Sensitivity Curves for Searches for Gravitational-Wave Backgrounds

Sensitivity curves for searches for gravitational-wave backgrounds Eric Thrane1, a and Joseph D. Romano2, b 1LIGO Laboratory, California Institute of Technology, MS 100-36, Pasadena, California 91125, USA 2Department of Physics and Astronomy and Center for Gravitational-Wave Astronomy, University of Texas at Brownsville, Texas 78520, USA (Dated: December 4, 2013) 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. I. INTRODUCTION When discussing the feasibility of detecting gravitational waves using current or planned detectors, one often plots characteristic strain hc(f) curves of predicted signals (defined below in Eq. 5), and compares them to sensitivity curves for different detectors. The sensitivity curves are usually constructed by taking the ratio of the detector’s noise power spectral density Pn(f) to its sky- and polarization-averaged response to a gravitational wave (f), defining Sn(f) Pn(f)/ (f) and an effective characteristicR strain noise≡ amplitudeR h (f) n ≡ fSn(f). If the curve corresponding to a predicted signal hc(f) lies above the detector sensitivity curve hn(f) inp some frequency band, then the signal has signal-to- noise ratio >1. An example of such a plot is shown in Fig. 1, which is taken from [1]. FIG. 1: Sensitivity curves for gravitational-wave observa- For stochastic gravitational waves, which are typically tions and the predicted spectra of various gravitational-wave searched for by cross-correlating data from two or more sources, taken from [1]. detectors, one often adjusts the height of a sensitivity curve to take into account the total observation time (e.g., T = 1 yr or 5 yr). For uncorrelated detector noise, the expected (power) signal-to-noise ratio of a cross- broadband nature of the signal. The integrated signal- correlation search for a gravitational-wave background to-noise ratio ρ (see Eq. 21) also scales like √Nbins = for frequencies between f and f +δf scales like √Tδf. So ∆f/δf, where Nbins is the number of frequency bins the effective characteristic strain noise amplitude hn(f) of width δf in the total bandwidth ∆f. As we shall should be multiplied by a factor of 1/(Tδf)1/4. Also, seep below, the actual value of the proportionality con- instead of characteristic strain, one often plots the pre- stant depends on the spectral shape of the background dicted fractional energy density in gravitational waves and on the detector geometry (e.g., the separation and Ωgw(f) as a function of frequency, which is proportional relative orientation of the detectors), in addition to the 2 2 to f hc(f) (see Eq. 6). An example of such a plot is individual detector noise power spectral densities. Since shown in Fig. 2, which is taken from [2]. this improvement to the sensitivity is signal dependent, it But for stochastic gravitational waves, plots such as is not always folded into the detector sensitivity curves, Figs. 1 and 2 do not always tell the full story. Searches even though the improvement in sensitivity can be sig- for gravitational-wave backgrounds also benefit from the nificant.1 And when it is folded in, as in Fig 2, a single aElectronic address: [email protected] 1 To be clear, integration over frequency is always carried out bElectronic address: [email protected] in searches for stochastic gravitational-wave backgrounds, even 2 −2 −4 10 10 LIGO S4 CMB & Matter BBN Spectra −6 10 Planck −4 LIGO S5 10 Pulsar −8 Cosmic Strings 10 Limit −6 gw (f) Ω AdvLIGO 10 −10 Ω 10 LISA CMB Large Pre−Big−Bang −12 Angle −8 10 10 Inflation −14 10 −10 10 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 1 2 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) f (Hz) FIG. 2: Plot showing strengths of predicted gravitational- FIG. 3: Ωgw(f) sensitivity curves from different stages in a po- wave backgrounds in terms of Ωgw(f) and the corresponding tential future Advanced LIGO Hanford-LIGO Livingston cor- sensitivity curves for different detectors, taken from [2]. Up- relation search for power-law gravitational-wave backgrounds. per limits from various measurements, e.g., S5 LIGO Hanford- The top black curve is the single-detector sensitivity curve, as- Livingston and pulsar timing, are shown as horizontal lines sumed to be the same for both H1 or L1. The red curve shows in the analysis band of each detector. The upper limits take the sensitivity of the H1L1 detector pair to a gravitational- into account integration over frequency, but only for a single wave background, where the spikes are due to zeros in the spectral index. Hanford-Livingston overlap reduction function (see left panel, Fig. 5). 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 spectral index is assumed, making it difficult to compare obtained by integrating over frequency for different power law published limits with arbitrary models. In other cases, indices, assuming a signal-to-noise ratio ρ = 1. Finally, the limits are given as a function of spectral index, but the blue power-law integrated sensitivity curve is the envelope of constrained quantity depends on an arbitrary reference the black lines. See Sec. III, Fig. 7 for more details. frequency; see Eq. 7. To illustrate the improvement in sensitivity that comes from integrating over frequency, consider the simple case and interpretation of these curves will be given in Sec III, of a white gravitational-wave background signal in white Fig. 7. We show this figure now for readers who might uncorrelated detector noise. In this case, ρ increases by be anxious to get to the punchline. precisely √Nbins compared to the single bin analysis. For In Sec. II, we briefly review the fundamentals of cross- ground-based detectors like LIGO, typical values2 of ∆f correlation searches for gravitational-wave backgrounds, and δf are ∆f 100 Hz and δf 0.25 Hz, leading to defining an effective strain noise power spectral density ≈ ≈ Nbins 400, and a corresponding improvement in ρ of Seff (f) for a network of detectors. For simplicity, we about≈ 20; see, e.g., [2]. For colored spectra and non- consider cross-correlation searches for unpolarized and trivial detector geometry the improvement will be less, isotropic stochastic backgrounds using two or more de- but a factor of 5-10 increase in ρ is not unrealistic. tectors. In Sec. III we present a graphical method for con- ∼ In this paper, we propose a relatively simple way to structing sensitivity curves for power-law backgrounds graphically represent this improvement in sensitivity for based on the expected signal-to-noise ratio for the search, gravitational-wave backgrounds that have a power-law and we apply our method to construct new power-law in- frequency dependence in the sensitivity band of the de- tegrated sensitivity curves for correlation measurements tectors. An example of such a “power-law integrated involving second-generation ground-based detectors such sensitivity curve” is given in Fig. 3 for a correlation mea- as Advanced LIGO, space-based detectors such as the Big surement between the Advanced LIGO detectors in Han- Bang Observer (BBO), and a pulsar timing array. For ford, WA and Livingston, LA. Details of the construction completeness, we also construct a power-law integrated sensitivity curve for an autocorrelation measurement using LISA. We conclude with a brief discussion in Sec. IV. though this is not always depicted in sensitivity curves. 2 The 0.25 Hz bin width typical of LIGO stochastic analyses is II. FORMALISM chosen to be sufficiently narrow that one can approximate the signal and noise as constant across the width of the bin, yet sufficiently wide that the noise can be approximated as stationary In this section, we summarize the fundamental prop- over the duration of the data segment. erties of a stochastic background and the correlated re- 3 sponse of a network of detectors to such a background. In power-law spectra: order to keep track of the many different variables neces- sary for this discussion, we have included Table I, which f β Ω (f)=Ω , (7) summarizes key variables. gw β f ref where β is the spectral index and fref is a reference fre- A. Statistical properties quency, typically set to 1 yr−1 for pulsar-timing observa- tions and 100 Hz for ground-based detectors. The choice In transverse-traceless coordinates, the metric pertur- of fref, however, is arbitrary and does not affect the de- bations hab(t, ~x) corresponding to a gravitational-wave tectability of the signal. background can be written as a linear superposition of It follows trivially that the characteristic strain also sinusoidal plane gravitational waves with frequency f, has a power-law form: propagation direction kˆ, and polarization A: f α hc(f)= Aα , (8) hab(t, ~x)= fref ∞ 2 ˆ A ˆ i2πf(t−kˆ·~x/c) (1) df d Ωkˆ hA(f, k)eab(k) e , where the amplitude Aα and spectral index α are related −∞ 2 S A to Ω and β via: Z Z X β where eA (kˆ) are the gravitational-wave polarization ten- 2 ab 2π 2 2 sors and A =+, (see e.g., [3]).

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