Sensitivity curves for searches for gravitational-wave backgrounds

TL;DR

This work introduces power-law integrated sensitivity curves to visualize detector sensitivity to stochastic gravitational-wave backgrounds while incorporating broadband integration over frequency. It formalizes the cross-correlation framework, defines an effective strain spectrum $S_{eff}(f)$ and energy-density spectrum $\Omega_{gw}(f)$ for detector networks, and prescribes a practical construction to obtain envelopes that account for power-law spectra. The authors demonstrate the method by deriving PI curves for Advanced LIGO networks, BBO, LISA (autocorrelation), and a pulsar timing array, highlighting how frequency integration enhances detectability beyond time integration alone. The paper also provides public code and a clear interpretive framework, enabling researchers to compare models against detector capabilities and to visualize upper limits in a broadband, model-agnostic manner.

Abstract

We propose a graphical representation of detector sensitivity curves for stochastic gravitational-wave backgrounds that takes into account the increase in sensitivity that comes from integrating over frequency in addition to integrating over time. This method is valid for backgrounds that have a power-law spectrum in the analysis band. We call these graphs "power-law integrated curves." For simplicity, we consider cross-correlation searches for unpolarized and isotropic stochastic backgrounds using two or more detectors. We apply our method to construct power-law integrated sensitivity curves for second-generation ground-based detectors such as Advanced LIGO, space-based detectors such as LISA and the Big Bang Observer, and timing residuals from a pulsar timing array. The code used to produce these plots is available at https://dcc.ligo.org/LIGO-P1300115/public for researchers interested in constructing similar sensitivity curves.

Figures (11)

• Figure 1: Sensitivity curves for gravitational-wave observations and the predicted spectra of various gravitational-wave sources, taken from hobbs.
• Figure 2: Plot showing strengths of predicted gravitational-wave backgrounds in terms of $\Omega_{\rm gw}(f)$ and the corresponding sensitivity curves for different detectors, taken from stoch-S5. Upper limits from various measurements, e.g., S5 LIGO Hanford-Livingston and pulsar timing, are shown as horizontal lines in the analysis band of each detector. The upper limits take into account integration over frequency, but only for a single spectral index.
• Figure 3: $\Omega_{\rm gw}(f)$ sensitivity curves from different stages in a potential future Advanced LIGO Hanford-LIGO Livingston correlation search for power-law gravitational-wave backgrounds. The top black curve is the single-detector sensitivity curve, assumed to be the same for both H1 or L1. The red curve shows the sensitivity of the H1L1 detector pair to a gravitational-wave background, where the spikes are due to zeros in the Hanford-Livingston overlap reduction function (see left panel, Fig. \ref{['f:LIGO-BBO']}). The green curve shows the improvement in sensitivity that comes from integration over an observation time of 1 year for a frequency bin size of $0.25$ Hz. The set of black lines are obtained by integrating over frequency for different power law indices, assuming a signal-to-noise ratio $\rho =1$. Finally, the blue power-law integrated sensitivity curve is the envelope of the black lines. See Sec. \ref{['s:sensitivity']}, Fig. \ref{['f:graphical_construction']} for more details.
• Figure 4: A plot of the transfer function ${\cal R}_I(f)=\gamma_{II}(f)$ normalized to unity for the strain response of an equal-arm Michelson interferometer. The dips in the transfer function occur around integer multiples of $c/(2L)$, where $L$ is the arm length of the interferometer.
• Figure 5: Left panel: Normalized overlap reduction function for the LIGO detectors located in Hanford, WA and Livingston, LA. Right panel: Normalized overlap reduction function for two mini LISA-like Michelson interferometers located at opposite vertices of the BBO hexagram configuration.