Detecting Scale-Dependence of Bias from APM-BGC Galaxies

We present an investigation of the scale-dependence of bias described by the linear model: $(\delta \rho({\bf x})/\bar{\rho})_{g} = b (\delta \rho(x)/\bar{\rho})_{m}$, $b$ being the bias parameter, and $\rho({\bf x})_{g}$ and $\rho({\bf x})_{m}$ are the galaxy number density and mass density, respectively. Using a discrete wavelet decomposition, we show that the behavior of bias scale-dependence cannot be described by one parameter $b$. In the linear bias model the scale-dependence should be measured by the $j$-spectra of wavelet-coefficient-represented bias parameters $\tilde{b}^{(n)}_j$ and $b_j^{(n)}$, $n$ being positive integers. Because $\tilde{b}^{(n)}_j$ with different $n$ are independent from each other, a systematic analysis of the $j$-spectra of $\tilde{b}^{(n)}_j$ and $b_j^{(n)}$ is necessary. We performed a $j$-spectrum analysis for samples of elliptical and lenticular (EL), and spiral (SP) galaxies listed in the APM bright galaxy catalog. We found that, for statistics of two-point correlation functions or DWT power spectrum, the scale-independence holds within 1 $\sigma$. However, the bias scale-dependence becomes substantial when phase-sensitive statistics (e.g. $\tilde{b}^{(n)}_j$ with $n>2$ or $b_j^{(n)}$) are applied. These results indicate that the bias scale-dependence has the same origin as the non-Gaussianity of galaxy distributions. This is generally consistent with the explanation that the bias scale-dependence originated from non-linear and non-local relationship between galaxy formation and their environment.