Optimized numerical evolution of perturbations across sharp background trajectory turns in multifield inflation

Abstract

Features in the primordial power spectrum require numerical methods that are both accurate and scalable across the wide class of multifield inflationary models that produce them. Sharp turns in the background trajectories, induced by either potential or geometric effects, render these computations particularly challenging. In this work, we introduce an efficient method for evolving primordial scalar fluctuations, requiring timesteps comparable to those used for the background evolution. We demonstrate that the method accurately tracks perturbations through rapidly turning trajectories in arbitrary field-space geometries, enabling systematic exploration of spectral features across diverse multifield scenarios. Our approach scales robustly to large numbers of degrees of freedom, providing a reliable computational framework for probing regimes that significantly depart from slow-roll dynamics.

Figures (9)

• Figure 1: Background field trajectories in two-field inflationary models, illustrating the effects of field-space geometries and structures in the potential. Left panel: Field trajectories for the nonlinear smooth potential $V(\varphi^1,\varphi^2)=\frac{\lambda}{4}(\varphi^1)^4+\frac{g}{2}(\varphi^2)^2(\varphi^1)^2$, deformed by curvature effects arising from a nontrivial metric (see \ref{['eq:fld_met_ex']}). The trajectories in solid red illustrate how field-space geometry induces sharp, nearly non-differentiable turns in the trajectories, leading to features in the primordial curvature spectrum. For comparison, the curve in dashed black lines displays the background trajectory in flat field-space. Right panel: Field trajectories exhibiting sharp turns generated by the scalar potential in \ref{['eq:potential_st']}. The solid red curves consider initial conditions in the high-velocity regime in which trajectories are not constrained to move along the minimum potential, while the trajectories in dashed black lines correspond to trajectories with zero initial velocity.
• Figure 2: Evolution of constant-$k$ curvature modes for different values of the mass ratio $m_2/m_1$. Solid lines show the evolution obtained using the Cholesky decomposition scheme presented in GalvezGhersi:2016wbu, with dots at the end of the solid lines marking the point at which the evolution breaks down. Dotted lines depict the full mode evolution obtained using the evolution scheme developed in this paper. Instabilities manifest in the Cholesky scheme as interruptions in the evolution following sharp turns in the background trajectories. The inset in the upper-left corner shows that, as the mass ratio grows, the post-turn mode evolution becomes increasingly nontrivial.
• Figure 3: Mode injection scheme and evolution of the inflationary horizon (in the red solid curve). Time-translation invariance allows us to initialize each constant-$k$ mode (in light blue dashed lines) from a common initial surface of fixed physical wavelength (in a dashed black line). The inset in the upper-left corner illustrates variations in the horizon scale, which arise as a consequence of the sharp turns in the background trajectory shown in the right panel of Figure \ref{['fig:def_geom_V_traj']}.
• Figure 4: Validation of the amplitude-phase separation method. Left panel: Evolution of the eigenvalues of the effective oscillation frequency before (in solid red) and after (in solid blue) applying the scale separation procedure. The eigenvalues are reduced by several orders of magnitude, allowing for larger evolution timesteps and significantly improving the computational efficiency. This supports the conclusion that the procedure described in section \ref{['sec:method']} effectively separates the largest oscillation scales from the system. Right panel: Power spectrum of primordial curvature fluctuations in an inflationary scenario sourced by the potential $V(\varphi^1,\varphi^2)=\frac{\lambda}{4}(\varphi^1)^4+\frac{g}{2}(\varphi^1)^2(\varphi^2)^2$. For sufficiently smooth potentials, the power spectrum shows that the results obtained using the scale-separation method agree with those from the Cholesky decomposition approach of GalvezGhersi:2016wbu across all wavenumbers $k$.
• Figure 5: Evolution of adiabatic, cross-correlation and isocurvature power spectra at fixed wavenumber in the presence of sharp turns in the background field evolution. The upper panels show the effects of progressively larger geometric deformations parameterized according to \ref{['eq:fld_met_ex']} considering a smooth potential. The lower panels illustrate the deformations of the primordial spectra induced by increasingly large deviations in the potential, parameterized according to \ref{['eq:potential_st']}. In both cases, darker colors correspond to larger departures from flat field-space geometry or from the quadratic potential, while dashed lines denote the undeformed cases where $\Delta V=0$ and $\Delta h_{AB}=0$. During mode evolution, these deviations can enhance the cross-correlation power to values comparable to those of the adiabatic and isocurvature modes. In contrast to the behavior observed in the Cholesky decomposition approach of GalvezGhersi:2016wbu (see Figure \ref{['fig:cholesky_problem']}), the mode evolution here proceeds without interruption from dynamical instabilities.