Mimetic-$f(Q)$ gravity: cosmic reconstruction and energy conditions

In this study, we present a novel approach to mimetic gravity incorporating a non-zero nonmetricity tensor with vanishing torsion and curvature, establishing a generalized mimetic-$f(Q)$ gravity framework. Using the Lagrange multiplier method, we have obtained and discussed characteristics of the theory’s field equations. In order to study cosmic evolution given by the hybrid scale factor, we implemented the reconstruction method in two different ways. In the first case, we have obtained corresponding Lagrange multiplier $\eta$ and potential $U$ for the specific $f(Q)=f(Q)=Q-6\lambda M^2(\frac{Q}{6M^2}{)}^{\alpha }$ function, while in the second scenario we have recovered $f(Q)$ functional and mimetic potential for the given Lagrange multiplier ${\eta }_0+\gamma H^2$. Subsequently, we explore the fundamental properties of the $f(Q)=Q-6\lambda M^2(\frac{Q}{6M^2}{)}^{\alpha }$ model and analyse the energy conditions to establish its validity. Our findings indicate that the framework introduced herein allows for the derivation of a wide range of viable cosmological models that satisfy energy constraints that are necessary in description of the accelerated expansion.