Cosmological constraints on the very low frequency gravitational-wave background

Abstract

The curl modes of cosmic microwave background polarization allow one to indirectly constrain the primordial background of gravitational waves with frequencies around 10^(-18) to 10^(-16) Hz. The proposed high precision timing observations of a large sample of millisecond pulsars with the pulsar timing array or with the square kilometer array can either detect or constrain the stochastic gravitational-wave background at frequencies greater than roughly 0.1 yr^(-1). While existing techniques are limited to either observe or constrain the gravitational-wave background across six or more orders of magnitude between 10^(-16) and 10^(-10) Hz, we suggest that the anisotropy pattern of time variation of the redshift related to a sample of high-redshift objects can be used to study the background around a frequency of 10^(-12) Hz. Useful observations to detect an anisotropy signal in the 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 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). For reasonable observations expected in the future involving extragalactic sources, we find limits at the level of Omega(GW)< 10^(-6) at a frequency around 10^(-12) Hz while the ultimate limit is likely to be around Omega(GW)< 10^(-11). On the other hand, if there is a background of gravitational waves at 10^(-12) Hz with an amplitude larger than this limit, its presence will be visible as a measurable anisotropy in the time-evolving redshift of extragalactic sources.