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Monthly Notices of the Royal Astronomical Society
, Volume 476 (1) – May 1, 2018

7 pages

/lp/ou_press/statistical-inconsistencies-in-the-kids-450-data-set-yQ0M063k8i

- Publisher
- Oxford University Press
- Copyright
- © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society
- ISSN
- 0035-8711
- eISSN
- 1365-2966
- D.O.I.
- 10.1093/mnras/sty099
- Publisher site
- See Article on Publisher Site

Abstract The Kilo-Degree Survey (KiDS) has been used in several recent papers to infer constraints on the amplitude of the matter power spectrum and matter density at low redshift. Some of these analyses have claimed tension with the Planck Λ cold dark matter cosmology at the ∼2σ–3σ level, perhaps indicative of new physics. However, Planck is consistent with other low-redshift probes of the matter power spectrum such as redshift-space distortions and the combined galaxy-mass and galaxy–galaxy power spectra. Here, we perform consistency tests of the KiDS data, finding internal tensions for various cuts of the data at ∼2.2σ–3.5σ significance. Until these internal tensions are understood, we argue that it is premature to claim evidence for new physics from KiDS. We review the consistency between KiDS and other weak lensing measurements of S8, highlighting the importance of intrinsic alignments for precision cosmology. cosmic background radiation, cosmological parameters, large-scale structure of Universe, cosmology: observations 1 INTRODUCTION Precision observations of the cosmic microwave background radiation (CMB) by the Planck satellite (Planck Collaboration XVI 2014a; Planck Collaboration XIII 2016a, hereafter P16) and other experiments (Hinshaw et al. 2013; Sievers et al. 2013; Story et al. 2013) have shown that the Λ cold dark matter (ΛCDM) cosmology, with nearly scale invariant, adiabatic, Gaussian initial perturbations, provides an excellent description of our Universe. Measurements of weak lensing of the CMB (Planck Collaboration XV 2016b) show further that the ΛCDM model remains a good description of the Universe down to a redshift of z ∼ 2, where the CMB lensing kernel peaks. It is, nevertheless, important to test the model at lower redshifts, particularly at redshifts $$z {\, \buildrel< \over \sim \,}1$$ when the Universe becomes dominated by dark energy. Deviations from the ΛCDM model at low redshift could potentially reveal evidence for dynamical dark energy or modifications to General Relativity (see Amendola et al. 2016, for a review). Weak galaxy lensing is an important probe of the matter power spectrum at low redshifts (Blandford et al. 1991; Miralda-Escude 1991; Kaiser 1992). Several ambitious deep imaging projects have reported results recently. These include the Canada–France–Hawaii Telescope Lensing Survey (CFHTLenS; Heymans et al. 2012, 2013; Joudaki et al. 2017a), Deep Lens Survey (DLS; Jee et al. 2016), Dark Energy Survey (DES; Abbott et al. 2016; DES Collaboration et al. 2017; Troxel et al. 2017), and Kilo-Degree Survey (KiDs; Hildebrandt et al. 2017; Köhlinger et al. 2017). Weak lensing analysis of these surveys can be used to constrain the parameter combination1S8 = σ8(Ωm/0.3)0.5, which can be compared to the Planck value from P16,2S8 = 0.825 ± 0.016 derived from the Planck temperature power spectrum, low multipole polarization, and Planck lensing (TT+lowTEB+lensing, in the notation of P16). However, the weak galaxy lensing results span a range of values. The reanalysis of CFHTlenS by Joudaki et al. (2017a) finds $$S_8 = 0.732^{+0.029}_{-0.031}$$; Jee et al. (2016) find $$S_8 = 0.818^{+0.034}_{-0.026}$$ from DLS; Abbott et al. (2016) find S8 = 0.81 ± 0.06 from the DES Science Verification data; (Hildebrandt et al. 2017, hereafter H17) find S8 = 0.745 ± 0.039 from a tomographic correlation function analysis of KiDs while (Köhlinger et al. 2017, hereafter K17) find S8 = 0.651 ± 0.058 from a tomographic power spectrum analysis of KiDs. The DES Year 1 weak lensing analysis3 (Troxel et al. 2017) gives $$S_8 =0.789^{+0.024}_{-0.026}$$. Some of these values are in tension with Planck. For example, H17 find a 2.3σ discrepancy between KiDs and Planck, while K17 find a 3.2σ discrepancy. However, the results from these different surveys do not agree particularly well with each other (even when using the same shear catalogue), showing differences in the value of S8 at the ∼2σ–2.5σ level. A statistically significant tension between the Planck ΛCDM cosmology and weak galaxy lensing could have important consequences for fundamental physics (e.g. Joudaki et al. 2017b). But how seriously should we take the weak lensing results? A minimal requirement is that a cosmic shear data set should be internally self-consistent. The main purpose of this paper is to show that this does not seem to be the case with KiDS. Before we begin, we make a few remarks concerning cosmic shear analysis. Most analyses involve estimation of correlation functions ξ+ and ξ− as a function of relative angular separation θ, or of the cosmic shear E-mode power spectrum Pκ(ℓ) as a function of multipole ℓ. These are related by \begin{eqnarray} \xi _{\pm } = {1 \over 2 \pi } \int {\rm d}\ell \ell P_\kappa (\ell )J_{0,4} (\ell \theta ). \end{eqnarray} (1)For a cross-power spectrum between redshift bins i and j, the shear power spectrum is related to the non-linear matter power spectrum Pδ by \begin{eqnarray} P^{ij}_\kappa (\ell ) = \int _0^{\chi _H} {\rm d}\chi {q_i(\chi ) q_j(\chi ) \over [f_K(\chi )]^2} P_\delta \left({(\ell +1/2) \over f_K(\chi )}, \chi \right), \end{eqnarray} (2)where (following the notation of H17) χ is the comoving radial distance, fK(χ) is the comoving angular diameter distance to distance χ, and qi(χ) is the lensing efficiency for tomographic redshift bin i: \begin{eqnarray} q_i(\chi ) = {3 H_0^2 \Omega _{\rm m} \over 2 c^2} {f_K(\chi ) \over a(\chi )} \int _\chi ^{\chi _{\rm H}} {\rm d}\chi ^\prime n_i(\chi ^\prime ) {f_K(\chi ^\prime - \chi ) \over f_K(\chi ^\prime )}, \end{eqnarray} (3)where χH is the comoving Hubble distance and ni(χ) is the effective (weighted) number density galaxies in redshift bin i normalized so that ∫ni(χ)dχ = 1. Even if the image analysis is assumed to be free of systematic errors and biases, inferences on cosmology require an accurate model of the redshift distribution ni(χ), which in turn requires accurate calibration of the photometric redshifts used to define the redshift bin i. A key test of the accuracy of the photometric redshift calibrations would be to demonstrate consistency between distinct cross-correlations i, j. However, this is not straightforward because of intrinsic ellipticity alignments between neighbouring galaxies (II term) and between gravitation shear and intrinsic shear (GI term). The power spectra4 of these terms are usually modelled as (Hirata & Seljak 2004; Bridle & King 2007) \begin{eqnarray} & & P^{ij}_{II}(\ell ) =\int _0^{\chi _{\rm H}} {\rm d}\chi F^2(z) {n_i(\chi ) n_j(\chi ) \over [f_K(\chi )]^2} P_\delta \left({(\ell +1/2) \over f_K(\chi )}, \chi \right), \end{eqnarray} (4) \begin{eqnarray} P^{ij}_{GI}(\ell ) &=&\int _0^{\chi _H} {\rm d}\chi F(z) {(q_i(\chi ) n_j(\chi ) + n_i(\chi )q_j(\chi )) \over [f_K(\chi )]^2} \nonumber \\ &&\times P_\delta \left({(\ell +1/2) \over f_K(\chi )}, \chi \right). \end{eqnarray} (5)In these equations, \begin{eqnarray} F(z) = -A_{IA} C\rho _c {\Omega _{\rm m} \over D(z)}, \end{eqnarray} (6)where D(z) is the linear growth rate of perturbations normalized to unity at the present day, and C is a normalizing constant, usually chosen to be $$C = 5 \times 10^{-14}\,h^{-2}\,{\rm M}_{\odot }^{-1}\,{\rm Mpc}^3$$. With this choice, the intrinsic alignment amplitude is expected to be of the order of unity (and positive if intrinsic ellipticities are aligned with the stretching axis of the tidal field). This model of intrinsic alignments is heuristic and simplified (see Blazek et al. 2017 for a more complex alignment model). Even in the context of this model, the intrinsic alignment amplitude may vary with redshift, luminosity, and galaxy type. For current weak lensing surveys, intrinsic alignments are not benign. The contributions of equations (4) and (5) are comparable to any claimed tensions between the Planck value of S8 and those inferred from cosmic shear surveys (with positive AIA tending to raise the value of S8 and negative values lowering S8). How can we test the intrinsic alignment model? The conventional solution is to introduce additional nuisance parameters to characterize uncertainties in the intrinsic alignment model (e.g. Kirk et al. 2012), relying on the redshift dependence of the measured signals to disentangle true cosmic shear from intrinsic alignments. This, of course, requires accurate knowledge of the redshift distributions and their errors. Current cosmic shear data are still relatively sparse, with a small number of measurements in coarse redshift bins. The number of internal consistency checks of the data and the various components of the model (including nuisance parameters) are therefore limited.5 In Section 2, we perform consistency tests of the KiDS data from H17. In Section 3, we compare the KiDS results with Planck and measurements of redshift-space distortions (RSDs) and rich cluster abundances, which provide independent measures of the amplitude of the matter fluctuations at similar redshifts to those of the KiDS galaxies. Section 4 compares the results from various weak lensing analyses. Our main conclusions are presented in Section 5. 2 TESTS OF THE KIDS DATA We use the KiDS cross-correlation measurements of ξ+ and ξ− in four tomographic redshift bins as reported by H17 together with the associated cosmomc likelihood module and covariance matrix.6 For reference, the four redshift bins span the following ranges in photometric redshift zB: 0.1 < zB ≤ 0.3 (bin 1), 0.3 < zB ≤ 0.5 (bin 2), 0.5 < zB ≤ 0.7 (bin 3), and 0.7 < zB ≤ 0.9 (bin 4). We used the same angular ranges, photometric redshift calibrations and errors, nuisance parameters, and priors as in ‘fiducial’ analysis in H17 (first entry in their table 4) and verified that we recovered the identical best-fitting χ2 (162.8) and constraint on S8 (S8 = 0.745 ± 0.039). We then removed all cross-correlations involving one of the photometric redshift bins. The results are summarized in Table 1 and in Fig. 1. Figure 1. View largeDownload slide The upper panel shows the posteriors for the intrinsic alignment parameter AIA (equation 6) as we remove all cross-correlations involving a particular redshift bin. The lower panel shows the 68 and 95 per cent constraints on S8 for the data minus redshift bin 3 (orange) and minus redshift bin 4 (grey). The blue contours show the Planck constraints from the TT+lowTEB+lensing data combination as given in P16. Figure 1. View largeDownload slide The upper panel shows the posteriors for the intrinsic alignment parameter AIA (equation 6) as we remove all cross-correlations involving a particular redshift bin. The lower panel shows the 68 and 95 per cent constraints on S8 for the data minus redshift bin 3 (orange) and minus redshift bin 4 (grey). The blue contours show the Planck constraints from the TT+lowTEB+lensing data combination as given in P16. Table 1. Conditional χ2 tests removing photometric redshift bins. $${\boldsymbol y}^D$$ S8 AIA $$\chi ^2_{\rm cond}$$ $$N_{\sigma _{\rm cond}}$$ minus z-bin 1 0.745 ± 0.040 1.14 ± 0.85 61.0 (52) 0.89 minus z-bin 2 0.754 ± 0.042 1.24 ± 0.80 66.3 (52) 1.40 minus z-bin 3 0.771 ± 0.039 1.25 ± 0.57 78.2 (52) 2.60 minus z-bin 4 0.684 ± 0.071 −0.1 ± 1.7 87.9 (52) 3.52 minus ξ− 0.778 ± 0.040 1.10 ± 0.73 89.7 (60) 2.71 minus ξ+ 0.705 ± 0.048 0.92 ± 0.97 84.1 (70) 1.20 $${\boldsymbol y}^D$$ S8 AIA $$\chi ^2_{\rm cond}$$ $$N_{\sigma _{\rm cond}}$$ minus z-bin 1 0.745 ± 0.040 1.14 ± 0.85 61.0 (52) 0.89 minus z-bin 2 0.754 ± 0.042 1.24 ± 0.80 66.3 (52) 1.40 minus z-bin 3 0.771 ± 0.039 1.25 ± 0.57 78.2 (52) 2.60 minus z-bin 4 0.684 ± 0.071 −0.1 ± 1.7 87.9 (52) 3.52 minus ξ− 0.778 ± 0.040 1.10 ± 0.73 89.7 (60) 2.71 minus ξ+ 0.705 ± 0.048 0.92 ± 0.97 84.1 (70) 1.20 Notes. The first column defines the portion of the data vector ($${\boldsymbol y}^D$$) used to fit the model. The second and third columns give the marginalized mean values of S8, AIAins, and their 1σ errors. The fourth column gives the conditional $$\chi ^2_{\rm cond}$$, as defined in equation (15), for the rest of data vector, $${\boldsymbol x}^D$$. The numbers in parentheses list the length, Nx, of the vector $${\boldsymbol x}^D$$. The fifth column gives the number of standard deviations by which $$\chi ^2_{\rm cond}$$ differs from Nx, $$N_{\sigma _{\rm cond}} =(\chi ^2_{\rm cond} - N_{\rm x})/\sqrt{2N_{\rm x}}$$. View Large The first point to note is that the intrinsic alignment amplitude is reasonably stable to the removal of photometric redshift bins. All of the posteriors shown in Fig. 1 are consistent with the intrinsic alignment solution from the full data set ($$A_{{\rm IA}} = 1.10^{+0.68}_{-0.54}$$). However, it is also clear that redshift bin 4 carries a high weight in fixing AIA. With redshift bin 4 removed, the posterior distribution develops a long tail to negative values that is cut-off by the lower end of the AIA prior (uniform between −6 < AIA < 6). As a consequence of this long tail, the best-fitting value of S8 with bin 4 removed is driven to lower values and its error increases substantially compared to the full sample (lower panel of Fig. 1 and Table 1). Redshift bin 4 is therefore critical in pinning down the intrinsic alignment solution and reducing the error on S8. If redshift bin 3 is removed, S8 rises and the constraints in the S8–Ωm plane become compatible with Planck (Fig. 1). This is not unexpected, because one can see from fig. 5 of H17 that the best-fitting fiducial model tends to sit high for all cross-spectra involving tomographic redshift bin 3 (particularly for ξ−). With redshift bin 3 removed, there is substantial overlap in the posteriors in the S8–Ωm plane with those from the full sample and with the other subsets of the data summarized in Table 1. However, these various estimates of S8 are highly correlated since they share common data. Are the parameter shifts seen in these subsets statistically reasonable? We turn to this question next. We can perform a more elaborate statistical consistency test by dividing the data vector into two components: \begin{eqnarray} {\boldsymbol z}^D = ({\boldsymbol x}^D, {\boldsymbol y}^D). \end{eqnarray} (7)We can then fit $${\boldsymbol y}^D$$ to a model (including nuisance parameters), $$\hat{\boldsymbol y}$$. The model parameters also make a theory prediction for the data partition $${\boldsymbol x}^D$$, which we denote $${\hat{\boldsymbol x}}$$. We can then write the theory vector for $${\boldsymbol z}^D$$ as \begin{eqnarray} {\hat{\boldsymbol z}} = (\lambda {\hat{\boldsymbol x}}, {\hat{\boldsymbol y}}), \end{eqnarray} (8)introducing a new parameter λ. Evidently, if the data partitions and model are consistent, the new parameter λ should be consistent with unity. The tests described in this section are all based on the ΛCDM model, but with a free amplitude. Since cosmic shear measurements have very limited ability to fix shape parameters, and the data cuts that we apply cover similar redshift ranges, it seems reasonable to interpret differences in λ as indicative of systematic errors in the data. To recap, we run Markov chain Monte Carlo (MCMC) chains to determine the model parameters from a data partition $${\boldsymbol y}^D$$ and determine a single amplitude parameter λ by fitting to the rest of the data $${\boldsymbol x}_D$$. The posterior distributions of λ for the data cuts of Table 1 are shown in Fig. 2. Figure 2. View largeDownload slide Posterior distributions of the parameter λ defined in equation (8). The upper figure shows the distributions if the model parameters are fitted to ξ+ (denoted λ−) and to ξ− (denoted λ+). The lower figure shows the posterior distributions of λ for partitions of the data in which all cross-correlations involving a particular tomographic redshift bin are removed from the fit to the theoretical model (e.g. λ3 corresponds to a theoretical model fitted to all cross-correlations that do not involve tomographic redshift bin 3). Figure 2. View largeDownload slide Posterior distributions of the parameter λ defined in equation (8). The upper figure shows the distributions if the model parameters are fitted to ξ+ (denoted λ−) and to ξ− (denoted λ+). The lower figure shows the posterior distributions of λ for partitions of the data in which all cross-correlations involving a particular tomographic redshift bin are removed from the fit to the theoretical model (e.g. λ3 corresponds to a theoretical model fitted to all cross-correlations that do not involve tomographic redshift bin 3). The upper plot in Fig. 2 compares the amplitudes λ− (fitting the model parameters to ξ+) and λ+ (fitting the model parameters to ξ−). This agrees with the visual impression given by fig. 5 of H17, namely that ξ− wants a low amplitude while ξ+ prefers a high amplitude. Integrating these distributions \begin{eqnarray} \int _0^{1} P(\lambda _-) {\rm d}\lambda _- = 2.9\times 10^{-3}, \end{eqnarray} (9) \begin{eqnarray} \int _1^\infty P(\lambda _+) {\rm d}\lambda _+ = 4.2 \times 10^{-2}. \end{eqnarray} (10)A value of λ = 1 therefore lies in the tails of both posterior distributions. These results show that ξ− sits about 2.8σ low compared to the best-fitting ΛCDM cosmology determined from ξ+. The lower plot in Fig. 2 tests consistency between photometric redshift bins including both ξ+ and ξ− in the fits. The parameters λi (with i running from 1 to 4) are computed for data partitions in which $${\boldsymbol y}^D$$ excludes all cross-correlations involving photometric redshift bin i. In this test, photometric redshift bin 3 is an outlier with \begin{eqnarray} \int _0^{1} P(\lambda _3) {\rm d}\lambda _3 = 1.3\times 10^{-2}, \end{eqnarray} (11)suggesting that the data involving photometric redshift bin 3 are inconsistent with the rest of the data at about the 2.2σ level. Again, this accords with the visual impression from fig. 5 of H17, which shows that cross-correlations in both ξ+ and ξ− involving photometric redshift bin 3 tend to lie below their best-fitting model. Instead of using an amplitude parameter λ, we can make a prediction for the vector $${\boldsymbol x}^D$$ conditional on the fit to $${\boldsymbol y}^D$$ \begin{eqnarray} {\boldsymbol x}^{\rm cond} = {\hat{\boldsymbol x}} + \boldsymbol{\sf C}_{xy} \boldsymbol{\sf C}_{yy}^{-1} ({\boldsymbol y}^D - {\hat{\boldsymbol y}}). \end{eqnarray} (12)If the best-fitting model is known exactly, the covariance of $${\boldsymbol x}^{\rm cond}$$ is \begin{eqnarray} {\boldsymbol{\sf C}}_{xx}^{\rm cond} = {\boldsymbol{\sf C}}_{xx}-{\boldsymbol{\sf C}}_{xy}{\boldsymbol{\sf C}}_{yy}^{-1}{\boldsymbol{\sf C}}_{yx}. \end{eqnarray} (13)However, in our application the best-fitting model is determined by fitting the data vector $${\boldsymbol y}^D$$ and so the uncertainty in the best-fitting model contributes an additional variance to $$\boldsymbol{\sf C}_{xx}^{\rm cond}$$: \begin{eqnarray} {\boldsymbol {\sf C}^{\prime}}_{xx}^{\rm cond}={\boldsymbol {\sf C}}_{xx}^{\rm cond}+{\Delta }{\boldsymbol {\sf C}}_{xx}^{\rm cond}, \end{eqnarray} (14)which we determine empirically by sampling over the MCMC chains. In our application, $${\bf \Delta C}_{xx}^{\rm cond}$$ is a small correction to $$\boldsymbol{\sf C}_{xx}^{\rm cond}$$. As a test of the consistency of the data, we compute a conditional χ2: \begin{eqnarray} \chi _{\rm cond}^{2}=(x^{D}-x^{\rm cond})^{T}({\boldsymbol {\sf C}^{\prime}}_{xx}^{\rm cond})^{-1}(x^{D}-x^{\rm cond}). \end{eqnarray} (15)The results of these tests are summarized in Table 1 and are consistent with the λ-tests shown in Fig. 2. Eliminating ξ− leads to a substantial increase in S8 that is incompatible with ξ− at about 2.7σ. The redshift bin 3 component of the data vector is inconsistent with the rest of the data vector at about 2.6σ. However, the $$\chi ^2_{\rm cond}$$ reveals a new inconsistency: the redshift bin 4 component of the data vector is inconsistent with the rest of the data vector at about 3.5σ. The origin of the high values of $$\chi ^2_{\rm cond}$$ for these various partitions of the data vector is clear from Fig. 3. The figure shows the data vector (red points) for all cross-correlations involving redshift bin 3 (upper two panels) and those involving redshift bin 4 (lower two panels) compared to the expectations $${\boldsymbol x}^{\rm cond}$$ conditional on the rest of the data (equation 12). The grey bands show ±1σ and ±2σ ranges around $${\boldsymbol x}^{\rm cond}$$ computed from the diagonal components of equation (14). The top two panels of Fig. 3 show that cross-correlations involving redshift bin 3 want a lower amplitude than the rest of the data. This problem is particularly acute for ξ− for the (3, 3) and (3, 4) redshift bin cross-correlations. These two cross-correlations carry quite high weight in fits to the full data vector (driving S8 down), yet they are inconsistent at nearly ∼2.6σ with the rest of the data. A possible explanation for this discrepancy is an inaccuracy in the calibration of the photometric redshifts for bin 3. In fact, van Uitert et al. (2017) present evidence for a 2.3σ negative shift of Δz ≈ −0.06 for this redshift bin. They find no evidence for significant shifts in the other redshift bins. Figure 3. View largeDownload slide The upper two panels show cross-correlations ξ+ and ξ− involving tomographic redshift bin 3 (red points). The numbers in each plot to identify the cross-correlation (e.g. 1, 3 denotes redshift bin 1 crossed with redshift bin 3). The grey bands show the allowed ±1σ (dark grey) and ±2σ (light grey) ranges allowed by the fits to the rest of the data. The lower two panels show the equivalent plots, but for cross-correlations involving tomographic redshift bin 4. Figure 3. View largeDownload slide The upper two panels show cross-correlations ξ+ and ξ− involving tomographic redshift bin 3 (red points). The numbers in each plot to identify the cross-correlation (e.g. 1, 3 denotes redshift bin 1 crossed with redshift bin 3). The grey bands show the allowed ±1σ (dark grey) and ±2σ (light grey) ranges allowed by the fits to the rest of the data. The lower two panels show the equivalent plots, but for cross-correlations involving tomographic redshift bin 4. As summarized in Table 1, removing redshift bin 4 lowers the value of S8 but increases the errors on S8 substantially because the intrinsic alignment amplitude is less well constrained. From Fig. 3, this low amplitude solution appears to match reasonably well with the general shape of the rest of the data vector, but now we see a high value of $$\chi ^2_{\rm cond}$$ arising from outliers. In the lower two panels of this figure, 8 out of 52 data points sit outside the conditional ±2σ range.7 Several of these outliers are at large angular scales and are not obvious in plots using errors computed from the diagonals of the full covariance matrix (e.g. fig. 5 of H17). However, the KiDS covariance matrix tells us that the data vector should be correlated across different tomographic redshift bins. What Fig. 3 shows is that the KiDS correlation functions display significantly higher variance than expected from the KiDS covariance matrix, particularly at large angular scales and for correlations involving redshift bin 4. This excess variance is a serious problem because it means that the KiDS errors on cosmological parameters are systematically underestimated, especially if data at small angular scales are excluded. Our analysis shows strong evidence for a statistical inconsistency between the KiDS estimates of ξ+ and ξ−. H17 and van Uitert et al. (2017) find evidence for non-zero B modes in the KiDS data at small angular scales (θ < 4.2 arcmin), indicative of systematics. If systematic errors contribute equally to the tangential and cross distortions (and this has not been demonstrated for KiDS), then the B modes will affect ξ+, but not ξ−. Eliminating ξ+ entirely from the fits lowers S8 to 0.705 ± 0.048 (see Table 1) with χ2 = 82.2 for 50 degrees of freedom (a 3.2σ excess). In other words, if one argues that the difference between ξ+ and ξ− is indicative of systematic errors in ξ+, then the tension between KiDS and Planck is exacerbated. 3 COMPARISON WITH OTHER TECHNIQUES FOR MEASURING THE AMPLITUDE OF THE FLUCTUATION SPECTRUM The results of the previous section show that there are some worrying internal inconsistencies in the KiDS data set as analysed in H17. These inconsistencies suggest that we should be cautious in interpreting the KiDS constraints on cosmology. However, the tests in themselves do not tell us the causes of the inconsistencies, or their impact on the estimates of S8. Is the amplitude of the matter fluctuations at redshifts $$z {\, \buildrel< \over \sim \,}1$$ really lower than expected in the Planck ΛCDM cosmology? Another way of studying the amplitude of the matter power spectrum is via RSDs (Kaiser 1987). RSDs provide a measurement of the parameter combination fσ8, where f is the logarithmic derivative of the linear growth rate with respect to the scale factor \begin{eqnarray} f = {{ \rm d \ ln} D \over { \rm d \ ln} a}, \end{eqnarray} (16)and a = (1 + z)−1. In the ΛCDM model, f ≈ Ωm(z)0.55 and so RSDs measure the parameter combination $$\sigma _8 \Omega _{\rm m}^{0.55}$$, i.e. similar to the parameter combination S8 up to a known constant. Measurements of RSD from the DR12 analysis of the Baryon Oscillation Spectroscopy Survey (BOSS) have been reported by Alam et al. (2017). These measurements are for three redshift slices with effective redshifts zeff = 0.38, 0.51, and 0.61, substantially overlapping with the redshift range of the KiDS survey. Huterer et al. (2017) have recently used the Supercal Type Ia supernova compilation (Scolnic et al. 2015) together with independent distance measurements of galaxies (Springob et al. 2014) to measure fσ8 at zeff = 0.02. The Planck ΛCDM cosmology is in excellent agreement with these measurements of fσ8 over the entire redshift range z = 0.02–0.61. The consistency between Planck and the RSD measurements is illustrated in Fig. 4, where we have combined the BOSS and Supercal RSD measurements to produce constraints in the σ8–Ωm plane8. The RSD constraints are in mild tension with the KiDS correlation function analysis of H17, and in even greater tension with the tomographic power-spectrum analysis of KiDS described by K17using the same shear catalogue. Figure 4. View largeDownload slide Constraints in the σ8–Ωm plane assuming the spatially flat ΛCDM cosmology. The 68 and 95 per cent contours from Planck are shown in blue. The constraints from the H17 fiducial KiDS analysis are shown in green. The grey contours show the constraints from the power-spectrum analysis of KiDS reported by K17. The red contours show the constraints from RSDs as discussed in the text. Figure 4. View largeDownload slide Constraints in the σ8–Ωm plane assuming the spatially flat ΛCDM cosmology. The 68 and 95 per cent contours from Planck are shown in blue. The constraints from the H17 fiducial KiDS analysis are shown in green. The grey contours show the constraints from the power-spectrum analysis of KiDS reported by K17. The red contours show the constraints from RSDs as discussed in the text. The abundance of rich clusters of galaxies (selected at various wavelengths) has been used in a number of studies to constrain the amplitude of the fluctuations spectrum at low redshift (e.g. Vikhlinin et al. 2009; Rozo et al. 2010; Hasselfield et al. 2013; Planck Collaboration XX 2014b; Mantz et al. 2015; de Haan et al. 2016; Planck Collaboration XXIV 2016c). As summarized in several of these papers, calibration of cluster masses is a major source of uncertainty in this type of analysis. Two recent studies (Mantz et al. 2015; de Haan et al. 2016) use weak gravitational lensing mass estimates from the ‘Weighing the Giants’ programme (Applegate et al. 2014; Kelly et al. 2014; von der Linden et al. 2014) to calibrate cluster scaling relations. Mantz et al. (2015) use an X-ray selected sample of clusters from the ROSAT All-Sky Survey covering the redshift range 0 < z < 0.5, finding σ8(Ωm/0.3)0.17 = 0.81 ± 0.03. de Haan et al. (2016) use a sample of clusters identified with the South Pole Telescope with median redshift zmed = 0.53 to infer σ8(Ωm/0.27)0.3 = 0.797 ± 0.031. Both of these estimates are consistent with the PlanckP16 ΛCDM cosmology: σ8(Ωm/0.3)0.17 = 0.818 ± 0.009, σ8(Ωm/0.27)0.3 = 0.848 ± 0.012. Thus, there is no convincing evidence for any discrepancy between rich cluster counts and the expectations from the Planck–ΛCDM cosmology. The de Haan et al. (2016) study is particularly interesting because it covers a similar redshift range to those of the BOSS RSD and KiDS measurements, yet is consistent with Planck and RSD. 4 COMPARISON OF WEAK LENSING ESTIMATES OF S8: THE IMPORTANCE OF INTRINSIC ALIGNMENTS Fig. 4 shows a discrepancy between the H17 and K17 analyses, which are based on the same shear catalogue. There is little doubt that the H17 and K17 analyses are incompatible, since not one of the 14 469 samples in the K17 MCMC likelihood chain9 has parameters close to those of the best fit found by H17. In fact, van Uitert et al. (2017, hereafter vU17) have computed cross power spectra from ξ+ and ξ− for the KiDS data using the identical redshift bins to those used in K17. Their autospectrum for the highest redshift bin differs substantially from the quadratic estimate of K17. The origin of this difference is not understood.10 Another pointer that the K17 results are affected by systematic errors comes from the intrinsic alignment solution. K17 find $$A_{{\rm IA}} = -1.72^{+1.49}_{-1.25}$$ which has the opposite (and from the theoretical perspective, counterintuitive) sign to that found by H17. This difference drives down the amplitude of S8 in the K17 analysis. Both the direct comparison of spectra reported by van Uitert et al. (2017) and the shift to a negative intrinsic alignment amplitude suggest that the K17 analysis is suspect. The key point that we want to emphasize here is that the intrinsic alignment parameter AIA is not a benign ‘nuisance’ parameter (for reviews, see e.g. Joachimi et al. 2015; Troxel & Ishak 2015). The modelling of intrinsic alignments is degenerate with the cosmological parameters of interest, σ8, Ωm, and S8, and so the model and associated parameters matter. Systematic errors in the data can be absorbed by the intrinsic alignment model and this will have an impact on cosmology. For example, van Uitert et al. (2017) have noted that the parameter AIA can absorb systematic errors in the calibrations of photometric redshift distributions (this can also be inferred from Fig. 1 which shows the sensitivity of the intrinsic alignment solution for the KiDS data to the highest photometric redshift bin). Implausible (e.g. strongly negative) values of AIA suggest systematic errors and should therefore be followed up. As an example, one of the lowest weak lensing determinations of S8 comes from the reanalysis of the revised CFHTLenS data (Joudaki et al. 2017b). However, these authors find a strongly negative value of AIA = −3.6 ± 1.6, a value which seems unlikely for any reasonable mix of galaxy types. The recent DES analysis of Troxel et al. (2017) uses a redshift-dependent amplitude: AIA[(1 + z)/(1.62)]η, finding $$A_{{\rm IA}} = 1.3^{+0.5}_{-0.6}$$, $$\eta =3.7^{+1.0}_{-2.3}$$.11 Troxel et al. (2017) also test a more elaborate ‘mixed’ alignment model based on the work of Blazek et al. (2017). This model leads to a downward shift of S8 by about 1σ, demonstrating that uncertainties in the modelling of intrinsic alignments make a non-negligible contribution to the errors in cosmological parameters. Returning to the KiDS survey, one way of achieving better control of intrinsic alignments and photometric redshift calibration errors is to add additional types of data. vU17 have analysed the shear power spectra from KiDS, PE (constructed by integrating over ξ+ and ξ−). In addition, they use the Galaxies Mass Assembly (GAMA) redshift survey (Driver et al. 2011; Liske et al. 2015) to compute the galaxy-mass power spectra, Pgm by cross-correlating the KiDS shear measurements with GAMA galaxies, and the galaxy–galaxy power spectra Pgg. From Pgm + Pgg, they find S8 = 0.853 ± 0.042. Combining with PE, they find AIA = 1.30 ± 0.40 and S8 = 0.801 ± 0.032 (consistent with the Planck and RSD results shown in Fig. 4). Fig. 5 gives a summary of the results discussed in this section. The two analyses that are most discrepant with the S8 value from Planck (CFHTLenS and K17) both have strongly negative intrinsic alignment solutions. The H17 results are in tension with Planck but become consistent with Planck with the addition of galaxy–galaxy and galaxy-mass data (vU17). The DES year 1 analyses plotted in Fig. 5 are both consistent with Planck. The intrinsic alignment solutions of vU17 and DES Collaboration et al. (2017, i.e. AIA ∼ 1) seem physically plausible given the mix of galaxy types expected in these surveys. Figure 5. View largeDownload slide S8 plotted against the intrinsic alignment amplitude for various surveys together with 1σ errors on S8 and AIA. The grey bands show the 1σ and 2σ constraints from Planck. The data points are as follows: CFHTLens (Joudaki et al. 2017a); DLS (Jee et al. 2016)12; K17 shows the power spectrum analysis of KiDS (K17); H17 shows the correlation function analysis of KiDS (H17); vU17 shows the constraints from combining Pgg, Pgm, and PE measurements from KiDS and GAMA data (vU17); DES17 shows the cosmic shear constraints from DES year 1 data (Troxel et al. 2017, note that the DES analyses use a redshift-dependent model of intrinsic alignments, as described in the text); DES+17 shows the combination of DES year 1 cosmic shear results with galaxy–galaxy and galaxy-shear measurements (DES Collaboration et al. 2017). Figure 5. View largeDownload slide S8 plotted against the intrinsic alignment amplitude for various surveys together with 1σ errors on S8 and AIA. The grey bands show the 1σ and 2σ constraints from Planck. The data points are as follows: CFHTLens (Joudaki et al. 2017a); DLS (Jee et al. 2016)12; K17 shows the power spectrum analysis of KiDS (K17); H17 shows the correlation function analysis of KiDS (H17); vU17 shows the constraints from combining Pgg, Pgm, and PE measurements from KiDS and GAMA data (vU17); DES17 shows the cosmic shear constraints from DES year 1 data (Troxel et al. 2017, note that the DES analyses use a redshift-dependent model of intrinsic alignments, as described in the text); DES+17 shows the combination of DES year 1 cosmic shear results with galaxy–galaxy and galaxy-shear measurements (DES Collaboration et al. 2017). 5 CONCLUSIONS The main purpose of this paper has been to highlight and quantify internal inconsistencies in the KiDS cosmic shear analysis. Our main conclusion is that more effort is needed to resolve inconsistencies in the KiDS data. This includes understanding the origin of the B modes, systematic differences between ξ+ and ξ−, the parameter shifts seen by excluding photometric redshift bin 3, the large excess χ2 and scatter at large angular scales. Until this is done, it seems premature to draw inferences on new physics from KiDS. A comparison of Planck with other measures of the amplitude of the mass fluctuations, principally RSDs from BOSS, reveals no evidence for any inconsistencies with the Planck-based ΛCDM cosmology. We have also reviewed cosmic shear constraints on S8, emphasizing the degeneracy between intrinsic alignments and cosmology. As summarized in Fig. 5, the two analyses which yield the lowest values of S8 both have strongly negative values of AIA. The DES 1-yr analyses are consistent with the Planck ΛCDM value for S8 (DES Collaboration et al. 2017; Troxel et al. 2017) and give physically plausive values for AIA. The H17 value of S8 from KiDS sits about 2.3σ low compared to Planck, but is pulled upwards with the addition of galaxy–galaxy, galaxy-mass data (vU17). Overall, we conclude there is no strong evidence for any inconsistency between the Planck ΛCDM cosmology and measures of the amplitude of the fluctuation spectrum at low redshift. ACKNOWLEDGEMENTS We thank Hiranya Peiris, Benjamin Joachimi, Fergus Simpson, and the referee for helpful comments on the pre-print version of this paper. We thank Frankie Nobis-Efstathiou for help with the early stages of this project. We also thank Anthony Challinor, Steven Gratton, and members of the KiDS team for comments on aspects of this analysis. We also thank members of the Planck Parameters team. PL acknowledges support from an Isaac Newton Studentship at the University of Cambridge and from the Science and Technologies Facilities Council. Footnotes 1 Where σ8 is the present-day linear theory root-mean-square amplitude of the matter fluctuation spectrum averaged in spheres of radius 8 h−1Mpc, Ωm is the present-day matter density in units of the critical density ρc, and h is the Hubble constant in units of 100 km s−1 Mpc−1. 2 Unless stated otherwise, we quote ±1σ errors on parameters. 3 DES Year 1 results (DES Collaboration et al. 2017; Troxel et al. 2017) appeared after the submission of this paper and so will not be discussed in detail. 4 Neglecting B modes. 5 The situation is very different to the CMB, where there is a large amount of information to separate a high-amplitude frequency-independent cosmological signal with a distinctive power spectrum from low-amplitude foregrounds with smooth power spectra. 6 Downloaded from http://kids.strw.leidenuniv.nl. 7 Assuming Gaussian statistics, the p-value for this is about 2.4 × 10−3. 8 This is done using the final_consensus_dV_FAP_fsig data files and covariance matrix downloaded from https://sdss3.org/science/boss_publications.php. We then scanned the likelihood, using uniform priors in H0 and Ωm h2 to rescale the BOSS distance DV and Alcock–Paczynski (Alcock & Paczynski 1979) parameter FAP to the fiducial sound horizon used in the BOSS analysis, fixing Ωb h2 to the P16 ΛCDM value. 9 KiDS450_QE_EB_4bins_3zbins_basez_ia_bary_nu.txt, downloaded from http://kids.strw.leidenuniv.nl. 10 Note that the quadratic estimator used by K17 is sensitive to noise estimation, particularly if there are B-mode systematics (which are known to be present in the KiDS data). Inaccurate noise estimation would primarily affect the autospectra, where the noise levels are high compared to the cosmological signal (see fig. 4 of H17). 11 These constraints become $$A_{{\rm IA}} = 0.5^{+0.32}_{-0.38}$$, $$\eta = 0^{+2.7}_{-2.8}$$ with the addition of galaxy–galaxy and galaxy-shear data, DES Collaboration et al. (2017). 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Monthly Notices of the Royal Astronomical Society – Oxford University Press

**Published: ** May 1, 2018

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