# The stochastic background: scaling laws and time to detection for pulsar timing arrays

**Authors:** Xavier Siemens, Justin Ellis, Fredrick Jenet, Joseph D. Romano  
**arXiv:** [1305.3196](https://arxiv.org/abs/1305.3196)  
**Date:** 2013-05-14  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/1305.3196v1/

## Key points

- Siemens et al. investigate scaling laws and time to detection for pulsar timing arrays in relation to the stochastic background
- They challenge the commonly assumed power-law increase of signal-to-noise ratio as a function of observation time (T)
- After a certain amount of time, pulsar timing arrays enter a new regime where signal-to-noise ratio scales only as the square root of T
- Pulsar timing data quality and cadence become relatively unimportant in this regime due to gravitational wave domination at lowest frequencies
- Pulsar timing arrays transition into this regime more quickly than expected, with T=10 years and typical stochastic background amplitudes
- Increasing the number of pulsars in an array is proposed as the most effective strategy for enhancing detectability in this new regime
- Realistic simulations using NANOGrav pulsar timing array data support their claims
- Detection of the stochastic background is possible within a decade and could potentially occur as early as 2016

## Summary

In their study, Siemens et al. investigate the scaling laws and time to detection for pulsar timing arrays in relation to the stochastic background. They derive scaling laws for the signal-to-noise ratio of the optimal cross-correlation statistic and challenge the commonly assumed power-law increase of the signal-to-noise ratio as a function of observation time (T). Their findings reveal that this power-law increase only holds true at early times. After a certain amount of time has elapsed, pulsar timing arrays enter a new regime where the signal-to-noise ratio scales only as the square root of T.

Furthermore, they demonstrate that in this regime, the quality of pulsar timing data and cadence become relatively unimportant due to gravitational wave domination at the lowest frequencies of pulsar timing residuals. Interestingly, pulsar timing arrays transition into this regime more quickly than initially expected. For observations with T=10 years and typical stochastic background amplitudes, pulsars with residual root mean square values below approximately 1 microsecond are already operating within this regime. To enhance detectability in this new regime, Siemens et al. propose increasing the number of pulsars in an array as the most effective strategy. To support their claims, they conduct realistic simulations using NANOGrav (North American Nanohertz Observatory for Gravitational Waves) pulsar timing array data. Through an aggressive survey campaign that regularly adds new millisecond pulsars to its array, they demonstrate that detection of the stochastic background is possible within a decade and could potentially occur as early as 2016. Overall, their study provides valuable insights into understanding the behavior and detectability of gravitational waves using pulsar timing arrays; it highlights important considerations regarding observation time, signal-to-noise ratio scaling laws and strategies for maximizing detection capabilities in different regimes.