Preheating and oscillon formation in Einstein-scalar-Gauss-Bonnet gravity

Technical Summary: Preheating and Oscillon Formation in Einstein-scalar-Gauss-Bonnet Gravity

Problem Statement
Non-perturbative processes in the early universe, specifically during the preheating phase following inflation, can generate overdense structures in scalar fields known as oscillons. These objects are cosmologically significant as potential seeds for primordial black holes (PBHs) and sources of stochastic gravitational waves. While preheating has been extensively studied within General Relativity (GR) and standard scalar-tensor theories, the impact of leading-order higher-derivative corrections—motivated by the expectation that GR is a low-energy limit of a UV-complete theory—remains less explored in fully non-perturbative regimes.

This work investigates whether Einstein-scalar-Gauss-Bonnet (EsGB) gravity, a specific class of Horndeski theories, significantly alters the formation and growth of oscillons. Crucially, the authors examine the limits of the Effective Field Theory (EFT) description during these high-curvature events. The central question is whether the local backreaction and non-linear dynamics of oscillon formation can drive the system out of the EFT's regime of validity, thereby rendering standard approximations (which often neglect local curvature feedback) insufficient.

Methodology
The authors perform numerical relativity (NR) simulations of the preheating phase using the Einstein-scalar-Gauss-Bonnet action. The setup follows the $\alpha$-attractor inflationary model with a potential $V(\phi) \propto (1 - e^{\phi/\mu})^2$, focusing on the specific case $\mu = 0.06 M_{Pl}$.

• Action and Coupling: The study considers the action $S = \int d^4x \sqrt{-g} [\frac{M_{Pl}^2}{2}R - V(\phi) + X + f(\phi)\mathcal{L}_{GB}]$, where $X$ is the kinetic term and $\mathcal{L}_{GB}$ is the Gauss-Bonnet invariant. The coupling function is chosen as $f(\phi) = \frac{\lambda}{2\beta}(1 - e^{-\beta\phi^2})$, designed to suppress modifications at the end of inflation ($\phi_{reh}$) while allowing them to become active during the oscillon formation phase. The coupling strength is parameterized by the dimensionless curvature scale $\tilde{L} \equiv \sqrt{|\lambda|}H_{reh}$.
• Initial Data: Simulations begin at the end of inflation with initial conditions consistent with Planck 2018 CMB observations. The scalar field includes sub-horizon Gaussian perturbations derived from quantum vacuum fluctuations. The initial data satisfies the Hamiltonian and momentum constraints using the Conformal Transverse Traceless (CTTK) method, ensuring compatibility with both GR and EsGB at $t=0$ (since $f'(\phi_{reh}) \approx 0$).
• Numerical Evolution: The system is evolved using the GRFolres code, which implements a well-posed initial value formulation for EsGB theory. The authors employ a modified puncture gauge and constraint-damping terms to ensure numerical stability. The evolution tracks the fully non-linear dynamics, including gravitational backreaction, without assuming a homogeneous background.
• Diagnostics: To assess the validity of the EFT and the properties of the resulting structures, the authors track:
  1. Density Contrast ($\delta$): To identify oscillon formation ($\delta \sim 1$).
  2. Compactness ($C$): Defined as $M/(8\pi R M_{Pl}^2)$, to assess proximity to black hole formation.
  3. Weak Coupling Condition (WCC): A non-covariant criterion requiring $\sqrt{|f'(\phi)|}/L \ll 1$, where $L$ is the minimum of relevant length scales (curvature, field gradients, potential). Violation indicates the EFT has left its regime of validity.
  4. Loss of Hyperbolicity: The breakdown of the Cauchy evolution is detected when the determinant of the effective metric associated with the quadratic cone becomes negative, signaling a loss of predictivity.

Key Contributions and Results
The study systematically varies the coupling strength $\tilde{L}$ to determine the threshold where EsGB modifications become dominant or destabilizing.

• Oscillon Properties: For couplings below a critical threshold ($\tilde{L} \lesssim 0.180$), the formation and properties of oscillons are not significantly modified compared to the GR case. The compactness of the formed oscillons remains low ($C \sim 10^{-3}$), consistent with previous findings that single-field models generally do not produce PBHs. The effective mass receives corrections of order $10^{-2}$, localized to the oscillon walls where field gradients are steep.
• EFT Breakdown: For couplings exceeding the critical value ($\tilde{L} \gtrsim 0.180$), the dynamics of oscillon formation drive the system out of the EFT's regime of validity. The breakdown is triggered by large curvature terms ($R_{ij}$) and second derivatives of the scalar field ($\nabla_\mu \nabla_\nu \phi$) within the densest regions of the oscillon.
• Loss of Predictivity: In the super-critical regime, the Weak Coupling Condition is violated, and the system experiences a loss of hyperbolicity (the effective metric determinant becomes negative). Consequently, the Cauchy evolution breaks down, and the simulation cannot predict the final state of the oscillon. This indicates that the derivative expansion of the EFT fails, and higher-order terms neglected in the action would become relevant.
• Stability: If oscillons survive the formation phase without triggering an EFT breakdown, they tend to be stable, and the EFT corrections remain bounded.

Significance and Claims
The paper claims that while modest EsGB couplings do not qualitatively alter the standard picture of oscillon formation or PBH production in single-field models, large couplings present a fundamental limitation on the predictive power of the theory during preheating.

The authors emphasize that the "large coupling" regime is defined relative to the local curvature scales of the forming oscillons, not the background Hubble scale. This distinction is critical: approximations that neglect local backreaction and non-linear dynamics may miss the onset of EFT breakdown. The results suggest that cosmological observations of the early universe (specifically preheating dynamics) could provide constraints on higher-curvature corrections in gravitational EFTs that are more stringent than those derived from compact objects (like black holes or neutron stars), provided the dynamics are well understood. However, the authors caution that constraints should not be naively extrapolated across regimes, as bounds depend on characteristic frequencies and background configurations.

In summary, the work demonstrates that the non-linear dynamics of preheating can act as a probe for the validity of gravitational EFTs, revealing that for sufficiently strong couplings, the theory itself loses predictivity before the formation of stable compact objects or black holes can be fully resolved.