Cosmological Constraints on the Very Low Frequency Gravitational-Wave Background

While there are no strong observational constraints on the gravitational wave background across six or more orders of magnitude between 10^{-16} Hz and 10^{-10} Hz and it is difficult to get a constraint below 10^{-12}Hz using objects in our Galaxy, we suggest that the anisotropy pattern of time variation of the redshift related to a sample of high redshift objects can be used to constrain the gravitational wave background around 10^{-12} Hz. Useful observations for the monitoring of an anisotropy signal in a global redshift change include spectroscopic observations of the Ly-alpha forest in absorption towards a sample of quasars, redshifted 21 cm line observations either in absorption or emission towards a sample of neutral HI regions before or during reionization, and high frequency (0.1 Hz to 1 Hz) gravitational wave analysis of a sample of neutron star--neutron star binaries detected with gravitational wave instruments such as the Decihertz Interferometer Gravitational Wave Observatory (DECIGO). The low frequency background can also be constrained by arcsecond-scale anisotropy observations of the CMB. For reasonable observations in the future involving extragalactic sources, we find best limits at the level of \Omega_{GW} < 10^{-5} at a frequency around 10^{-12} Hz while the eventual ultimate limit one cannot beat is \Omega_{GW} < 10^{-11}.