Tracing the Redshift Evolution of Hubble Parameter with Gravitational-Wave Standard Sirens

Proposed space-based gravitational-wave detectors such as BBO and DECIGO can detect ~ 106 neutron star (NS) binaries and determine the luminosity distance to the binaries with high precision. Combining the luminosity distance and electromagnetically derived redshift, one would be able to probe cosmological expansion out to high redshift. In this paper, we show that the Hubble parameter as a function of redshift can be directly measured with monopole and dipole components of the luminosity distance on the sky. As a result, the measurement accuracies of the Hubble parameter in each redshift bin up to z=1 are 3-14%, 1.5-8%, and 0.8-4% for the observation time 1 yr, 3 yr, and 10 yr, respectively

All-order evaluation of weak measurements: --- The cases of an operator A{\bf A} which satisfies the property A2=1{\bf A}^{2}=1 ---

Some exact formulae of the expectation values and probability densities in a weak measurement for an operator ${\bf A}$ which satisfies the property ${\bf A}^{2}=1$ are derived. These formulae include all-order effects of the unitary evolution due to the von-Neumann interaction. These are valid not only in the weak measurement regime but also in the strong measurement regime and tell us the connection between these two regime. Using these formulae, arguments of the optimization of the signal amplification and the signal to noise ratio are developed in two typical experimental setups.Comment: 17 pages, 10 figures (v1); Fig.3 and some typos are corrected (v2); Comments and references are added and some typos are corrected (v3