Directly inferring cosmology and the neutron-star equation of state from gravitational-wave mergers

Upgrades to existing gravitational-wave observatories have the potential to simultaneously constrain the nuclear equation of state and Hubble's constant $H_0$ to percent level with merging neutron star binaries. In practice, performing simultaneous inference of $H_0$ and the equation of state is limited computationally by the requirement to solve the equations of general-relativistic hydrostatic equilibrium millions of times. We develop a machine-learning model to solve the Tolman-Oppenheimer-Volkoff equations in less than a millisecond, and demonstrate its utility by performing direct inference of both equation of state and Hubble's constant for synthetic neutron star merger signals with LIGO-Virgo-KAGRA operating at A+ sensitivities. We show that a population of fifteen mergers observed with A+ allows for the radius of a $1.4\,M_{\odot}$ neutron star and $H_0$ to be constrained to $R_{1.4} = 11.74^{+0.35}_{-0.28}$ km and $H_0 = 68^{+17}_{-13} \rm \ km \ s^{-1} \ Mpc^{-1}$, at 90% credible interval and 68% credible interval respectively. These constraints utilise only the gravitational-wave information to infer cosmological parameters; such numbers will be further improved with the addition of electromagnetic counterparts and/or galaxy catalogues.