Improved Lemaitre-Tolman model and the mass and turn-around radius in group of galaxies II: the role of dark energy

In this paper, we extend our previous study \cite{DelPopolo2021} on the Lemaitre-Tolman (LT) model showing how the prediction of the model changes when the equation of state parameter ($w$) of dark energy is modified. In the previous study, it was considered that dark energy was merely constituted by the cosmological constant. In this paper, as in the previous study, we also took into account the effect of angular momentum and dynamical friction ($J\eta$ LT model) that modifies the evolution of a perturbation, initially moving with the Hubble flow. As a first step, solving the equation of motion, we calculated the relationship between mass, $M$, and the turn-around radius, $R_0$. If one knows the value of the turn-around radius $R_0$, it is possible to obtain the mass of the studied objects. As a second step, we build up, as in the previous paper, a relationship between the velocity, $v$, and radius, $R$. The relation was fitted to data of groups and clusters. Since the relationship $v-R$ depends on the Hubble constant and the mass of the object, we obtained optimized values of the two parameters of the objects studied. The mass decreases of a factor of maximum 25% comparing the $J\eta$ LT results (for which $w=-1$) and the case $w=-1\mathrm{/}3$, while the Hubble constant increases going from $w=-1$ to $w=-1\mathrm{/}3$. Finally, the obtained values of the mass, $M$, and $R_0$ of the studied objects can put constraints to the dark energy equation of state parameter, $w$.