Non-Local Metric-Affine Gravity

• Non-Local Metric-Affine Theories are gravitational models where the metric and affine connection are independent, and non-local operators act on curvature, torsion, and non-metricity.
• They address both ultraviolet and infrared issues of General Relativity by smearing interactions and modifying the traditional gravitational particle spectrum.
• These theories enable novel cosmological dynamics and the potential avoidance of singularities by dynamically activating torsional and non-metric degrees of freedom.

Non-Local Metric-Affine Theories form a broad class of gravitational models in which the metric and affine connection are promoted to independent dynamical variables and the action functional incorporates non-local operators such as the inverse d'Alembertian $\Box^{-1}$ or entire analytic form factors in $\Box$. This approach generalizes the standard framework of local metric-affine gravity by introducing extended, non-local interactions among curvature invariants, torsion, and non-metricity, often motivated by the need to provide ultraviolet (UV) completions, suppress classical singularities, or generate novel infrared (IR) cosmological dynamics. Recent developments have produced both non-local metric-affine versions of well-known models (Capozziello et al., 2 Oct 2025, Golovnev et al., 2015) and novel classes of non-local metric-affine theories without Riemannian analogues, yielding dynamics unattainable in purely metric or Palatini gravity.

1. Foundations of Non-Local Metric-Affine Gravity

In the traditional metric-affine framework, the action $S$ is constructed as a diffeomorphism-invariant integral over a scalar density built from the metric $g_{\mu\nu}$ and an independent affine connection $\Gamma^\lambda_{\ \mu\nu}$. The most general local action at lowest order includes the Ricci scalar $R$ (built with $\Gamma$), quadratic torsion $S^\lambda_{\ [\mu\nu]}$, and non-metricity $Q_{\lambda\mu\nu} = -\nabla_\lambda g_{\mu\nu}$ invariants (Vitagliano et al., 2010, Vitagliano, 2013). At this level, whenever the action contains only terms with at most two derivatives, the independent connection is generically non-propagating (auxiliary) and can be eliminated algebraically, reducing the theory to General Relativity (GR) plus non-minimal matter couplings.

Non-local generalizations modify the gravitational action with operators that act over extended spacetime regions. A representative action (in $n$ dimensions) has the schematic form

$$S = \int d^n x\, \sqrt{-g} \left[ R + R\, F(\Box) R + \cdots \right]$$

where $F(\Box)$ is an analytic function of the d'Alembertian, typically expanded as a power series,

$$F(\Box) = \sum_{k=0}^\infty f_k \Box^k, \quad \Box \equiv g^{\mu\nu} \nabla_\mu \nabla_\nu.$$

Alternatively, IR modifications may use $F(\Box^{-1})$. In the metric-affine framework, non-locality is extended beyond curvature scalars, incorporating non-local operators acting on torsion and non-metricity invariants, e.g., $S_\mu \Box S^\mu$ or $Q_{\lambda\mu\nu} F(\Box) Q^{\lambda\mu\nu}$ (Capozziello et al., 2 Oct 2025).

This construction can address both high-energy (UV) and large-scale (IR) limitations of GR, such as ghost instabilities, cosmological singularities, and cosmic acceleration, by smearing interactions and degrees of freedom across spacetime (Capozziello et al., 2 Oct 2025, Golovnev et al., 2015, Olmo et al., 2015).

2. Structure and Dynamics of Non-Local Metric-Affine Models

The key technical innovation in non-local metric-affine gravity lies in the independent variation of action functionals containing non-local operators with respect to both $g_{\mu\nu}$ and $\Gamma^\lambda_{\ \mu\nu}$ (Capozziello et al., 2 Oct 2025, Golovnev et al., 2015). For example, in models with a Lagrangian $R + R F(\Box) R$, the field equations for the connection can often be solved exactly or order-by-order:

• The Palatini tensor $P_\lambda^{\ \mu\nu}$, encoding the $\Gamma$-variation, leads to equations involving derivatives of the scalar field $\phi = 1 + 2 F(\Box) R$. This yields algebraic relations for the distortion tensor $N^\lambda_{~\mu\nu}$, which can be substituted back into the metric field equations.
• The result is that the non-local metric-affine theory is generally on-shell equivalent to a higher-derivative (or non-local) scalar-tensor theory, with the order of derivatives often reduced compared to its metric counterpart due to the structure of the connection field equations—e.g., reducing from sixth- to fourth-order in derivatives (Capozziello et al., 2 Oct 2025).

A crucial feature is that in the presence of non-local kinetic terms for torsion and non-metricity (such as $S_\mu \Box S^\mu$), these geometric fields can become genuinely dynamical even in vacuum, as opposed to remaining auxiliary (Capozziello et al., 2 Oct 2025, Vitagliano, 2013, Vitagliano et al., 2010). This creates new particle degrees of freedom and possible new dispersion relations, altering the spectrum of metric-affine gravity.

Dynamics: Key Equations

A typical dynamical system for a non-local metric-affine gravity model reads as follows: $$\begin{aligned} &\phi = 1 + 2\,F(\Box) R \ &R_{(\mu\nu)} - \frac{1}{2} R g_{\mu\nu} + \text{non-local corrections} = 0 \ &\phi\, P_\lambda^{\ \mu\nu} + \delta^\nu_\lambda\, \partial^\mu \phi - g^{\mu\nu}\, \partial_\lambda \phi = 0 \end{aligned}$$ where $P$ is the Palatini tensor; solutions for $N^\lambda_{\ \mu\nu}$ (distortion) or other non-Riemannian geometrical fields follow (Capozziello et al., 2 Oct 2025). For torsion sector, terms like $S_\mu \Box S^\mu$ generate propagating equations for $S_\mu$; their algebraic solution requires solving nonlocal, possibly coupled, field equations.

3. Connections to Local and Non-Local Unified Approaches

Non-local metric-affine theories connect to several other established frameworks:

• Local Limit: In the absence of non-local operators, the algebraic elimination of the connection reduces the theory to GR or to scalar-tensor gravity, with no propagating non-metric or torsional modes at second order (Vitagliano et al., 2010, Vitagliano, 2013).
• Pure Metric Case: For actions depending solely on $g_{\mu\nu}$, the Ricci scalar $R$ and its curvature invariants, non-local modifications reproduce familiar higher-derivative models with ghost issues generally resolved by careful analytic structure for $F(\Box)$ (Golovnev et al., 2015).
• Emergent Non-locality: The C-theory approach explicitly demonstrates that even local metric-affine models, when rewritten in a single-metric formulation, yield non-local actions involving inverse-d'Alembertian operators, with non-locality interpreted as an emergent property of the underlying connection–metric structure (Golovnev et al., 2015).
• No Riemannian Analogue: When non-local operators act on torsion or non-metricity invariants, truly novel models arise with no reduction to Riemannian (or even Palatini) sectors. The dynamics and spectrum of such models can depart radically from the standard theories (Capozziello et al., 2 Oct 2025).

4. Physical Implications and Applications

Spectrum and Propagation

Non-local metric-affine models generically modify the gravitational particle spectrum. Analysis of the weak-field regime using spin-parity projection operators allows a classification of propagating states, identification of massless and massive modes (e.g., graviton, spin-1 massive vector, or other higher-spin fields), and determination of whether new degrees of freedom are ghost-free and tachyon-free (Mikura et al., 18 Jan 2024, Barker et al., 12 Feb 2024). The inclusion of non-local operators, when combined with appropriate parameter tuning, can preserve unitarity and avoid the proliferation of unphysical states.

Cosmological Scenarios

Applications to Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology demonstrate that non-local metric-affine corrections can drive superexponential or inflationary expansion, with effective Friedmann equations of the form

$$3H^2 = -\frac{1}{4}\left(\frac{\dot\phi}{\phi}\right)^2 - 3H \frac{\dot\phi}{\phi} + \cdots$$

where $\phi$ incorporates non-local scalar dynamics (Capozziello et al., 2 Oct 2025). Theories with torsion or non-metricity kinetic terms yield further modifications, potentially contributing to both early-universe inflation and late-universe acceleration. Importantly, in the large-time limit, the non-local corrections typically decay and standard GR cosmology is restored.

Novel Models and No Riemannian Counterpart

Non-local terms involving torsion and non-metricity invariants, e.g.,

$$S \supset \int d^n x\, \sqrt{-g}\, [\alpha\, S_\mu S^\mu + \beta\, S_\mu \Box S^\mu],$$

can make torsion (and analogously, non-metricity) dynamical even in vacuum—a behavior impossible to realize in purely metric gravity. These models possess unique scalar, vector, or tensor modes whose cosmological and astrophysical signatures must be directly investigated within the metric-affine framework (Capozziello et al., 2 Oct 2025).

5. Theoretical Constraints and Ghost-Free Structure

The construction of viable non-local metric-affine gravity theories necessitates careful tuning of the analytic structure of non-local operators to avoid tachyons and ghosts. The analysis of the full quadratic action in momentum space, with invertible kinetic operators and non-trivial projectors for each spin-parity sector, provides systematic constraints on allowed parameter space (Mikura et al., 18 Jan 2024, Barker et al., 12 Feb 2024). For example, the avoidance of massive spin-2 ghosts and the controlled propagation of only a healthy vector or scalar mode imply relationships among the coefficients of quadratic and non-local invariants; only a narrow subset of parameter space yields healthy, unitary models.

Transformations and equivalence between different formulations—single-metric, connection-compatible-metric, or field-redefinition frames—are subtle in the presence of non-local operators; equivalence at the level of equations of motion requires careful handling of nonlocal field redefinitions and surface terms, especially under varying boundary conditions (Golovnev et al., 2015).

6. Open Problems and Future Perspectives

• Stability and Health: While most local metric-affine models are free of ghosts at second order, the introduction of non-local operators, especially in the torsion and non-metricity sectors, demands further analysis of stability and the absence of Ostrogradsky instabilities.
• Phenomenology and Observational Tests: Quantitative predictions, such as gravitational wave propagation, cosmic microwave background signatures, or the evasion of singularities, remain to be fully developed for these theories in realistic settings.
• Comparison with Riemannian and Teleparallel Extensions: The existence of geometrically distinct dynamical modes (e.g., kinetic torsion in teleparallel or symmetric teleparallel analogues) motivates further unification and classification of all possible metric-affine, non-local completions (Iosifidis et al., 2018).
• Non-Locality as Emergent or Fundamental: Whether the observed non-local modifications are fundamental, emergent from the "integration out" of non-Riemannian degrees of freedom, or required by consistency with quantum gravitational effects remains an open conceptual question (Golovnev et al., 2015).

7. Summary Table of Model Features

Model Type	Non-local Operator	Extra D.o.F.	Riemannian Limit
$R + R\,F(\Box)R$ (metric-affine)	$F(\Box)$ on $R$	Scalar (ghost-free)	$f(R)$ scalar-tensor
Torsion kinetic $S_\mu \Box S^\mu$	$F(\Box)$ on $S_\mu$	Dynamical torsion	Absent
Non-metricity kinetic $Q \Box Q$	$F(\Box)$ on $Q$	Dynamical nonmetricity	Absent
C-theory non-local metric-affine	$F(\Box)$ via field redefinition	Emergent	Riemannian for $D=0$

In all models, the independent affine connection is essential for the appearance and propagation of non-metric degrees of freedom. Novel non-local structure can be consistently incorporated in metric-affine gravity, yielding healthy, ghost-free propagation provided the analytic structure of non-locality is judiciously selected, and opening up qualitatively new gravitational dynamics absent from Riemannian approaches (Capozziello et al., 2 Oct 2025, Golovnev et al., 2015, Mikura et al., 18 Jan 2024, Barker et al., 12 Feb 2024).