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).