KiDS-450: The tomographic weak lensing power spectrum and constraints on cosmological parameters

TL;DR

KiDS-450 analyzes the tomographic weak-lensing power spectrum using a quadratic estimator to extract band powers across two and three redshift bins. The cosmology is inferred in a Bayesian framework within flat $\Lambda$CDM, marginalizing over intrinsic alignments, baryon feedback, massive neutrinos, excess-noise power, and calibration/redshift uncertainties; theoretical predictions use CLASS with HALOFIT. The results yield $S_8 = 0.651 \pm 0.058$ (3 z-bins), indicating a tension with Planck constraints, and provide competitive upper limits on the total neutrino mass from lensing alone. The analysis includes an analytical covariance with SSC and a thorough treatment of systematics, offering an independent cross-check of prior KiDS-450 cosmic-shear analyses and guiding future, computation-heavy tomographic estimations for upcoming surveys.

Abstract

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 $Λ$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 $S_8 \equiv σ_8 \sqrt{Ω_{\rm m}/0.3}$ yielding $S_8 = \ 0.651 \pm 0.058$ (3 z-bins), confirming the recently reported tension in this parameter with constraints from Planck at $3.2σ$ (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/

Figures (18)

• Figure 1: Extracted B-mode band powers as a function of multipole and redshift correlation (from left to right) from 50 Gaussian random field realizations for three different noise levels each. Crosses (red) correspond to $\sigma_\epsilon = 0.10$, triangles (blue) to $\sigma_\epsilon = 0.19$, and circles (black) to $\sigma_\epsilon = 0.28$ for fixed number densities of $n_{\rm eff}(z_1) = 2.80 \, {\rm arcmin}^{-2}$ and $n_{\rm eff}(z_2) = 2.00 \, {\rm arcmin}^{-2}$. Crosses and circles are plotted with constant multiplicative offset in multipoles for illustrative purposes. The vertical dashed lines (grey) indicate the borders of the band power intervals (Table \ref{['tab:bp_intervals']}). The errors are derived from the run-to-run scatter and divided by $\sqrt{50}$ to represent the error on the mean.
• Figure 2: The same as in Fig. \ref{['fig:signals_EE_2zbins_GRFs']} but for E-modes. The grey solid line in each panel shows the input power spectrum used for the creation of the Gaussian random fields (GRFs). A quantitative comparison between input power and extracted power for the highest noise sample is presented in Fig. \ref{['fig:comparison_E_modes_GRFs']}. Note that the first and last band powers are not expected to recover the input power (Section \ref{['sec:band_power_select']}).
• Figure 3: The normalised redshift distributions for the full sample, two and three tomographic bins employed in this study and estimated from the weighted direct calibration scheme ('DIR') presented in Hildebrandt2016. The dashed vertical lines mark the median redshift per bin (Table \ref{['tab:n_eff']}) and the (grey) shaded regions indicate the target redshift selection by cutting on the Bayesian point estimate for photometric redshifts $z_{\rm B}$. The (coloured) regions around each fiducial $n(z)$ per bin shows the $1\sigma$-interval estimated from $1000$ bootstrap realisations of the redshift catalogue. Lower panel: the summed and re-normalised redshift distribution over all tomographic bins.
• Figure 4: Measured E-mode band powers in three tomographic bins averaged with the effective area per patch over all 13 KiDS-450 subpatches. On the diagonal we show from the top-left to the bottom-right panel the auto-correlation signal of the low-redshift bin (blue), the intermediate-redshift bin (orange), and the high-redshift bin (red). The unique cross-correlations between these redshift bins are shown in the off-diagonal panels (grey). Note that negative band powers are shown at their absolute value with an open symbol. The redshift bins targeted objects in the range $0.10 < z_1 \leq 0.30$ for the lowest bin, $0.30 < z_2 \leq 0.60$ for the intermediate bin, and $0.60 < z_3 \leq 0.90$ for the highest bin. The $1\sigma$-errors in the signal are derived from the analytical covariance convolved with the averaged band window matrix (Section \ref{['sec:cov']}), whereas the extent in $\ell$-direction is the width of the band. Band powers in the shaded regions (grey) to the left and right of each panel are excluded from the cosmological analysis (Section \ref{['sec:band_power_select']}). The solid line (black) shows the power spectrum for the best-fitting cosmological model (Section \ref{['sec:cosmo_inference']}). Moreover, we show the intrinsic alignment contributions, i.e. $C^{\rm GG}$ as dotted black line, $|C^{\rm GI}|$ as dash-dotted blue line, and $C^{\rm II}$ as dashed purple line. In addition to that, we also show $C^{\rm GG}$ without baryon feedback as a dashed black line. Note that for an accurate comparison of theory to data such as presented in Section \ref{['sec:cosmo_inference']}), the theoretical power spectrum must be transformed to band powers (equation \ref{['eq:conv_window_func']}). The dashed grey lines in the redshift auto-correlation models indicate the noise power spectrum in the data (Table \ref{['tab:n_eff']}), which does not contribute to the redshift cross-correlations. Note, however, that the band powers are centred at the naïve $\ell$-bin centre and thus the convolution with the band window function is not taken into account in this figure, in contrast to the cosmological analysis.
• Figure 5: Same as Fig. \ref{['fig:signals_EE_3zbins']} but for B-mode band powers corrected for the 'resetting bias' introduced by the algorithm (Section \ref{['sec:fiducial_B_modes']}). Note, however, the different scale (linear) and normalization used here with respect to Fig. \ref{['fig:signals_EE_3zbins']}; for reference we also plot the best-fitting E-mode power spectrum as solid line (black). We show the measured B-modes as (black) dots with $1\sigma$-errors derived from the averaged shape-noise contribution to the analytical covariance convolved with the B-mode part of the averaged band window matrix. Note that the last band at high multipoles in each panel is designed to sum up the oscillating part of the pixel-window function and hence intrinsically biased.