Bootstrapping LSS perturbation theory beyond third order

Bootstrap techniques relying on the constraints imposed by Extended Galilean Invariance (EGI), have proved to be very useful in the context of perturbation theory of the Large Scale Structure (LSS). It has been formulated in both the Eulerian as well as Lagrangian space. While the Eulerian bootstrap formalism has been successfully applied to both tracer and matter kernels, the application of bootstrap methods in Lagrangian space has so far been restricted to matter. Up to third order, it has been shown that implementing EGI constraints in Eulerian space fully reproduces the bias expansion for tracers. Previous studies have demonstrated that time non-locality affects the bias expansion in a non-trivial way starting from fifth order. Motivated by this fact, we extend the bootstrap approach upto fifth order in both Eulerian and Lagrangian space and demonstrate that it fully captures the time non-local effects. For this, we generalize the Lagrangian bootstrap for tracers, and found that it agrees with the corresponding results obtained in Eulerian space. One of the major challenges in implementing EGI constraints in Eulerian space, is to systematically find out all the "spurious poles" and make them vanish. We have proposed a method that bypasses this difficulty making the procedure tractable at higher orders. From Lagrangian perspective, we have identified coefficients in the tracer kernel whose ratios are independent of tracer properties and may serve as direct probes of the underlying cosmology.