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

TL;DR

This paper develops a covariant Boltzmann framework for cosmology within a homogeneous K-essence emergent spacetime, where the scalar field disforms the gravitational metric and tilts the light cone. The authors derive a modified mass-shell condition, geodesic equations, and collision terms, showing that photons remain massless in the emergent frame but experience geometry-driven rescalings of energy, temperature, and interaction rates. In the pre-recombination era, they show that photon-baryon acoustic oscillations and diffusion damping acquire new scalings, notably with $n_{e}\\sigma_{T}^{\\rm eff}a\\propto a^{-8}$ and $1/k_{D}^{2}\\propto a^{29/2}$ for a DBI-like kinetic K-essence, implying enhanced coupling and altered damping tails in the CMB. The line-of-sight formalism remains applicable, but all quantities are evaluated in the emergent geometry, making CMB observables a potential probe of emergent gravity scenarios and non-canonical scalar field dynamics.

Abstract

Recent cosmological tensions, notably the Hubble and $S_{8}$ tensions, necessitate extensions of the conventional $Λ$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 $n_{e}σ_{T}^{\rm eff}a\propto a^{-8}$, 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.