LIGO: The Laser Interferometer Gravitational-Wave Observatory
LLIGO: The Laser InterferometerGravitational-Wave Observatory
The LIGO Scientific CollaborationB. P. Abbott , R. Abbott , R. Adhikari , P. Ajith , B. Allen , ,G. Allen , R. S. Amin , S. B. Anderson , W. G. Anderson ,M. A. Arain , M. Araya , H. Armandula , P. Armor , Y. Aso ,S. Aston , P. Aufmuth , C. Aulbert , S. Babak , P. Baker ,S. Ballmer , C. Barker , D. Barker , B. Barr , P. Barriga ,L. Barsotti , M. A. Barton , I. Bartos , R. Bassiri ,M. Bastarrika , B. Behnke , M. Benacquista , J. Betzwieser ,P. T. Beyersdorf , I. A. Bilenko , G. Billingsley , R. Biswas ,E. Black , J. K. Blackburn , L. Blackburn , D. Blair ,B. Bland , T. P. Bodiya , L. Bogue , R. Bork , V. Boschi ,S. Bose , P. R. Brady , V. B. Braginsky , J. E. Brau ,D. O. Bridges , M. Brinkmann , A. F. Brooks , D. A. Brown ,A. Brummit , G. Brunet , A. Bullington , A. Buonanno ,O. Burmeister , R. L. Byer , L. Cadonati , J. B. Camp ,J. Cannizzo , K. C. Cannon , J. Cao , L. Cardenas ,S. Caride , G. Castaldi , S. Caudill , M. Cavagli`a ,C. Cepeda , T. Chalermsongsak , E. Chalkley , P. Charlton ,S. Chatterji , S. Chelkowski , Y. Chen , , N. Christensen ,C. T. Y. Chung , D. Clark , J. Clark , J. H. Clayton ,T. Cokelaer , C. N. Colacino , R. Conte , D. Cook ,T. R. C. Corbitt , N. Cornish , D. Coward , D. C. Coyne ,J. D. E. Creighton , T. D. Creighton , A. M. Cruise ,R. M. Culter , A. Cumming , L. Cunningham ,S. L. Danilishin , K. Danzmann , , B. Daudert , G. Davies ,E. J. Daw , D. DeBra , J. Degallaix , V. Dergachev ,S. Desai , R. DeSalvo , S. Dhurandhar , M. D´ıaz ,A. Dietz , F. Donovan , K. L. Dooley , E. E. Doomes ,R. W. P. Drever , J. Dueck , I. Duke , J. -C. Dumas ,J. G. Dwyer , C. Echols , M. Edgar , A. Effler , P. Ehrens ,E. Espinoza , T. Etzel , M. Evans , T. Evans , S. Fairhurst ,Y. Faltas , Y. Fan , D. Fazi , H. Fehrmenn , L. S. Finn ,K. Flasch , S. Foley , C. Forrest , N. Fotopoulos , a r X i v : . [ g r- q c ] M a y IGO A. Franzen , M. Frede , M. Frei , Z. Frei , A. Freise ,R. Frey , T. Fricke , P. Fritschel , V. V. Frolov , M. Fyffe ,V. Galdi , J. A. Garofoli , I. Gholami , J. A. Giaime , ,S. Giampanis , K. D. Giardina , K. Goda , E. Goetz ,L. M. Goggin , G. Gonz´alez , M. L. Gorodetsky , S. Goßler ,R. Gouaty , A. Grant , S. Gras , C. Gray , M. Gray ,R. J. S. Greenhalgh , A. M. Gretarsson , F. Grimaldi ,R. Grosso , H. Grote , S. Grunewald , M. Guenther ,E. K. Gustafson , R. Gustafson , B. Hage , J. M. Hallam ,D. Hammer , G. D. Hammond , C. Hanna , J. Hanson ,J. Harms , G. M. Harry , I. W. Harry , E. D. Harstad ,K. Haughian , K. Hayama , J. Heefner , I. S. Heng ,A. Heptonstall , M. Hewitson , S. Hild , E. Hirose ,D. Hoak , K. A. Hodge , K. Holt , D. J. Hosken ,J. Hough , D. Hoyland , B. Hughey , S. H. Huttner ,D. R. Ingram , T. Isogai , M. Ito , A. Ivanov , B. Johnson ,W. W. Johnson , D. I. Jones , G. Jones , R. Jones , L. Ju ,P. Kalmus , V. Kalogera , S. Kandhasamy , J. Kanner ,D. Kasprzyk , E. Katsavounidis , K. Kawabe ,S. Kawamura , F. Kawazoe , W. Kells , D. G. Keppel ,A. Khalaidovski , F. Y. Khalili , R. Khan , E. Khazanov ,P. King , J. S. Kissel , S. Klimenko , K. Kokeyama ,V. Kondrashov , R. Kopparapu , S. Koranda , D. Kozak ,B. Krishnan , R. Kumar , P. Kwee , P. K. Lam ,M. Landry , B. Lantz , A. Lazzarini , H. Lei , M. Lei ,N. Leindecker , I. Leonor , C. Li , H. Lin , P. E. Lindquist ,T. B. Littenberg , N. A. Lockerbie , D. Lodhia , M. Longo ,M. Lormand , P. Lu , M. Lubinski , A. Lucianetti ,H. L¨uck , , B. Machenschalk , M. MacInnis , M. Mageswaran ,K. Mailand , I. Mandel , V. Mandic , S. M´arka , Z. M´arka ,A. Markosyan , J. Markowitz , E. Maros , I. W. Martin ,R. M. Martin , J. N. Marx , K. Mason , F. Matichard ,L. Matone , R. A. Matzner , N. Mavalvala , R. McCarthy ,D. E. McClelland , S. C. McGuire , M. McHugh ,G. McIntyre , D. J. A. McKechan , K. McKenzie ,M. Mehmet , A. Melatos , A. C. Melissinos ,D. F. Men´endez , G. Mendell , R. A. Mercer , S. Meshkov ,C. Messenger , M. S. Meyer , J. Miller , J. Minelli ,Y. Mino , V. P. Mitrofanov , G. Mitselmakher ,R. Mittleman , O. Miyakawa , B. Moe , S. D. Mohanty ,S. R. P. Mohapatra , G. Moreno , T. Morioka , K. Mors ,K. Mossavi , C. MowLowry , G. Mueller , IGO H. M¨uller-Ebhardt , D. Muhammad , S. Mukherjee ,H. Mukhopadhyay , A. Mullavey , J. Munch ,P. G. Murray , E. Myers , J. Myers , T. Nash , J. Nelson ,G. Newton , A. Nishizawa , K. Numata , J. O’Dell ,B. O’Reilly , R. O’Shaughnessy , E. Ochsner , G. H. Ogin ,D. J. Ottaway , R. S. Ottens , H. Overmier , B. J. Owen ,Y. Pan , C. Pankow , M. A. Papa , , V. Parameshwaraiah ,P. Patel , M. Pedraza , S. Penn , A. Perraca , V. Pierro ,I. M. Pinto , M. Pitkin , H. J. Pletsch , M. V. Plissi ,F. Postiglione , M. Principe , R. Prix , L. Prokhorov ,O. Punken , V. Quetschke , F. J. Raab , D. S. Rabeling ,H. Radkins , P. Raffai , Z. Raics , N. Rainer ,M. Rakhmanov , V. Raymond , C. M. Reed , T. Reed ,H. Rehbein , S. Reid , D. H. Reitze , R. Riesen , K. Riles ,B. Rivera , P. Roberts , N. A. Robertson , , C. Robinson ,E. L. Robinson , S. Roddy , C. R¨over , J. Rollins ,J. D. Romano , J. H. Romie , S. Rowan , A. R¨udiger ,P. Russell , K. Ryan , S. Sakata , L. Sancho de la Jordana ,V. Sandberg , V. Sannibale , L. Santamar´ıa , S. Saraf ,P. Sarin , B. S. Sathyaprakash , S. Sato , M. Satterthwaite ,P. R. Saulson , R. Savage , P. Savov , M. Scanlan ,R. Schilling , R. Schnabel , R. Schofield , B. Schulz ,B. F. Schutz , , P. Schwinberg , J. Scott , S. M. Scott ,A. C. Searle , B. Sears , F. Seifert , D. Sellers ,A. S. Sengupta , A. Sergeev , B. Shapiro , P. Shawhan ,D. H. Shoemaker , A. Sibley , X. Siemens , D. Sigg ,S. Sinha , A. M. Sintes , B. J. J. Slagmolen , J. Slutsky ,J. R. Smith , M. R. Smith , N. D. Smith , K. Somiya ,B. Sorazu , A. Stein , L. C. Stein , S. Steplewski ,A. Stochino , R. Stone , K. A. Strain , S. Strigin ,A. Stroeer , A. L. Stuver , T. Z. Summerscales , K. -X. Sun ,M. Sung , P. J. Sutton , G. P. Szokoly , D. Talukder ,L. Tang , D. B. Tanner , S. P. Tarabrin , J. R. Taylor ,R. Taylor , J. Thacker , K. A. Thorne , A. Th¨uring ,K. V. Tokmakov , C. Torres , C. Torrie , G. Traylor ,M. Trias , D. Ugolini , J. Ulmen , K. Urbanek ,H. Vahlbruch , M. Vallisneri , C. Van Den Broeck ,M. V. van der Sluys , A. A. van Veggel , S. Vass , R. Vaulin ,A. Vecchio , J. Veitch , P. Veitch , C. Veltkamp , A. Villar ,C. Vorvick , S. P. Vyachanin , S. J. Waldman , L. Wallace ,R. L. Ward , A. Weidner , M. Weinert , A. J. Weinstein ,R. Weiss , L. Wen , , S. Wen , K. Wette , J. T. Whelan , , IGO S. E. Whitcomb , B. F. Whiting , C. Wilkinson ,P. A. Willems , H. R. Williams , L. Williams , B. Willke , ,I. Wilmut , L. Winkelmann , W. Winkler , C. C. Wipf ,A. G. Wiseman , G. Woan , R. Wooley , J. Worden ,W. Wu , I. Yakushin , H. Yamamoto , Z. Yan , S. Yoshida ,M. Zanolin , J. Zhang , L. Zhang , C. Zhao , N. Zotov ,M. E. Zucker , H. zur M¨uhlen , and J. Zweizig LIGO - California Institute of Technology, Pasadena, CA 91125, USA Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik, D-30167Hannover, Germany University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA Stanford University, Stanford, CA 94305, USA Louisiana State University, Baton Rouge, LA 70803, USA University of Florida, Gainesville, FL 32611, USA University of Birmingham, Birmingham, B15 2TT, United Kingdom Leibniz Universit¨at Hannover, D-30167 Hannover, Germany Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik, D-14476Golm, Germany Montana State University, Bozeman, MT 59717, USA LIGO - Hanford Observatory, Richland, WA 99352, USA University of Glasgow, Glasgow, G12 8QQ, United Kingdom University of Western Australia, Crawley, WA 6009, Australia LIGO - Massachusetts Institute of Technology, Cambridge, MA 02139, USA Columbia University, New York, NY 10027, USA The University of Texas at Brownsville and Texas Southmost College, Brownsville,TX 78520, USA San Jose State University, San Jose, CA 95192, USA Moscow State University, Moscow, 119992, Russia LIGO - Livingston Observatory, Livingston, LA 70754, USA Washington State University, Pullman, WA 99164, USA University of Oregon, Eugene, OR 97403, USA Syracuse University, Syracuse, NY 13244, USA Rutherford Appleton Laboratory, HSIC, Chilton, Didcot, Oxon OX11 0QX, UnitedKingdom University of Maryland, College Park, MD 20742 USA University of Massachusetts - Amherst, MA 01003 USA NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA University of Michigan, Ann Arbor, MI 48109, USA University of Sannio at Benevento, I-82100 Benevento, Italy The University of Mississippi, University, MS 38677, USA Charles Sturt University, Wagga Wagga, NSW 2678, Australia Caltech-CaRT, Pasadena, CA 91125, USA Carleton College, Northfield, MN 55057, USA The University of Melbourne, Parkville VIC 3010, Australia Cardiff University, Cardiff, CF24 3AA, United Kingdom E¨otv¨os University, ELTE 1053 Budapest, Hungary University of Salerno, 84084 Fisciano (Salerno), Italy The University of Sheffield, Sheffield S10 2TN, United Kingdom The Pennsylvania State University, University Park, PA 16802, USA Inter-University Centre for Astronomy and Astrophysics, Pune - 411007, India Southern University and A&M College, Baton Rouge, LA 70813, USA California Institute of Technology, Pasadena, CA 91125, USA University of Rochester, Rochester, NY 14627, USA The University of Texas at Austin, Austin, TX 78712, USA Australian National University, Canberra, 0200, Australia Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA University of Minnesota, Minneapolis, MN 55455, USA University of Adelaide, Adelaide, SA 5005, Australia University of Southampton, Southampton, SO17 1BJ, United Kingdom Northwestern University, Evanston, IL 60208, USA National Astronomical Observatory of Japan, Tokyo 181-8588, Japan Institute of Applied Physics, Nizhny Novgorod, 603950, Russia University of Strathclyde, Glasgow, G1 1XQ, United Kingdom Loyola University, New Orleans, LA 70118, USA Hobart and William Smith Colleges, Geneva, NY 14456, USA Louisiana Tech University, Ruston, LA 71272, USA Andrews University, Berrien Springs, MI 49104, USA Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain Sonoma State University, Rohnert Park, CA 94928, USA Trinity University, San Antonio, TX 78212, USA Rochester Institute of Technology, Rochester, NY 14623, USA Southeastern Louisiana University, Hammond, LA 70402, USAE-mail: [email protected]
Abstract.
The goal of the Laser Interferometric Gravitational-Wave Observatory(LIGO) is to detect and study gravitational waves of astrophysical origin. Directdetection of gravitational waves holds the promise of testing general relativity in thestrong-field regime, of providing a new probe of exotic objects such as black holesand neutron stars, and of uncovering unanticipated new astrophysics. LIGO, a jointCaltech-MIT project supported by the National Science Foundation, operates threemulti-kilometer interferometers at two widely separated sites in the United States.These detectors are the result of decades of worldwide technology development, design,construction, and commissioning. They are now operating at their design sensitivity,and are sensitive to gravitational wave strains smaller than one part in 10 . Withthis unprecedented sensitivity, the data are being analyzed to detect or place limits ongravitational waves from a variety of potential astrophysical sources. Submitted to:
Rep. Prog. Phys. . Introduction
The prediction of gravitational waves (GWs), oscillations in the space-time metric thatpropagate at the speed of light, is one of the most profound differences between Einstein’sgeneral theory of relativity and the Newtonian theory of gravity that it replaced. GWsremained a theoretical prediction for more than 50 years until the first observationalevidence for their existence came with the discovery and subsequent observations of thebinary pulsar PSR 1913+16, by Russell Hulse and Joseph Taylor. This is a system oftwo neutron stars that orbit each other with a period of 7.75 hours. Precise timing ofradio pulses emitted by one of the neutron stars, monitored now over several decades,shows that their orbital period is slowly decreasing at just the rate predicted for thegeneral-relativistic emission of GWs [1]. Hulse and Taylor were awarded the Nobel Prizein Physics for this work in 1993.In about 300 million years, the PSR 1913+16 orbit will decrease to the point wherethe pair coalesces into a single compact object, a process that will produce directlydetectable gravitational waves. In the meantime, the direct detection of GWs willrequire similarly strong sources – extremely large masses moving with large accelerationsin strong gravitational fields. The goal of LIGO, the Laser Interferometer Gravitational-Wave Observatory [2] is just that: to detect and study GWs of astrophysical origin.Achieving this goal will mark the opening of a new window on the universe, withthe promise of new physics and astrophysics. In physics, GW detection could provideinformation about strong-field gravitation, the untested domain of strongly curved spacewhere Newtonian gravitation is no longer even a poor approximation. In astrophysics,the sources of GWs that LIGO might detect include binary neutron stars (like PSR1913+16 but much later in their evolution); binary systems where a black hole replacesone or both of the neutron stars; a stellar core collapse which triggers a Type IIsupernova; rapidly rotating, non-axisymmetric neutron stars; and possibly processesin the early universe that produce a stochastic background of GWs [3].In the past few years the field has reached a milestone, with decades-old plans tobuild and operate large interferometric GW detectors now realized in several locationsworldwide. This article focuses on LIGO, which operates the most sensitive detectorsyet built. We aim to describe the LIGO detectors and how they operate, explain howthey have achieved their remarkable sensitivity, and review how their data can be usedto learn about a variety of astrophysical phenomena.
2. Gravitational waves
The essence of general relativity is that mass and energy produce a curvature offour-dimensional space-time, and that matter moves in response to this curvature.The Einstein field equations prescribe the interaction between mass and space-time curvature, much as Maxwell’s equations prescribe the relationship betweenelectric charge and electromagnetic fields. Just as electromagnetic waves are time-ependent vacuum solutions to Maxwell’s equations, gravitational waves are time-dependent vacuum solutions to the field equations. Gravitational waves are oscillatingperturbations to a flat, or Minkowski, space-time metric, and can be thought ofequivalently as an oscillating strain in space-time or as an oscillating tidal force betweenfree test masses.As with electromagnetic waves, gravitational waves travel at the speed of light andare transverse in character – i.e. , the strain oscillations occur in directions orthogonalto the direction the wave is propagating. Whereas electromagnetic waves are dipolar innature, gravitational waves are quadrupolar: the strain pattern contracts space alongone transverse dimension, while expanding it along the orthogonal direction in thetransverse plane (see Fig. 1). Gravitational radiation is produced by oscillating multipolemoments of the mass distribution of a system. The principle of mass conservation rulesout monopole radiation, and the principles of linear and angular momentum conservationrule out gravitational dipole radiation. Quadrupole radiation is the lowest allowedform, and is thus usually the dominant form. In this case, the gravitational wave fieldstrength is proportional to the second time derivative of the quadrupole moment ofthe source, and it falls off in amplitude inversely with distance from the source. Thetensor character of gravity – the hypothetical graviton is a spin-2 particle – means thatthe transverse strain field comes in two orthogonal polarizations. These are commonlyexpressed in a linear polarization basis as the ‘+’ polarization (depicted in Fig. 1) andthe ‘ × ’ polarization, reflecting the fact that they are rotated 45 degrees relative to oneanother. An astrophysical GW will, in general, be a mixture of both polarizations.Gravitational waves differ from electromagnetic waves in that they propagateessentially unperturbed through space, as they interact only very weakly with matter.Furthermore, gravitational waves are intrinsically non-linear, because the wave energydensity itself generates additional curvature of space-time. This phenomenon is onlysignificant, however, very close to strong sources of waves, where the wave amplitudeis relatively large. More usually, gravitational waves distinguish themselves fromelectromagnetic waves by the fact that they are very weak. One cannot hope todetect any waves of terrestrial origin, whether naturally occurring or manmade; insteadone must look to very massive compact astrophysical objects, moving at relativisticvelocities. For example, strong sources of gravitational waves that may exist in ourgalaxy or nearby galaxies are expected to produce wave strengths on Earth that donot exceed strain levels of one part in 10 . Finally, it is important to appreciate thatGW detectors respond directly to GW amplitude rather than GW power; therefore thevolume of space that is probed for potential sources increases as the cube of the strainsensitivity.
3. LIGO and the worldwide detector network
As illustrated in Fig. 1, the oscillating quadrupolar strain pattern of a GW is wellmatched by a Michelson interferometer, which makes a very sensitive comparison of imeh
Figure 1.
A gravitational wave traveling perpendicular to the plane of the diagramis characterized by a strain amplitude h . The wave distorts a ring of test particlesinto an ellipse, elongated in one direction in one half-cycle of the wave, and elongatedin the orthogonal direction in the next half-cycle. This oscillating distortion can bemeasured with a Michelson interferometer oriented as shown. The length oscillationsmodulate the phase shifts accrued by the light in each arm, which are in turn observedas light intensity modulations at the photodetector (green semi-circle). This depictsone of the linear polarization modes of the GW. the lengths of its two orthogonal arms. LIGO utilizes three specialized Michelsoninterferometers, located at two sites (see Fig. 2): an observatory on the Hanfordsite in Washington houses two interferometers, the 4 km-long H1 and 2 km-long H2detectors; and an observatory in Livingston Parish, Louisiana, houses the 4 km-long L1detector. Other than the shorter length of H2, the three interferometers are essentiallyidentical. Multiple detectors at separated sites are crucial for rejecting instrumental andenvironmental artifacts in the data, by requiring coincident detections in the analysis.Also, because the antenna pattern of an interferometer is quite wide, source localizationrequires triangulation using three separated detectors.The initial LIGO detectors were designed to be sensitive to GWs in the frequencyband 40 – 7000 Hz, and capable of detecting a GW strain amplitude as small as 10 − [2].With funding from the National Science Foundation, the LIGO sites and detectors weredesigned by scientists and engineers from the California Institute of Technology and theMassachusetts Institute of Technology, constructed in the late 1990s, and commissionedover the first 5 years of this decade. From November 2005 through September 2007,they operated at their design sensitivity in a continuous data-taking mode. The datafrom this science run, known as S5, are being analyzed for a variety of GW signals bya group of researchers known as the LIGO Scientific Collaboration [4]. At the mostsensitive frequencies, the instrument root-mean-square (rms) strain noise has reachedan unprecedented level of 3 × − in a 100 Hz band.Although in principle LIGO can detect and study GWs by itself, the potential to igure 2. Aerial photograph of the LIGO observatories at Hanford, Washington (top)and Livingston, Louisiana (bottom). The lasers and optics are contained in the whiteand blue buildings. From the large corner building, evacuated beam tubes extend atright angles for 4 km in each direction (the full length of only one of the arms is seenin each photo); the tubes are covered by the arched, concrete enclosures seen here. do astrophysics can be quantitatively and qualitatively enhanced by operation in a moreextensive network. For example, the direction of travel of the GWs and the completepolarization information carried by the waves can only be extracted by a network ofdetectors. Such a global network of GW observatories has been emerging over the pastdecade. In this period, the Japanese TAMA project built a 300 m interferometer outsideTokyo, Japan [5]; the German-British GEO project built a 600 m interferometer nearHanover, Germany [6]; and the European Gravitational Observatory built the 3 km-longinterferometer Virgo near Pisa, Italy [7]. In addition, plans are underway to develop alarge scale gravitational wave detector in Japan sometime during the next decade [8].arly in its operation LIGO joined with the GEO project; for strong sources theshorter, less sensitive GEO 600 detector provides added confidence and directional andpolarization information. In May 2007 the Virgo detector began joint observationswith LIGO, with a strain sensitivity close to that of LIGO’s 4 km interferometersat frequencies above ∼
4. Detector description
Figure 1 illustrates the basic concept of how a Michelson interferometer is used tomeasure a GW strain. The challenge is to make the instrument sufficiently sensitive: atthe targeted strain sensitivity of 10 − , the resulting arm length change is only ∼ − m,a thousand times smaller than the diameter of a proton. Meeting this challenge involvesthe use of special interferometry techniques, state-of-the-art optics, highly stable lasers,and multiple layers of vibration isolation, all of which are described in the sections thatfollow. And of course a key feature of the detectors is simply their scale: the arms aremade as long as practically possible to increase the signal due to a GW strain. SeeTable 1 for a list of the main design parameters of the LIGO interferometers. The LIGO detectors are Michelson interferometers whose mirrors also serve asgravitational test masses. A passing gravitational wave will impress a phase modulationon the light in each arm of the Michelson, with a relative phase shift of 180 degreesbetween the arms. When the Michelson arm lengths are set such that the un-modulatedlight interferes destructively at the antisymmetric (AS) port – the dark fringe condition –the phase modulated sideband light will interfere constructively, with an amplitudeproportional to GW strain and the input power. With dark fringe operation, the fullpower incident on the beamsplitter is returned to the laser at the symmetric port.Only differential motion of the arms appears at the AS port; common mode signals arereturned to the laser with the carrier light.Two modifications to a basic Michelson, shown in Fig. 3, increase the carrierpower in the arms and hence the GW sensitivity. First, each arm contains a resonantFabry-Perot optical cavity made up of a partially transmitting input mirror and a highreflecting end mirror. The cavities cause the light to effectively bounce back and forthmultiple times in the arms, increasing the carrier power and phase shift for a givenstrain amplitude. In the LIGO detectors the Fabry-Perot cavities multiply the signalby a factor of 100 for a 100 Hz GW. Second, a partially-reflecting mirror is placedbetween the laser and beamsplitter to implement power recycling [9]. In this technique,an optical cavity is formed between the power recycling mirror and the Michelsonsymmetric port. By matching the transmission of the recycling mirror to the optical
TMITMBSPRM 4 kmIO MC ASPOREF W W W k W RF length detectorRF alignment detectorQuadrant detectorLaser FI Reflection portREF Pick-off portPO Anti-symmetric portAS Input opticsIO Faraday isolatorFI Mode cleanerMCPRM Power recycling mirrorBS 50/50 beamsplitterITM Input test massETM End test mass
Figure 3.
Optical and sensing configuration of the LIGO 4 km interferometers (thelaser power numbers here are generic; specific power levels are given in Table 1). TheIO block includes laser frequency and amplitude stabilization, and electro-optic phasemodulators. The power recycling cavity is formed between the PRM and the two ITMs,and contains the BS. The inset photo shows an input test mass mirror in its pendulumsuspension. The near face has a highly reflective coating for the infrared laser light,but transmits visible light. Through it one can see mirror actuators arranged in asquare pattern near the mirror perimeter. losses in the Michelson, and resonating this recycling cavity, the laser power storedin the interferometer can be significantly increased. In this configuration, known as apower recycled Fabry-Perot Michelson, the LIGO interferometers increase the power inthe arms by a factor of ≈ ,
000 with respect to a simple Michelson.
The laser source is a diode-pumped, Nd:YAG master oscillator and power amplifiersystem, and emits 10 W in a single frequency at 1064 nm [10]. The laser power andfrequency are actively stabilized, and passively filtered with a transmissive ring cavity(pre-mode cleaner, PMC). The laser power stabilization is implemented by directinga sample of the beam to a photodetector, filtering its signal and feeding it back tothe power amplifier; this servo stabilizes the relative power fluctuations of the beamto ∼ − / √ Hz at 100 Hz [11]. The laser frequency stabilization is done in multiplestages that are more fully described in later sections. The first, or pre-stabilizationstage uses the traditional technique of servo locking the laser frequency to an isolatedreference cavity using the Pound-Drever-Hall (PDH) technique [12], in this case viaeedback to frequency actuators on the master oscillator and to an electro-optic phasemodulator. The servo bandwith is 500 kHz, and the pre-stabilization achieves a stabilitylevel of ∼ − Hz / √ Hz at 100 Hz. The PMC transmits the pre-stabilized beam,filtering out both any light not in the fundamental Gaussian spatial mode and lasernoise at frequencies above a few MHz [13]. The PMC output beam is weakly phase-modulated with two radio-frequency (RF) sine waves, producing, to first-order, twopairs of sideband fields around the carrier field; these RF sideband fields are used in aheterodyne detection system described below.After phase modulation, the beam passes into the LIGO vacuum system. All themain interferometer optical components and beam paths are enclosed in the ultra-high vacuum system (10 − – 10 − torr) for acoustical isolation and to reduce phasefluctuations from light scattering off residual gas [14]. The long beam tubes areparticularly noteworthy components of the LIGO vacuum system. These 1.2 m diameter,4 km long stainless steel tubes were designed to have low-outgassing so that the requiredvacuum could be attained by pumping only from the ends of the tubes. This wasachieved by special processing of the steel to remove hydrogen, followed by an in-situ bakeout of the spiral-welded tubes, for approximately 20 days at 160 C.The in-vacuum beam first passes through the mode cleaner (MC), a 12 m long,vibrationally isolated transmissive ring cavity. The MC provides a stable, diffraction-limited beam with additional filtering of laser noise above several kilohertz [15], andit serves as an intermediate reference for frequency stabilization. The MC length andmodulation frequencies are matched so that the main carrier field and the modulationsideband fields all pass through the MC. After the MC is a Faraday isolator and areflective 3-mirror telescope that expands the beam and matches it to the arm cavitymode.The interferometer optics, including the test masses, are fused-silica substrateswith multilayer dielectric coatings, manufactured to have extremely low scatter andlow absorption. The test mass substrates are polished so that the surface deviationfrom a spherical figure, over the central 80 mm diameter, is typically 5 angstroms orsmaller, and the surface microroughness is typically less than 2 angstroms [16]. Themirror coatings are made using ion-beam sputtering, a technique known for producingultralow-loss mirrors [17, 18]. The absorption level in the coatings is generally a fewparts-per-million (ppm) or less [19], and the total scattering loss from a mirror surfaceis estimated to be 60 – 70 ppm.In addition to being a source of optical loss, scattered light can be a problematicnoise source, if it is allowed to reflect or scatter from a vibrating surface (such as avacuum system wall) and recombine with the main beam [20]. Since the vibrating,re-scattering surface may be moving by ∼
10 orders of magnitude more than the testmasses, very small levels of scattered light can contaminate the output. To control this,various baffles are employed within the vacuum system to trap scattered light [20, 21].Each 4 km long beam tube contains approximately two hundred baffles to trap lightscattered at small angles from the test masses. These baffles are stainless steel truncated
Laser type and wavelength Nd:YAG, λ = 1064 nmArm cavity finesse 220Arm length 3995 m 3995 m 2009 mArm cavity storage time, τ s φ
25 cm ×
10 cm, 10.7 kgBeam radius (1 /e power) ITM/ETM 3.6 cm / 4.5 cm 3.9 cm / 4.5 cm 3.3 cm / 3.5 cmTest mass pendulum frequency 0.76 Hz Table 1.
Parameters of the LIGO interferometers. H1 and H2 refer to theinterferometers at Hanford, Washington, and L1 is the interferometer at LivingstonParish, Louisiana. cones, with serrated inner edges, distributed so as to completely hide the beam tubefrom the line of sight of any arm cavity mirror. Additional baffles within the vacuumchambers prevent light outside the mirror apertures from hitting the vacuum chamberwalls.
Starting with the MC, each mirror in the beam line is suspended as a pendulum by a loopof steel wire. The pendulum provides f − vibration isolation above its eigenfrequencies,allowing free movement of a test mass in the GW frequency band. Along the beamdirection, a test mass pendulum isolates by a factor of nearly 2 × at 100 Hz. Theposition and orientation of a suspended optic is controlled by electromagnetic actuators:small magnets are bonded to the optic and coils are mounted to the suspensionsupport structure, positioned to maximize the magnetic force and minimize groundnoise coupling. The actuator assemblies also contain optical sensors that measure theposition of the suspended optic with respect to its support structure. These signals areused to actively damp eigenmodes of the suspension.The bulk of the vibration isolation in the GW band is provided by four-layer mass-spring isolation stacks, to which the pendulums are mounted. These stacks provideapproximately f − isolation above ∼
10 Hz [22], giving an isolation factor of about 10 at 100 Hz. In addition, the L1 detector, subject to higher environmental ground motionthan the Hanford detectors, employs seismic pre-isolators between the ground and theisolation stacks. These active isolators employ a collection of motion sensors, hydraulicctuators, and servo controls; the pre-isolators actively suppress vibrations in the band0 . −
10 Hz, by as much as a factor of 10 in the middle of the band [23].
The two Fabry-Perot arms and power recycling cavities are essential to achieving theLIGO sensitivity goal, but they require an active feedback system to maintain theinterferometer at the proper operating point [24]. The round trip length of each cavitymust be held to an integer multiple of the laser wavelength so that newly introducedcarrier light interferes constructively with light from previous round trips. Under theseconditions the light inside the cavities builds up and they are said to be on resonance. Inaddition to the three cavity lengths, the Michelson phase must be controlled to ensurethat the AS port remains on the dark fringe.The four lengths are sensed with a variation of the PDH reflection scheme [25]. Instandard PDH, an error signal is generated through heterodyne detection of the lightreflected from a cavity. The RF phase modulation sidebands are directly reflected fromthe cavity input mirror and serve as a local oscillator to mix with the carrier field.The carrier experiences a phase-shift in reflection, turning the RF phase modulationinto RF amplitude modulation, linear in amplitude for small deviations from resonance.This concept is extended to the full interferometer as follows. At the operating point,the carrier light is resonant in the arm and recycling cavities and on a Michelson darkfringe. The RF sideband fields resonate differently. One pair of RF sidebands (fromphase modulation at 62.5 MHz) is not resonant and simply reflects from the recyclingmirror. The other pair (25 MHz phase modulation) is resonant in the recycling cavitybut not in the arm cavities. ‡ The Michelson mirrors are positioned to make one arm30 cm longer than the other so that these RF sidebands are not on a Michelson darkfringe. By design this Michelson asymmetry is chosen so that most of the resonatingRF sideband power is coupled to the AS port.In this configuration, heterodyne error signals for the four length degrees-of-freedomare extracted from the three output ports shown in Fig. 3 (REF, PO and AS ports).The AS port is heterodyned at the resonating RF frequency and gives an error signalproportional to differential arm length changes, including those due to a GW. ThePO port is a sample of the recycling cavity beam, and is detected at the resonatingRF frequency to give error signals for the recycling cavity length and the Michelsonphase (using both RF quadratures). The REF port is detected at the non-resonatingRF frequency and gives a standard PDH signal proportional to deviations in the laserfrequency relative to the average length of the two arms.Feedback controls derived from these errors signals are applied to the two endmirrors to stabilize the differential arm length, to the beamsplitter to control theMichelson phase, and to the recycling mirror to control the recycling cavity length. Thefeedback signals are applied directly to the mirrors through their coil-magnet actuators, ‡ These are approximate modulation frequencies for H1 and L1; those for H2 are about 10% higher.
ASER PC AOM VCO pha s ep i e z o s l o w fast Pre-Stabilized Laser Mode Cleaner Common Arm Length f ~ 700 kHz f ~ 100 kHz f ~ 20 kHz
PRMMCRef.Cavity t he r m a l Figure 4.
Schematic layout of the frequency stabilization servo. The laser is lockedto a fixed-length reference cavity through an AOM. The AOM frequency is generatedby a Voltage Controlled Oscillator (VCO) driven by the MC, which is in turn drivenby the common mode arm length signal from the REF port. The laser frequency isactuated by a combination of a Pockels Cell (PC), piezo actuator, and thermal control. with slow corrections for the differential arm length applied with longer-range actuatorsthat move the whole isolation stack.The common arm length signal from the REF port detection is used in the final levelof laser frequency stabilization [26] pictured schematically in Fig. 4. The hierarchicalfrequency control starts with the reference cavity pre-stabilization mentioned in Sec. 4.2.The pre-stabilization path includes an Acousto-Optic Modulator (AOM)driven bya voltage-controlled oscillator, through which fast corrections to the pre-stabilizedfrequency can be made. The MC servo uses this correction path to stabilize the laserfrequency to the MC length, with a servo bandwidth close to 100 kHz. The moststable frequency reference in the GW band is naturally the average length of the twoarm cavities, therefore the common arm length error signal provides the final level offrequency correction. This is accomplished with feedback to the MC, directly to the MClength at low frequencies and to the error point of the MC servo at high frequencies,with an overall bandwidth of 20 kHz. The MC servo then impresses the correctionsonto the laser frequency. The three cascaded frequency loops – the reference cavity pre-stabilization; the MC loop; and the common arm length loop – together provide 160 dBof frequency noise reduction at 100 Hz, and achieve a frequency stability of 5 µ Hz rmsin a 100 Hz bandwidth.The photodetectors are all located outside the vacuum system, mounted on opticaltables. Telescopes inside the vacuum reduce the beam size by a factor of ∼
10, andthe small beams exit the vacuum through high-quality windows. To reduce noise fromscattered light and beam clipping, the optical tables are housed in acoustical enclosures,and the more critical tables are mounted on passive vibration isolators. Any back-scattered light along the AS port path is further mitigated with a Faraday isolatormounted in the vacuum system.The total AS port power is typically 200 – 250 mW, and is a mixture of RF sidebandocal oscillator power and carrier light resulting from spatially imperfect interferenceat the beamsplitter. The light is divided equally between four length photodetectors,keeping the power on each at a detectable level of 50 – 60 mW. The four length detectorsignals are summed and filtered, and the feedback control signal is applied differentiallyto the end test masses. This differential-arm servo loop has a unity-gain bandwidth ofapproximately 200 Hz, suppressing fluctuations in the arm lengths to a residual level of ∼ − m rms. An additional servo is implemented on these AS port detectors to cancelsignals in the RF-phase orthogonal to the differential-arm channel; this servo injects RFcurrent at each photodetector to suppress signals that would otherwise saturate thedetectors. About 1% of the beam is directed to an alignment detector that controls thedifferential alignment of the ETMs.Maximal power buildup in the interferometer also depends on maintaining stringentalignment levels. Sixteen alignment degrees-of-freedom – pitch and yaw for each of the6 interferometer mirrors and the input beam direction – are controlled by a hierarchyof feedback loops. First, orientation motion at the pendulum and isolation stackeigenfrequencies is suppressed locally at each optic using optical lever angle sensors.Second, global alignment is established with four RF quadrant photodetectors atthe three output ports as shown in Fig. 3. These RF alignment detectors measurewavefront misalignments between the carrier and sideband fields in a spatial versionof PDH detection [27, 28]. Together the four detectors provide 5 linearly independentcombinations of the angular deviations from optimal global alignment [29]. These errorsignals feed a multiple-input multiple-output control scheme to maintain the alignmentwithin ∼ − radians rms of the optimal point, using bandwidths between ∼ . ∼ lockacquisition algorithm. So far this section has described how the interferometer ismaintained at the operating point. The other function of the control system is toacquire lock: to initially stabilize the relative optical positions to establish the resonanceconditions and bring them within the linear regions of the error signals. Before lock thesuspended optics are only damped within their suspension structures; ground motionnd the equivalent effect of input-light frequency fluctuations cause the four (real orapparent) lengths to fluctuate by 0.1 – 1 µ m rms over time scales of 0.5 – 10 sec. Theprobability of all four degrees-of-freedom simultaneously falling within the ∼ At full power operation, a total of 20 – 60 mW of light is absorbed in the substrateand in the mirror surface of each ITM, depending on their specific absorption levels.Through the thermo-optic coefficient of fused silica, this creates a weak, though notinsignificant thermal lens in the ITM substrates [31]. Thermo-elastic distortion of thetest mass reflecting surface is not significant at these absorption levels. While the ITMthermal lens has little effect on the carrier mode, which is determined by the arm cavityradii of curvature, it does affect the RF sideband mode supported by the recyclingcavity. This in turn affects the power buildup and mode shape of the RF sidebandsin the recycling cavity, and consequently the sensitivity of the heterodyne detectionsignals [32, 33]. Achieving maximum interferometer sensitivity thus depends criticallyon optimizing the thermal lens and thereby the mode shape, a condition which occursat a specific level of absorption in each ITM (approximately 50 mW). To achieve thisoptimum mode over the range of ITM absorption and stored power levels, each ITMthermal lens is actively controlled by directing additional heating beams, generatedfrom CO lasers, onto each ITM [34]. The power and shape of the heating beams arecontrolled to maximize the interferometer optical gain and sensitivity. The shape can beselected to have either a Gaussian radial profile to provide more lensing, or an annularradial profile to compensate for excess lensing. The GW channel is the digital error point of the differential-arm servo loop. In principlethe GW channel could be derived from any point within this loop. The error point ischosen because the dynamic range of this signal is relatively small, since the largelow-frequency fluctuations are suppressed by the feedback loop. To calibrate the errorpoint in strain, the effect of the feedback loop is divided out, and the interferometerresponse to a differential arm strain is factored in [35]; this process can be done eitherin the frequency domain or directly in the time domain. The absolute length scale is igure 5.
Antenna response pattern for a LIGO gravitational wave detector, inthe long-wavelength approximation. The interferometer beamsplitter is located atthe center of each pattern, and the thick black lines indicate the orientation of theinterferometer arms. The distance from a point of the plot surface to the center ofthe pattern is a measure of the gravitational wave sensitivity in this direction. Thepattern on the left is for + polarization, the middle pattern is for × polarization, andthe right-most one is for unpolarized waves. established using the laser wavelength, by measuring the mirror drive signal required tomove through an interference fringe. The calibration is tracked during operation withsine waves injected into the differential-arm loop. The uncertainty in the amplitudecalibration is approximately ± ± µ sec.The response of the interferometer output as a function of GW frequency iscalculated in detail in references [36, 37, 38]. In the long-wavelength approximation,where the wavelength of the GW is much longer than the size of the detector, theresponse R of a Michelson-Fabry-Perot interferometer is approximated by a single-poletransfer function: R ( f ) ∝
11 + if /f p , (1)where the pole frequency is related to the storage time by f p = 1 / πτ s . Above the polefrequency ( f p = 85 Hz for the LIGO 4 km interferometers), the amplitude responsedrops off as 1 /f . As discussed below, the measurement noise above the pole frequencyhas a white (flat) spectrum, and so the strain sensitivity decreases proportionally tofrequency in this region. The single-pole approximation is quite accurate, differing fromthe exact response by less than a percent up to ∼ .7. Environmental Monitors To complete a LIGO detector, the interferometers described above are supplementedwith a set of sensors to monitor the local environment. Seismometers and accelerometersmeasure vibrations of the ground and various interferometer components; microphonesmonitor acoustic noise at critical locations; magnetometers monitor fields that couldcouple to the test masses or electronics; radio receivers monitor RF power around themodulation frequencies. These sensors are used to detect environmental disturbancesthat can couple to the GW channel.
5. Instrument performance
During the commissioning period, as the interferometer sensitivity was improved,several short science runs were carried out, culminating with the fifth science run(S5) at design sensitivity. The S5 run collected a full year of triple-detector coincidentinterferometer data during the period from November 2005 through September 2007.Since the interferometers detect GW strain amplitude, their performance is typicallycharacterized by an amplitude spectral density of detector noise (the square root of thepower spectrum), expressed in equivalent GW strain. Typical high-sensitivity strainnoise spectra are shown in Fig. 6. Over the course of S5 the strain sensitivity of eachinterferometer was improved, by up to 40% compared to the beginning of the run througha series of incremental improvements to the instruments.The primary noise sources contributing to the H1 strain noise spectrum are shownin Fig. 7. Understanding and controlling these instrumental noise components has beenthe major technical challenge in the development of the detectors. The noise terms canbe broadly divided into two classes: displacement noise and sensing noise. Displacementnoises cause motions of the test masses or their mirrored surfaces. Sensing noises, onthe other hand, are phenomena that limit the ability to measure those motions; theyare present even in the absence of test mass motion. The strain noises shown in Fig. 6consists of spectral lines superimposed on a continuous broadband noise spectrum. Themajority of the lines are due to power lines (60 , , , ... Hz), “violin mode” mechanicalresonances (340 , , ... Hz) and calibration lines (55 , , and 1100 Hz). These high Qlines are easily excluded from analysis; the broadband noise dominates the instrumentsensitivity. Sensing noises are shown in the lower panel of Fig. 7. By design, the dominant noisesource above 100 Hz is shot noise, as determined by the Poisson statistics of photondetection. The ideal shot-noise limited strain noise density, ˜ h ( f ), for this type of −23 −22 −21 −20 −19 Frequency (Hz) E qu i v a l en t s t r a i n no i s e ( H z − / ) Figure 6.
Strain sensitivities, expressed as amplitude spectral densities of detectornoise converted to equivalent GW strain. The vertical axis denotes the rms strainnoise in 1 Hz of bandwidth. Shown are typical high sensitivity spectra for each of thethree interferometers (red: H1; blue: H2; green: L1), along with the design goal forthe 4-km detectors (dashed grey). interferometer is [9]: (cid:101) h ( f ) = (cid:115) π (cid:126) ληP BS c (cid:112) πf τ s ) πτ s , (2)where λ is the laser wavelength, (cid:126) is the reduced Planck constant, c is the speed of light, τ s is the arm cavity storage time, f is the GW frequency, P BS is the power incidenton the beamsplitter, and η is the photodetector quantum efficiency. For the estimatedeffective power of ηP BS = 0 . ·
250 W, the ideal shot-noise limit is (cid:101) h = 1 . × − / √ Hzat 100 Hz. The shot-noise estimate in Fig. 7 is based on measured photocurrents in theAS port detectors and the measured interferometer response. The resulting estimate, (cid:101) h (100Hz) = 1 . × − / √ Hz, is higher than the ideal limit due to several inefficiencies inthe heterodyne detection process: imperfect interference at the beamsplitter increasesthe shot noise; imperfect modal overlap between the carrier and RF sideband fieldsdecreases the signal; and the fact that the AS port power is modulated at twice the RFphase modulation frequency leads to an increase in the time-averaged shot noise [39].Many noise contributions are estimated using stimulus-response tests, where a sine-wave or broadband noise is injected into an auxiliary channel to measure its couplingto the GW channel. This method is used for the laser frequency and amplitude noise −24 −23 −22 −21 −20 −19 Frequency (Hz) S t r a i n no i s e ( H z − / ) c p p pMIRRORTHERMAL SEISMICSUSPENSIONTHERMAL ANGLECONTROL AUXILIARYLENGTHSACTUATOR
100 100010 −24 −23 −22 −21 −20 Frequency (Hz) S t r a i n no i s e ( H z − / ) p p p s s s sc c mSHOTDARK LASER AMPLITUDELASERFREQUENCY RF LOCALOSCILLATOR Figure 7.
Primary known contributors to the H1 detector noise spectrum. Theupper panel shows the displacement noise components, while the lower panel showssensing noises (note the different frequency scales). In both panels, the black curve isthe measured strain noise (same spectrum as in Fig. 6), the dashed gray curve is thedesign goal, and the cyan curve is the root-square-sum of all known contributors (bothsensing and displacement noises). The labelled component curves are described in thetext. The known noise sources explain the observed noise very well at frequenciesabove 150 Hz, and to within a factor of 2 in the 40 – 100 Hz band. Spectral peaksare identified as follows: c, calibration line; p, power line harmonic; s, suspension wirevibrational mode; m, mirror (test mass) vibrational mode. stimates, the RF oscillator phase noise contribution, and also for the angular controland auxiliary length noise terms described below. Although laser noise is nominallycommon-mode, it couples to the GW channel through small, unavoidable differences inthe arm cavity mirrors [40, 41]. Frequency noise is expected to couple most stronglythrough a difference in the resonant reflectivity of the two arms. This causes carrierlight to leak out the AS port, which interferes with frequency noise on the RF sidebandsto create a noise signal. The stimulus-response measurements indicate the coupling isdue to a resonant reflectivity difference of about 0.5%, arising from a loss difference oftens of ppm between the arms. Laser amplitude noise can couple through an offset fromthe carrier dark fringe. The measured coupling is linear, indicating an effective staticoffset of ∼ Displacement noises are shown in the upper panel of Fig. 7. At the lowest frequencies thelargest such noise is seismic noise – motions of the earth’s surface driven by wind, oceanwaves, human activity, and low-level earthquakes – filtered by the isolation stacks andpendulums. The seismic contribution is estimated using accelerometers to measure thevibration at the isolation stack support points, and propagating this motion to the testmasses using modeled transfer functions of the stack and pendulum. The seismic wallfrequency, below which seismic noise dominates, is approximately 45 Hz, a bit higherthan the goal of 40 Hz, as the actual environmental vibrations around these frequenciesare ∼
10 times higher than was estimated in the design.Mechanical thermal noise is a more fundamental effect, arising from finite lossespresent in all mechanical systems, and is governed by the fluctuation-dissipation theorem[42, 43]. It causes arm length noise through thermal excitation of the test masspendulums ( suspension thermal noise ) [44], and thermal acoustic waves that perturbthe test mass mirror surface ( test mass thermal noise ) [45]. Most of the thermal energyis concentrated at the resonant frequencies, which are designed (as much as possible) tobe outside the detection band. Away from the resonances, the level of thermal motionis proportional to the mechanical dissipation associated with the motion. Designing themirror and its pendulum to have very low mechanical dissipation reduces the detection-band thermal noise. It is difficult, however, to accurately and unambiguously establishthe level of broadband thermal noise in-situ ; instead, the thermal noise curves in Fig. 7are calculated from models of the suspension and test masses, with mechanical lossparameters taken from independent characterizations of the materials.For the pendulum mode, the mechanical dissipation occurs near the ends of thesuspension wire, where the wire flexes. Since the elastic energy in the flexing regionsdepends on the wire radius to the fourth power, it helps to make the wire as thin aspossible to limit thermal noise. The pendulums are thus made with steel wire for itsstrength; with a diameter of 300 µ m the wires are loaded to 30% of their breaking stress.The thermal noise in the pendulum mode of the test masses is estimated assuming arequency-independent mechanical loss angle in the suspension wire of 3 × − [46].This is a relatively small loss for a metal wire [47].Thermal noise of the test mass surface is associated with mechanical damping withinthe test mass. The fused-silica test mass substrate material has very low mechanicalloss, of order 10 − or smaller [48]. On the other hand, the thin-film, dielectric coatingsthat provide the required optical reflectivity – alternating layers of silicon dioxide andtantalum pentoxide – have relatively high mechanical loss. Even though the coatingsare only a few microns thick, they are the dominant source of the relevant mechanicalloss, due to their level of dissipation and the fact that it is concentrated on the test massface probed by the laser beam [43]. The test mass thermal noise estimate is calculatedby modeling the coatings as having a frequency-independent mechanical dissipation of4 × − [45]. The auxiliary length noise term refers to noise in the Michelson and power recyclingcavity servo loops which couple to the GW channel. The former couples directlyto the GW channel while the latter couples in a manner similar to frequency noise.Above ∼
50 Hz the sensing noise in these loops is dominated by shot noise; since loopbandwidths of ∼
100 Hz are needed to adequately stabilize these degrees of freedom, shotnoise is effectively added onto their motion. Their noise infiltration to the GW channel,however, is mitigated by appropriately filtering and scaling their digital control signalsand adding them to the differential-arm control signal as a type of feed-forward noisesuppression [24]. These correction paths reduce the coupling to the GW channel by10 – 40 dB.We illustrate this more concretely with the Michelson loop. The shot-noise-limitedsensitivity for the Michelson is ∼ − m / √ Hz. Around 100 Hz, the Michelson servoimpresses this sensing noise onto the Michelson degree-of-freedom (specifically, onto thebeamsplitter). Displacement noise in the Michelson couples to displacement noise in theGW channel by a factor of π/ ( √ F ) = 1 / F is the arm cavity finesse. TheMichelson sensing noise would thus produce ∼ − m / √ Hz of GW channel noise around100 Hz, if uncorrected. The digital correction path subtracts the Michelson noise fromthe GW channel with an efficiency of 95% or more. This brings the Michelson noisecomponent down to ∼ − m / √ Hz in the GW channel, 5 – 10 times below the GWchannel noise floor.Angular control noise arises from noise in the alignment sensors (both optical leversand wavefront sensors), propagating to the test masses through the alignment controlservos. Though these feedback signals affect primarily the test mass orientation, thereis always some coupling to the GW degree-of-freedom because the laser beam is notperfectly aligned to the center-of-rotation of the test mass surface [49]. Angular controlnoise is minimized by a combination of filtering and parameter tuning. Angle controlbandwidths are 10 Hz or less and strong low-pass filtering is applied in the GW band.n addition, the angular coupling to the GW channel is minimized by tuning the center-of-rotation, using the four actuators on each optic, down to typical residual couplinglevels of 10 − − − m/rad. The actuator noise term includes the electronics that produce the coil currents keepingthe interferometer locked and aligned, starting with the digital-to-analog converters(DACs). The actuation electronics chain has extremely demanding dynamic rangerequirements. At low frequencies, control currents of ∼
10 mA are required to provide ∼ µ m of position control, and tens of mA are required to provide ∼ . / √ Hz above 40 Hz. The relatively limited dynamic range of the DACs is managedwith a combination of digital and analog filtering: the higher frequency components ofthe control signals are digitally emphasized before being sent to the DACs, and then de-emphasized following the DACs with complementary analog filters. The dominant coilcurrent noise comes instead from the circuits that provide the alignment bias currents,followed closely by the circuits that provide the length feedback currents.
In the 50 – 100 Hz band, the known noise sources typically do not fully explainthe measured noise. Additional noise mechanisms have been identified, thoughnot quantitatively established. Two potentially significant candidates are nonlinearconversion of low frequency actuator coil currents to broadband noise (upconversion),and electric charge build-up on the test masses. A variety of experiments have shownthat the upconversion occurs in the magnets (neodymium iron boron) of the coil-magnet actuators, and produces a broadband force noise, with a f − spectral slope;this is the phenomenon known as Barkhausen noise [50]. The nonlinearity is small butnot negligible given the dynamic range involved: 0.1 mN of low-frequency (below afew Hertz) actuator force upconverts of order 10 − N rms of force noise in the 40 –80 Hz octave. This noise mechanism is significant primarily below 80 Hz, and varies inamplitude with the level of ground motion at the observatories.Regarding electric charge, mechanical contact of a test mass with its nearby limit-stops, as happens during a large earthquake, can build up charge between the twoobjects. Such charge distributions are not stationary; they tend to redistribute on thesurface to reduce local charge density. This produces a fluctuating force on the testmass, with an expected f − spectral slope. Although the level at which this mechanismoccurs in the interferometers is not well-known, evidence for its potential significancecomes from a fortuitous event with L1. Following a vacuum vent and pump-out cyclepartway through the S5 science run, the strain noise in the 50 – 100 Hz band went downby about 20%; this was attributed to charge reduction on one of the test masses.n addition to these broadband noises, there are a variety of periodic or quasi-periodic processes that produce lines or narrow features in the spectrum. The largest ofthese spectral peaks are identified in Fig. 7. The groups of lines around 350 Hz, 700 Hz, et cetera are vibrational modes of the wires that suspend the test masses, thermallyexcited with kT of energy in each mode. The power line harmonics, at 60 Hz, 120 Hz,180 Hz, et cetera infiltrate the interferometer in a variety of ways. The 60 Hz line, forexample, is primarily due to the power line’s magnetic field coupling directly to the testmass magnets. As all these lines are narrow and fairly stable in frequency, they occupyonly a small fraction of the instrument spectral bandwidth. While Figs. 6 and 7 show high-sensitivity strain noise spectra, the interferometers exhibitboth long- and short-term variation in sensitivity due to improvements made to thedetectors, seasonal and daily variations in the environment, and the like. One indicatorof the sensitivity variation over the S5 science run is shown in Fig. 8: histograms of therms strain noise in the frequency band of 100 – 200 Hz.To get a sense of shorter term variations in the noise, Fig. 9 shows the distribution ofstrain noise amplitudes at three representative frequencies where the noise is dominatedby random processes. For stationary, Gaussian noise the amplitudes would follow aRayleigh distribution, and deviations from that indicate non-Gaussian fluctuations. AsFig. 9 suggests, the lower frequency end of the measurement band shows a higher levelof non-Gaussian noise than the higher frequencies. Some of this non-Gaussianity isdue to known couplings to a fluctuating environment; much of it, however, is due toglitches – any short duration noise transient – from unknown mechanisms. Additionalcharacterizations of the glitch behavior of the detectors can be found in reference [51].Another important statistical figure-of-merit is the interferometer duty cycle, thefraction of time that detectors are operating and taking science data. Over the S5period, the individual interferometer duty cycles were 78%, 79%, and 67% for H1, H2,and L1, respectively; for double-coincidence between L1 and H1 or H2 the duty cyclewas 60%; and for triple-coincidence of all three interferometers the duty cycle was 54%.These figures include scheduled maintenance and instrument tuning periods, as well asunintended losses of operation.
6. Data Analysis Infrastructure
While the LIGO interferometers provide extremely sensitive measurements of the strainat two distant locations, the instruments constitute only one half of the “Gravitational-wave Observatory” in LIGO. The other half is the computing infrastructure and dataanalysis algorithms required to pull out gravitational wave signals from the noise.Potential sources and the methods used to search for them are discussed in the nextsection. First, we discuss some features of the LIGO data and their analysis that are −22
RMS Strain [100 − 200 Hz] T i m e pe r R M S [ a . u .] Figure 8.
Histograms of the RMS strain noise in the band 100 −
200 Hz, computedfrom the S5 data for each of the LIGO interferometers (red: H1; green: L1; blue: H2).Each RMS strain value is calculated using 30 minutes of data. Much of the higherRMS portions of each distribution date from the first ∼
100 days of the run, aroundwhich time sensitivity improvements were made to all interferometers. Typical RMSvariations over daily and weekly time scales are ±
5% about the mean. With the halfarm-length of H2, its RMS strain noise in this band is expected to be about two timeshigher than that of H1 and L1. common to all searches.The raw instrument data are collected and archived for off-line analysis. For eachdetector, approximately 50 channels are recorded at a sample rate of 16,384 Hz, 550channels at reduced rates of 256 to 4,096 Hz, and 6000 digital monitors at 16 Hz. Theaggregate rate of archived data is about 5 MB/s for each interferometer. Computerclusters at each site also produce reduced data sets containing only the most importantchannels for analysis.The detector outputs are pre-filtered with a series of data quality checks to identifyappropriate time periods to analyze. The most significant data quality (DQ) flag,“science mode”, ensures the detectors are in their optimum run-time configuration; itis set by the on-site scientists and operators. Follow-up DQ flags are set for impendinglock loss, hardware injections, site disturbances, and data corruptions. DQ flags arealso used to mark times when the instrument is outside its nominal operating range,for instance when a sensor or actuator is saturating, or environmental conditions areunusually high. Depending on the specific search algorithm, the DQ flags introduce aneffective dead-time of 1% to 10% of the total science mode data. −6 −5 −4 −3 −2 −1 RMS Strain [x 10 ] P r obab ili t y d i s t r i bu t i on f un c t i on ( a . u . )
80 Hz150 Hz850 HzGaussian Noise
Figure 9.
Distribution of strain noise amplitude for three representative frequencieswithin the measurement band (data shown for the H1 detector). Each curve isa histogram of the spectral amplitude at the specified frequency over the secondhalf of the S5 data run. Each spectral amplitude value is taken from the Fouriertransform of 1 second of strain data; the equivalent noise bandwidth for each curveis 1.5 Hz. For comparison, the dashed grey lines are Rayleigh distributions, whichthe measured histograms would follow if they exhibited stationary, Gaussian noise.The high frequency curve is close to a Rayleigh distribution, since the noise thereis dominated by shot noise. The lower frequency curves, on the other hand, exhibitnon-Gaussian fluctuations.
Injections of simulated gravitational wave signals are performed to test thefunctionality of all the search algorithms and also to measure detection efficiencies.These injections are done both in software, where the waveforms are added to thearchived data stream, and directly in hardware, where they are added to the feedbackcontrol signal in the differential-arm servo. In general the injected waveforms simulatethe actual signals being searched for, with representative waveforms used to test searchesfor unknown signals.As described in the section on instrument performance, the local environment andthe myriad interferometer degrees-of-freedom can all couple to the gravitational wavechannel, potentially creating artifacts that must be distinguished from actual signals.Instrument-based vetoes are developed and used to reject such artifacts [51]. Thevetoes are tested using hardware injections to ensure their safety for gravitational wavedetections. The efficacy of these vetoes depends on the search type. . Astrophysical Reach and Search Results
LIGO was designed so that its data could be searched for GWs from many differentsources. The sources can be broadly characterized as either transient or continuousin nature, and for each type, the analysis techniques depend on whether thegravitational waveforms can be accurately modeled or whether only less specific spectralcharacterizations are possible. We therefore organize the searches into four categoriesaccording to source type and analysis technique:(i) Transient, modeled waveforms: the compact binary coalescence search. The namefollows from the fact that the best understood transient sources are the final stagesof binary inspirals [52], where each component of the binary may be a neutron star(NS) or a black hole (BH). For these sources the waveform can be calculated withgood precision, and matched-filter analysis can be used.(ii) Transient, unmodeled waveforms: the gravitational-wave bursts search. Transientsystems such as core-collapse supernovae [53], black-hole mergers, and neutron starquakes, may produce GW bursts that can only be modeled imperfectly, if at all,and more general analysis techniques are needed.(iii) Continuous, narrow-band waveforms: the continuous wave sources search. Anexample of a continuous source of GWs with a well-modeled waveform is a spinningneutron star (e.g., a pulsar) that is not perfectly symmetric about its rotation axis[54].(iv) Continuous, broad-band waveforms: the stochastic gravitational-wave background search. Processes operating in the early universe, for example, could have produceda background of GWs that is continuous but stochastic in character [55].In the following sections we review the astrophysical results that have beengenerated in each of these search categories using LIGO data; reference [56] containslinks to all the LIGO observational publications. To date, no GW signal detections havebeen made, so these results are all upper limits on various GW sources. In those caseswhere the S5 analysis is not yet complete, we present the most recent published resultsand also discuss the expected sensitivity, or astrophysical reach, of the search based onthe S5 detector performance.
Binary coalescences are unique laboratories for testing general relativity in the strong-field regime [57]. GWs from such systems will provide unambiguous evidence for theexistence of black holes and powerful tests of their properties as predicted by generalrelativity [58, 59]. Multiple observations will yield valuable information about thepopulation of such systems in the universe, up to distances of hundreds of megaparsecs(Mpc, 1 parsec = 3.3 light years). Coalescences involving neutron stars will provideinformation about the nuclear equation of state in these extreme conditions. Suchystems are considered likely progenitors of short-duration gamma ray bursts (GRBs)[60]. Post-Newtonian approximations to general relatively accurately model a binarysystem of compact objects whose orbit is adiabatically tightening due to GW emission[61]. Several examples of such binary systems exist with merger times less than theage of the universe, most notably the binary pulsar system PSR 1913+16 describedpreviously. After an extended inspiral phase, the system becomes dynamically unstablewhen the separation decreases below an innermost stable circular orbit (approximately25 km for two solar-mass neutron stars) and the objects plunge and form a single blackhole in the merger phase. The resulting distorted black hole relaxes to a stationaryKerr state via the strongly damped sinusoidal oscillations of the quasi-normal modes inthe ringdown phase. The smoothly evolving inspiral and ringdown GW waveforms canbe approximated analytically, while the extreme dynamics of the merger phase requirenumeric solutions to determine the GW waveform [62]. Collectively, the inspiral, mergerand ringdown of a binary system is termed a Compact Binary Coalescence (CBC).The waveform for a compact binary inspiral is a chirp: a sinusoid increasing infrequency and amplitude until the end of the inspiral phase. The inspiral phase of aneutron star binary (BNS, with each mass assumed to be 1.4 M (cid:12) ) will complete nearly2,000 orbits in the LIGO band over tens of seconds before merger, and emit a maximumGW frequency of about 1500 Hz. Higher mass inspirals terminate at proportionally lowerGW frequencies. For non-spinning objects, the inspiral waveform is uniquely determinedby the two component masses m and m of the system [63]. No analytic waveformsexist for the merger phase; calculating these waveforms is one of the primary goalsof numerical relativity [64, 65]. The ringdown phase is described by an exponentially-damped sinusoid, determined by the quasi-normal mode frequency and the quality factorof the final black hole [66]. Since the inspiral and ringdown waveforms for a given masspair ( m , m ) are accurately known, searches for them are performed using optimalmatched filtering employing a bank of templates covering the desired ( m , m ) parameterspace. An optimized algorithm generates the template bank, minimizing the number oftemplates while allowing a maximum Signal to Noise Ratio (SNR) loss of 3% [67, 68, 69].In practice approximately seven thousand templates are used to cover total massesbetween 2 and 35 M (cid:12) .The matched filtering process generates a trigger when the SNR of the filter outputexceeds a threshold. The threshold is set by balancing two factors: it must be lowenough so that a good estimation can be made of the background due to detector noise,and it must be high enough to keep the number of triggers manageable. Associated witheach trigger is a specific template, or mass pair, and a coalescence time which maximizethe SNR for that signal event [70].Triggers are first generated independently for each detector. The number of falsetriggers created by detector noise is then greatly reduced by finding the set of coincidentriggers – those that correspond to similar template masses and coalescence times, withinappropriate windows, between at least two LIGO detectors. Coincident triggers aresubject to additional consistency checks, such as the χ [71] and r [72] tests.Typically many thousands of coincident triggers per month remain at the end of thepipeline. These surviving triggers are compared with the background from accidentalcoincidences of triggers due to detector noise. Time shift trials are used to estimate thebackground: the analysis is repeated with the triggers from different detectors shiftedin time relative to each other by an amount large compared to the coincidence window.A hundred such trials are typically made. For each region of mass parameter space, thetime shift trials establish a false alarm rate as a function of SNR. In-time coincidenttriggers with the smallest false alarm rate are potential detection candidates [73].A large number of software injections is made to tune the analysis pipeline andevaluate its detection efficiency. The injected waveforms cover the largest practicalrange of parameter space possible (component masses, spins, orientations, sky locationsand distances). The resulting detection efficiency is combined with simple models ofthe astrophysical source distribution to arrive at an estimated cumulative luminosity towhich the search is sensitive. These models [74, 75] predict that the rate of CBCs shouldbe proportional to the stellar birth rate in nearby spiral galaxies. This birth rate can beestimated from a galaxy’s blue luminosity § , so we express the cumulative luminosity inunits of L , where L is 10 times the blue solar luminosity (the Milky Way contains ∼ . L ). To date, the detection candidates resulting from the analysispipeline are consistent with the estimated background and thus are likely accidentalcoincidences. In the absence of detection, mass-dependent upper limits are set on therate of CBCs in the local universe. These rate limits are expressed per unit L .An inspiral search with total masses between 2 and 35 M (cid:12) has been completedusing the first calendar year of S5 data [73]. Figure 10 shows the resulting rate upperlimit for low mass binary coalescences as a function of the total mass (left), and as afunction of the mass of a black hole in a black hole-neutron star system with a neutronstar mass of 1 . M (cid:12) (right). The same analysis set a binary neutron star coalescencerate upper limit of 3 . × − yr − L − . This upper limit is still significantly higher thanrecent CBC rate estimates derived from the observed BNS population – approximately5 × − yr − L − for NS/NS binaries [75].Since the LIGO sensitivities improved as S5 progressed, analysis of the full dataset should provide significantly more interesting coalescence results. In the meantime,the astrophysical reach for these sources can be estimated from the detector noiseperformance. The minute-by-minute strain noise spectra for each detector are usedto calculate the horizon distance : the maximum distance at which an inspiral could bedetected with an SNR of 8. For BNS inspirals, the horizon distance was 30 – 35 Mpc § Blue luminosity is short for B-band luminosity, signifying one of a standard set of optical filters usedin measuring the luminosity of galaxies. igure 10.
S5 year 1 upper limits on the binary coalescence rate per year and per L10as a function of total mass of the binary system assuming a uniform distribution in themass ratio (left) and as a function of the mass of a black hole in a BHNS system witha neutron star mass of 1 . M (cid:12) (right). The darker area shows the excluded regionwhen accounting for marginalization over systematic errors. The lighter area shows theadditional region that would have been excluded if systematic errors had been ignored.From reference [73]. each for L1 and H1, and about 17 Mpc for H2. Based on the increased horizon distancesand extrapolations from the first-year search results, we expect to achieve better thana factor of two increase in cumulative exposure with the full run analysis.The sensitivity to black hole ringdowns is similarly estimated using the S5 detectorstrain noise. Figure 11 shows the single detector range for black hole ringdowns averagedover sky position and spin orientation. The range estimate assumes 1% of total mass isradiated as gravitational waves, in rough agreement with numerical simulations. Unlikeneutron star inspirals, the abundance of such “intermediate mass black holes” and hencetheir merger rate is difficult to predict [62].Searches are also in progress for GWs from CBCs with total masses up to 100 solarmasses, and from CBCs coincident with short-hard GRBs observed during the S5 run.In addition, procedures are being developed for establishing confidence in candidatedetection events, and for extracting the physical parameters of detected events. In addition to the well-modelled signals described in previous sections, we searchfor gravitational-wave “bursts”, defined as any short-duration signal ( t (cid:46) ≤ f ≤ ,
000 Hz).For example, the collapsing core of a massive star (the engine that powers a type IIsupernova) can emit GWs through a number of different mechanisms [76]. A compactbinary merger – discussed in the earlier section about CBC searches – may be considereda burst, especially if the mass is large so that the bulk of the long inspiral signal isbelow the sensitive frequency band of the detectors, leaving only a short signal from theactual merger to be detected. Cosmic strings, if they exist, are generically expected to
100 200 300 400 500 6000100200300400500600 M / M sun H o r i z on d i s t an c e ( M p c ) Figure 11.
S5 sensitivity to binary black hole ringdowns for the H1 (red), L1 (green)and H2 (blue) detectors. When the ringdown frequency coincides with a spectral linethe sensitivity is much reduced (300 M / M sun corresponds to 60 Hz). bend into cusps and kinks which are efficient radiators of beamed GWs. There may wellbe other astrophysical sources, since any energetic event that involves an asymmetricreshaping or re-orientation of a significant amount of mass will generate GWs.Many energetic gravitational events will also emit electromagnetic radiation and/orenergetic particles that can be observed with telescopes and other astronomicalinstruments, as in the case of supernovae. Thus, besides searching for GW signalsalone, we can search for a class of joint emitters and use information from conventionalobservations to constrain the GW event time and sky position, allowing a more sensitive“externally triggered” search. For example, Gamma-ray bursts (GRBs), and softgamma-ray repeater (SGR) flares are highly energetic events that make excellent targetsfor externally-triggered GW burst searches. While the progenitor(s) of GRBs are notentirely clear, most if not all short-duration GRBs are thought be produced by mergersof neutron stars or of a neutron star with a black hole, which would radiate a great deal ofenergy in GWs. Similarly, SGRs are believed to be neutron stars with very high magneticfields ( i.e. magnetars) that sporadically produce flares of electromagnetic radiation. Theflares may be related to deformations of the neutron star crust which could couple to GWemission. If an associated GW signal for these progenitors is detected, the combined GWand EM/particle data will reveal complementary information about the astrophysics ofthe event. A number of robust burst detection methods have beendeveloped that do not rely on knowledge of the signal waveform. Most fit into oneof three general categories: excess power, cross-correlation, or coherent. xcess power methods decompose the data into different frequency components,either with a Fourier basis or with some family of wavelets, and look for signal power thatis significantly above the baseline detector noise level in some time-frequency region. Anexcess power method typically generates triggers from each detector, and then appliesa coincidence test to find consistent event candidates with excess power in two or moredetectors.
Cross-correlation methods directly compare the data streams from a pair ofdetectors to look for a common signal within uncorrelated noise. A cross-correlationstatistic is calculated by integrating over a short time window – ideally, comparable inlength to the duration of the signal – with a range of relative time delays correspondingto different GW arrival directions. The cross-correlation is insensitive to the relativeamplitude of the common signal in the two data streams which may be different due tothe antenna response of the detectors.
Coherent methods generalize the concepts of excess power and cross-correlationto take full advantage of having three or more data streams. Detectors at each sitesee a different linear combination of the same two time-dependent GW polarizationcomponents, so a network of detectors at three sites (e.g. the two LIGO sites plus Virgo)has enough information to over-determine the waveform and provide a consistencytest for each hypothetical arrival direction. This is essentially a maximum likelihoodapproach on the space of possible GW signals, except that a “regulator” or Bayesianprior is used to disfavor physically unlikely scenarios [77, 78]. If only two sites areavailable, the use of this regulator allows a somewhat weaker coherent analysis to beperformed on data from only two detectors. In externally triggered searches, coherentanalysis is simpler because the sky location of the potential signal is already known. Inthis case two sites are sufficient to fully determine the GW signal.Each of these analysis methods produces a statistic (or more than one) thatdescribes the “strength” of the event candidate. The strength statistics are compared tothe background distribution using time shift analysis, like the CBC searches. Externallytriggered searches also determine the background from time shifted off-source data.These search methods generally work well for a wide range of signals, with somewaveform-dependent variation between methods. They are less sensitive than matchedfiltering for a known signal but are computationally efficient and are often within afactor of 2 in sensitivity.
The most general searches are thosethat look for GW bursts coming from any sky position at any time. Because there is nomorphological distinction between a GW burst signal and an instrumental glitch, these“all-sky” searches place stringent demands on data quality evaluation, instrumentalveto conditions, and consistency tests among detectors. The primary S4 all-sky burstsearch [79] was designed to detect signals with frequency content in the range 64 –1600 Hz and durations of up to ∼ .
15 per day.To interpret a null result such as this one, a Monte Carlo method evaluates whatsignals could have been detected by the search. The data are re-processed with simulatedGW signals using the same analysis pipeline to measure the detection efficiency in thepresence of actual detector noise. The intrinsic amplitude of a simulated burst signal ischaracterized by a model-independent quantity, the “root-sum-square” GW strain, h rss ,that expresses the amplitude of the GW signal arriving at the Earth without regard tothe response of any particular detector. It has units of Hz − / , allowing it to be directlyrelated to the amplitude spectral density of the detector noise as shown in Fig. 12.In principle, the efficiency of a burst search pipeline can be evaluated for anymodeled GW waveform, e.g. from a core collapse simulation or a binary merger signalgenerated using numerical relativity. In practice, the search efficiency is evaluated for acollection of ad hoc waveforms that have certain general features but do not correspondto any particular physical model. One of our standard waveforms is a “sine-Gaussian”, asinusoidal signal with central frequency f within a Gaussian envelope with dimensionlesswidth parameter Q . Evaluating the detection efficiency as a function of frequency,Fig. 12 shows the effective rate limit as a function of signal strength using an “exclusiondiagram”.To understand the reach of the analysis in astrophysical terms, the search sensitivityin terms of h rss can be related to a corresponding energy emitted in gravitational waves, E GW . As discussed in the S4 all-sky burst search paper [79], for sine-Gaussians andother quasiperiodic signals, E GW ∼ r c G (2 πf ) h (3)where the GW energy emission is assumed to be isotropic. GW emission is not isotropic,but the energy flux varies by a factor of no more than 4. Using the fact that the S5data has lower noise than S4 by approximately a factor of two, sources at a typicalGalactic distance of 10 kpc could be detected for energy emission in GWs as low as ∼ × − M (cid:12) . For a source in the Virgo galaxy cluster, approximately 16 Mpc away,GW energy emission as low as ∼ . M (cid:12) could be detected.We can draw more specific conclusions about detectability for models ofastrophysical sources that predict the absolute energy and waveform emitted. Followingthe discussion in [79], we estimate that a similar burst search using S5 data coulddetect the core-collapse signals modeled by Ott et al. [80] out to 0 . M (cid:12) non-spinning progenitor (model s11WW) and to 16 kpc for their 15 M (cid:12) spinningprogenitor (model m15b6). The latter of these would be detectable throughout most ofour Galaxy. A merger of two 10 M (cid:12) black holes would be detectable out to a distance Hz [strain/ rss h -22 -21 -20 -19 -18 -17 -16 r a t e [ e v en t s / da y ] -1
10 110
70 Hz153 Hz235 Hz554 Hz849 Hz1053 Hz S1 S2S4
Figure 12.
Exclusion diagram (rate limit at 90% confidence level, as a function ofsignal amplitude) for sine-Gaussian signals with Q = 8 .
9. Search results from the S1,S2 and S4 science runs are shown. (A burst search was also performed for S3, butit used only 8 days of data and systematic studies were not not carried through toproduce a definitive rate limit.) of approximately 3 Mpc, while a merger of two 50 M (cid:12) black holes could be detected asfar away as ∼
120 Mpc.
The exceptionally intenseGRB 070201 was a particularly interesting event for a triggered burst search becausethe sky position, determined from the gamma-ray data, overlapped one of the spiralarms of the large, nearby galaxy M31 (Andromeda). An analysis of GW data [81] foundno evidence of an inspiral or a more general burst signal; that finding ruled out (at the ∼
99% level) the possibility of a binary merger in M31 being the origin of GRB 070201.We have searched for GW bursts associated with the giant flare of SGR 1806 − −
20 and 1900+14 during the S5 run [82]. No GW signals were identified.Energy emission limits were established for a variety of hypothetical waveforms, manyof them within the energy range allowed by some models, and some as low as 3 × J. Future observations – especially for giant flares, and flares of the recently-discoveredSGR 0501+4516 which is closer to Earth – will be sensitive to GW energy emission ator below the level of observed electromagnetic energies.Externally triggered GW burst searches are in progress or planned usingobservations of supernovae, anomalous optical transients, radio bursts, and neutrinoss triggers. In general, the constraints on event time, sky position, and (possibly)signal properties provided by the external triggers make these searches a few timesmore sensitive in amplitude than all-sky searches. It is thus possible to investigatea rich population of energetic transient events that may plausibly produce detectablegravitational waves.
Continuous GW signals may be generated by rotating neutron star such as thosepowering millisecond radio pulsars. In these systems, a quadrupole mass asymmetry, orellipticity, (cid:15) , radiates GWs at twice the neutron star rotation frequency. The maximumsustainable ellipticity, and hence the maximum GW emission, is a function of the neutronstar’s internal structure and equation of state. Current limits on the ellipticity are basedon the change in frequency of the radio pulsar signal, the spin down rate, assuming thatthe spin down is entirely due to GW emission. An especially interesting candidate isthe Crab pulsar, for which the spin-down bound on ellipticity is (cid:15) ≤ . × − andfor which the bound on detectable strain is h ≤ . × − at about 59.6 Hz, twiceits spin frequency [83]. “Standard” neutron star equations of state predict (cid:15) ≤ − ,while exotic pulsars such as quark stars may have (cid:15) ≤ − [84]. For most radio pulsars,however, the spin down limit overestimates the ellipticity and associated GW emissionbecause of electromagnetic damping of the rotation.Compared with CBCs or bursts, neutron star powered millisecond radio pulsars area weak source of GWs which LIGO can detect only if the source is within a few hundredparsecs. Nonetheless, there are dozens of known sources within this range that may bedetected if they have sufficiently high ellipticity. Furthermore, millisecond pulsars areattractive sources of continuous GWs since the stable rotation periods allow coherentintegration over many hours, weeks and months to improve the signal to noise ratio. The shape of a rotating neutron star’s detected GW waveformis a function of at least six source parameters: two each for the pulsar position andorientation on the sky, and at least two for the spin frequency and frequency drift(1st time derivative). The intrinsic phase of a spinning neutron star waveform asmeasured in the neutron star’s rest frame, Φ( T ), is modelled as an approximate sinusoidat instantaneous frequency ν and spin-down rate ˙ ν . The observed phase in the detectorframe, φ ( t ), is in general a more complicated function of time due to the variable timedelay δt = T − t . The delay δt contains components arising from the Earth’s orbitalmotion (for which | δt | ≤ . | δt | ≤ µ s), andfrom the general relativistic Shapiro delay ( | δt | ≤ µ s) for signals passing close to thesun [85].The six-dimensional parameter space and long duration of the S5 run makes all-sky coherent searches for unknown neutron stars, for which the amplitude and phasevariations are tracked throughout the observation time, computationally prohibitive.hree techniques that trade off between sensitivity and computation have beenimplemented: 1) semi-coherent, long duration all-sky searches sensitive only to powerand neglecting phase using the entire data set [86]; 2) coherent, short-duration all-skysearches sensitive to amplitude and phase but computationally limited to ≈ > F -statistic in the frequency domain [54]. Each CPU in the distributednetwork calculates the coherent signal for each frequency bin and sky position for a 30-hour contiguous segment. The loudest frequency bins are followed up with coincidencestudies between detectors and continuity studies with adjacent time segments.The deepest searches are performed for millisecond radio pulsars with well-characterized, stable ephemerides. The 154 pulsars with spin frequencies greater than25 Hz are selected from the Australian Telescope National Facility online catalogue [91].Of these, 78 have sufficient stability and timing resolution to make knowledge of theirwaveform improve the detection SNR over the long observation time. To consistentlyincorporate the prior information, the targeted search uses a time-domain, Bayesiananalysis in which the detection likelihood is calculated for each detector. Informationfrom multiple detectors is combined to form a joint likelihood assuming the detectors’noises are independent. This procedure allows upper limits from successive science runsto be combined and provides a natural framework for incorporating uncertainties in theephemerides. .3.2. Analysis results Analyses of the full S5 data are underway using the techniquesdescribed above. An all-sky search using the PowerFlux technique on the first 8 monthsof S5 with the H1 and L1 detectors has been completed [89]. This produced upperlimits on strain amplitude in the band 50 – 1100 Hz. For a neutron star with equatorialellipticity of 10 − , the search was sensitive to distances as great as 500 pc. −27 −26 −25 −24 −23 Frequency (Hz) S ou r c e s t r a i n a m p li t ude , h Figure 13.
Limits on GW strain from rotating neutron stars. Upper curve (blackpoints): all-sky strain upper limits on unknown neutron stars for spindown rates ashigh as 5 × − Hz s − and optimal orientation, from analysis of the first 8 months ofS5 data [89]. Middle curve (gray points): expected sensitivity for the Einstein@Homesearch with 5280 hrs of S5 data. Lower curve (gray band): expected range for 95%confidence level Bayesian upper limits on radio pulsars with known epherimides, usingthe full S5 data. Black triangles: upper limits on GW emission from known radiopulsars based on their observed spin-down rates. Because of the narrow bandwidth (10 − Hz) and complicated frequency modulationof pulsar signals, instrument artifacts do not significantly contribute to the noise inpulsar searches. The few exceptions – non-stationary noise near 60 Hz harmonics,wandering lines, etc. – have been easy to identify and remove. Consequently we canpredict the astrophysical reach of the full S5 data set with a high degree of confidencebased on the performance of previous searches and the S5 noise performance. Figure 13shows the projected S5 strain amplitude sensitivity for the more sensitive searches, alongwith the upper limits established by the PowerFlux analysis.
A stochastic background of gravitational waves could result from the randomsuperposition of an extremely large number of unresolved and independent GW emissionevents [92]. Such a background is analogous to the cosmic microwave backgroundadiation, though its spectrum is unlikely to be thermal. The emission events could bethe result of cosmological phenomena, such as the amplification of vacuum fluctuationsduring inflation or in pre-big-bang models; phase transitions in the history of theuniverse; or cosmic strings, topological defects that may have been formed duringsymmetry-breaking phase transitions in the early universe. Or a detectable backgroundcould result from many unresolved astrophysical sources, such as rotating neutron stars,supernovae, or low-mass X-ray binaries.Theoretical models of such sources are distinguished by the power spectra theypredict for the stochastic background production. The spectrum is usually described bythe dimensionless quantity Ω GW ( f ), which is the GW energy density per unit logarithmicfrequency, divided by the critical energy density ρ c to close the universe:Ω GW ( f ) = fρ c dρ GW df . (4)In the LIGO frequency band, most of the model spectra are well approximated by apower-law: Ω GW ( f ) ∝ f α . LIGO analyses consider a range of values for α , though inthis review we will focus on results for a frequency independent Ω GW ( α = 0), since manyof the cosmological models predict a flat or nearly flat spectrum over the LIGO band.The strain noise power spectrum for a flat Ω GW falls as f − , with a strain amplitudescale of + : h GW = 4 × − (cid:112) Ω GW (cid:18)
100 Hz f (cid:19) / Hz − / . (5) Unlike the cosmic microwave background, any GW stochasticbackground will be well below the noise floor of a single detector. To probe below thislevel, we cross-correlate the output of two detectors [93]. Assuming the detector noisesare independent of each other, in the cross-correlation measurement the signal – due toa stochastic background present in each output – will increase linearly with integrationtime T , whereas the measurement noise will increase only with the square root of T .Similarly the signal will increase linearily with the effective bandwidth (∆ f ) of thecorrelation, and the noise as (∆ f ) / . Thus with a sufficiently long observation timeand wide bandwidth, a small background signal can in principle be detected beneaththe detector noise floor.The assumption of independent detector noise is crucial, and it is valid whencomparing L1 with either of the Hanford detectors due to their wide physical separaion.But this separation also extracts a price: the coherent cross-correlation of a stochasticbackground signal is reduced by the separation time delay between the detectors andthe non-parallel alignment of their arms. These effects are accounted for by the overlapreduction function γ ( f ), which is unity for co-located and co-aligned detectors, anddecreases below unity when they are shifted apart or mis-aligned. For a Livingston-Hanford detector pair, the overlap is on average (cid:104) γ (cid:105) ∼ . + We assume here and in the rest of this paper a Hubble expansion rate of 72 km/sec/Mpc. he low frequency noise floor of a single S5 LIGO detector is roughly equivalentto a stochastic background spectrum with Ω GW = 0 .
01 ( h GW = 4 × − Hz − / at100 Hz). The cross-correlation measurement will be sensitive to a background Ω GW lower than this noise floor by a factor of (cid:104) γ (cid:105) ( T ∆ f ) / . With a year of observation timeand an effective bandwidth of 100 Hz this is a factor of several thousand, so we expectto probe for a stochastic background in the range Ω GW ∼ − − − . Since the SNR for a search on Ω GW grows inversely with theproduct of the strain noise amplitude spectra of the two detectors, the sensitivity ofthis search grew quickly as the detectors improved. Analysis of the S4 data yielded aBayesian 90% upper limit of Ω GW < . × − for a flat spectrum in the 51 – 150 Hzband [94]. Projecting for the S5 data, the lower strain noise and longer integration timeshould improve on this by an order of magnitude. While the cross-correlation analysisfor S5 is still in progress, it is straightforward to calculate the expected variance of thecross-correlation using only the interferometers’ strain noise spectra over the run. Thispredicts that the potential upper limit on Ω GW will be in the range 4 − × − .Such a result would be the first direct measurement to place a limit on Ω GW morestringent than the indirect bound set by Big-Bang-Nucleosynthesis (BBN). The BBNbound, currently the most constraining bound in the band around 100 Hz, derives fromthe fact that a large GW energy density present at the time of BBN would have alteredthe abundances of the light nuclei in the universe [92, 55]. For the BBN model to beconsistent with observations of these abundances, the total GW energy density at thetime of nucleosynthesis is thus constrained. In the limiting case that the GW energy wasconfined to LIGO’s sensitive band of 50 – 150 Hz, the BBN bound is: Ω GW < . × − [55, 95].The LIGO results are also being used to constrain the parameter space of modelspredicting a stochastic GW background, such as cosmic string models and pre-big-bangmodels [94]. The gamut of theoretical models and observations pertaining to a stochasticGW background spans an impressively wide range of frequencies and amplitudes. Theseare displayed in the landscape plot of Fig. 14.The analysis described so far searches for an isotropic background of GWs. Thecross-correlation method has also been extended to search for spatial anisotropies, asmight be produced by an ensemble of astrophysical sources [96]. Such a GW radiometerrequires spatially separated interferometers in order to point the multi-detector antennaat different locations on the sky. The result is a map of the power distribution of GWsconvolved with the antenna lobe of the radiometer, with an uncertainty determined bythe detector noise. Radiometer analysis of the S4 data yielded upper limits on the GWstrain power from point sources in the range of ∼ − Hz − to ∼ − Hz − , dependingon sky position and the GW power spectrum model [97]. The S5 analysis should improveon the strain power sensitivity by a factor of 30. The corresponding GW energy fluxdensity that the search will be sensitive to is ∼ − Watt / m / Hz ( f /
100 Hz) . − − − − − − − − − − − − − − − − COBE PulsarLimit DopplerTracking LIGO S3LIGO S4LIGO S5BBNCMB & MatterSpectraInflation Pre − Big − BangCosmicStrings
Frequency (Hz) ! G W Figure 14.
Observational limits and potential sources for a stochastic background ofgravitational waves. The LIGO S5 curve refers to the potential upper limit from theS5 run, based on strain noise data. The curves corresponding to inflationary, cosmic-string and pre-big-bang models are examples; the model parameters allow significantvariation in the predicted spectra. The BBN and CMB & Matter Spectra boundsapply to the total GW energy over the frequency range spanned by the correspondinglines. See reference [94] for more details.
8. Future directions
From its inception, LIGO was envisioned not as a single experiment, but as an on-goingobservatory. The facilities and infrastructure construction were specified, as much aspossible, to accommodate detectors with much higher sensitivity. We have identifieda set of relatively minor improvements to the current instruments [98] that can yielda factor of 2 increase in strain sensitivity and a corresponding factor of 8 increase inthe probed volume of the universe. The two most significant enhancements are higherlaser power and a new, more efficient readout technique for the GW channel. Higherpower is delivered by a new master oscillator-power amplifier system, emitting 35 Wof single frequency 1064 nm light [99], 3.5 times more power than the initial LIGOlasers. For the readout, a small mode-cleaner cavity is inserted in the AS beam path,between the Faraday isolator and the length photodetectors. This cavity filters out RFsidebands and the higher-order mode content of the AS port light, reducing the shot-noise power. Instead of RF heterodyning, signal detection is done by slightly offsettingthe differential arm length from the dark fringe, and using the resulting carrier field asthe local oscillator in a DC homodyne detection scheme. These improvements (knownollectively as Enhanced LIGO) are currently being implemented and commissioned onH1 and L1, and a one-to-two year science run with these interferometers is expected tobegin in mid-2009.Significantly greater sensitivity improvements are possible with more extensiveupgrades. Advanced LIGO will replace the exisiting interferometers with significantlyimproved technology to achieve a factor of at least 10 in sensitivity over the initialLIGO interferometers and to lower the seismic wall frequency down to 10 Hz [100, 101].Advanced LIGO has been funded by the National Science Foundation, begining in April2008. Installation of the Advanced LIGO interferometers is planned to start in early-2011.The Advanced LIGO interferometers are configured like initial LIGO – a power-recycled Fabry-Perot Michelson – with the addition of a signal recycling mirror at theanti-symmetric output. Signal recycling gives the ability to tune the interferometerfrequency response, so that the point of maximum response can be shifted away fromzero frequency [9]. The laser wavelength stays at 1064 nm, but an additional high-powerstage brings the laser power up to 200 W [102]. The test masses will be significantlylarger – 40 kg – in order to reduce radiation pressure noise and to allow larger beamsizes. Larger beams and better dielectric mirror coatings combine to reduce the testmass thermal noise by a factor of 5 compared to initial LIGO [103].The test mass suspensions become significantly more intricate to provide muchbetter performance. They incorporate four cascaded stages of passive isolation, insteadof just one, including vertical isolation comparable to the horizontal isolation at allstages except one [104]. The test mass is suspended at the final stage with fused silicafibers rather than steel wires; these fibers have extremely low mechanical loss and willreduce suspension thermal noise nearly a hundred-fold [105]. The current passive seismicisolation stacks that support the suspensions are replaced with two-stage active isolationplatforms [106]. These stages are designed to actively reduce the ground vibration by afactor of ∼ −
10 Hz band, and provide passive isolation at higher frequencies.The combination of the isolation platforms and the suspensions will reduce seismic noiseto negligible levels above approximately 10 Hz.The successful operation of Advanced LIGO is expected to transform the field fromGW detection to GW astrophysics. We illustrate the potential using compact binarycoalescences. Detection rate estimates for CBCs can be made using a combination ofextrapolations from observed binary pulsars, stellar birth rate estimates, and populationsynthesis models. There are large uncertainties inherent in all of these methods, however,leading to rate estimates that are uncertain by several orders of magnitude. We thereforequote a range of rates, spanning plausible pessimistic and optimistic estimates, as wellas a likely rate. For a NS mass of 1 . M (cid:12) and a BH mass of 10 M (cid:12) , these rate estimatesfor Advanced LIGO are: 0 . − yr − , with a likely rate of 40 yr − for NS-NS binaries;0 . − yr − , with a likely rate of 10 yr − for NS-BH binaries; 2 − yr − , with alikely rate of 30 yr − for BH-BH binaries. cknowledgments The authors gratefully acknowledge the support of the United States National ScienceFoundation for the construction and operation of the LIGO Laboratory and the ParticlePhysics and Astronomy Research Council of the United Kingdom, the Max-Planck-Society and the State of Niedersachsen/Germany for support of the construction andoperation of the GEO 600 detector. The authors also gratefully acknowledge the supportof the research by these agencies and by the Australian Research Council, the NaturalSciences and Engineering Research Council of Canada, the Council of Scientific andIndustrial Research of India, the Department of Science and Technology of India,the Spanish Ministerio de Educacion y Ciencia, The National Aeronautics and SpaceAdministration, the John Simon Guggenheim Foundation, the Alexander von HumboldtFoundation, the Leverhulme Trust, the David and Lucile Packard Foundation, theResearch Corporation, and the Alfred P. Sloan Foundation.
References γ -ray bursts. Nature, 437:845–850, October 2005.[61] Luc Blanchet. Gravitational radiation from post-Newtonian sources andinspiralling compact binaries. Living Rev. Rel., 5:3, 2002.[62] Eanna E. Flanagan and Scott A. Hughes. Measuring gravitational waves frombinary black hole coalescences. I: Signal to noise for inspiral, merger, and ringdown.Phys. Rev., D57:4535–4565, 1998.[63] Luc Blanchet, Thibault Damour, Bala R. Iyer, Clifford M. Will, and Alan. G.Wiseman. Gravitational radiation damping of compact binary systems to secondpostNewtonian order. Phys. Rev. Lett., 74:3515–3518, 1995.[64] Mark Hannam. Status of black-hole-binary simulations for gravitational- wavedetection. arXiv, gr-qc:0901.2931, 2009.[65] Benjamin Aylott et al. Testing gravitational-wave searches with numericalrelativity waveforms: Results from the first Numerical INJection Analysis(NINJA) project. arXiv, gr-qc:0901.4399, 2009.[66] Jolien D. E. Creighton. Search techniques for gravitational waves from black-holeringdowns. Phys. Rev. D, 60:022001, 1999.[67] Benjamin J. Owen and B. S. Sathyaprakash. Matched filtering of gravitationalwaves from inspiraling compact binaries: Computational cost and templateplacement. Phys. Rev. D, 60:022002, 1999.68] S. Babak, R. Balasubramanian, D. Churches, T. Cokelaer, and B. S.Sathyaprakash. A template bank to search for gravitational waves from inspirallingcompact binaries. I: Physical models. Class. Quant. Grav., 23:5477–5504, 2006.[69] Thomas Cokelaer. Gravitational waves from inspiralling compact binaries:hexagonal template placement and its efficiency in detecting physical signals.Phys. Rev. D, 76:102004, 2007.[70] B. Abbott et al. Search for gravitational waves from binary inspirals in S3 and S4LIGO data. Phys. Rev. D, 77:062002, 2008.[71] Bruce Allen. A χ2