Cosmic voids evolution in modified gravity via hydrodynamics

We present a hydrodynamical description of spherical void evolution in modified gravity (MG), extending the standard General Relativity (GR) and dynamical dark energy treatment by encoding gravity modifications into effective couplings that enter the Euler and Poisson equations. This yields a compact non-linear evolution equation for the Eulerian density contrast, controlled by a time- and density-dependent effective gravitational strength, and provides a direct map between model functions and void observables. We apply the framework to the luminal Galileon class of models, where derivative self-interactions generate Vainshtein screening and might lead to a breakdown of the physical branch in sufficiently underdense regions. Exploiting this feature, we apply the void-informed viability requirement that translates into bounds on the theory parameter space and, equivalently, on the minimum attainable void depth as a function of redshift. For viable parameters of a concrete model, we quantify the impact of MG on isolated void evolution, the Lagrangian to Eulerian mapping, and the shell-crossing threshold. Relative to GR, we find a clear hierarchy of MG effects, with ${\cal O}(10%)$ modifications in the gravitational couplings, percent-level shifts in the void density evolution, and sub-percent deviations in both the mapping and the shell-crossing thresholds. Moreover, within the adopted parametrization, we show analytically that voids always lie in an unscreened regime on the physical branch. Overall, the formalism provides a self-consistent route to predict void dynamics and consistency constraints in a broad class of MG models.

💡 Research Summary

This paper develops a fully self‑consistent hydrodynamical framework for the non‑linear evolution of spherical cosmic voids in modified‑gravity (MG) theories. Starting from the Newtonian continuity and Euler equations for a pressure‑less fluid, the authors replace the standard Poisson and lensing relations with two effective, time‑ and density‑dependent coupling functions, μ_NL(a,R) and Σ_NL(a,R). In General Relativity (GR) both functions equal unity, but in screened MG models they encode the strength of the fifth force and its environmental dependence.

The resulting non‑linear evolution equation for the Eulerian density contrast δ_E reads

δ_E’’ + (2+H’/H) δ_E’ – (4/3)(δ_E’)²/(1+δ_E) – (3/2) Ω_m μ_NL(a,R) (1+δ_E) δ_E = 0,

where primes denote derivatives with respect to ln a. Linearizing gives the familiar growth equation with an effective linear coupling μ_L. The authors adopt an inverse top‑hat initial profile and integrate the equations from a_in = 10⁻⁷ (deep matter domination) with the early‑time conditions δ_E = δ_L and δ_E’ = δ_E, neglecting the decaying mode.

For concreteness they focus on the luminal Galileon class, a scalar‑tensor theory that exhibits Vainshtein screening. In underdense regions the screening radius becomes smaller than the void radius, driving μ_NL toward unity. However, the non‑linear Galileon field equation can lose its real solution when the void depth exceeds a critical value, leading to a “physical branch” breakdown (imaginary fifth force). The authors turn this pathology into a diagnostic: they derive a “void‑informed viability condition” that translates into a redshift‑dependent upper bound on the attainable void density contrast. This condition carves out an allowed region in the model’s parameter space (e.g., the Galileon coefficients c₂, c₃), complementing standard stability constraints.

Numerical integration shows that the effective gravitational coupling deviates from GR by at most ~10 % across the void’s evolution. The resulting change in the Eulerian density contrast is at the percent level, while the mapping from Lagrangian radius (initial comoving coordinate) to Eulerian radius (observable size) and the shell‑crossing threshold (the point where inner shells overtake outer ones) differ from the GR predictions by less than 0.5 %. Importantly, the analysis proves analytically that, on the physical branch, all viable voids lie in an unscreened regime: the fifth force is essentially active throughout the void interior.

The paper therefore provides three major contributions: (1) a compact, non‑linear void evolution equation that directly incorporates MG effects via effective couplings; (2) a novel, structure‑formation‑based viability criterion that restricts MG parameter space using void depth limits; and (3) quantitative predictions showing that, for the luminal Galileon, MG modifies the gravitational strength at the ~10 % level but leaves observable void properties (density evolution, radius mapping, shell‑crossing) virtually unchanged at the sub‑percent level.

These results suggest that cosmic voids are excellent laboratories for testing MG: they are naturally unscreened, highly sensitive to fifth‑force strength, yet their basic geometric and dynamical observables remain robust enough to allow precise theoretical predictions. Future work should extend the framework to non‑spherical voids, include baryonic physics (gas pressure, radiative cooling), and confront the predictions with upcoming large‑scale surveys such as DESI, Euclid, and the Vera C. Rubin Observatory, where the percent‑level effects identified here may become observationally detectable.