Universal Nonminimal Coupling-to-Starobinsky Matching and a Single-Field Attractor

TL;DR

The paper develops a frame-independent, off-shell framework for 4D parity-even quadratic gravity by proving that Hubbard–Stratonovich lifts, auxiliary eliminations (including Palatini Γ), and Jordan–Einstein Weyl rescalings commute at the action level. It shows that generic heavy nonminimal couplings universally generate a positive $R^2$ sector, yielding an emergent scalaron with mass tied to the heavy-sector projection and a frame-universal matching. In the Einstein frame, a Weyl uplift and a Doob-based selection stabilize isocurvature modes, enforcing a single-field attractor with a calculable bound and sharp Starobinsky-like CMB targets (notably $n_s eq 1$ and $r eq 0$). A key phenomenological consequence is that a measured tensor-to-scalar ratio $r$ directly constrains the heavy-sector nonminimal couplings via $oldsymbol{oldsymbol{\nu}}$-type matching, providing a falsifiable link between UV completions and observable cosmology.

Abstract

We establish an off-shell commutativity theorem in 4D parity-even quadratic gravity that the Hubbard-Stratonovich/Legendre lifts, algebraic elimination of auxiliaries, including the torsionless Palatini connection, and Jordan-Einstein Weyl rescalings commute at the action level up to boundary terms. This yields a frame-independent characterization of the propagating degrees of freedom and isolates a universal scalaron EFT in the metric branch, while clarifying the algebraic nature of the Palatini $f(R)$ scalar. We obtain, as a result, a frame-universal matching from the generic nonminimal couplings to a positive $R^2$ sector and a quantitative single-field attractor bound, enhanced by a $1/ΔN$ selection term, providing sharp and falsifiable CMB targets.

Figures (3)

• Figure 1: The commuting diagram of off-shell, local maps.
• Figure 2: The single-field stability map. The solid blue curve marks the attractor threshold $m_{s,\rm eff}^2/H^2=1$. Regions above this curve are safe from isocurvature due to the combined effect of the Weyl uplift ($K_{\min}$) and Doob selection ($3/\Delta N$). Dashed lines show isocurvature suppression levels ($\mathcal{P}_S/\mathcal{P}_\zeta$). Panels correspond to different baseline stabilities ($c_* \equiv 3\,V_{;ss}/V$) of the Jordan-frame potential.
• Figure 3: Stability map for the Linear-NMC Starobinsky model. The vertical axis is the NMC strength $\mu$, and the horizontal axis is remaining $e$-folds $\Delta N$. Regions above the solid blue line ($m_{s,\rm eff}^2/H^2 = 1$) are dynamically stable single-field attractors. The dotted contours show isocurvature suppression levels.