ from the simulation, compared with the purely homogeneous prediction (dashed lines). Different colors correspond to different $\mathcal{P}_{\zeta,\rm tree}^{\rm max}$ in the range $10^{-4 \div 0}$, like in all other plots.

Phase space trajectory $\dot\phi(\phi)$ from the simulation, compared with the purely homogeneous prediction (dashed lines). Different colors correspond to different $\mathcal{P}_{\zeta,\rm tree}^{\rm max}$ in the range $10^{-4 \div 0}$, like in all other plots.

Relative contributions to the energy density in the simulation as a function of $e$-folds time $N$. Full lines represent the potential energy $V_{\phi}$, while dashed lines and dotted line are respectively the kinetic ($K_{\phi}$) and gradient energy ($G_{\phi}$) in the simulation. Different colors correspond to different $\mathcal{P}_{\zeta,\rm tree}^{\rm max}$ in the range $10^{-4 \div 0}$.

Relative contributions to the energy density in the simulation as a function of $e$-folds time $N$. Full lines represent the potential energy $V_{\phi}$, while dashed lines and dotted line are respectively the kinetic ($K_{\phi}$) and gradient energy ($G_{\phi}$) in the simulation. Different colors correspond to different $\mathcal{P}_{\zeta,\rm tree}^{\rm max}$ in the range $10^{-4 \div 0}$.

Relative contributions to the energy density in the simulation as a function of $e$-folds time $N$. Full lines represent the potential energy $V_{\phi}$, while dashed lines and dotted line are respectively the kinetic ($K_{\phi}$) and gradient energy ($G_{\phi}$) in the simulation. Different colors correspond to different $\mathcal{P}_{\zeta,\rm tree}^{\rm max}$ in the range $10^{-4 \div 0}$.

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