Dual Effective Field Theory formulation of Metric--Affine and Symmetric Teleparallel Gravity

TL;DR

The paper introduces a unified effective field theory (EFT) framework for non-Riemannian gravities with independent connections, encompassing metric--affine $f(R,Q)$, EiBI, and symmetric teleparallel $f(Q)$ models. By exploiting exact matrix solutions and convergent matrix-binomial expansions, it rewrites Einstein-like equations in terms of an auxiliary metric $h_{\mu\nu}$ and a local correction $(\Delta T)_{\mu\nu}$ built from stress-tensor invariants $\{T, T^2, T_{\mu\nu}T^{\mu\nu},\ldots\}$, organized by a characteristic density $\rho_\star$. The work extends this dual EFT to EiBI as a determinantal resummation of the same $T$-tower, and to $f(Q)$ gravity through a constitutive map relating non-metricity to matter, with background matching linking EFT coefficients to quadratic non-metricity corrections. It provides explicit examples for perfect fluids, electromagnetic fields, and compact stars, illustrating how the leading corrections modify $\rho_{\rm eff}$, $p_{\rm eff}$ and the Friedmann or TOV equations. Collectively, the results yield a common algebraic language for curvature-based Palatini theories, EiBI, and non-metricity gravities, offering a practical, perturbative path toward a unified framework for non-Riemannian gravity.

Abstract

We develop a unified algebraic and effective field theory (EFT) formulation for non--Riemannian extensions of General Relativity with an independent connection. For metric--affine $f(R,Q)$ gravity we show that the connection equations admit an exact matrix solution, whose square--root structure generates a convergent binomial/Neumann expansion in powers of the stress tensor $T_{μν}$. For the Eddington--inspired Born--Infeld (EiBI) theory we show that the connection can be solved algebraically as well, and that its determinantal field equations produce a parallel Neumann expansion with coefficients fixed by the underlying determinant operator. This allows us to rewrite the Einstein--like equations in the auxiliary metric as an effective Einstein equation for $g_{μν}$ with a local algebraic correction $(ΔT)_{μν}$ that follows from a dual EFT built from the invariants $\{T,\,T^2,\,T_{μν}T^{μν},\ldots\}$, organised by a characteristic density scale. We prove a convergence criterion based on the spectral radius of $\hat T^μ_ν$ and interpret EiBI gravity as a determinantal resummation of the same $T$--tower. Extending the framework to symmetric teleparallel $f(Q)$ gravity, we identify the EFT coefficients in terms of $f_Q$ and $f_{QQ}$ and present a background matching for $f(Q)=Q+αQ^2$. The resulting dual EFT provides a common algebraic language for metric--affine, Born--Infeld and non--metricity gravities.

Figures (3)

• Figure 1: Structure of the dual EFT expansion. The analytic map $\Sigma(T)$ induces a local effective action $S_{\rm eff}[g,T]=S_{\rm GR}+\sum_{n\ge1} a_n\,T^n/\rho_\star^{\,n-1}$. For perfect-fluid sources, the linear term ($\propto T$) governs leading departures from GR, while trace-free sectors (e.g. electromagnetism) start at quadratic order ($\propto T^2$). Quadratic Palatini models $f(R,Q)$ effectively populate the first two orders, whereas Eddington-inspired Born--Infeld corresponds to a resummation to all orders in $T$. Convergence holds in the controlled regime $\rho(T)/\rho_\star\ll1$ (see App. \ref{['app:rigorous-convergence']}).
• Figure 2: Cosmological application (schematic). Effective Hubble rate versus normalized density $x=\rho/\rho_\star$. General Relativity gives $H^2/H_\star^2=x$ (black). The dual EFT truncation up to $T^2$ (blue, dashed) is shown with a representative coefficient $a_2$. A Born--Infeld--like resummation (orange) captures non-linear saturation and a bounce at $x=x_b$. Parameters $(a_2,x_b)$ are illustrative for visualization; the formalism fixes their mapping to $(f_R,f_Q;\alpha,\beta)$.
• Figure 3: Schematic flow of the dual EFT construction. Starting from a matter source $T_{\mu\nu}$, the algebraic matrix map $\Sigma$ encodes the nonlinear gravitational response of the connection. From $\Sigma$ one obtains the auxiliary metric $h_{\mu\nu}$ satisfying Einstein--like equations. The relation between $h_{\mu\nu}$ and the physical metric $g_{\mu\nu}$ defines an effective local action $S_{\rm eff}[g,T]$ expanded in powers of $T_{\mu\nu}/\rho_\star$. This unified algebraic framework applies to cosmological fluids, stellar interiors, and electromagnetic fields.

Theorems & Definitions (12)

• Lemma 3.1: Absolute convergence of the matrix binomial series
• proof : Sketch of proof
• Theorem 3.2: Convergence for the Palatini map
• proof
• Corollary 3.3: Practical criterion for fluid sources
• Lemma C.1: Power--series functional calculus
• proof
• Theorem C.2: Convergence and identification with the principal square root
• proof
• Proposition C.3: Absolute convergence and remainder bound