Ambiguity in matter sector for modified gravity involving $$\delta ^2 \mathcal {L}_{m}/\delta g^{\mu u }\delta g^{\alpha \beta }$$ and its implications to astrophysics and cosmology

Abstract Matter density ( $$\rho $$ ρ ) and radial pressure ( p ) are often used as the matter Lagrangian density ( $$\mathcal {L}_{m}$$ L m ) because both are thermodynamically consistent and produce the same Einstein field equation (EFE) in general relativity (GR). New gravity models with explicit links between matter and geometry instead involve second-order derivatives of $$\mathcal {L}_{m}$$ L m relative to the metric tensor. So, picking either p or $$-\rho $$ - ρ for $$\mathcal {L}_{m}$$ L m gives different effective EFEs. This confusion appears because one usually treats the four-velocity ( $$u_\mu $$ u μ ) and the metric tensor ( $$g_{\mu \nu }$$ g μ ν ) as independent. Here, we revisit the basics and offer a consistent framework by relaxing that assumption, thereby making the modified gravity theory independent of the choice of $$\mathcal {L}_{m}$$ L m . Finally, we test this approach on neutron and quark stars (ultraviolet region) and on cosmological situations with matter-dominated or dust ( $$p=0$$ p = 0 ), radiation-dominated ( $$p=\rho /3$$ p = ρ / 3 ), and vacuum energy ( $$p=-\rho $$ p = - ρ ) equations of state (infrared region), showing how it clarifies the ambiguity in picking $$\mathcal {L}_{m}$$ L m for gravity models.