$\phi$CDM models provide an alternative to the standard $\Lambda$CDM paradigm, while being physically better motivated. These models lead to a time-dependent speed of sound for dark energy that is difficult to replicate by $w$CDM parametrizations. We review the most up-to-date status of observational evidence for the $\phi$CDM models in this paper. We start with an overview of the motivation behind these classes of models, the basic mathematical formalism, and the different classes of models. We then present a compilation of recent results of applying different observational probes to constraining $\phi$CDM model parameters. Over the last twenty years, the precision of observational data has increased immensely, leading to ever tighter constraints. A combination of the recent measurements favors the spatially flat $\Lambda$CDM model, but a large class of $\phi$CDM models is still not ruled out.

We investigate how well a simple leading order perturbation theory model of the bispectrum can fit the BAO feature in the measured bispectrum monopole of galaxies. Previous works showed that perturbative models of galaxy bispectrum start failing at the wavenumbers of k ~ 0.1 Mpc/h. We show that when the BAO feature in the bispectrum is separated it can be successfully modeled up to much higher wavenumbers. We validate our modeling on GLAM simulations that were run with and without the BAO feature in the initial conditions. We also quantify the amount of systematic error due to BAO template being offset from the true cosmology. We find that the systematic errors do not exceed 0.3 per cent for reasonable deviations from the true cosmology.

We investigate the potential of machine learning (ML) methods to model small-scale galaxy clustering for constraining Halo Occupation Distribution (HOD) parameters. Our analysis reveals that while many ML algorithms report good statistical fits, they often yield likelihood contours that are significantly biased in both mean values and variances relative to the true model parameters. This highlights the importance of careful data processing and algorithm selection in ML applications for galaxy clustering, as even seemingly robust methods can lead to biased results if not applied correctly. ML tools offer a promising approach to exploring the HOD parameter space with significantly reduced computational costs compared to traditional brute-force methods if their robustness is established. Using our ANN-based pipeline, we successfully recreate some standard results from recent literature. Properly restricting the HOD parameter space, transforming the training data, and carefully selecting ML algorithms are essential for achieving unbiased and robust predictions. Among the methods tested, artificial neural networks (ANNs) outperform random forests (RF) and ridge regression in predicting clustering statistics, when the HOD prior space is appropriately restricted. We demonstrate these findings using the projected two-point correlation function ($w_\mathrm{p}(r_\mathrm{p})$), angular multipoles of the correlation function ($\xi_\ell(r)$), and the void probability function (VPF) of Luminous Red Galaxies from Dark Energy Spectroscopic Instrument mocks. Our results show that while combining $w_\mathrm{p}(r_\mathrm{p})$ and VPF improves parameter constraints, adding the multipoles $\xi_0$, $\xi_2$, and $\xi_4$ to $w_\mathrm{p}(r_\mathrm{p})$ does not significantly improve the constraints.

The cosmological Mass Varying Neutrino model beyond the standard $\Lambda$CDM scenario is considered. The interaction of the fermionic field and the scalar field with the inverse power law Ratra-Peebles potential via the Yukawa coupling is studied in detail. Depending on the model parameter $\alpha$ of the Ratra-Peebles potential of the scalar field, the expansion rate of the universe, the mass equation, the mass of the scalar field, the sum of neutrino masses, the mutual influence of the sum of neutrino masses and the value of the scalar field Ratra-Peebles potential as well as the total density of the thermodynamic potential of the coupled fermionic and scalar fields at the critical point are explored. The values of the sum of neutrino masses $m_\nu(a_0)\leq0.07~{\rm eV}$ calculated for values of the model parameter $\alpha$ of the Ratra-Peebles potential 0&lt;\alpha\leq0.016 are consistent with the constraint m_\nu(a_0)&lt;0.071~{\rm eV} of the cosmological DESI measurements, while the values $m_\nu(a_0)\leq0.45~{\rm eV}$ for 0&lt;\alpha\leq0.143 are consistent with the upper limit $m_\nu(a_0)<0.45~{\rm eV}$ obtained in the KATRIN experiment.