GW150914: Implications for the Stochastic Gravitational-Wave Background from Binary Black Holes
B. P. Abbott et al.*
(LIGO Scientific Collaboration and Virgo Collaboration) (Received 13 February 2016; published 31 March 2016)
The LIGO detection of the gravitational wave transient GW150914, from the inspiral and merger of two black holes with masses≳30M⊙, suggests a population of binary black holes with relatively high mass. This observation implies that the stochastic gravitational-wave background from binary black holes, created from the incoherent superposition of all the merging binaries in the Universe, could be higher than previously expected. Using the properties of GW150914, we estimate the energy density of such a background from binary black holes. In the most sensitive part of the Advanced LIGO and Advanced Virgo band for stochastic backgrounds (near 25 Hz), we predict ΩGWðf¼25 HzÞ ¼1.1þ2.7−0.9×10−9 with 90% confidence. This prediction is robustly demonstrated for a variety of formation scenarios with different parameters. The differences between models are small compared to the statistical uncertainty arising from the currently poorly constrained local coalescence rate. We conclude that this background is potentially measurable by the Advanced LIGO and Advanced Virgo detectors operating at their projected final sensitivity.
DOI:10.1103/PhysRevLett.116.131102
Introduction.—On September 14, 2015 the Advanced LIGO[1–3]Hanford and Livingston detectors observed the gravitational-wave (GW) event GW150914 with a signifi- cance in excess of 5.1σ [4]. The observed signal is consistent with a binary black hole waveform with com- ponent masses of m1¼36þ5−4M⊙ and m2¼29þ4−4M⊙, as measured in the source frame, and coalescing at a lumi- nosity distance of410þ160−180 Mpc, corresponding to a redshift of z¼0.09þ0.03−0.04 [4,5].
For every event like GW150914 observed by advanced gravitational-wave detectors, there are many more too distant to be resolved. The gravitational waves from these unresolvable events combine to create a stochastic back- ground, which can be detected by correlating the signals from two or more gravitational-wave detectors[6]. While it has long been known that the advanced detectors could observe such a background, the detection of GW150914 suggests that the binary black hole background level is likely to be at the higher end of previous predictions (see, e.g.,[7–14]).
Heavy black holes like GW150914 are predicted to form in low-metallicity stellar environments, lower than about half of solar metallicity, and in the presence of relatively weak massive-star winds [15]. These masses are also larger than the masses inferred from reliable dynamical measurements in black hole x-ray binaries [15]. More massive binaries emit more energy in gravitational waves.
Hence, the measurement of the component masses of GW150914 favors a higher amplitude of the corresponding gravitational-wave background.
In addition, the coalescence rate of binary black holes like GW150914 in the local Universe is estimated to be 16þ38−13 Gpc−3yr−1 (Table I [16]) median with a 90%
credible interval. This rate excludes the lower end of predetection rate estimates [15], while being consistent with the higher end. A higher coalescence rate also implies a brighter stochastic background.
Currently, there are two possible formation channels that are consistent with the GW150914 event[15]. Binary black holes may be formed from isolated binaries of massive stars in galactic fields, or through dynamical interactions in dense stellar environments such as globular clusters[15].
The evolution of the merger rate with redshift depends, in part, on the assumed formation scenario.
In this Letter, we discuss the detectability of the stochastic background produced by binary black holes throughout the Universe based on the measured properties of GW150914.
Binary black hole background.—The energy density spectrum of gravitational waves is described by the following dimensionless quantity[6]:
ΩGWðfÞ ¼ f ρc
dρGW
df ; ð1Þ
wheredρGWis the energy density in the frequency interval ftofþdf,ρc ¼3H20c2=8πGis the critical energy density required to close the Universe, and H0¼67.8 0.9km=s Mpc is the Hubble constant[17].
A population of binary black holes is characterized by the distribution of the intrinsic source parametersθ(usually the component masses and spin). Since this distribution is unknown at present, following[16]and[18], we divide the
*Full author list given at the end of the article.
distribution into distinct classes corresponding to the observed candidates. If binary black holes in some class k, with source parametersθk, merge at a rateRmðz;θkÞper unit comoving volume Vc per unit source time, then the total gravitational-wave energy density spectrum from all the sources in this class is given by (see, e.g.,[7–14]) ΩGWðf;θkÞ ¼ f
ρcH0 Z z
max
0 dzRmðz;θkÞdEdfGW
s ðfs;θkÞ ð1þzÞEðΩM;ΩΛ; zÞ ; ð2Þ and the final energy density spectrum is the sum of ΩGWðf;θkÞ from each class. (When the distribution of the source parameters is better understood after multiple detections, the discrete sum can be replaced by a continu- ous integral.) In Eq.(2),dEGW=dfsðfs;θkÞis the spectral energy density of a source of class k at the frequency fs¼fð1þzÞ, which depends on the source parametersθk; EðΩM;ΩΛ; zÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ΩMð1þzÞ3þΩΛ
p captures the depend-
ence of the comoving volume on redshift for the standard flat cosmology model, withΩM¼0.31andΩΛ¼1−ΩM. The (1þz) factor in the denominator of Eq.(2)corrects for the cosmic expansion, converting time in the source frame to the detector frame. The parameter zmax corresponds to the time of the first coalescences. We setzmax¼10, noting, however, that sources abovez∼5contribute very little to the total background (see, e.g.,[7–14]).
The merger rateRmðz;θkÞis a convolution of the binary formation rate Rfðz;θkÞwith the distribution of the time delaysPðtd;θkÞbetween binary black hole formation and merger (see, e.g., [19])
Rmðz;θkÞ ¼ Z tmax
tmin
Rfðzf;θkÞPðtd;θkÞdtd; ð3Þ
wheretdis the time delay,zfis the redshift at the formation timetf¼tðzÞ−td, andtðzÞ is the age of the Universe at merger.
Inference on GW150914 [5], along with expectations that gravitational-wave emission is efficient in circularizing the orbit [15], allows us to restrict our models for dEGW=dfs to circular orbits. Measurements do not con- strain the component spins in the orbital plane [5]; there- fore, we restrict our model to spins (anti-)aligned with the orbital angular momentum, and use the functional form of dEGW=dfs derived in [20]. In addition to the component masses, this model depends on the effective spin parameter along the direction of the orbital angular momentumχeff, which takes values between−1(in which both black holes have maximal spins antialigned with respect to the orbital angular momentum) andþ1(assuming maximally aligned spins) [5].
Fiducial model.—The GW150914 event appears con- sistent with both the dynamic and field formation channels [15]; however, the field channel is currently better described in the stochastic background literature. Thus,
ourFiducialmodel is inspired by population synthesis studies of field binaries (see[14]).
We assume that the binary black hole formation rate is proportional to the star formation rate (SFR) at metallicity Z≤Z⊙=2[15], whereZ⊙ is the solar metallicity. That is, to compute the binary black hole formation rate, the SFR is multiplied by the fraction of star formation occurring below the metallicity thresholdZc ¼Z⊙=2. For the SFR, we use the recent model [21], referred to, here, as “Vangioni,” based on theγ-ray burst rate of[22]and on the normali- zation described in[23,24]. We adopt the mean metallicity- redshift relation of[25], rescaled upwards by a factor of 3 to account for local observations [21,26]. In addition, we assume the metallicity is log10-normally distributed with a standard deviation of 0.5 around the mean at each redshift [27]. We further assume that the time delay distribution followsPðtdÞ∝tαd, withα¼−1fortd> tmin [19,28–34], where tmin¼50Myr is the minimum delay time for a massive binary to evolve until coalescence (e.g.,[35]), and a maximum time delaytmax equal to the Hubble time.
The rest of theFiducialmodel parameters correspond to the median inferred parameters of GW150914: the chirp massMc¼28M⊙, the symmetric mass ratioη∼0.25, and the effective spin parameterχeff ¼−0.06. We normalize the overall merger rate so that the local merger rate atz¼0 matches the most conservative median inferred rate, 16Gpc−3yr−1 [16].
Results.—We plot ΩGWðfÞ for the Fiducial model as a solid blue curve in Fig. 1(a). The curve is shown against the pink shaded region, which represents the 90%
credible interval statistical uncertainty in the local rate.
Considering this uncertainty, we predictΩGWðf¼25HzÞ¼ 1.1þ2.7−0.9×10−9. The spectrum is well approximated by a power law ΩGWðfÞ∝f2=3 at low frequencies where the contribution from the inspiral phase is dominant and the spectral energy density is dEGW=dfs≃½ðGπÞ2=3=3× M5=3c f−1=3s . This power law remains a good approximation until the spectrum reaches a maximum atf∼100Hz. The shape is in agreement with previous predictions (see, e.g., [8–14]), except that the maximum is shifted to lower frequencies, due to the higher mass considered.
This calculation of ΩGWðfÞ captures the total energy density in gravitational waves generated by binary black hole coalescences. In practice, some of these sources will be individually detected as resolved binaries. We define
“the residual background”as the energy density spectrum that excludes potentially resolvable binaries. While the total background is a property of the Universe, the residual background is detector-dependent. As sensitivity improves, the surveyed volume increases, more binaries are resolved, and the residual background decreases.
The dashed blue curve in Fig.1(a)represents the residual background calculated for the network of the Advanced LIGO[1,2]and Advanced Virgo[37,38]detectors at final sensitivity, assuming that a binary black hole signal is
detected if it is associated with a single-detector matched filter signal-to-noise ratio ofρ>8in at least two detectors [39]. The difference between the two curves is about 30%
in the sensitive frequency band (10–50 Hz), indicating that the residual background carries complementary informa- tion about the binary black hole population. Binaries with the same component masses as GW150914 can be detected at a redshift up toz≲1.3by advanced detectors operating at design sensitivity if optimally located and oriented (see Fig. 4 of[15]). However, most sources atz≳0.4will not be individually resolvable because of unfavorable location and orientation.
The sensitive frequency band of the Advanced LIGO- Advanced Virgo network to a gravitational-wave back- ground produced by binary black holes is 10–50 Hz, where ΩGW∼f2=3. It corresponds to more than 95% of the accumulated sensitivity [13,14,40]. The black curves shown in Fig. 1(a) are power-law integrated curves [41], which represent the expected1σsensitivity of the standard cross-correlation search [6] to power-law gravitational- wave backgrounds, of which theΩGWðfÞ∝f2=3spectrum for binary inspirals is an example. A power-law integrated curve is calculated by taking the locus of power-law spectra that have expected SNR¼1, where [6]
SNR¼ 3H20 10π2
ffiffiffiffiffiffi p2TZ ∞
0 dfXn
i¼1
X
j>i
γ2ijðfÞΩ2GWðfÞ f6PiðfÞPjðfÞ
1=2
; ð4Þ
for a network of detectors i¼1;2;…; n. Hence, if the spectrum of an astrophysical background intersects a black curve, then it has an expected SNR≥1. In Eq.(4),PiðfÞ and PjðfÞ are the one-sided strain noise power spectral densities of two detectors;γijðfÞis the normalized isotropic overlap reduction function [42,43]; andT is the accumu- lated coincident observation time. While Eq.(4)is derived by assuming a Gaussian background [6], it can also be applied to non-Gaussian backgrounds (with signals that are clearly separated in time) such as the binary black hole background considered here [44]. The different black curves shown in this plot illustrate the improvement in expected sensitivity in the coming years.
Following [36,40], we consider five different phases, denoted O1 to O5, corresponding to the first five observing runs, summarized in TableI. For clarity, we show only the O1, O2, and O5 power-law integrated curves since the differences between the projected sensitivities for O3, O4, and O5 are relatively small. In Fig. 1(b), we plot the expected accumulated SNR for theFiducialmodel as a function of total observation time. For both the sensitivity curves and the accumulated SNR, we assume a coincident duty cycle for each pair of detectors of 33% for O1 (actual) and 50% for all other runs (predicted). The total back- ground associated with the Fiducial model could be identified with SNR¼3, corresponding to false alarm probability <3×10−3, after approximately 6 years of observing. In the most optimistic scenario given by statistical uncertainties, the total background could be FIG. 1. Expected sensitivity of the network of Advanced LIGO and Virgo detectors to theFiducialfield model. Left panel: Energy density spectra are shown in blue (solid for the total background; dashed for the residual background, excluding resolved sources, assuming final Advanced LIGO and Virgo [1,2] sensitivity). The pink shaded region “Poisson” shows the 90% C.L. statistical uncertainty on the total background, propagated from the local rate measurement. The black power-law integrated curves show the1σ sensitivity of the network expected for the two first observing runs O1 and O2, and for 2 years at the design sensitivity in O5. (O3 and O4 are not significantly different than O5; see TableI.) If the astrophysical background spectrum intersects a black line, it has expected SNR≥1. In both panels we assume a coincident duty cycle of 33% for O1 (actual) and 50% for all other runs (predicted). Right panel:
Predicted SNR as a function of total observing time. The blue lines and pink shaded region have the same interpretation as in the left panel. Each observing run is indicated by an improvement in the LIGO-Virgo network sensitivity[36], which results in a discontinuity in the slope. The thresholds for SNR¼1, 3 (false-alarm probability<3×10−3) and 5 (false-alarm probability<6×10−7) are indicated by horizontal lines.
identified after 1.5 years with SNR¼3and after approx- imatively 2 years with SNR¼5, which is even before design sensitivity is reached. It would take about 2 years of observing to achieve SNR¼3 and about 3.5 years for SNR¼5for the optimistic residual background. The most pessimistic case considered here is out of reach of the advanced detector network but is in the scope of third generation detectors, such as the proposed Einstein Telescope [45] whose sensitivity would enable to reach ΩGW∼10−12 after a year of observation[46,47].
Alternative models.—We now investigate the impact of possible variations on theFiducialmodel. We consider the following alternative models: (i)AltSFRdiffers from the Fiducial model in assuming a different SFR proposed by Tornatore et al. [48], who combined obser- vations and simulations at higher redshift; the formation rate is assumed to be proportional to the SFR, with no metallicity threshold. We also considered the Madau and Dickinson SFR[25]and found that it produces an energy density spectrum that is essentially indistinguishable from theFiducialmodel. (ii)LongDelayis identical to the Fiducial model but assumes a significantly longer minimum time delay tmin¼5Gyr, potentially consistent with binary black hole formation via the chemically homogeneous evolution of rapidly rotating massive stars in very tight binaries[49]. (iii)LowMetallicityis the same asFiducial, but assumes that a significantly lower metallicity is required to form heavy black holes, with a threshold ofZc ¼Z⊙=10[15]. (iv)FlatDelayassumes a flat time delay distribution, α¼0, with tmin¼50Myr and tmax¼1Gyr. This is inspired by the supposition that dynamical formation of the most massive binaries is likely to happen fairly early in the history of the host environment. (v) ConstRatefollows the assumption of [4] in considering a redshift-independent merger rate, RmðzÞ ¼16Gpc−3yr−1. (vi) LowMass is the same as the Fiducial model except we add a second class of lower-mass binary black hole sources corresponding to a smaller range for individual detections during O1. As an example, we assume a chirp mass of half the mass of
GW150914, Mc¼15M⊙ and a local merger rate of 61Gpc−3yr−1 [5,16], corresponding to the second most significant event (LVT151012) identified in [4,39] with insufficient significance to decisively claim a detection.
Here, we assume that the metallicity threshold isZc ¼Z⊙. Figure 2 shows the impact of alternative models described above. The differences in the spectra of alter- native models are not negligible. However, all models considered here fall within the range of statistical uncer- tainty in the local merger rate estimate relative to the Fiducialmodel in the sensitive frequency band.
The impact of an alternative star formation rate, as examined through model AltSFR, is particularly small, indicating that the accuracy of SFR models is not a significant source of systematic error in predicting the strength of the gravitational-wave background.
TABLE I. Different phases in the evolution of the Advanced LIGO and Advanced Virgo detector network over the next several years. The Advanced LIGO and Advanced Virgo noise curves corresponding to high-sensitivity versions of“early-,” “mid-,” “late-,”and“design-”spectra are taken from[36]. Note that AdVirgo did not participate in the O1 observing run, so is not included in the first phase.“Duration”refers to the planned calendar time as opposed to the amount of accumulated data, for which we assume a duty cycle of 33% for O1 (actual) and 50% for all other runs (predicted). The last column indicates the signal-to-noise ratio at the end of each phase assuming the Fiducialmodel and accounting for the range of uncertainty in the rate.
Observing run Epoch Duration (months) aLIGO sensitivity AdVirgo sensitivity SNR (90% C.L.)
O1 2015–2016 4 Early 0–0.14
O2 2016–2017 6 Mid Early 0.046–0.81
O3 2017–2018 9 Late Mid 0.19–3.4
O4 2019 12 Design Late 0.31–5.6
O5 2020+ Design Design 0.67–12
FIG. 2. Energy density spectra for the different models sum- marized in the text. The pink shaded region Poisson shows the 90% C.L. statistical uncertainty propagated from the local rate measurement, on theFiducialmodel. The black dashed curve shows the design sensitivity of the network of Advanced LIGO [1,2]and Advanced Virgo[37,38]; see TableI. If the astrophysi- cal background spectrum intersects with the dashed black line, it has an expected SNR≥1.
Relative to the Fiducial model, the LongDelay, FlatDelay, and ConstRate models all predict fewer binaries at z >0, even though all of these models are constrained to have the same local merger rate (z¼0).
Consequently, these latter three models yield a lower energy density. TheLowMetallicitymodel is characterized by a greater high-redshift merger rate than the Fiducial model, with significant merger rates extending out to z∼5–6. This is because very little of the local Universe has the required low metallicity, so the local mergers come from the long time-delay tail of a large high-redshift population. Consequently, theLowMetallicitymodel has a higher overall normalization as well as a different spectral shape at frequencies above 100 Hz due to the redshifting of the dominant high-z contribution to the gravitational-wave background to lower frequencies.
Relative to theFiducialmodel, theLowMassmodel shows a greater energy density at all frequencies, particu- larly at high frequencies due to the signals from lower-mass binaries. This model indicates that if there is a significant rate of mergers of binaries with smaller masses than GW150914, their contribution to the gravitational-wave energy density spectrum could be significant. The delta- function mass distributions assumed in all models are motivated by the observed candidates, but are not realistic.
We have analyzed two alternative broad mass distributions considered in [16], flat in the log-mass of the component black holes and a Salpeter-like mass function for the larger black hole with a flat mass ratio; these yield broadly consistent energy densities. We have not carried out a systematic study of black hole spin. Measurements of GW150914 prefer small values of spin in the direction of orbital momentum, but spins in the orbital plane are not constrained. Preliminary studies carried out as part of this investigation suggest thatΩGWðfÞcould change by a factor of ≲2for models including spin.
Conclusions and discussion.—The detection of gravita- tional waves from GW150914 is consistent with the existence of high-mass binary black hole mergers with a coalescence rate of tens per Gpc3 per year. As a conse- quence, the stochastic background from binary black holes is expected to be at the higher end of previous predictions (see, e.g.,[8–14]). We have shown that, for theFiducial field model, the energy density spectrum is ΩGWðf¼ 25HzÞ ¼1.1þ2.7−0.9×10−9 with 90% confidence. This, in turn, implies that the background may be measured by the network of Advanced LIGO and Virgo detectors operating at or near their final sensitivity. The uncertainty in this prediction arises from the statistical uncertainty in the local merger rate estimate.
Our predictions are subject to statistical fluctuations in the observed ΩGWðfÞ due to random realizations of the binaries that coalesce during the observing run. These fluctuations are much smaller than the current local merger uncertainty[44]. The predictions may also be conservative.
Throughout, we have assumed the use of the standard cross-correlation statistic, which is known to be suboptimal for non-Gaussian backgrounds [50]. The development of more sensitive non-Gaussian pipelines may hasten the detection of the binary black hole background[51–53].
We have examined several alternative models for the merger rate evolution with redshift, representative of the uncertainties in the formation channels for high-mass binary black holes. We find that all of these variations lie within the envelope of the uncertain local rate normali- zation in the 10–50 Hz band, as illustrated in Fig. 2. The power-law slope of the spectrum in this frequency band is not expected to deviate from 2=3 unless there is a significant contribution from sources with high total mass merging at high redshift,Mð1þzÞ≳200M⊙. This illus- trates the robustness of the predicted amplitude and power- law slope of the energy density spectrum.
However, this also implies that the stochastic background measurement with Advanced LIGO and Advanced Virgo detectors can only constrain the amplitude of the background power law in the 10–50 Hz sensitive frequency band. The sensitivity of this search at the2σlevel will correspond to ΩGW∼10−9 at 25 Hz with the full- sensitivity network of the Advanced LIGO and Advanced Virgo detectors. Therefore, the stochastic search alone will not be able to distinguish between different model varia- tions that have a similar effect on the spectrum in the 10–50 Hz band. Future measurements of individual binary coalescences will help break at least some of these degeneracies, by providing a better estimate of the local merger rate and chirp mass distribution. Combining the two types of measurements (stochastic and individual coales- cence event) could, therefore, help distinguish between different astrophysical formation scenarios for binary black holes[54], but the full potential of this approach may only be reached using the third generation of gravitational-wave detectors.
Finally, gravitational waves from distant binary black hole mergers may be a foreground noise source for the detection of a cosmological background from the early epochs of the Universe in the frequency band of ground- based detectors. However, this astrophysical background has a different spectral shape and different statistical properties (noncontinuous and non-Gaussian) that could be used, in principle, to distinguish it from the primordial background.
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, 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 computa- tional resources.
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M. E. Zucker,1,10S. E. Zuraw,101 and J. Zweizig1
(LIGO Scientific Collaboration and Virgo Collaboration)
1LIGO, California Institute of Technology, Pasadena, California 91125, USA
2Louisiana State University, Baton Rouge, Louisiana 70803, USA
3Università di Salerno, Fisciano, I-84084 Salerno, Italy
4INFN, Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo, I-80126 Napoli, Italy
5University of Florida, Gainesville, Florida 32611, USA
6LIGO Livingston Observatory, Livingston, Louisiana 70754, USA
7Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Université Savoie Mont Blanc, CNRS/IN2P3, F-74941 Annecy-le-Vieux, France
8Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany
9Nikhef, Science Park, 1098 XG Amsterdam, Netherlands
10LIGO, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
