Non-Local Palatini Models in Modified Gravity

• Non-local Palatini models are gravitational theories that incorporate analytic non-local curvature functions by treating metric and connection as independent variables.
• They derive modified field equations with complex form-factor structures that aim to maintain ghost stability while preserving classic vacuum solutions under certain conditions.
• These models impact inflationary dynamics by enabling multi-field kinetically coupled regimes similar to Starobinsky inflation, though singularity resolution remains an open challenge.

Non-local Palatini models constitute a class of gravitational theories in which infinite-derivative, non-local modifications to gravity are formulated in the Palatini approach: the metric and affine connection are treated as independent dynamical variables. These models generalize the standard Einstein–Hilbert action by promoting curvature invariants to analytic functions of the covariant d'Alembertian operator, leading to dynamics non-local in spacetime. Key research themes include the explicit derivation of dynamical equations, field-space structure in equivalence frames, ghost stability, and the implications for singularity resolution and cosmological dynamics, especially inflation (Briscese et al., 2015, Bombacigno et al., 2024).

1. Gravitational Action and Non-local Extensions

The cornerstone of non-local Palatini gravity is an action of the form: $$S_{\rm grav}[g,\Gamma] = S_{\rm EH} + S_{\rm Scalar} + S_{\rm Ricci} + S_{\rm Riemann}$$ with independent metric $g_{\mu\nu}$ and symmetric affine connection $\Gamma^{\alpha}_{\;\;\mu\nu}$. The terms are:

• Einstein–Hilbert term:

$$S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)$$

• Analytic non-local curvature extensions:

$$S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R$$

$$S_{\rm Ricci} = \int d^4x\,\sqrt{-g}\; R_{\mu\nu}\,h_2(-\Box/\Lambda^2)\,R^{\mu\nu}$$

$$S_{\rm Riemann} = \int d^4x\,\sqrt{-g}\; R_{\mu\nu\rho\sigma}\,h_3(-\Box/\Lambda^2)\,R^{\mu\nu\rho\sigma}$$

where each $h_i(z)$ is analytic around $z=0$ and expands into an infinite sum of derivatives, making the model genuinely non-local. The d'Alembertian $\Box$ is constructed using the affine connection, distinct from purely metric non-local theories (Briscese et al., 2015).

2. Palatini Variational Principle and Field Equations

Variation of the total action proceeds independently in $g_{\mu\nu}$0 and $g_{\mu\nu}$1:

• The metric variation gives modified Einstein equations with an effective non-local stress-energy tensor $g_{\mu\nu}$2, incorporating the analytic form-factors and their derivative structure.
• The connection variation yields a generalized compatibility condition:

$g_{\mu\nu}$3

where $g_{\mu\nu}$4 encodes the non-trivial non-local dependence from the series expansions of the $g_{\mu\nu}$5 (Briscese et al., 2015).

Unlike the Einstein–Palatini system, the connection equation cannot, in generic non-local cases, be rewritten as compatibility with a conformally rescaled metric, due to the presence of double sums over non-local terms.

3. Vacuum Solutions, Singularities, and the Role of Form-factors

In the regime where the Riemann tensor contributions are absent ($g_{\mu\nu}$6), all classical vacuum solutions of Einstein gravity with (A)dS asymptotics remain exact solutions of the full non-local Palatini system. Explicitly, for vacuum with vanishing cosmological constant and Levi-Civita connection, $g_{\mu\nu}$7, all non-local corrections vanish identically. Thus, the Schwarzschild, Kerr, and their (A)dS generalizations survive unmodified (Briscese et al., 2015).

The crucial implication is that non-locality in this formulation fails to regularize black-hole singularities or similar curvature divergences: $g_{\mu\nu}$8 still diverges at $g_{\mu\nu}$9 in the Schwarzschild metric. Even inclusion of the $\Gamma^{\alpha}_{\;\;\mu\nu}$0 (Riemann$\Gamma^{\alpha}_{\;\;\mu\nu}$1) term introduces a far more non-trivial connection equation, and no universal argument guarantees singularity regularization. In contrast, certain purely metric non-local models with suitable exponential form-factors can smooth singularities, but these results do not directly transfer to the Palatini formalism (Briscese et al., 2015).

4. Hybrid Metric–Palatini Non-locality and Ghost Structure

Recent developments incorporate non-locality through the inverse d'Alembertian acting on scalars such as $\Gamma^{\alpha}_{\;\;\mu\nu}$2, particularly in hybrid metric–Palatini frameworks. An example is the action: $\Gamma^{\alpha}_{\;\;\mu\nu}$3 Auxiliary fields ($\Gamma^{\alpha}_{\;\;\mu\nu}$4) and Lagrange multipliers localize the action, yielding a two-field scalar–tensor theory in the Jordan frame. Conformal transformation to the Einstein frame exposes a non-trivial kinetic structure: $\Gamma^{\alpha}_{\;\;\mu\nu}$5 where $\Gamma^{\alpha}_{\;\;\mu\nu}$6 and $\Gamma^{\alpha}_{\;\;\mu\nu}$7 parameterize the fields. In purely metric $\Gamma^{\alpha}_{\;\;\mu\nu}$8-models, the kinetic matrix is degenerate ($\Gamma^{\alpha}_{\;\;\mu\nu}$9), signaling the presence of a ghost, i.e., no choice of $$S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)$$0 ensures complete ghost-freedom unless the action is degenerate or additional couplings are engineered. Hybrid metric–Palatini actions allow the possibility of coupling non-local terms to distinct curvature invariants, such that the resulting multi-field Einstein-frame action removes the ghost via non-degeneracy of $$S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)$$1 (Bombacigno et al., 2024). This construction is essential for physical viability.

5. Inflationary Dynamics in Non-local Palatini and Hybrid Models

In the Einstein frame, the dynamics of the multi-field system is governed by: $$S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)$$2 Inflationary solutions are studied in a Friedmann–Robertson–Walker (FRW) background. The non-minimal kinetic mixing term $$S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)$$3 implies an initial kick to the spectator-like field $$S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)$$4, generating a turn in field-space trajectory. Subsequently, Hubble damping freezes $$S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)$$5, and inflation proceeds via canonical slow-roll of the inflaton $$S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)$$6 down a plateau potential, formally equivalent to Starobinsky inflation plus an inert spectator field. Calculations show that key cosmological observables, such as scalar tilt $$S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)$$7 and tensor ratio $$S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)$$8, align with plateau inflation predictions, with only sub-leading isocurvature corrections from the spectator (Bombacigno et al., 2024).

6. Open Problems: Singularity Resolution and Consistency

A key motivation for non-local gravity is the hope of resolving curvature singularities. In metric non-local models with exponential form-factors (e.g., $$S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)$$9), the Newtonian potential becomes non-singular, but in the Palatini formulation, these desirable features do not automatically persist, due to independent connection dynamics and the altered structure of the field equations.

If $$S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R$$0, the full set of Palatini equations becomes highly non-trivial, and it remains an open problem whether singularity resolution is possible for generic non-local analytic forms. The necessary criteria likely involve the detailed analytic structure of the form-factors $$S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R$$1 and the specific way non-local operators are coupled to independent curvature components (Briscese et al., 2015).

7. Summary Table: Key Features of Non-local Palatini Models

Feature	Pure Metric Non-local Model	Palatini Non-local Model	Hybrid Metric–Palatini Non-local Model
Dynamical variables	Metric $$S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R$$2	$$S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R$$3, $$S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R$$4	$$S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R$$5, $$S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R$$6
Singularities in vacuum	Removable for some $$S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R$$7	Not removed for $$S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R$$8	Depends on coupling structure
Ghost freedom	Typically has a ghost	Ghost content dictated by $$S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R$$9	Ghost-free with suitable coupling
Inflationary dynamics	Starobinsky plus spectator field	As above if localized, but details model-dependent	Multi-field kinetic, single effective inflaton
Connection–metric relation	Levi–Civita	Nontrivial non-local relation	Model-dependent

The references for these results are (Briscese et al., 2015, Bombacigno et al., 2024). The general expectation is that non-local Palatini models extend the landscape of viable modifications to gravity, but pose significant challenges in achieving ghost freedom and resolving curvature singularities without introducing new pathologies.