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

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.

💡 Research Summary

The paper addresses a long‑standing computational bottleneck in multifield inflationary scenarios where the background trajectory in field space undergoes sharp turns. Such rapid changes, whether induced by non‑trivial field‑space geometry or by localized features in the scalar potential, force the perturbation modes to oscillate on very short time‑scales. Traditional numerical approaches—most notably the Cholesky‑decomposition scheme introduced in earlier work—must resolve these fast phases directly, which forces the time step to be orders of magnitude smaller than that used for the background evolution. Consequently, the cost of computing the two‑point correlation functions becomes prohibitive, especially when the number of fields is large.

The authors propose a two‑pronged solution. First, they introduce a parallel‑transport gauge in field space. By redefining the perturbation variables so that they are parallel‑transported along the background trajectory, the covariant derivative terms involving the field‑space connection are shifted entirely into the phase equation of the mode functions. This gauge preserves the symplectic structure and leaves the canonical commutation relations intact, while simplifying the form of the equations of motion.

Second, they perform an amplitude‑phase decomposition of each Fourier mode: (u_k(t)=A_k(t),e^{i\theta_k(t)}). Substituting this ansatz into the gauge‑fixed equations yields a coupled system of nonlinear second‑order ordinary differential equations for the amplitude (A_k) and the phase (\theta_k). Crucially, the effective frequency governing the amplitude dynamics, (\Omega_k), is suppressed by several orders of magnitude relative to the original physical frequency (\omega_k). The suppression arises because the rapid oscillatory part is now captured entirely by the phase equation, which can be integrated analytically or with a very coarse step. As a result, the amplitude evolves on a slow timescale, allowing the numerical integrator to use timesteps comparable to those required for the background fields without loss of accuracy.

The paper validates the method on several illustrative models. In the first example, a two‑field system is equipped with a non‑trivial metric (h_{AB}=\delta_{AB}+\Delta h_{AB}) containing thirty Gaussian “bumps” that generate a series of abrupt turns in the background trajectory. The traditional Cholesky scheme fails at the first turn, whereas the new amplitude‑phase method tracks the evolution continuously through all turns. In the second example, a smooth quadratic potential is perturbed by a sinusoidal deformation (\Delta V) that creates multiple sharp bends in the trajectory. Again, the new method remains stable while the older approach breaks down. The authors also demonstrate that the computational cost scales roughly as (N_f^2) with the number of fields, confirming that the algorithm remains tractable for models with dozens of fields.

Beyond numerical stability, the method enables systematic exploration of the phenomenology associated with sharp turns. The authors compute curvature power spectra that exhibit localized features (peaks, oscillations) and quantify the transfer of power between adiabatic and isocurvature modes. Because the algorithm can handle arbitrary field‑space curvature and potential‑induced bends, it opens the door to comprehensive scans of multifield models that were previously inaccessible due to computational constraints.

In conclusion, the paper delivers a robust, efficient framework for evolving primordial scalar perturbations through sharply turning background trajectories in arbitrary field‑space geometries. By separating fast phase dynamics from slow amplitude dynamics within a parallel‑transport gauge, the authors achieve accuracy comparable to full‑mode integration while using timesteps similar to those of the background evolution. This advancement paves the way for high‑precision predictions of spectral features, non‑Gaussian signatures, and cross‑correlations in a broad class of multifield inflationary theories, aligning theoretical capabilities with the upcoming era of high‑resolution CMB and large‑scale‑structure observations.