Influence of scalar fields on the approach to a cosmological singularity

In their long-term study of the approach to the singularity in generic cosmological spacetimes [1–5], Belinskii, Khalatnikov, and Lifshitz (BKL) concluded that generic cosmological spacetimes approach the singularity as a (different) Mixmaster universe [6] at every spatial point. The Mixmaster universe is characterized by “oscillations” from one Kasner solution [7] to another. When BKL considered the influence of a classical minimally coupled scalar field on the approach to the singularity, they found that it can suppress Mixmaster oscillations [8]. While this result is now well-known, the mechanism by which it happens is not widely appreciated. I wish to address this issue here. The role of the scalar field may be easily clarified using the method of consistent potentials (MCP) originally due to Grubĭsić and Moncrief [9]. The MCP has been applied to a variety of cosmological spacetimes to explain the nature of the (numerically observed) approach to the singularity in spatially inhomogeneous cosmologies [10]. The MCP first assumes that the approach to the singularity is asymptotically velocity term dominated (AVTD) [11] and then looks for a contradiction. If the model is AVTD, it approaches arbitrarily closely to a Kasner solution with a possibly different Kasner solution at every spatial point. For any model, an asymptotic velocity term dominated (VTD) solution may be found by neglecting all terms in Einstein’s equations containing spatial derivatives and taking the limit as τ →∞. (The MCP includes the implicit assumption that strong cosmic censorship holds for these cosmological models — i.e. there exists a foliation labeled by τ such that some curvature invariant blows up as τ →∞.) If the model is actually AVTD, then substitution of the VTD solution into the full Einstein equations will be consistent — asymptotically, all terms neglected in obtaining the VTD solution will be exponentially small. For convenience, the MCP will be applied to the Hamiltonian whose variation yields the relevant Einstein equations since exponentially small (growing) terms in the Hamiltonian will yield exponentially small (growing) terms in Einstein’s equations upon variation. Any terms which cannot be made consistent with the VTD solution indicate, within the MCP, that the model is not AVTD. Here I shall consider essentially the same models as in [10] but with the addition of a scalar field. I shall explore the influence of a massless, minimally coupled scalar field in some detail. The possibly interesting dynamics which may result from exponentially coupled scalar fields will be discussed elsewhere. For each cosmology, I shall use the MCP to show how the scalar field yields an asymptotically velocity term dominated (AVTD) approach to the singularity. Consider first the primary Mixmaster model — the spatially homogeneous vacuum Bianchi Type IX cosmology [4,6]. In the presence of a spatially homogeneous scalar field, φ(τ), Einstein’s equations are obtained from the variation of the Hamiltonian constraint H = 0 where