Cosmological Parameter Estimation from the Two-Dimensional Genus Topology -- Measuring the Shape of the Matter Power Spectrum

We present measurements of the two-dimensional genus of the SDSS-III BOSS catalogs to constrain cosmological parameters governing the shape of the matter power spectrum. The BOSS data are divided into twelve concentric shells over the redshift range $0.2<z<0.6$, and we extract the genus from the projected two-dimensional galaxy density fields. We compare the genus amplitudes to their Gaussian expectation values, exploiting the fact that this quantity is relatively insensitive to non-linear gravitational collapse. The genus amplitude provides a measure of the shape of the linear matter power spectrum, and is principally sensitive to ${\mathrm{\Omega }}_{\mathrm{c}}{h}^2$ and scalar spectral index $n_{\mathrm{s}}$. A strong negative degeneracy between ${\mathrm{\Omega }}_{\mathrm{c}}{h}^2$ and $n_{\mathrm{s}}$ is observed, as both can increase small scale power by shifting the peak and tilting the power spectrum respectively. We place a constraint on the particular combination $n_{\mathrm{s}}^{3\mathrm{/}2}{\mathrm{\Omega }}_{\mathrm{c}}{h}^2$ – we find $n_{\rm s}^{3/2} \Omega_{\rm c}h^{2} = 0.1121 \pm 0.0043$ after combining the LOWZ and CMASS data sets, assuming a flat $\mathrm{\Lambda }$CDM cosmology. This result is practically insensitive to reasonable variations of the power spectrum amplitude and linear galaxy bias. Our results are consistent with the Planck best fit $n_{\mathrm{s}}^{3\mathrm{/}2}{\mathrm{\Omega }}_{\mathrm{c}}{h}^2=0.1139\pm 0.0009$.