Noether Symmetries in $f(Q)-$Cosmology

TL;DR

The paper investigates Noether symmetries in $f(Q)$-cosmology within symmetric teleparallel gravity, focusing on FLRW spacetimes and four distinct flat, symmetric connections. Using a minisuperspace formulation with a scalar field $\,\phi=f'(Q)$ and potential $V(\phi)=f(Q)-Qf'(Q)$, the authors apply Noether's theorem to constrain admissible $f(Q)$ forms and associated conservation laws. They show that point Noether symmetries occur only for power-law potentials, leading to a power-law form $f(Q)\propto Q^{\frac{n}{n-1}}$; for each connection, they derive corresponding invariants and, where possible, exact or similarity solutions. The results provide a geometric selection rule for viable $f(Q)$ models and illuminate integrable structures in symmetric teleparallel cosmology, including de Sitter attractor behavior in certain limits.

Abstract

We apply the Noether symmetry analysis in $f\left( Q\right)$-Cosmology to determine invariant functions and conservation laws for the cosmological field equations. For the FLRW background and the four families of connections, it is found that only power-law $f\left( Q\right)$ functions admit point Noether symmetries. Finally, exact and analytic solutions are derived using the invariant functions.

Figures (1)

• Figure 1: Qualitative evolution of the effective equation of state parameter $w_{eff}\left( z\right)$ as it is given by the numerical simulations of the dynamical system (\ref{['ll.03']}), (\ref{['ll.04']}) where $z$ is the redshift, $1+z=a^{-1}$. Plots are for initial conditions We have assumed the initial conditions $t_{0}$ is the present time, $a\left( t_{0}\right) =1$, $\phi \left( t_{0}\right) =1\,$. Left Fig. is for $I_{1}=0.01$, $\sqrt{-6V_{0}}=0.2$, while Right Fig. is for $I_{1}=-0.01$ and $\sqrt{-6V_{0}}=0.2$. The lines are for different values of parameter $n$ as they are given in the legends.