We prove that a homogeneous and isotropic universe containing a scalar field with a power-law potential, V (φ) = Aφ n , with 0 < n < 1 and A > 0 always develops a finite-time singularity at which the Hubble rate and its first derivative are finite, but its second derivative diverges. These are the first examples of cosmological models with realistic matter sources that possess weak singularities of 'sudden' type. We also show that a large class of models with even weaker singularities exist for noninteger n > 1. More precisely, if k < n < k + 1 where k is a positive integer then the first divergence of the Hubble rate occurs with its (k + 2)th derivative. At early times these models behave like standard large-field inflation models but they encounter a singular end-state when inflation ends. We term this singular inflation.

We introduce and study extensions of the varying alpha theory of Bekenstein-Sandvik-Barrow-Magueijo to allow for an arbitrary coupling function and self-interaction potential term in the theory. We study the full evolution equations without assuming that variations in alpha have a negligible effect on the expansion scale factor and the matter density evolution, as was assumed in earlier studies. The background FRW cosmology of this model in the cases of zero and non-zero spatial curvature is studied in detail, using dynamical systems techniques, for a wide class of potentials and coupling functions. All the asymptotic behaviours are found, together with some new solutions. We study the cases where the electromagnetic parameter, zeta, is positive and negative, corresponding to magnetic and electrostatic energy domination in the non-relativistic matter. In particular, we investigate the cases where the scalar field driving alpha variations has exponential and power-law self-interaction potentials and the behaviour of theories where the coupling constant between matter and alpha variations is no longer a constant.

It has been well known since the 1970s that stationary black holes do not generically support scalar hair. Most of the no-hair theorems which support this depend crucially upon the assumption that the scalar field has no time dependence. Here we fill in this omission by ruling out the existence of stationary black hole solutions even when the scalar field may have time dependence. Our proof is fairly general, and in particular applies to non-canonical scalar fields and certain non-asymptotically flat spacetimes. It also does not rely upon the spacetime being a black hole.

Vacuum solutions admitting a hypersurface-orthogonal repeated principal null direction are an important class of 4d algebraically special spacetimes. We investigate the 5d analogues of such solutions: vacuum spacetimes admitting a hypersurface-orthogonal multiple Weyl aligned null direction (WAND). Such spacetimes fall into 4 families determined by the rank of the 3×3 matrix that defines the expansion and shear of the multiple WAND. The rank 3 and rank 0 cases have been studied previously. We investigate the 2 remaining families. We show how to define coordinates which lead to a considerable simplification of the Einstein equation with cosmological constant. The rank 2 case gives warped product and Kaluza-Klein versions of the 4d Robinson-Trautman solutions as well as some new solutions. The rank 1 case gives product, or analytically continued Schwarzschild, spacetimes. * hsr1000@cam.ac.uk

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)These equations permit there to be a finite time, t s , at which a, H, and ρ all remain finite, in accord with Eq. (1), but where p,ρ andä 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 singularityä → ∞ to arise at finite time as t → t s because of a divergence in the matter pressure, p → ∞. Here is an explicit example. On the time interval 0 ≤ t ≤ t s , we can choose a solution for the scale factor a(t) of the formwhere a s ≡ a(t s ), q and n are positive constants. If t → t s from below then a → a s , H → H s and ρ → ρ s > 0, where a s , H s , and ρ s are all finite, but p → ∞ andä → −∞ whenever 1 < n < 2 and 0 < q ≤ 1. As t → 0 we *