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 percentlevel 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 (∝ v 2 r ) 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 σ, fitting for these effects as additional parameters increases the error bar by less than 30% across a wide range of bias values.
Linear Equations of Motion in Lagrangian SpaceIn the Lagrangian picture, fluid dynamics is encoded in the displacements Ψ σ (q) of fluid elements of each species, σ, originally situated at Lagrangian positions q, such that their Eulerian positions at conformal time τ (dτ = a −1 dt) are given by [28,37,38] x σ (q, τ ) = q + Ψ σ (q, τ ).(2.1)The subscript σ = {c, b} denotes the species, either cold dark matter (CDM) or baryons, respectively, whose motion we are tracking. Assuming that initial displacements are infinitesimally small compared to those at the redshifts of interest, the overdensity, δ σ , of each species at Eulerian