3D cosmic shear: Numerical challenges, 3D lensing random fields generation, and Minkowski functionals for cosmological inference

Cosmic shear—the weak gravitational lensing effect generated by fluctuations of the gravitational tidal fields of the large-scale structure—is one of the most promising tools for current and future cosmological analyses. The spherical-Bessel decomposition of the cosmic shear field (3D cosmic shear) is one way to maximize the amount of redshift information in a lensing analysis and therefore provides a powerful tool to investigate in particular the growth of cosmic structure that is crucial for dark energy studies. However, the computation of simulated 3D cosmic shear covariance matrices presents numerical difficulties, due to the required integrations over highly oscillatory functions. We present and compare two numerical methods and relative implementations to perform these integrations. We then show how to generate 3D Gaussian random fields on the sky in spherical coordinates, starting from the 3D cosmic shear covariances. To validate our field-generation procedure, we calculate the Minkowski functionals associated with our random fields, compare them with the known expectation values for the Gaussian case and demonstrate parameter inference from Minkowski functionals from a cosmic shear survey. This is a first step towards producing fully 3D Minkowski functionals for a lognormal field in 3D to extract Gaussian and non-Gaussian information from the cosmic shear field, as well as towards the use of Minkowski functionals as a probe of cosmology beyond the commonly used two-point statistics.

Figure 1

Comparison of the differential signal-to-noise curve [Eq. (37)] as a function of the angular multipole. The two curves have been obtained from the signal and noise parts of the covariance matrices produced with GLaSS (red) and the Levin method (blue).

Figure 2

Relative difference of the signal-to-noise curve calculated with the GLaSS and Levin method, as a function of the multipole $\ell$.

Figure 3

Comparison of the diagonal elements of the signal part of the covariance matrices [Eq. (8)] for two multipoles $\ell =100$ and $\ell =500$, produced with GLaSS (solid lines, cyan and red for $\ell =100$ and $\ell =500$, respectively) and the Levin method (dashed lines, blue and black for $\ell =100$ and $\ell =500$, respectively). All curves have been plotted without performing any interpolation. We show the same curves using a linear (upper panel) and a logarithmic (bottom panel) scale on the $y$ axis. The differences at higher $k$ ($k\gtrsim 0.2h/\mathrm{Mpc}$ for $\ell =100$, $k\gtrsim 0.4h/\mathrm{Mpc}$ for $\ell =500$) arise from the higher numerical noise present in the GLaSS computations in that $k$ regime. However, these contributions are many orders of magnitude smaller than the main contributions around the peaks of the curves, and much smaller than the contributions from the noise (cf. Fig. 4), therefore can be safely neglected. We demonstrate this point in the upper panel by indicating the shaded region for each multipole $\ell$ where the signal represents a fraction $\le 1/1000$ of the noise: these regions correspond to the $k$-ranges where the GLaSS and Levin predictions for the signal are in apparent disagreement (cf. also Fig. 5). In both panels, the curves for $\ell =500$ have been multiplied by a factor 1000 for easier visualization.

Figure 4

Comparison of the diagonal elements of the noise part of the covariance matrices [Eq. (12)] for the same two multipoles $\ell =100$ and $\ell =500$ of Fig. 3, produced with GLaSS (solid, cyan for $\ell =100$ and red for $\ell =500$) and the Levin method (dashed, blue for $\ell =100$ and black for $\ell =500$). All curves have been plotted without performing any interpolation and multiplied by a factor $k^2$. In the case $\ell =500$, the curves produced by both methods have also been multiplied by a factor 20 for easier visualization.

Figure 5

Differences between the predictions for the signal (left panels) and noise (right panels) contributions to the covariance matrices for multipoles $\ell =100$ (top panels) and $\ell =500$ (bottom panels), normalized to their sum. We stress here again that the discrepancies at high $k$ should not be a concern because the $k$-regimes where they originate produce contributions very much subdominant with respect to the peaks of the signal curves, and also with respect to the relevant contributions from the noise (cf. Figs. 3 and 4). In the signal plots, we shade the regions where the signal is a fraction $\le 1/1000$ of the noise (cf. Fig. 3): these regions correspond to the $k$ values where the differences between the two codes are bigger; however, since the signal contributions from these regions are negligible, this discrepancy can be safely ignored.

Figure 6

Convergence field sampled at three different values of the radius $\chi$, with the observer situated in the center. A section of the outer and middle sphere has been removed to facilitate visualization. The lensing covariance matrix which we used for sampling the random field is given by Eq. (8). We consider only contributions from the signal part of the covariance matrix, and use $30\ell$ modes ranging between 10 and 1000. We use a linear matter power spectrum for the calculation of the covariance.

Figure 7

Numerical estimations of the first MF $V_0^G$ (dots), calculated on our generated Gaussian fields at different values of the radius (represented by different colors), compared with the theoretical predictions given by Eq. (54) (joined by lines), as a function of the threshold $\nu$. The range of the thresholds always varies between $-4\sqrt{\sigma }$ and $+4\sqrt{\sigma }$, where $\sigma$ is the (average) variance of the lensing field at a fixed radius.

Figure 8

Numerical estimations of the second MF $V_1^G$ (dots), calculated on our generated Gaussian fields at different values of the radius (represented by different colors), compared with the theoretical predictions. The color scheme is the same as in Fig. 7.

Figure 9

Numerical estimations of the third MF $V_2^G$ (dots), calculated on our generated Gaussian fields at different values of the radius (represented by different colors), compared with the theoretical predictions. The color scheme is the same as in Fig. 7.

Figure 10

Covariance matrix between different MFs at different radii. We consider correlations between all three MFs $V_0$, $V_1$, $V_2$, all functions of the threshold $\nu$ (ranging from ${\nu }_1$ to ${\nu }_{\max }$), as calculated at different radii (labeled by different $\chi$ values, ranging from ${\chi }_1$ to ${\chi }_{\max }$ and specifically equal to 1000, 2000, 3000, 4000 and $5000\mathrm{Mpc}/h$, as in Figs. 7, 8, 9). In the matrix, we indicate the block submatrices that represent the covariance between the three MFs. We used a logarithmic scale for both positive and negative values to highlight the many orders of magnitude spanned by the entries of the matrix and the different contibutions given by the three MFs.

Figure 11

Correlation matrix between different MFs at different radii; the matrix entries represent the Pearson correlation coefficient, obtained from the covariance matrix entries (the same plotted in Fig. 10) following Eq. (58). We consider correlations between all three MFs $V_0$, $V_1$, $V_2$, all functions of the threshold $\nu$ (ranging from ${\nu }_1$ to ${\nu }_{\max }$), as calculated at different radii (labeled by different $\chi$ values, ranging from ${\chi }_1$ to ${\chi }_{\max }$ and specifically equal to 1000, 2000, 3000, 4000 and $5000\mathrm{Mpc}/h$, as in Figs. 7, 8, 9). In the matrix, we indicate the block submatrices that represent the correlation between the three MFs.

Figure 12

${\chi }^2$ obtained considering the covariance of different combinations of MFs, i.e., considering the three MFs singularly ($⟨V_0,V_0⟩$ (blue), $⟨V_1,V_1⟩$ (green) and $⟨V_2,V_2⟩$ (red) ). Our fiducial model is represented by the choice ${\mathrm{\Omega }}_{\mathrm{m}}=0.3$.