Joint constraints on galaxy bias and $\sigma_8$ through the N-pdf of the
  galaxy number density
  We present a full description of the N-probability density function of the
galaxy number density fluctuations. This N-pdf is given in terms, on the one
hand, of the cold dark matter correlations and, on the other hand, of the
galaxy bias parameter. The method relies on the assumption commonly adopted
that the dark matter density fluctuations follow a local non-linear
transformation of the initial energy density perturbations. The N-pdf of the
galaxy number density fluctuations allows for an optimal estimation of the bias
parameter (e.g., via maximum-likelihood estimation, or Bayesian inference if
there exists any a priori information on the bias parameter), and of those
parameters defining the dark matter correlations, in particular its amplitude
($\sigma_8$). It also provides the proper framework to perform model selection
between two competitive hypotheses. The parameters estimation capabilities of
the N-pdf are proved by SDSS-like simulations (both ideal log-normal
simulations and mocks obtained from Las Damas simulations), showing that our
estimator is unbiased. We apply our formalism to the 7th release of the SDSS
main sample (for a volume-limited subset with absolute magnitudes $M_r \leq
-20$). We obtain $\hat{b} = 1.193 \pm 0.074$ and $\hat{\sigma_8} = 0.862 \pm
0.080$, for galaxy number density fluctuations in cells of a size of
$30h^{-1}$Mpc. Different model selection criteria show that galaxy biasing is
clearly favoured.
