GW170817: Implications for the Stochastic Gravitational-Wave Background from Compact Binary Coalescences
B. P. Abbott et al.*
(LIGO Scientific Collaboration and Virgo Collaboration)
(Received 16 October 2017; revised manuscript received 16 January 2018; published 28 February 2018) The LIGO Scientific and Virgo Collaborations have announced the event GW170817, the first detection of gravitational waves from the coalescence of two neutron stars. The merger rate of binary neutron stars estimated from this event suggests that distant, unresolvable binary neutron stars create a significant astrophysical stochastic gravitational-wave background. The binary neutron star component will add to the contribution from binary black holes, increasing the amplitude of the total astrophysical background relative to previous expectations. In the Advanced LIGO-Virgo frequency band most sensitive to stochastic backgrounds (near 25 Hz), we predict a total astrophysical background with amplitude ΩGWðf¼25HzÞ¼1.8þ2.7−1.3×10−9 with 90% confidence, compared withΩGWðf¼25HzÞ¼1.1þ1.2−0.7×10−9 from binary black holes alone. Assuming the most probable rate for compact binary mergers, we find that the total background may be detectable with a signal-to-noise-ratio of 3 after 40 months of total observation time, based on the expected timeline for Advanced LIGO and Virgo to reach their design sensitivity.
DOI:10.1103/PhysRevLett.120.091101
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, six high confidence events detections and one sub- threshold 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, Washington, and Livingston, Louisiana, 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 back- ground, 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 astro- physical backgrounds, see[12–18]. 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 Advanced Virgo after a few years of their operation at design sensitivity. In the most optimistic case, a detection is possible even before instru- mental design sensitivity is reached.
In this Letter, 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 back- ground 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 back- ground can be described by the dimensionless quantity
ΩGWðfÞ ¼ f ρc
dρGW
df ; ð1Þ
which represents the fractional contribution of gravitational waves to the critical energy density of the Universe[10].
HeredρGWis the energy density in the frequency intervalf
*Full author list given at the end of the article.
Editors' Suggestion
to fþdf, ρc¼3H20c2/ð8πGÞ is the critical energy density of the Universe, and the Hubble parameter H0¼ 67.90.55km s−1Mpc−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 parametersθ(such as component masses or spins). If such a population merges at a rateRmðz;θÞper unit comoving volume per unit source time at a given redshiftz, then the total gravitational-wave energy density spectrum from all the sources is given by (see, e.g.[12,20])
ΩGWðf;θÞ ¼ f ρcH0
Z z
max
0 dzRmðz;θÞdEGWðfs;θÞ/dfs
ð1þzÞEðΩM;ΩΛ; zÞ : ð2Þ Here dEGWðfs;θÞ/dfs is the energy spectrum emitted by a single source evaluated in terms of the source frequencyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifs¼ ð1þzÞf. The function EðΩM;ΩΛ; zÞ ¼
ΩMð1þzÞ3þΩΛ
p accounts for the dependence of
comoving volume on redshift assuming the best-fit cos- mology from Planck[22], whereΩM¼1−ΩΛ¼0.3065. We choose to cut off the redshift integral at zmax¼10. Redshifts larger thanz¼5contribute little to the integral because of the½ð1þzÞEðzÞ−1factor in Eq.(2), as well as the small number of stars formed at such high redshift; see, for example, [12–18,23].
The energy spectrumdEGW/dfs 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 whichdE/dfs∝Mc5/3f−1/3, whereMc¼ ðm1m2Þ3/5/ðm1þm2Þ1/5is the chirp mass for a binary system with component massesm1andm2. In the BNS case we only consider the inspiral phase, since neutron stars merge at ∼2kHz, 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 rateRmðz;θÞis given by Rmðz;θÞ ¼
Z t
max
tmin
Rfðzf;θÞpðtd;θÞdtd; ð3Þ
wheretdis the time delay between formation and merger of a binary, pðtd;θÞ is the time delay distribution given parameters θ, zf is the redshift at the formation time tf¼tðzÞ−td, andtðzÞis the age of the Universe at merger.
We assume that the binary formation rateRfðzf;θÞ 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 pðtdÞ∝1/td, for tmin< td< tmax. Here tmin is the minimum delay time between the binary formation and merger. We assumetmin¼20Myr[28]. The maximum time delaytmaxis set to the Hubble time[29–37].
We also need to consider the distribution of the component masses to calculate ΩGW. We assume that each mass is drawn from uniform distribution ranging from 1 to2M⊙. The value of Rm at z¼0 is normalized to the median BNS merger rate implied by GW170817, which is 1540þ3200−1220 Gpc−3yr−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 tmin¼50 Myr for the minimum time delay[20,37]. Massive black holes are formed preferentially in low-metallicity environments.
For binary systems with at least one black hole more massive than 30M⊙, we therefore reweight the star formation rateRfðzÞby the fraction of stars with metal- licities 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) componentpðm1Þ∝m−2.351 and a uni- form distribution of the secondary[6,7]. In addition, we require that the component masses take values in the range 5–95M⊙ with m1þm2<100M⊙ and m2< m1, 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 103þ110−63 Gpc−3yr−1 [7,40]. As shown in [21], using a flat-log mass distribution instead of the power-law only affects ΩGWðfÞ 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 ΩGW
in the low frequency part of the spectrum. We estimate the magnitude of the effect on the rateRusing the approximate scaling relationship R∼hVTi−1∼hM5/2c i−1, where VT is the sensitive spacetime volume of the instrument[41,42]. In the BBH case, for example, imposing a mass cutoff ofm1, m2<50M⊙ 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 approx- imately 10%. Both adjustments are well within the dom- inant 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 ofndetectors, assuming an isotropic, unpolarized, Gaussian, and stationary back- ground, the optimal signal-to-noise ratio (SNR) of a cross-correlation search is given by
SNR¼ 3H20 10π2
ffiffiffiffiffiffi p2TZ ∞
0 dfXn
i¼1
X
j>i
γ2ijðfÞΩ2GWðfÞ f6PiðfÞPjðfÞ
1/2
; ð4Þ in which i,jrun over detector pairs, PiðfÞandPjðfÞare the one-sided strain noise power spectral densities of the two detectors, andγijðfÞis the normalized isotropic over- lap 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 densityΩGWðfÞfor 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 ΩtotGWðf¼25HzÞ ¼1.8þ2.7−1.3×10−9.
The spectrum is well approximated by a power law ΩGWðfÞ∝f2/3 at low frequencies where the contribution from the inspiral phase is dominant. The power law remains a good approximation for f≲100Hz. Indeed, previous work suggests that the deviation from power-law behavior 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 mergers½ΩBBHGWðf¼25HzÞ ¼ 1.1þ1.2−0.7×10−9is 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 ½ΩGWBNSðf¼25HzÞ ¼0.7þ1.5−0.6×10−9. 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 TableI.
The estimates on the background energy density shown in Fig. 1 are compared with power-law integrated (PI) curves[45]at various observing sensitivities (O2, O3, and design). The PI curves represent the expected1σsensitivity of the standard cross-correlation search to power-law gravitational-wave backgrounds [45], for example, the ΩGWðfÞ∝f2/3 spectrum expected from the inspiral phase of binary mergers. Hence, if a power-law spectrum
FIG. 1. Left: 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 (gray shaded region) for the total background. We also show expected PI curves for observing runs O2, O3, and design sensitivity (see 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. Right: 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 SNR¼3after 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[44]).
intersects a PI curve, then it has SNR≥1 for the corre- sponding observing run.
Although the stochastic background is dominated by unresolvable sources, the energy-density spectra in Fig. 1 includecontributions from the loudest, individually detect- able 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 dis- tributions that enter the calculation ofΩGW[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[46–48].
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 used 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[44], 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 [44], although 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 SNR¼3, corre- sponding 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.2and show an example BNS (red) and BBH (green) background. The BNS events create an approx- imately continuous background consisting of a superposi- tion of overlapping sources, since the duration of the TABLE I. Estimates of the background energy densityΩGWðfÞ
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 timeτbetween events as seen by a detector in the frequency band above 10 Hz, and the number of overlapping sources at a given timeλ. We quote the number given the median rate and associated Poisson error bounds.
ΩGWð25HzÞ τ[s] λ BNS 0.7þ1.5−0.6×10−9 13þ49−9 15þ30−12 BBH 1.1þ1.2−0.7×10−9 223þ352−115 0.06þ0.06−0.04
Total 1.8þ2.7−1.3×10−9 1244−8 15þ31−12
FIG. 2. We present a simulated time series of duration 104 s illustrating the character of the BBH and BNS signals in the time domain. In red we show a simulated BNS background corre- sponding to the median rate as shown in Fig.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 to 2600 s, 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 popcornlike, and the signals do not overlap. Remark- ably, even though the backgrounds have very different structure in the time domain, the energy in both backgrounds is comparable below 100 Hz, as seen in Fig.1.
waveform (190 s on average in the frequency band above 10 Hz) is long compared to the average time interval between two successive events (13þ49−9 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 nonstationary and non-Gaussian background (sometimes referred to as a popcorn back- ground 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 (223þ352−115 s on average) resulting in rare overlaps.
TableIshows the estimated energy density at 25 Hz for each of the BNS, BBH, and total backgrounds. We also show the average time between eventsτ 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 contribu- tion to the stochastic gravitational-wave background from BNS mergers. Assuming the median merger rates, the background may be detected with SNR¼3after 40 months of accumulated observation time, during the design phase (2022+)[44]. 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 observ- ing 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 analysis we have presented here assumes the standard cross-correlation search. Specialized non-Gaussian searches may be more sensitive, particularly to the BBH background [52–54].
Unlike a standard matched filter search, non-Gaussian pipelines do not attempt to find individual events, but rather to measure the rate of subthreshold events inde- pendently 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[55–57], unpolarized[58], and consistent with general relativity[59]. Finally, understand- ing the astrophysical background is crucial to subtract it and enable searches for a background of cosmological origin[49].
The authors gratefully acknowledge the support of the U.S. National Science Foundation (NSF) for the construc- tion 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 Vicepresid`encia 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 Innovation 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 Communi- cations, 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 acknowl- edge 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.
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