FLRW solutions in $f(Q)$ theory: The effect of using different connections

We study a Friedmann-Lemaître-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.

Figure 1

Plots of the Hubble function in $f(Q)=Q+\alpha Q^{\mu }$ theory for $\mu =5$ and $\mu =6$ and for different orders of magnitude of the coupling constant $\alpha$ in the cosmic time, $\tau$, gauge. In both graphs the corresponding GR solution is displayed with the dotted $\alpha =0$ line. For the equation of state parameter we have considered $w=\frac{1}{3}$.

Figure 2

The behavior of the Hubble function in $f(Q)=Q+\alpha Q^5$ theory and in General Relativity for various values of the perfect fluid equation of state constant $w$, with respect to the cosmic time $\tau$. The plots are given for $\alpha =1$.

Figure 3

Parametric plot that depicts the bouncing solution of the scale factor as a function of the cosmic time $\tau$, as defined by (44). The graph shows the existing discontinuity in the first derivative. The values that have been used for the involved parameters are ${\rho }_0=\alpha =1$, $w=\frac{1}{3}$.

Figure 4

Parametric plot of the Hubble function with respect to the cosmic time for an equation of state parameter $w=\frac{1}{3}$. The first graph includes positive values of $q$ and the second negative. For comparison, the corresponding GR solution ($q=0$) is given by the dotted line.

Figure 5

Graphs of the Hubble function with respect to the cosmic time for two different pairs of $\kappa$, $\lambda$ and for various different values of the power $\mu$ of $f(Q)=Q^{\mu }$.