B P Abbott

Upper Limits on the Stochastic Gravitational-Wave Background from Advanced LIGO's First Observing Run

A wide variety of astrophysical and cosmological sources are expected to contribute to a stochastic gravitational-wave background. Following the observations of GW150914 and GW151226, the rate and mass of coalescing binary black holes appear to be greater than many previous expectations. As a result, the stochastic background from unresolved compact binary coalescences is expected to be particularly loud. We perform a search for the isotropic stochastic gravitational-wave background using data from Advanced LIGO's first observing run. The data display no evidence of a stochastic gravitational-wave signal. We constrain the dimensionless energy density of gravitational waves to be Ω0 < 1.7 × 10 -7 with 95% confidence, assuming a flat energy density spectrum in the most sensitive part of the LIGO band (20 -86 Hz). This is a factor of ∼33 times more sensitive than previous measurements. We also constrain arbitrary power-law spectra. Finally, we investigate the implications of this search for the background of binary black holes using an astrophysical model for the background.

Introduction.-Many astrophysical and cosmological phenomena are expected to contribute to a stochastic gravitational-wave background, henceforth, simply refered to as a "background". These include unresolved compact binary coalescences of both black holes and neutron stars [1][2][3][4][5], rotating neutron stars [6][7][8], supernovae [9][10][11][12], cosmic strings [13][14][15][16], inflationary models [17][18][19][20][21][22][23][24], phase transitions [25][26][27], and the pre-Big Bang scenario [28][29][30][31]. The variety of mechanisms potentially contributing to the background provides the opportunity to study a number of different environments within the Universe.

The recent detections of binary black hole (BBH) coalescences by Advanced LIGO [32,33] suggest that the Universe may be rich with coalescing BBHs. While events like GW150914 and GW151226 are loud enough to be clearly detected, we expect there to be many more events that are too far away to be individually resolved and that contribute to the background. Since this BBH population originates from sources that are too distant to be individually detected, the stochastic search probes a distinct population of binaries compared to nearby sources [34]. The background from these binaries provides complementary information to individually resolved binary coalescences [35].

In this Letter, we report on the search for an isotropic background using data from Advanced LIGO's first observing run O1. We search for the background by crosscorrelating data streams from the two separate LIGO detectors and looking for a coherent signal. We find no evidence for the background and place the best upper limits to date on the energy density of the background in the LIGO frequency band. We also update the impli- * Full author list given at the end of the article. cations for a BBH background using all the data from O1.

Data.-Before this analysis, the best limits on the background from Initial LIGO and Virgo data were obtained using 2009-2010 [36] and 2005-2007 data [37]. In this work we use data from the upgraded Advanced LIGO observatories in Hanford, WA (H1) and Livingston, LA (L1) [38]. We analyze O1 data from September 18, 2015 15:00 UTC-January 12, 2016 16:00 UTC.

Method.-We define the background energy density spectrum as [39] Ω

GW (f ) = f ρ c dρ GW df , (1)
where f is the frequency, ρ c = 3c 2 H 2 0 /(8πG) is the critical energy density to close the Universe (numerically, ρ c = 7.8 × 10 -9 erg/cm 3 using the Hubble constant H 0 = 68 km s -1 Mpc -1 from [40,41]), and dρ GW is the gravitational-wave energy density in the frequency range from f to f + df . For the LIGO frequency band, most theoretical models for Ω GW (f ) can be approximated as a power law in frequency [39,42,43]:

Ω GW (f ) = Ω α f f ref α . (2)
Following [35], we assume a reference frequency of 25 Hz, which corresponds to the most sensitive band of the LIGO stochastic search for a detector network operating at design sensitivity. The variable Ω α characterizes the background amplitude across the sensitive frequency band. Past analyses have used α = 0 and α = 3 to represent cosmologically and astrophysically motivated background models respectively [36,[42][43][44][45]. In this analysis we use these two spectral indices but also include limits on the background spectrum assuming α = 2/3, which describes the background dominated by compact binary inspirals [35,46]. This choice of spectral index is especially interesting given the loud background from BBHs inferred from recent Advanced LIGO detections in O1 [32,33,35,47].

Our search uses a cross-correlation method optimized to search for the background using the pair of LIGO detectors [39]. As discussed for instance in [START_REF] Romano | Living Reviews In Relativity[END_REF], crosscorrelation is preferred to auto-correlation methods because the noise variances in each detector are not known sufficiently well to perform subtraction of the noise autopower. We define the estimator

Ŷα = ∞ -∞ df ∞ -∞ df δ T (f -f )s * 1 (f )s 2 (f ) Qα (f ) (3)
with variance

σ 2 Y ≈ T 2 ∞ 0 df P 1 (f )P 2 (f )| Qα (f )| 2 , (4)
where s1,2 (f ) are the Fourier transforms of the strain time series data from the two detectors, δ T (f -f ) is a finite-time approximation to the Dirac delta function, T is the observation time, P 1,2 are the one-sided power spectral densities for the detectors, and Qα (f ) is a filter function to optimize the search [49],

Qα (f ) = λ α γ(f )H 2 0 f 3 P 1 (f )P 2 (f ) f f ref α . (5)
The spatial separation and relative orientation of the two detectors are accounted for in the overlap reduction function, γ(f ) [START_REF] Christensen | [END_REF] and the normalization constant λ α is chosen such that Ŷα = Ω α . Data Quality.-For this analysis, the strain time series data are down-sampled to 4096 Hz from 16384 Hz and separated into 50%-overlapping 192 s segments, as in [42]. The segments are Hann-windowed and high-pass filtered with a 16 th order Butterworth digital filter with knee frequency of 11 Hz. The data are coarse-grained to a frequency resolution of 0.031 Hz. This is a finer frequency resolution than was used in previous analyses due to the need to remove many finely spaced lines at low frequencies.

We apply cuts in the time and frequency domains, following [36]. The total live time after all time domain vetoes have been applied was 29.85 days. These cuts remove 35% of the time-series data. The frequency domain cuts remove 21% of the observing band. In the Supplementary Matrial [START_REF] Abbott | LIGO Scientific Collaboration and[END_REF], which includes Refs. [START_REF] Abbot | [END_REF][53][54][55], we discuss in more detail the removed times and frequencies, the recovery of hardware and software injections, and an analysis of correlated noise due to geophysical Schumann resonances.

Results.-Our search finds no evidence of the background, and the data are consistent with statistical fluctuations, assuming Gaussian noise. The integrand of Frequency (Hz)

Ω 0 Ω 0 ± 2 σ Ω 0 FIG. 1.
We show the estimator for Ω0 in each frequency bin, along with ±2σ error bars, in the frequency band that contains 99% of the sensitivity for α = 0. The loss of sensitivity at around 65 Hz is due to a zero in the overlap reduction function. There are several lines associated with known instrumental artifacts which do not lead to excess cross-correlation.

The data are consistent with Gaussian noise, as described in the Results section.

Equation 3, multiplied by df = 0.031 Hz, gives an estimator for Ω 0 in each frequency bin. We plot this quantity, along with ±2σ error bars, in Figure 1. To check for Gaussianity, we employ a noise model that the estimator in each frequency bin is drawn from a Gaussian distribution with zero mean with the standard deviation of that frequency bin. We obtain a χ 2 per degree of freedom of 0.92, indicating that the data are consistent with Gaussian noise. Consequently, we are able to place upper bounds on the energy density present in the background. For α = 0, we place the bound Ω 0 < 1.7 × 10 -7 at 95% confidence, where 99% of the sensitivity comes in the frequency band 20 -86 Hz. This is a factor of 33 times more sensitive than the previous best limit at these frequencies [36].

Following [56], we show 95% confidence contours in the Ω α -α plane in Figure 2 by computing the joint posterior for Ω α and α. In addition, in Table I, we report upper limits on the energy density for specific fixed values of the spectral index, marginalizing over amplitude calibration uncertainty [57] using the conservative estimates of 11.8% for H1 and 13.4% for L1. Phase calibration uncertainties are negligible.

We also compare our results with the limits placed at high frequencies from the two co-located detectors at the Hanford site (H1 and H2). In [37], the limit Ω 3 < 7.7 × 10 -4 in the frequency band 460 -1000 Hz was obtained for the spectral index α = 3 and f ref = 900 Hz. Using Spectral index α Frequency band with 99% sensitivity Amplitude Ωα 95% CL upper limit Previous limits [ The frequency band with 99% of the sensitivity are shown, along with the point estimate and standard deviation for the amplitude of the background, and 95% confidence level upper limits using O1 data for three values of the spectral index, α = 0, 2/3, 3. We also show the previous upper limits using Initial LIGO-Virgo data.

- 2. Following [56], we present 95 % confidence contours in the Ωα -α plane. The region above these curves is excluded at 95% confidence. We show the constraints coming from the final science run of Initial LIGO-Virgo [36] and from O1 data. Finally, we display the projected (not observed) design sensitivity to Ωα and α for Advanced LIGO and Virgo [58].

this same frequency band, and using the cross-correlated data between the Hanford and Livingston detectors, we place a limit Ω 3 < 1.7 × 10 -2 for f ref = 900 Hz. This is about a factor of 22 larger than the limit from the colocated detectors, in part due to the loss in sensitivity of a stochastic search from cross-correlating detectors at different spatial locations.

In Figure 3, we show the constraints from this analysis and from previous analyses using other detectors, theoretical predictions, and the expected sensitivity of future measurements by LIGO-Virgo and by the Laser Interferometer Space Antenna (LISA). Where applicable, we show constraints using power-law integrated curves (PI curves) [59], which account for the broadband nature of the search by integrating a range of power-law signals over the sensitive frequency band of the detector. By construction, any power-law spectrum which crosses a PI curve is detectable with SNR ≥ 2.

The blue curve labeled 'aLIGO O1' in Figure 3 shows the measured O1 PI curve. We also display the PI curve for the final science run of Initial LIGO and Virgo [36], H1-H2 [37], as well as the projected design sensitivity for the advanced detector network. The curve labeled 'Design' assumes 2 years of co-incident data taken with both Advanced LIGO and Virgo operating at design sensitivity, using the projections in [58]. For the sake of comparison, the measured O1 PI curve at α = 0 is 1.6 times larger than the projected PI curve at α = 0 using the projections in [58] and 29.85 days of live time, which is fairly good agreement between predicted and achieved sensitivity. Finally, in red we present the projected sensitivity of a space-based detector with similar sensitivity to LISA, using the PI curve presented in [59] computed using the projections in [64,65].

We compare these constraints with direct limits from the ringing of Earth's normal modes [63], indirect limits from the Cosmic Microwave Background (CMB) and Big Bang Nucleosynthesis (BBN) [61], and limits from pulsar timing arrays [62] and CMB measurements at low multipole moments [60].

In addition, we give examples of several models which can contribute to the background. We show the background expected from slow-roll inflation with a tensor-toscalar-ratio r = 0.11 (the upper limit allowed by Planck [40]). We also show examples of the BBH coalescence model, and the binary neutron star (BNS) coalescence model, which we describe below. As noted in [66], LISA is likely to be able to detect the BBH background of the size considered here.

Astrophysical Implications.-In order to model the background from binary systems we will follow the approach of [35]. We divide the compact binary population into classes labeled by k [67,68]. Each class has distinct values of source parameters (for example the masses), which we denote by θ k . The total astrophysical background is a sum over the contributions in each class. The contribution of class k to the background may be written in terms of an integral over the redshift z as [1,5,[69][70][71][72][73][74]]

Ω GW (f ; θ k ) = f ρ c H 0 zmax 0 dz R m (z; θ k ) dEGW df (f s ; θ k ) (1 + z)E(Ω M , Ω Λ , z) , (6)
where R m (z; θ k ) is the binary merger rate per unit comoving volume per unit time, dE GW /df (f s , θ k ) is the energy spectrum emitted by a single binary evaluated in terms of the source frequency f s = (1 + z)f , and 3 + Ω Λ accounts for the de-FIG. 3. Presented here are constraints on the background in PI form [59], as well as some representative models, across many decades in frequency. We compare the limits from ground-based interferometers from the final science run of Initial LIGO-Virgo, the co-located detectors at Hanford (H1-H2), Advanced LIGO (aLIGO) O1, and the projected design sensitivity of the advanced detector network assuming two years of coincident data, with constraints from other measurements: CMB measurements at low multipole moments [60], indirect limits from the Cosmic Microwave Background (CMB) and Big-Bang Nucleosynthesis [61,62], pulsar timing [62], and from the ringing of Earth's normal modes [63]. We also show projected limits from a space-based detector such as LISA [59,64,65], following the assumptions of [59]. We extend the BNS and BBH distributions using an f 2/3 power-law down to low frequencies, with a low-frequency cut-off imposed where the inspiral time-scale is of order the Hubble scale. In Figure 5, we show the region in the black box in more detail.

E(Ω M , Ω Λ , z) = Ω M (1 + z)
pendence of comoving volume on cosmology. We use cosmological parameters from Planck [40], and Ω M = 1 -Ω Λ = 0.308.

The energy spectrum dE GW /df is determined from the strain waveform of the binary system. The dominant contribution to the background comes from the inspiral phase, however for BBH we include the merger and ringdown phases using the waveforms from [5,75] with the modifications from [76]. We choose to cut off the redshift integral at z max = 10. Redshifts larger than five contribute little to the integral due to the small number of stars formed at such high redshift [1,5,34,[69][70][71][72][73][74].

To compute the binary merger rate R m (z; θ k ), we use the same assumptions as in [35], unless stated otherwise. For the BNS case, we assume that the minimal time between the formation and the coalescence of the binary is t min = 20 Myr, following for instance [46]. This is to be compared to t min = 50 Myr for BBH [35,77].

As was emphasized in [78], heavy stellar mass black holes are expected to form in regions of low metallicity, which are associated with weaker stellar winds. To account for this effect, following [35], for binary systems with chirp masses larger than 30 M , we use only the fraction of stars that form in an environment with metallicity Z < Z /2. For BBH (and BNS) systems with smaller masses, we do not use a cutoff. However, we note that it makes little difference whether or not the cutoff is applied to high masses.

With the model defined above, the free parameters are the local merger rate R local = R m (0; θ k ) and the average chirp mass M c . The distribution of the chirp mass has little effect on the spectrum for a fixed average chirp mass [5].

We place upper limits at 95% confidence in the M c -R local plane, which are shown in Figure 4. Alongside the O1 results, we show the limits using Initial LIGO-Virgo data, as well as projected sensitivity of the advanced detector network. The limits presented here are about 10 times more sensitive than those placed with Initial LIGO-Virgo data. Furthermore, the future runs of the advanced detectors are expected to yield another factor of 100 improvement in sensitivity in R local for a given average chirp mass. We also show the local rate and chirp mass inferred from direct detections of BBH mergers during O1 [47,68]. Comparing the projected design sensitivity on R local and M c , with the values inferred from BBH observations in O1, suggests that it may be possible for the advanced detector network to detect the astrophysical BBH background.

Finally, instead of treating the chirp mass and local merger rate as free parameters, we can use the information from individually observed BBHs to compute the corresponding background, see Figure 5. To do this, we use the same model described above and we adopt the three rate models described in [47]. Specifically, we consider the three-events-based, power-law, and flat-log distributions of component masses. In each case, the rate at redshift z = 0 is normalized to the local rate derived from the O1 detections. With these assumptions we compute the background, including statistical uncertainty bands showing the 90% uncertainty in the local rate. The three rate models agree well in the sensitive frequency band of Initial LIGO-VIRGO aLIGO O1 Design FIG. 4. Displayed here are the 95% confidence contours on the local rate and average chirp mass parameters, using the model described in the Astrophysical Implications section. In addition to the constraint from Advanced LIGO (aLIGO) O1 data, we show the constraint from the final science run of Initial LIGO-Virgo, and the projected design sensitivity of Advanced LIGO-Virgo. We also show the median rate with 90% uncertainty inferred from O1 data for the power-law and flat-log mass distributions [47], along with the band containing 68% of the chirp mass for each distribution. The gray band separates BNS from BBH backgrounds. The dip at 30 M is due to the metallicity cutoff, as described in the Astrophysical Implications section.

advanced detectors (10-100 Hz). Note also that the final sensitivity of the advanced detectors may be sufficient to detect this background.

Conclusions.-The results presented here represent the first search for the stochastic gravitational-wave background made with the Advanced LIGO detectors. With no evidence of a stochastic signal, we place an upper limit of Ω 0 < 1.7 × 10 -7 on the GW energy density, for a spectral index α = 0. This is ∼33 times more sensitive than previous direct measurements in this frequency band. We also constrain the binary coalescence parameters of chirp mass and local merger rate. For fixed chirp mass below the high mass threshold of 30 M , the constraint on the merger rate is improved by a factor of ∼ 24, while for fixed merger rate, the constraint on the chirp mass is improved by a factor of ∼ 7, as can be seen from Figure 4. Finally, we update the background predictions due to BBH coalescences using data from O1. In this work we have focused the implications of our results for an astrophysical BBH background, as this provides the most promising candidate for first detecting the background. The implications of our search for other astrophysical and cosmological models can be seen in Figure 3. There is also an upcoming publication that will study implications for cosmic string models in more detail. These O1 results are a glimpse of the improvements in sensitivity to be seen in upcoming years. As the advanced detectors reach design sensitivity, there is a reasonable possibility of detecting the background due to BBHs. Even if no detection is made with these future searches, the searches will be able to constrain important cosmological and astrophysical background models.

FIG. 5. We present a range of potential spectra for a BBH background, using the flat-log, power-law, and 3-delta mass distribution models described in [47,78], with the local rate inferred from the O1 detections [47]. For the flat-log and power-law distributions, we show the 90% Poisson uncertainty band due to the uncertainty in the local rate measurement. In addition, we show the measured O1 PI curve and the projected PI curve for Advanced LIGO-Virgo operating at design sensitivity.

  FIG.2. Following[56], we present 95 % confidence contours in the Ωα -α plane. The region above these curves is excluded at 95% confidence. We show the constraints coming from the final science run of Initial LIGO-Virgo[36] and from O1 data. Finally, we display the projected (not observed) design sensitivity to Ωα and α for Advanced LIGO and Virgo[58].

TABLE I .

 I

Acknowledgments.-The authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, Department of Science and Technology, India, Science & Engineering Research Board (SERB), India, Ministry of Human Resource Development, India, the Spanish Ministerio de Economía y Competitividad, the Conselleria d'Economia i Competitivitat and Conselleria d'Educació, Cultura i Universitats of the Govern de les Illes Balears, the National Science Centre of Poland, the European Commission, the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the Lyon Institute of Origins (LIO), the National Research Foundation of Korea, Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation, the Natural Science and Engineering Research Council Canada, Canadian Institute for Advanced Research, the Brazilian Ministry of Science, Technology, and Innovation, Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Russian Foundation for Basic Research, the Leverhulme Trust, the Research Corporation, Ministry of Science and Technology (MOST), Taiwan and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, MPS, INFN, CNRS and the State of Niedersachsen/Germany for provision of computational resources. This article has been assigned the document number LIGO-P1600258-v18.


Benjamin P Abbott

Rich Abbott

Thomas D Abbott

Fausto Acernese

Kendall Ackley

Carl Adams

Thomas Adams

Paolo Addesso

Rana X Adhikari

Vaishali B Adya

Implications for the stochastic gravitationnal-wave background from compact binary coalescences

Keywords:

  L'archive ouverte pluridisciplinaire

Introduction -On 17 August 2017, the Laser Interferometer Gravitational-Wave Observatory (LIGO) [1] Scientific and Virgo [2] Collaborations detected a new gravitational-wave source: the coalescence of two neutron stars [3]. This event, GW170817, comes almost two years after GW150914, the first direct detection of gravitational waves from the merger of two black holes [4]. In total, five high confidence detections and one subthreshold candidate from binary black hole merger events have been reported: GW150914 [4], GW151226 [5], LVT151012 [6], GW170104 [7], GW170608 [8], and GW170814 [9]. The last of these events was reported as a three-detector observation by the two Advanced LIGO detectors located in Hanford, WA and Livingston, LA in the United States and the Advanced Virgo detector, located in Cascina, Italy.

In addition to loud and nearby events that are detectable as individual sources, there is also a population of unresolved events at greater distances. The superposition of these sources will contribute an astrophysical stochastic background, which is discernible from detector noise by cross-correlating the data streams from two or more detectors [10,11]. For a survey of the expected signature of astrophysical backgrounds, see [12][13][14][15][16][17][18]. See also [19] for a recent review of data analysis tools for stochastic backgrounds.

Following the first detection of gravitational waves from GW150914, the LIGO and Virgo Collaborations calculated the expected stochastic background from a binary black hole (BBH) population with similar masses to the components of GW150914 [20]. These calculations were updated at the end of the first Advanced LIGO observing run (O1) [21] to take into account the two other black hole merger events observed during the run: GW151226, a high-confidence detection; and LVT151012, an event of lower significance. The results indicated that the background from black hole mergers would likely be detectable by Advanced LIGO and Ad-vanced Virgo after a few years of their operation at design sensitivity. In the most optimistic case, a detection is possible even before instrumental design sensitivity is reached.

In this paper, we calculate the contribution to the background from a binary neutron star (BNS) population using the measured BNS merger rate derived from GW170817 [3]. We also update the previous predictions of the stochastic background from BBH mergers taking into account the most recent published statements about the rate of events [7]. We find the contributions to the background from BBH and BNS mergers to be of similar magnitude. As a consequence, an astrophysical background (including contributions from BNS and BBH mergers) may be detected earlier than previously anticipated.

Background from compact binary mergers -The energy-density spectrum of a gravitational wave background can be described by the dimensionless quantity

Ω GW (f ) = f ρ c dρ GW df , (1)
which represents the fractional contribution of gravitational waves to the critical energy density of the Universe [10]. Here dρ GW is the energy density in the frequency interval f to f + df , ρ c = 3H 2 0 c 2 /(8πG) is the critical energy density of the Universe, and the Hubble parameter H 0 = 67.9±0.55 km s -1 Mpc -1 is taken from Planck [22].

In order to model the background from all the binary mergers in the Universe, we follow a similar approach to [20]. A population of binary merger events may be characterized by a set of average source parameters θ (such as component masses or spins). If such a population merges at a rate R m (z; θ) per unit comoving volume per unit source time at a given redshift z, then the total gravitational-wave energy density spectrum from all the sources is given by (see, e.g. [12,20])

Ω GW (f, θ) = f ρ c H 0 zmax 0 dz R m (z; θ)dE GW (f s ; θ)/df s (1 + z)E(Ω M , Ω Λ , z) .
(2) Here dE GW (f s , θ)/df s is the energy spectrum emitted by a single source evaluated in terms of the source frequency f s = (1 + z)f . The function E(Ω M , Ω Λ , z) = Ω M (1 + z) 3 + Ω Λ accounts for the dependence of comoving volume on redshift assuming the best-fit cosmology from Planck [22], where Ω M = 1 -Ω Λ = 0.3065. We choose to cut off the redshift integral at z max = 10. Redshifts larger than z = 5 contribute little to the integral because of the [(1 + z)E(z)] -1 factor in Eq. 2, as well as the small number of stars formed at such high redshift, see for example [12][13][14][15][16][17][18]23].

The energy spectrum dE GW /df s is determined from the strain waveform of the binary system. The dominant contribution to the background comes from the inspiral phase of the binary merger, for which dE/df s ∝ M 5/3 c f -1/3 , where M c = (m 1 m 2 ) 3/5 /(m 1 +m 2 ) 1/5 is the chirp mass for a binary system with component masses m 1 and m 2 . In the BNS case we only consider the inspiral phase, since neutron stars merge at ∼ 2 kHz, well above the sensitive band of stochastic searches. We introduce a frequency cutoff at the innermost stable circular orbit. For BBH events, we include the merger and ringdown phases using the waveforms from [13,24] with the modifications from [25].

The merger rate R m (z; θ) is given by

R m (z; θ) = tmax tmin R f (z f ; θ)p(t d ; θ)dt d , (3)
where t d is the time delay between formation and merger of a binary, p(t d ; θ) is the time delay distribution given parameters θ, z f is the redshift at the formation time t f = t(z) -t d , and t(z) is the age of the Universe at merger. We assume that the binary formation rate R f (z f ; θ) scales with the star formation rate. For the BNS background, we make similar assumptions to those used in [20], which are outlined in what follows below.

We adopt the star formation model of [26], which produces very similar results as compared to the model described by [27]. We assume a time delay distribution p(t d ) ∝ 1/t d , for t min < t d < t max . Here t min is the minimum delay time between the binary formation and merger. We assume t min = 20 Myr [28]. The maximum time delay t max is set to the Hubble time [29][30][31][32][33][34][35][36][37]. We also need to consider the distribution of the component masses to calculate Ω GW . We assume that each mass is drawn from uniform distribution ranging from 1 to 2 M . The value of R m at z = 0 is normalized to the median BNS merger rate implied by GW170817, which is 1540 +3200 -1220 Gpc -3 yr -1 [3]. This rate is slightly higher than the realistic BNS merger rate predictions of [38], and those adopted in previous studies (e.g. [12,14]), but is consistent with optimistic predictions.

The calculation of the BBH background is similar, with the following differences. We assume t min = 50 Myr for the minimum time delay [20,37]. Massive black holes are formed preferentially in low-metallicity environments. For binary systems with at lease one black hole more massive than 30M , we therefore re-weight the star formation rate R f (z) by the fraction of stars with metallicities Z ≤ Z /2. Following [20], we adopt the mean metallicity-redshift relation of [27], with appropriate scalings to account for local observations [26,39]. For a consistent computation, a mass distribution needs to be specified. We use a power-law distribution of the primary (i.e., larger mass) component p(m 1 ) ∝ m -2. 35 1 and a uniform distribution of the secondary [6,7]. In addition, we require that the component masses take values in the range 5 -95 M with m 1 + m 2 < 100 M and m 2 < m 1 , in agreement with the observations of BBHs to date [7]. For the rate of BBH mergers, we use the most recent published result associated with the power-law mass distribution 103 +110 -63 Gpc -3 yr -1 [7,40]. As shown in [21], using a flat-log mass distribution instead of the powerlaw only affects Ω GW (f ) at frequencies above 100 Hz, which has very little impact on the detectability of the stochastic background with LIGO and Virgo. Frequencies below 100 Hz contribute to more than 99% of the sensitivity of the stochastic search [21].

The choices affecting the mass distribution have a minimal effect on the background energy density Ω GW in the low frequency part of the spectrum. We estimate the magnitude of the effect on the rate R using the approximate scaling relationship R ∼

V T -1 ∼ M 5/2 c -1
, where V T is the sensitive spacetime volume of the instrument [41,42]. In the BBH case for example, imposing a mass cutoff of m 1 , m 2 < 50 M to the power-law mass distribution reduces the estimate by less than 15%. Similarly, using the same scaling law, we estimate that replacing a uniform mass distribution for BNS with a Gaussian mass distribution centered around 1.4M leads to an increase of approximately 10%. Both adjustments are well within the dominant statistical Poisson uncertainty.

Predictions and detectability -A stochastic background of gravitational-waves introduces a correlated signal in networks of terrestrial detectors. This signal, expected to be much weaker than the detector noise, can be distinguished from noise by cross-correlating the strain data from two or more detectors. For a network of n detectors, assuming an isotropic, unpolarized, Gaussian, and stationary background, the optimal signal-to-noise ratio (SNR) of a cross-correlation search is given by

SNR = 3H 2 0 10π 2 √ 2T   ∞ 0 df n i=1 j>i γ 2 ij (f )Ω 2 GW (f ) f 6 P i (f )P j (f )   1/2 , ( 4
)
TABLE I. Estimates of the background energy density ΩGW(f ) at 25 Hz for each of the BNS, BBH and total background contributions, along with the 90% Poisson error bounds. We also show the average time τ between events as seen by a detector in the frequency band above 10 Hz, and the number of overlapping sources at a given time λ. We quote the number given the median rate and associated Poisson error bounds.

ΩGW( 25 in which i, j run over detector pairs, P i (f ) and P j (f ) are the one-sided strain noise power spectral densities of the two detectors, and γ ij (f ) is the normalized isotropic overlap reduction function between the pair [10,20].

While the cross correlation search is not optimal for non-Gaussian backgrounds, Eq. 4 gives the correct expression for the cross-correlation signal-to-noise ratio irrespective of the Gaussianity of the background [28,43].

On the left hand panel of Fig. 1, we show the estimates on the background energy density Ω GW (f ) for the BNS and BBH merger populations described in the previous section (red and green curves, respectively). The total (combined) background from BBH and BNS mergers is also plotted (solid blue curve) along with the 90% credible Poisson uncertainties in the local rate (indicated by the grey shaded region). Considering this uncertainty, we predict Ω tot GW (f = 25 Hz) = 1.8 +2.7 -1.3 × 10 -9 . The spectrum is well approximated by a power law Ω GW (f ) ∝ f 2/3 at low frequencies where the contribution from the inspiral phase is dominant. The power law remains a good approximation for f 100 Hz. Indeed, previous work suggests that the deviation from powerlaw behaviour is probably inaccessible by advanced detectors [23]. The frequency region between 10 Hz and 100 Hz accounts for more than 99% of the accumulated SNR for the Advanced LIGO-Virgo network [21]. The median value of the background from BBH mergers [Ω BBH GW (f = 25 Hz) = 1.1 +1.2 -0.7 × 10 -9 ] is comparable with our previous predictions [20] but the statistical uncertainty has decreased by a factor of 32% due to an increase in the number of observed BBH mergers. The median contribution from BNS systems is of the same order [Ω GW BNS (f = 25 Hz) = 0.7 +1.5 -0.6 × 10 -9 ]. The statistical uncertainty of the BNS background is larger than that of the BBH background, since we have only one BNS detection. These results are summarized in Table I.

The estimates on the background energy density shown in Fig. 1 are compared with power-law integrated (PI) curves [44] at various observing sensitivities (O2, O3 and Design). The PI curves represent the expected 1σ sensitivity of the standard cross-correlation search to powerlaw gravitational-wave backgrounds [44], for example the Ω GW (f ) ∝ f 2/3 spectrum expected from the inspiral phase of binary mergers. Hence, if a power-law spectrum intersects a PI curve, then it has SNR ≥ 1 for the corresponding observing run.

Although the stochastic background is dominated by unresolvable sources, the energy-density spectra in Fig. 1 include contributions from the loudest, individually detectable events. Simulations of the astrophysical background given the inferred rate and mass distribution indicate that removing sources that are individually detectable by the LIGO-Virgo network with a combined SNR > 12 has a very small impact on the results. The detectable sources have an even smaller effect than shown by the analysis in [20], which considered only the population of loud, high mass sources. This highlights the fact that the spectrum is dominated by low-mass systems, which are less likely to be detected individually.

Different assumptions are possible on the various distributions that enter the calculation of Ω GW (Eq. 2), such as star formation rate, metallicity evolution and delay times. However, we have verified that variations on our assumptions are contained within the Poisson band shown in Fig. 1, consistent with the detailed study in [20]. Additionally, if some of the observed BBH systems were of primordial (rather than stellar) origin, with a different redshift distribution, their contribution to the stochastic background spectrum could be weaker [45][46][47].

The right hand panel of Fig. 1 shows the expected accumulated SNR as a function of total observation time, updated from [20]. For O1, which lasted approximately 4 months, we use the actual instrumental sensitivities of the two LIGO detectors (Virgo was not yet operating at that time) [21]. The second observing run (O2) was recently completed and ran for approximately 9 months. For this run we use typical sensitivities of 100 Mpc for Livingston and 60 Mpc for Hanford, assuming a duty cycle of 50%. We do not include Virgo as it does not contribute significantly to the sensitivity of stochastic searches in O2 due to the lower range and one month integration time. For the next planned observing run (O3), and the following stages of sensitivity improvements, we include Virgo and assume a 50% duty cycle for each detector. Following [48], we assume O3 will be 12 months long (2017 -2018). We define the "Near Design" phase (2019+) to be a 12 month run where Hanford and Livingston operate at design sensitivity and Virgo at late sensitivity. Lastly, we assume that the "Design" phase (2022+) will be 24 months and will have Hanford, Livingston, and Virgo operating at design sensitivity. These assumptions are broadly consistent with [48], though we make specific assumptions about the duration of each observing run for concreteness.

The median total background from a combined BBH and BNS background may be identified with SNR = 3, corresponding to false alarm probability < 3×10 -3 , after approximately 40 months of observing. In the most optimistic scenario allowed by statistical uncertainties, the total background could be identified after 18 months or as early as O3. The most pessimistic case considered here is out of reach of the advanced detector network but is in the scope of third generation detectors [49].

Although the BNS and BBH backgrounds have similar energy-densities, they have extremely different statistical properties. To illustrate this, we plot a simulated strain time-series in Fig. 2 and show an example BNS (red) and BBH (green) background. The BNS events create an approximately continuous background consisting of a superposition of overlapping sources, since the duration of the waveform (190 s on average in the frequency band above 10 Hz) is long compared to the average time interval between two successive events (13 +49 -9 s on average). Considering the uncertainty on the rate, the average number of overlapping sources can vary between 3 and 45. In Fig. 2, 14 sources overlap on average.

The BBH background is different in nature even though the resulting energy density spectrum is similar. BBH events create a highly non-stationary and non-Gaussian background (sometimes referred to as a popcorn background in the literature), i.e. individual events are well separated in time, on top of the continuous background from contributed BNS inspirals. The duration of the waveform is much smaller for these massive sources (14 s on average in the band above 10 Hz, considering both the power law mass distribution and the distribution in redshift [49]) and much less than the time interval between events (223 +352

-115 s on average) resulting in rare overlaps.

Table I shows the estimated energy density at 25 Hz for each of the BNS, BBH and Total backgrounds. We also show the average time between events τ for each of these backgrounds as well as the average number of overlapping sources at any time λ, and the associated Poisson error bounds. The inverse of τ gives the rate of events in the Universe in s -1 .

Conclusion -The first gravitational wave detection of a binary neutron star system implies a significant contribution to the stochastic gravitational wave background from BNS mergers. Assuming the median merger rates, the background may be detected with SNR = 3 after 40 months of accumulated observation time, during the Design phase (2022+) [48]. In the most optimistic case, an astrophysical background may be observed at a level of 3σ after only 18 months of observation, during O3, the next observing run.

There are additional factors which may lead to an even earlier detection. First, the presence of additional sources, for example black hole-neutron star (BHNS) systems, will further add to the total background. Even small contributions to the background can decrease the time to detection. Previous estimates in the literature, for example [50,51], suggest that the BHNS will contribute roughly a few percent to the total background, although the uncertainty is large. Second, the analy-sis we have presented here assumes the standard crosscorrelation search. Specialized non-Gaussian searches may be more sensitive, particularly to the BBH background [52][53][54]. Unlike a standard matched filter search, non-Gaussian pipelines do not attempt to find individual events, but rather to measure the rate of sub-threshold events independently of their distribution.

A detection of the astrophysical background allows for a rich set of follow-up studies to fully understand its composition. The difference in the time-domain structure of the BBH and BNS signals may allow the BNS and BBH backgrounds to be measured independently. After detecting the background, stochastic analyses can address whether the background is isotropic [55][56][57], unpolarized [58], and consistent with general relativity [59]. Finally, understanding the astrophysical background is crucial to subtract it and enable searches for a background of cosmological origin [49].

FIG. 1. The left panel shows the predicted median background for the BNS (red) and BBH (green) models described in the text, the total combined background (blue), and the Poisson error bars (grey shaded region) for the total background. We also show expected PI curves for observing runs O2, O3, and design sensitivity (see the main text for details about the assumptions made for these observing runs). Virgo is included in O3 and beyond. The PI curves for O3 and beyond cross the Poisson error region, indicating the possibility of detecting this background or placing interesting upper limits on the evolution of the binary merger rates with redshift. In the right panel, we plot the signal-to-noise ratio as a function of cumulative observing time for the median total background (blue curve) and associated uncertainty (shaded region). The median of the predicted total background can be detected with SNR = 3 after 40 months of observation time, with LIGO-Virgo reaching design sensitivity (2022 -2024). The markers indicate the transition between observing runs. We only show 12 months of the Design phase for clarity, however this phase is 24 months long for the calculation of the PI curves (see [48]). In red we show a simulated BNS background corresponding to the median rate as shown in Figure 1, and in green we display the median BBH background. We do not show any detector noise, and do not remove some loud and close events that would be detected individually. The region in the black box, from 1800 -2600 seconds, is shown in greater detail in the inset. The BNS time series is continuous as it consists of a superposition of overlapping signals. On the other hand the BBH background (in green) is popcorn-like, and the signals do not overlap. Remarkably, even though the backgrounds have very different structure in the time domain, the energy in both backgrounds are comparable below 100 Hz, as seen in Figure 1.

novation

  Office Hungary (NKFI), the National Research Foundation of Korea, Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation, the Natural Science and Engineering Research Council Canada, the Canadian Institute for Advanced Research, the Brazilian Ministry of Science, Technology, Innovations, and Communications, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the Ministry of Science and Technology (MOST), Taiwan and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, MPS, INFN, CNRS and the State of Niedersachsen/Germany for provision of computational resources.This article has been assigned the document number LIGO-P1700272.

FIG. 2 .

 2 FIG.2. We present a simulated time series of duration 10 4 seconds illustrating the character of the BBH and BNS signals in the time domain. In red we show a simulated BNS background corresponding to the median rate as shown in Figure1, and in green we display the median BBH background. We do not show any detector noise, and do not remove some loud and close events that would be detected individually. The region in the black box, from 1800 -2600 seconds, is shown in greater detail in the inset. The BNS time series is continuous as it consists of a superposition of overlapping signals. On the other hand the BBH background (in green) is popcorn-like, and the signals do not overlap. Remarkably, even though the backgrounds have very different structure in the time domain, the energy in both backgrounds are comparable below 100 Hz, as seen in Figure1.

Acknowledgments -

The authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigación, the Vicepresidència i Conselleria d'Innovació, Recerca i Turisme and the Conselleria d'Educació i Universitat del Govern de les Illes Balears, the Conselleria d'Educació, Investigació, Cultura i Esport de la Generalitat Valenciana, the National Science Centre of Poland, the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Regional Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the Lyon Institute of Origins (LIO), the National Research, Development and In-