Statistics of peaks of weakly non-Gaussian random fields: Effects of bispectrum in two- and three-dimensions

Analytic expressions for the statistics of peaks of random fields with weak non-Gaussianity are provided. Specifically, the abundance and spatial correlation of peaks are represented by formulas which can be evaluated only by virtually one-dimensional integrals. We assume the non-Gaussianity is weak enough such that it is represented by linear terms of the bispectrum. The formulas are formally given in $N$-dimensional space, and explicitly given in the case of $N=1$, 2, 3. Some examples of peak statistics in cosmological fields are calculated for the cosmic density field and weak lensing field, assuming the weak non-Gaussianity is induced by gravity. The formulas of this paper would find a fit in many applications to statistical analyses of cosmological fields.

Figure 1

The differential number density of peaks in three dimensions. In the upper panel, predictions of Gaussian (dashed line) and non-Gaussian (solid line) fields are shown, where the number density is measured in units of the smoothing radius $R=20h^{-1}\mathrm{Mpc}$. In the lower panel, the ratio of the non-Gaussian prediction to the Gaussian prediction is plotted.

Figure 2

The differential number density of peaks in two-dimensional weak lensing field. In the upper panel, predictions of Gaussian (dashed line) and non-Gaussian (solid line) fields are shown, where the number density is measured in units of the smoothing angle $\vartheta =10$ arc min. In the lower panel, the ratio of the non-Gaussian prediction to the Gaussian prediction is plotted. Nonlinearity and noise effects are not included.

Figure 3

The power spectrum of peaks in three-dimensional density field with a smoothing radius $R=20h^{-1}\mathrm{Mpc}$. Predictions of Gaussian field with first-order and second-order approximations are shown in dashed and dotted lines, respectively (the second-order approximation contains the first-order and second-order contributions). The component of non-Gaussian correction is shown in a dot-dashed line. The total correlation function is shown in a solid line. The scaled power spectrum of the underlying smoothed density field, ${b_{10}}^2{P}_{\mathrm{L}}(k)W^2(kR)$, is also plotted in a lower solid line.

Figure 4

The correlation function of peaks in three-dimensional density field with a smoothing radius $R=20h^{-1}\mathrm{Mpc}$. Predictions of Gaussian field with first-order and second-order approximations are shown in dotted and dashed lines, respectively. The component of non-Gaussian correction is shown in a dot-dashed line. The total correlation function is shown in a solid line.

Figure 5

Same as Fig. 4, but the underlying power spectrum is given by that of a CDM power spectrum without baryons.