Quintaverse-Phantiverse Cosmologies in $f(Q,C)$ Gravity: a Non-coincident Geometrical Origin of Cosmic Acceleration

We investigate the asymptotic dark energy sectors of a quadratic form of $f(Q,C)$ Lagrangian. We reformulate the original equations of motion to obtain the equivalent dynamical system, for two non-trivial classes of affine connections allowed by symmetries of $f(Q,C)$ gravity. In these scenarios, the late-time accelerated expansion emerges as an intrinsic geometrical effect of the $f(Q,C)$ sector, without invoking an independent dark energy component. We find that an effective dark matter solution emerges for one class, whereas for the other class, the positivity of the matter density and the conservation of the momentum tensor cannot be simultaneously satisfied. This leads to the conclusion that there exists an effective dark matter solution for which the matter contribution is a combination of a pressureless matter as well as a stiff fluid. Furthermore, in both cases there always exists a universal stable accelerated expansion solution. Besides, the matter solutions are saddle points for those values of the model parameters that ensure the conservation of the energy momentum. In addition, we introduce a parameterization in which the equation of state parameter at the fixed point corresponding to accelerated expansion is characterized by a free parameter that quantifies its deviation from $-1$. We thereby inspired to invent the terms Quintaverse'' and Phantiverse’', corresponding to quintessence-like and phantom-like attractor solutions. Also, we numerically illustrate the evolution of different cosmological quantities and the phase-space diagrams. Particularly, the behavior of the Universe using two kinematic toolkits, namely, the cosmographic and statefinder variables, is discussed.