Large-Scale Biasing and the Primordial Gravitational Potential

We modified the Press-Schechter (PS) formalism and then analytically deriveda constrained mass distribution function n(M|varphi) for the regions havingsome specified value of the primordial gravitational potential, varphi. Theresulting modified PS theory predicts that gravitationally bound clumps withmasses corresponding to rich clusters are significantly biased toward theregions of negative primordial potential - the troughs of the potential. Theprediction is quantitative, depending on the mass and the depth of the troughs,which can be tested in large N-body simulations. As an illustration of themagnitude of the effect we calculate the constrained mass function for the CDMmodel with Gamma = Omega h = 0.25 normalized to sigma_{8} = 1. In particular,we show that the probability of finding a clump of mass 10^{14} -10^{15}h^{-1}M_{odot} in the region of negative initial potential is 1.3 - 3times greater (depending on the mass) than that in the region ofpositiveinitial potential. The scale of the potential fluctuationsR_{varphi}=sqrt{3} sigma_{varphi}/sigma_{varphi'} is shown to be approx 120h^{-1} Mpc for the spectrum in question. The rms mass density contrast on thisscale is only about sigma_{delta}(R_{varphi}) approx 0.03. Assuming that themodified PS theory is statistically correct, we conclude that clusters aresignificantly biased (b ge 10, b is a bias factor defined by Delta n_{cl}/n_{cl} = b Delta rho_m/rho_m) toward the regions having negative initialpotential.Comment: Latex file,15 pages,2 postscript figures, accepted by ApJ Lette