Directional limits on persistent gravitational waves from Advanced LIGO's first observing run
LIGO Scientific Collaboration, Virgo Collaboration, B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, C. Affeldt, M. Agathos, K. Agatsuma, N. Aggarwal, O. D. Aguiar, L. Aiello, A. Ain, P. Ajith, B. Allen, A. Allocca, P. A. Altin, A. Ananyeva, S. B. Anderson, W. G. Anderson, S. Appert, K. Arai, M. C. Araya, J. S. Areeda, N. Arnaud, K. G. Arun, S. Ascenzi, G. Ashton, M. Ast, S. M. Aston, P. Astone, P. Aufmuth, C. Aulbert, A. Avila-Alvarez, S. Babak, P. Bacon, M. K. M. Bader, P. T. Baker, F. Baldaccini, G. Ballardin, S. W. Ballmer, J. C. Barayoga, S. E. Barclay, B. C. Barish, D. Barker, F. Barone, B. Barr, L. Barsotti, M. Barsuglia, D. Barta, J. Bartlett, I. Bartos, R. Bassiri, A. Basti, J. C. Batch, C. Baune, V. Bavigadda, M. Bazzan, C. Beer, M. Bejger, I. Belahcene, M. Belgin, A. S. Bell, B. K. Berger, G. Bergmann, C. P. L. Berry, D. Bersanetti, A. Bertolini, J. Betzwieser, S. Bhagwat, R. Bhandare, I. A. Bilenko, G. Billingsley, C. R. Billman, J. Birch, R. Birney, O. Birnholtz, S. Biscans, A. S. Biscoveanu, A. Bisht, M. Bitossi, C. Biwer, M. A. Bizouard, J. K. Blackburn, J. Blackman, C. D. Blair, D. G. Blair, R. M. Blair, S. Bloemen, O. Bock, M. Boer, G. Bogaert, A. Bohe, et al. (896 additional authors not shown)
DDirectional limits on persistent gravitational wavesfrom Advanced LIGO’s first observing run
B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, , K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, C. Affeldt, M. Agathos, K. Agatsuma, N. Aggarwal, O. D. Aguiar, L. Aiello, , A. Ain, P. Ajith, B. Allen, , , A. Allocca, , P. A. Altin, A. Ananyeva, S. B. Anderson, W. G. Anderson, S. Appert, K. Arai, M. C. Araya, J. S. Areeda, N. Arnaud, K. G. Arun, S. Ascenzi, , G. Ashton, M. Ast, S. M. Aston, P. Astone, P. Aufmuth, C. Aulbert, A. Avila-Alvarez, S. Babak, P. Bacon, M. K. M. Bader, P. T. Baker, F. Baldaccini, , G. Ballardin, S. W. Ballmer, J. C. Barayoga, S. E. Barclay, B. C. Barish, D. Barker, F. Barone, , B. Barr, L. Barsotti, M. Barsuglia, D. Barta, J. Bartlett, I. Bartos, R. Bassiri, A. Basti, , J. C. Batch, C. Baune, V. Bavigadda, M. Bazzan, , C. Beer, M. Bejger, I. Belahcene, M. Belgin, A. S. Bell, B. K. Berger, G. Bergmann, C. P. L. Berry, D. Bersanetti, , A. Bertolini, J. Betzwieser, S. Bhagwat, R. Bhandare, I. A. Bilenko, G. Billingsley, C. R. Billman, J. Birch, R. Birney, O. Birnholtz, S. Biscans, , A. S. Biscoveanu, A. Bisht, M. Bitossi, C. Biwer, M. A. Bizouard, J. K. Blackburn, J. Blackman, C. D. Blair, D. G. Blair, R. M. Blair, S. Bloemen, O. Bock, M. Boer, G. Bogaert, A. Bohe, F. Bondu, R. Bonnand, B. A. Boom, R. Bork, V. Boschi, , S. Bose, , Y. Bouffanais, A. Bozzi, C. Bradaschia, P. R. Brady, V. B. Braginsky ∗ , M. Branchesi, , J. E. Brau, T. Briant, A. Brillet, M. Brinkmann, V. Brisson, P. Brockill, J. E. Broida, A. F. Brooks, D. A. Brown, D. D. Brown, N. M. Brown, S. Brunett, C. C. Buchanan, A. Buikema, T. Bulik, H. J. Bulten, , A. Buonanno, , D. Buskulic, C. Buy, R. L. Byer, M. Cabero, L. Cadonati, G. Cagnoli, , C. Cahillane, J. Calder´on Bustillo, T. A. Callister, E. Calloni, , J. B. Camp, W. Campbell,
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LIGO, California Institute of Technology, Pasadena, CA 91125, USA Louisiana State University, Baton Rouge, LA 70803, USA American University, Washington, D.C. 20016, USA Universit`a di Salerno, Fisciano, I-84084 Salerno, Italy INFN, Sezione di Napoli, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy University of Florida, Gainesville, FL 32611, USA LIGO Livingston Observatory, Livingston, LA 70754, USA Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP),Universit´e Savoie Mont Blanc, CNRS/IN2P3, F-74941 Annecy-le-Vieux, France University of Sannio at Benevento, I-82100 Benevento,Italy and INFN, Sezione di Napoli, I-80100 Napoli, Italy Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik, D-30167 Hannover, Germany Nikhef, Science Park, 1098 XG Amsterdam, The Netherlands LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Instituto Nacional de Pesquisas Espaciais, 12227-010 S˜ao Jos´e dos Campos, S˜ao Paulo, Brazil INFN, Gran Sasso Science Institute, I-67100 L’Aquila, Italy INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA Leibniz Universit¨at Hannover, D-30167 Hannover, Germany Universit`a di Pisa, I-56127 Pisa, Italy INFN, Sezione di Pisa, I-56127 Pisa, Italy Australian National University, Canberra, Australian Capital Territory 0200, Australia California State University Fullerton, Fullerton, CA 92831, USA LAL, Univ. Paris-Sud, CNRS/IN2P3, Universit´e Paris-Saclay, F-91898 Orsay, France Chennai Mathematical Institute, Chennai 603103, India Universit`a di Roma Tor Vergata, I-00133 Roma, Italy Universit¨at Hamburg, D-22761 Hamburg, Germany INFN, Sezione di Roma, I-00185 Roma, Italy Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik, D-14476 Potsdam-Golm, Germany APC, AstroParticule et Cosmologie, Universit´e Paris Diderot,CNRS/IN2P3, CEA/Irfu, Observatoire de Paris,Sorbonne Paris Cit´e, F-75205 Paris Cedex 13, France West Virginia University, Morgantown, WV 26506, USA Universit`a di Perugia, I-06123 Perugia, Italy INFN, Sezione di Perugia, I-06123 Perugia, Italy European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy Syracuse University, Syracuse, NY 13244, USA SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom LIGO Hanford Observatory, Richland, WA 99352, USA Wigner RCP, RMKI, H-1121 Budapest, Konkoly Thege Mikl´os ´ut 29-33, Hungary Columbia University, New York, NY 10027, USA Stanford University, Stanford, CA 94305, USA Universit`a di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova, Italy INFN, Sezione di Padova, I-35131 Padova, Italy Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, 00-716, Warsaw, Poland Center for Relativistic Astrophysics and School of Physics,Georgia Institute of Technology, Atlanta, GA 30332, USA University of Birmingham, Birmingham B15 2TT, United Kingdom Universit`a degli Studi di Genova, I-16146 Genova, Italy INFN, Sezione di Genova, I-16146 Genova, Italy RRCAT, Indore MP 452013, India Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom Caltech CaRT, Pasadena, CA 91125, USA University of Western Australia, Crawley, Western Australia 6009, Australia Department of Astrophysics/IMAPP, Radboud University Nijmegen,P.O. Box 9010, 6500 GL Nijmegen, The Netherlands Artemis, Universit´e Cˆote d’Azur, CNRS, Observatoire Cˆote d’Azur, CS 34229, F-06304 Nice Cedex 4, France Institut de Physique de Rennes, CNRS, Universit´e de Rennes 1, F-35042 Rennes, France Washington State University, Pullman, WA 99164, USA Universit`a degli Studi di Urbino ’Carlo Bo’, I-61029 Urbino, Italy INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy University of Oregon, Eugene, OR 97403, USA Laboratoire Kastler Brossel, UPMC-Sorbonne Universit´es, CNRS,ENS-PSL Research University, Coll`ege de France, F-75005 Paris, France Carleton College, Northfield, MN 55057, USA Astronomical Observatory Warsaw University, 00-478 Warsaw, Poland VU University Amsterdam, 1081 HV Amsterdam, The Netherlands University of Maryland, College Park, MD 20742, USA Laboratoire des Mat´eriaux Avanc´es (LMA), CNRS/IN2P3, F-69622 Villeurbanne, France Universit´e Claude Bernard Lyon 1, F-69622 Villeurbanne, France Universit`a di Napoli ’Federico II’, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA RESCEU, University of Tokyo, Tokyo, 113-0033, Japan. University of Adelaide, Adelaide, South Australia 5005, Australia Tsinghua University, Beijing 100084, China Texas Tech University, Lubbock, TX 79409, USA The University of Mississippi, University, MS 38677, USA The Pennsylvania State University, University Park, PA 16802, USA National Tsing Hua University, Hsinchu City, 30013 Taiwan, Republic of China Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia University of Chicago, Chicago, IL 60637, USA Kenyon College, Gambier, OH 43022, USA Korea Institute of Science and Technology Information, Daejeon 305-806, Korea University of Cambridge, Cambridge CB2 1TN, United Kingdom Universit`a di Roma ’La Sapienza’, I-00185 Roma, Italy University of Brussels, Brussels 1050, Belgium Sonoma State University, Rohnert Park, CA 94928, USA Montana State University, Bozeman, MT 59717, USA Center for Interdisciplinary Exploration & Research in Astrophysics (CIERA),Northwestern University, Evanston, IL 60208, USA Universitat de les Illes Balears, IAC3—IEEC, E-07122 Palma de Mallorca, Spain The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA Bellevue College, Bellevue, WA 98007, USA Institute for Plasma Research, Bhat, Gandhinagar 382428, India The University of Sheffield, Sheffield S10 2TN, United Kingdom California State University, Los Angeles, 5154 State University Dr, Los Angeles, CA 90032, USA Universit`a di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy INFN, Trento Institute for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy Cardiff University, Cardiff CF24 3AA, United Kingdom Montclair State University, Montclair, NJ 07043, USA National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan Canadian Institute for Theoretical Astrophysics,University of Toronto, Toronto, Ontario M5S 3H8, Canada MTA E¨otv¨os University, “Lendulet” Astrophysics Research Group, Budapest 1117, Hungary School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
University and Institute of Advanced Research, Gandhinagar, Gujarat 382007, India
IISER-TVM, CET Campus, Trivandrum Kerala 695016, India
University of Szeged, D´om t´er 9, Szeged 6720, Hungary
Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA
Tata Institute of Fundamental Research, Mumbai 400005, India
INAF, Osservatorio Astronomico di Capodimonte, I-80131, Napoli, Italy
University of Michigan, Ann Arbor, MI 48109, USA
Rochester Institute of Technology, Rochester, NY 14623, USA
NCSA, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
University of Bia(cid:32)lystok, 15-424 Bia(cid:32)lystok, Poland
SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom
University of Southampton, Southampton SO17 1BJ, United Kingdom
University of Washington Bothell, 18115 Campus Way NE, Bothell, WA 98011, USA
Institute of Applied Physics, Nizhny Novgorod, 603950, Russia
Seoul National University, Seoul 151-742, Korea
Inje University Gimhae, 621-749 South Gyeongsang, Korea
National Institute for Mathematical Sciences, Daejeon 305-390, Korea
Pusan National University, Busan 609-735, Korea
NCBJ, 05-400 ´Swierk-Otwock, Poland
Institute of Mathematics, Polish Academy of Sciences, 00656 Warsaw, Poland
The School of Physics & Astronomy, Monash University, Clayton 3800, Victoria, Australia
Hanyang University, Seoul 133-791, Korea
The Chinese University of Hong Kong, Shatin, NT, Hong Kong
University of Alabama in Huntsville, Huntsville, AL 35899, USA
ESPCI, CNRS, F-75005 Paris, France
University of Minnesota, Minneapolis, MN 55455, USA
Universit`a di Camerino, Dipartimento di Fisica, I-62032 Camerino, Italy
Southern University and A&M College, Baton Rouge, LA 70813, USA
The University of Melbourne, Parkville, Victoria 3010, Australia
College of William and Mary, Williamsburg, VA 23187, USA
Instituto de F´ısica Te´orica, University Estadual Paulista/ICTP SouthAmerican Institute for Fundamental Research, S˜ao Paulo SP 01140-070, Brazil
Whitman College, 345 Boyer Avenue, Walla Walla, WA 99362 USA
Universit´e de Lyon, F-69361 Lyon, France
Hobart and William Smith Colleges, Geneva, NY 14456, USA
Janusz Gil Institute of Astronomy, University of Zielona G´ora, 65-265 Zielona G´ora, Poland
King’s College London, University of London, London WC2R 2LS, United Kingdom
IISER-Kolkata, Mohanpur, West Bengal 741252, India
Indian Institute of Technology, Gandhinagar Ahmedabad Gujarat 382424, India
Andrews University, Berrien Springs, MI 49104, USA
Universit`a di Siena, I-53100 Siena, Italy
Trinity University, San Antonio, TX 78212, USA
University of Washington, Seattle, WA 98195, USA
Abilene Christian University, Abilene, TX 79699, USA
We employ gravitational-wave radiometry to map the gravitational waves stochastic backgroundexpected from a variety of contributing mechanisms and test the assumption of isotropy us-ing data from Advanced LIGO’s first observing run. We also search for persistent gravita-tional waves from point sources with only minimal assumptions over the 20 - 1726 Hz frequencyband. Finding no evidence of gravitational waves from either point sources or a stochastic back-ground, we set limits at 90% confidence. For broadband point sources, we report upper limitson the gravitational wave energy flux per unit frequency in the range F α, Θ ( f ) < (0 . − × − erg cm − s − Hz − ( f/
25 Hz) α − depending on the sky location Θ and the spectral power in-dex α . For extended sources, we report upper limits on the fractional gravitational wave energydensity required to close the Universe of Ω( f, Θ) < (0 . − . × − sr − ( f/
25 Hz) α depending onΘ and α . Directed searches for narrowband gravitational waves from astrophysically interesting ob-jects (Scorpius X-1, Supernova 1987 A, and the Galactic Center) yield median frequency-dependentlimits on strain amplitude of h < (6 . , . , and 7 . × − respectively, at the most sensitivedetector frequencies between 130 – 175 Hz. This represents a mean improvement of a factor of 2across the band compared to previous searches of this kind for these sky locations, considering thedifferent quantities of strain constrained in each case. Introduction. —A stochastic gravitational-wave back-ground (SGWB) is expected from a variety of mech-anisms [1–5]. Given the recent observations of binaryblack hole mergers GW150914 and GW151226 [6, 7], weexpect the SGWB to be nearly isotropic [8] and domi-nated [9] by compact binary coalescences [10–12]. TheLIGO and Virgo Collaborations have pursued the searchfor an isotropic stochastic background from LIGO’s firstobservational run [13]. Here, we adopt an eyes-wide-openphilosophy and relax the assumption of isotropy in orderto allow for the greater range of possible signals. Wesearch for an anisotropic background, which could indi-cate a richer, more interesting cosmology than currentmodels. We present the results of a generalized searchfor a stochastic signal with an arbitrary angular distri-bution mapped over all directions in the sky.Our search has three components. First, we utilize abroadband radiometer analysis [14, 15], optimized for de-tecting a small number of resolvable point sources. Thismethod is not applicable to extended sources. Second,we employ a spherical harmonic decomposition [16, 17],which can be employed for point sources but is bettersuited to extended sources. Last, we carry out a nar-rowband radiometer search directed at the sky positionof three astrophysically interesting objects: Scorpius X-1(Sco X-1) [18, 19], Supernova 1987 A (SN 1987A) [20, 21],and the Galactic Center (GC) [22].These three search methods are capable of detecting awide range of possible signals with only minimal assump-tions about the signal morphology. We find no evidence of persistent gravitational waves, and set limits on broad-band emission of gravitational waves as a function of skyposition. We also set narrowband limits as a function offrequency for the three selected sky positions.
Data. —We analyze data from Advanced LIGO’s 4 kmdetectors in Hanford, WA (H1) and Livingston, LA (L1)during the first observing run (O1), from 15:00 UTC,Sep 18, 2015 – 16:00 UTC, Jan 12, 2016. During O1,the detectors reached an instantaneous strain sensitivityof 7 × − Hz − / in the most sensitive region between100 – 300 Hz , and collected 49.67 days of coincident H1L1data. The O1 observing run saw the first direct detectionof gravitational waves and the first direct observation ofmerging black holes [6, 7].For our analysis, the time-series data are down-sampled to 4096 Hz from 16 kHz, and divided into 192 s,50% overlapping, Hann-windowed segments, which arehigh-pass filtered with a 16th order Butterworth digitalfilter with knee frequency of 11 Hz (following [13, 23]).We apply data quality cuts in the time domain in or-der to remove segments associated with instrumental ar-tifacts and hardware injections used for signal valida-tion [24, 25]. We also exclude segments containing knowngravitational-wave signals. Finally, we apply a standardnon-stationarity cut (see, e.g., [26]), to eliminate seg-ments that do not behave as Gaussian noise. These cutsremove 35% of the data. With all vetoes applied, thetotal live time for 192 s segments is 29.85 days.The data segments are Fourier transformed and coarse-grained to produce power spectra with a resolutionof 1 /
32 Hz. This is a finer frequency resolution thanthe 1 / Method. — The main goal of a stochastic search is toestimate the fractional contribution of the energy densityin gravitational waves Ω gw to the total energy densityneeded to close the Universe ρ c . This is defined byΩ gw ( f ) = fρ c dρ gw df (1)where f is frequency and dρ gw represents the energy den-sity between f and f + df [29]. For a stationary andunpolarized signal, ρ gw can be factored into an angularpower P (Θ) and a spectral shape H ( f ) [30], such thatΩ gw ( f ) = 2 π H f H ( f ) (cid:90) d Θ P (Θ) , (2)with Hubble constant H = 68 km s − Mpc − from [31].The angular power P (Θ) represents the gravitationalwave power at each point in the sky. To express thisin terms of the fractional energy density, we define theenergy density spectrum as a function of sky positionΩ( f, Θ) = 2 π H f H ( f ) P (Θ) . (3)We define a similar quantity for the energy flux, where F ( f, Θ) = c π G f H ( f ) P (Θ) (4)has units of erg cm − s − Hz − sr − [15, 16], c is thespeed of light and G is Newton’s gravitational constant. Point sources versus extended sources. —We employtwo different methods to estimate P (Θ) based on thecross-correlation of data streams from a pair of detectors[17, 29]. The radiometer method [14, 15] assumes thatthe cross-correlation signal is dominated by a small num-ber of resolvable point sources. The point source poweris given by P Θ and the angular power spectrum is then P (Θ) ≡ P Θ δ (Θ , Θ ) . (5) Although the radiometer method provides the opti-mal method for detecting resolvable point sources, itis not well-suited for describing diffuse or extendedsources, which may have an arbitrary angular distribu-tion. Hence, we also implement a complementary spher-ical harmonic decomposition (SHD) algorithm, in whichthe sky map is decomposed into components Y lm (Θ) withcoefficients P lm [16]: P (Θ) ≡ (cid:88) lm P lm Y lm (Θ) . (6)Here, the sum over l runs from 0 to l max and − l ≤ m ≤ l .We discuss the choice of l max below. While the SHD algo-rithm has comparably worse sensitivity to point sourcesthan the radiometer algorithm, it accounts for the detec-tor response, producing more accurate sky maps. Spectral models. —In both the radiometer algorithmand the spherical harmonic decomposition algorithm, wemust choose a spectral shape H ( f ). We model the spec-tral dependence of Ω gw ( f ) as a power law: H ( f ) = (cid:18) ff ref (cid:19) α − , (7)where f ref is an arbitrary reference frequency and α isthe spectral index (see also [13]). The spectral modelwill also affect the angular power spectrum, so P (Θ) isimplicitly a function of α .We can rewrite the energy density map Ω( f, Θ) to em-phasize the spectral properties, such thatΩ( f, Θ) = Ω α (Θ) (cid:18) ff ref (cid:19) α , (8)where Ω α (Θ) = 2 π H f P (Θ) (9)has units of fractional energy density per steradianΩ gw sr − . The spherical harmonic analysis presentsskymaps of Ω α (Θ). Note that when P (Θ) = P (themonopole moment), we recover a measurement for theenergy density of the isotropic gravitational wave back-ground. Similarly, the gravitational wave energy flux canbe expressed as F ( f, Θ) = F α (Θ) (cid:18) ff ref (cid:19) α − , (10)where F α (Θ) = c π G f P (Θ) . (11)In the radiometer case we calculate the flux in each di-rection F α, Θ = c π G f P Θ , (12)which is obtained by integrating Equation 11 overthe sphere for the point-source signal model de-scribed in Equation 5. This quantity has units oferg cm − s − Hz − . Following [13, 32], we choose f ref =25 Hz, corresponding to the most sensitive frequency inthe spectral band for a stochastic search with the Ad-vanced LIGO network at design sensitivity.We consider three spectral indices: α = 0, correspond-ing to a flat energy density spectrum (expected frommodels of a cosmological background), α = 2 /
3, cor-responding to the expected shape from a population ofcompact binary coalescences, and α = 3, correspondingto a flat strain power spectral density spectrum [17, 32].The different spectral models are summarized in Table I. Cross Correlation. — A stochastic background wouldinduce low-level correlation between the two LIGO de-tectors. Although the signal is expected to be buried inthe detector noise, the cross-correlation signal-to-noiseratio (SNR) grows with the square root of integrationtime [29]. The cross correlation between two detectors,with (one-sided strain) power spectral density P i ( f, t ) fordetector i , is encoded in what is known as “the dirtymap” [16]: X ν = (cid:88) ft γ ∗ ν ( f, t ) H ( f ) P ( f, t ) P ( f, t ) C ( f, t ) . (13)Here, ν is an index, which can refer to either individualpoints on the sky (the pixel basis) or different lm indices(the spherical harmonic basis). The variable C ( f, t ) isthe cross-power spectral density measured between thetwo LIGO detectors at some segment time t . The sumruns over all segment times and all frequency bins. Thevariable γ ν ( f, t ) is a generalization of the overlap reduc-tion function, which is a function of the separation andrelative orientation between the detectors, and charac-terizes the frequency response of the detector pair [33];see [16] for an exact definition.We can think of X ν as a sky map representation ofthe raw cross-correlation measurement before deconvolv-ing the detector response. The associated uncertainty isencoded in the Fisher matrix:Γ µν = (cid:88) ft γ ∗ µ ( f, t ) H ( f ) P ( f, t ) P ( f, t ) γ ν ( f, t ) , (14)where ∗ denotes complex conjugation.Once X ν and Γ µν are calculated, we have the ingredi-ents to calculate both the radiometer map and the SHDmap. However, the inversion of Γ µν is required to cal-culate the maximum likelihood estimators of GW powerˆ P µ = Γ − µν X ν [16]. For the radiometer, the correlationsbetween neighbouring pixels can be ignored. The ra-diometer map is given byˆ P Θ =(Γ ΘΘ ) − X Θ σ radΘ =(Γ ΘΘ ) − / , (15) where the standard deviation σ radΘ is the uncertainty as-sociated with the point source amplitude estimator ˆ P Θ ,and Γ ΘΘ is a diagonal entry of the Fisher matrix for apointlike signal. For the SHD analysis, the full Fishermatrix Γ µν must be taken into account, which includessingular eigenvalues associated with modes to which thedetector pair is insensitive. The inversion of Γ µν is simpli-fied by a singular value decomposition regularization. Inthis decomposition, modes associated with the smallesteigenvalues contribute the least sensitivity to the detec-tor network. Removing a fraction of the lowest eigen-modes “regularizes” Γ µν without significantly affectingthe sensitivity (see [16]). The estimator for the SHD andcorresponding standard deviation are given byˆ P lm = (cid:88) l (cid:48) m (cid:48) (Γ − R ) lm,l (cid:48) m (cid:48) X l (cid:48) m (cid:48) σ SHD lm = (cid:2) (Γ − R ) lm,lm (cid:3) / , (16)where Γ R is the regularized Fisher matrix. We remove1 / Angular scale. —In order to carry out the calculation inEq. 16, we must determine a suitable angular scale, whichwill depend on the angular resolution of the detector net-work and vary with spectral index α . The diffraction-limited spot size on the sky θ (in radians) is given by θ = c df ≈
50 Hz f α , (17)where d = 3000 km is the separation of the LIGO detec-tors. The frequency f α corresponds to the most sensitivefrequency in the detector band for a power law with spec-tral index α given the detector noise power spectra [15].In order to determine f α we find the frequency at which apower-law with index α is tangent to the single-detector“power-law integrated curve” [34]. The angular resolu-tion scale is set by the maximum spherical harmonic or-der l max , which we can express as a function of α since l max = πθ ≈ πf α . (18)The values of f α , θ , and l max for three different values of α are shown in Table I. As the spectral index increases,so does f α , decreasing the angular resolution limit, thusincreasing l max . Angular power spectra. —For the SHD map, we calcu-late the angular power spectra C l , which describe theangular scale of structure in the clean map, using an un-biased estimator [16, 17]ˆ C l ≡ l + 1 (cid:88) m (cid:104) | ˆ P lm | − (Γ − R ) lm,lm (cid:105) . (19) Narrowband radiometer. —The radiometer algorithmcan be applied to the detection of persistent gravitationalwaves from narrowband point sources associated with a
All-sky (broadband) Results
Max SNR (% p -value) Upper limit range α Ω gw H ( f ) f α (Hz) θ (deg) l max BBR SHD BBR ( × − ) SHD ( × − )0 constant ∝ f − / ∝ f / ∝ f − / ∝ f constant 256.50 11 16 3.43 (47) 3.86 (11) 0.1 – 0.9 0.4 – 2.8TABLE I. Values of the power-law index α investigated in this analysis, the shape of the energy density and strain powerspectrum. The characteristic frequency f α , angular resolution θ (Eq. 17), and corresponding harmonic order l max (Eq. 18)for each α are also shown. The right hand section of the table shows the maximum SNR, associated significance ( p -value) andbest upper limit values from the broadband radiometer (BBR) and the spherical harmonic decomposition (SHD). The BBRsets upper limits on energy flux [erg cm − s − Hz − ( f/
25 Hz) α − ] while the SHD sets upper limits on the normalized energydensity [sr − ( f/
25 Hz) α ] of the SGWB. given sky position [15, 17]. We “point” the radiometer inthe direction of three interesting sky locations: Sco X-1,the Galactic Center, and the remnant of supernova SN1987A.Scorpius X-1 (Sco X-1) is a low-mass X-ray binary be-lieved to host a neutron star that is potentially spun upthrough accretion, in which gravitational wave emissionmay provide a balancing spin-down torque [18, 19, 35,36]. The frequency of the gravitational wave signal isexpected to spread due to the orbital motion of the neu-tron star. At frequencies below ∼
930 Hz this Dopplerline broadening effect is less than 1/4 Hz, the frequencybin width selected in past analyses [15, 17]. At higherfrequencies, the signal is certain to span multiple bins.We therefore combine multiple 1/32 Hz frequency binsto form optimally sized combined bins at each frequency,accounting for the expected signal broadening due to thecombination of the motion of the Earth around the Sun,the binary orbital motion, and any other intrinsic modu-lation. For more detail on the method of combining bins,see the technical supplement to this paper [37].The possibility of a young neutron star in SN 1987A[20, 21] and the likelihood of many unknown, isolatedneutron stars in the Galactic Center region [22] indicatepotentially interesting candidates for persistent gravita-tional wave emission. We combine bins to include thesignal spread due to Earth’s modulation. For SN 1987A,we choose a combined bin size of 0.09 Hz. We would besensitive to spin modulations up to | ˙ ν | < × − Hz s − within our O1 observation time spanning 116 days. TheGalactic Center is at a lower declination with respectto the orbital plane of the Earth. The Earth modulationterm is therefore more significant so for the Galactic Cen-ter we choose combined bins of 0.53 Hz across the band.In this case we are sensitive to a frequency modulationin the range | ˙ ν | < . × − Hz s − . Significance. —To assess the significance of the SNR inthe combined bins of the narrowband radiometer spectra,we simulate many realizations of the strain power consis-tent with Gaussian noise in each individual frequency bin.Combining these in the same way as the actual analysisleaves us with a distribution of maximum SNR values across the whole frequency band for many simulations ofnoise.For a map of the whole sky, the distribution of maxi-mum SNR is complicated by the many dependent trialsdue to covariances between different pixels (or patches)on the sky. We calculate this distribution numerically bysimulating many realizations of the dirty map X ν withexpected covariances described by the Fisher matrix Γ µν (cf. Eqs. 13 and 14, respectively). This distribution isthen used to calculate the significance (or p-value) of agiven SNR recovered from the sky maps [17]. We takea p-value of 0.01 or less to indicate a significant result.The absence of any significant events indicates the dataare consistent with no signal being detected, in whichcase we quote Bayesian upper limits at 90% confidence[15, 17] Results —The search yields four data products:
Radiometer sky maps , optimized for broadbandpoint sources, are shown in Fig. 1. The top row showsthe SNR. Each column corresponds to a different spec-tral index, α = 0 , / .
32, 3 . .
43 corresponding to false-alarm probabilities typi-cal of what would be expected from Gaussian noise; seeTable I. We find no evidence of a signal, and so set limitson gravitational-wave energy flux, which are provided inthe bottom row of Fig. 1 and summarized in Table I.
SHD sky maps , suitable for characterizing ananisotropic stochastic background, are shown in Fig. 2.The top row shows the SNR and each column corre-sponds to a different spectral index ( α = 0 , / .
96, 3 .
06, and3 .
86 corresponding to false-alarm probabilities typical ofthose expected from Gaussian noise; see Table I. Failingevidence of a signal, we set limits on energy density perunit solid angle, which are provided in the bottom row ofFig. 2 and summarised in Table I. Interactive visualiza-tions of the SNR and upper limit maps are also availableonline [38].
Angular power spectra are derived from the SHDsky maps. We present upper limits at 90% confidence onthe angular power spectrum indices C l from the spherical0 FIG. 1. All-sky radiometer maps for point-like sources showing SNR (top) and upper limits at 90% confidence on energy flux F α, Θ [erg cm − s − Hz − ] (bottom) for three different power-law indices, α = 0 , / p = 7% , p = 12% , p = 47% (see Table I).FIG. 2. All-sky spherical harmonic decomposition maps for extended sources showing SNR (top) and upper limits at 90 %confidence on the energy density of the gravitational wave background Ω α [ sr − ] (bottom) for three different power-law indices α = 0 , / p = 18% , p = 11% , p = 11% (see Table I). harmonic analysis in Figure 3. FIG. 3. Upper limits on C l at 90% confidence vs l for theSHD analyses for α = 0 (top, blue squares), α = 2 / α = 3 (bottom, green triangles). Radiometer spectra , suitable for the detection ofa narrowband point source associated with a given skyposition, are given in Fig. 4, the main results of which aresummarised in Table II. For the three sky locations (ScoX-1, SN 1987A and the Galactic Center), we calculatethe SNR in appropriately sized combined bins across the LIGO band. For Sco X-1, the loudest observed SNR is4.58, which is consistent with Gaussian noise. For SN1987A and the Galactic Center, we observe maximumSNRs of 4.07 and 3.92 respectively, corresponding to falsealarm probabilities consistent with noise; see Table II.Since we observe no statistically significant signal, weset 90% confidence limits on the peak strain amplitude h for each optimally sized frequency bin. Upper limits wereset using a Bayesian methodology with the constraintthat h > h < . × − at 134 Hz , h < . × Narrowband Radiometer Results
Direction Max SNR p-value (%) Frequency band (Hz) Best UL ( × − ) Frequency band (Hz)Sco X-1 4.58 10 616 −
617 6.7 134 − −
196 5.5 172 − − − − at 172 Hz and h < . × − at 172 Hz from ScoX-1, SN 1987A and the Galactic Center respectively inthe most sensitive part of the LIGO band Conclusions.
We find no evidence to support the de-tection of either point-like or extended sources and setupper limits on the energy flux and energy density of theanisotropic gravitational wave sky. We assume three dif-ferent power law models for the gravitational wave back-ground spectrum. Our mean upper limits present an im-provement over initial LIGO results of a factor of 8 in fluxfor the α = 3 broadband radiometer and factors of 60 and4 for the spherical harmonic decomposition method for α = 0 and 3 respectively [17, 40]. We present the first up-per limits for an anisotropic stochastic background dom-inated by compact binary inspirals (with an Ω gw ∝ f / spectrum) of Ω / (Θ) < − × − sr − dependingon sky position. We can directly compare the monopolemoment of the spherical harmonic decomposition to theisotropic search point estimate Ω / = (3 . ± . × − from [13]. We obtain Ω / = (2 π / H ) f √ π P =(4 . ± . × − . The two results are statistically con-sistent. Our spherical harmonic estimate of Ω / has alarger uncertainty than the dedicated isotropic search be-cause of the larger number of (covariant) parameters es-timated when l max >
0. We also set upper limits onthe gravitational wave strain from point sources locatedin the directions of Sco X-1, the Galactic Center andSupernova 1987A. The narrowband results improve onprevious limits of the same kind by more than a factorof 10 in strain at frequencies below 50 Hz and above 300Hz, with a mean improvement of a factor of 2 across theband [17].
Acknowledgments. — The authors gratefully acknowledgethe support of the United States National Science Founda-tion (NSF) for the construction and operation of the LIGOLaboratory and Advanced LIGO as well as the Science andTechnology Facilities Council (STFC) of the United Kingdom,the Max-Planck-Society (MPS), and the State of Niedersach-sen/Germany for support of the construction of AdvancedLIGO and construction and operation of the GEO600 detec-tor. Additional support for Advanced LIGO was providedby the Australian Research Council. The authors gratefullyacknowledge the Italian Istituto Nazionale di Fisica Nucle-are (INFN), the French Centre National de la Recherche Sci-entifique (CNRS) and the Foundation for Fundamental Re- search on Matter supported by the Netherlands Organisationfor Scientific Research, for the construction and operation ofthe Virgo detector and the creation and support of the EGOconsortium. The authors also gratefully acknowledge researchsupport from these agencies as well as by the Council of Scien-tific and Industrial Research of India, Department of Scienceand Technology, India, Science & Engineering Research Board(SERB), India, Ministry of Human Resource Development,India, the Spanish Ministerio de Econom´ıa y Competitividad,the Conselleria d’Economia i Competitivitat and Conselleriad’Educaci´o, Cultura i Universitats of the Govern de les IllesBalears, the National Science Centre of Poland, the EuropeanCommission, the Royal Society, the Scottish Funding Council,the Scottish Universities Physics Alliance, the Hungarian Sci-entific Research Fund (OTKA), the Lyon Institute of Origins(LIO), the National Research Foundation of Korea, IndustryCanada and the Province of Ontario through the Ministry ofEconomic Development and Innovation, the Natural Scienceand Engineering Research Council Canada, Canadian Insti-tute for Advanced Research, the Brazilian Ministry of Science,Technology, and Innovation, Funda¸c˜ao de Amparo `a Pesquisado Estado de S˜ao Paulo (FAPESP), Russian Foundation forBasic Research, the Leverhulme Trust, the Research Corpo-ration, Ministry of Science and Technology (MOST), Taiwanand the Kavli Foundation. The authors gratefully acknowl-edge the support of the NSF, STFC, MPS, INFN, CNRS andthe State of Niedersachsen/Germany for provision of compu-tational resources. This is LIGO document LIGO-P1600259.[1] M. Maggiore, Phys. Rep. , 283 (2000).[2] B. Allen, in
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In this supplement we describe how we use the nar-rowband, directed radiometer search[14] to make a state-ment on the gravitational wave (GW) strain amplitude h of a persistent source given some power described byour cross correlation statistic. We take into account theexpected modulation of the quasi-monochromatic sourcefrequency over the duration of the observation. We dothis by combining individual search frequency bins into combined bins that cover the extent of the possible mod-ulation. Source Model. —We can relate GW frequency emittedin the source frame f s to the observed frequency in thedetector frame f d using the relation f d = [1 − A ( t ) − B ( t ) − C ( t )] f s (20)where A ( t ) takes into account the modulation of the sig-nal due to Earth’s motion with respect to the source, B ( t )takes into account the orbital modulation for a source ina binary orbit, and C ( t ) takes into account any othermodulation due to intrinsic properties of the source (forexample any spin-down terms for isolated neutron stars).The Earth modulation term is given by A ( t ) = (cid:126)v E ( t ) · ˆ kc (21)where (cid:126)v E is the velocity of the Earth. In equatorial co-ordinates: (cid:126)v E ( t ) = ωR [sin θ ( t )ˆ u − cos θ ( t ) cos φ ˆ v − cos θ ( t ) sin φ ˆ w ] , (22)in which R is the mean distance between Earth and theSun, ω is the angular velocity of the Earth around theSun and φ = 23 ◦ , 26 min, 21.406 sec is the obliquity ofthe ecliptic. The time dependent phase angle θ ( t ) is givenby θ ( t ) = 2 π ( t − T VE ) /T year , where T year is the numberof seconds in a year and T VE is the time at the Vernalequinox. The unit vector ˆ k pointing from the source tothe earth is given by ˆ k = − cos δ cos α ˆ u − cos δ sin α ˆ v − sin δ ˆ w , where δ is the declination and α is the right as-cension of the source on the sky.In the case of a source in a binary system, the binaryterm (for a circular orbit) is given by B ( t ) = 2 πP orb a sin i × cos (cid:18) π t − T asc P orb (cid:19) (23)where a sin i is the projection of the semi-major axis (inunits of light seconds) of the binary orbit on the line ofsight, T asc is the time of the orbital ascending node and P orb is the binary orbital period.In the case of an isolated source we set B ( t ) = 0, while C ( t ) can take into account any spin modulation expected to occur during an observation time. In the absence ofa model for this behaviour, a statement can be madeon the maximum allowable spin modulation that can betolerated by our search. Search. —The narrowband radiometer search is runwith 192 s segments and 1/32 Hz frequency bins. Foreach 1/32 Hz frequency bin we combine the number ofbins required to account for the extent of any signal fre-quency modulation The source frequency f s is taken asthe center of a frequency bin. We calculate the minimumand maximum detector frequency f d over the time of theanalysis corresponding to the respective edges of the binin order to define our combined bins.We combine the detection statistic Y i and variance σ Y,i into a new combined statistic Y c for each representativefrequency bin via Y c = a (cid:88) i = − b Y i and σ Y,c = a (cid:88) i = − b σ Y,i , (24)where i represents the index for each of the frequency binswe want to combine. If we assign i = 0 to the bin wherethe source frequency falls, then a and b are the number offrequency bins we want to combine above and below thesource frequency bin, respectively. The overlapping bins,which ensure we do not lose signal due to edge effects,create correlations between our combined bins. Significance. —To establish significance, we assumethat the strain power in each frequency bin is consis-tent with Gaussian noise and simulate > Y i in each frequency bin i by drawing from a Gaus-sian distribution with σ = σ Y,i . We then combine thesebins into combined bins as we do in the actual analysisand calculate the maximum of the signal to noise ratio,SNR = Y c /σ Y,c , across all of the combined bins. Weuse the distribution of maximum SNR to establish thesignificance of our results.
Upper limits. —In the absence of a significant detectionstatistic, we set upper limits on the tensor strain ampli-tude h of a gravitational wave source with frequency f s .To take into account the unknown parameters of the sys-tem, such as the polarization ψ and inclination angle ι ,and consider reduced sensitivity to signals that are notcircularly polarized, we calculate a direction-dependentand time-averaged value µ ι,ψ . This value represents ascaling between the true value of the amplitude h andwhat we would measure with our search, and is given by µ ι,ψ = (cid:80) Mj =1 (cid:104) ( A + /h ) F +d j + ( A × /h ) F × d j (cid:105) (cid:16) F +d j + F × d j (cid:17)(cid:80) Mj =1 (cid:16) F +d j + F × d j (cid:17) (25)for each time segment j . Here A + = 12 h (1 + cos ι ) and A × = h cos ι , (26)4and ψ dependence is implicit in F A d j = F A j F A j , where A indicates (+ or × ) polarization and the response func-tions F A j and F A j for the LIGO detectors are defined in[29] (see also [41]). We calculate µ ι,ψ many times for auniform distribution of cos ι and ψ , and then marginalizeover it. We also marginalize over calibration uncertainty,where we assume (as in the past) that calibration uncer-tainty is manifest in a multiplicative factor ( l + 1) > l is normally distributed around 0 with uncertaintygiven by the calibration uncertainty, σ l = 0 .