Assessment of the information content of the power spectrum and bispectrum

The covariance matrix of the matter and halo power spectrum and bispectrum are studied. Using a large suite of simulations, we find that the non-Gaussianity in the covariance is significant already at mildly nonlinear scales. We compute the leading disconnected non-Gaussian correction to the matter bispectrum covariance using perturbation theory, and find that the corrections result in good agreement in the mildly nonlinear regime. The shot noise contribution to the halo power spectrum and bispectrum covariance is computed using the Poisson model, and the model yields decent agreement with simulation results. However, when the shot noise is estimated from the individual realization, which is usually done in reality, we find that the halo covariance is substantially reduced and gets close to the Gaussian covariance. This is because most of the non-Gaussianity in the covariance arises from the fluctuations in the Poisson shot noise. We use the measured non-Gaussian covariance to access the information content of the power spectrum and bispectrum. The signal-to-noise ratio (S/N) of the matter and halo power spectrum levels off in the mildly nonlinear regime, $k\sim 0.1–0.2{\mathrm{Mpc}}^{-1}h$. In the nonlinear regime the S/N of the matter and halo bispectrum increases but much slower than the Gaussian results suggest. We find that both the S/N for power spectrum and bispectrum are overestimated by the Gaussian covariances, but the problem is much more serious for the bispectrum. Because the bispectrum is affected strongly by nonlinearity and shot noise, inclusion of the bispectrum only adds a modest amount of S/N compared to that of the power spectrum.

Figure 1

A diagrammatic representation of the leading contributions to the covariance of the matter power spectrum. $C_{\mathrm{G}}^{\mathrm{NL}}$ is the Gaussian term (top left), while the rest are non-Gaussian contributions. The set of black dots on the left and right of each diagram represent the two $\delta$’s in each power spectrum estimator. The legs branching from each dot represent the perturbation theory kernel, $F_1$, $F_2$, and $F_3$, respectively. Each wavy line represents the linear power spectrum. In $C_{\mathrm{G}}^{\mathrm{NL}}$, the filled circle indicates that the linear power spectrum is replaced by the one-loop power spectrum.

Figure 2

Various contributions to the diagonal elements of the covariance of the halo power spectrum for two selected halo groups. The Gaussian covariance (solid black) and the non-Gaussian terms proportional to $1/{\tilde{n}}^3$ (dashed green) and $P/{\tilde{n}}^2$ (dashed red).

Figure 3

The diagonal elements of the dark matter power spectrum covariance at $z=1$, 0.5, and 0 (from top to bottom). The results are normalized with respect to the Gaussian covariance. The results from the large (blue circles), small (red triangles), and hires (green squares) sets are shown. The perturbation theory predictions are overplotted (black solid line).

Figure 4

The correlation coefficients of the dark matter power spectrum covariance $r$ at $z=1$, 0.5, and 0 (from left to right). $r(k_i,k_j)$ as a function of $k_j$ for a list of fixed $k_i=0.076$, 0.19, and $0.40{\mathrm{Mpc}}^{-1}h$ (top to bottom). The results from the large (blue circles), small (red triangles), and hires (green squares) sets are shown. The perturbation theory predictions are overplotted (black solid line).

Figure 5

The diagonal elements of the halo power spectrum covariance. The mean Poisson shot noise is subtracted. The results from large group 4 (blue circles), small group 4 (red triangles), and hires group 1 (violet squares) and group 4 (green squares) are shown. The predictions are computed using Eq. (12) (black lines).

Figure 6

Similar to Fig. 4, except for the halo power spectrum covariance. The mean Poisson shot noise is subtracted. The results from large group 4 (blue circles), small group 4 (red triangles), and hires group 1 (violet squares) and group 4 (green squares) are shown. The predictions are computed using Eq. (12) (black lines). At $z=0$, the lower curve is the theory prediction for hires group 1.

Figure 7

The diagonal elements of the halo power spectrum with the mean shot noise subtracted (circles, upper data set) and the shot noise estimated and subtracted from each realization (triangles, lower data set). The data from the small set are used. The prediction using Eq. (16) is overplotted (solid blue).

Figure 8

The correlation coefficients for the halo power spectrum with the mean shot noise subtracted (circles, upper data set) and the shot noise estimated and subtracted from each realization (triangles, lower data set). The data from the small set are used. The prediction using Eq. (16) is overplotted (solid blue).

Figure 9

A diagrammatic representation of the covariance of the matter bispectrum to the tree-level order. $C_{\mathrm{G}}^{\mathrm{NL}}$ is the Gaussian term, while the rest are non-Gaussian contributions. The set of black dots on the left and right of each diagram represents the three $\delta$’s in each bispectrum estimator. The legs branching from each dot represent the perturbation theory kernel, $F_1$, $F_2$, and $F_3$, respectively. Each wavy line represents the linear power spectrum. In $C_{\mathrm{G}}^{\mathrm{NL}}$, the filled circle indicates that the linear power spectrum is replaced by the nonlinear power spectrum. The Gaussian term couples only triangles of the same shape, while the non-Gaussian terms couple triangles with at least one side equal to each other.

Figure 10

The leading tree-level contributions to the diagonal elements of the dark matter bispectrum covariance matrix at $z=0$ for the equilateral triangle configuration. The binning $\mathrm{\Delta }k=0.019{\mathrm{Mpc}}^{-1}h$ is used here and all the plots for the bispectrum. Upper panel: the Gaussian contributions $C_{\mathrm{G}}^{\mathrm{L}}$ (solid blue) and $C_{\mathrm{G}}^{\text{Loop}}$ (red, solid for the positive part, dashed for the negative part), the non-Gaussian $BB$ contribution (solid, cyan) and $PT$ contributions (solid, green), and the sum of all the contributions (solid black). Lower panel: the ratio of various terms normalized with respect to $C_{\mathrm{G}}^{\mathrm{L}}$. The violet line includes all the high order correction terms.

Figure 11

Various shot noise contributions to the covariance of the halo bispectrum. The diagonal elements for the equilateral triangle configuration are shown. We have plotted the results for two selected groups with the parameters used written on the plot: the Gaussian term (dotted-dashed, blue), the non-Gaussian contribution $C_{\mathrm{NG}}^{(1{)}^{\prime }}$ (dashed, red) and $C_{\mathrm{NG}}^{(0{)}^{\prime }}$ (dashed, green), and the sum of the covariances (solid, black). In the low panel the Gaussian covariance overlaps with the total covariance.

Figure 12

The diagonal elements of the dark matter bispectrum covariance matrix for equilateral triangle configuration at $z=1$, 0.5, and 0 (from top to bottom). The results from the small (red triangles), large (blue circles), and hires (green squares) simulation sets are shown. The perturbation theory predictions (solid black lines) are also overplotted.

Figure 13

The correlation coefficient for the dark matter bispectrum. The equilateral triangle configurations are used. Results at $z=1$, 0.5, and 0 (left to right columns) are shown. The three rows correspond to the results obtained with $k_i$ fixed to be 0.076, 0.19, and $0.40{\mathrm{Mpc}}^{-1}h$, respectively. The results from the small (red triangles), large (blue circles), and hires (green squares) simulation sets and the prediction (solid black) are shown.

Figure 14

The diagonal elements of the halo bispectrum for the equilateral triangle configuration. The mean Poisson shot noise is subtracted. The halo groups from large (blue circles), small (red triangles), and hires [violet squares (group 1) and green squares (group 4)] simulation sets are shown. The theory prediction (solid black lines) includes the Gaussian covariance and the non-Gaussian terms $C_{\mathrm{NG}}^{(0{)}^{\prime }}$ and $C_{\mathrm{NG}}^{(1{)}^{\prime }}$.

Figure 15

The correlation coefficient for the halo bispectrum of the equilateral triangle shape. The mean Poisson shot noise is subtracted. The halo groups from large (blue circles), small (red triangles), and hires [violet squares (group 1) and green squares (group 4)] simulation sets are shown. The predictions (solid black lines) include Gaussian covariance and non-Gaussian corrections $C_{\mathrm{NG}}^{(0{)}^{\prime }}$ and $C_{\mathrm{NG}}^{(1{)}^{\prime }}$.

Figure 16

The diagonal elements of equilateral triangle halo bispectrum obtained with the mean Poisson shot noise subtraction (blue circles) and individual shot noise subtraction (violet triangles) are compared. The variance of the Poisson shot noise, $\mathrm{var}(B_{\text{Pois}})$ (solid green line), the covariance between the raw halo bispectrum and the Poisson shot noise, $\mathrm{cov}(B_{\mathrm{raw}},B_{\text{Pois}})$ (solid red line), and the prediction using the ansatz in Eq. (45) (cyan squares) are also plotted.

Figure 17

The correlation coefficients for the equilateral triangle halo bispectrum with the mean shot noise subtraction (blue circles) and individual shot noise subtraction (violet triangles). The black solid line shows the prediction using the ansatz in Eq. (45). Only the results for $k_i=0.076$ and $0.19{\mathrm{Mpc}}^{-1}h$ are shown.

Figure 18

The trace of the precision matrix as a function of $n/p$, where $n$ is the number of realizations used to estimate the covariance matrix and $p$ is the number of the bispectrum bins, which is equal to 429 here. Both the results from dark matter and halo (group 4) at $z=0$ are shown. The results from the naive estimate $C_{\text{sample}}^{-1}$ (blue triangles) and bias-corrected estimate $C_{\text{unbiased}}^{-1}$ (green circles) are compared.

Figure 19

The signal-to-noise ratio for dark matter and halo power spectrum and bispectrum. Dark matter and halo group 4 data from the small simulation set are used. The signal-to-noise ratio of the power spectrum (blue) and bispectrum (red), and the sum of them (yellow) are shown. The results obtained with the full non-Gaussian covariance results (circles, solid line) are compared with the Gaussian covariance (triangles, dotted line) ones.

Figure 20

The combined S/N ratio obtained with the cross covariance between $P$ and $B$ properly taken into account and the cross covariance ignored. The upper panel is for dark matter, while the lower one is for halo group 4. The results at $z=1$ (blue), 0.5 (red), and 0 (green) are compared. Both the narrow bispectrum binning results ($2k_{\text{F}}$, solid line) and the wide binning one ($8k_{\text{F}}$, dashed line) are shown.

Figure 21

The distribution of the bispectrum estimator. The results from the small simulation set at $z=1$ (blue), 0.5 (green) and 0 (red) are shown. The upper panels correspond to the results from dark matter while the lower panels are for halo group 4. The equilateral triangle configuration is used. Results at three different wave numbers $k=0.06$, 0.25 and $0.44{\mathrm{Mpc}}^{-1}h$ (from left to right) are displayed. The data have been transformed to the standard variable by Eq. (a1). The Gaussian distribution with zero mean and unity variance is overplotted (solid black line).

Figure 22

The skewness of the bispectrum estimator. The results from the dark matter (upper panel) and halo group 4 (lower panel) of the small simulation set are shown. Only the equilateral triangle configuration is used. The results from $z=1$ (blue circles), 0.5 (red triangles) and 0 (green squares) are compared.

Figure 23

Similar to Fig. 22 except for kurtosis.

Figure 24

Diagrammatic representation of the two-point, three-point and four-point correlation including the shot noise contributions.

Figure 25

Diagrammatic representation of the six-point correlation including the shot noise contributions.