FLRW solutions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>Q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> theory: The effect of using different connections
N. Dimakis, Andronikos Paliathanasis, M. Roumeliotis, T. Christodoulakis
Abstract
We study a Friedmann-Lema\^{\i}tre-Robertson-Walker spacetime in the theory of $f(Q)$ gravity, where $Q$ denotes the nonmetricity 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 $Q=\mathrm{const}$ case, which just reproduces General Relativity with a cosmological constant, and derive novel vacuum solutions for a power-law $f(Q)$ function.