Effects of biasing on the galaxy power spectrum at large scales

In this paper we study the effect of biasing on the power spectrum at large scales. We show that even though nonlinear biasing does introduce a white noise contribution on large scales, the $P(k)\propto k^n$ behavior of the matter power spectrum on large scales may still be visible and above the white noise for about one decade. We show, that the Kaiser biasing scheme which leads to linear bias of the correlation function on large scales, also generates a linear bias of the power spectrum on rather small scales. This is a consequence of the divergence on small scales of the pure Harrison-Zeldovich spectrum. However, biasing becomes $k$ dependent if we damp the underlying power spectrum on small scales. We also discuss the effect of biasing on the baryon acoustic oscillations.

Figure 1

In the left panel we show the underlying $\Lambda \mathrm{CDM}$ power spectrum (top red line) and the corresponding biased power spectra (lower green lines) for $\nu =5$, 4, 3, 2, and 1 from top to bottom. As mentioned in the text, the amplitude of the biased power spectra is unphysical. In the right panel we have normalized the power spectra to their maximum value after which they all collapse to $P_1(k)$ as discussed in the main text. We have also indicated the scales corresponding to the horizon scale at equality, $k_{\mathrm{eq}}$ (blue vertical line), and the present horizon scale, $k_0$ (black vertical line).

Figure 2

The normalized correlation function ${\xi }_c=\xi /{\sigma }^2$ is shown. Since the spectral index is close to 1, ${\sigma }^2$ is very large and ${\xi }_c(r)$ is small already on small scales.

Figure 3

We show the biased power spectrum for a smoothed underlying power spectrum as explained in the text for several smoothing scales (indicated in the corresponding panels) and for $\nu =5$, 4, 3, 2, and 1 from top to bottom. Note also here that the amplitude of the biased power spectra is unphysical. Actually, the value $P(0)$ is (up to a factor ${\nu }^2$) the fraction of the volume for which ${\xi }_c(r)=\xi (r)/{\sigma }^2$ is larger than $\nu$ which becomes large for larger damping scale $R$ since damping reduces the variance ${\sigma }^2$.

Figure 4

In this plot we show the biased power spectra after small scale damping. The top red line corresponds to the underlying power spectrum, the red-dashed line is the smoothed underlying power spectrum and the lower green lines show the biased power spectra after applying smoothing for $\nu =5$, 4, 3, 2, 1 from top to bottom. On large scales, $k<1/R$, the biased spectra are not affected and correspond to the ones shown in Fig. 1.

Figure 5

Biasing in the presence of baryons. For this plot we choose ${\Omega }_b{h}^2=0.025$, ${\Omega }_M{h}^2=0.05$ and, $h=0.5$. With this too large ratio of ${\Omega }_b/{\Omega }_M$ the BAO’s are well visible. In the left panel we show the underlying $\Lambda \mathrm{CDM}$ power spectrum (red-dashed line) that includes the effects of baryons and the corresponding biased power spectra (green solid lines) for $\nu =5$, 4, 3, 2, and 1 from top to bottom. In the right panel we have normalized the power spectra to their maximum value that clearly shows the two effects explained in the main text.

Figure 6

The parameters are like for Fig. 5. We show the underlying $\Lambda \mathrm{CDM}$ power spectrum smoothed with a Gaussian window function (red-dashed line) and the corresponding biased power spectra (green solid lines) for $\nu =5$, 4, 3, 2, and 1 from top to bottom for the different smoothing scales indicated in the panels.

Figure 7

In this plot we show the biased power spectra after applying smoothing. The top red line corresponds to the underlying power spectrum, the red-dashed line is the smoothed underlying power spectrum and the green lines show the biased power spectra after applying smoothing for $\nu =5$, 4, 3, 2, 1 from top to bottom.