Shear-free gravitational collapse of dust cloud and dark energy

The present paper deals with the gravitational collapse of shear-free inhomogeneous spherical star consisting of dust fluid (Formula: see text) in the background of dark energy components (Formula: see text) with equation of state Formula: see text (Formula: see text being a nonzero constant parameter describing dark energy component). We discussed the development of apparent horizon to investigate the black hole formation in gravitational collapsing process. The collapsing process is examined first separately for dust cloud and dark energy and then under the combined effect of dust interacting with dark energy. It is obtained that when only dust cloud or dark energy is present the collapse leads to the formation of black hole under certain conditions. When both of them are present, the collapsing star does not form black hole. However, when dark energy is considered as cosmological constant Formula: see text, the collapse leads to black hole formation.