In this study, we explore several new characteristics of a static anisotropic hybrid star with strange quark matter (SQM) and ordinary baryonic matter (OBM) distribution. Here, we use the MIT bag model equation of state to connect the density and pressure of SQM inside stars, whereas the linear equation of state $$p_r =\alpha \rho -\beta $$p r = α ρ - β connects the radial pressure and matter density caused by baryonic matter. The stellar model was developed under a background of f(Q) gravity using the quadratic form of f(Q). We utilized the Tolman–Kuchowicz ansatz (Tolman in Phys. Rev. 55:364–373, 1939; Kuchowicz in Acta Phys Pol 33: 541, 1968) to find the solutions to the field equations under modified gravity. We have matched the interior solution to the external Schwarzschild spacetime in order to acquire the numerical values of the model parameters. We have selected the star Her X-1 to develop various profiles of the model parameters. Several significant physical characteristics have been examined analytically and graphically, including matter densities, tangential and radial pressures, energy conditions, anisotropy factor, redshirt, compactness, etc. The main finding is that there is no core singularity present in the formations of the star under investigation. The nature of mass and the bag constant $$B_g$$B g have been studied in details through equi-mass and equi- $$B_g$$B g contour. The maximum allowable mass and the corresponding radius have been obtained via $$M-R$$M - R plots.

Using the Noether symmetry approach, we investigate \(f( \mathcal{R} , \varphi ,\chi)\) theories of gravity, where \( \mathcal{R} \) is the scalar curvature, \( \varphi \) is the scalar field, and \(\chi\) is the kinetic term of \( \varphi \). Based on the Lagrangian for \(f( \mathcal{R} , \varphi ,\chi)\) gravity, we obtain the determining equations. We consider \(f( \mathcal{R} , \varphi ,\chi)\) models of a flat Friedmann–Robertson–Walker universe. Using the obtained solutions, we find conserved quantities. In the framework of this scenario, the continuity equation is extremely important for analyzing the energy density and pressure. Using the first integral of motion, we present a graphical analysis of the energy density, pressure component, and parameter of the equation of state. The negativity of the pressure observed in the considered cases in fact suggests that this theory can describe a Noether universe with dark matter.

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.