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Direct constraints on the biasing field should be provided by the data themselves, of galaxy density (e.g., from redshift surveys) versus mass density (from peculiar velocity surveys, gravitational lensing, etc.). A hint of scatter in the biasing relation is the fact that the smoothed density peaks of the Great Attractor (GA) and Perseus Pisces (PP) are of comparable height in the mass distribution as recovered by POTENT from observed velocities [15, 13, 19], while PP is higher than GA in the galaxy maps [33, 52]. Another piece of indirect evidence for scatter comes from a linear regression of the 1200km s-1-smoothed density fields of POTENT mass and optical galaxies in our cosmological neighborhood, which yields a chi2 ~ 2 per degree of freedom [33]. One way to obtain a more reasonable chi2 ~ 1 is to assume a biasing scatter of sigmab ~ 0.5 (while sigma ~ 0.3 at that smoothing). With b1 ~ 1, one has sigmab2 / b12 ~ 0.25. This is only a crude estimate; there is yet much to be done with future data along the lines of reconstructing the ``biasing field" in a given region of space.

We have recently worked out a promising way to recover the mean biasing function b(delta) and its associated parameters bhat and btilde from a measured PDF of the galaxy distribution [53]. This method is inspired by a ``de-biasing" technique by Narayanan & Weinberg [44]. If the biasing relation g(delta) were deterministic and monotonic, then it could be derived directly from the cumulative PDFs of galaxies and mass, Cg(g) and Cd(delta), via

Equation 21 (21)

We find, using halos in N-body simulations, that this is a good approximation for <g|delta> despite the significant scatter about it. This is demonstrated in Figure 2.

Figure 2a
Figure 2b

Figure 2. The PDFs and the mean biasing function, from a cosmological N-body simulation of sigma8 = 0.3 (z = 1) with top-hat smoothing of 8 h Mpc and for halos of M > 2 x 1012 Msun Left: the cumulative probability distributions C of density fluctuations of halos (g) and of mass (delta). A log-normal distribution is shown for comparison. The errors are by bootstrap re-sampling of halos. The horizontal separation between the curves approximates the mean biasing function <g|delta> = b(delta)sigma2 at the corresponding value of delta. Right: the density fields of halos and mass compared at grid points. The symbols describe the mean biasing function as derived from this data in bins. The solid curve is derived from the PDFs via Eq. (21). The dotted lines mark the error corresponding to the error in the PDF. (Based on [53].)

The other key point is that the cumulative PDF of mass density is relatively insensitive to the cosmological model or the power spectrum of density fluctuations [4, 5]. We find [53], using a series of N-body simulations of the CDM family of models in a flat or an open universe with and without a tilt in the power spectrum, that, compared to the differences between Cg and C, the latter can always be properly approximated by a cumulative log-normal distribution of 1 + delta with a single parameter sigma. Deviations may show up in the extreme tails of the distribution [5], which may affect the skewness and higher moments but are of little concern for our purpose here. This means that in order to evaluate b(delta) one only needs to measure Cg(g) from a galaxy density field, and add the rms sigma of mass fluctuations at the same smoothing scale. Since the redshift surveys are by far richer and more extended than peculiar-velocity samples, this method will allow a much better handle on b(delta) than the local comparison of density fields of galaxies and mass.

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