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.
