# Biased Tracers of Two Fluids in the Lagrangian Picture

## TL;DR

This paper develops a Lagrangian perturbation theory framework for biased tracers in a two-fluid universe, quantifies corrections to bias and advection, and assesses their impact on BAO measurements, finding modest biases and manageable uncertainties.

## Contribution

It introduces a novel two-fluid Lagrangian bias expansion, computes linear and nonlinear corrections, and evaluates their effects on BAO analysis.

## Key findings

- Percent-level two-fluid corrections at low redshift.
- Relative velocity effects can bias BAO scale by up to 0.5 sigma.
- Fitting for two-fluid parameters increases BAO measurement errors by less than 30%.

## Abstract

We explore Lagrangian perturbation theory (LPT) for biased tracers in the presence of two fluids, focusing on the case of cold dark matter (CDM) and baryons. The presence of two fluids induces corrections to the Lagrangian bias expansion and tracer advection, both of which we formulate as expansions in the three linear modes of the Lagrangian equations of motion. We compute the linear-order two-fluid corrections in the Zeldovich approximation, finding that modifications to the bias expansion and tracer advection both enter as percent-level corrections over a large range of wavenumbers at low redshift and draw parallels with the Eulerian formalism. We then discuss nonlinear corrections in the two-fluid picture, and calculate contributions from the relative velocity effect ($\propto v_r^2$) at one loop order. Finally, we conduct an exploratory Fisher analysis to assess the impact of two-fluid corrections on baryon acoustic oscillations (BAO) measurements, finding that while modest values of the relative bias parameters can introduce systematic biases in the measured BAO scale of up to $0.5\, \sigma$, fitting for these effects as additional parameters increases the error bar by less than $30\%$ across a wide range of bias values.