Fundamental Cosmic Anisotropy and its Ramifications I: Killing vector fields and constructing their metric

TL;DR

The paper presents an algorithmic framework to construct a spacetime metric whose Killing vector fields realize a given 3D Lie algebra, enabling genuinely homogeneous yet anisotropic cosmologies. By introducing the notions of a pre-KLA and an invariant frame, it shows how a GL$(3)$ transform can align a KVF set with canonical structure constants, after which differential equations for frame components $p$, $q$, and $r$ yield a frame $\{X_i\}$ whose dual $\{e^i\}$ satisfies $g=g_{\mu\nu}(t)\,e^\mu e^\nu$ with KVFs as (subset) of the Killing algebra. The method is demonstrated through explicit Bianchi III and twice-Bianchi II examples, including the separation of time dependence in spatially homogeneous metrics and the resulting explicit spatial metrics. The approach generalizes prior tabulations by providing a practical, coordinate-friendly recipe applicable to any suitable 3D Lie algebra, with potential applications to CMB signatures in anisotropic cosmologies and extensions to other symmetry classes.

Abstract

On the largest scales, the universe appears to be almost homogeneous and isotropic, adhering to the cosmological principle. In contrast, on smaller scales inhomogeneities and anisotropy become increasingly prominent, reflecting the origin, emergence, and formation of structure in the universe. Moreover, a range of tensions between various cosmological observations may suggest it necessary to explore the consequences of departure from the ideal, uniform universe on the fundamental level. Thus, in this work, the foundation of spatially homogeneous yet anisotropic universes is studied. Specifically, when given a 3D Lie algebra of \emph{desired} Killing vector fields (as would be the case for a homogeneous yet anisotropic universe), we provide an explicit construction for the metric that has exactly those as its Killing vector fields. This construction is presented accessibly, in a directly-usable, algorithmic fashion. Some examples demonstrating the construction are worked out, including a constructive method to separate out (cosmic) time dependence in spatially homogeneous, anisotropic cosmologies.

Figures (3)

• Figure 1: Illustration of effect of Lie dragging, in two dimensions. The dashed loops around $p$, $q$, and $q'$ are level sets of the distance. Dragging the metric from $p$ to $q$ via $\xi$ leaves the level sets intact, so $\xi$ represents an (infinitesimal) isometry of the metric. In contrast, when we drag from $p$ to $q'$ via $\eta$, the level sets are perturbed. This shows the metric has been altered, and hence that $\eta$ is not an (infinitesimal) isometry.
• Figure 2: Image displaying the construction. We have the path from $p$ to $q$, following the various characteristics, indicated by different dashing patterns, for the specified lengths $s_I$, $I=1,2,3$. In this case, we follow the characteristic of $\xi_1$ for distance $s_1$, then the characteristic of $\xi_2$ for $s_2$, and finally the characteristic of $\xi_3$ for $s_3$, ending up at $q$. Along each characteristic we know how to evaluate the invariant frame components $\tensor{X}{_i^I}$, namely by means of \ref{['eq:evolutionAlongCharacteristic']}.
• Figure 3: Simulation of a CMB (Sachs-Wolfe effect) in a Bianchi V universe: a superposition of wave modes permissible in this universe. The "washing out" towards the bottom is predicted from the form of the Bianchi V metric.

Theorems & Definitions (13)

• Theorem 3.1
• proof
• Definition 4.1: Isometry
• Definition 4.2: Transitivity
• Lemma 4.3
• proof
• Definition 4.4: Spatial homogeneity
• Definition 4.5: Bianchi model
• Remark 4.6: Differing nomenclature
• Remark 4.7: Different choices $q$, $s$ in \ref{['eq:total-isom-dim']}