One speculation in cosmology is that the dark energy may take the form of ``phantom energy'' with an equation-of-state parameter w<−1. Suppose that the universe is spatially flat and contains matter with a density parameter Ωm,0, and phantom energy with a density parameter Ωp,0=1−Ωm,0 and equation-of-state parameter wp<−1. At what scale factor amp are the energy density of phantom energy and matter equal? Write down the Friedmann equation for this universe in the limit that a≫amp. Integrate the Friedmann equation to show that the scale factor a goes to infinity at a finite cosmic time trip, given by the relation
H0(trip−t0)≈3∣1+wp∣2(1−Ωm,0)−1/2.(5.119)
This fate for the universe is called the ``Big Rip.'' Current observations of our own universe are consistent with H0=68 km s−1 Mpc−1, Ωm,0=0.3, and wp=−1.1. If these numbers are correct, how long do we have remaining until the ``Big Rip''?