Mapping the Real Space Distributions of Galaxies in SDSS DR7: II. Measuring the growth rate, clustering amplitude of matter and biases of galaxies at redshift $0.1$

We extend the real-space mapping method developed in Shi et at. (2016) so that it can be applied to flux-limited galaxy samples. We use an ensemble of mock catalogs to demonstrate the reliability of this extension, showing that it allows for an accurate recovery of the real-space correlation functions and galaxy biases. We also demonstrate that, using an iterative method applied to intermediate-scale clustering data, we can obtain an unbiased estimate of the growth rate of structure $f{\sigma }_8$, which is related to the clustering amplitude of matter, to an accuracy of $\sim 10\mathrm{\%}$. Applying this method to the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7), we construct a real-space galaxy catalog spanning the redshift range $0.01\le z\le 0.2$, which contains 584,473 galaxies in the north Galactic cap (NGC). Using this data, we infer $\text{fss}$ at a median redshift $z=0.1$, which is consistent with the WMAP9 cosmology at the $1\sigma$ level. By combining this measurement with the real-space clustering of galaxies and with galaxy-galaxy weak lensing measurements for the same sets of galaxies, we are able to break the degeneracy between $f$, ${\sigma }_8$, and $b$. From the SDSS DR7 data alone, we obtain the following cosmological constraints at redshift $z=0.1$: $f=$$0.464_{-0.040}^{+0.040}$, ${\sigma }_8=0.769_{-0.089}^{+0.121}$, and $b=1.910_{-0.268}^{+0.234}$, $1.449_{-0.196}^{+0.194}$, $1.301_{-0.177}^{+0.170}$, and $1.196_{-0.161}^{+0.159}$ for galaxies within different absolute magnitude bins $^{0.1}{\rm M}_r-5\log h=[-23,0, -22.0], [-22,0, -21.0], [-21.0, -20.0]$and$[-20.0, -19.0]$, respectively.