The universe is not a Lie, but actually an Hopf, algebra

TL;DR

This work investigates how κ-Poincaré Hopf-algebraic structures can illuminate classical cosmology by mapping momentum-space deformations to deSitter spacetime features. The author develops a dual picture linking deSitter redshift and κ-Poincaré lateshift, interprets nontrivial coproducts as momentum-based composition laws, and analyzes backreaction within both κ-Poincaré and deSitter frameworks. A central contribution is showing that deSitter Hopf algebra, including its coalgebra and antipode, can be meaningfully interpreted in a curved-spacetime context and used to reproduce photon propagation and redshift along cosmological worldlines. The slicing approach extends these ideas to more realistic cosmologies (matter-dominated and ΛCDM), suggesting a practical route to apply Hopf-algebra techniques to non-maximally-symmetric universes and to extract phenomenological consequences from observational data.

Abstract

In this paper I would like to show how the Deformed Special Relativity family of models - developed to approach spacetime quantization - can actually be applied to the description of classical cosmology. I use the bicrossproduct basis of $κ$-Poincaré algebra to describe photon propagation in deSitter classical General Relativity. I show the Hopf algebraic aspects of deSitter model, and give an explicit physical interpretation of $κ$-Poincaré Hopf algebraic features in spacetime. Such an approach allows to unravel some not yet known General Relativistic relations of deSitter cosmology, and reinterpret many features of Quantum Gravity phenomenology as classical properties of maximally symmetric spacetime models. In the last section of the paper I give a first example on how to apply this mathematical framework to more realistic (non maximally symmetric) spacetime models, such as $Λ\text{CDM}$ and matter-dominated universe.

Figures (6)

• Figure 1: Alice tries to express the components of vector $\vec{k_{@A}}$ in terms of vector $\vec{p}_{@A}$ which lies in her reference frame and $\vec{q}_{@B}$ whose components are defined by the translated observer Bob.
• Figure 2: The vector sum $\vec{k}=\vec{p}\oplus \vec{q}$ as expressed in the frame at rest (in black) and in the boosted one (in red). The Magenta vector represents $q'$ without taking into account the backreaction on the rapidity parameter $\xi$. The values fixed for the parameters are $H=1$, $a^0=b^0=1$ and $\xi=0.5$.
• Figure 3: On top $n=13$ locally Special relativistic vectors in a $z$ vs. $x^i$ graphic are composed according to nontrivial coproducts \ref{['zxcopr']}. Of course there is no need for a large precision, since a straight line is easy to reproduce, however an interesting feature to be noticed is that the vectors are wider and wider, due to the nontrivial composition laws of both variables. In the graphic below the reconstruction of the deSitter photons worldlines, with respect to the photon's time of emission ($-t$). The larger $n$, the more accurate is the reconstruction of the deSitter worldlines $x^i(t)$. The values of $n$ represented are (4,6,20,30,1000).
• Figure 4: An observer in its origin receives photon worldlines emitted in a distant past $-t$ for different cosmological models.
• Figure 5: Phenomenological coproduct deformed composition of Special Relativistic vectors $(\epsilon^0,\epsilon^1)$, along the worldlines shown in Fig.\ref{['fig:BBAlice']}. The observer in $(0,0)$ reconstructs the photon paths as an exchange of signals between $n=100$ intermediate observers.