Searching For Stochastic Gravitational Waves Below a Nanohertz

The stochastic gravitational-wave background is imprinted on the times of arrival of radio pulses from millisecond pulsars. Traditional pulsar timing analyses fit a timing model to each pulsar and search the residuals of the fit for a stationary time correlation. This method breaks down at gravitational-wave frequencies below the inverse observation time of the array; therefore, existing analyses restrict their searches to frequencies above 1 nHz. An effective method to overcome this challenge is to study the correlation of secular drifts of parameters in the pulsar timing model itself. In this paper, we show that timing model correlations are sensitive to sub-nanohertz stochastic gravitational waves and perform a search using existing measurements of pulsar spin-decelerations and pulsar binary orbital decay rates. We do not observe a signal at our present sensitivity, constraining the stochastic gravitational-wave relic energy density to $\Omega_\text{GW} ( f ) < 3.8 \times 10 ^{ - 9} $ at 450~pHz with sensitivity which scales as the frequency squared until approximately 10 pHz. We place additional limits on the amplitude of a power-law spectrum of $A_\star \lesssim 1.8\times10^{-14}$ for a reference frequency of $f_* = 1~{\rm year} ^{-1} $ and the spectral index expected from supermassive black hole binaries, $\gamma = 13/3$. If detection of a supermassive black hole binary signal above 1 nHz is confirmed, this search method will serve as a critical complementary probe of the dynamics of galaxy evolution.