Bias, redshift space distortions and primordial nongaussianity of
  nonlinear transformations: application to Lyman alpha forest
  On large scales a nonlinear transformation of matter density field can be
viewed as a biased tracer of the density field itself. A nonlinear
transformation also modifies the redshift space distortions in the same limit,
giving rise to a velocity bias. In models with primordial nongaussianity a
nonlinear transformation generates a scale dependent bias on large scales. We
derive analytic expressions for these for a general nonlinear transformation.
These biases can be expressed entirely in terms of the one point distribution
function (PDF) of the final field and the parameters of the transformation. Our
analysis allows one to devise nonlinear transformations with nearly arbitrary
bias properties, which can be used to increase the signal in the large scale
clustering limit. We apply the results to the ionizing equilibrium model of
Lyman-alpha forest, in which Lyman-alpha flux F is related to the density
perturbation delta via a nonlinear transformation. Velocity bias can be
expressed as an average over the Lyman-alpha flux PDF. At z=2.4 we predict the
velocity bias of -0.1, compared to the observed value of -0.13 +/- 0.03. Bias
and primordial nongaussianity bias depend on the parameters of the
transformation. Measurements of bias can thus be used to constrain these
parameters, and for reasonable values of the ionizing background intensity we
can match the predictions to observations. Matching to the observed values we
predict the ratio of primordial nongaussianity bias to bias to have the
opposite sign and lower magnitude than the corresponding values for the highly
biased galaxies, but this depends on the model parameters and can also vanish
or change the sign.
