6.2 Astrophysics results

Prior to the advent of the large scale interferometric detectors there had been some limited effort to produce astrophysical results with the prototype interferometers. The Caltech 40 m detector was used to search for, and set an upper limit on, the gravitational wave emission from pulsar PSR J1939+2134 [171], and on the rate of neutron star binary inspirals in our galaxy, using coincident observations with the University of Glasgow prototype [290] and, more recently, on its own [74]. Coincident observations using the prototype detectors at the University of Glasgow and Max Planck Institute for Quantum Optics, in Garching, Germany, were used to set an upper limit on the strain of gravitational wave bursts [243]. The Garching detector was used to search for periodic signals from pulsars, and in particular set a limit on a potential source in SN 1987A [244]. However since the start of science data taking for the large scale detectors there has been a rapid rise in the number, and scope, of science result papers being published. With the vastly improved sensitivities pushing upper limits on source populations and strengths towards astrophysically interesting areas.

The recent analysis efforts have generally been split into four broad areas depending on the expected signal type: unmodelled transients or bursts, e.g., supernovae; modelled transients, e.g., inspirals and ring-downs (or more specifically compact binary coalescences, CBC); continuous sources; stochastic sources. Within each area a variety of different sources could exist and a variety of analysis techniques have been developed to search for them. Some electromagnetic sources, such as radio pulsars and γ-ray bursts, are also used to enhance searches. A good review of the data analysis methods used in current searches, and the astrophysical consequences of some of the results described below, can be found in [273].

Here we will briefly summarise the main astrophysics results from the science runs. We will mainly focus on those produced by the LIGO Scientific Collaboration [215] detectors LIGO and GEO600 from S1 to the S5 run. At the time of writing not all of the data from the S6 run had been fully analysed, with more results expected over the next year. Reviews of some early S5, and prior science run, results can also be found in [254, 139]. In none of the searches so far has convincing evidence for a gravitational-wave signal been seen.

6.2.1 Unmodelled bursts

Searches for unmodelled bursts, e.g., from supernova core-collapse, are based on looking for short duration periods of excess power in the detectors. Transients are common features in the data, so to veto these events from true gravitational-wave signals they must be coincident in time, and to some extent amplitude and waveform, between multiple detectors. Various methods to assess instrumental excess power, and inter-detector correlations, are used, some examples of which can be found here [202, 77, 284, 228, 107, 112, 113]. These algorithms will produce triggers, which are periods of excess power that cross a predetermined signal-to-noise ratio threshold (determined by tuning the algorithms on a section of playground data, so that the output produces a desired false alarm rate). The number of triggers are then compared to a background rate. Real signals cannot be turned off, and detectors cannot be shielded from them, so the background rate has to be approximated by time shifting one detector’s data stream with respect the the others. Time shifts should only leave triggers due to random coincidences in detector noise and there should be no contribution from real signals. Once a background is calculated, the statistical significance of the foreground rate can be compared to it. To assess the sensitivity of these searches, hardware (the interferometer mirrors are physically moved via the control system) and software signals are injected into the data stream at various strengths and the efficiency of the algorithms at detecting them is measured. A good description of some of these techniques can be found in [13Jump To The Next Citation Point] and [24Jump To The Next Citation Point].

Data from the LIGO S1 run was searched for gravitational-wave bursts of between 4 to 100 ms, and within the frequency band 150 to 3000 Hz [13]. Triple coincident data from all three detectors was used for the analysis. No plausible candidate event was found, but a 90% confidence upper limit on the event rate of 1.6 events per day was set. The search was typically sensitive, at a ≳ 50% detection efficiency, to bursts with amplitudes of h ∼ 10−19– 10−17 Hz− 1∕2 rss (defined in terms of ∘ -------- h ≡ ∫ |h |2dt rss, which is the root-sum-squared strain amplitude spectral density). Due, in part, to its lower sensitivity GEO600 data was not used in this analysis. The S2 data’s improved sensitivity, and advances in the analysis techniques, allowed a sensitivity to signals (in the frequency range 100 – 1100 Hz) in the amplitude range −20 − 19 −1∕2 hrss ∼ 10 –10 Hz [22]. Interpreting the best sensitivities astrophysically gave order of magnitude estimates on the visible range of ∼ 100 pc for a class of theoretical supernova waveforms, and 1 Mpc for the merger of 50M ⊙ black holes. Again no signal was seen, but a 90% upper limit of 0.26 events per day was set for strong bursts. In the frequency range 700 – 2000 Hz TAMA300 data was also used in the search giving amplitude sensitivities of − 19 −1∕2 hrss ∼ 1 –3 × 10 Hz and decreasing the rate upper limit to 0.12 events per day [20].

The S3 run produced two searches for burst sources. One used the 8 days of triple coincidence data from the three LIGO detectors to search for sub-second bursts in the frequency range 100 – 1100 Hz [24Jump To The Next Citation Point]. The search was sensitive to signals with amplitudes over −20 − 1∕2 hrss ∼ 1 × 10 Hz, but did not include an astrophysical interpretation of the limit or event rate upper limit. This run included coincident operation with the Italian AURIGA bar detector and this data has been analysed [90]. This search looked for short bursts of less than 20 ms within the 850 – 950 Hz band (around the bar’s sensitive resonant frequency). This had comparable sensitivity to the LIGO-only S2 search and produced a 90% confidence rate upper limit of 0.52 events/day.

For S4 15.5 days of LIGO data were searched for sub-second bursts in the frequency range 64 – 1600 Hz [27Jump To The Next Citation Point]. This was sensitive to signals with hrss ≲ 10 −20 and set a 90% confidence rate upper limit of 0.15 per day. The search results are also cast as astrophysical limits on source ranges and energetics. These show that there would be a 50% detection efficiency to signals of sine-Gaussian nature (at the most sensitive frequency of 153 Hz and quality factor Q = 8.9) at a distance of 10 kpc for an energy of 10 −7M c2 ⊙, and would be sensitive to signals out to the Virgo cluster (∼ 16 Mpc) for an energy release of 2 0.25M ⊙c. See [27] for a comparison of previous burst searches. There was also a burst search combining S4 GEO600 and LIGO data for the first time. This searched data between 768 – 2048 Hz where the sensitivities were most comparable and used 257 hours of quadruple coincidence between the detectors and saw no gravitational wave events [35].

For the analysis on the first year of S5 data the frequency range for the all-sky burst search was split – a low frequency search covered the most sensitive region between 60 – 2000 Hz [46], and a high frequency search covering 1 – 6 kHz (this being the first time an untriggered burst search looked at frequencies above 3 kHz) [50]. The high frequency search set a 90% upper limit on the rate of 5.4 events per year for strong events. The low frequency search analysed more data than the high frequency and set an event rate limit of 3.6 events per year. The second year of S5 LIGO data was analysed with GEO600 and Virgo VSR1 data [1] to search for bursts over the whole 50 – 6000 Hz band. Combining this with the earlier S5 searches gave hrss upper limits for a variety of simulated waveforms of 6 × 10–22 Hz–1/2 to 2 × 10–20 Hz–1/2, and a 90% confidence event rate for signals between 64 – 2048 Hz of less than two per year.

6.2.2 Modelled bursts – compact binary coalescence

Modeled bursts generally mean the inspiral and coalescence stage of binaries consisting of compact objects, e.g., neutron stars and black holes. The signals are generally well approximated by post-Newtonian expansions of the Einstein equation, which give the amplitude and phase evolution of the orbit. More recently signal models have started to include numerical relativity simulations of the merger stage [89]. As mentioned in Section 6 the best estimate of the number of signals observable with initial LIGO at design sensitivity (i.e. during S5) would be 0.02 per year (based on an event rate of 1 × 10–6 per year per MWEG).

The majority of inspiral searches make use of matched filtering in which a template bank of signal models is built [250, 252], with a maximum mismatch between templates that is generally of order ∼ 10%. These templates are then cross-correlated with the data and statistically significant triggers (i.e. times when the template and data are highly correlated) from this are looked for. Triggers must be coincident between detectors and the significance of any trigger is judged against a background calculated in the same way as described in Section 6.2.1. See [18Jump To The Next Citation Point] for a good description of the search method.

The first search for an inspiral signal with data from the LIGO S1 run looked for compact object coalescences with component masses between 1 –3M ⊙ and was sensitive to such sources within the Milky Way and Magellanic Clouds [11]. It gave a 90% confidence level upper limit on the rate of 170 per year per MWEG.

For the S2 LIGO analysis the search was split into 3 areas covering neutron-star binaries, black-hole binaries and primordial black-hole binaries in the galactic halo. The neutron-star–binary search [18] used 15 days of data with coincidence between either H1 and L1 or H2 and L1. It had a range of ∼ 1.5 Mpc, which spanned the Local Group of galaxies, and gave a 90% event rate upper limit on systems with component masses of 1 –3M ⊙ of 47 per year per MWEG. The black-hole–binary search looked for systems with component masses in the 3– 20M ⊙ range using the same data set as the neutron-star–binary search [24]. This search had a 90% detection efficiency for sources out to 1 Mpc and set a 90% rate upper limit of 38 per year per MWEG. The third search looked for low mass (0.2– 1M ⊙) primordial black-hole binaries in a 50 kpc radius halo surrounding the Milky Way [19]. This placed a 90% confidence-rate upper limit of 63 events per year per Milky Way halo. The S2 search was performed in coincidence with the TAMA300 DT8 period and an inspiral search for neutron-star binaries was performed on data when TAMA300 and at least one of the LIGO sites was operational. This gave a total of 584 hours of data for the analysis, which set a 90% rate upper limit of 49 per year per MWEG, although this search was only sensitive to sources within the majority of the Milky Way [23].

The search for neutron-star–black-hole binaries in S3 LIGO data used techniques designed specifically for systems with spinning components. It searched for systems with component masses in the range 1 –20M ⊙ and analysed 167 hours of triple coincident data and 548 hours of H1-H2 data to set the upper limits [40]. For a typical system with neutron-star and black-hole mass distributions centred on 1.35M ⊙ and 5M ⊙ (from the population statistics discussed in [39Jump To The Next Citation Point]) this search produced a 90% confidence-rate upper limit of 15.9 per year per L 10.

The search for a wide range of binary systems with components consisting of primordial black holes, neutron stars, and black holes with masses in the ranges given above was conducted on the combined S3 and S4 data [39]. 788 hours of S3 data and 576 hours of S4 data were used and no plausible gravitational-wave candidate was found. The highest mass range for the black-hole–binary search was set at 40M ⊙ for S3 and 80M ⊙ for S4. At peak in the mass distribution of these sources 90% confidence-rate upper limits were set at 4.9 per year per L10 for primordial black holes, 1.2 per year per L10 for neutron-star binaries, and 0.5 per year per L10 for black-hole–binaries. S4 data has also been used to search for ring-downs from perturbed black holes, for example following black-hole-binary coalescence [47]. The search was sensitive to ring-downs from 10 –500M ⊙ black holes out to a maximum range of 300 Mpc, and produced a best 90% confidence upper limit on the rate of ring-downs to be 1.6 × 10–3 per year per L10 for the mass range 85 –390M ⊙.

One other kind of modeled burst search is that looking for gravitational waves produced by cusps in cosmic (super)strings. Just over two weeks of LIGO S4 data were used to search for such signals [44]. This was used to constrain the rate and parameter space (string tension, reconnection probability, and loop sizes), but was not able to beat limits set by Big Bang nucleosynthesis.

Data from the first [48] and second year of S5 (prior to Virgo joining with VSR1) [49] have been searched for low-mass binary coalescences with total masses in the range 2 –35M ⊙. The second year search results have produced the more stringent upper limits with 90% confidence rates for neutron-star-binaries, black-hole-binaries and neutron-star–black-hole systems respectively of 1.4 × 10–2, 7.3 × 10–4 and 3.6 × 10–3 per year per L10. Five months of overlapping S5 and VSR1 data were also searched for the same range of signals [5] giving 90% confidence upper rates of 8.7 × 10–3 per year per L 10, 2.2 × 10–3 per year per L 10, and 4.4 × 10–4 per year per L10. The whole 2 years of LIGO S5 data were also used to search for higher mass binary coalescences with component mass between 1 –99M ⊙ and total masses of 25– 100M ⊙. No signal was seen, but a 90% confidence upper limit rate on mergers of black-hole–binary systems with component masses between 19 and 28M ⊙, and with negligible spin, was set at 2.0 Mpc–3 Myr–1 [9].

6.2.3 Externally-triggered burst searches

Many gravitational wave burst sources will be associated with electromagnetic (or neutrino) counterparts, for example short γ-ray bursts (GRBs) are potentially caused by black-hole and neutron-star coalescences. Joint observation of a source as both a gravitational wave and electromagnetic event also greatly increases the confidence in a detection. Therefore many searches have been performed to look for bursts coincident (temporally and spatially) with external electromagnetic triggers, such as GRBs observed by Swift for example. These searches have used both excess power and modeled matched-filter methods to look for signals.

During S2 a particularly bright γ-ray burst event (GRB 030329) occurred and was specifically targeted using data from H1 and H2. The search looked for signals with duration less than ∼ 150 ms and in the frequency range 80 – 2048 Hz [17]. This produced a best strain upper limit for an unpolarised signal around the most sensitive region at ∼ 250 Hz of hrss = 6 × 10 −21 Hz −1∕2.

For S4 there were two burst searches targeting specific sources. The first target was the hyperflare from the Soft γ-ray Repeater SGR 1806–20 (SGRs are thought to be “magnetars”, neutron stars with extremely large magnetic fields of order 1015 Gauss) on 27 December 2004 [186] (this actually occurred before S4 in a period when only the H1 detector was operating). The search looked for signals at frequencies corresponding to short duration quasi-periodic oscillations (QPOs) observed in the X-ray light curve following the flare [28]. The most sensitive 90% upper limit was for the 92.5 Hz QPO at h = 4.5 × 10− 22 Hz −1∕2 rss, which corresponds to an energy emission limit of −8 2 4.3 × 10 M ⊙c (of the same order as the total electromagnetic emission assuming isotropy). The other search used LIGO data from S2, S3 and S4 to look for signals associated with 39 short duration γ-ray bursts (GRBs) that occurred in coincidence with these runs [38]. The GRB triggers were provided by IPN, Konus-Wind, HETE-2, INTEGRAL and Swift as distributed by the GRB Coordinate Network [149]. The search looked in a 180-second window around the burst peak time (120 seconds before and 60 seconds after) and for each burst there were at least two detectors contributing data. No signal coincident with a GRB was observed and the sensitivities were not enough to give any meaningful astrophysical constraints, although simulations suggest that for S4, as in the general burst search, it would have been sensitive to sine-Gaussian signals out to tens of Mpc for an energy release of order a solar mass.

The first search of Virgo data in coincidence with a GRB was performed on data from a commissioning run in September 2005. The long duration GRB 050915a was observed by Swift on 15 September 2005 and Virgo data was used to search for an unmodelled burst in a window of 180 seconds around (120 s before and 60 s after) the GRB peak time [63]. The search produced a strain upper limit of order 10–20 in the frequency range 200 – 1500 Hz, but was mainly used as a test-bed for setting up the methodology for future searches, including coincidence analysis with LIGO.

Data from the S5 run has been used to search for signals associated with even more γ-ray bursts. One search looked specifically for emissions from GRB 070201 [156, 155], which showed evidence of originating in the nearby Andromeda galaxy (M31). The data around the time of this burst was used to look for an unmodelled burst and an inspiral signal as might be expected from a short GRB. The analysis saw no gravitational-wave event associated with the GRB, but ruled out the event being a neutron-star–binary inspiral located in M31 with a 99% confidence [36]. Again, assuming a neutron-star–binary inspiral, but located outside M31, the analysis set a 90% confidence limit that the source must be at a distance greater than 3.5 Mpc. Assuming a signal again located in M31, the unmodelled burst search set an upper limit on the energy emitted via gravitational waves of 4.4 × 10−4M c2 ⊙, which was well within the allowable range for this being an SGR hyper-flare in M31. Searches for 137 GRBs (both short and long GRBs) that were observed, mainly with the Swift satellite, during S5 and VSR1 have been performed again using unmodelled burst methods [53] and for (22 short bursts) inspiral signals [4]. No evidence for a gravitational-wave signal coincident with these events was seen. The unmodeled burst observations were used to set lower limits on the distance to each GRB, with typical limits, assuming isotropic emission, at iso 21∕2 D ∼ 15 Mpc (E GW ∕0.01M ⊙c ). The inspiral search, which was sensitive to CBCs with total system masses between 2M ⊙ and 40M ⊙, was able to exclude with 90% confidence any bursts being neutron-star–black-hole mergers within 6.7 Mpc, although the peak distance distribution of GRBs is well beyond this.

Another search has been to look for gravitational waves associated with flares from known SGRs and anomalous X-ray pulsars (AXPs), both of which are thought to be magnetars. During the first year of S5 there were 191 (including the December 2004 SGR 1806–20 event) observed flares from SGRs 1806–20 and 1900+14 for which at least one LIGO detector was online [37Jump To The Next Citation Point], and 1279 flare events if extending that to six known galactic magnetars and including all S5 and post-S5 Astrowatch data including Virgo and GEO600 [7Jump To The Next Citation Point]. The data around each event was searched for ring-down signals in the frequency range 1 – 3 kHz and with decay times 100 – 400 ms as might be expected from f -mode oscillations in a neutron star. It was also searched for unmodeled bursts in the 100 – 1000 Hz range. No gravitational bursts were seen from any of the events. For the earlier search [37Jump To The Next Citation Point] the lowest 90% upper limit on the gravitational-wave energy from the ring-down search was E90G%W = 2.4 × 1048 erg for an SGR 1806–20 burst on 24 August 2006. The lowest 90% upper limit on the unmodeled search was E9G0W% = 2.9 × 1045 erg for an SGR 1806–20 burst on 21 July 2006. The smallest limits on the ratio of energy emitted via gravitational waves to that emitted in the electromagnetic spectrum were of order 10 – 100, which are into a theoretically-allowed range. The latter search [7] gave the lowest gravitational-wave emission-energy upper limits for white noise bursts in the detector-sensitive band, and for f -mode ring-downs (at 1090 Hz), of 3.0 × 1044 erg and 1.4 × 1047 erg respectively, assuming a distance of 1 kpc. The f -mode energy limits approach the range seen emitted electromagnetically during giant flares. One of these flares, on 29 March 2006, was actually a “storm” of many flares from SGR 1900+14. For this event a more sensitive search has been performed by stacking data around the time of each flare [51]. Waveform dependent upper limits of the gravitational-wave energy emitted were set between 2 × 1045 erg and 6 × 1050 erg, which are an order of magnitude lower than the previous upper limit for this storm (included in the search of [37]) and overlap with the range of electromagnetic energies emitted in SGR giant flares.

Another possible source of gravitational waves associated with electromagnetically-observed phenomenon are pulsar glitches. During these it is possible that various gravitational-wave–emitting vibrational modes of the pulsar may be excited. A search has been performed for fundamental modes (f -modes) in S5 data following a glitch observed in the timing of the Vela pulsar in August 2006 [8]. Over the search frequency range of 1 – 3 kHz this provided upper limits on the peak strain of 0.6 – 1.4 × 10–20 depending on the spherical harmonic that was excited.

Already efforts are under way to invert this process of searching gravitational-wave data for external triggers, and instead supplying gravitational-wave burst triggers for electromagnetic follow-up. This is being investigated across the range of the electromagnetic spectrum from radio [259], through optical (e.g., [198, 119]) and X-ray/γ-ray, and even looking for coincidence with neutrino detectors [86, 258, 111]. Having multi-messenger observations can have a large impact on the amount of astrophysical information that can be learnt about an event [255].

6.2.4 Continuous sources

Searches for continuous waves focus on rapidly-spinning neutron stars as sources. There are fully targeted searches, which look for gravitational waves from known radio pulsars in which the position and spin evolution of the objects are precisely known. There are semi-targeted searches, which look at potential sources in which some, but not all, the source signal parameters are known, for example neutron stars in X-ray binary systems, or sources in supernova remnants where no pulses are seen, which have known position, but unknown frequency. Finally, there are all-sky broadband searches in which none of the signal parameters are known. The targeted searches tend to be most sensitive as they are able to perform coherent integration over long stretches of data with relatively low computational overheads, and have a much smaller parameter space leading to fewer statistical outliers. Due to various neutron-star population statistics, creation rates and energetics arguments, there is an estimate that the amplitude of the strongest gravitational-wave pulsar observed at Earth will be h0 ≲ 4 × 10−24 [29Jump To The Next Citation Point] (a more thorough discussion of this argument can be found in [203]), although this does not rule out stronger sources.

The various search techniques used to produce these results all look for statistically-significant excess power in narrow frequency bins that have been Doppler demodulated to take into account the signal’s shifting frequency caused by the Earth’s orbital motion with respect to the source (or also including the modulations to the signal caused by the source’s own motion relative to the Earth, such as for a pulsar in a binary system). The statistical significance of a measured level of excess power is compared to what would be expected from data that consisted of Gaussian noise alone. A selection of the searches are summarised in [260], but for more detailed descriptions of the various methods see [100, 206, 188, 33Jump To The Next Citation Point, 29Jump To The Next Citation Point, 132].

In S1 a fully-coherent targeted search for gravitational waves from the then-fastest millisecond pulsar J1939+2134 was performed [14]. This analysis and the subsequent LSC known-pulsar searches assume that the star is triaxial and emitting gravitational waves at exactly twice its rotation frequency. All the data from LIGO and GEO600 was analysed and no evidence of a signal was seen. A 95% degree-of-belief upper limit on the gravitational-wave strain amplitude was set using data from the most sensitive detector, L1, giving a value of 1.4 × 10–22. This result was also interpreted as an ellipticity of the star given a canonical moment of inertia of 1038 kg m2 at 𝜖 = 2.9 × 10–4. However, this was still of order 100 000 times higher than the limit that can be set by equating the star’s rate of loss of rotational kinetic energy with that emitted via gravitational radiation – called the “spin-down limit”.

In S2 the number of known pulsar sources searched for with LIGO data increased from 1 to 28, although all of these were isolated pulsars (i.e. not in binary systems, although potentially still associated with supernova remnants or globular clusters). This search used pulsar timing data supplied by Lyne and Kramer from Jodrell Bank Observatory to precisely reconstruct the phase of the gravitational-wave signal over the period of the run. The lowest 95% upper limit on gravitational-waves amplitude was 1.7 × 10–24 for PSR J1910–5959D, and the smallest upper limit on ellipticity (again assuming the canonical moment of inertia) was 4.5 × 10–6 for the relatively-close pulsar PSR J2124–3358 [16], at a distance of 0.25 kpc. The pulsar closest to its inferred spin-down limit was the Crab pulsar (PSR J0534+2200) with an upper limit 30 times greater than that from spin-down. S2 also saw the use of two different all-sky–wide frequency band searches that focused on isolated sources, but also including a search for gravitational waves from the low mass X-ray binary Scorpius X1 (Sco-X1). The first search used a semi-coherent technique to search ∼ 60 days of S2 data in the frequency band between 200 – 400 Hz and with signal spin-downs between –1.1 × 10–9 and 0 Hz s–1 [15]. This gave a lowest gravitational-wave strain 95% upper limit of 4.4 × 10–23 for the L1 detector at around 200 Hz. The other all-sky search was fully coherent and as such was computationally limited to only use a few hours of the most sensitive S2 data. It searched frequencies between 160 – 728.8 Hz and spin-downs less than –4 × 10–10 Hz s–1 for isolated sources and gave a 95% upper limit across this band from 6.6 × 10–23 to 1 × 10–21 [29]. The search for gravitational waves from Sco-X1 used the same period of data. It did not have to search over sky position as this is well known, but did have to search over two binary orbital parameters – the projected semi-major axis and the orbital phase reference time. The frequency ranges of this search relied on estimates of the spin-frequency from quasi-periodic oscillations in the X-rays from the source and covered two 20 Hz bands from 464 – 484 Hz and 604 – 624 Hz (it should be noted that it is now thought that these estimates of the spin-frequency are unreliable). In these two ranges upper limits of 1.7 × 10–22 and 1.3 × 10–21 were found respectively.

One search that was carried out purely on LIGO S3 data was the coherent all-sky wide-band isolated pulsar search using the distributed computing project Einstein@Home [135Jump To The Next Citation Point]. The project is built upon the Berkeley Open Infrastructure for Network Computing [97] and allows the computational workload to be distributed among many computers generally contributed by the general public who sign up to the project. This used the most sensitive 600 hours of data from H1 and cut it into 60 ten hour stretches on each of which a coherent search could be performed. The data was farmed out to computers owned by participants in the project and ran as a background process or screen saver. The search band spanned the range from 50 – 1500.5 Hz. The search saw no plausible gravitational-wave candidates and the result is described at [25], but it was not used to produce an upper limit.

In the known pulsar search the number of sources searched for using the combined LIGO data from S3 and S4 was increased to 78. This included many pulsars within binary systems. For many of the pulsars that overlapped with the previous S2 analysis results were improved by about an order of magnitude. The lowest 95% upper limit on gravitational-waves amplitude was 2.6 × 10–25 for PSR  J1603–7202, and the smallest ellipticity was again for PSR J2124–3358 at just less than 10–6 [32]. The upper limit for the Crab pulsar was found to be only 2.2 times above that from the spin-down limit. Three different, but related, semi-coherent all-sky continuous wave searches were performed on S4 LIGO data, looking for isolated neutron stars in the frequency range from 50 – 1000 Hz and the spin-down range from –1 × 10–8 to 0 Hz s–1 [33]. The best 95% upper limit based on an isotropically-distributed, randomly-oriented, population of neutron stars was 4.3 × 10–24 near 140 Hz. This is approaching the amplitude of the strongest potential signal discussed above. For one of the searches, which combined data from the different detectors, an isolated pulsar emitting at near 100 Hz, and with an extreme ellipticity of 10–4 could have been seen at a distance of 1 kpc, although for a more realistic ellipticity of 10–8 only a distance of less than 1 pc would be visible over the entire LIGO band. The Einstein@Home project [135Jump To The Next Citation Point] was also used to search the most sensitive data from S4, which consisted of 300 hours of H1 data and 210 hours of L1 data. The search performed a coherent analysis on 30-hour stretches of this data and covered the frequency range of 50 – 1500 Hz [43]. The range of spin-downs ˙f was chosen by using a minimum spin-down age τ and having − f∕τ < f˙< 0.1f ∕τ (small spin-ups are allowed as some pulsars in globular clusters exhibit this due to their Doppler motions within the clusters), with τ = 1000 years for signals below 300 Hz and τ = 10 000 years above 300 Hz. Approximately 6000 years of computational time spread over about 100 000 computers were required to perform the analysis. No plausible gravitational-wave candidates were found, although the results suggest that 90% of sources with strain amplitudes greater than 10–23 would have been detected by the search. A search designed to produce a sky map of the stochastic background was also used to search for gravitational waves from Sco-X1 using a method of cross-correlating H1 and L1 data [31Jump To The Next Citation Point]. This produced a 90% root-mean-squared upper limit on gravitational wave strain of h = 3.4 × 10− 24(f ∕200 Hz) for frequencies greater than 200 Hz.

The first 8 months of S5 have been used to perform an all-sky search for periodic gravitational waves. This search used a semi-coherent method to look in the frequency range 50 – 1100 Hz and spin-down range –5 × 10–9 – 0 Hz s2 and used data from the H1 and L1 detectors [41]. It obtained 95% strain upper limits of less than 10–24 over a frequency band of 200 Hz. The search would have been sensitive to a neutron star with equatorial ellipticity greater than 10–6 within around 500 pc. Einstein@Home [135] has been used to search for periodic waves of 50 – 1500 Hz in 860 hours of data from a total span of 66 days of S5 data [42]. This search looked for young pulsars, but saw no significant candidates. It would have been sensitive to 90% of sources in the 125 – 225 Hz band with amplitudes greater than 3 × 10–24. The first approximately 9 months of S5 data was used for a coherent search for gravitational waves from the Crab pulsar [34]. In this search two methods were used: the first followed the method of the targeted search and assumed that the gravitational waves are phase locked to the electromagnetic pulses; the second allowed for some mechanism, which would cause a small mismatch between the two phases. Two 95% upper limits were set, one using astrophysical constraints on the pulsar orientation angle and polarisation angle [241] and the other applying no such constraints. With the first method these 95% upper limits were 3.4 × 10–25 and 2.7 × 10–25 respectively, which correspond to ellipticities of 1.8 × 10–4 and 1.4 × 10–4 (assuming the canonical moment of inertia). These beat the Crab pulsar’s spin-down limit by 4 to 5 times and can be translated into the amount of the available spin-down power that is emitted via gravitational waves, with the lower of these limits showing that less than 4% of power is going into gravitational waves. For the second search the uniform and restricted prior analyses gave upper limits of 1.7 × 10–24 and 1.2 × 10–24 respectively. The whole of S5 was used to search for emissions from 116 known pulsars [54]. During this search the Crab limit was further brought down to be less than a factor of 7 below the spin-down limit, and the spin-down limit is reached for one other pulsar PSR J0537–6910. Of the other pulsars, the best (lowest) upper limit on gravitational-wave amplitude was 2.3 × 10–26 for PSR J1603–7202 and our best (lowest) limit on the inferred pulsar ellipticity is 7.0 × 10–8 for PSR J2124–3358.

A semi-targeted search was performed with 12 days of S5 data, although this time searching for a source with a known position in the Cassiopeia A (Cas A) supernova remnant, but for which there is no known frequency. The search [2] looked in the frequency band between 100 – 300 Hz and covered a wide range of first and second frequency derivatives and no signal was seen, but it gave 95% amplitude and ellipticity upper limits over the band of (0.7 – 1.2) × 10–24 and (0.4 – 4) × 10–4 respectively. These results beat indirect limits on the emission based on energy-conservation arguments (similar, but not the same as the spin-down limits) and were also the first results to be cast as limits on the r-mode amplitude [251].

The Vela pulsar has a spin frequency of ∼ 11 Hz and was not accessible with current LIGO data. However, Virgo VSR2 data had sensitivity in the low frequency band that made a search for it worthwhile. Using ∼ 150 days of Virgo data, three semi-independent methods were used to search for the Vela pulsar [6]. No signal was seen, but a 95% upper limit on the amplitude of ∼ 2 × 10–24 was set, which beat the spin-down limit by ∼ 1.6 times. Other than the Crab pulsar, this is currently the only other object for which the spin-down limit has been beaten.

6.2.5 Stochastic sources

Searches are conducted for a cosmological, or astrophysical, background of gravitational waves that would show up as a coherent stochastic noise source between detectors. This is done by performing a cross-correlation of data from two detectors as described in [73].

In S1 the most sensitive detector pair for this correlation was H2–L1 (the H1–H2 pair are significantly hampered by local environmental correlations) and they gave a 90% confidence upper limit of Ωgw < 44 ± 92 within the 40 – 314 Hz band, where the upper limit is in units of closure density of the universe and for a Hubble constant in units of 72 km s–1 Mpc–1 [10]. This limit was several times better than previous direct-detector limits, but still well above the concordance ΛCDM cosmology value of the total energy density of the universe of Ω0 ≈ 1 (see, e.g., [189]).

No published stochastic background search was performed on S2 data, but S3 data was searched and gave an upper limit that improved on the S1 result by a factor of ∼ 105. The most sensitive detector pair for this search was H1–L1 for which 218 hours of data were used [21]. Upper limits were set for three different power-law spectra of the gravitational-wave background. For a flat spectra, as predicted by some inflationary and cosmic string models, a 90% confidence upper limit of Ωgw (f) = 8.4 × 10−4 in the 69 – 156 Hz range was set (again for a Hubble constant of 72 km s–1 Mpc–1). This is still about 60 times greater than a conservative bound on primordial gravitational waves set by big-bang nucleosynthesis (BBN). For a quadratic power law, as predicted for a superposition of rotating neutron-star signals, an upper limit of Ωgw(f ) = 9.4 × 10− 4(f∕100 Hz )2 was set in the range 73 – 244 Hz, and for a cubic power law, from some pre-Big-Bang cosmology models, an upper limit of Ωgw(f ) = 8.1 × 10− 4(f ∕100 Hz )3 in the range 76 – 329 Hz was produced.

For S4 ∼ 354 hours of H1–L1 data and ∼ 333 hours of H2–L1 data were used to set a 90% upper limit of Ωgw (f) < 6.5 × 10−5 on the stochastic background between 51 – 150 Hz, for a flat spectrum and Hubble constant of 72 km s–1 Mpc–1 [30]. This result is still several times higher than BBN limits. About 20 days of H1 and L1 S4 data was also used to produce an upper limit map on the gravitational wave background across the sky as would be appropriate if there was an anisotropic background dominated by distinct sources [31]. This search covered a frequency range between 50 – 1800 Hz and had spectral power limits (which come from the square of the amplitude h) ranging from 1.2 × 10−48 Hz−1(100 Hz ∕f)3 and 1.2 × 10 −47 Hz −1(100 Hz ∕f)3 for an f −3 source power spectrum, and limits of 8.5 × 10–49 Hz–1 and 6.1 × 10–48 Hz–1 for a flat spectrum.

Data from S4 was also used to perform the first cross-correlation between an interferometric and bar detector to search for stochastic backgrounds. L1 data and data from the nearby ALLEGRO bar detector were used to search in the frequency range 850 – 950 Hz, several times higher than the LIGO only searches [26]. A 90% upper limit on the closure density of Ωgw (f) ≤ 1.02 (for the above Hubble constant) was set, which beat previous limits in that frequency range by two orders of magnitude. This limit beats what would be achievable with LLO-LHO cross correlation of S4 data in this frequency range by a factor of several tens, due to the physical proximity of LLO and ALLEGRO.

The entire two years of S5 data from the LIGO detectors has been used to set a limit on the stochastic background around 100 Hz to be Ωgw(f) < 6.9 × 10−6 at 95% confidence (for a flat gravitational-wave spectrum) [52]. This now beats the indirect limits provided by BBN and cosmic microwave background observations.

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