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In this paper we show that the on-shell Lagrangian of a perfect fluid dependson microscopic properties of the fluid, giving specific examples of perfectfluids with different on-shell Lagrangians but with the same energy-momentumtensor. We demonstrate that if the fluid is constituted by localizedconcentrations of energy with fixed rest mass and structure (solitons) then theaverage on-shell Lagrangian of a perfect fluid is given by ${\mathcal L}_m=T$,where $T$ is the trace of the energy-momentum tensor. We show that our resultshave profound implications for theories of gravity where the matter Lagrangianappears explicitly in the equations of motion of the gravitational and matterfields, potentially leading to observable deviations from a nearly perfectcosmic microwave background black body spectrum: $n$-type spectral distortions,affecting the normalization of the spectral energy density. Finally, we putstringent constraints on $f(R,{\mathcal L}_m)$ theories of gravity using theCOBE-FIRAS measurement of the spectral radiance of the cosmic microwavebackground.

Perfect fluid Lagrangian and its cosmological implications in theories of gravity with nonminimally coupled matter fields