Formation and Decay of Oscillons in Einstein-Cartan Higgs Inflation

Abstract

We review recent progress in the understanding of the preheating stage of Higgs inflation formulated within the Einstein-Cartan framework of gravity. This setup smoothly interpolates between the metric and Palatini formulations of the theory, leading to a distinctive phenomenology in an intermediate regime. Following the end of inflation, the Higgs field undergoes a non-trivial out-of-equilibrium evolution driven by tachyonic instabilities and nonlinear self-interactions, which fragment the inflaton condensate and give rise to well-localized oscillon configurations. While early studies suggested the formation of long-lived oscillons and the possibility of an extended matter-dominated phase, more recent analyses show that self-interactions at small field values render these objects transient, eventually triggering their decay and the onset of radiation domination. We discuss the implications of this dynamics for the thermal history of the Universe, the inflationary observables, and the generation of stochastic gravitational waves.

Figures (13)

• Figure 1: Schematic map of metric--affine formulations of gravity in terms of non-metricity ($\nabla g\neq 0$), torsion ($T\neq 0$) and curvature ($R\neq 0$). GR sits at the intersection of vanishing torsion and non-metricity, while Palatini and EC correspond to different restrictions on $(\nabla g,\,T,\,R)$. Teleparallel and symmetric-teleparallel descriptions trade curvature for torsion or non-metricity, respectively.
• Figure 2: Geometric intuition for the emergence of an approximate shift symmetry in suitable canonical variables. A non-linear field redefinition can turn a scale-symmetric structure in the original variables into an approximately shift-symmetric regime in the canonically normalised field. In Higgs-inflation-like models this mechanism underlies the flattening of the Einstein-frame potential at large field values and the associated attractor behaviour of inflationary observables Rubio:2018ogq.
• Figure 3: Representative shape of the Einstein-frame potential across field ranges relevant for post-inflationary dynamics, with $\lambda = 0.001$, $\xi = 5 \cdot 10^4$, and $c = 1.21 \cdot 10^7$. At small amplitudes the potential is effectively quartic, at intermediate amplitudes it is approximately quadratic, and at large amplitudes it approaches a plateau. The relative extent of the intermediate quasi-quadratic regime depends on the underlying gravitational formulation, and plays a key role in determining whether the system efficiently fragments and forms oscillons during preheating. The arrows indicate that this regime becomes narrower when moving from metric Higgs inflation (MHI) to Palatini Higgs inflation (PHI).
• Figure 4: Qualitative comparison of the inflationary and post-inflationary behaviours across the metric and Palatini limits and the intermediate regime for a scenario with $c=\xi+6\xi_\eta^2$. The colour scale indicates the tensor-to-scalar ratio $r$ in the displayed region. The different damping rates of the homogeneous inflaton oscillations and the typical depth of plateau incursions control the strength and repetition of tachyonic amplification episodes. In the intermediate region the inflaton condensate can fragment efficiently, providing favourable conditions for oscillon formation before backreaction and eventual decay drive the system towards radiation domination. Adapted from Shaposhnikov:2020gts.
• Figure 5: Three-dimensional visualisation of the inhomogeneous field configuration in lattice simulations at representative times, with $\lambda = 0.001$, $\xi = 5 \cdot 10^4$, and $c = 1.21 \cdot 10^7$. Isosurfaces highlight compact, high-amplitude regions identified as oscillons. As the system evolves, the number and typical size of these objects change due to expansion and gradual radiative losses, ultimately leading to their dissolution and a redistribution of energy into relativistic modes. Adapted from Piani:2025dpy.