Biased Tracers of Two Fluids in the Lagrangian Picture
  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.
