Near-Horizon Solution for DGP Perturbations

TL;DR

This paper tackles the challenge that brane perturbations in DGP gravity do not close on the brane due to bulk couplings. It introduces a dynamical scaling ansatz for the bulk master variable $oldsymbol{ extOmega}$, $oldsymbol{ extOmega}=A(p) ext{a}^p G(x)$ with $x=rac{yH}{\xi}$, turning the bulk problem into an ordinary differential equation for $G(x)$ and a boundary-value problem anchored by $G(0)=1$ and $G(1)=0$. By iteratively solving for the scaling exponent $p$ and the off-brane gradient $R$, the authors obtain a self-consistent evolution of brane perturbations across radiation, matter, and de-Sitter epochs, comparing the dynamical-scaling (DS) solution to the quasi-static (QS) approximation. They find that QS is a good description for subhorizon modes well inside matter domination, but DS predicts additional decay of the gravitational potentials on large scales, strengthening the ISW effect relative to $oldsymbol{Lambda}$CDM, while the comoving density perturbation $oldsymbol{ extDelta}$ remains close to QS predictions (within a few percent). These results imply that DGP modifications imprint scale- and era-dependent signatures in the ISW and growth history, with the scaling framework providing a robust, near-horizon solution that can be used to test DGP against observations.

Abstract

We develop a scaling ansatz for the master equation in Dvali, Gabadadze, Porrati cosmologies, which allows us to solve the equations of motion for perturbations off the brane during periods when the on-brane evolution is scale-free. This allows us to understand the behavior of the gravitational potentials outside the horizon at high redshifts and close to the horizon today. We confirm that the results of Koyama and Maartens are valid at scales relevant for observations such as galaxy-ISW correlation. At larger scales, there is an additional suppression of the potential which reduces the growth rate even further and would strengthen the ISW effect.

Figures (10)

• Figure 1: Value of exponent $p$ in the scaling solution $\Omega|_{y=0} \propto a^{p}$ for superhorizon modes, plotted as a function of the scale factor. The $p^{(0)}$ calculation assumes that the density perturbation $\Delta$ follows Eq. \ref{['e:growth']} at all times. The $p^{(i)}$ result is the output of an iterative process, where $p$ is used to calculate $R$ and hence the evolution of $\Omega$ which is in turn used to derive a correction to $p$.
• Figure 2: Evolution of ratio of off-brane gradient to master variable, $R$, as defined in Eq. \ref{['e:R']}, for a selection of modes. On superhorizon scales, $R$ is constant whenever $p$ is a constant. Once the mode enters the horizon it rapidly approaches $R = -k/aH$. As the universe enters the de-Sitter phase, the modes again leave the horizon and $R$ asymptotes to $1$.
• Figure 3: Off-brane profile for $G(x)$ obtained by solving equation \ref{['e:ODE']} during matter domination ($\log a = -2$), compared to off-brane profiles for the quasi-static (QS) solution. For high $k/aH$ the profiles are very narrow and effectively independent of the position of the causal horizon: they penetrate very little into the bulk and the behavior of the solution is practically independent of the value of $p$. In this regime, the QS solution is practically coincident with the scaling solution. For modes with low $k/aH$, the solution is non-zero in the whole interval $x \in [0,1)$ and therefore it depends strongly on the value of $p$. QS severely underestimates the gradient of the profile in this regime.
• Figure 4: Ratio of master variable $\Omega$ for dynamic scaling and quasi-static solutions. In QS, $\Omega$ responds instantaneously to changes in $\Delta$. Fully dynamic solution to the Bianchi identity \ref{['e:bianchi']} requires time to respond and eventually decays to the QS solution. Initializing the calculation at earlier times changes neither the scale factor at which $\Omega$ responds nor its value today. Rapid growth occurs during the time before horizon crossing.
• Figure 5: Evolution of the principle gravitational observable $\Phi-\Psi$ for concordance $\Uplambda$CDM, the quasistatic (QS) solution, and our new dynamical scaling (DS) solution. For scales $k \gtrsim 0.01$ Mpc$^{-1}$ the scaling and QS solutions do not differ appreciably: the decay in the potentials is a little faster than $\Uplambda$CDM as a result of slower growth of density contrast. At larger scales, the DS solution exhibits significant additional decay owing to the different value of $\Omega$ at late times, as exhibited in Fig. \ref{['f:Omegacomp']}. All potentials normalized to 1 at $\log a = -2$.