SDSS galaxy bias from halo mass-bias relation and its cosmological implications

TL;DR

The paper tests a core prediction of structure formation: halo bias depends on halo mass relative to the nonlinear mass $M_{nl}$. It combines six-bin luminosity-dependent galaxy clustering with galaxy-galaxy weak lensing around the same SDSS galaxies to infer the full halo-mass distribution $p(M;L)$ for each luminosity bin and then predict the bias $b(L)$ from the halo-bias relation. The authors find remarkable agreement between the lensing-derived bias predictions and observations, enabling joint constraints on $\sigma_8$, $\,b_*$, and neutrino masses when combined with WMAP and SDSS power spectrum data. Their results yield $\,\sigma_8=0.88\pm0.06$ and $b_*=0.99\pm0.07$, with competitive neutrino-mass limits ($m_\nu<0.18$ eV for 3 degenerate families, no running; $m_\nu<0.24$ eV with running), demonstrating the value of integrating weak lensing with clustering to calibrate fluctuation amplitudes and break parameter degeneracies.

Abstract

We combine the measurements of luminosity dependence of bias with the luminosity dependent weak lensing analysis of dark matter around galaxies to derive the galaxy bias and constrain nonlinear mass and cosmological parameters. We take advantage of theoretical and simulation predictions that predict that while halo bias is rapidly increasing with mass for high masses, it is nearly constant in low mass halos. We use a new weak lensing analysis around the same SDSS galaxies to determine their halo mass probability distribution. These halo mass probability distributions are used to predict the bias for each luminosity subsample and we find an excellent agreement with observed values. The required nonlinear mass suggests slightly lower matter density than usually assumed, Omegam=0.25+/- 0.03 for the simplest models. We combine the bias constraints with those from the WMAP and the SDSS power spectrum analysis to derive new constraints on bias and sigma_8. For the most general parameter space we find sigma_8=0.88+/- 0.06 and b_*=0.99+/- 0.07. In the context of spatially flat models we improve the limit on the neutrino mass for the case of 3 degenerate families from m_nu<0.6eV without bias to m_nu<0.18eV with bias (95% c.l.), which is weakened to m_nu<0.24eV if running is allowed. The corresponding limit for 3 massless + 1 massive neutrino is 1.37eV.

Figures (6)

• Figure 1: Weak lensing signal $\Delta \Sigma (r)$ as a function of transverse separation $r$ as measured from SDSS data, together with the best fit 2-parameter model for each of 6 luminosity bins. Also shown are the best fit values for halo virial mass $M$ (in units of $10^{11}h^{-1}M_{\hbox{$\odot$}}$) and $\alpha$, the fraction of galaxies that are non-central, assuming $M_{\rm nl}=8\times 10^{12}M_{\hbox{$\odot$}}$. The model fits the data well in all bins. The mass fits are what comes from the fitting procedure and are a typical halo mass somewhere between mean and median. For bias calculations they are increased by varying amounts to account for the difference between fitted mass and mean mass, as described in the text.
• Figure 2: This figure shows the lensing-constrained model predictions for bias as a function of nonlinear mass using the 2-parameter models of the halo mass probability distribution. More general models of the halo probability distribution give very similar results and are not shown here. The nonlinear masses from top to bottom are $3.4 \times 10^{11}h^{-1}M_{\hbox{$\odot$}}$, $6.2 \times 10^{11}h^{-1}M_{\hbox{$\odot$}}$, $1.7 \times 10^{12}h^{-1}M_{\hbox{$\odot$}}$, $4.0 \times 10^{12}h^{-1}M_{\hbox{$\odot$}}$, $8.0 \times 10^{12}h^{-1}M_{\hbox{$\odot$}}$, $1.5 \times 10^{13}h^{-1}M_{\hbox{$\odot$}}$ and $2.4 \times 10^{13}h^{-1}M_{\hbox{$\odot$}}$. Errors have been suppressed (see Table 1).
• Figure 3: The halo bias predictions of galaxy fluctuation amplitude $\sigma_8^{\rm gal}$ as a function of luminosity varying linear matter amplitude $\sigma_8$: 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, from top to bottom on the right hand side. The remaining parameters have been fixed to $\Omega_m=0.3$ and $n_s=1$. Squares are for model with $\sigma_8=0.88$, $n_s=1.0$ and $\Omega_m=0.3$. For this model we show errors from theoretical modelling. Also shown as triangles are the observed values of $\sigma_8^{\rm gal}$.
• Figure 4: 68% (inner, blue) and 95% (outer, red) contours in $(\sigma_8,b_*)$ plane using SDSS+WMAP+bias measurements. The two parameters are strongly correlated because only their product is determined from SDSS clustering analysis. The additional bias constraint helps reduce the degeneracy.
• Figure 5: 68% (inner, blue) and 95% (outer, red) contours in $(\sigma_8,\tau)$ plane using SDSS+WMAP+bias measurements. There is a correlation between the two, so a better determination of optical depth $\tau$ from polarization data would help improve the constraints.