Nonlinear growth in modified gravity theories of dark energy

Theoretical differences in the growth of structure offer the possibility that we might distinguish between modified gravity theories of dark energy and $\Lambda \mathrm{CDM}$. A significant impediment to applying current and prospective large scale galaxy and weak lensing surveys to this problem is that, while the mildly nonlinear regime is important, there is a lack of numerical simulations of nonlinear growth in modified gravity theories. A major question exists as to whether existing analytical fits, created using simulations of standard gravity, can be confidently applied. In this paper we address this, presenting results of N-body simulations of a variety of models where gravity is altered including the Dvali, Gabadadze, and Porrati model. We consider modifications that alter the Poisson equation and also consider the presence of anisotropic shear stress that alters how particles respond to the gravitational potential gradient. We establish how well analytical fits of the matter power spectrum by Peacock and Dodds and Smith et al. are able to predict the nonlinear growth found in the simulations from $z=50$ up to today, and also consider implications for the weak lensing convergence power spectrum. We find that the analytical fits provide good agreement with the simulations, being within $1\sigma$ of the simulation results for cases with and without anisotropic stress and for scale-dependent and independent modifications of the Poisson equation. No strong preference for either analytical fit is found.

Figure 1

A two-dimensional description of cloud-in-cell density assignment. (a) The definition of the variables in relation to the particle’s actual position. The particle is the black dot, but it is extended to be a square particle denoted by the dotted lines, thus it lies in four cells. The sides of the cells and the size of the particle square are $L=D1+T1=D2+T2$. (b) The resultant mass distribution in each cell. Note that the mass is not retained in the original particle’s area, but rather smeared over the cell it occupies.

Figure 2

The ratio of the linear power spectrum in the modified theories to that for standard gravity for the models discussed in Sec. 2: the 5D gravity model of Uzan and Bernadeau (solid line), TM1 (dotted line), TM2 (dashed line), and DGP (dotted-dashed line).

Figure 3

Dimensionless matter power spectrum, ${\Delta }^2(k)\equiv k^3{P}_{\delta }(k)/2{\pi }^2$, for standard gravity. The full line and error bars show the average power spectrum and standard deviation for 24 simulations. The vertical dotted line represents $k_{Ny\mathrm{quist}}/2$, which is a conservative estimate for the largest $k$ at which we can believe the simulation results as in 30. The PD (dotted-dashed line) and SP (dashed line) analytical fits are also shown.

Figure 4

Ratios of the $z=0$ dimensionless matter power spectrum in the modified gravity model to that for standard gravity, for the 5D gravity model described in Sec. 2a for $r_s=20h^{-1}\mathrm{Mpc}$ (top, blue), $10h^{-1}\mathrm{Mpc}$ (middle, green), and $5h^{-1}\mathrm{Mpc}$ (bottom, red). The full line and error bars show the average of the ratios and standard deviation for 24 simulations. The vertical dotted line represents $k_{Ny\mathrm{quist}}/2$, which is a conservative estimate for the largest $k$ at which we can believe the simulation results as in 30. The PD (dotted-dashed line) and SP (dashed line) analytical fits agree with simulations at within $1\sigma$ for each $r_s$, in the region of interest, $k=0.1$ to $1{\mathrm{Mpc}}^{-1}$.

Figure 5

The ratios of the dimensionless matter power spectrum in modified to standard gravity, ${\Delta }_{\mathrm{alt}}^2(k)/{\Delta }_{\mathrm{std}}^2(k)$ as a function of redshift $50\le z\le 0$ for $k=0.53{\mathrm{Mpc}}^{-1}$. The color coding and lines styles are as in Fig. 4. The dotted lines show the ratios of the associated linear spectra. Note that the evolution is well tracked by the analytical fits, with both lying within $1\sigma$ for the simulations. At late times the SP fit drifts to around, or just over, the $1\sigma$ error.

Figure 6

The ratio of the weak lensing dimensionless convergence power spectrum, ${\Delta }^2(l)\equiv l^2{P}_{\kappa }(l)/2\pi$, for a $\delta$ function lensing source at $z_s=1$, as a function of multipole, $l$, for the 5D gravity model in Sec. 2a to that in standard gravity in comparison to the SP fit (left- hand panel) and PD fit (right-hand panel). The points and errors are the average and standard deviation of the ratios of the 24 simulations. The predicted spectra from the analytical fits (full lines) are wholly consistent with the simulations for all 3 modified gravity models with $r_s=20h^{-1}\mathrm{Mpc}$ (top, blue), $10h^{-1}\mathrm{Mpc}$ (middle, green), and $5h^{-1}\mathrm{Mpc}$ (bottom, red).

Figure 7

Ratios of the matter power spectrum in the DGP model with $r_c=6.1{\mathrm{Gpc}}^{-1}$ to that in standard gravity; both models have $H_0=70{\mathrm{kms}}^{-1}{\mathrm{Mpc}}^{-1}$ and ${\Omega }_m=0.3$. The full line is the average of the 24 realizations and errors represent the standard deviation of the simulations. The SP (dashed line) and PD (dotted-dashed line) analytic fits are in good agreement over the scales measured by the simulation, $k=0.1$ to $1{\mathrm{Mpc}}^{-1}$. The linear power spectrum ratio is shown by the dotted line.

Figure 8

The evolution of the ratio of the DGP matter power spectrum to standard gravity for $k=0.53{\mathrm{Mpc}}^{-1}$ as a function of scale factor, $a$. The full line is the average of the 24 realizations and errors represent the standard deviation of the simulations. The SP (dotted line) and PD (dotted-dashed line) analytic fits are good at predicting the transition and development of nonlinear growth at all epochs.

Figure 9

The ratios of matter power spectra at $a=1$ for modified gravity to standard gravity in the TM1 (left panel) and TM2 (right panel) models for ${\Sigma }_0=-0.016$ (dark blue, bottom), $-0.008$ (red), 0.008 (green), and 0.016 (light blue, top) as a function of scale, $k$. As in earlier figures, the full line represents the average of the 24 simulations, error bars represent 1 standard deviation, and $k_{Ny\mathrm{quist}}/2$ is indicated by the vertical dotted line. The predictions of the SP (dashed line) and PD (dotted-dashed line) fits are nearly identical, and are in excellent agreement with the simulations for both the weaker modifications with ${\Sigma }_0=\pm 0.008$ and the strong ones with ${\Sigma }_0=\pm 0.016$. The linear power spectra, showing the differences in linear growth factor arising from the modifications are shown by the dotted lines.

Figure 10

The evolution of the power spectrum over time for TM1 (left panel), and TM2 (right panel). Throughout the entire simulation the fits track the simulation results extremely well. The color coding and line styles are the same as in Fig. 9.

Figure 11

The ratios of modified convergence power to standard convergence power in the twin models TM1 (full triangles) and TM2 (empty triangles) for ${\Sigma }_0=0.016$ ($◃$, blue), $-0.008$ ($△$, red), 0.008 ($▿$, green), and 0.016 ($▹$, light blue) shown against the predicted spectrum using the SP fit (full line), as the predictions of SP and PD are virtually identical. As is to be expected, given the strong agreement between the fits and simulations of the matter power spectrum, the weak lensing spectra from the simulations are predicted well by the analytical fits.