Scale-dependent and background-preserving gravity from an action: cosmological tests Wiliam S. Hipólito-Ricaldi, Rodrigo von Marttens, Felipe de Melo-Santos and Davi C. Rodrigues Eur. Phys. J. C, 85 5 (2025) 599 Published online: 30 May 2025 DOI: 10.1140/epjc/s10052-025-14208-8