Cosmological solutions and growth index of matter perturbations in $f(Q)$ gravity

The present work studies one of Einstein's alternative formulations based on the non-metricity scalar $Q$ generalized as $f(Q)$ theory. More specifically, we consider the power-law form of $f(Q)$ gravity i.e. $f(Q)=Q+\alpha\, Q^n$. Here, we analyze the behavior of the cosmological model at the background and perturbation level. At the background level, we find the effective evolution of the model is the same as that of the $\Lambda$CDM for $|n|<1$. Interestingly, the geometric component of the theory solely determined the late-time acceleration of the Universe. We also examine the integrability of the model by employing the method of singularity analysis. In particular, we find the conditions under which field equations pass the Painlev\'{e} test and hence possess the Painlev\'{e} property. While the equations pass the Painlev\'{e} test in the presence of dust for any value of $n$, the test is valid after the addition of radiation fluid only for $n<1$. Finally, at the perturbation level, the behavior of matter growth index signifies a deviation of the model from the $\Lambda$CDM even for $|n|<1$.