Non-Local Gravitational Actions

• Non-local gravitational actions are theories where curvature couplings between distinct spacetime points are mediated by integral kernels and nonlocal operators.
• These models are localized using auxiliary fields, ensuring finite differential orders and preserving causal, diffeomorphism-invariant dynamics.
• They offer innovative approaches to cosmological challenges, including dark energy, dark matter simulation, and modifications to gravitational wave phenomena.

A non-local gravitational action introduces causal, diffeomorphism-invariant curvature couplings between spacetime points separated by non-zero geodesic distance, mediated by Green functions or integral kernels involving nonlocal operators such as the covariant inverse d’Alembertian or generalized pseudodifferential operators. Non-locality in gravitational actions arises both as a direct modification of the classical gravitational Lagrangian to encode IR/UV behavior, and as an inevitable structural feature of the quantum effective action (generating functional of 1PI diagrams) of quantum gravity, particularly beyond the perturbative regime. These theories admit classical and quantum versions and are realized in Riemannian, teleparallel, and metric–affine formulations. Their principal motivations include regularization/renormalization in the UV, infrared completion (cosmological constant problem, degravitation), and the simulation of dark-sector phenomena such as cosmic acceleration and dark matter.

1. Structural Forms of Non-Local Gravitational Actions

The generic non-local action takes the form

$$S[g] = \int d^4x\,\sqrt{|g|}\left[ L_{local}(g_{\mu\nu}, R_{\mu\nu\rho\sigma}, \ldots) + L_{nonlocal}(g,\Box^{-1},\ldots) \right]$$

where $L_{nonlocal}$ is built from curvature invariants and nonlocal operators (inverse d’Alembertian, nonpolynomial functions, or more general pseudodifferential structures). Notable subclasses include:

• Quadratic curvature non-local models: $R\,\Box^{-1}R$, $R_{\mu\nu}\Box^{-1}R^{\mu\nu}$, and their higher-order analogs ($R\Box^{-n}R$), often motivated by quantum effective action expansions and trace anomaly considerations (Donoghue, 2022, Kumar et al., 2018, Teimouri, 2017).
• General analytic functions: Extensions to $R\,f(\Box^{-1}R)$, $f(R,\Box^{-1}R)$, and similar functionals for the Gauss–Bonnet ($\mathcal{G}$) and torsion scalars ($T$), as in $f(\mathcal{G},\Box^{-1}\mathcal{G})$, $L_{nonlocal}$0 (Capozziello et al., 2022).
• Teleparallel nonlocality: Replacement of the local Weitzenböck torsion–superpotential constitutive relation by a nonlocal functional integral involving a causal kernel $L_{nonlocal}$1 over the past history of the torsion tensor (Mashhoon, 2011, Mashhoon, 2014, Mashhoon, 2022).

These nonlocal structures are encoded either directly in the action or are representable (often uniquely) in terms of auxiliary fields, thereby ensuring field equations remain of finite differential order after localization, at the cost of introducing new constraints.

2. Localization and Auxiliary Fields

Non-local actions with inverse derivatives are systematically “localized” by introducing auxiliary (scalar, vector, tensor) fields as dynamical variables, together with Lagrange multipliers enforcing the original nonlocal relations. For example, for

$L_{nonlocal}$2

auxiliary fields $L_{nonlocal}$3 and a multiplier $L_{nonlocal}$4 localize the action: $L_{nonlocal}$5 Variations w.r.t.\ auxiliary fields then enforce the nonlocal operator constraints (Kumar et al., 2018, Capozziello et al., 2022, Teimouri, 2017).

For higher nonlocality ($L_{nonlocal}$6), a tower of auxiliary fields $L_{nonlocal}$7 is introduced to localize the hierarchy of inverse d'Alembertian operators (Teimouri, 2017). In teleparallel models, the nonlocal constitutive relation is constructed by integrating the curvature/contorsion functions over spacetime with a causal kernel (Mashhoon, 2011, Mashhoon, 2022).

3. Field Equations, Causality, and Physical Degrees of Freedom

Variation of the localized action produces modified Einstein equations augmented by nonlocal corrections and additional constraint equations from the multipliers. The definition of the nonlocal operators—especially retarded/advanced boundary conditions enforced via specific Green functions—ensures causal (Schwinger–Keldysh/in-in) evolution in Lorentzian signature (Barvinsky, 2011). The use of retarded operators ensures that the equations at spacetime point $L_{nonlocal}$8 depend only on the past causal domain of $L_{nonlocal}$9.

Physical degrees of freedom are counted after taking into account the constraints from auxiliary fields and the boundary data for the Green functions. In the RR and related models, extra fields implementing nonlocality do not correspond to freely specifiable initial data or radiative quantum modes; their behavior is uniquely determined (modulo irrelevant attractors) by the history of the metric (Belgacem et al., 2017, Kumar et al., 2018).

In teleparallel and metric-affine nonlocal models, local Lorentz and diffeomorphism invariance are preserved, and the principle of equivalence holds since geodesics of the metric remain unchanged (Mashhoon, 2011, Mashhoon, 2014).

4. Flat, (A)dS Limits, and Infrared/Ultraviolet Behavior

The ultraviolet limit (flat space, small curvature) of well-constructed nonlocal actions typically reproduces the Einstein–Hilbert action with modified effective Planck mass and suppressed nonlocal terms. For example, in the model (Barvinsky, 2011),

$R\,\Box^{-1}R$0

with Planck mass $R\,\Box^{-1}R$1. Nonlocal terms become subleading, ensuring standard GR phenomenology in the short-distance regime.

In the infrared, nonlocal models can admit (anti-)de Sitter vacuum solutions, with the existence, degeneracy, and stability of vacua controlled by curvature-operator couplings and parameter constraints. The backgrounds are selected so that the (A)dS spacetime remains a solution to the modified equations, with the linearized spectrum containing only the two polarizations of the graviton (massless, ghost-free) (Barvinsky, 2011, Kumar et al., 2018). However, in some teleparallel nonlocal models, de Sitter spacetime may be excluded as a solution depending on the structure of the nonlocal kernel (Mashhoon, 2022).

The on-shell action vanishing for both vacua underlines the possibility of degenerate phases underpinning UV and IR behavior (Barvinsky, 2011).

5. Phenomenology: Cosmology, Astrophysics, and Observational Probes

Nonlocal modifications are leveraged primarily to address cosmological puzzles such as late-time acceleration (dark energy), dark matter simulation, and high-precision phenomenology in the strong and weak field regimes.

• Dark energy and cosmic acceleration: Nonlocal terms naturally provide effective dark energy, often driving a phantom or quasi-de Sitter equation of state without explicit cosmological constant (Belgacem et al., 2017, Capozziello et al., 2022, Modesto et al., 2013). The RR model ($R\,\Box^{-1}R$2) produces accelerated expansion with a phantom equation of state, fits CMB, BAO, SNe, Hubble, and structure formation data competitively with ΛCDM, and predicts a particular value for the sum of neutrino masses (Belgacem et al., 2017).
• Dark matter simulation: In teleparallel nonlocal gravity, the modified Poisson equation is

$R\,\Box^{-1}R$3

where $R\,\Box^{-1}R$4 introduces a logarithmic correction to the Newtonian potential, reproducing flat galaxy rotation curves on kpc scales without invoking elementary dark matter (Mashhoon, 2011, Mashhoon, 2014). The nonlocal length scale ℓ is determined empirically to be ℓ ~ kpc.

• Gravitational waves: Nonlocal modifications to the Einstein–Hilbert action alter the gravitational wave stress-energy pseudo-tensor, introducing new scalar breathing modes and modifying the quadrupole formula for binary inspirals (Capozziello et al., 2024, Capozziello et al., 2023). For $R\,\Box^{-1}R$5 corrections, the new breathing mode is strictly massless and undetectable except for fine-tuned couplings; the quadrupole luminosity receives dimensionless corrections O(a).
• Wormholes and exotic geometry: Nonlocal actions $R\,\Box^{-1}R$6 can furnish traversable, stable wormhole solutions without requiring any exotic (NEC-violating) matter; the nonlocal sector supplies the negative effective energy needed at the throat (Capozziello et al., 2022).
• Astrophysical tests: Corrections to the Newtonian potential, orbital precession, and periastron advance can be tested in systems such as the Galactic-center S2 star (Capozziello et al., 2022), constraining nonlocal model parameters on AU-parsec scales.

6. Quantum Effective Actions, Anomalies, and Trace Structure

Non-local gravitational corrections arise generically as finite, covariant terms in the quantum effective action, particularly from integrating out heavy or massless particles at one loop. A prototypical example is the “non-local partner” to the cosmological constant: $R\,\Box^{-1}R$7 which arises from one-loop diagrams for massive fields (Donoghue, 2022). This term is nonlocal, finite, fixed, and cannot be removed by local counterterms; its scale dependence naturally leads to decoupling at low energies, but can be dominant at and above the threshold scale.

In the effective action for massless fields, nonlocal logarithmic form-factors ($R\,\Box^{-1}R$8), $R\,\Box^{-1}R$9, and mixed $R_{\mu\nu}\Box^{-1}R^{\mu\nu}$0 terms exactly reproduce the QED trace anomaly and more general quantum-corrected Weyl transformations, confirming the central role of nonlocal actions in quantum field theory in curved spacetime (Donoghue et al., 2015).

7. Conservation Laws, Energy-Momentum Complex, and Noether Structure

Despite the nonlocality, both scalar-tensor and teleparallel nonlocal gravity models admit a locally conserved energy-momentum complex, constructed via Noether’s theorem for diffeomorphism or translation invariance of the action. The gravitational energy-momentum pseudotensor receives nonlocal corrections but the combined total (matter plus gravity) is locally conserved, supported by generalized contracted Bianchi identities even when nonlocal modifications are present (Capozziello et al., 2023).

The nonlocal corrections encode the “memory” of the full curvature history and lead to new radiative channels (scalar monopole/dipole GW emission), as well as new phenomenologies in dynamical gravitational systems. The weak-field expansion of the pseudotensor allows for direct confrontation of nonlocal gravitational predictions with high-precision observations, including gravitational wave luminosity and orbital dynamics.

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Major developments in the construction, interpretation, and application of nonlocal gravitational actions have established their technical consistency, rich phenomenology, and capability to address deep problems in gravitational physics and cosmology (Barvinsky, 2011, Belgacem et al., 2017, Mashhoon, 2014, Mashhoon, 2011, Donoghue, 2022, Kumar et al., 2018, Capozziello et al., 2022, Capozziello et al., 2023, Capozziello et al., 2024, Capozziello et al., 2022). Future research is expected to further constrain these models via cosmological, astrophysical, and gravitational wave observables, as well as to systematically develop their ultraviolet completions.