Self tuning scalar fields in spherically symmetric spacetimes

Abstract

We search for self tuning solutions to the Einstein-scalar field equations for the simplest class of 'Fab-Four' models with constant potentials. We first review the conditions under which self tuning occurs in a cosmological spacetime, and by introducing a small modification to the original theoryintroducing the second and third Galileon terms -show how one can obtain de Sitter states where the expansion rate is independent of the vacuum energy. We then consider whether the same self tuning mechanism can persist in a spherically symmetric inhomogeneous spacetime. We show that there are no asymptotically flat solutions to the field equations in which the vacuum energy is screened, other than the trivial one (Minkowski space). We then consider the possibility of constructing Schwarzschild de Sitter spacetimes for the modified Fab Four plus Galileon theory. We argue that the only model that can successfully screen the vacuum energy in both an FLRW and Schwarzschild de Sitter spacetime is one containing 'John' ∼ G µ ν ∂µφ∂ ν φ and a canonical kinetic term ∼ ∂αφ∂ α φ. This behaviour was first observed in [1]. The screening mechanism, which requires redundancy of the scalar field equation in the 'vacuum', fails for the 'Paul' term in an inhomogeneous spacetime.

Fab Four effective field theory treatment

“…Therefore only the John and Paul terms drive the screening. Finally, the Paul term demonstrates a pathological behavior in star-like objects [12,13]. Therefore, the John term is the most relevant term for the cosmological constant screening.…”

Lovelock Galileons and black holes

“…For a generic scalar we see that the scalar field equation can not be written as a first integral and we also end up with a PDE system, which even in the simpler case of [27], is not integrable [34]. Only when the scalar has a linear time dependence, we have a system of ODE s with no explicit time derivatives (but in the presence of the velocity charge q).…”

“…A different way to bifurcate no hair theorems is to involve scalar tensor interactions involving translational invariant Galileons such as the John term of Fab 4 which reads, G µ ν ∇ µ φ∇ ν φ where G µν is the 4 dimensional Einstein tensor. There when one considers additionally a linear time dependence of the scalar field [27] (see also [28][29][30][31][32][33][34], and [35][36][37][38][39][40][41]) it was shown that the time dependence yielded analytic GR like black holes with additionally well defined scalars on the black hole horizon. For a recent review on black holes and scalar fields see [42] and [43].…”

Cosmological self-tuning and local solutions in generalized Horndeski theories

“…One such example has been studied in [28] (see also [29,30]): a model containing the "John" term of the Fab Four not only provides an asymptotically de Sitter spacetime with an effective cosmological constant, independent from the bare vacuum constant, but it also gives a GRlike solution near a central source.…”