tokens: amplitude than a denser sample (gray region). Second, if the galaxy sample is sufficiently sparse, then the genus amplitude becomes sensitive to galaxy bias (or more precisely the combination b 2 A s )—the width of the filled regions indicates the sensitivity of to bias, for given number density and fixing all other cosmological parameters. For our fiducial number density a ∼10% variation in the galaxy bias generates a ∼0.5% uncertainty in the genus amplitude (see red shaded area). One can also observe a faint redshift evolution in the red shaded region—this is due to the amplitude of the matter power spectrum decreasing with redshift relative to the shot-noise term, which we have fixed to be constant at each redshift. However, if we increase the number density of the galaxy sample by an order of magnitude (gray shaded region), the sensitivity of the genus amplitude to the bias becomes negligible—as the shot noise decreases, the genus amplitude becomes insensitive to the amplitude of the power spectrum. The redshift evolution is also suppressed by selecting a denser sample. A more general word of caution is required: the shot-noise contribution is not perfectly represented by a white-noise power spectrum with . In fact, as we stray into scales at which shot noise affects our results, the expansion of the genus in Hermite polynomials will not provide a good representation of the Poissonian signal, as the Poisson distribution has a different moment-generating function. , where is the mean galaxy separation of the catalog, is an important condition on our analysis. Shot noise has been discussed in Appleby et al. ( 2017 ) and further in S. Appleby et al. (2020, in preparation). We note that both the redshift-space distortion effect and shot noise introduce a redshift dependence to the genus amplitude. For the smoothing scales and redshifts used in this work, the redshift-space distortion effect systematically decreases the genus amplitude with increasing redshift. Conversely, the shot-noise contribution increases the genus amplitude with redshift. Both effects are and will act to practically cancel one another. To completely suppress the shot noise and bias effects, the condition is required. For the smoothing scales and number densities considered in this work, the genus amplitude is only weakly sensitive to the combination of b , A s , and P SN . Nevertheless, we must include the P SN contribution to the power spectrum to avoid systematic bias in our cosmological parameter reconstruction. Appendix B Effect of Variation of Cosmological Parameters on Measured Genus Amplitudes At numerous points in our analysis when measuring the genus curves from the BOSS galaxy catalog, we have been forced to fix the distance–redshift relation. Specifically, we smoothed the field with constant comoving scale R G = 20 Mpc, which corresponds to angular smoothing on the unit sphere. We measured the genus per unit area, and so divided the genus by the total area
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