tokens: follows: the second Fisher matrix is computed from all ’s except for the two entries at the extrema of the anti-diagonal, which are set to zero; and so on until we compute the 10th Fisher matrix, where is non-zero only for i = j . The inset in Fig. 4 illustrates this procedure, with shading going from black (only diagonal elements considered) to the lightest of grey (all the cross-correlations between thin bins considered). The larger figure shows the structure of the full tomographic matrix of our hybrid method for this test of the importance of off-diagonal correlations. In this case, white means no correlation included. Figure 4. General structure of the full angular power spectrum tomographic matrix, , in our hybrid approach. Block diagonal matrices represent the thick bins, which we consider as independent. The inset illustrates the structure of one of those thick bins, with the thin bins shown. Shades of grey refer to the tests described in Section 4.5 . Now, we consider two extrema: the first and the last of the thick bins. Referring to the inset of Fig. 4 , for both of these thick bins, we start from the full angular power spectrum tomographic matrix and at each step we remove one off-diagonal, from the lightest grey until we remain only with the black, main diagonal. In Fig. 5 , we illustrate the result of this test, by plotting the absolute bias on cosmological parameters induced by neglecting lensing magnification (normalized to the reference values of Table 2 ), as a function of how many cross-correlations between thin bins we include. Apart from the first point(s), for which the bias may be lower, but just because the Fisher matrices are very noisy, it appears clear that the behaviour is very flat. As expected, the trend is flatter at low redshift in the first thick bin. Figure 5. Absolute bias on cosmological parameters induced by neglecting lensing magnification, as function a of off-diagonal elements included, for an analysis carried out either in the first or in the last of the thick redshift bins only, normalized to what is obtained by including all cross-correlations between thin bins in the given thick bin. Fig. 6 shows the bias on cosmological parameters due to neglecting lensing, as a function of how many off-diagonal spectra are considered in the full angular power spectrum, i.e. the larger matrix in Fig. 4 . On the horizontal axis, 1 means black elements in the larger matrix, all the way to 10, representing the inclusion of all elements up to the lightest grey shade. Each point is normalized to its previous value, e.g. the seventh blue bullet point tells us that the bias on Ω b will increase by if we include up to the seventh off-diagonal, with respect to what we would guess by considering only up to the sixth. We notice a net trend: the bias on cosmological parameter best-fit values induced by neglecting lensing may be significant if we considered only the main diagonal or the first few off-diagonals, but then it does
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