Scalar fields and the FLRW singularity

Abstract

The dynamics of multiple scalar fields on a flat FLRW spacetime can be described entirely as a relational system in terms of the matter alone. The matter dynamics is an autonomous system from which the geometrical dynamics can be inferred, and this autonomous system remains deterministic at the point corresponding to the singularity of the cosmology. We show the continuation of this system corresponds to a parity inversion at the singularity, and that the singularity itself is a surface on which the space-time manifold becomes non-orientable. * Electronic address: d.sloan@lancaster.ac.uk arXiv:1907.08287v1 [gr-qc]

New action for cosmology

“…As we have recently proven [11,31,32], the mathematical barrier to extending classical cosmological solutions beyond the initial singularity lies in the failure of the system of equations to be Lifschitz continuous, and hence satisfy the conditions of the Picard-Lindelöff theorem. This problem is alleviated when scale is removed from the system; it is only the evolution of the scale factor that is ill-defined.…”

“…Thus reports of general relativity predicting its own demise may be greatly exaggerated. This has been established through considering the relational evolution of Bianchi systems [11], including those with inflationary matter [31] and FLRW cosmologies with scalar fields [32]. The initial singularity may be thought of as a point at which dynamical evolution reaches the boundary of the description of space-time in terms of a four dimensional Riemannian manifold.…”

Total Collisions in the N-Body Shape Space

“…In this formulation, as was shown in [17], the total-collision solutions can be characterized neatly as solutions that end at a central configuration with zero dilatational momentum and zero shape momenta. The question then arises, of whether these solutions can be regularized in the manner of two-body collisions, or continued through the singularity similarly to what was done for cosmological solutions of general relativity in [18][19][20]. Regardless of whether the system has positive or zero energy, the asymptotics of the total-collision solutions is universal, and it is captured by Equation (94), which are completely determined by the eigenvalues of the Hessian matrix of the (log of) the shape potential at the central configuration.…”

Scale Symmetry and Friction

“…and hence, should we choose to reintroduce the scale factor a, thenż i ∝ a −3w . However, this is not strictly necessary for the evolution of the system; the system itself can be completely integrated without ever referring to the scale factor and can be shown to be integrable even in places where the symplectic system with the scale factor is not [17,46]. As we saw in the Kepler example, the scale itself is not a necessary quantity to include in our treatment.…”

“…Symmetries from this general set are referred to as Dynamical Similarities [12][13][14]. It has been shown that such symmetries are commonplace in physical systems and have particularly important philosophical and mathematical implications in the cosmological sector [15][16][17][18]. In previous work [12], we have dealt with the case where a single force was acting and chosen the appropriate symmetry that kept the coupling of the relevant potential fixed.…”