tokens: obtained by using (green), (red), and (blue). We make the following observations from this plot: β BF versus f / b falls nicely on a straight line with slope 1 (the solid line). However, note that the relation β BF = f / b (the dashed line) does not exactly hold: we have checked that the actual value of β BF depends on the s min used, decreasing as smaller s min is used; the results here are for . The trend suggests that, roughly speaking, the reconstruction simply spits out the ‘input’ β value, and therefore has no predictive power regarding β, unless we look beyond R ( s ). The two ( f, b ) pairs that give similar β = f / b ≈ 0.375 produce very similar β BF : this is a consequence of the f – b degeneracy mentioned above. For comparison, we include the best-fitting β BF values obtained using R ′ ( s ) as filled circles, using the same colour scheme. Because there is no reconstruction and therefore no input ( f, b ), we simply plot the points at a fixed horizontal coordinate f / b = 0.376 (the theoretical value). We note that β BF obtained in this way is indeed larger than the theoretical value, but this is well within the 1σ uncertainty indicated by the grey error bar; at this stage, we cannot accurately assess the impact on β BF due to the small sample size or the simulation resolution, which is beyond the scope of this work. As mentioned above, a small (per cent-level) error in R ′ ( s ) leads to a larger error in β BF . Figure 12. Upper left : The best-fitting β BF values obtained using the estimator R ( s ) and the reconstructed galaxy catalogues, for seven choices of input parameters f and b supplied to the reconstruction pipeline (crosses). R ( s ) is measured as the weighted average between and s max = 98.9 (green), 90.2 (red), and (blue), respectively, and the error bars are the standard deviations obtained from 15 realizations for the case of as the error bars for remaining s max values are similar. The filled circles are the same, but obtained from the estimator R ′ ( s ): since there is no reconstruction in this case, these data are plotted against the theoretical value β = f / b = 0.376. The dashed line represents β = f / b , whilst the solid line is a line of slope 1.0 to guide the eyes. Lower left : , the constant value fitting ξ 2 [ x ( k ) ]( s ) between and the three s max values s max = 98.9 (green, densely dashed), 90.2 (red, dashed), and (blue, sparesly dashed), respectively, for the same seven choices of input ( f, b ). The vertical dotted lines mark the best-fitting β BF at which is closest to zero, and the error bars are again the standard deviations of the 15 realizations. Right-hand panels : the estimator E4 (upper) and E3 (lower) for the seven choices of reconstruction parameters ( f, b ), with their corresponding values of β indicated in the legend and increase with the sparsenesses of the curves [β increases from to the top (bottom) to the bottom (top) line in the upper (lower) right-hand panels]. Note that
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