Gaia 400,894 QSO constraint on the energy density of low-frequency gravitational waves

Low frequency gravitational waves (GWs) are keys to understanding cosmological inflation and super massive blackhole (SMBH) formation via blackhole mergers, while it is difficult to identify the low frequency GWs with ground-based GW experiments such as the advanced LIGO (aLIGO) and VIRGO due to the seismic noise. Although quasi-stellar object (QSO) proper motions produced by the low frequency GWs are measured by pioneering studies of very long baseline interferometry (VLBI) observations with good positional accuracy, the low frequency GWs are not strongly constrained by the small statistics with 711 QSOs (Darling et al. 2018). Here we present the proper motion field map of 400,894 QSOs of the Sloan Digital Sky Survey (SDSS) with optical {\it Gaia} EDR3 proper motion measurements whose positional accuracy is $< 0.4$ milli-arcsec comparable with the one of the radio VLBI observations. We obtain the best-fit spherical harmonics with the typical field strength of $\mathcal{O}(0.1)\, \mu$arcsec, and place a tight constraint on the energy density of GWs, $\Omega_{\rm gw}=(0.964 \pm 3.804) \times 10^{-4}$ (95 \% confidence level), that is significantly stronger than the one of the previous VLBI study by two orders of magnitude at the low frequency regime of $f <10^{-9}\,{\rm [Hz]}\simeq (30\,{\rm yr})^{-1}$ unexplored by the pulsar timing technique. Our upper limit rules out the existence of SMBH binary systems at the distance $r < 400$ kpc from the Earth where the Milky Way center and local group galaxies are included. Demonstrating the limit given by our optical QSO study, we claim that astrometric satellite data including the forthcoming {\it Gaia} DR5 data with small systematic errors are powerful to constrain low frequency GWs.