Increasing the lensing figure of merit through higher order convergence moments

The unprecedented quality, the increased data set, and the wide area of ongoing and near future weak lensing surveys allows one to move beyond the standard two points statistics, thus making it worthwhile to investigate higher order probes. As an interesting step toward this direction, we explore the use of higher order moments (HOM) of the convergence field as a way to increase the lensing figure of merit (FoM). To this end, we rely on simulated convergence to first show that HOM can be measured and calibrated so that it is indeed possible to predict them for a given cosmological model provided suitable nuisance parameters are introduced and then marginalized over. We then forecast the accuracy on cosmological parameters from the use of HOM alone and in combination with standard shear power spectra tomography. It turns out that HOM allow one to break some common degeneracies, thus significantly boosting the overall FoM. We also qualitatively discuss possible systematics and how they can be dealt with.

Figure 1

Convergence of the HOM for different values of the survey area $\mathrm{\Omega }$. Top: Percentage deviation $\mathrm{\Delta }⟨{\kappa }^n⟩(\mathrm{\Omega })/⟨{\kappa }^n⟩({\mathrm{\Omega }}_{\mathrm{tot}})$ as a function of the smoothing angle $\theta$ for $\mathrm{\Omega }=\{100,500,1500,2500\}{\mathrm{deg}}^2$ (green, blue, orange, red lines) and ${\mathrm{\Omega }}_{\mathrm{tot}}=3500{\mathrm{deg}}^2$. Bottom: Same as before but with the difference normalized with respect to the statistical error. Solid (dashed) lines refer to the results for the Gaussian (top hat) smoothing filter.

Figure 2

Scaling of the HOM $⟨{\kappa }^2⟩$ (left), $⟨{\kappa }^3⟩$ (center), $⟨{\kappa }^4⟩$ (right) signal-to-noise (S/N) ratio ${\nu }_{\mathrm{SN}}$ with smoothing angle $\theta$ and survey area $\mathrm{\Omega }$. Top: S/N ratio as a function of $\theta$ for $\mathrm{\Omega }=\{100,500,1500,2500\}{\mathrm{deg}}^2$ (green, blue, orange, red lines). Center: S/N ratio as a function of $\mathrm{\Omega }$ for $\theta =\{4,7,10,13\}\mathrm{arcmin}$ (green, blue, orange, red lines). Bottom: Percentage deviation of the rescaled S/N ratio (see text) from the value estimated from the full survey area. In all panels, solid (dashed) lines refer to the result for the Gaussian (top hat) smoothing filter.

Figure 3

Testing the convergence of the correlation matrix $\mathbf{R}$. Top: Average value and 68% confidence range of the independent element of $\mathbf{R}$ for the Gaussian (left) and top hat (right) filters. Bottom: Average value and 68% confidence range of the percentage deviation of the $\mathbf{R}$ elements from the value they get for $\mathrm{\Omega }=3500{\mathrm{deg}}^2$. Dashed (dotted) lines mark the $\pm 10\%(\pm 5)\%$ limits.

Figure 4

HOM as estimated on the noiseless convergence map for Gaussian (top) and top hat (bottom) filters. Best fit curves are superimposed to the data with red, green, blue, orange, purple lines referring to the best fit parameters to the smoothing angle range (2,20), (2,12), (4,14), (6,16), $(8,18)\mathrm{arcmin}$, respectively.

Figure 5

Same as Fig. 4 but now we show the error normalized fit residuals (in absolute value).

Figure 6

Same as Fig. 4 but for moments from the map reconstructed from noisy shear data.

Figure 7

Fisher matrix forecast of the $1\sigma$ uncertainty on ${\mathrm{\Omega }}_M$ (top) and ${\sigma }_8$ (bottom) as a function of the prior parameter ${\sigma }_p$ from the fit to data covering the smoothing angle range $({\theta }_{\min },{\theta }_{\max })=(2,20)\mathrm{arcmin}$ sampled with step $d\theta =1,2,3,4\mathrm{arcmin}$ (from left to right). A Gaussian filter has been used for smoothing, while black, green, cyan, blue, orange, magenta, red lines refer to the results using all moments, second order only, third order only, fourth order only, second and third, second and fourth, third and fourth, respectively.

Figure 8

Same as Fig. 7 but all moments are used, while black, green, blue, orange, red lines refer to using data over the ranges (2, 20), (2, 12), (4, 14), (6, 16), (8, 18) arcmin, respectively, sampled with step $d\theta =1,2,3,4\mathrm{arcmin}$ from left to right.

Figure 9

FoM ratio as a function of the prior parameter ${\sigma }_p$. Black, green, cyan, blue, orange, magenta, red lines refer to using all HOM, second order only, third only, fourth only, second and third, second and fourth, third and fourth in combination with cosmic shear tomography based on three (top) and six (bottom) redshift bins. A Gaussian filter is used to smooth the convergence map with smoothing angles spanning the range (2, 20) arcmin with sampling step $d\theta =1,2,3,4\mathrm{arcmin}$ from left to right.

Figure 10

Same as Fig. 9 but changing the smoothing angle range and using all HOM in combination with shear tomography (with three and six redshift bin results plotted in top and bottom panels). Black, green, blue, orange, red lines refer to the cases with $({\theta }_{\min },{\theta }_{\max })=(2,20),(2,12),(4,14),(6,16);(8,18)\mathrm{arcmin}$, respectively, with sampling $d\theta =1,2,3,4\mathrm{arcmin}$ from left to right.

Figure 11

Ratio between the marginalized errors on the cosmological parameters for the case with the full HOM data set is used and shear tomography is computed splitting galaxies into six redshift bins. Same color code as Fig. 10 is used to denote the smoothing angle range used, while the sampling step changes from left to right as in that same figure. Panels are for $({\mathrm{\Omega }}_M,w_0,w_a,w_p,{\mathrm{\Omega }}_b,h,n_s,{\sigma }_8)$ from top left to bottom right, with $w_p$ the dark energy EoS at the pivot redshift.

Figure 12

The 68%, 95%, 99% confidence ranges in the $({\mathrm{\Omega }}_M,{\sigma }_8)$ plane as forecasted from the shear only (blue lines) and $\text{shear}+\mathrm{HOM}$ (red lines) Fisher matrix analysis. Top (bottom) panels refer to the case when three (six) redshift bins are used for tomography, while we use all HOM for the Gaussian case, the full smoothing angle range, a step $d\theta =1\mathrm{arcmin}$ and ${\sigma }_p=1$, 0.1, 0.01 from left to right.

Figure 13

Comparing convergence HOM measured on maps reconstructed from unmasked (blue) and masked (red) shear data. First three panels show the second, third, and fourth order moments, while the fourth one plots the difference between the two data sets normalized with respect to the statistical error (with blue, orange, red points referring to second, third, fourth order moments).