Interacting holographic dark energy, cosmic coincidence and the future singularity of the closed FRW universe

• Decelerating as well as accelerating phases of the universe are studied.
• Consistency of the results with the current observational data is shown.
• Fraction of coincidental time to the total lifetime of the universe is calculated.
• Correspondence between the holographic dark energy model and phantom is presented.

In this paper, we have considered the closed FRW universe filled with two interacting fluids; dark matter and holographic dark energy components. Under certain conditions, this dark energy model is characterised by a big rip type future singularity and therefore a finite life time of the universe (Cruz et al., 2008). As the universe passes through a significant time when the matter and the dark energy densities are roughly comparable between the matter and the dark energy dominated era. So, we calculated the fraction of total life time of the universe when the universe passes through the coincidental stage for this future singularity by considering 1ro