Unified Lagrangian Framework for Galaxy Clustering: Consistent Modeling of Bias, Redshift-Space Distortions, and Reconstruction

We present \emph{Unified Lagrangian Perturbation Theory} (ULPT), a perturbative framework for consistently modeling galaxy density fluctuations across real space, redshift space, and post-reconstruction fields. Unlike existing approaches that treat these cases separately, ULPT provides a single theoretical structure that incorporates the three essential coordinate mappings: the Lagrangian-to-Eulerian transformation, the real-to-redshift mapping induced by peculiar velocities, and the remapping from pre to post reconstruction. A key feature of our formulation is the explicit decomposition of the density field into two physically distinct components: the \emph{Jacobian deviation}, which encodes intrinsic linear and nonlinear growth, and the \emph{displacement-mapping effect}, which captures large-scale convective distortions. This separation enables a fully analytic and infrared (IR)-safe resummation, ensuring exact IR cancellation, a consistent Gaussian description of baryon acoustic oscillation (BAO) damping, and the correct residual structure in cross spectra between fields with distinct IR behavior. The perturbative expansion of ULPT naturally generates Galileon-type operators, thereby providing a compact and physically motivated operator basis for nonlinear and nonlocal Lagrangian bias, and allowing for a renormalization-free treatment of biased tracers. Within this framework, we derive a unified expression for the power spectrum that applies equally to dark matter, biased tracers, redshift-space distortions, and reconstructed fields. ULPT thus offers a robust and extensible foundation for precision modeling of large-scale structure, with potential extensions to higher-order statistics, such as the bispectrum, and to other two-point observables, such as galaxy--galaxy lensing.