Determine the galaxy bias factors on large scales using bispectrum method

We study whether the bias factors of galaxies can be unbiasedly recovered from their power spectra and bispectra. We use a set of numerical N-body simulations and construct large mock galaxy catalogs based upon the semi-analytical model of Croton et al. (2006). We measure the reduced bispectra for galaxies of different luminosity, and determine the linear and first nonlinear bias factors from their bispectra. We find that on large scales down to that of the wavenumber k=0.1h/Mpc, the bias factors b1 and b2 are nearly constant, and b1 obtained with the bispectrum method agrees very well with the expected value. The nonlinear bias factor b2 is negative, except for the most luminous galaxies with M<-23 which have a positive b2. The behavior of b2 of galaxies is consistent with the b2 mass dependence of their host halos. We show that it is essential to have an accurate estimation of the dark matter bispectrum in order to have an unbiased measurement of b1 and b2. We also test the analytical approach of incorporating halo occupation distribution to model the galaxy power spectrum and bispectrum. The halo model predictions do not fit the simulation results well on the precision requirement of current cosmological studies.Comment: 9 pages, 8 figures, accepted for publication in Ap

Rational points near planar curves and Diophantine approximation

In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich-Dickinson-Velani and Vaughan-Velani. Furthermore, we complete the Lebesgue theory of Diophantine approximation on weakly non-degenerate planar curves that was initially developed by Beresnevich-Zorin in the divergence case.Comment: 27 pages, corrected typos, to appear in Adv. Mat