Milky Way Structure from Double White Dwarf Gravitational-Wave Sources

The millihertz gravitational-wave sky will be dominated by $\sim {10}^4$–${10}^7$ resolved double white dwarf binaries in the Milky Way, whose three-dimensional spatial distribution traces the Galaxy’s structural parameters. We present $\text{textsc}Galena$, a hierarchical Bayesian pipeline that recovers these parameters from a double white dwarf catalogue through an inhomogeneous Poisson-process likelihood. Each source’s $3\times 3$ Galactic position covariance is computed from the waveform Fisher information matrix and propagated analytically through a $3\times 5$ Jacobian; the resulting measurement-error convolution is factored into a pre-computed sparse weight matrix, rendering the nested-sampling inference tractable. Applied to GBSIEVER-reported LISA Data Challenge sources ($N=2151$ after quality cuts), $\text{textsc}Galena$ recovers the disk scale length $R_d=2175_{-52}^{+51}$~pc and scale height $z_d=282_{-7}^{+7}$~pc, consistent with a canonical thin disk, together with the bulge fraction $A=0.187_{-0.012}^{+0.011}$ and bulge scale radius $R_b=773_{-25}^{+26}$~pc; the bulge fraction, now recovered much closer to the literature value of $0.25$ than in our earlier exact-position fit, is measured at $\sim 6\mathrm{\%}$ statistical precision. An independent particle-swarm optimisation of the same likelihood reproduces these values to within a fraction of a percent, supporting the attribution of the improvement to the Fisher-matrix error propagation rather than to the sampler. The search-stage astrophysical prior improves per-source distance estimates but leaves the hierarchical inference essentially unchanged.