Multiscale Cosmic Curvature: from Local Structures to Cosmology

TL;DR

The paper addresses how Dark Energy (DE) influences dynamics from local to cosmological scales by employing the McVittie spacetime to unify local gravity and cosmic expansion. It uses curvature invariants $\mathcal{R}$, $\mathcal{C}$, and $\mathcal{K}$ and introduces a universal dimensionless parameter $\kappa$ to compare regimes across galaxies, halos, and binaries, mapping these into a curvature phase space. The authors derive exact geodesic equations, present analytic expressions for the invariants, and demonstrate DE dominance near turnaround in groups and clusters, as well as in spherical density and binary systems, with validation from IllustrisTNG simulations and observational data. This geometric framework provides a scale-spanning diagnostic of DE effects, linking microphysical local dynamics to the large-scale cosmic acceleration and offering practical guidance for interpreting galaxy dynamics and structure formation in a $\Lambda$CDM context.

Abstract

This study tackles the impact Dark Energy (DE) in different systems by a simple unifying formalism. We introduce a parameter space that compare gravity across all cosmic scales, using the McVittie spacetime (McV) and connects spherically symmetric solutions with cosmological solutions. By analyzing the invariant scalars: the Ricci, Weyl, and Kretschmann scalars, we develop a phase-space description that predicts the dominance of the Cosmological Constant. We explore three cases: (1) the local Hubble flow around galaxy groups and clusters, (2) spherical density distributions and (3) binary motion. Our results show that the Kretschmann scalar of galaxy groups and clusters in their turnaround is $2Λ^2$ which is three times the Kretschmann scalar of the Cosmological Consonant. This quantifies the DE domination in local structures.

Figures (4)

• Figure 1: The normalized velocity-distance relation from the analytical solution (\ref{['eq:sol_new_v_r']}). The solution is scaled with $r_0$ and $r_0 /t_U$, where $t_U$ is the age of the Universe. The turnaround $r_0$ is related to the enclosed mass. The colored lines corresponds to $e = 1$. For different $e$ the area between the colored lines is also covered. The dashed line marks the asymptotic behavior for the unbound solution (which in the case of $H \neq 0$ corresponds to the Hubble flow).
• Figure 2: Left: The velocity vs. distance for an isolated pair from the IllustrisTNG simulation. Right: The radial velocity of galaxies as a function of their distance from the halo CoM. A green dashed line illustrates the theoretical Hubble Law with a slope of $70\, \text{km/s/Mpc}$, representing the expected outward velocity for galaxies due to the universe's expansion.
• Figure 3: The enclosed mass versus turnaround radius (following Pavlidou:2013zha) for galaxy groups and clusters. The data points are taken from the references listed in Table under Fig. \ref{['fig:cur_phase_space']}, including Kim:2020gaibib:Sorce2016Benisty:2025tctMakarov:2025Karachentsev:2008stPenarrubia:2014odaNasonova:2011mdkarachentsev2002Karachentsev:2006wwDelPopolo:2021hkz. The red curve indicates the Zero Gravity Surface, defined by the balance condition $r_\Lambda$ which separates gravitationally bound systems from those dominated by cosmic acceleration.
• Figure 4: The curvature phase space for the potential $\Phi$ versus the scalar $\sqrt{\mathcal{K}}$ in a de-Sitter $w = -1$ universe. The Solar System and S-stars around the galactic center are positioned much higher than galaxy groups and clusters, which lie at the transition to local expansion, indicating the dominance of $\Lambda$ in larger structures. The table below gives the characteristic turnaround radii $r_{ta}$ and masses ($M$) of galaxy groups and clusters. Another Ref. for the turnaround and enclosed mass is taken from Kim:2020gaibib:Sorce2016Benisty:2025tctMakarov:2025Karachentsev:2008stPenarrubia:2014odaNasonova:2011mdkarachentsev2002Karachentsev:2006wwDelPopolo:2021hkz