Cosmological perturbations in the DGP braneworld: numeric solution

Abstract

We solve for the behaviour of cosmological perturbations in the Dvali-Gabadadze-Porrati (DGP) braneworld model using a new numerical method. Unlike some other approaches in the literature, our method uses no approximations other than linear theory and is valid on large scales. We examine the behaviour of late-universe density perturbations for both the self-accelerating and normal branches of DGP cosmology. Our numerical results can form the basis of a detailed comparison between the DGP model and cosmological observations.

Figures (10)

• Figure 1: Typical computational grids used to solve the perturbation equations.
• Figure 2: Grid geometry used to derive evolution formulae in §\ref{['sec:triangle evolution']} and §\ref{['sec:diamond evolution']}. The principal brane nodes A, E, H and M are separated by a brane time interval $\delta x = h$. We have also introduced half-step nodes D, G and L, which are separated from the adjacent principal nodes by $\delta x = h/2$. The half-step nodes are needed because of the non-local nature of the boundary condition in the DGP model.
• Figure 3: Numeric solutions for the amplitude of tensor mode perturbations $E_\text{b}$ on a pure tension brane (left and center) and comparison of the late time power law index derived from analytic and simulation results (right). Note that our late time simulation results for $\epsilon = -1$ are all very similar to the $Hr_\text{c} = 5$ case shown here, and are all consistent with the analytic $\gamma = 0$ expectation.
• Figure 4: The results of our simulations on the brane for several choices of $k$. We have normalized the value of $\Phi$ to be unity at early times. Also note that the lower left panel shows the dimensionless bulk master variable $\hat{\Omega}_\text{b}$, as defined in Eq. (\ref{['eq:Omega hat def']}), divided by $\hat{a}^2$. All simulations are performed with $\hat{\rho}_* > 6$, which means that all modes enter the horizon at $\hat{a} = 1$, or when $a = a_*$.
• Figure 5: Behaviour of the $\zeta$ curvature perturbation on large scales for the self-accelerating branch (we have normalized $\zeta = 1$ at early times). Note that the curvature perturbation is conserved when the modes are superhorizon; i.e., at both early and late times. This is to be expected for any conservative theory of gravity, such as the DGP model.