Abundance of cosmic voids in EFT of dark energy

TL;DR

This work analyzes how modified gravity in the EFT of dark energy, encapsulated by the kinetic braiding parameter $α_B$ and a time-dependence controlled by $p$, alters cosmic voids using a spherical-shell model on a ΛCDM background. The authors derive the shell dynamics under the EFT, obtain the modified linear growth via an effective gravitational strength $μ$, and compute the void-formation threshold $δ_{sc}$ alongside the collapse threshold $δ_c$, finding that changes in $δ_{sc}$ are suppressed compared to $α_B$ due to competing effects. Using the Sheth–van de Weygaert framework, they show the void size function acquires scale-dependent modifications: small-scale changes are driven mainly by the altered linear power spectrum, while large-scale changes arise from a combination of the spectrum and $δ_{sc}$. These results highlight the potential of void statistics as probes of gravity on cosmological scales and motivate full N-body simulations to test the model's predictions and extend to broader Horndeski/DHOST theories.

Abstract

Cosmic voids in the large-scale structure are among the useful probes for testing gravity on cosmological scales. In this paper, we investigate the evolution of voids in the Horndeski theory using the effective field theory (EFT) of dark energy. Modeling the void formation with the dynamics of spherical mass shells, we study how modifications of gravity encoded into the EFT of dark energy change the linearly extrapolated critical density contrast that is relevant for the criterion for void formation, with particular focus on the time-dependent parameter characterizing the effect of kinetic braiding. It is found that the change in the critical density contrast is one order of magnitude smaller than the dimensionless EFT parameter because of a slight imbalance between two compensating effects. We then compute the void abundance using the Sheth--van de Weygaert void size function and demonstrate that it exhibits scale-dependent modifications. It is shown that the modifications to the void size function on small scales are almost entirely determined by the modified linear matter power spectrum, while the modifications on large scales are dominated by the contributions from the linear matter spectrum and the critical density contrast.

Figures (8)

• Figure 1: Shell-crossing time of each shell $t_*$ versus the initial physical radius $R_i$ in the $\Lambda$CDM model in general relativity, assuming $\Omega_{\mathrm{m}0}=0.3153$ and the initial profile characterized by $R_0=10^{-16}H_0^{-1}$, $s=3\times10^4$, and $\delta_0=3\times10^{-7}$.
• Figure 2: Linear density contrast at the shell-crossing time in the EFT of dark energy with $p=1$ and $p=0.85$. The parameter is given by $\alpha_{B0}=10^{-2}$ (solid blue line) and $\alpha_{B0}=10^{-1}$ (solid orange line). The dashed and dotted lines represent the results in the $\Lambda$CDM model and Einstein-de Sitter model, respectively.
• Figure 3: Expansion factor in the EFT of dark energy with $p=1$ and $p=0.85$. The parameter is given by $\alpha_{B0}=10^{-2}$ (solid blue line) and $\alpha_{B0}=10^{-1}$ (solid orange line). The dashed and dotted lines represent the results in the $\Lambda$CDM and Einstein-de Sitter models, respectively.
• Figure 4: Linear density contrast at the collapse time in the EFT of dark energy with $p=1$ and $p=0.85$. The parameter is given by $\alpha_{B0}=10^{-2}$ (solid blue line) and $\alpha_{B0}=10^{-1}$ (solid orange line). The dashed and dotted lines represent the results in the $\Lambda$CDM and Einstein-de Sitter models, respectively.
• Figure 5: Void size function (at $z=0$) in the EFT of dark energy (left) and the relative differences from that in the $\Lambda$CDM model (right), with the parameter being given by $\alpha_{B0}=10^{-2}$ (solid blue line) and $\alpha_{B0}=10^{-1}$ (solid orange line). The time-dependence of the EFT parameters is described by the model with $p=1$. The dashed lines represent the result in the $\Lambda$CDM model.