More General Sudden Singularities

We present a general form for the solution of an expanding generalrelativistic Friedmann universe that encounters a singularity at finite future time. The singularity occurs in the material pressure and acceleration whilst the scale factor, expansion rate and material density remain finite and the strong energy condition holds. We also show that the same phenomenon occurs, but under different conditions, for Friedmann universes in gravity theories arising from the variation of an action that is an arbitrary analytic function of the scalar curvature. There has been considerable interest in the possibility of finite and infinite future-time cosmological singularities in general relativity under the headings of ’big-rip’ singularities and associated ’phantom matter’ evolution [1][2]. These studies have also extended to several quantum and brane-world generalisations [3], [4][5], [6], [7, 8]. A demarcation of these and related finite-time singularities has been given in [9]. We re consider the most general form of sudden, or finite-time, singularity of the sort discussed in [10] that can arise in an Friedmann expanding universe. In [10] we showed that singularities of this sort can arise from a divergence of the pressure, p, at finite time despite the scale factor, a(t), the material density, ρ, and the Hubble expansion rate, H = ȧ/a remaining finite. The pressure singularity is accompanied by a divergence in the acceleration of the universe, ä, at finite time. Remarkably, these singularities occur without violating the strong energy conditions ρ > 0 and ρ + 3p > 0. They can even prevent a closed Friedmann universe that obeys these energy conditions from attaining an expansion maximum [11]. In order to prevent the occurrence of these sudden singularities it is necessary to bound the pressure. A sufficient condition is to require dp/dρ to be continuous or p/ρ to be finite [10]. Consider the Friedmann universe with curvature parameter k, and 8πG = c = 1, the Einstein equations reduce to