Scalar perturbations on the normal and self-accelerating branch of a DGP brane and $σ_8$

TL;DR

This work probes scalar perturbations in the DGP brane-world for both the normal and self-accelerating branches by constraining the background with Pantheon+ SN data and mock gravitational wave standard sirens, then solving the master equation for perturbations using a dynamical scaling solution with $p=4$ to obtain the evolution of the matter density contrast $Δ_m$ and the growth observable $fσ_8$. A closed second-order equation for $Δ_m$ is obtained by expressing the brane metric potentials in terms of a master variable $Ω$ and applying the DS scaling boundary, allowing computation of $σ_8$ from the growth history. The normal branch yields $σ_8≈0.71$–$0.77$ depending on priors on $Ω_{r0}$, while the self-accelerating branch gives $σ_8≈0.91$, making the latter observationally disfavored; the normal branch remains a viable modified-gravity option that could help address the $σ_8$ tension. Overall, the study demonstrates how combining SN, GW, and RSD data can tighten constraints on modified gravity models and reveal distinctive growth signatures relative to ΛCDM.

Abstract

In this work we constrain the value of $σ_8$ for the normal and self-accelerating branch of a DGP brane embedded in a five-dimensional Minkowski space-time. For that purpose we first constrain the model parameters $H_0$, $Ω_{m0}$, $Ω_{r0}$ and $M$ by means of the Pantheon+ catalog and a mock catalog of gravitational waves. Then, we solve numerically the equation for dark matter scalar perturbations using the dynamical scaling solution for the master equation and assuming that $p=4$ for the matter dominated era. Finally, we found that the evolution of matter density perturbations in both branches is different from the $Λ$CDM model and that the value of $σ_8=0.774\pm0.027$ for the normal branch and $σ_8=0.913\pm0.032$ for the self-accelerating branch.

Figures (6)

• Figure 1: $2\sigma$ C.L. constraints for the background parameters using standard sirens mock data GW (green) and SN Pantheon+ (blue), and the total sample GW+Pantheon+ (red) for the normal branch using the priors shown in Table \ref{['tab:nor2']}.
• Figure 2: Left side: a) Evolution of $\triangle_m/a$ for the normal branch using the values obtained for the background parameters shown in Table \ref{['tab:nor1']}: $\Omega_{m0}=0.546$, $H_0=69.58$ Km s$^{-1}$ Mpc$^{-1}$ and $\Omega_{r0}=0.155$ for $k=0.002h$Mpc$^{-1}$ (orange) and $k=0.01h$Mpc$^{-1}$ (green). b) Difference of branch using the values obtained for the background parameters shown in Table \ref{['tab:nor2']}: $\Omega_{m0}=0.470$, $H_0=69.94$ Km s$^{-1}$ Mpc$^{-1}$ and $\Omega_{r0}=0.029$ for $k=.002h$Mpc$^{-1}$ (red) and $k=.01h$Mpc$^{-1}$ (purple). Right side: Evolution of the growth factor for the self-accelerating branch using the values obtained for the background parameters shown in Table \ref{['tab:self_acc']}, $\Omega_{m0}=0.286$, $H_0=69.08$ Km s$^{-1}$ Mpc$^{-1}$ for $k=0.0005h$Mpc$^{-1}$ (red), $k=0.002h$Mpc$^{-1}$ (green) and $k=0.01h$Mpc$^{-1}$ (red). We also show the evolution of the growth factor in the $\Lambda$CDM model with the parameter values inferred from Planck $\Omega_{m0}=0.315$ and $H_0=67.4$ Km s$^{-1}$ Mpc$^{-1}$.
• Figure 3: Left side: a) Difference of $\triangle_m(a,k)/a-\triangle_m(a,k=0.125h$Mpc$^{-1}$)$/a$ for different values of $k$ for the normal branch. The difference for these values of $k$ is approximately 0. We use for the background the values shown in Table \ref{['tab:nor1']}: $\Omega_{m0}=0.546$, $H_0=69.58$ Km s$^{-1}$ Mpc$^{-1}$ and $\Omega_{r0}=0.155$ for $k=0.002h$Mpc$^{-1}$ (blue dot) and $k=0.01h$Mpc$^{-1}$ (orange line). b) And for $k=0.002h$Mpc$^{-1}$ (green dot) and $k=0.01h$Mpc$^{-1}$ (red line), we use the background values shown in Table \ref{['tab:nor2']}: $\Omega_{m0}=0.470$, $H_0=69.94$ Km s$^{-1}$ Mpc$^{-1}$ and $\Omega_{r0}=0.029$. Right side: Difference of $\triangle_m(a,k)/a-\triangle_m(a,k=0.125h$Mpc$^{-1}$)$/a$ for different values of $k$ for the self-accelerating branch, using the background parameters shown in Table \ref{['tab:self_acc']}, $\Omega_{m0}=0.286$, $H_0=69.08$ Km s$^{-1}$ Mpc$^{-1}$ for $k=0.0005h$Mpc$^{-1}$ (blue dot), $k=0.01h$Mpc$^{-1}$ (orange dash-dot line).
• Figure 4: $2\sigma$ C.L. constraints for the background parameters using standard sirens mock data GW (green) and SN Pantheon+ (blue), and the total sample GW+Pantheon+ (red) for the self-accelerating branch using the priors shown in Table \ref{['tab:self_acc']}.
• Figure 5: Left side: Posterior of $\sigma_8$ for the normal branch using data of Pantheon + and a flat prior for $0<\Omega_{r0}<0.25$. Right side: Posterior of $\sigma_8$ for the normal branch using a flat prior for $0<\Omega_{r0}<0.05$. We use a uniform prior for $\sigma_8\epsilon(0,1)$.