Sound-Horizon-Agnostic Forecast for Constraints on $H_0$ and $\mathrm{\Sigma }{m}_{\nu }$ from BAO, CMB Lensing, Supernovae and LSST-like Galaxy Lensing

We present a sound-horizon-agnostic forecast for constraints on the Hubble constant $H_0$ and the sum of the neutrino masses $\mathrm{\Sigma }{m}_{\nu }$ from a combination of LSST-like galaxy weak lensing, Simons-Observatory-like CMB lensing, the completed DESI BAO program, and expanded supernova samples. Treating the sound horizon at baryon drag $r_d$ as a free parameter, the method provides a consistency test of $\mathrm{\Lambda }$CDM without relying on the standard early-Universe calibration of $r_d$. The analysis incorporates realistic galaxy weak-lensing systematics, explicitly marginalizing over photometric-redshift calibration, multiplicative shear biases, intrinsic alignments, and baryonic feedback, and extends the CMB-lensing analysis to $L_{\max }=5000$. For fixed $\mathrm{\Sigma }{m}_{\nu }$, we forecast uncertainties of $\sigma (H_0)\simeq 1.5$ km/s/Mpc with our least informative prior set, $\sigma (H_0)\simeq 1.3$ km/s/Mpc with more restrictive priors, and $\sigma (H_0)\simeq 0.7$ km/s/Mpc with an additional informative prior on the amplitude of primordial fluctuations $A_s$. Allowing $\mathrm{\Sigma }{m}_{\nu }$ to vary weakens the $H_0$ constraints, while still yielding a forecast 95% CL upper limit of about 0.4 eV on $\mathrm{\Sigma }{m}_{\nu }$. We also clarify the scope of the $r_d$-agnostic approach through validation tests using mock data from modified recombination and early dark energy models. The method recovers unbiased constraints on $H_0$ for the former but not the latter, highlighting its dependence on assumptions about early-Universe physics.