Symmetries, Invariants and Generating Functions: Higher-order Statistics
  of Biased Tracers
  Gravitationally collapsed objects are known to be biased tracers of an
underlying density contrast. Using symmetry arguments, generalised biasing
schemes have recently been developed to relate the halo density contrast
$\delta_h$ with the underlying density contrast $\delta$, divergence of
velocity $\theta$ and their higher-order derivatives. This is done by
constructing invariants such as $s, t, \psi,\eta$. We show how the generating
function formalism in Eulerian standard perturbation theory (SPT) can be used
to show that many of the additional terms based on extended Galilean and
Lifshitz symmetry actually do not make any contribution to the higher-order
statistics of biased tracers. Other terms can also be drastically simplified
allowing us to write the vertices associated with $\delta_h$ in terms of the
vertices of $\delta$ and $\theta$, the higher-order derivatives and the bias
coefficients. We also compute the cumulant correlators (CCs) for two different
tracer populations. These perturbative results are valid for tree-level
contributions but at an arbitrary order. We also take into account the
stochastic nature bias in our analysis. Extending previous results of a local
polynomial model of bias, we express the one-point cumulants ${\cal S}_N$ and
their two-point counterparts, the CCs i.e. ${\cal C}_{pq}$, of biased tracers
in terms of that of their underlying density contrast counterparts. As a
by-product of our calculation we also discuss the results using approximations
based on Lagrangian perturbation theory (LPT).
