A designer approach to $f(Q)$ gravity and cosmological implications

We investigate the evolution of linear perturbations in the Symmetric Teleparallel Gravity, namely $f(Q)$ gravity, for which we design the $f(Q)$ function to match specific expansion histories. We consider different evolutions of the effective dark energy equation of state, $w_Q(a)$, which includes $w_Q=-1$, a constant $w_Q \neq -1$ and a fast varying equation of state. We identify clear patterns in the effective gravitational coupling, which accordingly modifies the linear growth of large scale structures. We provide theoretical predictions for the product of the growth rate $\tilde{f}$ and the root mean square of matter fluctuations $\sigma_8$, namely $\tilde{f}\sigma_8$ and for the sign of the cross-correlation power spectrum of the galaxy fluctuations and the cosmic microwave background radiation anisotropies. These properties can be used to distinguish the $f(Q)$ gravity from the standard cosmological model using accurate cosmological observations.