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 $\sim {10}^6$ 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.

Figure 1

Ratio of $d_L^{(1)}(z)$ to $d_L^{(0)}(z)$, in which the CMB dipole of $|v_0|=369.1\pm 0.9\mathrm{km}/\sec$ measured by Wilkinson Microwave Anisotropy Probe (WMAP) [16] is used.

Figure 2

Sky-averaged DECIGO noise curve. (Arm angle 60° is taken into account.) Diagonal lines represent frequency evolutions of an NS-NS binary at $z=5$ (solid red line), $z=1$ (dotted green line), and $z=0.1$ (dashed blue line). Diamonds on the lines from the right to the left denote the frequency of the binary 1 yr, 3 yr, and 10 yr before the merger.

Figure 3

Measurement accuracy of the luminosity distance with a single binary as a function of redshifts. The curves tagged ${\sigma }_{\mathrm{inst}}$ are those determined only by instrumental noise and with the observation time 1 yr (solid red curve), 3 yr (dotted green curve), and 10 yr (short-dashed blue curve), respectively. The lensing error and the peculiar-velocity error are represented by magenta (long-dashed) and light blue (dash-dotted) curves.

Figure 4

Number of NS-NS binaries (in the unit of ${10}^4$) that would be observed by DECIGO in each redshift bin of $\Delta z=0.1$ at a redshift $z$ during 3 yr observation. As is manifest from Eq. (18), the number of the binaries scales linearly with $T_{\mathrm{obs}}$.

Figure 5

The Hubble parameter calculated with our fiducial cosmological parameters (solid curve) and $1\sigma$-error bars estimated in the cases that we use all binaries observed by DECIGO during the observation time, 1 yr (red), 3 yr (green), and 10 yr (blue). Long observation time corresponds to the smaller error bar.

Figure 6

Measurement accuracy of the Hubble parameter with all observed binaries. We plot the measurement accuracies, including the lensing error, with the observation time $T_{\mathrm{obs}}=1\mathrm{yr}$, 3 yr, and 10 yr from the top to the bottom, respectively (solid curves), and those without the lensing error (dotted curves).