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Structure formation in the DGP cosmological model

Kazuya Koyama, Roy Maartens

TL;DR

The paper addresses how to correctly predict structure formation in the DGP brane-world cosmology by incorporating full 5D perturbations. Using the Mukohyama master variable and brane junction conditions, it derives a unique, Bianchi-consistent set of on-brane metric perturbations and Weyl stresses in the subhorizon regime, showing that the growth of matter perturbations is suppressed relative to LCDM due to gravity leakage into the bulk. The key result is a modified growth equation with a time-dependent factor $\beta$ that also yields a gravitational slip between $\Phi$ and $\Psi$, distinguishing DGP from dark-energy models with identical expansion histories. The work clarifies misconceptions from 4D-only analyses and provides a robust framework for testing DGP against structure-formation observations, enabling a more definitive confrontation with data.

Abstract

The DGP brane-world model provides an alternative to the standard LCDM cosmology, in which the late universe accelerates due to a modification of gravity rather than vacuum energy. The cosmological constant $Λ$ in LCDM is replaced by a single parameter, the crossover scale $r_c$, in DGP. The Supernova redshift observations can be fitted by both models, with $Λ\sim H_0^2$ and $r_c \sim H_0^{-1}$. This degeneracy is broken by structure formation, which is suppressed in different ways in the two models. There is some confusion in the literature about how the standard linear growth factor is modified in DGP. While the luminosity distance can be computed purely from the modified 4-dimensional Friedman equation, the evolution of density perturbations requires an analysis of the 5-dimensional gravitational field. We show that if the 5-dimensional effects are inappropriately neglected, then the 4-dimensional Bianchi identities are violated and the computed growth factor is incorrect. By using the 5-dimensional equations, we derive the correct growth factor.

Structure formation in the DGP cosmological model

TL;DR

The paper addresses how to correctly predict structure formation in the DGP brane-world cosmology by incorporating full 5D perturbations. Using the Mukohyama master variable and brane junction conditions, it derives a unique, Bianchi-consistent set of on-brane metric perturbations and Weyl stresses in the subhorizon regime, showing that the growth of matter perturbations is suppressed relative to LCDM due to gravity leakage into the bulk. The key result is a modified growth equation with a time-dependent factor that also yields a gravitational slip between and , distinguishing DGP from dark-energy models with identical expansion histories. The work clarifies misconceptions from 4D-only analyses and provides a robust framework for testing DGP against structure-formation observations, enabling a more definitive confrontation with data.

Abstract

The DGP brane-world model provides an alternative to the standard LCDM cosmology, in which the late universe accelerates due to a modification of gravity rather than vacuum energy. The cosmological constant in LCDM is replaced by a single parameter, the crossover scale , in DGP. The Supernova redshift observations can be fitted by both models, with and . This degeneracy is broken by structure formation, which is suppressed in different ways in the two models. There is some confusion in the literature about how the standard linear growth factor is modified in DGP. While the luminosity distance can be computed purely from the modified 4-dimensional Friedman equation, the evolution of density perturbations requires an analysis of the 5-dimensional gravitational field. We show that if the 5-dimensional effects are inappropriately neglected, then the 4-dimensional Bianchi identities are violated and the computed growth factor is incorrect. By using the 5-dimensional equations, we derive the correct growth factor.

Paper Structure

This paper contains 6 sections, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Perturbation equations on the brane
  3. The 5-dimensional equations
  4. The growth factor
  5. conclusion
  6. Perturbation equations

Figures (2)

  • Figure 1: The growth history $g(a)=\bigtriangleup(a)/a$ is shown for LCDM (long dashed) and DGP (solid, thick). The growth history for a dark energy model (short dashed) with the identical expansion histories with DGP is also shown (see Fig. 2) Linder. Due to the time variation of Newton's constant through $\beta$ in Eq. (\ref{['dpe']}), the growth factor $g(a)$ receives an additional suppression compared with the dark energy model. DGP-4D (solid, thin) shows the incorrect result in which the inconsistent assumption $\delta E_{\mu \nu}=0$ is adopted. We set the desity parameter for matter today as $\Omega_{m0}=0.3$.
  • Figure 2: The expansion history $r(z)=\int^z_0 dz H(z)^{-1}$ is shown for LCDM (dashed) and DGP (solid) where $1+z=a^{-1}$. Expansion history for DGP can be mimicked by a dark energy model with the equation of state $w(a)=w_0 +w_a (1-a)$ with $w_0=-0.78$ and $w_a=0.32$Linder.