In order to explain clustering of galaxies and its evolution, a linear bias factor, a parameter representing the difference between galaxy distribution and underlying matter distribution, is often introduced. The linear bias factor b of sgzKs at a scale of 8 h−1 Mpc is expressed as follows:
(15)$$\begin{eqnarray}
b = \frac{\sigma _{8, \rm sgzK}}{\sigma _{8}(z)}.
\end{eqnarray}$$The rms fluctuation of sgzK distribution $\sigma _{8, \rm sgzK}$ (Peebles 1980) and that of underlying matter distribution at a given redshift σ8(z) can be expressed as follows:
(16)$$\begin{eqnarray}
\sigma _{8, \rm sgzK} = \sqrt{J_{2}(\gamma) \biggl (\frac{r_{0}}{8\,h^{-1} ~\rm Mpc}\biggr)^{\gamma } }
\end{eqnarray}$$and
(17)$$\begin{eqnarray}
\sigma _{8}(z) = \sigma _{8}(0)D(z),
\end{eqnarray}$$where
(18)$$\begin{eqnarray}
D(z) = g(z)/[g(0)(1+z)],
\end{eqnarray}$$(19)$$\begin{eqnarray}
g(z) = \frac{5}{2}\Omega _{\mathrm{ m}}\biggl [\Omega _{\mathrm{ m}}^{4/7}-\Omega _{\Lambda }+\biggl (1+\frac{\Omega _{\mathrm{ m}}}{2}\biggr)\biggl (1+\frac{\Omega _{\Lambda }}{70}\biggr)\biggr ]^{-1},
\end{eqnarray}$$(20)$$\begin{eqnarray}
\Omega _{\mathrm{ m}} = \frac{\Omega _{\mathrm{ m},0}(1+z)^3}{E^{2}(z)},~~~ \Omega _{\Lambda } = \frac{\Omega _{\Lambda ,0}}{E^{2}(z)};
\end{eqnarray}$$the present rms mass fluctuation is σ8(0) = 0.83 (Planck Collaboration XVI 2014) and J2(γ) = 72/[2γ(3 − γ)(4 − γ)(6 − γ)]. Fig. 8 shows the bias factor of sgzKs along with those of other populations in the literature.2 The bias of sgzKs with Ks  21.5 is larger than those of SMGs with z  2, while it is similar to those of ULIRGs at comparable redshift, EROs, DOGs with z ∼ 1.2, and SMGs with z = 2–3. Galaxies and their host dark matter haloes can be connected through the correlation length or the bias factor under the assumption that each dark matter halo is occupied by a single galaxy. Therefore, in order to find present-day descendants of high-redshift galaxy populations, we can use a redshift evolution of the bias for dark matter haloes hosting high-redshift galaxies. For example, Hickox et al. (2012) and Wilkinson et al. (2017) investigated the evolution of either correlation length or bias for the haloes of SMGs to find an evolutionary link between high-redshift SMGs and local galaxies. Seo et al. (2019) also used the same method to find descendants of EROs. Thus, we investigated the evolution of bias for dark matter haloes hosting sgzKs with Ks  21.5. By the ellipsoidal collapse model, Mo & White (2002) estimated the bias for dark matter haloes with mass M at a given redshift as follows:
(21)$$\begin{eqnarray}
b (M, z) = 1 + \frac{1}{\delta _{\mathrm{ c}}}~\biggl [\nu ^{\prime 2}+b\nu ^{\prime 2(1-c)}-\frac{\nu ^{\prime 2c}/\sqrt{a}}{\nu ^{\prime 2c}+b(1-c)(1-c/2)}\biggr ], \\
\end{eqnarray}$$where
(22)$$\begin{eqnarray}
\nu ^{\prime } = \sqrt{a}\nu ~~~ \mathrm{with} ~~~ \nu = \frac{\delta _{\mathrm{ c}}}{\sigma (R)D(z)},
\end{eqnarray}$$a = 0.707, b = 0.5, c = 0.6, the critical overdensity is δc = 1.686 (Sheth, Mo & Tormen 2001), and σ(R) is the variance of density field fluctuation.3 To trace the bias of haloes over redshift, we calculated halo mass M at a given redshift using the formalism for mass growth rate of a halo. Fakhouri, Ma & Boylan-Kolchin (2010) suggested two kinds of mass growth rates, mean growth rate $\langle \dot{M}\rangle _{\mathrm{mean}}$ and median growth rate $\langle \dot{M}\rangle _{\mathrm{median}}$, as follows:
(23)$$\begin{eqnarray}
\langle \dot{M}\rangle _{\mathrm{mean}} = 46.1~\mathrm{M_{\odot }\,yr^{-1}}\biggl (\frac{M}{10^{12}~\mathrm{M_{\odot }}}\biggr)^{1.1} \,\,\,\,(1+1.11z)E(z)
\end{eqnarray}$$and
(24)$$\begin{eqnarray}
\langle \dot{M}\rangle _{\mathrm{median}} = 25.3~\mathrm{M_{\odot }\,yr^{-1}}\biggl (\frac{M}{10^{12}~\mathrm{M_{\odot }}}\biggr)^{1.1} \,\,\,\,(1+1.65z)E(z),
\end{eqnarray}$$where the former is larger than the latter. We estimated each mass growth rate separately, and took their average as the mass growth rate of the halo with M at a given z. In Fig. 8, dashed lines represent the evolution of bias for haloes, which are the results considering the mass growth of haloes. On the other hand, solid lines show the evolution of bias for haloes with fixed halo mass.
Citation: Planck Collaboration XVI 2014
Citation function: Uses