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.

In this work, we have studied the quasinormal modes of a black hole in a model of the type $$f(Q)=\underset{n}{\sum }a_{n}\left( Q-Q_{0}\right) ^{n} $$f ( Q ) = ∑ n a n Q - Q 0 n in f(Q) gravity by using a recently introduced method known as Bernstein spectral method and confirmed the validity of the method with the help of well known Padé averaged higher order WKB approximation method. Here we have considered scalar perturbation and electromagnetic perturbation in the black hole spacetime and obtained the corresponding quasinormal modes. We see that for a non-vanishing nonmetricity scalar $$Q_0$$Q 0 , quasinormal frequencies in scalar perturbation are greater than those in electromagnetic perturbation scenarios. On the other hand, the damping rate of gravitational waves is higher for electromagnetic perturbation. To confirm the quasinormal mode behaviour, we have also investigated the time domain profiles for both types of perturbations.

We propose a logarithmic parametrization form of energy density for the scalar field dark energy in the framework of the standard theory of gravity, which supports the necessary transition from the decelerated to the accelerated behavior of the Universe. The model under consideration is constrained by available observational data, including cosmic chronometers data-sets (CC), Baryonic Acoustic Oscillation (BAO) data-sets, and Supernovae (SN) data-sets, consisting of only two parameters $$\alpha $$α and $$\beta $$β . The combined CC+BAO+SN data-sets yields a transition redshift of $$z_{tr}=0.79^{+0.02}_{-0.02}$$z tr = 0 . 79 - 0.02 + 0.02 , where the model exhibits signature-flipping and is consistent with recent observations. For the combined data-sets, the present value of the deceleration parameter is calculated to be $$q_{0}=-0.43^{+0.06}_{-0.06}$$q 0 = - 0 . 43 - 0.06 + 0.06 . Furthermore, the analysis yields constraints on both the parameter density value for matter and the present value of the Hubble parameter, with values of $$\Omega _{m0}=0.25849^{+0.00026}_{-0.00025}$$Ω m 0 = 0 . 25849 - 0.00025 + 0.00026 and $$H_{0}=67.79_{-0.59}^{+0.59}$$H 0 = 67 . 79 - 0.59 + 0.59 km/s/Mpc, respectively, consistent with the results obtained from Planck 2018. Finally, the study investigates how the mass of a black hole evolves over time in a Universe with both matter and dark energy. It reveals that the black hole mass increases initially but stops increasing as dark energy dominates.