[Paper Review] Minisuperspace description of $f(Q)$-cosmology

Andronikos Paliathanasis, N. Dimakis+1 more|arXiv (Cornell University)|Aug 29, 2023

Cosmology and Gravitation TheoriesPhysics and Astronomy3 citations

TL;DR

This paper establishes a minisuperspace description for $f(Q)$-gravity in homogeneous cosmological models, showing that field equations are second-order in the coincident gauge and sixth-order in the non-coincident gauge, with two scalar degrees of freedom. It derives point-like Lagrangians for FLRW, Kantowski-Sachs, and Bianchi III geometries, and presents a vacuum solution in the non-coincident gauge, demonstrating integrability and providing a framework for Hamiltonian and quantum cosmology.

ABSTRACT

We investigate the existence of minisuperspace description for the homogeneous cosmological field equations within the framework of symmetric teleparallel $f(Q)$-gravity. We consider the background space to be described by the isotropic Friedmann--Lema\^ıtre--Robertson--Walker geometry, the anisotropic Kantowski-Sachs and the anisotropic Bianchi III geometries. Across all these models, we establish that the field equations in $f(Q)$-cosmology exhibit second-order characteristics in the coincident gauge and those of a sixth-order theory in the non-coincident gauge. Specifically, within the latter scenario, the dynamic degrees of freedom are attributed to two scalar fields. Finally, as an example of integrability, we derive a vacuum cosmological solution within the non-coincident gauge.

Motivation & Objective

• To develop a minisuperspace description of $f(Q)$-gravity for homogeneous cosmological models.
• To analyze the field equations in both coincident and non-coincident gauges across isotropic and anisotropic geometries.
• To derive point-like Lagrangians for FLRW, Kantowski-Sachs, and Bianchi III spacetimes.
• To identify the number and nature of dynamic degrees of freedom in the non-coincident gauge.
• To construct an exact vacuum solution as a proof of integrability in the non-coincident gauge.

Proposed method

• Derive the field equations for $f(Q)$-gravity in the Friedmann–Lemaître–Robertson–Walker, Kantowski-Sachs, and Bianchi III geometries using symmetric teleparallelism.
• Apply the coincident and non-coincident gauge conditions to classify the order of the field equations: second-order in the former, sixth-order in the latter.
• Use a Lagrange multiplier method to reformulate the action and derive point-like Lagrangians for each connection family.
• Identify dynamic degrees of freedom as two scalar fields in the non-coincident gauge via the structure of the derived Lagrangians.
• Perform integration by parts on boundary terms to express the Lagrangians in terms of canonical variables and their time derivatives.
• Construct a vacuum solution in the non-coincident gauge to demonstrate the integrability of the system.

Experimental results

Research questions

• RQ1 What is the minisuperspace structure of $f(Q)$-gravity in homogeneous cosmological models?
• RQ2 How do the field equations differ in form and order between the coincident and non-coincident gauges?
• RQ3 What are the dynamic degrees of freedom in $f(Q)$-cosmology under the non-coincident gauge?
• RQ4 Can a point-like Lagrangian be consistently derived for $f(Q)$-gravity in isotropic and anisotropic homogeneous spacetimes?
• RQ5 Is there an integrable vacuum solution in the non-coincident gauge of $f(Q)$-gravity?

Key findings

• The field equations in $f(Q)$-cosmology are second-order in the coincident gauge, simplifying the dynamics and enabling a standard minisuperspace formulation.
• In the non-coincident gauge, the field equations are sixth-order, indicating a more complex structure with two scalar degrees of freedom.
• Point-like Lagrangians are derived for all four connection families ($\Gamma_1$ to $\Gamma_4$) in FLRW, Kantowski-Sachs, and Bianchi III geometries.
• The non-coincident gauge formulation reveals that the dynamic degrees of freedom arise from scalar fields associated with the connection's non-trivial components.
• A vacuum solution is explicitly constructed in the non-coincident gauge, confirming the integrability of the system in this framework.
• The derived Lagrangians include terms involving $f'(Q)$, $f''(Q)$, and canonical variables such as $a$, $N$, and $\psi$, with proper surface terms handled via integration by parts.