Sudden Future Singularities

We show that a singularity can occur at a finite future time in an expanding Friedmann universe even when ρ > 0 and ρ+ 3p > 0. Explicit examples are constructed and a simple condition is given which can be used to eliminate behaviour of this sort if it is judged to be unphysical. There have been many recent investigations into the theoretical possibility that expanding universes can come to a violent end at a finite future time, without experiencing an expansion maximum or subsequent collapse to an allencompassing ’big crunch’ singularity. It has been shown that in general relativity and related metric theories of gravity this looming ’big rip’ in spacetime can be precipitated by the presence of matter fields which violate the dominant energy condition [1]. Hence, it can occur when matter exists with the extreme constitutional property that ρ+ p < 0, (1) where ρ is the fluid density and p is the pressure. Such extremal fluids have been dubbed ’phantom’ or ’ghost’ fields and their behaviour is counter intuitive in many respects. Here, we will show that although the extreme ’phantom’ equation of state ρ+p < 0 is sufficient to produce a singularity in the future of a non-contracting universe, as discussed in refs.[1], it is by no means necessary. A finite-time singularity can arise in the expanding phase of a Friedmann universe without requiring ρ + p < 0. In fact, we show that such a singularity can arise even though the strong-energy condition holds and the matter in the universe obeys ρ > 0 and ρ+ 3p > 0, regardless of the sign of the 3-curvature of the universe. Consider the Friedmann universe with expansion scale factor a(t), curvature parameter k, and Hubble expansion rate H = ȧ/a; then (8πG = c = 1) the Einstein equations reduce to