Deser-Woodard Nonlocal Gravity

• Deser-Woodard nonlocal gravity is an infrared modification of General Relativity where the action is deformed by a term R f(□⁻¹R) to account for late-time cosmic acceleration without a cosmological constant.
• The model reconstructs the distortion function to match ΛCDM background expansion while yielding distinct signatures in the cosmic growth rate, gravitational lensing, and gravitational-wave friction.
• Extensions of the DW model include bounce cosmologies and modified compact objects, though challenges remain regarding solar-system constraints and the development of a robust screening mechanism.

The Deser-Woodard (DW) nonlocal gravity model is a class of infrared modifications of General Relativity (GR) wherein the Einstein–Hilbert action is deformed by functionals involving the covariant inverse d'Alembertian acting on the Ricci scalar, typically of the form $R f(\Box^{-1}R)$. The distortion function $f$ is model-dependent and chosen to match observed cosmological histories, notably enabling late-time cosmic acceleration without a cosmological constant or introducing new matter fields. Over the past decade, this framework has driven significant activity in nonlocal gravity, including detailed studies of theoretical consistency, cosmological background and perturbative dynamics, static-vacuum solutions, black holes, and local (solar-system) constraints.

1. Theoretical Foundations and Action Structure

The original DW model modifies the gravitational action as follows: $$S_{\rm DW} = \frac{M_{\rm Pl}^2}{2}\int d^4x\sqrt{-g}\left[ R + R\,f(\Box^{-1}R) \right],$$ where $\Box = g^{\mu\nu}\nabla_\mu\nabla_\nu$ is the covariant d'Alembertian and $f$ is a free dimensionless function, typically reconstructed to enforce a desired Hubble expansion history (Amendola et al., 2019, Nersisyan et al., 2017, Park et al., 2012).

To render the nonlocality tractable, auxiliary scalar fields are introduced: $$X \equiv \Box^{-1}R, \quad U \equiv \Box^{-1}[f'(X) R],$$ which localizes the equations of motion. Variation yields the modified Einstein field equations: $$G_{\mu\nu} + \Delta G_{\mu\nu} = 8\pi G\,T_{\mu\nu},$$ with $\Delta G_{\mu\nu}$ algebraically expressed in terms of $(f, X, U)$ and their derivatives. This structure generalizes directly to other formulations, including the so-called "DW II" model, in which the distortion is promoted to a function $f(Y)$ where $$Y = \Box^{-1}[g^{\mu\nu}\partial_\mu X \partial_\nu X]$$, thereby incorporating nonlinear arguments and enlarging the model space (Chu et al., 31 Oct 2024, D'Agostino et al., 21 Feb 2025, Neves, 1 Jul 2025).

2. Cosmological Reconstruction and Observational Dynamics

A central feature of the DW model is the reconstruction of $f(X)$ to yield a background expansion $H_{\rm DW}(z)$ identical to $\Lambda$CDM: $$h_\Lambda^2(z) = \Omega_m(1+z)^3 + \Omega_r(1+z)^4 + \Omega_\Lambda,$$ The function $f(X)$ is determined via nested integral equations in redshift (or scale factor), ensuring that the model's background dynamics are indistinguishable from standard cosmology (Amendola et al., 2019, Bouchè et al., 2023, Nersisyan et al., 2017). The linear perturbation sector, implemented in Einstein-Boltzmann codes (e.g., CLASS), reveals deviations relative to GR:

• The scalar sector modifies the effective gravitational coupling $G_{\rm eff}$ and the slip and lensing parameters $(\eta, \mu, \Sigma)$, leading to a suppressed linear growth rate $f\sigma_8$ and a higher lensing power $C_\ell^{\phi\phi}$ compared to $\Lambda$CDM (Amendola et al., 2019).
• The tensor sector exhibits modified gravitational-wave friction: $$h''_{ij} + 2H\left[1-\frac{1}{2}H^{-1}\partial_\tau\ln(G_{\rm eff,gw}/G)\right]h'_{ij} + k^2h_{ij} = 16\pi G_{\rm eff,gw} a^2\pi_{ij},$$ where $G_{\rm eff,gw} = [1+f(X)+U]^{-1}$, allowing for variation in gravitational-wave luminosity distances (Amendola et al., 2019).

Cosmological fits using Planck CMB+lensing, JLA SNIa, and redshift-space-distortion (RSD) growth data determine that the DW model:

• Predicts $\sigma_8 = 0.7530\pm0.0096$ (vs $\Lambda$CDM's $0.8171\pm0.0089$), a reduction of $\sim7.8\%$ (Amendola et al., 2019).
• The linear growth rate $f\sigma_8$ for $k\sim 10^{-2}\,{\rm Mpc}^{-1}$ is $\sim10\%$ below $\Lambda$CDM at $z \lesssim 1$.
• Lensing potential $C_\ell^{\phi\phi}$ is enhanced by $\sim10\%$ for $40 \lesssim \ell \lesssim 400$.

However, adding RSD data opens a $\sim1.8\sigma$ tension between RSD-preferred growth and Planck lensing (Amendola et al., 2019, Nersisyan et al., 2017). The model is not observationally ruled out, yielding only "weak" statistical evidence in favor of $\Lambda$CDM.

3. Extensions: Bounce Cosmologies and Compact Objects

The model's flexibility allows it to accommodate alternative early-universe scenarios and strong-field solutions.

• Bounce Cosmologies: By reconstructing $f(X)$ for nonsingular backgrounds (e.g., symmetric, oscillatory, or matter bounces), the DW model can suppress cosmological singularities and ameliorate the BKL anisotropy instability, with nonlocal terms decaying after the bounce to recover GR in late times (Chen et al., 2019, Jackson et al., 2021).
• Black Holes and Wormholes: Revised formulations ("DW II") with $f(Y)$ and four auxiliary scalars yield analytic solutions for static spherically symmetric spacetimes, including black holes featuring power-law corrections to Schwarzschild, regular black holes, and traversable wormholes. The nonlocal sector plays the role usually attributed to exotic matter, sustaining traversable throats and regularizing or modifying horizons (D'Agostino et al., 21 Feb 2025, Neves, 1 Jul 2025, D'Agostino et al., 29 Jan 2025). Quasinormal mode analyses show isospectrality is broken only if background auxiliary fields are excited (Chen et al., 2021).

A sample of these strong-field solutions is outlined below:

Solution Type	Phenomenology	Key Modification
Black Hole (DW II)	Inverse power-law $g_{tt}$	$A(r)=1-2/r-\alpha/r^n$ [(D'Agostino et al., 21 Feb 2025); $n>1$]
Traversable Wormhole	Morris–Thorne/analytic $b(r)$	Supported by pure gravity with $f(Y)$ (D'Agostino et al., 29 Jan 2025, Neves, 1 Jul 2025)
Regular (extremal) Black Hole	No curvature singularity, auxiliary fields diverge at horizon	$A(r)=(1-M/r)^2$ (Reissner–Nordström–like) (Neves, 1 Jul 2025)

4. Theoretical Consistency and Screening

While the DW model is designed to recover GR in the ultraviolet and explain late-time cosmic acceleration, several theoretical issues remain:

• Solar-System Constraints: Nonlocal modifications generally induce time-variation in $G_{\rm eff}$ at late times, challenging compliance with experimental bounds (e.g., Lunar Laser Ranging $|\dot{G}/G|<10^{-13}\,{\rm yr}^{-1}$) (Bouchè et al., 2023).
• Screening: The standard prescription does not include a robust Vainshtein-like mechanism. Certain models (notably DW-2019) exploit spatial splitting of nonlocal source fields to restore $G_{\rm eff}\simeq G$ in strongly bound systems, potentially evading solar-system and binary pulsar constraints (Ding et al., 2019). However, most variants lack a complete screening mechanism, marking a significant open problem.
• Gravitational-Wave Flux and Divergences: A general challenge in nonlocal gravity is the appearance of $1/r$ spatial momentum density tails in the GW pseudo-tensor. In both DW I and DW II, these lead to divergent fluxes unless the distortion function satisfies $f'(0)=0$ (and, in DW I, also $f''(0)=0$), forcing $f$ to be at least quadratic in its argument and restricting nonlocal effects in the weak-field regime (Chu et al., 31 Oct 2024, Chu et al., 2018). Without this, energy conservation at asymptotic infinity is violated.

Model Variant	Divergence avoided if
DW I	$f'(0)=0=f''(0)$
DW II	$f'(0)=0$
Vardanyan-Akrami-Amendola-Silvestri (VAAS)	Divergence persists for generic parameters

5. Related Theories and Model Space

The DW framework constitutes a prototypical nonlocal theory but stimulates further generalizations:

• Scalar-Tensor Nonlocal Gravity: Promoting the operator to include tensors (e.g., $R_{\mu\nu}\Box^{-1}R^{\mu\nu}$) can yield theories with the same cosmological background expansion as DW but with improved phenomenology (e.g., identical gravitational slip $\eta=1$, exact compliance with GW speed constraints, or fixed parameter choices by theoretical arguments) (Tian, 2018).
• Alternative Distortion Functions: DW models with $f(Y)$ allow for further tailoring of growth and screening features (Ding et al., 2019, Jackson et al., 2021).
• Minimal m$^2\Box^{-1}R$ Theories: Models inspired by but distinct from DW, such as $m^2\Box^{-1}R$ or $m^2R\Box^{-2}R$, show different background and perturbation behavior, sometimes closer to $\Lambda$CDM, with altered late-time equation-of-state (Vardanyan et al., 2017).

6. Observational Constraints and Outlook

Empirical analyses reveal several key features of the DW model space:

• Linear growth suppression brings predictions for $S_8 = \sigma_8\sqrt{\Omega_m/0.3}$ and $f\sigma_8(z)$ into better agreement with lensing observations (KiDS, DES) than $\Lambda$CDM, partially mitigating the so-called "growth tension" (Bouchè et al., 2023, Nersisyan et al., 2017).
• The background expansion (Hubble function) is exactly matched to $\Lambda$CDM for standard forms of $f$, so the model does not resolve the $H_0$ tension.
• The absence of a robust screening mechanism remains a principal theoretical deficit. Solar-system tests, including high-precision measurements of light deflection, Shapiro time delay, perihelion advance, and geodetic precession, yield strong constraints on the amplitude and scale-dependence of nonlocal modifications in parameterizations such as $F(Y) = \zeta Y$, often requiring $|\zeta| \lesssim 2 \times 10^{-5}$ for $b \gtrsim 1.96$ and $|\zeta| \lesssim 3 \times 10^{-10}$ for $b \simeq 1.06$ (Liu et al., 11 Nov 2025).
• Next-generation observational probes—in particular, precise weak-lensing, galaxy clustering, and GW standard siren measurements—are forecast to decisively test DW model predictions, including distinctive GW-to-electromagnetic luminosity distance ratios, deviation in the lensing parameter $\Sigma(z)$, and potential strong-field signatures in compact-object spectra and shadow images (Amendola et al., 2019, Bouchè et al., 2023, D'Agostino et al., 21 Feb 2025, Neves, 1 Jul 2025).

7. Summary Table: Central Model Ingredients and Observables

Feature	DW Model Specification	Observational Signature
Nonlocal Action	$$S = \frac{M_{\rm Pl}^2}{2}\int \sqrt{-g}[R + R f(\Box^{-1}R)]$$	Late-time acceleration, no explicit $\Lambda$
Distortion Function $f$	Reconstructed from $\Lambda$CDM background	Suppressed $f\sigma_8$, enhanced $C_\ell^{\phi\phi}$
Gravitational Slip $\eta$	Nonzero; $$\eta(a) = \frac{1 + U + f - 4f_X}{1 + U + f - 6f_X}$$	Deviations in weak-lensing and ISW cross-correlations
GW Friction	$G_{\rm eff,gw}(a) = [1 + f(X) + U]^{-1}$	$D_L^{\rm gw}(z) \neq D_L^{\rm em}(z)$
Screening	Incomplete except for specific $f$ or background splitting	Solar-system and binary pulsar constraints
Black holes/wormholes	Analytic, horizon-regular/irregular geometries (DW II, $f(Y)$)	Deviations in shadow, ISCO, and QNM spectra

The Deser-Woodard nonlocal gravity model remains a mathematically and phenomenologically rich alternative to standard and scalar-tensor cosmologies, yielding distinctive, testable deviations at cosmological and strong-field scales, while attracting ongoing scrutiny regarding its screening mechanisms and compatibility with local gravity tests. Its future empirical viability will be determined by next-generation cosmological, GW, and solar-system experiments, alongside continued formal paper of the model's theoretical constraints and possible ultraviolet completions (Amendola et al., 2019, Liu et al., 11 Nov 2025, Nersisyan et al., 2017, Bouchè et al., 2023).