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Register a new userFitting formulae of the reduced-shear power spectrum for weak lensing
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Martin Kilbinger
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Context. Weak gravitational lensing is a powerful probe of large-scale structure and cosmology. Most commonly, second-order correlations of observed galaxy ellipticities are expressed as a projection of the matter power spectrum, corresponding to the lowest-order approximation between the projected and 3d power spectrum. Aims. The dominant lensing-only contribution beyond the zero-order approximation is the reduced shear, which takes into account not only lensing-induced distortions but also isotropic magnification of galaxy images. This involves an integral over the matter bispectrum. We provide a fast and general way to calculate this correction term. Methods. Using a model for the matter bispectrum, we fit elementary functions to the reduced-shear contribution and its derivatives with respect to cosmological parameters. The dependence on cosmology is encompassed in a Taylor-expansion around a fiducial model. Results. Within a region in parameter space comprising the WMAP7 68% error ellipsoid, the total reduced-shear power spectrum (shear plus fitted reduced-shear correction) is accurate to 1% (2%) for l<10^4 (l<2x10^5). This corresponds to a factor of four reduction of the bias compared to the case where no correction is used. This precision is necessary to match the accuracy of current non-linear power spectrum predictions from numerical simulations.
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The significant increase in precision that will be achieved by Stage IV cosmic shear surveys means that several currently used theoretical approximations may cease to be valid. An additional layer of complexity arises from the fact that many of these approximations are interdependent; the procedure to correct for one involves making another. Two such approximations that must be relaxed for upcoming experiments are the reduced shear approximation and the effect of neglecting magnification bias. Accomplishing this involves the calculation of the convergence bispectrum; typically subject to the Limber approximation. In this work, we compute the post-Limber convergence bispectrum, and the post-Limber reduced shear and magnification bias corrections to the angular power spectrum for a Euclid-like survey. We find that the Limber approximation significantly overestimates the bispectrum when any side of the bispectrum triangle, $ell_i<60$. However, the resulting changes in the reduced shear and magnification bias corrections are well below the sample variance for $ellleq5000$. We also compute a worst-case scenario for the additional biases on $w_0w_a$CDM cosmological parameters that result from the difference between the post-Limber and Limber approximated forms of the corrections. These further demonstrate that the reduced shear and magnification bias corrections can safely be treated under the Limber approximation for upcoming surveys.
We investigate the impact of a common approximation on weak lensing power spectra: the use of single-epoch matter power spectra in integrals over redshift. We disentangle this from the closely connected Limbers approximation. We derive the unequal-time matter power spectrum at one-loop in standard perturbation theory and effective field theory to deal with non-linear physics. We compare these formalisms and conclude that the unequal-time power spectrum using effective field theory breaks for larger scales. As an alternative, we introduce the midpoint approximation. We also provide, for the first time, a fitting function for the time evolution of the effective field theory counterterms based on the Quijote simulations. Then we compute the angular power spectrum using a range of approaches: the Limbers approximation, and the geometric and midpoint approximations. We compare our results with the exact calculation at all angular scales using the unequal-time power spectrum. We use DES Y1 and LSST-like redshift distributions for our analysis. We find that the use of the Limbers approximation in weak lensing diverges from the exact calculation of the angular power spectrum on large-angle separations, $ell < 10$. Even though this deviation is of order $2%$ maximum for cosmic lensing, we find the biggest effect for galaxy clustering and galaxy-galaxy lensing. We show that not only is this true for upcoming galaxy surveys, but also for current data such as DES Y1. Finally, we make our pipeline and analysis publicly available as a Python package called unequalpy.
We introduce the skew-spectrum statistic for weak lensing convergence $kappa$ maps and test it against state-of-the-art high-resolution all-sky numerical simulations. We perform the analysis as a function of source redshift and smoothing angular scale for individual tomographic bins. We also analyse the cross-correlation between different tomographic bins. We compare the numerical results to fitting-functions used to model the bispectrum of the underlying density field as a function of redshift and scale. We derive a closed form expression for the skew-spectrum for gravity-induced secondary non-Gaussianity. We also compute the skew-spectrum for the projected $kappa$ inferred from Cosmic Microwave Background (CMB) studies. As opposed to the low redshift case we find the post-Born corrections to be important in the modelling of the skew-spectrum for such studies. We show how the presence of a mask and noise can be incorporated in the estimation of a skew-spectrum.
We present measurements of the weak gravitational lensing shear power spectrum based on $450$ sq. deg. of imaging data from the Kilo Degree Survey. We employ a quadratic estimator in two and three redshift bins and extract band powers of redshift auto-correlation and cross-correlation spectra in the multipole range $76 leq ell leq 1310$. The cosmological interpretation of the measured shear power spectra is performed in a Bayesian framework assuming a $Lambda$CDM model with spatially flat geometry, while accounting for small residual uncertainties in the shear calibration and redshift distributions as well as marginalising over intrinsic alignments, baryon feedback and an excess-noise power model. Moreover, massive neutrinos are included in the modelling. The cosmological main result is expressed in terms of the parameter combination $S_8 equiv sigma_8 sqrt{Omega_{rm m}/0.3}$ yielding $S_8 = 0.651 pm 0.058$ (3 z-bins), confirming the recently reported tension in this parameter with constraints from Planck at $3.2sigma$ (3 z-bins). We cross-check the results of the 3 z-bin analysis with the weaker constraints from the 2 z-bin analysis and find them to be consistent. The high-level data products of this analysis, such as the band power measurements, covariance matrices, redshift distributions, and likelihood evaluation chains are available at http://kids.strw.leidenuniv.nl/
Lensing peaks have been proposed as a useful statistic, containing cosmological information from non-Gaussianities that is inaccessible from traditional two-point statistics such as the power spectrum or two-point correlation functions. Here we examine constraints on cosmological parameters from weak lensing peak counts, using the publicly available data from the 154 deg$^2$ CFHTLenS survey. We utilize a new suite of ray-tracing N-body simulations on a grid of 91 cosmological models, covering broad ranges of the three parameters $Omega_m$, $sigma_8$, and $w$, and replicating the Galaxy sky positions, redshifts, and shape noise in the CFHTLenS observations. We then build an emulator that interpolates the power spectrum and the peak counts to an accuracy of $leq 5%$, and compute the likelihood in the three-dimensional parameter space ($Omega_m$, $sigma_8$, $w$) from both observables. We find that constraints from peak counts are comparable to those from the power spectrum, and somewhat tighter when different smoothing scales are combined. Neither observable can constrain $w$ without external data. When the power spectrum and peak counts are combined, the area of the error banana in the ($Omega_m$, $sigma_8$) plane reduces by a factor of $approx2$, compared to using the power spectrum alone. For a flat $Lambda$ cold dark matter model, combining both statistics, we obtain the constraint $sigma_8(Omega_m/0.27)^{0.63}=0.85substack{+0.03 -0.03}$.
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