Revealing modified gravity signals in matter and halo hierarchical clustering

We use a set of N-body simulations employing a modified gravity (MG) model with Vainshtein screening to study matter and halo hierarchical clustering. As test-case scenarios we consider two normal branch Dvali-Gabadadze-Porrati (nDGP) gravity models with mild and strong growth rate enhancement. We study higher-order correlation functions ${\xi }_n(R)$ up to $n=9$ and associated reduced cumulants $S_n(R)\equiv {\xi }_n(R)/\sigma (R{)}^{2n-2}$. We find that the matter probability distribution functions are strongly affected by the fifth force on scales up to $50h^{-1}\mathrm{Mpc}$, and the deviations from general relativity (GR) are maximized at $z=0$. For reduced cumulants $S_n$, we find that at small scales $R\le 6h^{-1}\mathrm{Mpc}$ the MG is characterized by lower values, with the deviation growing from 7% in the reduced skewness up to even 40% in $S_5$. To study the halo clustering we use a simple abundance matching and divide haloes into thee fixed number density samples. The halo two-point functions are weakly affected, with a relative boost of the order of a few percent appearing only at the smallest pair separations ($r\le 5h^{-1}\mathrm{Mpc}$). In contrast, we find a strong MG signal in $S_n(R)$’s, which are enhanced compared to GR. The strong model exhibits a $>3\sigma$ level signal at various scales for all halo samples and in all cumulants. In this context, we find that the reduced kurtosis to be an especially promising cosmological probe of MG. Even the mild nDGP model leaves a $3\sigma$ imprint at small scales $R\le 3h^{-1}\mathrm{Mpc}$, while the stronger model deviates from a GR signature at nearly all scales with a significance of $>5\sigma$. Since the signal is persistent in all halo samples and over a range of scales, we advocate that the reduced kurtosis estimated from galaxy catalogs can potentially constitute a strong MG-model discriminatory as well as GR self-consistency test.

Figure 1

The contributions of the cosmic variance and the shot noise to the correlation functions estimators we use in this work.

Figure 2

Comparison of matter density variance for $z=0$ estimated from N-body simulations (points) with the PT predictions of Eq. (22). The lower panel shows the fractional difference of both nDGP models taken with respect to the GR case. The shaded region illustrates the cosmic variance error for the GR fiducial case.

Figure 3

Same as the previous figure, but for the reduced density skewness $S_3$ compared against the estimator of Eq. (24).

Figure 4

The distribution functions for our models computed for density fields smoothed at $R=0.5$, 25 and $50h^{-1}\mathrm{Mpc}$ (panels from left to right). For each panel the GR case is marked by solid blue line, nDGP ${\mathrm{\Omega }}_{rc}=0.0124$ is dashed orange and nDGP ${\mathrm{\Omega }}_{rc}=0.438$ is marked by dot-dashed line. In each upper panel three groups of lines correspond to three redshifts $z=0$, 0.5 and 1, where the data for latter two was scaled down for clarity. For each case the smaller bottom panel illustrates the $p(\delta )$ ratio taken with respect to the GR case only for $z=0$ case. Mark the change to linear scaling for the $\delta +1$ axis in the most left panel.

Figure 5

The matter density power spectrum computed at $z=0$ for our fiducial GR model (solid line) and two nDGP flavors (dotted and dashed-dotted lines). The shaded region illustrate the cosmic variance error. The bottom panel illustrates the fractional difference of both MG models with respect to the GR case.

Figure 6

Hierarchy of $n$-point volume averaged correlation functions ${\overline{\xi }}_n(R)$ computed for smoothed dark matter density fields at $z=0$ (left panel) and $z=1$ (right panel). Shaded regions (hardly visible) mark a combination of sampling and cosmic variance errors for the GR cases.

Figure 7

Hierarchical clustering amplitudes $S_n(R)$ computed for smoothed dark matter density fields at $z=0$ (left panel) and $z=1$ (right panel). Shaded regions (hardly visible) mark a combination of sampling and cosmic variance errors for the GR cases.

Figure 8

Fractional differences from the GR taken across three epochs $z=0$, 0.5, 1 (rows of panels from top to bottom) of the first three matter density reduced cumulants $S_3$, $S_4$ and $S_5$ (columns of panels from left to right). Shaded regions illustrate the total error budget ($\text{cosmic variance}+\text{shot noise}$) on the ratios.

Figure 9

Two-point correlation functions computed as a pair separation function R. Left panel: Two-point statistics for subsampled dark matter particles three our models: GR (solid blue), nDGP ${\mathrm{\Omega }}_{rc}=0.0124$ (dashed orange) and nDGP ${\mathrm{\Omega }}_{rc}=0.438$ (dashed-dotted red). The shaded regions mark illustrative cosmic variance error for the GR case. Three distinctive epochs are shown: $z=0$, 0.5 and 1. Consecutive values of $r^2{\xi }_2(r)$ were scaled down to allow better presentation. Middle panel: Same as the left panel but for all DM haloes (centrals and satellites) identified in our simulations. Right panel: Same as the middle panel but only for main haloes. In all three cases the lower subpanels illustrate the fractional departure from the fiducial GR case (${\xi }_2/{\xi }_2^{GR}-1$) at $z=0$. Here the error bars mark the Poisson sampling errors for the pair-number counts in each $r+\mathrm{\Delta }r$ separation bin. In contrast the shaded region (shown only for $z=0$ for clarity) reflect the cosmic variance error contribution.

Figure 10

The hierarchical amplitudes tiered from the reduced skewness, $S_3$, (the most bottom group of lines in each panel) to $S_9$ (the top group of lines in each panel) calculated for the DM halo overdensity field at $z=0$. In the top panel we show the data for all haloes found in our simulations ($\text{centrals}+\text{satellites}$), while in the bottom panel we use main haloes only (centrals).

Figure 11

Fractional differences from the GR taken at $z=0$ for three halo samples: $⟨n⟩=1.4\times {10}^{-3}{h}^3/{\mathrm{Mpc}}^3$ ($\text{centrals}+\text{satellites}$), $⟨n⟩=9\times {10}^{-4}{h}^3/{\mathrm{Mpc}}^3$ and $⟨n⟩=1\times {10}^{-4}{h}^3/{\mathrm{Mpc}}^3$ (rows of panels from top to bottom) of the first three reduced cumulants $S_3$, $S_4$ and $S_5$ (columns of panels from left to right). Shaded regions illustrate the total error budget ($\text{cosmic variance}+\text{shot noise}$ for each halo sample) on the ratios. The data for $S_5$ and $⟨n⟩=1\times {10}^{-4}{h}^3{\mathrm{Mpc}}^3$ is not shown due to large scatter and noise.