Fast generation of weak lensing maps by the inverse-Gaussianization method

To take full advantage of the unprecedented power of upcoming weak lensing surveys, understanding the noise, such as cosmic variance and geometry/mask effects, is as important as understanding the signal itself. Accurately quantifying the noise requires a large number of statistically independent mocks for a variety of cosmologies. This is impractical for weak lensing simulations, which are costly for simultaneous requirements of large box size (to cover a significant fraction of the past light cone) and high resolution (to robustly probe the small scale where most lensing signal resides). Therefore, fast mock generation methods are desired and are under intensive investigation. We propose a new fast weak lensing map generation method, named the inverse-Gaussianization method, based on the finding that a lensing convergence field can be Gaussianized to excellent accuracy by a local transformation [Y. Yu, P. Zhang, W. Lin, W. Cui, and J. N. Fry, Phys. Rev. D 84, 023523 (2011).]. Given a simulation, it enables us to produce as many as infinite statistically independent lensing maps as fast as producing the simulation initial conditions. The proposed method is tested against simulations for each tomography bin centered at lens redshift $z\sim 0.5$, 1, and 2, with various statistics. We find that the lensing maps generated by our method have reasonably accurate power spectra, bispectra, and power spectrum covariance matrix. Therefore, it will be useful for weak lensing surveys to generate realistic mocks. As an example of application, we measure the probability distribution function of the lensing power spectrum, from 16384 lensing maps produced by the inverse-Gaussianization method.

Figure 1

The local transforms $y({\delta }^{\mathrm{\Sigma }})$ obtained from 48 realizations for each redshift, and after the transformation, the y field has a Gaussian PDF. The local transforms diverse at both the high- and low-density ends, reflecting the rareness of the very high density and very low density regions. They correspond to Step 1 of the inverse-Gaussianization method, detailed in Sec. 2.

Figure 2

The projected density fields from simulation and the inverse-Gaussianization method are presented in the top and bottom rows and for $z=2.023$, 1.028, and 0.485 from left to right, respectively. Note that all the panels are different realizations. The color scale is logarithmic to better show the structures. The upper and lower plots are quite similar, except that the simulated ones contain many filamentary structures, while the mock ones do not. This difference is more obvious for lower redshift.

Figure 3

Left panels show the power spectra $C_y(\ell )$ of the Gaussianized $y$ fields (blue lines) at three redshifts. From top to bottom, the results are for $z=2.023$, 1.028 and 0.485, respectively. As a reminder, these redshifts correspond to lens redshifts instead of source redshifts. All the error bars are the r.m.s. among realizations. These results correspond to Step 2 of the inverse-Gaussianization method, detailed in § 2. Step 3 of the inverse-Gaussianization method generates Gaussian Random Fields (GRFs) with $C_y(\ell )$ as the input power spectrum. We plot the power spectra of the generated GRFs (red lines), just as a trivial test that the generated GRFs indeed process identical power spectra as the original $y$ fields. Step 4 of the inverse-Gaussianization method uses the Gaussianization function obtained in Step 1 (Fig. 1) to transform the $y$ maps into ${\delta }^{\mathrm{\Sigma }}$ maps. Right panels show the power spectra of these generated ${\delta }^{\mathrm{\Sigma }}$ maps (red lines). As comparisons, we show the power spectra of the original ${\delta }^{\mathrm{\Sigma }}$ maps (blue lines). At $z=1$ and $z=2$, our method well reproduces the power spectra at $\ell \lesssim 2000$ of particular interest to weak lensing cosmology. The performance degrades towards lower redshift. At $z=0.485$, the power is overproduced at $\ell \lesssim 600$, but underproduced at $\ell \gtrsim 1000$. Nevertheless, the error is usually smaller than 20% at $\ell \lesssim 10^4$.

Figure 4

Cross-correlation coefficients between power spectra of different $\ell$ bins, for the original ${\delta }^{\mathrm{\Sigma }}$ fields (top left panel), Gaussianized fields ($y$ fields, bottom left), Gaussian random $y$ fields (bottom right panel), and the ${\delta }^{\mathrm{\Sigma }}$ field generated by the inverse-Gaussianization method (top-right panel). For brevity, readers may only need to compare the top-left panel and the top-right panel. It shows that the inverse-Gaussianization method basically correctly captures the correlation in cosmic variance between power spectra of different $\ell$. This figure is for the lens redshift $z=2.023$, highly relevant to the CMB lensing analysis. It is also relevant to deep lensing surveys such as LSST and HSC, although to a weaker extent. Only the bin label (number) is presented. For the corresponding $\ell$ value, please refer to Fig. 3.

Figure 5

Identical to Fig. 4, but for the lens redshift $z=1.028$. Again, the similarity between the top-left panel (the true ${\delta }^{\mathrm{\Sigma }}$ field) and top right panel (the ${\delta }^{\mathrm{\Sigma }}$ field generated by our method) demonstrates the reasonably good performance of our method. Nevertheless, we notice off-diagonal elements of the large value in the Gaussianized fields (bottom left panel), which are statistically significant (compared to the Gaussian random realizations, bottom right panel). These significant nonzero off-diagonal elements reflect residual non-Gaussianity in the Gaussianized $y$ maps and therefore the imperfection of Gaussianization. These propagate (nonlinearly) into the final ${\delta }^{\mathrm{\Sigma }}$ maps we generate and cause a statistically significant difference between the top panels.

Figure 6

Identical to Figs. 4 and 5, but for the lens redshift $z=0.485$. Since this is the lowest redshift we investigate, the original ${\delta }^{\mathrm{\Sigma }}$ map shows the strongest non-Gaussianity and therefore the most significant off-diagonal elements (top-left panel). Nevertheless, our inverse-Gaussianization method still works and well reproduces the correlations between different $\ell$ bins.

Figure 7

The reduced bispectrum $Q(\ell ,u,\alpha )$ of the mocks produced by the inverse-Gaussianization method is presented in the solid line for various configurations at $z=1.028$. For comparison, the reduced bispectrum of simulation slices is presented in the dotted line. The overall behavior of the reduced bispectrum is reproduced well, except for the even weaker $\alpha$ dependence than the simulated one.

Figure 8

The solid lines in the left panel present the PDF of the band power at eight arbitrarily selected scales for $z=1.028$. The PDF is measured from 16384 mocks, and the bin size for the $x$ axis is 0.2. The corresponding ${\chi }^2$-distribution is plotted in the dotted line, while the Gaussian distribution is the black solid line. Increasing non-Gaussianity is observed toward decreasing $\ell$, but it is quite consistent with the ${\chi }^2$-distribution. The right panel presents the cumulants of band power distribution up to sixth order, along with the cumulants of the corresponding ${\chi }^2$-distribution. Toward large $\ell$, the band power cumulants reduce toward zero. However, residual skewness is still observed.