# Constraints on ultra-low-frequency gravitational waves with statistics of pulsar spin-down rates

## TL;DR

This study uses pulsar spin-down rate statistics to set new upper limits on ultra-low-frequency gravitational waves in a frequency range inaccessible to traditional methods, providing constraints on super-massive black hole binaries.

## Contribution

It introduces a novel statistical method analyzing pulsar spin-down rates to constrain ultra-low-frequency GWs, extending the accessible frequency range beyond conventional pulsar timing arrays.

## Key findings

- Upper bounds on GW amplitude derivatives in the Galactic Center and M87 directions.
- Derived constraints on GW strain amplitude at frequencies around 1/(100 years).
- Implications for the existence of super-massive black hole binaries.

## Abstract

We probe ultra-low-frequency gravitational waves (GWs) with statistics of spin-down rates of milli-second pulsars (MSPs) by a method proposed in our prevous work (Yonemaru et al. 2016). The considered frequency range is $10^{-12}{\rm Hz} \lesssim f_{\rm GW} \lesssim 10^{-10}$Hz, which cannot be accessed by the conventional pulsar timing array. The effect of such low-frequency GWs appears as a bias to spin-down rates which has a quadrupole pattern in the sky. We use the skewness of the spin-down rate distribution and the number of MSPs with negative spin-down rates to search for the bias induced by GWs. Applying this method to 149 MSPs selected from the ATNF pulsar catalog, we derive upper bounds on the time derivative of the GW amplitudes of $\dot{h} < 6.2 \times 10^{-18}~{\rm sec}^{-1}$ and $\dot{h} < 8.1 \times 10^{-18}~{\rm sec}^{-1}$ in the directions of the Galactic Center and M87, respectively. Approximating the GW amplitude as $\dot{h} \sim 2 \pi f_{\rm GW} h$, the bounds translate into $h < 3 \times 10^{-9}$ and $h < 4 \times 10^{-9}$, respectively, for $f_{\rm GW} = 1/(100~{\rm yr})$. Finally, we give the implications to possible super-massive black hole binaries at these sites.