Gauss–Bonnet–Chern approach to the averaged Universe

Abstract

The standard model of cosmology with postulated dark energy and dark matter sources may be considered as a fairly successful fitting model to observational data. However, this model leaves the question of the physical origin of these dark components open. Fully relativistic contributions that act like dark energy on large scales and like dark matter on smaller scales can be found through generalization of the standard model by spatially averaging the inhomogeneous Universe within general relativity. The spatially averaged 3 + 1 Einstein equations are effective balance equations that need a closure condition. Heading for closure we here explore topological constraints. Results are straightforwardly obtained for averaged 2 + 1 model universes. For the relevant 3 + 1 case, we employ a method based on the Gauss–Bonnet–Chern theorem generalized to Lorentzian spacetimes and implement a sandwich approach to obtain spatial average properties. The 3 + 1 topological approach supplies us with a new equation linking evolution of scalar invariants of the expansion tensor to the norm of the Weyl tensor. From this we derive general evolution equations for averaged scalar curvature and kinematical backreaction, and we discuss related evolution equations on this level of the hierarchy of averaged equations. We also discuss the relation between topological properties of cosmological manifolds and dynamical topology change, e.g. as resulting from the formation of black holes.

On non-Euclidean Newtonian theories and their cosmological backreaction

“…Then, formula (43) suggests that the structure formation might play a major role on the global expansion of our Universe if the topology is non-Euclidean. This behaviour is also present in general relativity [3], but in our case it arises at a non-relativistic level, as it shows that the Universe might have a local Newtonian dynamics, but a global dynamics which differs from the one of a homogeneous Universe. However, quantifying the mean specific energy remains difficult because it depends on the mesoscopic scale, which is not well defined in cosmology.…”

On average properties of inhomogeneous fluids in general relativity III: general fluid cosmologies

“…The rest mass M D of the fluid within D is thus more naturally defined in terms of the proper volume measure √ b d 3 x. Note that the two covariant 15 volume measures √ h d 3 x and √ b d 3 x coincide in the case of a flow-orthogonal foliation (possible for an irrotational fluid), which is the situation considered in Papers I and II. Hence, a degeneracy between both volumes is present in these papers, while they are distinct for any other choice of foliation.…”

“…We will also recover the expected relation between total rest mass and averaged rest mass density. 15 As h(t, x k ) d 3 x, the fluid-orthogonal volume 3−form b(t, x k ) d 3 x is also invariant under a change of spatial coordinates, as can be checked either directly or by rewriting it as…”

“…[31]; on specifications of potentials in the morphon analogy of backreaction acting as quintessence, inflation, or dark matter [28,32,89]; or on other assumptions of equations of state or models for bulk viscosity [77,7]. Closure may also be approached by topological considerations, as discussed in [18] and implemented for 2 + 1 spacetimes [68,15], and for 3 + 1 spacetimes leading to evolution equations for backreaction [15]. With this latter we enter a deeper level of the hierarchy of scalar-averaged equations.…”

Cauchy-compact flat spacetimes with extreme BTZ

“…On a side note, Mess results circulated as a draft for many years and piqued the interest of the 2+1 quantum gravity community in the 90's [Car98]. Since then, many works have focused on quantizations of Teichmüllerlike spaces [BW16, BW17, BW19, FG06, MW19] and their interpretation as quantizations for quantum gravity [Car98,MS16], relations with the cosmological microwave background [AAA `19] have been investigated [BMS14] and more recently link with averaged Einstein equations in relativistic inhomogeneous cosmology [Buc07] has been made [BB20].…”