Nonlocal and Infrared Modifications

Nonlocal gravity replaces a pointwise response by dependence on a spacetime history or an extended kernel. That move can represent calculable quantum effective dynamics, a phenomenological infrared modification, or an infinite-derivative fundamental proposal. The same symbol $\Box^{-1}$ can encode retarded, advanced, symmetric, or Feynman data, so this chapter begins with operator definition and ends with the conditions that turn a formal kernel into a falsifiable theory.

Helpful background. PDEs and Green Functions for Gravitation provides Green functions, zero modes, contours, and initial-value methods; Gravitational Actions and Boundary Terms provides functional variation and operator domains; Einstein Equations provides curvature dynamics and Bianchi constraints; Formulations of General Relativity provides mode, constraint, and well-posedness language.

Enter this chapter

Three expressions illustrate distinct theory classes:

$$R\Box_G^{-1}R, \qquad m^2R\Box_G^{-2}R, \qquad R\,\mathcal F(\Box/M^2)R.$$

The first two contain inverse differential operators and require a Green function $G$ plus homogeneous-data and zero-mode prescriptions. The third may use an entire analytic form factor with no inverse at its expansion point, but it still requires a functional domain and contour. None is fully defined by its compact algebraic appearance.

The six pages follow the order in which a calculation should be trusted.

Page	Central question	Result carried forward
Nonlocal Operators and Green-Function Prescriptions	Why is an unadorned inverse ambiguous?	A kernel, support rule, operator domain, zero-mode treatment, and initial or boundary prescription.
Localization with Auxiliary Fields	Which local auxiliary solutions belong to the original model?	Constrained local equations whose homogeneous data are fixed by the nonlocal inverse, with spurious modes excluded.
Infinite-Derivative and Entire-Function Gravity	What analytic property avoids extra flat-background poles?	A zero-free entire kinetic multiplier, exact-versus-truncated spectrum comparison, and initial-data count by propagator poles.
Infrared Nonlocal Cosmological Models	How do retarded data determine cosmological auxiliaries?	RR background equations and an executable local-versus-convolution benchmark with independent convergence diagnostics.
Causality and Initial Data in Nonlocal Gravity	Can a causal equation lack an ordinary single-field variational origin?	A comparison of kernel symmetrization, in-in evolution, equation-level prescriptions, and contour-defined data.
Predictivity and Regime of Validity	What turns a formal modification into a predictive theory?	A seven-part specification, comparison of results available at the stated date, radiative and secular checks, and connections to observations.

Localization follows prescription because the local equation alone has a larger solution space. Cosmological integration follows localization because otherwise a numerical code may evolve spurious auxiliary data. Causality and predictivity come last because their answer depends on which operator and foundation the preceding pages established.

The central equivalence and its restriction

For the representative RR model,

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

define

the auxiliary fields below. The displayed functional is shorthand for the effective equations obtained by formal variation followed by the retarded prescription, not a claim that ordinary variation of a retarded single-field action produces retarded evolution. With $M_{\rm Pl}^{-2}=8\pi G$, set

$$U=-\Box_{\mathrm{ret}}^{-1}R, \qquad S=-\Box_{\mathrm{ret}}^{-1}U.$$

The resulting local equations,

$$\Box U=-R, \qquad \Box S=-U,$$

are equivalent to the original operator only after retaining the same retarded data. On homogeneous FLRW,

$$\ddot U+3H\dot U=R, \qquad \ddot S+3H\dot S=U.$$

With zero data in an exactly radiation-dominated phase, both fields remain zero until the trace curvature turns on. Adding arbitrary homogeneous solutions solves the differential equations but defines another inverse and generally changes the metric. The localization and spurious-data distinction are derived in Belgacem et al. 2018, §§ 2.4.4–2.4.5, equations (2.30)–(2.42).

The computational page tests this equivalence on $a=t^{2/3}$. Fixed-step RK4 and an independent nested convolution agree for $U(4)$ and $S(4)$; endpoint errors decrease at fourth order, while a separately formed finite-difference residual decreases at second order. Those different rates make accidental self-comparison less likely.

Inverse kernels versus entire kinetic functions

An inverse operator emphasizes infrared history and is singular or ambiguous near zero modes unless its domain is fixed. An entire kinetic multiplier can instead suppress high Euclidean momentum. Around Minkowski, if

$$a(\Box)=e^{\gamma(\Box)}$$

with entire $\gamma$, then $a$ has no finite zero and introduces no additional propagator pole. A finite Taylor truncation does not preserve that property:

$$\frac{1}{k^2(1+k^2/M^2)} =\frac1{k^2}-\frac1{k^2+M^2}.$$

The truncated polynomial has an extra opposite-residue pole absent from the exact exponential. Barnaby and Kamran show why initial data are tied to the complete pole and contour structure rather than to the phrase “infinitely many derivatives” in Barnaby and Kamran 2008, §§ 2.1–2.3, 3.1, and 5.3.

This pole statement is background specific. Curved backgrounds, nonlinear interactions, causal support, strong coupling, and radiative preservation of the entire function require separate analysis.

Causality, variation, and well-posedness

For a single real field,

$$S=\frac12\int d^4x\,d^4y\, \phi(x)G_{\mathrm{ret}}(x,y)\phi(y)$$

varies to an equation containing

$$\frac12\left(G_{\mathrm{ret}}+G_{\mathrm{adv}}\right).$$

Ordinary functional variation symmetrizes the kernel. A causal nonlocal effective equation can nevertheless arise in the Schwinger–Keldysh in-in formalism, or a phenomenological model can impose a retarded prescription at equation level. The first has an underlying state and microscopic theory; the second must establish conservation and validity independently. Retarded support, a standard classical action, finite initial data, hyperbolicity, and microcausality are different properties.

Conventions and validity checks

All pages use signature $(-,+,+,+)$ and $\Box=g^{ab}\nabla_a\nabla_b$, so a homogeneous scalar obeys $\Box U=-\ddot U-3H\dot U$. A source using the opposite signature is translated with its Green function and action signs as one package.

Before interpreting a result, check:

• the kernel, contour, state, and initial surface;
• the physical versus auxiliary data count;
• Bianchi consistency and constraint propagation;
• kinetic and gradient signs on the selected background;
• zero modes, secular memory, and nonuniform GR limits;
• spatial and temporal numerical convergence plus direct convolution;
• the nonlocality scale, EFT cutoff, loop order, and first omitted effect.

Theory-side response functions, polarizations, background histories, and validity ranges pass to Tests of Gravity. Likelihoods, systematics, and bounds do not determine whether the inverse operator was well defined.

Choose a route

Implement an inverse-curvature model. Read the operator and localization pages, implement the same retarded data in local and convolution forms, then use the cosmology benchmark before coupling the Friedmann equation.

Study an entire-function proposal. Start with the complete quadratic propagator, locate every zero in the complex plane, state the contour and function space, and compare the exact operator with any numerical approximation.

Assess a causality claim. Identify whether the equation is fundamental, in-in effective, or imposed phenomenologically. Test kernel support, constraint preservation, and well-posedness separately.

Export a prediction. Complete the seven-part specification on the final page and label which background or function was calibrated rather than predicted.

Review the chapter

A numerical paper localizes $R\Box^{-2}R$ with two scalars, assigns random initial values to both, and observes late acceleration. Diagnose the result and design a calculation that tests the intended retarded model.

Solution

Write

$$U=-\Box_{\mathrm{ret}}^{-1}R, \qquad S=-\Box_{\mathrm{ret}}^{-1}U.$$

The local equations are second order, but their four scalar initial functions are not free. The retarded definition fixes them, commonly to zero on an initial surface in a phase where the source curvature vanishes. Random values add homogeneous solutions and therefore select another inverse or a different local scalar–tensor theory. Acceleration in those runs is not evidence for the declared retarded model.

Repeat the calculation with an initial surface in radiation domination and $U=\dot U=S=\dot S=0$. Integrate the local system while independently evaluating the nested retarded convolutions. Refine time step and tolerance, monitor the Friedmann constraint and auxiliary equation residuals, and move the initial surface earlier within the source-free regime. Local and convolution results must converge to one another and remain insensitive to that move.

Then perturb one auxiliary datum deliberately. Its effect measures the spurious homogeneous branch rather than numerical error. Finally test kinetic and gradient stability of the coupled background, state the nonlocality scale and effective origin, and take a GR limit at fixed finite history with the same kernel and data. Only after these checks is the acceleration a prediction of the specified model.

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.
• Maggiore, Michele, and Michele Mancarella (2014), “Nonlocal Gravity and Dark Energy,” Physical Review D 90, 023005. DOI; Open PDF.