Verification of perturbation corrections beyond the smooth-dark-energy approximation

Verify explicitly whether a fully consistent treatment of the modified perturbation equations in the Maggiore–Mancarella nonlocal $R\Box^{-2}R$ model produces corrections that remain modest in the observationally favored parameter region.

Background

The likelihood analysis implements the nonlocal background evolution but treats cosmological perturbations through an effective smooth-dark-energy approximation. Because CMB observables depend on perturbation dynamics, the authors state that the expected size of corrections from a fully consistent nonlocal perturbation treatment has not yet been established. The reported constraints are therefore conditional on the approximate perturbation prescription.

References

Nevertheless, since the observationally favored solutions exhibit a background evolution close to that of $\Lambda$CDM over the redshift range probed by current data, one may reasonably expect the corrections arising from a fully consistent treatment of nonlocal perturbations to remain relatively modest in this region of parameter space, although this remains to be verified explicitly.

— Nonlocal Gravity, Restricted CPL Dynamics and DESI Constraints  (2610.20720 - Amendola et al., 14 Oct 2026) in Section 4, Constraints from DESI and complementary datasets