In the past few years, f(Q) theories have drawn a lot of research attention in replacing Einstein’s theory of gravity successfully. The current study examines the novel cosmological possibilities emerging from two specific classes of f(Q) models using the parametrization form of the equation of state (EoS) parameter as $$\omega \left( z\right) =-\frac{1}{1+3\beta \left( 1+z\right) ^{3}}$$ω z = - 1 1 + 3 β 1 + z 3 , which displays quintessence behavior with the evolution of the Universe. We do statistical analyses using the Markov chain Monte Carlo (MCMC) method and background datasets like Type Ia Supernovae (SNe Ia) luminosities and direct Hubble datasets (from cosmic clocks), and Baryon Acoustic Oscillations (BAO) datasets. This lets us compare these new ideas about the Universe to the $$\Lambda $$Λ CDM model in a number of different possible ways. We have come to the conclusion that, at the current level of accuracy, the values of their specific parameters are the best fits for our f(Q) models. To conclude the accelerating behavior of the Universe, we further study the evolution of energy density, pressure, and deceleration parameter for these f(Q) models.

Despite having the somewhat successful description of accelerated cosmology, the early evolution of the universe always challenges mankind. Our promising approach lies in a new class of symmetric teleparallel theory of gravity named f(Q), where the non-metricity scalar Q is responsible for the gravitational interaction, which may resolve some of the issues. To study the early evolution of the universe, we presume an anisotropic locally rotationally symmetric (LRS) Bianchi-I spacetime and derive the motion equations. We discuss the profiles of energy density, equation of state and skewness parameter and observe that our models archive anisotropic spatial geometry in the early phase of the universe with a possible presence of anisotropic fluid and as time goes on, even in the presence of an anisotropic fluid, the universe could approach isotropy due to inflation and the anisotropy of the fluid fades away at the same time.

In general relativity, the contracted Bianchi identity makes the field equation compatible with the energy conservation, likewise in f(R) theories of gravity. We show that this classical phenomenon is not guaranteed in the symmetric teleparallel theory, and rather generally f(Q) model specific. We further prove that the energy conservation criterion is equivalent to the affine connection’s field equation of f(Q) theory, and except the f ( Q ) = α Q + β model, the non-linear f(Q) models do not satisfy the energy conservation or, equivalently the second field equation in every spacetime geometry; unless Q itself is a constant. So the problem is deep-rooted in the theory, several physically motivated examples are provided in the support.

The vanishing affine connections have been used solely while adopting the modified f(Q) gravity theory to the cosmology. Consequently, researchers could not get beyond what is already known in f(T) theory earlier. To alleviate this problem, in the present manuscript we investigate a recently proposed construction of f(Q) theory using non-vanishing affine connection in the spatially flat FLRW spacetime. We then investigate the cosmological solutions of f(Q) theory for a perfect fluid through the phase space analysis. We introduce few variables and dimensionless parameters to construct the corresponding equations suitable for the dynamical system approach. The conservation of the energy-momentum tensor leads to a constraint equation that relates the dynamical variables. Briefly, both unstable and stable de Sitter solutions appear which correspond to early and late times accelerated expansions. Also, unstable points corresponding to the matter dominated and radiation dominated eras have been found which do exist for every f(Q) function. As a result, the present discussion shows that f(Q) gravity endowed by non-vanishing affine connections is capable of explaining a true sequence of cosmic eras.