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

In this study, we present a approach to mimetic gravity incorporating a non-zero nonmetricity tensor with vanishing torsion and curvature, establishing a generalized mimetic-$f\left(Q\right)$ 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 η and potential U for the specific $f\left(Q\right)=f\left(Q\right)=Q-6\lambda M^2{\left(\frac{Q}{6M^2}\right)}^{\alpha }$ function, while in the second scenario we have recovered $f\left(Q\right)$ functional and mimetic potential for the given Lagrange multiplier ${\eta }_0+\gamma H^2$. Subsequently, we explore the fundamental properties of the $f\left(Q\right)=Q-6\lambda M^2{\left(\frac{Q}{6M^2}\right)}^{\alpha }$ model and analyse the energy conditions to establish its validity. Our findings indicate that the introduced framework allows for the derivation of cosmological models that satisfy necessary energy constraints. Specifically, we show that the considered model enters the quintessence region for the equation of state parameter w, simultaneously violating the strong energy condition (SEC), leading to repulsive behaviour consistent with accelerated expansion. Thus, the introduced extension demonstrates potential for accurately describing cosmological models.