From kinetic gases to an exponentially expanding universe - The Finsler-Friedmann equation

TL;DR

The paper argues that coupling the full 1PDF $\varphi$ of a kinetic gas to Finsler spacetime geometry via the Finsler gravity equation puts all velocity moments on equal footing in gravity, leading to the Finsler-Friedmann equation that governs both the scale factor and the dynamic causal structure through a velocity-dependent Lagrangian $L=\dot{\eta}^2 a(\eta)^2 f(s)^2$. It shows that all moments of the 1PDF contribute to gravity, yielding an infinite set of tensor-density equations on spacetime, not just a single 2-tensor equation. In vacuum, the framework admits an exponential expansion without a cosmological constant, with a causal structure that reduces to FLRW for small velocities but deviates for larger $s$, illustrating a mild velocity-dependent deformation of standard cosmology. These results suggest dark energy could emerge from phase-space geometry, motivating further study of non-vacuum cases and observational signatures of Finsler spacetimes sourced by kinetic gases.

Abstract

We investigate the gravitational field of a kinetic gas beyond its usual derivation from the second moment of the one-particle distribution function (1PDF), that serves as energy-momentum tensor in the Einstein equations. This standard procedure raises the question why the other moments of the 1PDF (which are needed to fully characterize the kinematical properties of the gas) do not contribute to the gravitational field and what could be their relevance in addressing the dark energy problem? Using the canonical coupling of the entire 1PDF to Finsler spacetime geometry via the Finsler gravity equation, we show that these higher moments contribute non-trivially. A Finslerian geometric description of our universe allows us to determine not only the scale factor but also of the causal structure dynamically. We find that already a Finslerian vacuum solution naturally permits an exponential expanding universe, without the need for a cosmological constant or any additional quantities. This solution possesses a causal structure which is a mild deformation of the causal structure of Friedmann-Lemaître-Robertson-Walker (FLRW) geometry; close to the rest frame defined by cosmological time (i.e., for slowly moving objects), the causal structures of the two geometries are nearly indistinguishable.

Figures (3)

• Figure 1: Numerical solution of equation \ref{['eq:s-dep2']}, with boundary conditions \ref{['eq:bdry']} and $c_1=1$ (blue line) compared to the approximate Taylor series solution up to $s^{20}$ with $c_1=1$ (orange line) and classical FLRW geometry (green line.)
• Figure 2: Numerical solution of equation \ref{['eq:s-dep2']}, with boundary conditions \ref{['eq:bdry']} and $c_1=1$ (blue line) compared to perturbative Taylor expansion solution of equation \ref{['eq:s-dep2']} (orange line). Left: Whole range. Right: Zoomed in.
• Figure 3: Left: The future pointing unit timelike directions $L=1$ of the Finsler FLRW geometry (blue) compared to FLRW geometry (orange), with two spatial directions suppressed. Right: The corresponding lightcones. For small values of $\dot x$ the causal structures are nearly undistinguishable. Differences only appear for larger $\dot x$, respectively for larger values of the symmetry adapted variable $s$.