Can Life develop in the expanded habitable zones around Red Giant Stars?

We present some new ideas about the possibility of life developing around sub-giant and red giant stars. Our study concerns the temporal evolution of the habitable zone. The distance between the star and the habitable zone, as well as its width, increases with time as a consequence of stellar evolution. The habitable zone moves outward after the star leaves the main sequence, sweeping a wider range of distances from the star until the star reaches the tip of the asymptotic giant branch. If life could form and evolve over time intervals from $5 \times 10^8$ to $10^9$ years, then there could be habitable planets with life around red giant stars. For a 1 M$_{\odot}$ star at the first stages of its post main-sequence evolution, the temporal transit of the habitable zone is estimated to be of several 10$^9$ years at 2 AU and around 10$^8$ years at 9 AU. Under these circumstances life could develop at distances in the range 2-9 AU in the environment of sub-giant or giant stars and in the far distant future in the environment of our own Solar System. After a star completes its first ascent along the Red Giant Branch and the He flash takes place, there is an additional stable period of quiescent He core burning during which there is another opportunity for life to develop. For a 1 M$_{\odot}$ star there is an additional $10^9$ years with a stable habitable zone in the region from 7 to 22 AU. Space astronomy missions, such as proposed for the Terrestrial Planet Finder (TPF) and Darwin should also consider the environments of sub-giants and red giant stars as potentially interesting sites for understanding the development of life