How Dark is Dark Energy?

Abstract

The nature of dark energy is one of the fundamental problems in cosmology. Introduced to explain the apparent acceleration of the Universe's expansion, its origin remains to be determined. In this paper, we illustrate a result that may significantly impact understanding the relationship between dark energy and structure formation in the late-epoch Universe. Our analysis exploits a scale-dependent energy functional, initially developed for image visualization, to compare the physical and geometrical data that distinct cosmological observers register on their celestial spheres. In the presence of late-epoch gravitational structures, this functional provides a non-perturbative technique that allows the standard Friedmann-Lemaître-Robertson-Walker (FLRW) observer to evaluate a measurable, scale-dependent difference between the idealized FLRW past light cone and the physical light cone. From the point of view of the FLRW observer, this difference manifests itself as a redshift-dependent correction $Λ^{(corr)}(z)$ to the FLRW cosmological constant $Λ^{(FLRW)}$. At the scale where cosmological expansion couples with the local virialized dynamics of gravitational structures, we get $Λ^{(corr)}(z)\sim 10^{-52}\,m^{-2}$, indicating that the late-epoch structures induce an effective cosmological constant that is of the same order of magnitude as the assumed value of the FLRW cosmological constant, a result that may lead to an interpretative shift in the very role of dark energy.

Figures (3)

• Figure 1: A pictorial rendering of the relation between the physical observer and her FLRW avatar. They share the observational event $p$ but generally have distinct 4-velocities. On the left part of the drawing, there is a representation of the light cones associated with the two observers and their respective celestial spheres. The right side of the picture provides a characterization of the FLRW celestial sphere (represented as a circumference for dimensional reason) in the local rest space $T_pM$ of the observational event (the tangent space to the FLRW spacetime $M$ at $p$). The vector $\textbf{n}$ defines the spatial direction pointing at the source as seen by the FLRW observer. The null vector $\textbf{l}$ is the associated past-directed null direction. The corresponding past-directed null geodesic reaching the source on the FLRW past light cone is described using the exponential map $\widehat{\exp}_p$, the map that to a (null) vector $\textbf{l}\in T_pM$ associates the corresponding spacetime null geodesic. A similar characterization holds for the physical celestial sphere.
• Figure 2: In the pre-homogeneity region, null geodesics develop conjugate and cut locus points with the ensuing formations of past light cone caustics. On the physical observer's celestial sphere $\mathbb{CS}_z$, this behavior gives rise to the various distortions familiar in gravitational lensing. Here, for instance, we have a pictorial rendering of the apparent double image of a single galaxy. Globally, these distortions take form in the metric geometry $h_{(z)}$ that the null geodesic flow induces (via the exponential map $\exp_p$) on $\mathbb{CS}_z$. The figure describes this geometry as the jagged celestial sphere; we can profitably handle its properties with the Lipschitzian methods described in CarFamCarFam2 .
• Figure 3: The construction of the conformal mapping between the physical and the FLRW celestial spheres directly results from characterizing the celestial spheres and the Lorentz transformation compensating for the observer's relative velocity. The picture shows the maps involved in this construction; their joint action (see the bottom of the drawing) acts between (constant redshift) sections of the light cones, an action that, under natural physical assumptions, characterizes the conformal transformation of the associated celestial spheres.