Large-scale clustering of cosmic voids

We study the clustering of voids using $N$-body simulations and simple theoretical models. The excursion-set formalism describes fairly well the abundance of voids identified with the watershed algorithm, although the void formation threshold required is quite different from the spherical collapse value. The void cross bias $b_{\mathrm{c}}$ is measured and its large-scale value is found to be consistent with the peak background split results. A simple fitting formula for $b_{\mathrm{c}}$ is found. We model the void auto-power spectrum taking into account the void biasing and exclusion effect. A good fit to the simulation data is obtained for voids with radii $\gtrsim 30\mathrm{Mpc}{h}^{-1}$, especially when the void biasing model is extended to 1-loop order. However, the best-fit bias parameters do not agree well with the peak-background results. Being able to fit the void auto-power spectrum is particularly important not only because it is the direct observable in galaxy surveys, but also our method enables us to treat the bias parameters as nuisance parameters, which are sensitive to the techniques used to identify voids.

Figure 1

The void-size distribution for various values of ${\delta }_{\mathrm{v}}$: $-2.8$ (blue), $-1.7$ (red), and $-1.0$ (green). For each value, the void-size distributions obtained using Eq. (5) by summing up the first 12 terms (dashed line), the approximate distribution from Eq. (7) (solid line), and the single barrier distribution from Eq. (8) (dotted-dashed line) are compared.

Figure 2

The void-size distribution at $z=1$, 0.5, and 0 (from left to right) measured in simulations. The data are obtained from L1500 with sampling density $0.02(\mathrm{Mpc}{h}^{-1}{)}^{-3}$ (star, black), and L250 with sampling densities 2 (triangle, gray), 0.2 (circle, gray), and $0.02(\mathrm{Mpc}{h}^{-1}{)}^{-3}$ (diamond, gray). The SvdW void-size distribution is shown with ${\delta }_{\mathrm{v}}$ equal to $-2.8$ (dashed line, green) and with the best-fit value ${\delta }_{\mathrm{v}}$ (solid line, red) as stated in the inset of each panel.

Figure 3

The PBS bias parameter $b_1$ (solid line), $b_2$ (dashed line), and $b_3$ (dotted-dashed line) as a function of void radius $R$. Two values of ${\delta }_{\mathrm{v}}$ are chosen: $-2.8$ (blue) and $-1.0$ (red).

Figure 4

Void-matter cross-bias $b_{\mathrm{c}}$ as a function of wave number $k$ for various void sizes $R$ at $z=0$. The results are shown for L1500 with sampling density $0.02(\mathrm{Mpc}{h}^{-1}{)}^{-3}$ (star, purple), L250 with sampling density 0.02 (diamond, blue), 0.2 (circle, green), and $2(\mathrm{Mpc}{h}^{-1}{)}^{-3}$ (triangle, red). Horizontal lines show best fits to the linear large-scale regime of $b_{\mathrm{c}}$ whenever the fit is feasible (solid line, black).

Figure 5

The large-scale best fit to the void-matter cross-bias $b_{\mathrm{c}}$ as a function of void radius $R$ from L1500 (circles), L250 with sampling density 2 (square) and $0.2(\mathrm{Mpc}{h}^{-1}{)}^{-3}$ (triangle) at redshifts $z=1$, 0.5, and 0 (from left to right). The curves show $b_1$ computed from Eq. (15) for various values of ${\delta }_{\mathrm{v}}$, in particular the best-fit ${\delta }_{\mathrm{v}}$ from the void-size distribution (solid line, red).

Figure 6

The void density profile in configuration space (left), its Fourier transform (middle), and its Fourier transform divided by the nonlinear matter power spectrum as predicted by Eq. (24). Here, $R=20\mathrm{Mpc}{h}^{-1}$, ${\delta }_{\text{cen}}=-0.8$, and $\alpha$ and $\beta$ are determined according to the parametrization in Ref. [28]. The values for $r_{\mathrm{s}}$ are shown in the inset.

Figure 7

Void density profiles for different void radii at $z=0$ from L1500 with sampling density $0.02(\mathrm{Mpc}{h}^{-1}{)}^{-3}$ (square, violet) and L250 with sampling densities 0.02 (circle, blue), 0.2 (star, green), and $2(\mathrm{Mpc}{h}^{-1}{)}^{-3}$ (triangle, red). Solid lines show the best fits using Eq. (24) (same colors).

Figure 8

Comparison of the void-matter cross-bias $b_{\mathrm{c}}$ from the L1500 simulation (blue with error bars) with the Fourier transform of the corresponding best-fit void density profile divided by the matter power spectrum (green lines) at $z=0$.

Figure 9

Void-matter cross-bias $b_c$ for different void sizes from L1500 (diamond, blue) and L250 with sampling density $2(\mathrm{Mpc}{h}^{-1}{)}^{-3}$ (circle, green) and its best fits using Eq. (29) (violet and red lines, respectively) at $z=0$.

Figure 10

Void auto-power spectrum from L1500 (square, violet) and L250 with sampling densities 0.02 (diamond, blue), 0.2 (circle, red), and $2(\mathrm{Mpc}{h}^{-1}{)}^{-3}$ (triangle, green) for voids of different size at $z=0$. Poisson shot noise has been subtracted.

Figure 11

Best fits from Eq. (40) (solid line, blue) to the void auto-power spectrum from L1500 at $z=0$ (gray data points with error bars). The individual components of the best fit are also shown: the linear bias term $P_{1,1}$ (dashed line, red) and the void-exclusion term $P_{\text{excl}}$ (dotted-dashed line, green). Here, model 1 is used for $P_{\text{excl}}$ and Poisson shot noise has been subtracted from the data.

Figure 12

The best-fit values for $b_1$ and $D$ using Eq. (40) to fit the void auto-power spectrum and the corresponding ${\chi }^2$ per degree of freedom. The results from model 1 (circle, blue), 2 (square, red), 3 (triangle, green), 4 (diamond, violet), and 5 (star, yellow) are shown. The solid line depicts $b_1$ as predicted from the PBS formalism. We also show the large-scale measurement of $b_{\mathrm{c}}$ (dotted line, black) for comparison.

Figure 13

Best fits from Eq. (46) (solid line, blue) to the void auto-power spectrum from L1500 at $z=0$ (gray data points with error bars). The individual components of the best fit are also shown: the linear bias term $P_{1,1}$ (dashed line, red), $P_{2,11}$ (dashed line, green), $P_{11,11}$ (dashed line, cyan), and the void-exclusion term $P_{\text{excl}}$ (dotted-dashed line, violet). Here, model 4 is used for $P_{\text{excl}}$ and Poisson shot noise has been subtracted from the data.

Figure 14

The best-fit values for $b_1$, $b_2$, and $D$ using Eq. (46) to fit the void auto-power spectrum and the corresponding ${\chi }^2$ per degree of freedom. The results from model 1 (circle, blue), 2 (square, red), 3 (triangle, green), 4 (diamond, violet), and 5 (star, yellow) are shown. The solid lines depict $b_1$ and $b_2$ as predicted from the PBS formalism. The large-scale measurement of $b_{\mathrm{c}}$ is also plotted (dotted line, black).