Einasto-inspired non-commutative black holes within dark matter galactic halos in f(Q) gravity

This manuscript aims to discuss the possible construction of non-commutative dark matter gravitationally confined configurations using the $f\left(Q\right)$ model of gravity. To achieve this goal, we consider a linear representation of $f\left(Q\right)$, given by $f\left(Q\right)={\sigma }_{0}Q+{\sigma }_1$, where ${\sigma }_0$ and ${\sigma }_1$ are free parameters, and $Q$ denotes the non-metricity associated with the theory. To model the non-commutative smeared sources of the gravitational field, we incorporate the Einasto density distribution, which generalizes mini black hole structures and also accounts for dark matter as a possible matter component of the structure. We show that the combination of a linear $f\left(Q\right)$ model with the Einasto model, through the mechanism of a de Sitter-inspired equation of state, enables the formulation of different dark matter galactic halos. Additionally, we demonstrate that using a nonlocalized equation of state leads to the formation of stellar configurations, smeared fuzzy droplets. However, the radial pressure within these diffused droplet stellar distributions remains negative. By developing diffused dark-matter fuzzy structures subject to the prevalent dark-matter models, we investigate the possible gravitational connection between black holes and supermassive astrophysical structures. We classify these stellar configurations into three distinct categories: fuzzy self-gravitating droplets with no event horizon, fuzzy black holes with a single horizon, and black hole objects with two horizons. Our investigation focuses on determining if these self-gravitating black hole structures can exist at galactic centers.