Cosmological constraints on the very low frequency gravitational-wave background

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 ${10}^{-16}\mathrm{Hz}$. 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 $0.1{\mathrm{\text{yr}}}^{-1}$. 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 ${10}^{-10}\mathrm{Hz}$, 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 ${10}^{-12}\mathrm{Hz}$. Useful observations to detect an anisotropy signal in the global redshift change include spectroscopic observations of the $\mathrm{Ly}\mathrm{\text{−}}\alpha$ 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 ${\Omega }_{\mathrm{GW}}<{10}^{-6}$ at a frequency around ${10}^{-12}\mathrm{Hz}$ while the ultimate limit is likely to be around ${\Omega }_{\mathrm{GW}}<{10}^{-11}$. On the other hand, if there is a background of gravitational waves at ${10}^{-12}\mathrm{Hz}$ with an amplitude larger than this limit, its presence will be visible as a measurable anisotropy in the time-evolving redshift of extragalactic sources.

Figure 1

Sensitivities for various current (thin lines) and future (thick lines) measurements. Curves labeled “WMAP” and “CMBpol” show the current and future constraints based on CMB anisotropies at large angular scales. The curve labeled “binary psr” is the limit based on a sample of binary pulsars, based on the time variation of their orbits, the curve labeled “msp” is the current limit based on a 17-year timing monitoring of 2 ms pulsars, while the curve labeled “PTA” is the limit based on proposed pulsar timing array network. The limit based on “QSO astrometry” involves proper motion observations of a sample of radio bright quasars while “redshift variation” involves time monitoring of redshift variations in the Lyman-$\alpha$ forest towards bright quasars, the 21 cm line at very low radio frequencies, and cosmological neutron star binaries with ultimate DECIGO. The curve labeled “21cm-ultimate” is the eventual ultimate limit for method with 21 cm line. The dotted line with “inflation” is the amplitude of the stochastic background constrained by WMAP and corresponds to the energy scale $3.4\times {10}^{16}\mathrm{GeV}$ [39]. The line with “MBH” is the estimated background made by merging massive black hole binaries given in 11.

Figure 2

The angular power spectrum $C_l$ in time-dependent redshift variations due to a low frequency gravitational-wave background with ${\Omega }_{\mathrm{GW}}={10}^{-4}$. The error bars assume the case when we can measure the monopole mode $R$ with sensitivity $4\times {10}^{-3}$ in a shell of sources center in redshift at $z=1$.