We study models of nonminimally coupled relativistic fluid and the $k$-essence scalar field in the background of a flat Friedmann-Lemaitre-Robertson-Walker universe. The nonminimal coupling term is introduced in the Lagrangian level. We employ the variational approach with respect to independent variables that produce modified $k$-essence field equations and the Friedmann equations. We have analyzed the coupled field-fluid framework explicitly using the dynamical system technique considering two different models based on inverse power-law potential. After examining these models it is seen that both models are capable of producing accelerating attractor solutions satisfying adiabatic sound speed conditions.

In this work, we present a new scheme to study the tachyon-dark-energy model using dynamical systems analysis by considering parametrization of the equation of state (EOS) of the dark energy. Both the canonical and phantom field dynamics are investigated. In our method, we do not require any explicit form of the tachyon potential. Instead of the potential, we start with an approximate form of the EOS of the tachyon field. This EOS is phenomenologically motivated and contains some dimensionless parameters. Using our method, we can construct the dynamical system which gives rise to the time evolution of the Universe. We have considered two different parametrizations of the EOS and studied the phase space dynamics in detail. Our analysis shows Taylor series parametrization of the EOS has serious cosmological limitations. We have also provided an example of how this method can be applied to coupled tachyon models with a specific form of interaction. Our proposal is generic in nature and can be applied to other scalar field dark energy models.

In this paper we investigate a non-minimal, space-time derivative dependent, coupling between the k-essence field and a relativistic fluid using a variational approach. The derivative coupling term couples the space-time derivative of the k-essence field with the fluid 4-velocity via an inner product. The inner product has a coefficient whose form specifies the various models of interaction. By introducing a coupling term at the Lagrangian level and using the variational technique we obtain the k-essence field equation and the Friedmann equations in the background of a spatially flat Friedmann–Lemaitre–Robertson–Walker (FLRW) metric. Explicitly using the dynamical analysis approach we analyze the dynamics of this coupled scenario in the context of two kinds of interaction models. The models are distinguished by the form of the coefficient multiplying the derivative coupling term. In the simplest approach we work with an inverse square law potential of the k-essence field. Both of the models are not only capable of producing a stable accelerating solution, they can also explain different phases of the evolutionary universe.

This work explores the dynamical stability of cosmological models where dark matter and dark energy can nonminimally couple to spacetime (scalar) curvature. Two different scenarios are presented here. In the initial case, only dark matter sector is coupled to curvature in the presence of a quintessence scalar field. In the second case both dark matter and the quintessence field are coupled to curvature. It is shown that one can get an accelerating expansion phase of the Universe in both the cases. The nature of the fixed points shows that there can be stable or unstable phases where the curvature coupling vanishes and dark energy and dark matter evolve independently. On the other hand there can be stable accelerating expansion phases where both the components are coupled to curvature.

We study the dynamics of the interacting models between the Gauss-Bonnet (GB) coupled scalar field and the dark matter fluid in a homogeneous and isotropic background. A key feature of GB coupling models is the varying speed of gravitational waves (GWs). We utilize recent constraints on the GW speed and conduct our analysis in two primary scenarios: model-dependent and model-independent. In the model-dependent scenario, where determining the GW speed requires a specific GB coupling functional form, we choose an exponential GB coupling. We adopt a dynamical system analysis to obtain the necessary constraints on the model parameters that describe different phases of the universe and produce a stable late-time accelerating solution following the GW constraint, and find that to satisfy all these constraints, fine-tuning of the free parameters involved in the models is often needed. In the model-independent scenario, the GW speed is fixed to one, and we construct the autonomous system to identify the late-time stable accelerating critical points. Furthermore, we adopt a Bayesian inference method using late-time observational data sets, including 31 data points from cosmic chronometer data (Hubble data) and 1701 data points from Pantheon+ and find that all the observational constraints can be satisfied without fine-tuning. In addition, we also utilize simulated binned Roman and LSST data to study the evolution of the universe in the model-independent scenario. We find that the model shows significant deviation at higher redshifts from ΛCDM and fits the current data much better than ΛCDM within the error bars.

We constrain the parameters of the k-essence scalar field model with inverse square and exponential potentials using data sets including Pantheon+SHOES and the Dark Energy Survey (DES) of Type Ia supernovae, Baryon Acoustic Oscillation (BAO) data from SDSS and DESI surveys, and direct measurements of the Hubble parameter and redshift obtained from the differential age method (CC). We also provide a brief perspective on the dynamical evolution of both models and derive stability constraints on the model parameters, which are then used to set appropriate priors. We adopt a Bayesian inference procedure to estimate the model parameters that best fit the data. A comprehensive analysis in light of observational data shows that the k-essence model fits well across all data combinations. However, according to the BIC criterion, the ΛCDM model provides a slightly better fit compared to the k-essence model.