Nonlinear growth in modified gravity theories of dark energy

TL;DR

This work addresses whether standard nonlinear fits for the matter power spectrum, PD and SP, remain valid in modified gravity theories that alter the Poisson equation and/or introduce anisotropic stress. Using N-body PM simulations for 5D gravity, DGP, and twin anisotropic-stress toy models, the authors compare the nonlinear evolution and resulting weak-lensing spectra to the PD and SP analytical predictions, finding within $1\sigma$ agreement over the mildly nonlinear regime ($k\sim0.1-1\ \mathrm{Mpc}^{-1}$) and redshifts up to today. The results support the robustness of these fits in MG scenarios and imply that the linear growth factor and spectral-index dependence largely capture MG-induced deviations in the mildly nonlinear transition, with no clear preference for either fit. These findings have practical implications for interpreting upcoming weak-lensing surveys and for constraining MG models without developing entirely new nonlinear prescriptions, though they caution against extrapolating to strongly nonlinear, subhalo-dominated scales. The study also highlights avenues for future work on scale-dependent MG, smaller scales, and comparisons with newer MG nonlinear ansätze.

Abstract

Theoretical differences in the growth of structure offer the possibility that we might distinguish between modified gravity theories of dark energy and \LambdaCDM. 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 1σof 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.

Figures (11)

• 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. \ref{['theories']}: the 5-D 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 errors bars show the average power spectrum and standard deviation for 24 simulations. The vertical dotted line represents $k_{Nyquist}/2$, which is a conservative estimate for the largest $k$ at which we can believe the simulation results as in Stab:2006yuk. The PD (dot-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. \ref{['5D']} for $r_{s}=20h^{-1}$ Mpc (top, blue), $10h^{-1}$ Mpc (middle, green) and $5h^{-1}$ Mpc (bottom, red). The full line and errors bars show the average of the ratios and standard deviation for 24 simulations. The vertical dotted line represents $k_{Nyquist}/2$, which is a conservative estimate for the largest $k$ at which we can believe the simulation results as in Stab:2006yuk. The PD (dot-dashed line) and SP (dashed line) analytical fits agree with simulations to within 1$\sigma$ for each $r_s$, in the region of interest, $k=0.1$ to $1$ Mpc$^{-1}$.
• Figure 5: The ratios of the dimensionless matter power spectrum in modified to standard gravity, $\Delta_{alt}^2(k)/\Delta^2_{std}(k)$ as a function of redshift $50\le z \le 0$ for $k=0.53$ Mpc$^{-1}$. The color coding and lines styles are as in Fig. \ref{['fig:UABratios']}. 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.