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Gravitationally collapsed objects are known to be biased tracers of anunderlying density contrast. Using symmetry arguments, generalised biasingschemes have recently been developed to relate the halo density contrast$\delta_h$ with the underlying density contrast $\delta$, divergence ofvelocity $\theta$ and their higher-order derivatives. This is done byconstructing invariants such as $s, t, \psi,\eta$. We show how the generatingfunction formalism in Eulerian standard perturbation theory (SPT) can be usedto show that many of the additional terms based on extended Galilean andLifshitz symmetry actually do not make any contribution to the higher-orderstatistics of biased tracers. Other terms can also be drastically simplifiedallowing us to write the vertices associated with $\delta_h$ in terms of thevertices of $\delta$ and $\theta$, the higher-order derivatives and the biascoefficients. We also compute the cumulant correlators (CCs) for two differenttracer populations. These perturbative results are valid for tree-levelcontributions but at an arbitrary order. We also take into account thestochastic nature bias in our analysis. Extending previous results of a localpolynomial model of bias, we express the one-point cumulants ${\cal S}_N$ andtheir two-point counterparts, the CCs i.e. ${\cal C}_{pq}$, of biased tracersin terms of that of their underlying density contrast counterparts. As aby-product of our calculation we also discuss the results using approximationsbased on Lagrangian perturbation theory (LPT).

Symmetries, Invariants and Generating Functions: Higher-order Statistics of Biased Tracers