This research study investigates Barrow holographic dark energy with an energy density of ${\rho }_{\Lambda }=C{H}^{2-\Delta }$ by considering the Hubble horizon as the IR cut-off in the $f(R,T)$ gravity framework. We employ Barrow holographic dark energy to obtain the equation of the state for the Barrow holographic energy density in a flat FLRW Universe. Concretely, we study the correspondence between quintessence, k-essence, and dilation scalar field models with the Barrow holographic dark energy in a flat $f(R,T)$ Universe. Furthermore, we reconstruct the dynamics and potential for all these models for different values of the Barrow parameter: $\Delta$. Via this study, we can show that for Barrow holographic quintessence, k-essence, and dilation scalar field models, if the corresponding model parameters satisfy some limitations, the accelerated expansion can be achieved.

In this paper, we investigate a scalar field cosmological model of accelerating Universe with the simplest parametrization of the equation of state parameter of the scalar field. We use $H\left(z\right)$ data, pantheon compilation of SN Ia data and BAO data to constrain the model parameters using the ${\chi }^2$ minimization technique. We obtain the present values of Hubble constant $H_0$ as $66.2_{-1.34}^{+1.42}$, $70.7_{-0.31}^{+0.32}$ and $67.{74}_{-1.04}^{+1.24}$ for $H\left(z\right)$, $H\left(z\right)$ + Pantheon and $H\left(z\right)$ + BAO respectively. In addition, we estimate the present age of the Universe in a derived model $t_0=14.{38}_{-0.64}^{+0.63}$ for joint $H\left(z\right)$ and pantheon compilation of SN Ia data which has only $0.88\sigma$ tension with its empirical value obtained in Plank collaboration. Moreover, the present values of the deceleration parameter $q_0$ come out to be $-0.{55}_{-0.038}^{+0.031}$, $-0.{61}_{-0.021}^{+0.030}$ and $-0.{627}_{-0.025}^{+0.022}$ by bounding the Universe in the derived model with $H\left(z\right)$, $H\left(z\right)$ + Pantheon compilation of SN Ia and $H\left(z\right)$ + BAO data sets, respectively. We also have performed the state-finder diagnostics to discover the nature of dark energy.

We investigate the Barrow holographic dark energy (BHDE) model using IR cutoff as the Hubble horizon in the background of a flat Friedmann–Lemaître–Robertson–Walker universe. The deceleration parameter exhibits the universe evolution from decelerated to accelerated phase. The EoS parameter of the BHDE model presents a nice behavior as it lies in the phantom era (\(\omega_{B}<-1\)) and the quintessence era (\(\omega_{B}\geq-1\)) for different values of the Barrow parameter \(\Delta\). The squared sound velocity \(v_{s}^{2}\) has been investigated for the stability of the model. In addition, a correspondence with the phantom and quintessence scalar fields has been studied for the model, which helps us to describe the accelerated expansion of the universe.