Boltzmann Dynamics in K-essence Cosmology: Photon Propagation in an Emergent Spacetime

Recent cosmological tensions, notably the Hubble and $S_{8}$ tensions, necessitate extensions of the conventional $\Lambda$CDM framework, wherein additional dynamical fields alter the effective spacetime encountered by matter and radiation. In K-essence cosmology, the scalar field induces an emergent FLRW geometry that is disformally linked to the gravitational metric, resulting in a \emph{tilted causal structure} where the light cone propagation differs from that of gravity. This study develops a covariant Boltzmann formalism inside a homogeneous K-essence framework and derives the modified mass-shell condition, geodesic equations, and collision integrals for both massless and massive particles. We demonstrate that the photon distribution retains its thermal properties in the emergent frame, while it seems geometrically rescaled in the gravitational frame. The Thomson and Compton processes maintain their microscopic structure while obtaining effective masses and interaction rates governed by the scalar field. During the tightly coupled epoch, the photon-baryon fluid experiences acoustic oscillations characterized by a modified sound horizon. For the kinetic K-essence DBI-type Lagrangian, the interaction rate scales as $n_{e}\sigma_{T}^{\rm eff}a\propto a^{-8}$, indicating a strong coupling in the early universe. Additionally, the diffusion damping scale scales as $k_{D}^{-2}\propto a^{29/2}$, indicating that small-scale anisotropies become increasingly sensitive to the evolving geometry. The results provide a coherent kinetic description of particle transport in a tilted spacetime and demonstrate that CMB propagation effects may serve as an observational probe of K-essence and emergent gravity frameworks.