Galaxy bias and non-linear structure formation in general relativity

Abstract

Length scales probed by the large scale structure surveys are becoming closerand closer to the horizon scale. Further, it has been recently understood thatnon-Gaussianity in the initial conditions could show up in a scale dependence ofthe bias of galaxies at the largest possible distances. It is thereforeimportant to take General Relativistic effects into account. Here we providea General Relativistic generalization of the bias that is valid both forGaussian and for non-Gaussian initial conditions. The collapse of objectshappens on very small scales, while long-wavelength modes are always in thequasi linear regime. Around every small collapsing region, it is thereforepossible to find a reference frame that is valid for arbitrary times and wherethe space time is almost flat: the Fermi frame. Here the Newtonianapproximation is applicable and the equations of motion are the ones of thestandard N-body codes. The effects of long-wavelength modes are encoded in themapping from the cosmological frame to the local Fermi frame. At the level ofthe linear bias, the effect of the long-wavelength modes on the dynamics of theshort scales is all encoded in the local curvature of the Universe, which allowsus to define a General Relativistic generalization of the bias in the standardNewtonian setting. We show that the bias due to this effect goes tozero as the square of the ratio between the physical wavenumber and the Hubblescale for modes longer than the horizon, confirming the intuitive picture thatmodes longer than the horizon do not have any dynamical effect. On the otherhand, the bias due to non-Gaussianities does not need to vanish for modes longerthan the Hubble scale, and for non-Gaussianities of the local kind it goes to aconstant. As a further application of our setup, we show that it is notnecessary to perform large N-body simulations to extract information aboutlong-wavelength modes: N-body simulations can be done on small scales andlong-wavelength modes are encoded simply by adding curvature to thesimulation, as well as rescaling the time and the scale.