Comment on "Test of the Gravitational Force Law on Cosmological Scales Using the Kinematic Sunyaev--Zeldovich Effect''

Abstract

Gallardo et al. [Phys. Rev. Lett. \textbf{136}, 151002 (2026)] constrain the index of the force law $g\propto r^{-n}$ with pairwise kinematic Sunyaev--Zeldovich measurements, $n=2.1\pm0.3$, and exclude the deep-MOND value $n=1$ at $3.3σ$. The exclusion does not follow from their calculation. Their templates for $n\neq2$ [their Eq.~(8)] raise the pair-averaged Newtonian acceleration to the power $n/2$, whereas the pair average of the pointwise law [their Eq.~(30) for deep MOND] is the average of the transformed field; the step that identifies the two attributes to the three-point function a correction that is of second order and not small. In a Gaussian linear field with linearly biased tracers, Gaussian integration by parts shows that the correctly averaged template is exactly proportional to the Newtonian one, with a constant that the analysis absorbs into the mean optical depth. Gaussian realizations with peak-selected tracers find the same proportionality to within 2\% (0.5\% for the fiducial selection), and refitting the data of the Letter with the resulting template gives $χ^2=20.1$--$20.2$, against $20.1$ for the Newtonian template and $33.8$ for Eq.~(8): in every controlled case examined the data do not discriminate between $n=1$ and $n=2$, and the $3.3σ$ exclusion, with the $10σ$ forecast that rescales it, rests on the order in which the average and the power were taken. What a self-consistent MOND cosmology predicts for this observable is a separate question, which the Letter does not address.