Efficient simulations of large-scale structure in modified gravity cosmologies with comoving Lagrangian acceleration

We implement an adaptation of the cola approach, a hybrid scheme that combines Lagrangian perturbation theory with an N-body approach, to model nonlinear collapse in chameleon and symmetron modified gravity models. Gravitational screening is modeled effectively through the attachment of a suppression factor to the linearized Klein-Gordon equations. The adapted cola approach is benchmarked, with respect to an N-body code both for the $\mathrm{\Lambda }$ cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$) scenario and for the modified gravity theories. It is found to perform well in the estimation of the dark matter power spectra, with consistency of 1% to $k\sim 2.5\mathrm{h}/\mathrm{Mpc}$. Redshift space distortions are shown to be effectively modeled through a Lorentzian parametrization with a velocity dispersion fit to the data. We find that cola performs less well in predicting the halo mass functions but has consistency, within $1\sigma$ uncertainties of our simulations, in the relative changes to the mass function induced by the modified gravity models relative to $\mathrm{\Lambda }\mathrm{CDM}$. The results demonstrate that cola, proposed to enable accurate and efficient, nonlinear predictions for $\mathrm{\Lambda }\mathrm{CDM}$, can be effectively applied to a wider set of cosmological scenarios, with intriguing properties, for which clustering behavior needs to be understood for upcoming surveys such as LSST, DESI, Euclid, and WFIRST.

Figure 1

The fractional difference between the $\mathrm{\Lambda }\mathrm{CDM}$ power spectrum, $P_n$, for one realization as obtained using different choices of time steps, $n$, and the high resolution results, $P_{800}$ for pm (left) and $P_{400}$ for cola (right), respectively.

Figure 2

The ratio of the first (left) and second (right) order growth factors, $D_1$ and $D_2$, respectively, relative to the $\mathrm{\Lambda }\mathrm{CDM}$ growth factor in each case, for the $|f_{R_0}|={10}^{-4}$ model (solid lines) and the symmetron scenario (dashed lines) described in Sec. 2a2.

Figure 3

Left: Tracking a particle’s position throughout the simulation for the exact and approximate hybrid in the $|f_{R_0}|={10}^{-4}$ model. Right: Ratio of the power spectra obtained from the approximate and the exact method for the $|f_{R_0}|={10}^{-4}$ model today.

Figure 4

(Top) Power spectra benchmarking for $\mathrm{\Lambda }\mathrm{CDM}$ with the pm N-body code (red dashed line) and cola method (blue full line). The nonlinear power spectrum fit developed by Smith et al. (green dotted line) is also shown for comparison. (Bottom) Ratio between both the cola and pm code $\mathrm{\Lambda }\mathrm{CDM}$ results above to the fit by Smith et al. The numbers of time steps used for cola and pm are 50 and 500, respectively.

Figure 5

Fractional difference in the CDM power spectra for the MG scenario relative to the $\mathrm{\Lambda }\mathrm{CDM}$ model, $\frac{\mathrm{\Delta }P}{P_{\mathrm{\Lambda }\mathrm{CDM}}}$, at $a=1$, for the same initial conditions for (left) the $f(R)$ scenario, for $f_{R_0}={10}^{-4},{10}^{-5}$, and ${10}^{-6}$, and (right) the symmetron model with $a_{\text{ssb}}=0.5$. The averaged results, and standard deviations, from the simulations with the pm code (red dashed line) and the cola code (blue solid line) are presented in each case. The numbers of time steps used for cola and pm are 50 and 500, respectively.

Figure 6

(Top) Fractional difference in the CDM power spectra for the $|f_{R_0}|={10}^{-4}$ scenario relative to $\mathrm{\Lambda }\mathrm{CDM}$ for one realization, as obtained by cola using various choices of time steps. (Bottom) Ratio of the fractional difference $ra{t}_n={(\frac{\mathrm{\Delta }P}{P})}_n$ for each choice to the high resolution result using 400 steps ${\text{rat}}_{400}$.

Figure 7

The ratio of the redshift space power spectrum, $P_{\text{rsd}}$, to the real space equivalent, $P_{\text{real}}$, for the different models. (Top left) A side-by-side comparison of the suppression of the redshift space clustering by nonlinear velocity correlations for the cola model for $\mathrm{\Lambda }\mathrm{CDM}$ (black), $f(R)$ models with $f_{R_0}={10}^{-4},{10}^{-5}$, and ${10}^{-6}$ ([blue circle, red triangle, and green square, respectively), and the symmetron model with $a_{\mathrm{ssb}}=0.5$ (cyan diamonds). The remaining plots show the comparison of pm (red dashed line) and cola code (blue solid line) predictions for the RSD to real space power spectrum ratio for each model in turn: (top right) $\mathrm{\Lambda }\mathrm{CDM}$, (middle left) $f_{R_0}={10}^{-4}$, (middle right) $f_{R_0}={10}^{-5}$, (bottom left) $f_{R_0}={10}^{-6}$, (bottom right) symmetron. Each plot also shows with the linear theory prediction (green dot-dashed) and a fit to the nonlinear suppression using Eq. (43) and allowing ${\sigma }_p$ to vary as a free parameter. The numbers of time steps used for cola and pm are 50 and 500, respectively.

Figure 8

Halo mass function benchmarking for $\mathrm{\Lambda }\mathrm{CDM}$ with the pm N-body code (red dashed line) and cola method (blue solid line). The halo mass function fit developed by Murray et al. (black dotted line) is also shown for comparison. The numbers of time steps used for cola and pm are 50 and 500, respectively.

Figure 9

Fractional difference in the CDM halo mass function, $n(M)$, for halos of mass $M$, at $a=1$, for each MG scenario relative to the $\mathrm{\Lambda }\mathrm{CDM}$ model, for the same initial conditions: (left) the $f(R)$ scenario, for $f_{R_0}={10}^{-4},{10}^{-5}$, and ${10}^{-6}$, and (right) the symmetron model with $a_{\text{ssb}}=0.5$. The averaged results, and standard deviations, from the simulations with the pm code (red dashed line) and the cola code (blue solid line) are presented in each case. The numbers of time steps used for cola and pm are 50 and 500, respectively.