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The speed of light is a complicated synthesizer quantity with distinctiveorigins which lead to coincident values in the standard theory. Due to the factthat different aspects of speed of light do not coincide in the local inertialframe for different Palatini modified gravity theories, when deviating fromgeneral relativity, one should consider which aspect of speed of light shouldbe taken into account meticulously and unambiguously. The aim of this study ismainly investigating the modification of the SN Ia luminosity distance, for twof(R) and f(RR) extended theories in the local inertial frame in Palatiniformalism considering different aspects of the speed of light. Besides thelocal inertial frame itself should be determined in the Palatini formalism as aframe in which the Einstein Equivalence Principle is valid. SN Ia luminositydistance should be modified considering the variation of the space-time causalstructure constant for two well-known extended models in Palatini formalism.

Studying the Supernova luminosity distance in Palatini formalism considering the role of causal structure constant