M Punturo

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S Braccini

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Chassande Mottin

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A Chincarini

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V Fafone

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A Giazotto

K Glampedakis

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H Grote

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G Hammond

M Hannam

J Harms

D Heinert

M Hendry

I Heng

E Hennes

S Hild

J Hough

S Husa

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F Khalili

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B Krishnan

M Lorenzini

H L Ück

E Majorana

I Mandel

V Mandic

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C Michel

Y Minenkov

N Morgado

S Mosca

B Mours

M Üller-Ebhardt

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R Nawrodt

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A Rocchi

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A R Üdiger

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C Schwarz

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F Speirits

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A Th Üring

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The Einstein Telescope: a third generation gravitational wave observatory

Advanced gravitational wave interferometers, currently under realization, will soon permit the detection of gravitational waves from astronomical sources. To open the era of precision gravitational-wave astronomy, a further substantial improvement in sensitivity is required. The future space-based Laser Interferometer Space Antenna (LISA) and the third-generation ground-based observatory Einstein Telescope (ET) promise to achieve the required sensitivity improvements in frequency ranges. The vastly improved sensitivity of the third generation of gravitational-wave observatories could permit detailed measurements of the sources' physical parameters and could complement, in a multi-messenger approach, the observation of signals emitted by cosmological sources obtained through other kinds of telescopes. This paper describes the progresses of the ET project which is currently in its design study phase.

Introduction

Interferometric gravitational wave (GW) detectors have demonstrated the validity of their working principle by coming close to, or even exceeding, the design sensitivity of the initial instruments: LIGO [START_REF] Abbott | LIGO: the laser interferometer gravitational-wave observatory[END_REF], Virgo [START_REF] Accadia | Commissioning status of the Virgo interferometer[END_REF], GEO600 [START_REF]The GEO 600 status[END_REF] and TAMA [START_REF] Arai | Recent progress of TAMA300[END_REF]. In the same infrastructures, currently hosting the initial GW detectors (and their limited upgrades, called "enhanced" interferometers: eLIGO and Virgo+) a second generation of interferometers (so-called "advanced detectors": "Advanced LIGO" [START_REF] Gregory | Advanced LIGO: the next generation of gravitational wave detectors[END_REF], "Advanced Virgo" [START_REF] Acernese | Advanced Virgo baseline design[END_REF] and GEO-HF [START_REF]The GEO 600 status[END_REF]) will be implemented over the next few years. The Laser Interferometer Space Antenna (LISA), a joint ESA-NASA mission expected to fly around 2020, is a space-borne detector to observe in the frequency range of 0.1 mHz to 0.1 Hz -a frequency range that is not accessible from ground. These detectors, based on technologies currently available, and partly already tested in reducedscale prototypes, but still to be implemented in full scale, will show a sensitivity improved roughly by a factor of ten with respect to the initial interferometers. Hence, a detection rate about a factor of 1000 larger than with the initial interferometers is expected, strongly enhancing the probability of detecting the signals generated by astro-physical sources. In particular, considering the predicted detection rate of the GW signal generated by a binary system of coalescing neutron stars [START_REF] Abadie | Predictions for the rates of compact binary coalescences observable by ground-based gravitational-wave detectors[END_REF], the sensitivity of the advanced interferometers is expected to guarantee the detection within months to a year at most.

Apart from extremely rare events, the signal-to-noise ratio (SNR) of detections in the "advanced" detectors is likely to be still too low for precise astronomical studies of the GW sources and for complementing optical and X-ray observations in the study of fundamental systems and processes in the Universe. This consideration led the GW community to investigate the possibility of building a new (third) generation of detectors, permitting both to observe, with huge SNR, GW sources at distances similar to those detectable in the advanced detectors and to reveal GW signals at distances comparable with the sight distance of the electromagnetic telescopes. As LISA will do for the super-massive black holes (M 10 6 M Sun ), ET, thanks to this capability to inspect the GW signal in great detail, could herald a new era of routine GW astronomy for lighter astrophysical bodies.

To realise a third generation GW observatory, with a significantly enhanced sensitivity (considering a target of a factor of ten improvement over advanced detectors in a wide frequency range), several limitations of the technologies adopted in the advanced interferometers must be overcome and new solutions must be developed to reduce the fundamental and technical noises that will limit the next generation detectors. But, mainly, new research facilities hosting the third generation GW observatory apparatuses must be realised, to circumvent the limitations imposed by the current facilities. Hereafter, we will describe some of the possible scientific goals and some of the challenges of a third generation GW observatory, as evaluated within the framework of the Einstein Telescope (ET) design study [START_REF] Punturo | The third generation of gravitational wave observatories and their science reach[END_REF].

ET Science reach

In Fig. 1 we plot a possible sensitivity curve of a third generation gravitational wave detector [START_REF] Hild | Pushing towards the ET sensitivity using 'conventional' technology[END_REF]. This is by no means the final design goal but it sets the stage for studying what science questions can be addressed with a third generation detector. A detector with a sensitivity a factor ten better than an advanced detector will open a new avenue Figure 1. A possible sensitivity (solid curve) of an underground, long suspension, cryogenic, signal and power recycled single third generation gravitational wave observatory (see Table 1 in [START_REF] Hild | Pushing towards the ET sensitivity using 'conventional' technology[END_REF]) compared with a typical sensitivity curve of an advanced detector (dashed curve). It is worth underlining that the evaluation of the possible noise level of a third generation GW observatory is an ongoing activity, still far from being concluded within the ET design study. For this reasons the curves are updated regularly and labeled with progressive letters to be distinguished. In the solid curve (so-called ET-B), corresponding to a single wide-band detector, the suspension thermal noise contribution is not yet included.

for understanding the physical phenomena of extreme objects in the Universe. The study team has started putting together a Vision Document [START_REF] Amaro-Seoane | Einstein Telescope design study: Vision document[END_REF] detailing the scope of such a detector. Here we list a few examples of the science questions we can expect to pose with ET.

(i) Observation of compact binary coalescences would allow accurate measurement of the masses of neutron stars and masses and spins of black holes [START_REF] Van | Binary black hole detection rates in inspiral gravitational wave searches[END_REF][START_REF] Van | Phenomenology of amplitude-corrected postnewtonian gravitational waveforms for compact binary inspiral: I. signal-to-noise ratios[END_REF]. For instance, for binaries at a given distance, ET will measure masses and spins an order-of-magnitude better than advanced detectors. More importantly, it should be possible to determine the component masses of binaries to better than 5% (except when the component objects are of comparable masses) over a wide range of masses from a few solar masses to 100's of solar masses. From a population of such observations, it will be possible to infer the maximum mass of a neutron star (a long-standing open problem in theoretical physics) and constrain its equationof-state [START_REF] Amaro-Seoane | Einstein Telescope design study: Vision document[END_REF]. The way this can be done is as follows: It is widely believed that short, hard gamma-ray bursts (shGRBs) are triggered by the coalescence of a compact binary in which one of the stars is a neutron star and the other a neutron star or a black hole. If this is the case, then one can reliably assume that the lighter of the components of a binary coalescence observed in coincidence is definitely a neutron star. A large enough sample should then give the mass function of neutron stars and tell us where the cutoff in the mass distribution is. (ii) Advanced detectors should make first coincident observations of binary mergers and shGRBs. One might not accumulate a sufficiently large population of such events with advanced detectors to fully understand the population of GRBs and their precursors. Advanced detectors could shed light on GRB progenitors (an outstanding problem in astronomy) and ET would allow classification of different types of GRBs, their demography, relationship to masses and spins of component stars, etc. (iii) Simultaneous detection of neutrinos, electromagnetic and gravitational radiation from supernovae could help understand the mechanism behind type II supernovae and the astrophysics/physics of gravitational collapse. Furthermore, such observations also help to deduce or constrain the neutrino and graviton masses [START_REF] Arnaud | Gravity wave and neutrino bursts from stellar collapse: A sensitive test of neutrino masses[END_REF]. ET's range for supernovae is about 5 Mpc, within which one expect a supernova once every 2 to 5 years [START_REF] Shin'ichiro Ando | Detection of neutrinos from supernovae in nearby galaxies[END_REF]. (iv) Comparing observations of massive binary star systems with numerical relativity predictions should allow testing general relativity and constrain alternative theories of gravity (such as the Brans-Dicke theory) [START_REF] K G Arun | Bounding the mass of the graviton with gravitational waves: Effect of higher harmonics in gravitational waveform templates[END_REF]. (v) Inspiralling binary neutron stars (BNS) are ideal standard candles (or standard "sirens"). A population of BNS merger events observed in coincidence with short-hard GRBs can be used to measure cosmological parameters, GW observations helping to accurately estimate the luminosity distance and GRB hosts giving the source redshift [START_REF] Bernard | Determining the hubble constant from gravitational wave observations[END_REF][START_REF] Sathyaprakash | Cosmography with the Einstein Telescope[END_REF][START_REF] Nissanke | Exploring short gamma-ray bursts as gravitational-wave standard sirens[END_REF]. (vi) ET will be sensitive to intermediate mass black hole binaries of total mass up to about a few thousand solar masses depending on the lower frequency cutoff [START_REF] Gair | Exploring intermediate and massive black-hole binaries with the Einstein Telescope[END_REF][START_REF] Gair | Probing seed black holes using future gravitational-wave detectors[END_REF][START_REF] Amaro | Detection of IMBHs with ground-based gravitational wave observatories: A biography of a binary of black holes, from birth to death[END_REF]. The formation, abundance and coalescence rates of such systems is highly uncertain although it is plausible that intermediate mass black holes could be seeds of massive black holes that are now found at galactic nuclei, but they could also form in dense star clusters or by other means. If such systems exist, ET will provide an all sky survey of such objects up to red-shifts of 2 or more. (vii) Intermediate black holes irrespective of when and where they form, could grow by accreting other compact objects such as stellar mass black holes and neutron stars. Here again the rates are unknown, but ET will be sensitive to the merger of stellar mass objects onto intermediate black holes at red-shifts of z = 1. Such events will be an invaluable tool to study the structure of spacetime geometry near massive black holes [START_REF] Gair | Exploring intermediate and massive black-hole binaries with the Einstein Telescope[END_REF][START_REF] Gair | Probing seed black holes using future gravitational-wave detectors[END_REF], even though LISA's observation of stellar mass black holes inspiralling into supermassive black holes would be better at probing the space-time geometry. (viii) ET will be able to detect a stochastic background of gravitational waves at the level of Ω GW ∼ 10 -12 , where Ω GW is the energy density in stochastic background in units of the closure density of the Universe. This compares well with LISA's sensitivity of Ω GW ∼ 10 -11 in the frequency range of 2 -20 mHz. Although ET's sensitivity is a few orders of magnitude poorer than that required to detect backgrounds predicted by inflationary Universe models, there is the possibility that phase transitions in the early Universe and other processes could give rise to a detectable background [START_REF] Amaro-Seoane | Einstein Telescope design study: Vision document[END_REF]. (ix) At the higher end of its frequency range, ET could observe normal modes in neutron stars excited in a host of astronomical events such as pulsar glitches, magnetar flares, soft-gamma repeaters, etc. GW, optical, X-ray, radio and gammaray windows would be invaluable tools for asteroseismology and the best way to probe neutron star interiors and to understand the equation-of-state of matter at extreme conditions of density, pressure, temperature and magnetic fields [START_REF] Amaro-Seoane | Einstein Telescope design study: Vision document[END_REF].

m 2 =1 M O . m 2 =2 M O . m 2 =4 M O . ...

Challenges for data analysis and the need for new search algorithms

A detector with a sensitivity window and span as ET will pose new data analysis challenges. As in the case of LISA, there will be many classes of sources all visible at the same time, requiring a paradigm shift in the way data are currently being analyzed. Some types of sources that can be assumed to be transients in current detectors will be in ET's band for many hours or even days. For such signals detector motion can no longer be neglected, requiring greater computational costs and the development of new search algorithms. A careful and comprehensive study of the data analysis challenges is currently underway. Here we discuss some of the basic problems a search algorithm should address in the ET era. Most of the following issues are relevant whatever the data analysis method followed, but more so in the case of a matched filtering search, e.g. for binary inspiral signals with a bank of templates.

As far as we can guess, compact binary mergers will dominate the ET observation band. Extrapolating the nominal rate of about one neutron star binary merger event per year within a distance of 100 Mpc [START_REF] Abadie | Predictions for the rates of compact binary coalescences observable by ground-based gravitational-wave detectors[END_REF] to the distance reach of ET of about 20 Gpc, one expects to detect an event about every 6 seconds. This extrapolation is, of course, not quite correct as it assumes sources to be uniformly distributed in space. In reality, we know that the star formation rate peaks at z ∼ 1 and so the actual merger rate might be smaller by a moderate factor. Even so, merger signals from binary neutron stars and other compact binaries will be observed quite frequently in ET. We will, therefore, focus on the sort of problems this class of sources might pose.

(i) Figure 2 The fact that a source's location on the sky is changing has to be taken into account in any analysis. Note, however, that at a frequency of 1 Hz the resolution of the detectors, even considering the baseline from the Earth's motion around the sun over 10 days, is only about 1 str and so this is not likely to be a big problem. (vi) The biggest challenge might be matched filtering the data over the entire parameter space of binary systems to which the detector could be potentially sensitive. The number of templates grows roughly as f -11/3 s where f s is the frequency below which the detector accumulates negligible amount of signal-tonoise. The value of f s might be a factor 20 -40 smaller in going from initial to 3rd generation detectors, thereby leading to a massive increase in the number of search templates and a corresponding increase in the trigger rate. Since the event rate is quite high (an event every 10 or 15 seconds) it might be possible to use sub-optimal techniques to dig out most of the events, and these need to be explored. (vii) Neutron stars and stellar mass black holes falling into intermediate mass black holes could last for several days in the band of sensitivity and will have close to millions of cycles. The complex orbits of such systems would pose a serious challenge to the analysis. (viii) Amidst millions of binary inspiral signals we could have occasional burst signals from supernovae, neutron star quakes and the associated normal modes, continuous waves from spinning neutron stars, stochastic background of primordial or astrophysical origin, etc. How easy would it be to disentangle these interesting signals and characterize their properties?

The ET study team is working on a set of mock data challenges to test some of the questions posed above. These challenges are similar to the ones carried out in the context of LISA [START_REF] Babak | The Mock LISA Data Challenges: from Challenge 3 to Challenge 4[END_REF] and are open for anyone to participate. Our goal is to produce data sets of increasing complexity in order to provide an opportunity for us to address the data analysis and computational challenges posed by a third generation detector.

Technologies in ET

To provide the Einstein Telescope with a sensitivity of a factor of 10 better than that of the "advanced detectors", the relevant fundamental noise sources should be suppressed (neglecting the role of the so-called "technical" noises): the seismic and gravity gradient noise at very low frequency (1 Hz -10 Hz), the suspension thermal noise and quantum noise, related to the radiation pressure exerted on the suspended mirror by the photons in the main Fabry-Perot cavities (10 Hz -40 Hz), the thermal noise of the suspended mirrors (mainly the coating contribution, 40 Hz -200 Hz) and, finally, at higher frequencies, shot noise component of the quantum noise.

Seismic and gravity gradient noise reduction

The seismic noise affects the sensitivity at low frequency of the current GW interferometric detectors. In the Virgo detector, the so-called Super-Attenuator (SA) [START_REF] Braccini | Measurement of the seismic attenuation performance of the Virgo superattenuator[END_REF], has shown its capability to filter the seismic noise below the expected thermal noise. The performances of the SA have been confirmed to be compliant with the attenuation requirements in Advanced Virgo [START_REF] Acernese | Measurements of superattenuator seismic isolation by Virgo interferometer[END_REF] and, considering as reference a seismic noise linear spectral density of 5 × 10 -9 / f 2 m • Hz -1/2 , value measured in the Kamioka (Japan) mine, selected for the construction of the LCGT interferometer [START_REF] Uchiyama | Present status of large-scale cryogenic gravitational wave telescope[END_REF], it is expected to be easily re-scalable to be compliant with the more restrictive ET noise requirements at low frequency [START_REF] Acernese | Measurements of superattenuator seismic isolation by Virgo interferometer[END_REF].

The gravity gradient noise is due to the direct coupling of the suspended testmass displacement with the mass vibration in the soil layers, perturbed by seismic waves, via the mutual attraction force expressed by Newton's universal law of gravitation [START_REF] Peter | Terrestrial gravitational noise on a gravitational wave antenna[END_REF][START_REF] Beccaria | Relevance of newtonian seismic noise for the Virgo interferometer sensitivity[END_REF][START_REF] Hughes | Seismic gravity-gradient noise in interferometric gravitationalwave detectors[END_REF]. Obviously the importance of this disturbance depends on the seismic noise level and on the contribution of the other low-frequency noise sources to the noise budget. In the third generation of GW detectors, the more stringent requirements in terms of sensitivity at low frequency enhance the importance of this noise source and enforce the need to realise the Einstein Telescope in an underground and quiet site.

Thermal noise reduction

Under the "thermal noise" label are grouped all those processes that modulate the optical path of the light in the interferometer coupling it to the Brownian fluctuation or to the stochastic fluctuation of the temperature field in the optical components. Usually, one distinguishes between the suspension thermal noise, affecting the position of the test masses through the fluctuations of the suspension wires or fibres, and the mirror thermal noise, which is the sum of all the overlapping fluctuation and dissipation processes occurring in the test masses and in its high-reflectivity coatings. The strategies to reduce the thermal noise impact in the second generation GW detectors are essentially an evolution of what has been applied in the initial detectors and are based on the reduction of the dissipation in the suspension system, in order to concentrate all the fluctuation energy into the normal modes of the system, resulting in a low noise level off-resonance.

In addition to these strategies, in ET we propose to act directly on the total fluctuation energy, by reducing the operative temperature of the suspended optics. Hence, cryogenics is one of the most appealing technologies to reduce the thermal noise of the optics suspension in the third generation of GW observatories. The design of the cryogenic suspension and of its cooling system is progressing in the ET design study and possible material candidates for the test masses and suspension fibres have been identified in sapphire (as already done [START_REF] Uchiyama | Mechanical quality factor of a sapphire fiber at cryogenic temperatures[END_REF][START_REF] Tomaru | Maximum heat transfer along a sapphire suspension fiber for a cryogenic interferometric gravitational wave detector[END_REF] for LCGT) and silicon [START_REF] Alshourbagy | First characterization of silicon crystalline fibers produced with the µ-pulling technique for future gravitational wave detectors[END_REF][START_REF] Reid | Mechanical dissipation in silicon flexures[END_REF][START_REF] Punturo | The third generation of gravitational wave observatories and their science reach[END_REF].

Quantum Noise reduction

Quantum noise in interferometric gravitational wave detectors can be understood as the coupling of vacuum fluctuations with the optical readout fields inside the interferometer. This coupling causes an uncertainty in the phase and amplitude of the probe beam, which affects the interferometer output signal in two ways: the phase uncertainty directly spoils the phase measurement of the Michelson interferometer; this effect is called shot noise. The amplitude uncertainty, or in other words, the changing amplitude of the light, will cause the light pressure on the test masses to change, which correspondingly causes motions of the test masses; this effect is called radiation pressure noise. Quantum noise is the sum of shot noise and radiation pressure noise and in the classical Michelson interferometers poses a fundamental limit to the sensitivity of the detector, the so called Standard Quantum Limit (SQL).

Techniques to improve the sensitivity beyond the SQL are called Quantum Noise Reduction (QNR) or somewhat misleadingly Quantum Non-Demolition (QND) schemes. A more detailed introduction to this topic is given in [START_REF] Punturo | The third generation of gravitational wave observatories and their science reach[END_REF], see also [START_REF] Helge M Üller-Ebhardt | Review of quantum non-demolition schemes for the Einstein Telescope[END_REF] for a review of QND schemes discussed in the context of the Einstein Telescope. Currently we aim at using a Michelson interferometer with Signal Recycling and a squeezed light field injected into the interferometer output for reducing the quantum noise in the Einstein Telescope. This method would benefit greatly from a "xylophone" approach (see section 4.4). Alternatively a Sagnac topology is studied as a possible alternative; the Sagnac interferometer is insensitive to radiation pressure noise; so far, however, much less experimental expertise has been gained with this topology.

Multiple interferometer detector

As described in the previous subsections, to realise a third generation GW detector, the technologies currently operative in the initial and planned for the advanced detectors must be further advanced and new solutions must be adopted. The crosscompatibility between the different solutions becomes a crucial issue; for example, the requirements imposed by the reduction of the quantum noise conflicts with those imposed by the thermal noise suppression. This technological difficulty in realizing a single wide-band third generation detector can be avoided. The base line currently favoured in the ET design study [START_REF] Hild | A xylophone configuration for a third-generation gravitational wave detector[END_REF] is a combination of two interferometers, specialised on different frequency bands: the so-called xylophone philosophy [START_REF] Conforto | Proposal for lower frequency companions for the advanced LIGO gravitational wave interferometric detectors[END_REF].

Here the output of a low-frequency-specialised detector is combined with the output of a high-frequency machine. The former one could be a cryogenic interferometer at an underground site, with long suspensions, but moderate optical power, whereas [START_REF] Hild | A xylophone configuration for a third-generation gravitational wave detector[END_REF], curve socalled "ET-C"), with respect to a single wide frequency range interferometer ET implementation [START_REF] Hild | Pushing towards the ET sensitivity using 'conventional' technology[END_REF].

the high frequency interferometer could essentially be a long arm advanced detector, implementing squeezed light states, a very high power laser and large test masses. A possible realisation of such as xylophone strategy, evaluated in [START_REF] Hild | A xylophone configuration for a third-generation gravitational wave detector[END_REF] ("ET-C") for the ET design study, is plotted in Fig. 3 and compared with the single-interferometer implementation ("ET-B") of the ET observatory, described in [START_REF] Hild | Pushing towards the ET sensitivity using 'conventional' technology[END_REF].

Site and infrastructure

In Subsection 4.1 it has been assumed that the required seismic noise spectral density, compliant with the ET sensitivities shown in Fig. 3, corresponds to the noise measured in an underground site. In effect, one of the major activities to be accomplished in the ET design study phase is the identification of the noise requirements of the site hosting the observatory and the compilation of a candidate list in Europe. The first results of this study indicate that the site hosting the ET observatory should be located a few hundred meters underground, in order to reduce the dominant disturbance of the surface seismic waves, in a region with reduced anthropogenic activity, far from main natural noise sources such as the ocean. To reduce the gravity gradient noise a "soft" soil is recommendable, but it causes major construction difficulties and additional costs.

Another important characteristic of the site is the length of the tunnels hosting the main cavities arms. To gain a factor of ten with respect to the advanced detectors, the length of the ET arms should be about 10 km; as this length agrees well with the average lifetime of the tunnel boring machines, which need to be bought, it also optimizes the costs in this respect. In effect, the cost of the site excavation and of the hosting infrastructure, under evaluation in the ET project, will surely dominate the overall budget of the project. For this reason it will be mandatory to maximise the usage of the site, for example, installing multiple detectors in the tunnels.

Detector Geometry

All the currently active GW interferometric detectors are L-shaped, with orthogonal arms, since this geometry maximises the sensitivity of a single detector with respect to the arm length. But other geometries are possible, like triangular-shaped detectors already proposed in the past [START_REF] Winkler | Plans for a large gravitational wave antenna in Germany[END_REF], and could become preferable if the selection criteria are more complex than the simple sensitivity maximisation. As analysed in detail in [START_REF] Freise | Triple michelson interferometer for a third-generation gravitational wave detector[END_REF], a triangular-shaped observatory, composed of three co-located interferometric detectors, could present many advantages in terms of redundancy, signal reconstruction and cost/benefit minimisation, and this geometry is becoming the baseline option of the ET project.

ET Project evolution

The Einstein Telescope design study is supported for three years (2008-2011) within the European Community Seventh Framework Programme (FP7), having the aim of delivering the conceptual design of such a research infrastructure, investigating the technological feasibility, the science targets, the site requirements and prepare a costing draft for the infrastructure.

After this phase, a preparatory activity is expected to be necessary to define the technological details, and the legal and organisational issues. The start of construction (2018-2019) is expected to occur after the first detection of GWs, which is reckoned to happen within at most one year after the advanced detectors will have reached their nominal sensitivity. The construction and commissioning timeline of ET is under study, but about 7-8 years are expected to be necessary before we have the first data available.

Figure 2 .

 2 Figure 2. Duration of inspiral signals from binaries of total mass (as given by the x-axis) and mass m 2 of one of the component stars (as in the legend): Signals from binary neutron stars will last for about nine days in the detector band. Many signals will last for more than a day but we will also have very short duration signals from intermediate mass black hole binaries. Curves are shown for mass of one of the components varying from 1M to 1024M , increasing by a factor of 2 at each step.

Figure 3 .

 3 Figure 3. Sensitivity of a third generation GW observatory implemented by two frequency-specialised (LF & HF) detectors (xylophone topology[START_REF] Hild | A xylophone configuration for a third-generation gravitational wave detector[END_REF], curve socalled "ET-C"), with respect to a single wide frequency range interferometer ET implementation[START_REF] Hild | Pushing towards the ET sensitivity using 'conventional' technology[END_REF].

  plots the duration of binary inspiral signals for systems with various masses, all starting at a frequency of 1 Hz. Remarkably, inspiral signals could stay in the sensitive frequency band for as long as 10 days, for the lightest systems, to as briefly as only a few 100 milliseconds, for the heaviest ones. (ii) The preponderance of signals in ET, as opposed to their rare occurrence in the "advanced detectors", and their long duration means that signals will inevitably overlap with one another and that might cause confusion noise. It is necessary to evaluate the efficiency of the current algorithms in extracting overlapping signals buried in, say, Gaussian noise. (iii) The occurrence of many overlapping signals could cause significant degradation of the parameter accuracies and thereby compromise ET's science potential. Moreover, the presence of many signals invalidates the assumption of stationarity of the data. How reliably can we extract signals and what are the parameter accuracies? (iv) At 1 Hz the Doppler modulation due to Earth's rotation and revolution can be neglected for signals that last for consicerably less than one day. For longer signals Doppler modulations in signal amplitude and frequency become important. (v)

PACS numbers: 04.80.Nn Submitted to: Class. Quantum Grav.

Acknowledgments

The authors gratefully acknowledge the help of Albrecht R üdiger in correcting this paper. This work has been performed with the support of the European Commission under the Framework Programme 7 (FP7) "Capacities", project Einstein Telescope (ET) design study (Grant Agreement 211743), http://www.et-gw.eu/


Thierry Pradier

The Antares Neutrino Telescope and Multi-Messenger Astronomy

Antares is currently the largest neutrino telescope operating in the Northern Hemisphere, aiming at the detection of high-energy neutrinos from astrophysical sources. Such observations would provide important clues about the processes at work in those compact and energetic sources, and possibly help solve the puzzle of ultra-high energy cosmic rays. In this context, Antares is developing several programs to improve its capabilities of revealing possible spatial and/or temporal correlations of neutrinos with other cosmic messengers: photons, cosmic rays and gravitational waves. The neutrino telescope and its most recent results are presented, together with these multi-messenger programs.

Introduction

Astroparticle physics has entered an exciting period with the recent development of experimental techniques that have opened new windows of observation of the cosmic radiation in all its components: photons, cosmic rays, but also gravitational waves and high energy neutrinos, that could be detected both by IceCube [START_REF] Halzen | for IceCube[END_REF] and Antares.

The advantage of using neutrinos as new messengers lies firstly on their weak interaction cross-section ; unlike protons or γ, they provide a cosmological-range unaltered information from the heart of their sources. Secondly, charged particles are deflected by magnetic fields, while neutrinos point directly to their exact production site.

The neutrinos Antares is aiming at are typically TeV ν from AGNs or galactic sources (microquasars), 30 orders of magnitude lower in flux than solar neutrinos. The detection of those specific neutrinos requires under water/ice instruments, or alternatively acoustic/radio techniques in the PeV-EeV range and air showers arrays above 1 EeV. In spite of efforts in those various energy ranges, since the detection of the MeV neutrino burst from SN 1987A by Kamiokande/Baksan/imb/Mont-Blanc [2] no astrophysical source for neutrinos above a few GeV has ever been identified.

Sources for TeV ν are typically compact objects (neutron stars/black holes), from which often emerge relativistic plasma jets with a still unclear composition -purely leptonic or with some hadronic component. Most of these sources have already been extensively studied from radio wavelengths up to γ-rays. These photons can be produced by e -via inverse compton effect (on ambient photon field)/synchrotron radiation, or by protons/nuclei via photoproduction of π 0 /π ± :

p/A + p/γ -→ π 0 π ± where π 0 → γγ and π ± → ν µ µ, with µ → ν µ ν e e (1)
In the former scenario, no neutrinos are produced, whereas in the latter, the neutrino flux is directly related to the gamma flux: a TeV neutrino detection from gamma-ray sources would then yield a unique way to probe the inner processes of the most powerful events in the universe. Several hints exist which indicates that hadrons could be accelerated up to very high energies. Firstly, the combined radio, X-rays and γ-rays observations of the shell-type supernova remnant RX J1713.7-3946 [3] favour the production of photons via π 0 decay (figure 1, left). Secondly, the correlations between X and γ for the Blazar 1ES1959+650 [4] prove the existence of γ flares not visible in X (figure 1, right), which is difficult to account for in purely leptonic models. Finally, hadronic cosmic rays have been observed by experiments such as Auger or Kascade. The connection between high energy neutrino astronomy and both gamma-ray astronomy, charged cosmic rays and gravitational waves thus emphasizes that not only a multi-wavelength but also a multi-messenger approach, combining data from different observatories, is suited for the study of the most powerful astrophysical sources in the Universe.

Sections 2 and 3 first describe the Antares neutrino telescope and its latest physics results. Section 4 presents the different strategies imagined to detect neutrinos from transient sources, while section 5 details the correlations that can be performed with other observatories, such as Auger, Hess, or Virgo/LIGO.

The Antares Neutrino Telescope: principles & description

Antares is a three-dimensional grid of photomultiplier tubes, arranged in strings, able to detect the Čerenkov photons induced by the passage in sea water of relativistic charged particles, produced by the interaction of a cosmic neutrino in the Earth [START_REF] Markov | International Conference on High-Energy Physics[END_REF]. The measurements of the time of the hits (with a time resolution of the order of ns), their amplitude (with a resolution of about 30%) and position (by measuring the position of each PMT, to reach a resolution of about 10 cm) are needed to achieve the reconstruction of those signals with the desired resolution. Muon tracks initiated by the charged current interaction of a ν µ in the Earth are detected via their directional Čerenkov light and can be reconstructed with an angular resolution below 0.3 • above 10 TeV. The resolution below this energy is dominated by the kinematics of the interaction. The energy resolution is quite poor, a factor 2-3 on average, restricted by the granularity/density of the light sensors and the fact that the muon traverses the detector. Showers produced by ν e on the other hand emit quasi-isotropic light, and can be reconstructed with a better energy resolution (∼ 30 %) but with a poorer angular resolution (several degrees).

The main physical backgrounds are twofold. Atmospheric muons produced in the upper atmosphere by the interaction of cosmic rays can be strongly suppressed because of their downward direction. Upward-going atmospheric neutrinos on the other hand are more delicate to identify: they have exactly the same signature (an upward muon track) as the expected cosmic signal Antares awaits for, but could be distinguished from astrophysical neutrinos, e.g., with their energy signature.

The Antares neutrino telescope, deployed at 2500 m below sea surface, 40 km off the coast of Toulon (Southern France) is composed of 12 strings, with 25 storeys each containing a triplet of 10 " photomultipliers oriented at 45 degrees downward to be optimally sensitive to upward going muons [START_REF][END_REF]. Since May 2008, 12 lines are continuously taking data. A schematic description of the detector, together with the layout of the lines, and of one of the storey, can be seen in figure 2.

Selected results from Antares

The main results obtained recently by the Antares neutrino telescope, related to atmospheric muons and neutrino-induced upward-going muons, are presented in this section.

Atmospheric muons

The attenuation of the muon flux as a function of depth can be observed in figure 3 (left), as computed using the method described in [7]. Alternatively, the reconstructed zenith angle can be converted to an equivalent slant depth through the sea water, and a depth intensity relation can be extracted, also shown in figure 3 (right). The results are in agreement with previous measurements.

Neutrinos

The muons produced by the interaction of neutrinos can be isolated from the atmospheric muons by requiring that the muon trajectory is reconstructed as up-going. In figure 4 the zenith angular distribution of muons in the 2007+2008 data (5 lines and 9-12 lines) sample is shown. A total of 1062 up-going neutrino candidates are found, in good agreement with expectations from the atmospheric neutrino background.

An all sky search, independent of assumption on the source location, has been performed on the 5-line data. No significant cluster was found. A search amongst a pre-defined list of 24 of the most promising galactic and extra-galactic neutrino sources (e.g. supernova remnants, BL Lac objects) was performed. The corresponding flux expected sensitivities, assuming an E 2 flux, are also plotted in figure 4 and compared to published upper limits from other experiments (see [8] for Amanda). Also shown is the predicted upper limit for Antares after one full year of twelve line operation.

Searches for neutrinos from GRBs

Among all the possible astrophysical sources, transient sources offer one of the most promising perspectives for the detection of cosmic neutrinos thanks to the almost background free search. Several models predict the production of high-energy neutrinos by compact sources involving jets [9], in particular gamma-ray bursts (GRBs) [10] : the detection of these neutrinos would provide evidence for hadron acceleration in GRBs.

Two different methods to detect transient sources can be used. The triggered search is based on the search for neutrino candidates in conjunction with an accurate timing and positional information provided by an external source. The rolling search is based on the search for high energy or multiplet of neutrino events coming from the same position within a given time window.

Triggered search with the GCN

Classically, GRBs or flare of AGNs are detected by gamma-ray satellites (Swift, Fermi) which deliver in real time an alert to the Gamma-ray bursts Coordinates Network (GCN, figure 5). The characteristics (direction and time of the detection) of this alert are then distributed to the other observatories. The small difference in arrival time and position expected between photons and neutrinos allows a very efficient detection by reducing the associated background. This method has been implemented in Antares mainly for the GRB detection since the end of 2006. Data triggered by more than 500 alerts have been stored up to now. Based on the time of the external alert, in complement to the standard acquisition strategy, an on-line running program stores the data coming from the whole detector during 2 minutes without any filtering (not possible otherwise, due to the high data rate). This allows to lower the energy threshold of the event selection during the offline analysis with respect to the standard filtered data. Due to a continuous buffering of data (covering 60s) and thanks to the very fast response time of the GCN network (see figure 5), Antares is able to record data before the detection of the GRB by the satellite [START_REF] Bouwhuis | for Antares[END_REF]. The analysis of the data relying on those external alerts is on-going.

Due to the very low background rate, even the detection of a small number of neutrinos correlated with GRBs could set a discovery. But, due to the relatively small field of view of the gamma-ray satellites (e.g., Swift has a 1.4 sr field of view), only a small fraction of the existing bursts are triggered. Moreover, choked GRBs without photons counterpart can not be detected by this method. This justifies the use of a complementary rolling search strategy.

Rolling search: the TAToO project

This second method relies on the detection of a burst of neutrinos in temporal and directional coincidence. Applied to Antares, the detection of 2 neutrinos within a short time is almost statistically significant: the number of doublets due to atmospheric neutrino background events is of the order of 0.004 per year when a temporal window of 900 s and a directional cone of 2 • × 2 • are defined. It is also possible to search for single cosmic neutrino events by requiring that the reconstructed muon energy is higher than a given energy threshold (typically above a few tens of TeV), for which the atmospheric neutrino background is negligible (see figure 6). When the neutrino telescope is running, this method is almost 100% efficient and applies whenever the neutrinos are emitted with respect to the gamma-ray flash. The main drawback is that a detection is not automatically associated to an astronomical source. It is thus fundamental to organize a complementary follow-up program to confirm the detection.

In this context, Antares is organizing a follow-up program in collaboration with TAROT (Télescope à Action Rapide pour les Objets Transitoires, Rapid Action Telescope for Transient Objects), called TAToO (TAROT-Antares Target of Opportunity, figure 6), as already proposed for IceCube [START_REF] Kowalski | [END_REF]. The TAROT network is composed of two 25 cm optical robotic telescopes located at Calern (South of France) and La Silla (Chile). The main advantages of the TAROT instruments are the large field of view of 1.86 • × 1.86 • and their very fast positioning time (less than 10 s). A GRB afterglow requires a very fast observation strategy in contrast to a core-collapse supernovae for which the optical signal will appear several days after the neutrino signal. The observational strategy is composed of a real time observation followed by few observations during the following month. Depending on the neutrino trigger settings, an alert is sent to TAROT at a rate of about one or two per month. The total latency is quite impressive: less than 10 ms/event for reconstruction, with a pointing accuracy of the order of 0.6 • above 10 TeV, less than 1s for alert sending, with a positioning of the telescopes in less than 10 seconds.

A confirmation by an optical telescope of a neutrino alert will not only give the nature of the source but also allow to increase the precision of the source direction determination in order to trigger other observatories. The TAToO program is operational since February 2009, and several neutrino alerts have already been sent [START_REF] Dornic | for Antares, 31 st ICRC conference[END_REF]. A thorough analysis of these alerts is underway.

Correlations with other observatories

As already pointed out, high energy cosmic neutrinos could be produced in astrophysical sources that are also potential emitters of charged cosmic rays, gamma-rays or gravitational waves. Correlations with dedicated instruments could then bring a harvest of unique information about these sources.

Auger and ultra-high energy cosmic rays

The Pierre Auger Observatory reported an anisotropy in the arrival directions of ultrahigh energy cosmic rays (UHECR) [START_REF] Auger | [END_REF]. Correlation with Active Galactic Nuclei (AGN) from the Véron-Cetty catalog was the most significant for cosmic rays with energies higher than 57 EeV and AGNs at distances less than 75 Mpc. The anisotropy was estimated with a confidence level of 99%. The suggested correlation with local AGN sources mostly follows the location of the supergalactic plane. On the other side, no significant excess above the atmospheric neutrino flux was reported by the neutrino telescopes IceCube and Antares.

Instead of searching for such a localized excess, neutrino arrival direction can be correlated with the arrival directions of ultra high energy cosmic rays, as described in [15]. By obtaining the probability density function of the number of neutrino events within specific angular distance from observed UHECRs, the number of neutrino events in the vicinity of observed ultra-high energy cosmic rays, necessary to claim a discovery with a chosen significance, can be calculated. For example, for 27 UHECR, a correlation significance of 5σ is reached with 2%-25% of neutrino events falling in 1-10 • bins around the original UHECR direction. Possible observed correlation of the arrival directions of those two messengers would provide a strong indication of hadronic acceleration theories.

TeV γ sources observed by Hess

The exact composition of the emission of the sources observed by Hess [16] and other gamma-ray observatories is still an open issue. Some of these sources could have a non-negligible hadronic component, and could then be neutrino emitters. In the case of continuous sources (as opposed to transients), a stacking analysis can be performed, which consists in summing up all the events detected in a given angular region around the direction of the γ-ray source, when below the horizon. Extended sources can be divided in different regions of interest to increase the signal-to-noise ratio. Obviously, adding more and more sources to the analysis also increases the number of background events detected. It is found that an optimum of 25 sources is needed to reach a significance close to 3σ after 5 years of Antares data taking, assuming that all Hess sources are hadronic, after which the sensitivity decreases.

There have also been many discussions about the possibility to detect muons produced by high energy gamma-rays in underground, underice or underwater neutrino telescopes. In contrast to upward-going muons from neutrinos, downwardgoing muons from gamma rays suffer from a high atmospheric muon background. Therefore the sensitivity of a neutrino telescope to gamma ray induced muons is quite lower than atmospheric Čerenkov telescopes. However it monitors continuously all directions. There are at least three processes by which a photon can produce muons : photoproduction, muon pair production and charm decay. It is then possible to look for a global excess of the muon flux in a direction correlated with the position of the source (and within a given time window, if the source is transient). If the statistics is sufficient, which depends on the source spectral index and flux, accurately measured for a number of galactic sources, calibration of both the absolute pointing and the angular resolution of the neutrino telescope could be performed [START_REF] Guillard | 31 st ICRC conference[END_REF].

Gravitational Waves: the gwhen project

Coincident searches of high-energy neutrinos (HEN) and gravitational waves (GW) are also of great interest and are detailed in these proceedings [START_REF] Chassande-Mottin | Proceedings of GWDAW[END_REF]. Both GW and HEN are alternative cosmic messengers that carry information from the innermost regions of the astrophysical engines. Such messengers could also reveal new, hidden sources that were not observed by conventional photon-based astronomy. The observation of such a coincidence would be a landmark event and can also provide observational evidence that GW and HEN originate from a common astrophysical source. Similar coincident studies with GW interferometers have already been proposed in the context of lowenergy neutrinos from core-collapse supernovae [START_REF] Arnaud | [END_REF].

The network of GW detectors formed by the LIGO and Virgo interferometers (figure 7, left) can determine the direction/time of GW bursts in connection with neutrino events observed in Antares. The Virgo/LIGO network started a datataking phase mid-2009, when Antares had already started his data taking with 12 lines. Figure 7 (right) shows the daily averaged visibility of some sources by the network. Joint searches are ongoing within a dedicated gwhen working group: investigations [20] led for Antares indicate that a joint optimized Virgo+LIGO/Antares analysis is needed, in order to increase the coincident detection efficiency, while keeping the coincident false-alarm rate as low as possible. Left: the network of groundbased GW interferometers. The location of Antares is also shown.

Right: averaged visibility skymap for Antares/Virgo+LIGO, where the location of known microquasars/soft-gamma repeaters, potential emitters of GW bursts and HEN, is also shown. This daily (normalized) averaged visibility takes into account the interferometers' beam patterns, and the fact that a neutrino source can be detected only when below the horizon.

Conclusions

Antares is now taking data since 2008 and has demonstrated its ability to operate and get competitive physics results from a neutrino telescope under the sea in the Northern Hemisphere, in spite of its reduced size with respect to IceCube. Note that, because of its location, Antares can observe the Galactic Centre and most of Hess sources on the galactic plane.

Through multi-messenger programs described hereabove, the Antares detector not only enhances its own capabilities as a neutrino telescope, but also contributes to the global effort of understanding the most violent phenomena in our Universe. In addition to offline searches for spatio-temporal correlations with other cosmic messengers (photons, cosmic rays, GW), Antares has the capability to handle external alerts in real time and to trigger follow-up observations with the small latency time required for the study of transient sources. The possibility to store a few minutes of raw data in coincidence with a GCN alert also brings new opportunities for deeper offline analyses.

This could be extended in the future to handle alerts involving other messengers, e.g. gravitational waves, to complete the gwhen project. Finally, the extension of the follow-up programs to other instruments in different wavelengths (X, radio) would undoubtely contribute to the development of the astrophysical potential of Antares.

Figure 1 .

 1 Figure 1. Left: Multiwavelength observations of the SNR RXJ 1713.7-39; the solid curve at energies above 10 7 eV corresponds to π 0 -decay γ-ray emission, whereas the dashed and dash-dotted curves indicate the inverse Compton (IC) and Nonthermal Bremsstrahlung (NB) emissions, respectively. Right: Whipple vs RXTE flux, for the Blazar 1ES1959+650, which shows the existence of orphan γ-ray flares (crosses).

Figure 2 .

 2 Figure 2. The layout of the completed Antares detector. Top Left: acoustic storey (not described here). Bottom left: optical storey with its three photomultipliers.

Figure 3 .

 3 Figure 3. Left: Attenuation of the flux of muons as a function of depth, as extracted using the method described in [7]. Right: Vertical depth intensity relation of atmospheric muons with E µ > 20 GeV (black points).

Figure 4 .

 4 Figure 4. Left: Zenith distribution of reconstructed muons in the 2007+2008 data (90 • is downard). Right: expected sensitivity for Antares in the 5 line configuration as a function of the declination. The expected sensitivity of Antares for one year with twelve lines is also shown.

Figure 5 .

 5 Figure 5. Left: the Gamma-ray bursts Coordinates Network (GCN). Right: Antares response time to an alert.

Figure 6 .

 6 Figure 6. Left: a typical GRB neutrino spectrum. Right: Concept of TAToO.

Figure 7 .

 7 Figure 7.Left: the network of groundbased GW interferometers. The location of Antares is also shown.Right: averaged visibility skymap for Antares/Virgo+LIGO, where the location of known microquasars/soft-gamma repeaters, potential emitters of GW bursts and HEN, is also shown. This daily (normalized) averaged visibility takes into account the interferometers' beam patterns, and the fact that a neutrino source can be detected only when below the horizon.

Aknowledgements: Great thanks to the Organizing Committee, especially Fulvio Ricci, for trusting me for this talk.


T Accadia

F Acernese

F Antonucci

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J.-Y Vinet

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Noise from scattered light in Virgo's second science run data

Virgo is one of the large, ground-based interferometers aimed at detecting gravitational waves. One of the technical problems limiting its sensitivity is caused by light in the output beams which is back-scattered by seismically excited surfaces and couples back into the main beam of the interferometer. The resulting noise was thoroughly studied, measured and mitigated before Virgo's second science run (VSR2). The residual noise during VSR2, which increases in periods with large microseism activity, is accurately predicted by the theoretical model. The scattered light has been associated with transient events in the gravitational-wave signal of the interferometer.

Introduction

The Virgo detector, which is aimed at detecting gravitational waves, is a power-recycled Michelson interferometer with 3 km long Fabry-Perot cavities in its arms. It features an ultra-stable Nd-YAG laser that sends 25 W into the interferometer and test-mass mirrors that are seismically isolated from ground by means of a very efficient multistage suspension system (Super-Attenuators). All the core optics are located in an ultra-high vacuum system. Virgo measures signals in a frequency-band between 10 and 4000 Hz and reaches a strain sensitivity as low as 1 • 10 -22 / √ Hz. Virgo has recently concluded its second science run (VSR2) in coincidence with the LIGO and GEO detectors. The sensitivity reached during VSR2 was close to its design goal. Noticeably, all major fundamental and technical noise-sources have been modeled and their incoherent sum closely matches the measured residual noise [START_REF] Accadia | Commissioning status of the Virgo interferometer[END_REF]. One of the sources of technical noise is related to diffused light, which has been studied extensively and has been mitigated for the case of light scattered inside the vacuum system [START_REF] Vinet | Scattered light noise in gravitational wave interferometric detectors: Coherent effects[END_REF][START_REF] Vinet | Scattered light noise in gravitational wave interferometric detectors: A statistical approach[END_REF][START_REF] Takahashi | Direct measurement of the scattered light effect on the sensitivity in TAMA300[END_REF]. Similarly, light scattered from optics located on benches outside the vacuum system can introduce noise [START_REF] Lück | Opto-mechanical frequency shifting of scattered light[END_REF].

At Virgo, in-air benches are placed at the input port (External Injection Bench, EIB), output port (External Detection Bench, EDB) and behind the terminal mirrors (North and West-End Benches, NEB and WEB), which transmit just a tiny fraction of light for controlling the alignment of the interferometer. Optical powers between 100 mW and a few Watt are received by these benches. Inevitably, a tiny fraction of this light gets diffused by the optical components (lenses, mirrors, photo-detectors, beam dumps) on the benches, which move due to seismic excitation. Part of this light, which is phase-modulated by the motion of the optics, scatters backwards and recombines with the main optical beam resonating inside the interferometer.

Scattered light from external optics was found to be limiting the sensitivity of Virgo after its first science run (VSR1, 2007) [START_REF] Acernese | Noise studies during the first Virgo science run and after[END_REF]. Section 2 describes the model developed for scattered-light noise as well as measurements and mitigations performed on the interferometer. As predicted by the model, some residual noise was still present during VSR2. Section 3 shows how this noise causes transient events in case of bad weather.

Back-scattered light from in-air benches

This section will discuss the case of light scattered by Virgo's terminal benches. A more detailed analysis and results for the other benches can be found in reference [START_REF] Tournefier | Back-scattering by the optical benches: results from Virgo and constraints for AdV[END_REF].

Model

It is assumed that all optics move coherently with the bench, which is valid since only bench motion at frequencies below those of optical mount resonances is considered (e.g. f ≤ 100Hz).

Consider the light that is resonating in one of the long Fabry-Perot cavities with amplitude A 0 . A fraction T of the optical power (40 ppm for Virgo) is transmitted by the end-mirror to the terminal bench. Each optical component on this bench diffuses the impinging light according to its bidirectional reflectance distribution function (BRDF). Overall, a tiny fraction f sc of the optical power scatters exactly backwards and recouples to the optical field resonating inside the Fabry-Perot. This is the back-scattered beam with amplitude A sc . Its power is suppressed by the mirror transmission factor (T ) and then multiplied by the Fabry-Perot optical gain (2F/π), where F is the finesse. The amplitude of the scattered beam is therefore

A sc = A 0 T 2F π √ f sc .
The scattered field carries a phase noise with respect to the static Fabry-Perot field because its optical path length is modulated by the displacement of the scattering optics. Its phase angle is

φ sc (t) = 4π λ (x 0 + δx opt (t)) = φ 0 + δφ sc (t), (1)
where δx opt is the displacement of the optics along the direction of beam (essentially the displacement of bench in the horizontal plane), x 0 is the static optical path and λ is the optical wavelength (1064 nm). The total field inside the arm cavity is given by A tot = A 0 + A sc , its phase angle is the phase noise added to the interferometer beam: δΦ = Asc A 0 • sin φ sc . Since the measured gravitational-wave strain is proportional to this phase, the noise introduced by the scattered light h sc can be written as

h sc (t) = λ 8F L δΦ = K • f sc • sin(φ sc ) = G • sin 4π λ (x 0 + δx sc (t)) , ( 2
)
where L is the length of the cavity (3 km). The factor K is defined as λ

4L

T √ 2F π and the coupling factor G as K • √ f sc , which relates the motion of the scatterer (the bench) to strain noise. A similar model can be obtained for the other benches, but with K depending on different optical parameters of the interferometer.

In practice the beam optical path undergoes slow drifts caused by thermal effects, air flows and beam alignment fluctuations. These drifts are of the order of a few microns in a time scale of some seconds. The angle φ 0 can thus vary by more than 2π within this timescale. The following considerations apply to Eq. 2:

• for small bench motion (δx(t) << λ 4π 10 -7 m) the equation linearizes as

h sc (t) = G • cos φ 0 • 4π λ • δx sc (t).
The term cos φ 0 accounts for the slow drifts, it averages to 1/ √ 2 for time scales longer than one minute or so, while for shorter observation times the noise is non-stationary.

• for larger bench motion (δx(t) ≥ 10 -7 m) the induced strain noise is nonlinear.

This condition does occur in correspondence of mechanical resonances of the tables (bench and support structure, f res 10 -20Hz), and at the microseism peak frequency (f res = 0.3Hz), which can be as large as some micrometers in case of intense sea activity.

One consequence of the non-linearity is that intense low frequency seismic excitations (below 10Hz) can produce noise in the detection band (above 10Hz). This up-conversion has indeed been observed in VSR2 and will be discussed in the following sections.

Measurement

Using the noise model of Eq. 2, the contribution of scattered light from the external benches can be estimated once the bench displacement and coupling factor are measured. One tri-axial seismometer (Episensor model ES-T ) is placed on each optical bench, which measures the bench displacement in the frequency range from 0.2Hz to 100Hz. Figure 1 illustrates the method of the coupling factor measurement. A mechanical shaker (Brüel & Kjaer model 4809) is used to seismically excite the bench with a sinusoidal signal. The signal amplitude is A m 10 -6 m and the frequency is chosen close to the low frequency edge of the flat acceleration response function of the shaker (f m 20Hz). Alternatively, a narrow-band white noise is used to excite the table mechanical resonance (f m 15Hz).

The bench motion is measured by the seismic sensor (Figure 1(left)). The resulting noise (h tot ) in the interferometer is shown in Figure 1(right) compared to the quiet strain noise (h 0 ) when no extra excitation is applied to the bench. The coupling factor G of the model is adjusted to best reproduce the back-scattering noise (h 2 sc = h 2 tot -h 2 0 ). Note the characteristic shape of the noise shown in Figure 1(right), which shows bumps at multiples of the frequency of the excitation signal. This is typically observed when, as in this case, the scatterer has essentially an oscillatory motion: δx sc (t) A m sin(2πf m t). The back-scattering noise (Equation 2) is in this case a frequency modulated signal at f m with modulation index m = A m 4π λ . According to the Jacoby-Anger identity, the amplitude of the n th harmonics is given by the n th Bessel function. Harmonics are visible up to a maximum frequency f max m • f m .

Mitigation

Noise produced by back-scattered light from the external benches was limiting Virgo's sensitivity at the end of VSR1 (Figure 2, left). The contributions of the individual benches are projected using the model with the coupling factor G measured as explained in section 2.2, while using the displacement of the bench measured in quiet conditions. A lot of effort has been spent to mitigate this noise by acting in two ways [START_REF] Canuel | Diffused light mitigation in Virgo and constraints for Virgo+ and AdV[END_REF]:

• reducing the motion of the benches at their resonance frequency (around 15 Hz) by a factor 3: a resonant damper helped absorbing energy of the table mode, a new bench cover helped coupling less acoustic noise to the bench and a slowdown of the fans of the air-conditioning reduced seismic and acoustic noise emission.

• reducing the fraction of diffused light reflected by the benches.

Major sources of scattering on the bench were located by looking for bright spots with IR-cameras, as well as by tapping on optical elements and listening to the noise produced in the output signal of the interferometer. The amount of reflected light was reduced by misaligning lenses, dumping secondary beams, cleaning optics from dust particles and replacing standard mirrors by super-polished ones. For critical objects, the BRDF has been measured in a dedicated set-up, which allowed the estimation of the effective bench reflectivity f sc based on the optical geometry [START_REF] Canuel | Determination of back-scattering from single optics[END_REF][START_REF] Tournefier | Back-scattering by the optical benches: results from Virgo and constraints for AdV[END_REF]. Table 1 shows the coupling factors for the various benches measured before and after the mitigation using the method of Sec.2.2. On average a reduction of a factor 2 to 5 was achieved. The measured coupling factors G are in good agreement with values calculated using estimates for K and f sc . The large coupling of the West-End bench can be explained by the larger transmission of the West-End mirror (T = 40ppm) with respect to the North-End mirror (T = 10ppm). Figure 2(right) shows the projection of benches scattered light noise during VSR2.

Scattered-light noise effects in VSR2

Influence of bad weather

During VSR2, it was observed that the largest changes in the sensitivity of the interferometer correlate with the presence of bad weather. The most significant effect of the bad weather is an increase in sea-activity, which typically follows strong winds with a delay of a few hours. This causes a large excess of seismic noise between 0.2 and 1 Hz, see Fig. 3(left). Although the scattered light is not the only problem caused by bad weather, its effect on the output signal of the interferometer at low frequencies is the most obvious, see Fig. 3(right).

Noise in the time-domain

As can be seen in Eq. 2, the effect of a path-length change of the scattered light is periodic with a period of λ/2. For fast path-length changes, the effect of the scattered light thus causes fringes in the interferometer output with frequency f fringe with v sc the velocity of the scatterer (the derivative of δx sc ). This velocity can be measured by various sensors, since at the low frequency of the microseism, the earth and the optical bench move synchronously, while the mirror is kept fixed by the seismic isolation system and the control system of the interferometer. Best results were obtained by taking the derivative of a position sensor of the seismic isolation system. The low frequency oscillations caused by the microseism produce pretty typical arches in the spectrogram of the output of the interferometer, see Fig. 4(left). The frequency as a function of time is almost perfectly predicted by the model of Eq. 3 for the velocity of the West-End bench, whose noise contribution is dominating.

f fringe (t) = 2 v sc (t) λ , (3)
Also the second harmonic frequency of the arches can be observed in the spectrograms, see Fig. 4(left). This can be explained by scattered light that follows a double-bounce path, in which the light is transmitted by the end-mirror, reflects off some moving component on the bench, reflects off the high reflective end-mirror and scatters a second time on the bench before reentering the main beam. In this case, the optical path of the scattered light changes with 4 times the change in position of the ground, instead of 2 times as seen in Eq. 3. In this case, the induced contribution to the output of the interferometer is thus

h(t) = G 2 • sin 8π λ • x(t) (4)
Since this involves two spurious reflections, it would be expected that G 2 is many orders of magnitude lower than G. In reality it was found that G 2 6 × 10 -21 at the West-End bench, only a factor 4 lower than G. One possibility is that one of the scatterers has a very high reflection (e.g. a photo-diode or an uncoated glass surface). Most of these components are tilted to not scatter directly back into the beam, so there is probably some unfortunate geometrical path that allows such a reflection to reenter after a reflecting once off the end-mirror.

One problem with the double-bounce fringes is that they occur at twice the frequency of the normal fringes:

f fringe2 (t) = 2f fringe = 4 v sc (t) λ (5)
Even though G 2 is lower, these fringes have a larger impact on the sensitivity of the interferometer since the noise occurs at higher frequencies, where Virgo is more sensitive.

Transient events

One type of gravitational-wave signals that is sought in the output of the interferometer is short and burst-like. The data coming out of the interferometer is analyzed by various software pipelines to find these signals. Since the disturbances caused by scattered light can also be short and burst-like (the arches in the spectrogram described before), they are detected by the pipelines and cause a large amount of false triggers in case of bad weather.

One of these pipelines, called Omega, is looking for sine-Gaussian type bursts in the whitened output signal of the interferometer [START_REF] Chatterji | Multiresolution techniques for the detection of gravitational-wave bursts[END_REF]. For each trigger, it records the time, frequency and signal-to-noise-ratio (SNR). As indicated by Eq. 3 and Eq. 5, the fringe frequency depends on the velocity of the scatterer. To test if this relation also holds for the observed glitches, the reported frequency has been plotted against the velocity of the West-End bench, interpolated at the moment of the glitch, see Fig. 4(right). Indeed, a distinct cloud of high-SNR points is clearly visible, which are well predicted by the model for the double-bounce glitches. Note that the pipeline does not report at frequencies below 50 Hz, where the interferometer is less sensitive. A procedure to veto these glitches based on the measured velocity is being studied [START_REF] The | and the[END_REF].

Conclusions

Light back-scattered by in-air optics can be a critical source of noise to gravitationalwave interferometers, since the phase of the reflected light is modulated by the motion of the optical benches, which is driven by natural and anthropogenic seismic noise. In case of large motion of the optics, the low-frequency seismic noise can up-convert into the detection band of the interferometer and severely reduce the sensitivity of the instrument. This phenomena has been thoroughly studied in the last years and its effect on the sensitivity is now accurately modeled. The noise can be prevented by reducing either the amount of back-scattered light or the motion of the scatterer.

Procedures have been developed to measure the coupling factor of this noise to the gravitational-wave signal. This was essential to identify the problem and evaluate improvements. Due to a large effort to reduce the amount of scattered light and the environmental noise on the optical benches, the noise produced by scattered light is now not limiting Virgo's sensitivity in quiet conditions. It is, however, still an issue in periods with high microseism. This was evident during Virgo's second science run, which showed a large increase in the glitch rate in case of bad weather. This problem might be partly mitigated in the data analysis by implementing a veto based on the ground velocity.

Using the obtained experience, actions are planned to suppress the scattered-light noise to well below the design sensitivity of future upgrades (Virgo+ and Advanced Virgo). Direct improvements are expected from the increase of the finesse of the Fabry-Perot cavities, a reduction of the transmission of the end-mirrors and the addition of antireflection coating that was missing on some output windows. It will remain necessary to prevent as much as possible the reflection of light from the external benches. This will be done by careful design of the optical layout of the benches, a choice of low-scattering optical components and the efficient dumping of spurious reflections. For Advanced Virgo, the main photo-diodes will be located under vacuum on seismically isolated benches. An active seismic isolation system is considered for the external benches.

Figure 1 .

 1 Figure 1. Measurement of scattered light coupling at the West-End bench. Pictures show the spectral composition of bench displacement (left) and interferometer output (right) in the undisturbed case (cyan) and when one sinusoidal excitation is applied to the bench (black). The excess noise in the interferometer output is matched closely by the scattered light model (red).

Figure 2 .

 2 Figure 2. Projection of diffused light noise from external benches after VSR1 (left) and during VSR2 (right).

FrequencyFigure 3 .

 3 Figure 3. Comparison between good and bad weather. (left) Linear spectral density of the velocity of the West-End Bench along the beam axis, reconstructed using position sensors and accelerometers. The big increase below 1 Hz is caused by sea-activity, while the structure around 10 Hz is caused by bench resonances. (right) Measured sensitivity of the interferometer and projection of scattered-light noise. In bad weather conditions, this noise is limiting up to 100 Hz.

Figure 4 .

 4 Figure 4. (left) Spectrogram of the interferometer typical output signal in conditions of intense microseism activity, the color scale is logarithmic. Overlaid are curves calculated with Eq. 3 and Eq. 5 using the measured speed of the West-End Bench, without any adjustment of parameters. (right) Scatter-plot of glitch frequency vs ground velocity at the moment of a glitch. This plot displays the glitches collected by the Omega pipeline during one day with high microseism. The SNR of the glitch is indicated by the color scale. The colored lines indicate the relations of Eq. 3 and Eq. 5.

Table 1 .

 1 Measured coupling factors of external benches, before and after the diffused light mitigation campaign. Values in parenthesis are derived from estimated K and bench diffused light fractions.


Piotr Jaranowski

Andrzej Królak
email: krolak@impan.gov.pl

Searching for gravitational waves from known pulsars using the F and G statistics

Keywords: numbers: 95.55.Ym, 04.80.Nn, 95.75.Pq, 97.60.Gb

In searches for gravitational waves emitted by known isolated pulsars in data collected by a detector one can assume that the frequency of the wave, its spindown parameters, and the position of the source in the sky are known, so the almost monochromatic gravitational-wave signal we are looking for depends on at most four parameters: overall amplitude, initial phase, polarization angle, and inclination angle of the pulsar's rotation axis with respect to the line of sight. We derive two statistics by means of which one can test whether data contains such gravitational-wave signal: the G-statistic for signals which depend on only two unknown parameters (overall amplitude and initial phase), and the F -statistic for signals depending on all four parameters. We study, by means of the Fisher matrix, the theoretical accuracy of the maximum-likelihood estimators of the signal's parameters and we present the results of the Monte Carlo simulations we performed to test the accuracy of these estimators.

Introduction

We study the detection of almost monochromatic gravitational waves emitted by known single pulsars in data collected by a detector. Several such searches were already performed with data collected by the LIGO and GEO600 detectors [1,2,3,4,5]. We thus assume that the frequency of the wave (together with its time derivatives, i.e. the spindown parameters) and the position of the source in the sky are known. The gravitational-wave signal we are looking for depends on at most four (often called amplitude) parameters: overall amplitude, initial phase, polarization angle, and inclination angle (of the pulsar's rotation axis with respect to the line of sight).

In Sec. 2 we introduce three statistics by means of which one can test whether data contains a gravitational-wave signal: the H-statistic for completely known signals, the G-statistic for signals which depend on only two unknown parameters (overall amplitude and initial phase), and the F-statistic suitable for signals depending on all four amplitude parameters. Both statistics G and F are derived from the maximum Confidential: not for distribution. Submitted to IOP Publishing for peer review 27 May 2010 likelihood (ML) principle, and the statistic G is independently obtained using Bayesian approach and the composite hypothesis testing. In Sec. 3 we study, by means of the Fisher matrix, the theoretical accuracy of the ML estimators of the signal's parameters and in Sec. 4 we present the results of the Monte Carlo simulations we performed to test the accuracy of the ML estimators.

Using the F and G statistics to perform targeted searches for gravitational waves from pulsars

In the case when the signal s(t) we are looking for is completely known, the test that maximizes probability of detection subject to a certain false alarm probability is the likelihood-ratio test, i.e. we accept the hypothesis that the signal is present in detector's data x if

Λ(x) := p 1 (x) p 0 (x) ≥ λ 0 , (1)
where the likelihood function Λ(x) is the ratio of probability densities p 1 (x) and p 0 (x) of the data x when the signal is respectively present or absent. The parameter λ 0 is a threshold calculated from a chosen false alarm probability. Assuming stationary and additive Gaussian noise with one-sided spectral density constant (and equal to S 0 ) over the bandwidth of the signal, the log likelihood function is approximately given by [6] ln

Λ[x(t)] ∼ = 2 T o S 0 x(t)s(t) - 1 2 s(t) 2 , (2)
where T o is the observation time and the time-averaging operator • is defined as

g := 1 T o To 0 g(t) dt. (3)
Equation ( 2) implies that the likelihood-ratio test (1) can be replaced by the test

H[x(t)] := x(t)s(t) ≥ H 0 , (4)
where the optimal statistic H in this case is the matched filter and H 0 is the threshold for detection. Suppose now that the signal s(t; θ) depends on a set of unknown parameters θ, then a suitable test can be obtained using a Bayesian approach and composite hypothesis testing. The composite hypothesis in this case is the hypothesis that when a signal is present it can assume any values of the parameters. Assuming that the cost functions are independent of the values of the parameters, we obtain the following Bayesian decision rule to choose the hypothesis that the signal is present (see e.g. [START_REF] Whalen | Detection of Signals in Noise[END_REF], Chapter 5.9):

1 p 0 (x) Θ p 1 (x; θ)π(θ) dθ ≥ γ 0 , ( 5
)
where Θ is the parameter space on which θ is defined and π(θ) is the joint a priori distribution of θ. The expression on the left hand side of Eq. ( 5) is know as the Bayes factor and it is the ratio between the posterior probability distribution on the signal parameters marginalized over the parameters themselves (this is the signal model Bayesian evidence) and the noise model which has no defining parameters (this is the noise model Bayesian evidence).

As a template for the response of an interferometric detector to the gravitationalwave signal from a rotating neutron star we use the model derived in [6]. This template depends on the set of following parameters: θ = (h 0 , φ 0 , ψ, ι, f , δ, α), where h 0 is the dimensionless amplitude, φ 0 is an initial phase, ψ is the polarization angle, ι is the inclination angle, angles δ (declination) and α (right ascension) are equatorial coordinates determining the position of the source in the sky, and the 'frequency vector' f := (f 0 , f 1 , f 2 , . . . ) collects the frequency f 0 and the spindown parameters of the signal. In the case of pulsars known from radio observations we in general know the subset ξ = (f , δ, α) of the parameters θ.

Sometimes, like in the case of the Vela pulsar, we also know from X-ray observations the values of the angles ψ and ι (see [7,8] for observational results). We then have only two unknown parameters: h 0 and φ 0 . The response s(t) of the detector to the gravitational wave we can write in this case in the following form [6]:

s(t) = h 0 cos φ 0 h c (t) + h 0 sin φ 0 h s (t), (6)
where h c and h s are known functions of time,

h c (t) := A + cos 2ψ h 1 (t) + sin 2ψ h 2 (t) -A × sin 2ψ h 3 (t) -cos 2ψ h 4 (t) , h s (t) := -A × sin 2ψ h 1 (t) -cos 2ψ h 2 (t) -A + cos 2ψ h 3 (t) + sin 2ψ h 4 (t) . (7)
Here the constants A + and A × are

A + := 1 2 (1 + cos 2 ι), A × := cos ι, (8)
and the four functions of time h k (k = 1, . . . , 4) depend only on parameters ξ and are defined as follows

h 1 (t; ξ) := a(t; δ, α) cos φ(t; f , δ, α), h 2 (t; ξ) := b(t; δ, α) cos φ(t; f , δ, α), h 3 (t; ξ) := a(t; δ, α) sin φ(t; f , δ, α), h 4 (t; ξ) := b(t; δ, α) sin φ(t; f , δ, α), (9)
where a, b are the amplitude modulation functions and φ is the phase modulation function. Their explicit forms are given in [6].

Let us calculate the likelihood function for the signal (6). Observing that the amplitude modulation functions a and b vary much more slowly than the phase φ of the signal and assuming that the observation time is much longer than the period of the signal we approximately have [6]

h 1 h 3 ∼ = h 1 h 4 ∼ = h 2 h 3 ∼ = h 2 h 4 ∼ = 0, h 1 h 1 ∼ = h 3 h 3 ∼ = 1 2 A, h 2 h 2 ∼ = h 4 h 4 ∼ = 1 2 B, h 1 h 2 ∼ = h 3 h 4 ∼ = 1 2 C, (10)
where we have introduced the time averages

A := a 2 , B := b 2 , C := ab . ( 11
)
As a consequence of the above approximations we have the following approximate expressions for the time averaged products of the functions h c and h s ,

h 2 c ∼ = h 2 s ∼ = N, h c h s ∼ = 0, ( 12
)
where N is a constant defined as

N := 1 2 A(A 2 + cos 2 2ψ + A 2 × sin 2 2ψ) + B(A 2 + sin 2 2ψ + A 2 × cos 2 2ψ) + C(A 2 + -A 2 × ) sin 4ψ . ( 13
)
With the above approximations the likelihood function Λ for the signal (6) can be written as

ln Λ[x(t); φ 0 , h 0 ] ∼ = 2 T o S 0 h 0 cos φ 0 x(t)h c (t) + h 0 sin φ 0 x(t)h s (t) - 1 2 h 2 0 N . ( 14
)
Let us also note that the optimal signal-to-noise ratio (SNR) ρ for the signal (6) (see [6] for definition) can be approximately computed as

ρ ∼ = 2T o S 0 s(t) 2 ∼ = 2T o N S 0 h 0 . ( 15
)
It is natural to assume that the prior probability density of the phase parameter φ 0 is uniform over the interval [0, 2π) and that it is independent of the distribution of the amplitude parameter h 0 , i.e.

π(φ 0 ) = 1 2π , φ ∈ [0, 2π). ( 16
)
With the above assumptions the integral 2π 0 p 1 (x; φ 0 , h 0 )π(φ 0 )dφ 0 can be explicitly calculated (see [START_REF] Whalen | Detection of Signals in Noise[END_REF], Chapter 7.2) and we obtain the following decision criterion exp -

h 2 0 NT o S 0 I 0 2h 0 T o N S 0 G[x(t)] ≥ γ 0 , (17)
where I 0 is the modified Bessel function of zero order and the statistic G is defined as

G[x(t)] := T o NS 0 x(t)h c (t) 2 + x(t)h s (t) 2 . ( 18
)
The function on the left-hand side of Eq. ( 17) is a monotonically increasing function of G and it can be maximized if G is maximized independently of the value of h 0 . Thus the test

G[x(t)] ≥ G 0 , (19)
provides a uniformly most powerful test with respect to the amplitude h 0 . When we have no a priori information about the parameters a standard method is the maximum likelihood (ML) detection which consists of maximizing the likelihood function Λ[x(t); θ] with respect to the parameters of the signal. If the maximum of Λ exceeds a certain threshold we say that the signal is detected. The values of the parameters that maximize Λ are said to be the ML estimators of the parameters of the signal. For the case of signal (6) it is convenient to introduce new parameters

A c := h 0 cos φ 0 , A s := h 0 sin φ 0 . ( 20
)
Then one can find the ML estimators of the amplitudes A c and A s in a closed analytic form,

Âc ∼ = xh c N , Âs ∼ = xh s N . (21)
It is easy to find that the estimators Âc and Âs are unbiased and also that they are of minimum variance, i.e. their variances attain the lower Cramér-Rao bound determined by the Fisher matrix. The variances of both estimators are the same and equal to 1/N . Substituting the estimators Âc and Âs for the parameters A c and A s in the likelihood function one obtains a reduced likelihood function. This reduced likelihood function is precisely equal to the G-statistic given by Eq. ( 18), i.e. G[x(t)] = ln Λ[x(t); Âc , Âs ].

The formula for the G-statistic obtained without usage of the simplifying assumptions [START_REF] Jaranowski | Analysis of Gravitational-Wave Data[END_REF] is given in Appendix A.

When the all four parameters (h 0 , φ 0 , ψ, ι) are unknown one can introduce new parameters A k (k = 1, . . . , 4) that are functions of (h 0 , φ 0 , ψ, ι) such that the response s(t) takes the form

s(t) = A 1 h 1 (t) + A 2 h 2 (t) + A 3 h 3 (t) + A 4 h 4 (t), ( 22
)
where the functions h k are given by Eqs. ( 9) and the parameters A k read

A 1 := h 0+ cos 2ψ cos φ 0 -h 0× sin 2ψ sin φ 0 , A 2 := h 0+ sin 2ψ cos φ 0 + h 0× cos 2ψ sin φ 0 , A 3 := -h 0+ cos 2ψ sin φ 0 -h 0× sin 2ψ cos φ 0 , A 4 := -h 0+ sin 2ψ sin φ 0 + h 0× cos 2ψ cos φ 0 ; (23)
here h 0+ := h 0 A + and h 0× := h 0 A × [see Eq. ( 8)]. The ML estimators of A k can again be obtained in an explicit analytic form and the reduced likelihood function is the F-statistic given by (see [6] for details)

F[x(t)] := ln Λ[x(t); Â1 , . . . , Â4 ] ∼ = 2T o S 0 D B ( xh 1 2 + xh 3 2 ) + A ( xh 2 2 + xh 4 2 ) -2C ( xh 1 xh 2 + xh 3 xh 4 ) , (24)
where

D := AB -C 2 . The test F[x(t)] ≥ F 0 (25)
is not a uniformly most powerful test with respect to unknown parameters (h 0 , φ 0 , ψ, ι).

It was recently shown that uniform a priori distributions of (h 0 , φ 0 , ψ, cos ι) lead to a statistic that can be more powerful than F [START_REF] Prix | [END_REF]. In Fig. 1 we have plotted the receiver operating characteristics (ROC) for the three statistics H, G, and F considered in the present section.

The Fisher matrix

Using the Fisher matrix we can assess the accuracy of the parameter estimators. We have two theorems that can loosely be stated as follows.

Theorem 1 (Cramèr-Rao bound) The diagonal elements of the inverse of the Fisher matrix are lower bounds on the variances of unbiased estimators of the parameters.

Theorem 2 Asymptotically (i.e. when the SNR tends to infinity) the ML estimators are unbiased and their covariance matrix is equal to the inverse of the Fisher matrix. For an almost monochromatic signal s = s(t; θ), which depends on the parameters θ = (θ 1 , . . . , θ m ), the elements of the Fisher matrix Γ can be approximately calculated from the formula

Γ θiθj ∼ = 2T o S 0 ∂s ∂θ i ∂s ∂θ j , i,j = 1, . . . , m. (26)
In the case when only the parameters h 0 and φ 0 are unknown (G-statistic search), the Fisher matrix can be computed easily from Eqs. ( 6) and (26). It is diagonal and the standard deviations of the parameters defined as the square roots of the diagonal elements of the inverse of the Fisher matrix read:

σ h0 h 0 = 1 ρ , σ φ0 = 1 ρ , ( 27
)
where ρ is the optimal SNR [given in Eq. ( 15)]. When all the four amplitude parameters h 0 , φ 0 , ψ, and ι are unknown (F-statistic search), the Fisher matrix can be computed by means of formulas given in Appendix B. In this case it is not diagonal, indicating that the amplitude parameters are correlated. The quantities σ h0 /h 0 , σ φ0 , σ ψ , σ ι (where the standard deviations again are defined as square roots of diagonal elements of the inverse of the Fisher matrix) have rather complicated analytical form but they possess a number of simple properties. They are inversely proportional to the overall amplitude h 0 , independent on the initial phase φ 0 , and very weakly dependent on ψ, however there is a strong dependence on ι.

In Fig. 2 we have shown the dependence of the standard deviations on the cosine of the inclination angle ι. The time averages from Eqs. [START_REF] Prix | [END_REF] (needed to compute the Fisher matrix) were computed here for the location of the Virgo detector [START_REF] Acernese | Status of Virgo[END_REF] and for a randomly chosen position of the source in the sky. We have also taken h 0 = 6.0948 × 10 -2 , T o = 441610 s, and S 0 = 2 Hz were used in the Monte Carlo simulations described in Sec. 4. We see in Fig. 2 that the standard deviations become singular when cos ι = ±1. This singularity originates from the degeneracy of the amplitude parameters for cos ι = ±1. In this case the amplitude parameters from Eqs. (23) become

A 1 = h 0 cos(2ψ ± φ 0 ), A 2 = h 0 sin(2ψ ± φ 0 ), A 3 = ∓A 2 , A 4 = ±A 1 . (28)
Thus only two of them are independent. Therefore the determinant of the 4dimensional Fisher matrix is equal to zero at cos ι = ±1 and consequently its inverse does not exist in this case.

Monte Carlo simulations

We have performed two Monte Carlo simulations in order to test the performance of the ML estimators. We have compared the simulated standard deviations of the estimators with the ones obtained from the Fisher matrix. In particular we have investigated the behavior of the ML estimators near the Fisher matrix singularity at cos ι = ±1. In each simulation run we have generated the signal using Eq. ( 22), we have added it to a white Gaussian noise, and we have estimated the amplitude parameters using the F-statistic. Each simulation run was repeated 1000 times for different realizations of the noise.

In the first simulation we have investigated the bias and the standard deviation of the ML estimator of the amplitude parameter h 0 as functions of the SNR for the two cases: cos ι = 0.1 and cos ι = -0.93. The results are presented in Fig. 3. For the first case the ML estimator is nearly unbiased and its standard deviation is close to the one predicted by the Fisher matrix even for low SNRs. In the second case the simulation shows considerable bias of the estimator and its standard deviation lower than the one predicted by the Fisher matrix. However, Theorem 2 is satisfied in the second case. For cos ι close to ±1 we have to go to SNR ∼ 1000 in order for the ML estimator to be unbiased and its standard deviation close to the one given by the Fisher matrix.

In the second simulation, illustrated in Fig. 4, we have investigated the bias and the standard deviation of the ML estimators of the amplitude parameters h 0 and cos ι as functions of cos ι for the fixed SNR ρ = 15.6. We find that for | cos ι| < 0.5 the biases are less than 10% and the Fisher matrix overestimates the standard deviations also by less than 10%. We see that over the whole range of cos ι the standard deviations of the parameters are roughly constant whereas the biases increases as the | cos ι| increases. At cos ι ± 1 the amplitude h 0 is overestimated by almost a factor of 2.

One reason why Theorem 1 does not apply here is that it holds for unbiased estimators. Also a more precise statement of Theorem 1 (see e.g. Theorem 8 in [START_REF] Jaranowski | Analysis of Gravitational-Wave Data[END_REF]) requires that the Fisher matrix Γ is positive definite for all values of parameters. This last assumption is clearly not satisfied here as det Γ = 0 for cos ι = ±1. where the vector A collects the amplitude parameters, A := (A 1 , A 2 , A 3 , A 4 ), and the known functions h k (k = 1, . . . , 4) are given in Eqs. [START_REF] Whalen | Detection of Signals in Noise[END_REF]. We further assume, as in Sec.

2, that the noise spectral density is constant (and equal to S 0 ) over the bandwidth of the signal and that the approximations [START_REF] Jaranowski | Analysis of Gravitational-Wave Data[END_REF] Let us introduce new set of parameters θ := (h 0 , φ 0 , ψ, ι). Then the Fisher matrix Γ(θ) for these parameters can be computed as (T denotes here matrix transposition)

Γ(θ) = J T • Γ(A) • J, (B.4)
where the Jacobi 4 × 4 matrix J has elements ∂A i /∂θ j (i, j, = 1, . . . , 4), which can be computed by means of Eqs. (23).

Figure 1 .

 1 Figure 1. Receiver operating characteristic (ROC) for the statistics H, G, and F for the optimal signal-to-noise ratio ρ = 2.

- 1 ,Figure 2 .

 12 Figure 2. Dependence of standard deviations (calculated from the Fisher matrix) of the parameters h 0 , φ 0 , ψ, and cos ι on the cosine of the inclination angle ι. We have taken φ 0 = 4.03 and ψ = -0.22 (values of other parameters needed to perform the computation of the Fisher matrix are listed in the text of Sec. 3).

Figure 3 .

 3 Figure 3. Mean and normalized standard deviation of the ML estimator of the amplitude h 0 as a function of the SNR. The top two panels are the means of the estimator for the two values of cos ι. The continuous line is the true value and the circles are results of the simulation for 1000 realizations of the noise. The bottom two panels are the standard deviations. The continuous line is obtained form the Fisher matrix whereas the circles are results of the simulation. We have taken φ 0 = 4.03 and ψ = -0.22.

Figure 4 .

 4 Figure 4. Means and normalized standard deviations of the ML estimators of h 0 and cos ι as functions of cos ι. The top two panels are the means of the estimators. The continuous lines are the true values and the circles are results of the simulation for 1000 realizations of the noise. The bottom two panels are the standard deviations. The continuous lines are obtained form the Fisher matrix whereas the circles are results of the simulation. We have assumed φ 0 = 4.03, ψ = -0.22, and ρ = 15.6. Plots for 0 ≤ cos ι ≤ +1 (not shown here) are mirror images of the plots for -1 ≤ cos ι ≤ 0.

  k=1

A

  k h k (t), (B.1)

Γ

  are valid. Then the Fisher matrix for the signal's parameters A reads

Acknowledgments

This work was supported by the MNiSW grant no. N N203 387237. A.K. would like to acknowledge hospitality of the Max Planck Institute for Gravitational Physics in Hannover, Germany, where part of this work was done. We would like to thank members of the LSC-Virgo CW data analysis for helpful discussions.

Appendix A. The general form of the G-statistic

It is not difficult to obtain the G-statistic without simplifying assumptions [START_REF] Jaranowski | Analysis of Gravitational-Wave Data[END_REF]. The estimators of the amplitude parameters A c and A s are then given by

and the general form of the G-statistic reads

Appendix B. Fisher matrix for amplitude parameters

Let us consider the gravitational-wave signal s of the form s(t; A) =