Cosmological simulations of normal-branch braneworld gravity

We introduce a cosmological model based on the normal branch of Dvali-Gabadadze-Porrati (DGP) braneworld gravity with a smooth dark energy component on the brane. The expansion history in this model is identical to $\Lambda \mathrm{CDM}$, thus evading all geometric constraints on the DGP crossover scale $r_c$. This well-defined model can serve as a first approximation to more general braneworld models whose cosmological solutions have not been obtained yet. We study the formation of large-scale structure in this model in the linear and nonlinear regime using N-body simulations for different values of $r_c$. The simulations use the code presented in [25] and solve the full nonlinear equation for the brane-bending mode in conjunction with the usual gravitational dynamics. The brane-bending mode is attractive rather than repulsive in the DGP normal branch, hence the sign of the modified gravity effects is reversed compared to those presented in [25]. We compare the simulation results with those of ordinary $\Lambda \mathrm{CDM}$ simulations run using the same code and initial conditions. We find that the matter power spectrum in this model shows a characteristic enhancement peaking at $k\sim 0.7h/\mathrm{Mpc}$. We also find that the abundance of massive halos is significantly enhanced. Other results presented here include the density profiles of dark matter halos, and signatures of the brane-bending mode self-interactions (Vainshtein mechanism) in the simulations. Independently of the expansion history, these results can be used to place constraints on the DGP model and future generalizations through their effects on the growth of cosmological structure.

Figure 1

Behavior of the dark energy component in the $\mathrm{nDGP}+\mathrm{DE}$ models: density ${\rho }_{\mathrm{DE}}$ and ${\rho }_m$, as well as the resulting effective ${\rho }_{\Lambda }$ (left panel); equation of state $w_{\mathrm{DE}}$ of the dark energy (right panel). The model parameters are defined in Table .

Figure 2

Left panel: The $\beta$ function Eq. (2.9) in the $\mathrm{nDGP}+\mathrm{DE}$ models. We also show the function for the self-accelerating DGP model considered in 25 (note that $\beta <0$ for that model). Right panel: Linear growth factor $D(a)/a$ as a function of $a$, where $D(a)$ is normalized to $a$ in the matter-dominated regime, for $\Lambda \mathrm{CDM}$ and $\mathrm{nDGP}+\mathrm{DE}$ models in the quasistatic regime ($k\gtrsim 0.01h/\mathrm{Mpc}$).

Figure 3

Overdensity threshold for the $\phi$ self-interactions, defined as inverse of the nonlinearity parameter $g$ [Eq. (2.17)], as a function of scale factor in the different DGP cosmologies.

Figure 4

Left panel: Deviation of the $\mathrm{nDGP}-1$ matter transfer function from $\Lambda \mathrm{CDM}$ at the initial redshift of the simulations, $z_i=49$. The solid line shows the PPF calculation for DGP which is used to correct the power spectrum for the initial conditions. The red dashed line shows the same deviation calculated using the quasistatic calculation [Eqs. (2.8, 2.9)]. Right panel: Relative deviation of the matter power spectrum at $z=0$ measured in simulations with Gaussian smoothing of the r.h.s of Eq. (2.13) to simulations without smoothing. The thick lines show results for $L_{\mathrm{box}}=400\mathrm{Mpc}/h$, while the thin lines show those for $L_{\mathrm{box}}=128\mathrm{Mpc}/h$. All $k$ values were rescaled to those for a $256\mathrm{Mpc}/h$ box. The vertical dashed line shows the maximum wave number used in the analysis of the simulation results, $k_{\max }=k_{\mathrm{Ny}}/8$ (see text).

Figure 5

Relative deviation in the matter power spectrum from $\Lambda \mathrm{CDM}$ at $z=0$, $\Delta P(k)/P_{\Lambda \mathrm{CDM}}(k)\equiv (P_{\mathrm{DGP}}(k)-P_{\Lambda \mathrm{CDM}}(k))/P_{\Lambda \mathrm{CDM}}(k)$ for $\mathrm{nDGP}-1$ (left panel) and $\mathrm{nDGP}-2$ (right panel). The blue squares and red triangles show the measurements from the full and linearized DGP simulations, respectively. The short-dashed line shows the almost scale-invariant deviation predicted using linear perturbation theory (Sec. 2b), while the long-dashed line shows the deviation obtained when using the halofit prescription together with the linear prediction for DGP.

Figure 6

Relative deviation of the halo-mass function from $\Lambda \mathrm{CDM}$, $\Delta n_{\ln M}/n_{\ln M}(\Lambda \mathrm{CDM})$, where $\Delta n_{\ln M}\equiv n_{\ln M}(\mathrm{DGP})-n_{\ln M}(\Lambda \mathrm{CDM})$, at $z=0$ (top) and $z=1$ (bottom) for $\mathrm{nDGP}-1$ (left panel) and $\mathrm{nDGP}-2$ (right panel). Blue squares and red triangles denote the full and linearized DGP simulations, respectively.

Figure 7

Average halo density profiles measured in the full DGP simulations. Each figure corresponds to a fixed mass range: $\lg M/M_{\odot }/h=12–13$ (left), 13–14 (middle), 14–15 (right). The top panels show profiles of ${\delta }_{\rho }\equiv (\rho -{\overline{\rho }}_m)/{\overline{\rho }}_m$ vs $r$ in units of $r_{200}$, while the bottom panel shows the relative deviation from the GR simulation with the same expansion history in each case ($\Lambda \mathrm{CDM}$ for nDGP, QCDM for sDGP). For all figures, individual halo profiles were scaled to their respective $r_{200}$ before averaging.

Figure 8

Top panel: Average profiles of the radial gradient of $\phi$, i.e. the force modification, measured in the full DGP simulations and scaled by $3\beta /2$, for halos of mass $\lg M/M_{\odot }/h=14–15$. The profiles were averaged in a similar way as the density profiles. Bottom panel: Deviation of the dynamical potential $\Psi$ [Eq. (2.10)] from the Newtonian potential or lensing potential ${\Phi }_{-}={\Psi }_N$ [Eq. (2.12)] around the same dark matter halos. This quantity is probed by Solar System tests.

Figure 9

Ratio of the reduced bispectrum measured in full and linearized DGP simulations, vs the angle between the wave vectors ${\mathbf{k}}_1=0.5h/\mathrm{Mpc}$, ${\mathbf{k}}_2=1h/\mathrm{Mpc}$. This is a direct probe of the configuration dependence of the Vainshtein effect. For each model, the bispectra were measured in the $256\mathrm{Mpc}/h$ boxes. The errors are bootstrap error bars on the ratio determined from the different runs. We have omitted the error bars on the $\mathrm{nDGP}-2$ model for clarity—they are very similar to those for $\mathrm{nDGP}-1$.

Figure 10

Relative deviation of the power spectrum at $z=0$ measured in a simulation using the $G_{\mathrm{eff}}(\delta )$ approximation, $P_{G_{\mathrm{eff}}}$, from that of the full DGP simulation, $P_{\mathrm{DGP}}(k)$ for $\mathrm{nDGP}-1$ and $\mathrm{nDGP}-2$. In each case, results are shown for a “typical” run with $L_{\mathrm{box}}=64\mathrm{Mpc}/h$.