Turnaround radius in ΛCDM and dark matter cosmologies. II. The role of dynamical friction

Abstract

This paper is an extension of the paper by Del Popolo et al. [Phys. Rev. D 101, 083505 (2020)] to take account of the effect of dynamical friction. We show how dynamical friction changes the threshold of collapse ${\ensuremath{\delta}}_{c}$ and the turnaround radius ${R}_{t}$. We numerically determine the relationship between the turnaround radius ${R}_{\mathrm{t}}$ and mass ${M}_{\mathrm{t}}$ in $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$, dark energy scenarios, and an $f(R)$ modified gravity model. Dynamical friction gives rise to a ${R}_{\mathrm{t}}\text{\ensuremath{-}}{M}_{\mathrm{t}}$ relation differing from that of the standard spherical collapse. In particular, dynamical friction amplifies the effects of shear and vorticity already studied by Del Popolo et al. A comparison of the ${R}_{\mathrm{t}}\text{\ensuremath{-}}{M}_{\mathrm{t}}$ relationships for $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$, dark energy, and modified gravity models shows that the ${R}_{\mathrm{t}}\text{\ensuremath{-}}{M}_{\mathrm{t}}$ relationship of $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ is similar to that of the dark energy models, and small differences are seen when comparing with the $f(R)$ models. The effects of shear, rotation, and dynamical friction are particularly evident at galactic scales, giving rise to a difference between the ${R}_{\mathrm{t}}\text{\ensuremath{-}}{M}_{\mathrm{t}}$ relation of the standard spherical collapse of the order of $\ensuremath{\simeq}60%$. Finally, we show how the new values of the ${R}_{\mathrm{t}}\text{\ensuremath{-}}{M}_{\mathrm{t}}$ relation influence the constraints on the $w$ parameter of the equation of state.