"If the dark energy in the universe increases without limit, it could overcome all forces that hold the universe together. The key value is the equation of state parameter w, the ratio between the dark energy pressure and its energy density. If −1 < w < 0, the expansion of the universe tends to accelerate, but the dark energy tends to dissipate over time, and the Big Rip does not happen."
"According to the latest cosmological data available, the uncertainties are still too large to discriminate among the three cases w < −1, w = −1, and w > −1."
"In their paper, the authors consider a hypothetical example with w = −1.5, H0 = 70 km/s/Mpc, and Ωm = 0.3, in which case the Big Rip would happen approximately 22 billion years from the present. In this scenario, galaxies would first be separated from each other about 200 million years before the Big Rip. About 60 million years before the Big Rip, galaxies would begin to disintegrate as gravity becomes too weak to hold them together. Planetary systems like the Solar System would become gravitationally unbound about three months before the Big Rip, and planets would fly off into the rapidly expanding universe. In the last minutes, stars and planets would be torn apart, and the now-dispersed atoms would be destroyed about 10−19 seconds before the end. At the time the Big Rip occurs, even spacetime itself would be ripped apart and the scale factor would be infinity"
Only if we figure out how to get off this one solitary rock and expand beyond the solar system. The sun (Sol) will continue to go through its natural processes which will see it expand larger than the Earth's orbit. That will be the end of physical Earth, but life as we know it will have ended before then.
So, I'm not sure who the "we" will be that will have less than 14 billion years.
