Predictivity and Regime of Validity

A formal expression such as $R\Box^{-1}R$ is not yet a predictive theory. It becomes one only after the operator, causal prescription, solution space, coefficient origin, validity regime, propagating content, and observable map are fixed. Different choices can share the same compact notation while making different predictions. This page gives a minimum specification and assesses which parts are established for representative infrared and entire-function models.

Required background. PDEs and Green Functions for Gravitation supplies operator domains, zero modes, and causal kernels; Gravitational Actions and Boundary Terms supplies the variational and effective-equation distinctions.

Helpful background. Einstein Equations supplies constraints and the GR comparison; Formulations of General Relativity supplies mode and initial-data counting; Gravity Beyond General Relativity supplies power counting, branch labels, and comparison with observations.

A minimum predictive specification

Seven declarations are required.

1. Operator and domain. State whether the object is $\Box_{\mathrm{ret}}^{-1}$, a spectral logarithm, or an entire function of $\Box$; define zero-mode subtraction, contour, functional space, and boundary or initial surface.
2. Dynamical foundation. Distinguish a fundamental classical action, an in-in quantum effective equation, and a phenomenological equation-level prescription. They obey different variation and causality rules.
3. Fields and data. List physical fields and constrain every auxiliary homogeneous solution to the original nonlocal prescription. State the metric coupled to matter.
4. Modes and consistency. Count poles or constrained initial functions on the named background, then test kinetic signs, gradients, constraint propagation, and nonlinear well-posedness separately.
5. Scales and errors. Identify the nonlocality scale, any EFT cutoff, curvature and frequency ratios, memory or secular enhancement, loop ordering, and the first omitted effect.
6. Branch and GR limit. Specify cosmological history, state, Green function, and asymptotic branch. Demonstrate convergence of dimensionless observables rather than only sending a coefficient to zero.
7. Portable prediction. Export background functions, perturbation equations, polarizations, response kernels, normalization, and validity range without importing an empirical likelihood.

A model that leaves one of these blank may still be a useful ansatz, but its unanswered item is a research question rather than evidence of viability.

Scale separation in an inverse operator

For the RR expression

$$\Gamma_{\mathrm{RR}} =\frac{M_{\mathrm{Pl}}^2}{2}\int\sqrt{-g} \left[R-\frac{m^2}{6}R\Box_{\mathrm{ret}}^{-2}R\right]d^4x,$$

the formal modification is governed by $m^2/\Box$ twice integrated against curvature. The functional denotes formal variation followed by a retarded equation-level prescription; it is not a conventional retarded action. At a local frequency $\omega\gg m$, one might expect suppression by $m^2/\omega^2$. That estimate fails near a zero mode or over a history long enough for the inverse to accumulate secular growth. A usable expansion parameter must include the norm of the retarded convolution:

$$\epsilon_{\mathrm{hist}} \sim (mL)^2 \frac{\lVert\Box_{\mathrm{ret}}^{-2}R\rVert} {\lVert R\rVert L^4},$$

where $L$ names the spacetime scale and the norms and data class must be declared. The formula is schematic, but it exposes what a local $m^2/\omega^2$ estimate omits.

The controlled GR limit keeps the initial surface and retarded prescription fixed, sets auxiliary homogeneous data to their defining values, and takes $m\to0$ over a finite dimensionless history for which the convolutions remain bounded. Interchanging the late-time and $m\to0$ limits can be nonuniform.

Radiative stability and microscopic origin

Nonlocal form factors occur in quantum effective actions. For example, massless loops generate calculable curvature terms containing $\log(-\Box/\mu^2)$. Their coefficients and renormalization-scale dependence are tied to the underlying field content. This establishes that nonlocality itself is not exotic; it does not derive an arbitrary infrared inverse power with a cosmologically relevant coefficient. The distinction between ultraviolet-calculable logarithms and conjectured infrared mass generation is reviewed in Belgacem et al. 2018, §§ 2.2–2.3, equations (2.10)–(2.22).

Radiative stability asks whether quantum corrections preserve the functional relations used for the prediction. A zero-free entire multiplier $a(\Box)=e^{\gamma(\Box)}$ avoids additional Minkowski poles at tree level, but generic corrections to the quadratic kernel need not remain the exponential of the same entire function. A proposed symmetry, fixed point, or explicit renormalization calculation must control that claim. Otherwise the form factor is a specified effective input whose coefficients and precision are limited to the order calculated.

Similarly, an infrared scale $m$ can be treated phenomenologically, but a claim that it is dynamically generated requires a microscopic calculation that determines its magnitude, sign, tensor structure, state dependence, and renormalization. Background success does not supply that derivation.

Results and open questions through 2026-08-19

This discussion reflects results available through 2026-08-19. The comparison axes are operator definition, causal foundation, data count, established result, and missing validity condition.

Evidence or formulation	Supported result and competing interpretation	Scope, limitation, or unresolved status
Calculable nonlocal quantum effective terms	Massless loops generate nonlocal logarithmic curvature form factors; in-in evolution can make their response causal. Belgacem et al. 2018, §§ 2.1–2.2	This supports effective nonlocality but does not derive the inverse powers or new infrared scale used in late-time models. The deep infrared gravitational effective action remains incompletely controlled.
RR infrared model with retarded data	A single mass scale, fixed auxiliary data, FLRW background, and linear perturbation system form a concrete predictive model. Maggiore and Mancarella 2014, §§ II–III and VI; Belgacem et al. 2018, §§ 2.4–3.2	The permissive view treats this as a self-contained effective equation. The conservative view requires a microscopic derivation of the kernel and scale; nonlinear well-posedness beyond studied backgrounds remains unresolved.
Reconstructed $Rf(\Box^{-1}R)$ cosmology	The original model admits a free distortion function Deser and Woodard 2007, § 3, Eqs. (10)–(14). Reconstruction can fit a prescribed homogeneous history where $X(t)=\Box^{-1}R$ can be inverted and the causal initial data satisfy the Friedmann constraint Deffayet and Woodard 2009, § 3, Eqs. (19)–(23).	If $f$ is reconstructed from the target background, that background is not an independent prediction. Stability, structure formation, and the functional prior become the discriminating outputs.
Zero-free entire-function gravity	A complete zero-free kinetic multiplier can retain only the massless Minkowski graviton pole and soften Euclidean momentum. Biswas et al. 2012, Eqs. (10)–(13)	The result is quadratic and background specific. Nonlinear initial-value theory, causal support, curved-background poles, and radiative preservation of the form factor are not established generically.
Analytic infinite-derivative initial-value analysis	Propagator poles and the chosen contour, rather than derivative count alone, determine the number of analytic initial data. Barnaby and Kamran 2008, §§ 2.1–2.3, 3.1, and 5.3	Results assume a declared analytic function space and contour. They do not give a universal nonlinear gravity theorem or make different contour choices physically equivalent.

The strongest conservative position restricts confidence to nonlocal terms derived in a controlled effective action or to phenomenological models whose prescription, data, stability, and errors are completely exposed. The strongest permissive position accepts a compact, retarded, internally consistent cosmological equation as predictive before a microscopic derivation, provided it can be falsified by outputs not used to choose its kernel. These positions assign different evidential weight to origin; neither permits hidden homogeneous data or an unspecified inverse.

Negative results matter. Ordinary variation cannot yield a purely retarded single-field kernel; a polynomial truncation can introduce poles absent from an exact entire function; reconstructing a free function from a background removes that background from the prediction set; and flat-space pole absence does not prove nonlinear health. No universal theorem currently settles all nonlocal formulations.

The conclusions could change after a microscopic calculation derives an infrared kernel and scale, a nonlinear well-posedness or instability theorem changes an admissible data class, radiative corrections preserve or destroy a claimed zero-free form factor, or an independent branch calculation changes a portable theory prediction.

From formal model to discriminating output

A theory calculation should first reproduce its defining inverse in two ways: local auxiliary evolution with restricted data and direct convolution with the declared kernel. It should then verify the Bianchi identity, constraint propagation, convergence under time and spatial refinement, and insensitivity to moving the initial surface farther into a regime where the source vanishes. Deliberately adding a homogeneous auxiliary mode is an adversarial test: a changed observable confirms that the prescription is physically meaningful.

Next separate inputs from predictions. If a function $f$ is reconstructed to match $H(a)$, that background is an input. Perturbation growth, slip, gravitational-wave propagation, or another solution branch may remain predictions, but only inside the same operator domain and cutoff.

Finally export theory-side response functions and their covariance, normalization, scale range, and initial-state assumptions to Tests of Gravity. Dataset selection, nuisance parameters, likelihoods, numerical bounds, and empirical verdicts are handled there. A fit cannot repair an undefined Green function or an unstable branch.

Common pitfalls

A compact action is a complete model. The inverse, state, boundary data, and effective-equation foundation still need definitions.

Agreement with a reconstructed background is a prediction. It is a calibration condition. Independent perturbations or cross-regime results do the testing.

Tree-level pole avoidance is radiative stability. It is not. The functional relation among form factors must be protected or recomputed.

Exercises

A proposed theory uses

$$S=\frac{M_{\mathrm{Pl}}^2}{2}\int\sqrt{-g}\, R\left[1+f(\Box^{-1}R)\right]d^4x,$$

chooses $f$ to reproduce a target $H(a)$, and then reports that the background confirms the model. Apply the seven-part specification, identify what is input rather than prediction, and give three independent calculations that could make the proposal falsifiable.

Solution

First specify which inverse appears. A retarded choice needs its initial surface, zero-mode treatment, and auxiliary homogeneous data; a symmetric or Feynman inverse defines a different problem. State whether the equation is an in-in effective equation or a phenomenological retarded prescription, and identify the physical matter metric.

Localize with $U=\Box^{-1}R$ but restrict $U$ to the chosen kernel. Count metric and physical modes without promoting $U_{\mathrm{hom}}$ to arbitrary data. Check the Bianchi identity, kinetic and gradient signs, constraint propagation, and the domain in which the function $f$ and any derivative expansion are controlled. Define a GR limit that holds the kernel, branch, and finite observation interval fixed.

Because $f$ was selected to reproduce $H(a)$, the target background is an input and cannot confirm the model. Three discriminating calculations are: (1) derive and evolve scalar and tensor perturbations with retarded auxiliary data, including a convergence and direct-convolution check; (2) compute another background or strong-field branch not used in the reconstruction and test its stability; and (3) derive the form and coefficient of $f$ from an underlying quantum theory, or calculate radiative corrections and show that the assumed functional class is preserved to a stated accuracy. The resulting response functions can then be handed to empirical testing with their validity range intact.

References

• Barnaby, Neil, and Niky Kamran (2008), “Dynamics with Infinitely Many Derivatives: The Initial Value Problem,” Journal of High Energy Physics 2008(02), 008. DOI; Open PDF.
• Belgacem, Enis, Yves Dirian, Stefano Foffa, and Michele Maggiore (2018), “Nonlocal Gravity: Conceptual Aspects and Cosmological Predictions,” Journal of Cosmology and Astroparticle Physics 2018(03), 002. DOI; Open PDF.
• Biswas, Tirthabir, Erik Gerwick, Tomi Koivisto, and Anupam Mazumdar (2012), “Towards Singularity- and Ghost-Free Theories of Gravity,” Physical Review Letters 108, 031101. DOI; Open PDF.
• Deffayet, Cédric, and Richard P. Woodard (2009), “Reconstructing the Distortion Function for Nonlocal Cosmology,” Journal of Cosmology and Astroparticle Physics 2009(08), 023. DOI; Open PDF.
• Deser, Stanley, and Richard P. Woodard (2007), “Nonlocal Cosmology,” Physical Review Letters 99, 111301. DOI; Open PDF.
• Maggiore, Michele, and Michele Mancarella (2014), “Nonlocal Gravity and Dark Energy,” Physical Review D 90, 023005. DOI; Open PDF.