[Paper Review] FLRW solutions in $f(Q)$ theory: the effect of using different connections

N. Dimakis, A. Paliathanasis+2 more|arXiv (Cornell University)|May 10, 2022

Cosmology and Gravitation Theories4 citations

TL;DR

This paper investigates Friedmann–Lemaître–Robertson–Walker (FLRW) cosmologies in $f(Q)$ gravity using four distinct symmetric, flat connections compatible with FLRW isometries. By treating the non-metricity scalar $Q$ as the time variable, the authors derive general solutions for arbitrary $f(Q)$ with perfect fluid matter and novel vacuum solutions for power-law $f(Q)$, demonstrating dynamics beyond General Relativity, especially when $Q \neq$ const.

ABSTRACT

We study a Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) space-time in the theory of $f(Q)$-gravity, where $Q$ denotes the non-metricity scalar. It has been previously shown in the literature, that there exist four distinct families of connections, which are compatible with the isometries of the FLRW metric; three for the spatially flat case and one when the spatial curvature is present. In the spatially flat case, one connection is dynamically irrelevant and yields the dynamics of the coincident gauge in the Cartesian coordinates. For this, we obtain the general solution of an arbitrary $f(Q)$ theory with a perfect fluid matter content, and present various examples for specific choices of the $f(Q)$ function. We proceed by studying the effect of the rest of the connections, which are dynamical and affect the equations of the motion. We concentrate in scenarios that depart from the $Q=$const. case, which just reproduces General Relativity with a cosmological constant, and derive novel vacuum solutions for a power-law $f(Q)$ function.

Motivation & Objective

• To analyze the impact of different symmetric, flat connections on FLRW cosmology in $f(Q)$ gravity, particularly those compatible with spatial curvature and isometries.
• To resolve the ambiguity in gauge choice by systematically studying all four connections (three for flat, one for curved spatial sections) rather than assuming the coincident gauge a priori.
• To derive general analytical solutions for arbitrary $f(Q)$ theories with perfect fluid matter, especially by using $Q$ as the time variable to bypass coordinate-dependent complications.
• To explore non-GR dynamics by focusing on cases where $Q$ is not constant, thus avoiding dynamical equivalence to GR with a cosmological constant.
• To present new vacuum solutions for power-law $f(Q)$ functions, including a Milne-like solution in the non-flat case, and reduce complex equations to integrable forms.

Proposed method

• Identify and classify all symmetric, flat connections compatible with the isometries of the FLRW metric, distinguishing between spatially flat and curved cases.
• Use the non-metricity scalar $Q$ as the independent time variable to reparametrize the dynamical equations, simplifying the system and enabling analytical integration.
• Derive the field equations for $f(Q)$ gravity in each connection class, showing how $Q$ depends on the connection's functional form and the metric.
• For the flat case, solve the equations for arbitrary $f(Q)$ with a perfect fluid, identifying a degenerate solution where one connection's function $\gamma(t)$ does not affect dynamics.
• For the two non-degenerate connections, derive general vacuum solutions for power-law $f(Q) = \alpha Q^n$, obtaining exact or partially reduced solutions.
• Reduce the dynamical system for the third connection to an Abel-type differential equation, enabling further analysis despite non-integrability.

Experimental results

Research questions

• RQ1 How do different symmetric, flat connections compatible with FLRW isometries affect the cosmological dynamics in $f(Q)$ gravity?
• RQ2 What is the role of the non-metricity scalar $Q$ as a time variable in simplifying and solving the field equations for arbitrary $f(Q)$ theories?
• RQ3 Can non-constant $Q$ solutions in $f(Q)$ gravity yield dynamics distinct from General Relativity with a cosmological constant?
• RQ4 What are the exact or partially reduced solutions for vacuum $f(Q)$ models with power-law $f(Q)$, especially in the non-flat FLRW case?
• RQ5 How does the arbitrariness of the connection function $\gamma(t)$ in the coincident gauge affect particle motion and physical observables in non-metric theories?

Key findings

• For the spatially flat case, one connection yields a degenerate solution where the function $\gamma(t)$ in the connection does not affect the gravitational equations, corresponding to the coincident gauge in Cartesian coordinates.
• For the two dynamically active flat connections, the authors derive a general vacuum solution for power-law $f(Q) = \alpha Q^n$, showing non-GR dynamics when $Q \neq$ const.
• In the non-flat case, a special solution is found that resembles the Milne universe (Riemann flat), with $Q = \frac{24k\mu^{2}}{(1-2\mu)^{2}a_{0}^{2}}e^{\mp\frac{2\sqrt{-k}t}{2\mu-1}}$, valid for $\mu \neq 1$.
• The solution for the third flat connection reduces to solving an Abel equation of the second kind, indicating non-trivial but analyzable dynamics.
• The non-metricity scalar $Q$ remains non-constant even in the $\mu=1$ limit (GR limit), unless $\gamma(t) = \mp\sqrt{-k}$, showing that GR is not recovered in the usual way.
• The use of $Q$ as the time variable successfully generates solutions outside the GR regime, confirming that $f(Q)$ gravity can support rich cosmological dynamics beyond standard models.