Self-Consistent Cosmological Simulations of DGP Braneworld Gravity

TL;DR

The paper develops and executes cosmological N-body simulations of the self-accelerating DGP braneworld gravity by solving the full non-linear brane bending mode equation in the quasi-static limit. Through these simulations, it demonstrates how non-linear self-interactions of the brane bending mode induce Vainshtein screening, suppressing deviations from GR inside dense regions and altering structure formation on larger scales. The results show a suppressed non-linear power spectrum and halo mass function relative to GR with the same expansion history, with significant implications for weak lensing and cluster-based constraints that strongly disfavor the self-accelerating DGP model without a cosmological constant. The work provides a robust computational framework for studying non-linear braneworld gravity and sets the stage for applying these methods to more general models and observational tests.

Abstract

We perform cosmological N-body simulations of the Dvali-Gabadadze-Porrati braneworld model, by solving the full non-linear equations of motion for the scalar degree of freedom in this model, the brane bending mode. While coupling universally to matter, the brane-bending mode has self-interactions that become important as soon as the density field becomes non-linear. These self-interactions lead to a suppression of the field in high-density environments, and restore gravity to General Relativity. The code uses a multi-grid relaxation scheme to solve the non-linear field equation in the quasi-static approximation. We perform simulations of a flat self-accelerating DGP model without cosmological constant. The results of the DGP simulations are compared with standard gravity simulations assuming the same expansion history, and with DGP simulations using the linearized equation for the brane bending mode. This allows us to isolate the effects of the non-linear self-couplings of the field which are noticeable already on quasi-linear scales. We present results on the matter power spectrum and the halo mass function, and discuss the behavior of the brane bending mode within cosmological structure formation. We find that, independently of CMB constraints, the self-accelerating DGP model is strongly constrained by current weak lensing and cluster abundance measurements.

Figures (10)

• Figure 1: Left panel:$\varphi(r)$ measured in simulations with a $512^3$ grid for a spherical mass for different values of $\beta$, where $r$ is the distance from the center of the mass. $R=10$ grid cells, and $r_c = 800\,R$ is held fixed (see text). The solid lines show the full analytical solution for the spherical mass, while dotted lines show the linearized solution (Section \ref{['sec:philin']}). Right panel: Relative deviation of the acceleration measured in the spherical mass solution from the analytical Newtonian value vs. $r$, for the same parameters as in the left panel. Again, solid lines show the full exact solution, while dotted lines show the linearized solution.
• Figure 2: Left panel: Dimensionless RMS residual $\mathcal{L}=\sqrt{\langle r^2\rangle}$ and RMS of the $\varphi$ field vs. $a$ in cosmological simulations (top), and Vainshtein radius for a single particle in the simulations, in units of $r_{\rm cell}$ (bottom). Right panel: Relative deviation of linearized DGP power spectra smoothed with $r_s/r_{\rm cell}=1.0$ from the unsmoothed simulations (solid lines, for box sizes from $64\,{\rm Mpc}/h$ [thick] to 400$\,{\rm Mpc}/h$ [thin]), and relative deviation of power spectra for lower resolution simulations ($N_g=128$, left dashed curves, $N_g=256$, right dashed curves). Also shown as vertical dotted lines are the maximum wavenumbers considered for each simulation, $k_{\rm max}=k_{\rm Ny}/8$.
• Figure 3: The time derivative terms Eqs. (\ref{['eq:T0']})--(\ref{['eq:T2']}) of $\varphi$ measured in the simulations relative to the spatial Laplacian, as a function of scale factor $a$ for our largest and smallest simulation boxes. The time derivatives always remain $5-6$ orders of magnitude below the spatial derivatives.
• Figure 4: Left panel: Power spectrum measured in the QCDM effective dark energy cosmology simulations (top), and halofit predictions. The lower panel shows the deviations from the halofit prediction separately for each simulation box size. Right panel: Halo mass function $dn/d\ln M_{200}$ measured in the QCDM simulations (points) and the mass function prediction for QCDM after TinkerEtal. Also shown is the mass function for a $\Lambda$CDM model with the same primordial power spectrum. The lower panel shows the deviation from the prediction separately for each box size.
• Figure 5: Slices through a full DGP simulation with $L_{\rm box}=64\,{\rm Mpc}/h$ at $z=0$. Top left: density in logarithmic scale; top right: dynamical potential $\Psi$; bottom left: brane-bending mode $\varphi$; bottom right: non-linear suppression of $\varphi$ field: $\varphi_{\rm NL} \equiv \varphi - \varphi_{L}$ (see text).