Simulations of Galileon modified gravity: Clustering statistics in real and redshift space

We use N-body simulations to study the statistics of massive halos and redshift space distortions for theories with a standard $\Lambda \mathrm{CDM}$ expansion history and a Galileon-type scalar field. The extra scalar field increases the gravitational force, leading to enhanced structure formation. We compare our measurements of the real space matter power spectrum and halo properties with fitting formula for estimating these quantities analytically. We find that a model for power spectrum, halo mass function, and halo bias, derived from $\Lambda \mathrm{CDM}$ simulations, can fit the results from our simulations of modified gravity when ${\sigma }_8$ is appropriately adjusted. We also study the redshift space distortions in the two point correlation function measured from these simulations, finding a difference in the ratio of the redshift space to real space clustering amplitude relative to standard gravity on all scales. We find enhanced clustering on scales $r>10\mathrm{Mpc}/h$ and increased damping of the correlation function for scales $r<9\mathrm{Mpc}/h$. The boost in the clustering on large scales due to the enhanced gravitational forces cannot be mimicked in a standard gravity model by simply changing ${\sigma }_8$. This result illustrates the usefulness of redshift space distortion measurements as a probe of modifications to general relativity.

Figure 1

Results from linear theory. In the top panel, we show the $z=0$ amplitude of the dark matter linear power spectrum that would be inferred by standard methods for different values of $r_c$. This measure, apparent ${\sigma }_8$, is defined in the text and in the plot title where $D_{\mathrm{GR}}$ is the growth factor in standard gravity. In the lower panel, we plot the ratio of the strength (in linear theory) of the extra scalar force as compared with the Newtonian force; this appears as $g=1/(3B(a))$ in the text. Although the extra force has in principle a maximum strength of $1/3$ the Newtonian force, the phenomenological model we adopt suppresses this under the assumption that the background density of space will modulate the strength of the extra scalar’s force, as was found in the DGP model.

Figure 2

(Left): Halo mass function at $z=0$ measured from $\Lambda \mathrm{CDM}$ (red triangles) and Galileon modified gravity simulations (blue squares) with $r_c=1665\mathrm{Mpc}$ (upper panel) and 1089 Mpc (lower panel). The $\Lambda \mathrm{CDM}$ predictions of Tinker et al. [48] for ${\sigma }_8=0.8$ (solid curves) and ${\sigma }_8=0.88$, 0.92 (dashed curves) are also plotted. (Right): Percent difference for the mass function measured from modified gravity simulations relative to a $\Lambda \mathrm{CDM}$ simulation with ${\sigma }_8=0.8$ (triangles and squares) and Tinker predictions relative to ${\sigma }_8=0.8$ (dashed lines).

Figure 3

Linear halo bias at $z=0$ for the same cases shown in Fig. 2. The results for the halo bias are noisier than those for the mass function but similarly display the degeneracy between Galileon modified gravity and $\Lambda \mathrm{CDM}$ with larger ${\sigma }_8$ at single redshifts.

Figure 4

Power spectrum at $z=0$ for the same cases shown in Figs. 2, 3. The left panels show the simulated power spectra for the $\Lambda \mathrm{CDM}$ simulation (triangles) and for the Galileon simulations (squares) with $r_c=1665\mathrm{Mpc}$ (top) and $r_c=1089\mathrm{Mpc}$ (bottom). Also shown are the linear power spectrum for $\Lambda \mathrm{CDM}$ with ${\sigma }_8=0.8$ and the HALOFIT power spectra for ${\sigma }_8=0.8$ (solid line), 0.88 (dashed top), and 0.92 (dashed bottom). The right panels show relative deviations from these cases. On deeply nonlinear scales, the Vainshtein screening is effective, and it is not possible to fit the Galileon simulations using standard $\Lambda \mathrm{CDM}$ fits with higher values of ${\sigma }_8$.

Figure 5

The linear growth rate—both weighted by ${\sigma }_8(z)$ and alone—as a function of redshift, $z$, for $\Lambda \mathrm{CDM}$, the $r_c=1089\mathrm{Mpc}$ and $r_c=1665\mathrm{Mpc}$ Galileon models, plus a comparison model with $r_c=4000\mathrm{Mpc}$ are shown as a black dashed line, a solid black line, a dashed blue line, and a dashed-dotted red line, respectively. The grey band around the $\Lambda \mathrm{CDM}$ line represents the expected precision of future surveys, which hope to achieve $\pm 2\%$ accuracy in their measurement of the growth rate.

Figure 6

Upper panels: The dark matter two point correlation function measured in real and redshift space at $z=0$ are shown as red circles and blue squares for $\Lambda \mathrm{CDM}$ and the $r_c=1089\mathrm{Mpc}$ ($r_c=1665\mathrm{Mpc}$) model in the left (right) panel. Open symbols denote the correlation function measured in real space, while closed symbols represent the correlation function measured in redshift space. Lower panels: The relative difference in the ratio of the correlation function measured in redshift space to that in real space, in the $r_c=1089\mathrm{Mpc}$ ($r_c=1665\mathrm{Mpc}$) model, compared to $\Lambda \mathrm{CDM}$ is shown in the left (right) panel. The dotted grey line in both the right and left panels shows the Kaiser linear theory prediction for this ratio using the appropriate growth rate for each model.

Figure 7

The relative difference in the ratio of the correlation function measured in redshift space to that in real space, in the $r_c=1089\mathrm{Mpc}$ ($r_c=1665\mathrm{Mpc}$) model, compared to $\Lambda \mathrm{CDM}$ is shown in the left (right) panel. The upper, middle, and lower panels show this ratio measured from the simulations at $z=0.8$, $z=0.6$ and $z=0.4$, respectively with jackknife errors on the mean. The dotted grey line in both the right and left panels shows the Kaiser linear theory prediction for this ratio using the appropriate growth rate for each model at the redshift indicated by the legend.

Figure 8

The relative difference in the ratio of the correlation function measured in redshift space to that in real space, in the $r_c=1089\mathrm{Mpc}$ ($r_c=1665\mathrm{Mpc}$) model compared to $\Lambda \mathrm{CDM}$, is shown in the top (bottom) panel as a red solid line with error bars as in Fig. 6. The difference in the ratio ${\xi }_s/{\xi }_r$ measured from a $\Lambda \mathrm{CDM}$ simulation with ${\sigma }_8=0.92$ (${\sigma }_8=0.88$) to a $\Lambda \mathrm{CDM}$ simulation with ${\sigma }_8=0.8$ is shown as a black dashed line in the left (right) panel. The grey shaded regions represent the jackknife errors on the mean.

Figure 9

The ratio of ${\xi }_s/{\xi }_r$ measured using all halos with $\mathrm{\text{masses}}>{10}^{13}{M}_{\odot }/h$ in the $r_c=1089\mathrm{Mpc}$ ($r_c=1665\mathrm{Mpc}$) model compared to $\Lambda \mathrm{CDM}$ are shown in the left (right) panels as a green solid line. The results at $z=0$, $z=0.4$, and $z=0.6$ are shown in the bottom, middle, and top panels, respectively. The grey shaded region shows the Jackknife errors on the mean.

Figure 10

Spherically symmetric source code test: comparison of code output (red filled circles) with exact solution (solid red line). The exact solution with $r_c=0$ (i.e., with no nonlinear terms in the equation) is plotted as a dashed red line. This comparison was done for the spherical source of constant density with parameters chosen so that the Vainshtein radius of the source, including nonlinearities, would be 50 in grid units. This is reflected in the linear (dashed) and nonlinear (solid line and red circles) solutions converging just beyond a radius of 50.

Figure 11

Residual error vs scale factor, for runs with $L_{\mathrm{box}}=400\mathrm{Mpc}/\mathrm{h}$ and $r_c=1089$ (1665) Mpc in black (red). The convergence target for this box size is plotted as a horizontal dashed line.