New Singularities in Unexpected Places

Spacetime singularities have been discovered which are physically much weaker than those predicted by the classical singularity theorems. Geodesics evolve through them and they only display infinities in the derivatives of their curvature invariants. So far, these singularities have appeared to require rather exotic and unphysical matter for their occurrence. Here we show that a large class of singularities of this form can be found in a simple Friedmann cosmology containing only a scalar-field with a power-law self-interaction potential. Their existence challenges several preconceived ideas about the nature of spacetime singularities and impacts upon the end of inflation in the early universe. A striking feature of relativistic cosmology is the prediction that past and future singularities can occur. Originally, singularities were defined by the existence of incomplete geodesics, and a variety of sufficient conditions for geodesic incompleteness were established by a series of important theorems from 1965-1972 [1]. More recently, by using the Einstein equations, new types of physical singularities have been identified which can occur at finite time and are unaccompanied by geodesic incompleteness [2, 3, 4]. Many quantities, such as the density and the expansion rate, which diverge at traditional ’big bang’ singularities, remain finite whilst other physical quantities, like the pressure, diverge in finite proper time. The simplest example of what is termed a ’sudden’ singularity occurs in the zerocurvature Friedmann universe with scale factor a(t) and Hubble rate H = ȧ/a, containing matter with density ρ and pressure p. The field equations are (8πG = 1 = c) 3H = ρ, (1) ρ̇ = −3H(ρ+ p), (2) ä = − (ρ+ 3p)a 6 , (3) These equations permit there to be a finite time, ts, at which a,H, and ρ all remain finite, in accord with Eq. (1), but where p, ρ̇ and ä all become infinite, in accord with Eqs. (2)-(3). The key to their existence is in not assuming any functional link between p and ρ, nor any boundedness condition on p, and this freedom allows an acceleration singularity ä → ∞ to arise at finite time as t → ts because of a divergence in the matter pressure, p → ∞. Here is an explicit example. On the time interval 0 ≤ t ≤ ts, we can choose a solution for the scale factor a(t) of the form