R Flaminio

J Franc

C Michel

N Morgado

L Pinard

B Sassolas

Study of coating mechanical and optical losses in view of reducing mirror thermal noise in gravitational wave detectors

Keywords: numbers: 04.80.Nn, 62.40.+i, 68.60.-p, 95.55.Ym

Mirror coatings play a crucial role in the performance of laser interferometers devoted to gravitational wave detection such as Virgo and LIGO. Mechanical losses in the coating material limit the sensitivity of the detectors due to the associated mirror thermal noise. The absorption of light in the coating induces a thermal lens in the mirror substrate which reduces the quality of the optical interference and requires sophisticated thermal compensation systems. This paper describes the work ongoing at LMA in order to reduce mechanical losses and optical absorption in the coating. The results obtained by doping the Ta 2 O 5 layers and testing different high index materials are described. Finally the performances of different potential coatings are compared and the results obtained with a 40 kg mirror are reported. Titania doped Ta 2 O 5 shows mechanical losses of 2 10 -4 and absorption below 0.5 ppm.

Nb 2 O 5 appears to be the best competitor from the thermal noise point of view but it has an optical absorption 4 to 5 five times larger.

Introduction

Mirror coatings play a crucial role in the performance of laser interferometers devoted to gravitational wave detection such as Virgo [1] and LIGO [2]. Mechanical losses inside the coating limit the sensitivity of the detectors due to the associated mirror thermal noise. This noise is one of the limitations to the sensitivity of Virgo and LIGO and it is expected to remain the main limitation in the sensitivity of advanced detectors to gravitational waves with frequencies around 100 Hz [START_REF]Advanced Virgo Baseline Design[END_REF].

Another important aspect connected with the coating properties is the absorption of light inside the coating material. The absorption of light induces a thermal lens in the mirror substrates which reduces the quality of the optical interference. In the end this effect limits the amount of laser power that can be stored in the interferometer and, as a consequence, the sensitivity of the detectors at frequencies above a few hundreds Hz where the limitation comes from the shot noise. To correct for these effects sophisticated thermal compensation systems have been developed [START_REF] Willems | Proceedings of 'Topical Problems of Nonlinear Wave Physics[END_REF]. This has been one of the major difficulties encountered in the commissioning of LIGO and Virgo and will require the implementation of more complex thermal compensation devices in the advanced detectors [START_REF]Advanced Virgo Baseline Design[END_REF].

For several years the effect of the mechanical losses inside the coating was not considered and the focus was put on the optical absorption. Both initial LIGO and Virgo used multi-layers coatings made of silica as low index material and of tantalum pentoxide (or tantala) as high index material. The reason for this choice was mainly driven by the low absorption of the tantalum pentoxide at the used wavelength (1064 nm). The use of this material allows reducing the coating absorption to about 1 ppm [5].

At the end of the nineties it was realized that the mirror mechanical losses were limited by the mechanical dissipation in the coating [6,[START_REF] Gretarsson | 3rd Edoardo Amaldi Conference on Gravitational Waves[END_REF]. Moreover the test of different types of coatings showed that the losses came from the mechanical dissipation inside the high index material, namely the tantalum pentoxide [8,9,10]. The losses were measured at several different places and turned out to be in the range of 3 to 5 10 -4 depending on the coating manufacturer. These values are about one order of magnitude larger than those measured in the silica (i.e. the coating low index material).

For the above reasons it is important to minimize the mechanical losses in the high index material keeping the optical absorption as low as possible. These two parameters are the most important from the point of view of the gravitational wave detector sensitivity. But other parameters need to be kept under control. The ion beam sputtering technique is known to produce coatings which can have a high internal stress. This stress can induce a deformation on the substrate figure. It can also reduce the adherence of the coating on the substrate and be the cause of formation of defects inside the coating material. For these reasons it is important to measure the stress in the coating when new materials are tested.

The need to reduce the mechanical losses in the high index material was the main motivation for the development of titania doped tantalum pentoxide that was performed at LMA and that is reported in [11].

This paper describes the most recent development performed at LMA both on the tantala coatings and on different alternative materials. The experimental apparatus used to perform this development is described in section 2. Then the result obtained by doping the tantalum pentoxide layers are reported in section 3 together with the characterization performed on the best performing materials. Section 4 describes the results obtained with other high index materials tested at LMA. Finally the results are discussed in section 5.

Experimental apparatus

Coating chambers

In order to reduce the optical absorption it is important to reduce to a minimum the presence of impurities in the coating layers. Ion beam sputtering is the technique that allows producing the purest coating materials. An Ar ion source combined with an electron source (neutralizer) is used to produce a low energy neutral beam of Ar in a vacuum of 3 10 -4 Torr. The Ar beam is directed against a target made of the material to be deposited (typically Ta or Si). Oxidation is realized by injecting oxygen in the chamber during the process. The sputtered material reaches the substrate where it condenses thus producing the required coating layer. By changing the target material one can produce alternate layers of low index and high index materials and realize the required multi-layer structure.

Two coating chambers of this type are used at LMA. The first coating chamber has a volume of 0.5 m 3 and is equipped with a Kaufmann ion source. This chamber can coat substrates as large as 3 inches in diameter and is equipped with a planetary support for the substrates. The second is a 10 m 3 chamber equipped with two RF sources and able to coat substrates up to 1 m diameter. The latter is the coating chamber which is used to produce mirrors for the Virgo interferometer [START_REF] Beauville | Optical Systems Design[END_REF].

Optical loss measurements

The optical absorption of single coating layers and multi-layer structures are measured taking advantage of the so called mirage effect. A 30 W Nd:YAG laser is chopped at a few hundreds Hz and is sent to the coating layer to be measured. The laser power absorbed in the coating produces a thermal gradient in the silica substrate which varies at the chopper frequency. A second low power He-Ne laser beam (probe beam) crosses the thermal gradient induced by the high power beam. The gradient in the index of refraction produces a deflection of the probe beam which varies at the chopper frequency.

The position of the transmitted He-Ne laser is measured with a quadrant photodiode and a lock-in amplifier. This measurement combined with the proper calibration allows measuring coating absorptions with a sensitivity of 0.02 ppm. The same device can be used to measure bulk absorption with a sensitivity of 0.1 ppm/cm. By moving the sample across the two beams it is possible to produce maps of the absorption across samples as large as 400 mm. A more detailed description of this set-up can be found in [START_REF] Cimma | Optical Systems Design[END_REF].

Mechanical loss measurements

The mechanical losses of the coating are measured by studying their effect on the quality factor of thin silica cantilevers (Suprasil 311 from Heraeus). The cantilevers are 45 mm long, 5 mm wide and 110 µm thick. The cantilever geometry has been chosen such that its resonant modes are in the range of frequencies of interest for gravitational wave detectors. Once clamped at one end the fundamental resonant frequency is around 60 Hz, the 2 nd and 3 rd resonant modes are respectively at about 400 Hz and 1.1 kHz.

The mechanical quality factor of the cantilever resonant modes is measured before and after the cantilever is coated. By comparing these two measurements one can deduce the value of the mechanical loss of the coating. This requires knowing the ratio between the energy stored in the substrate and the energy stored in the coating that depends on their Young modulus and on their thickness.

In order to reduce the error on the coating loss measurement it is important to have the highest possible quality factor for the uncoated cantilever. To this purpose the cantilevers are annealed at 900 o C and cleaned. This process allows relaxing the stress, decreasing the internal density fluctuations and removing impurities left from the polishing process. It may be worth mentioning that the effect of annealing on silica mechanical loss has already been observed in the past [START_REF] Numata | [END_REF]15]. Particular care is taken in the clamping of the cantilever. The clamp itself is polished before mounting a new cantilever.

Both the surfaces in contact with the cantilever and at right angles are polished in order to have a sharp edge in contact with the cantilever and to reduce the Van der Walls forces between the clamp and the vibrating cantilever. Finally the cantilever is clamped very tightly. The cantilever and its support are placed in a vacuum chamber at 10 -6 Torr in order to eliminate any dissipation due to the residual gas.

The cantilever position is measured by means of an optical lever which uses a low power He-Ne laser.

The latter is reflected on the cantilever and is readout by a quadrant photodiode. With this bench it is possible to obtain a quality factor for the first cantilever mode of the order of 250000. The residual losses are due to the clamp and are in the range of 4 10 -6 .

The same cantilevers are used to measure the stress in the coating. The principle of the measurement is relatively simple [16]. The curvature of cantilever is measured before the coating is deposited by means of an optical interferometric profiler. The comparison with the cantilever curvature measured after the coating is deposited allows deducing the stress in the coating.

Doping of the Ta 2 O 5

The first class of attempts done to reduce the mechanical loss of the high index material consists in improving the tantalum pentoxide by doping it with other oxides i.e. materials that are known to have a good transparency to near infrared light. Several materials were tested to this purpose. Table 1 shows a summary of the measurements that were made on 500 nm thick monolayers of doped Ta 2 O 5 . In general the doping allows an increase of the refraction index of the material. This increases the contrast between the high and the low index materials thus reducing the amount of high index material that is needed to obtain a given reflectivity. As a consequence, if the mechanical losses are kept constant, the thermal noise would decrease. Unfortunately, as it is visible in the table, in most cases the optical and the mechanical losses tend to increase as well.

Refraction index

In the case of the doping with Co a very large increase of the optical absorption is measured. The doping with Ti+W shows a pretty low absorption and a promising mechanical loss but an attempt to coat a mirror with this material has shown a large optical absorption, probably due to some diffusion of material between the layers. The best results are obtained with Ti doping. In this case both the mechanical losses and the optical absorption decrease.

Optical index

A study performed in the small coating chamber showed that by increasing the doping level it was possible to increase the index of refraction while decreasing the optical absorption. Figure 1 shows the index of refraction and the optical absorption as a function of the amount of doping material used in the chamber (expressed in arbitrary units).

A more detailed optimization was performed on the larger coating machine as the large mirrors needed by laser interferometer gravitational wave detectors are produced in this machine. By adjusting the parameters of the coating process it was possible to decrease the mechanical loss to about 1.9-2.0 10 -4 .

The best value ever measured was around 1.5 10 -4 which is a factor of two better than the value of the undoped Ta 2 O 5 .

Two different levels of Ti doping were tested in the large coating machine.

A Rutheford backscattering analysis was carried on both materials to measure the different atoms concentration and thus the doping level. Samples of the coatings deposited on fused silica and silicon substrates were exposed to a 2.5 MeV He+ beam. The data were analyzed using the SIMNRA software [START_REF] Mayer | Proceedings of the 15th International Conference on the Application of Accelerators in Research and Industry[END_REF]. This program is a montecarlo simulation including non-Rutherford backscattering, nuclear reaction analysis and elastic recoil detection analysis with MeV ions. About 300 different non-Rutherford and nuclear reactions cross-sections are taken into account.

First the undoped tantala coating was analyzed. This analysis shows that the ratio Ta/O corresponds to the good stoichiometry. The coating density is found to be 7.2 g/cm 3 , which is about 12% smaller than expected for bulk Ta 2 O 5 . Subsequently the two Ti doped materials were analyzed: one of the resulting RBS spectra are shown in Figure 2. This analysis shows that the two samples have a ratio of Ta/Ti equal to 1.25 and 3. Their densities are equal to 6.4 g/cm3 and 5.5 g/cm3 respectively. These values are compatible with the measured atomic concentrations and with the density of the undoped tantala. The sample with more Ti has an index of refraction equal to 2.18 while the other has an index of refraction around 2.07 (for sake of comparison the un-doped tantala has n=2.035). On the other hand the coating with more Ti has also higher mechanical losses (~2.2 10 -4 compared to ~1.9 10 -4 ) so that in the end the two materials have comparable performances in terms of coating thermal noise. The RBS analysis also reveals the presence of a small fraction of Ar in both types of coating. The Ar total atomic concentration was found to be in the range between 3% and 4% (corresponding to about one Ar atom for 6 to 7 Ta atoms in the case of pure Tantala). This is not surprising as Ar is used as sputtering beam.

The material with less doping was used to coat the first prototype of the Advanced LIGO mirrors. This is a 34 cm diameter and 20 cm thick mirror made of fused silica and having a weight of 40 kg. The deposited coating had a diameter of 300 cm and was made of 18 doublets of Titania doped Ta 2 O 5 and SiO 2 in order to get a transmission of 10 ppm. The map of the coating absorption is shown in Figure 3.

The average absorption of the coating is about 0.25 ppm. For sake of comparison the initial Virgo mirrors produced with standard tantala have absorption of about 0.7 ppm.

Study of other high index materials

Another approach to reduce the mechanical losses in the coatings consists in replacing the Ta 2 O 5 with another kind of high index material. Two materials have been studied to this purpose: zirconium dioxide and niobium pentoxide. Both materials were produced in the smaller coating chamber.

The ZrO 2 was first produced in its undoped form and then doped with titanium and tungsten. The results obtained are shown in Table 2. The index of refraction of ZrO 2 is relatively high (n=2.1) and its mechanical losses are lower than standard Ta 2 O 5 . Unfortunately the stress in the coating is quite high and this makes the production of multi-layers structures more critical. In addition the optical absorption is about ten times larger than that of Ta By doping the Zr with tungsten it is possible to increase the index of refraction and strongly decrease the stress. Unfortunately both the mechanical losses and the absorption increase.

Doping with Ti gives intermediate results. The stress is decreased and the index of refraction is slightly increased. But the optical absorption remains quite higher than that of Ta 2 O 5 and, contrary to what happens in the case of Ta 2 O 5 , the mechanical losses are worst than those of the undoped ZrO 2 . In conclusion using ZrO 2 does not allow improving the performances compared to Ta 2 O 5 .

The Nb 2 O 5 was studied only in its un-doped form. The refraction index is 2.21, considerably higher than that of Ta 2 O 5 . According to the literature it is possible to achieve even higher values [18]. This is of help from the thermal noise point of view as less material is required to achieve the same reflectivity. The optical absorption is 2.2 ppm, about twice that of Ta

Discussion

In this section we compare the performances of different types of multi-layers coatings from the point of view of the mirror thermal noise. Four different types of high index materials have been considered: Ti: Ta 2 O 5 , Nb 2 O 5 , ZrO 2 and TiO 2 . These are combined with two different types of low index materials: silica (SiO 2 ) and alumina (Al 2 O 3 ). In all cases the multilayer coating design is tuned to have a transmission equal to about 4 ppm. This value has been chosen to be in the range of the transmission of the end mirrors that will be used by the advanced detectors. The mirror thermal noise is calculated assuming that the mirror displacement is sensed with a laser beam having a radius of 6 cm (defined as the distance form the axis at which the electric field goes to 1/e. This is approximately the maximum beam diameter one can use on a 35 cm diameter mirror such as those that will be used in advanced detectors. Finally, the mirror thermal noise has been calculated assuming that the coating is deposited on a substrate made of silica, since this is the material that will be used both by Advanced LIGO and Advanced Virgo.

Even if in most cases the coating thermal noise due to its internal mechanical losses is the dominant thermal noise, the simulation includes also other effects such as the substrate internal losses, the thermo-elastic damping in the coating and in the substrate and thermo-refractive effect in the coating (see [19] and reference therein for a short review of all these noise sources).

The coating parameters that have been used in the simulation are reported in the table below. Many of the parameters have been taken from the literature. The complete list of references used to build Expansion coefficient (K -1 ) 0.5 10 -6 8.4 10 -6 3.6 10 -6 0.5 10 -6 5.8 10 -6 10 -5

Thermo optic coefficient (K -1 ) 8 10 -6 1.3 10 -6 14 10 -6 -1.8 10 -4 1.43 10 gives the worst result. This is due to the larger mechanical losses of TiO 2 but also to the large thermoelastic dissipation. It should be noted that the mechanical losses of the TiO 2 coating reported in Table 1 were measured on a coating that was produced using evaporation. A test with TiO 2 coatings produced by ion beam sputtering needs to be done.

Figure 1 .

 1 Figure 1. Refraction index (left) and absorption (right) of Ti: Ta 2 O 5 as a function of the amount of dopant used in the chamber expressed in arbitrary units.

Figure 2 .

 2 Figure 2. RBS spectrum of a Ti doped Tantala coating. The channel corresponds to the energy of the backscattered ions. This is calibrated using pure elements targets of Al, Ti, W and Zr. The peaks widths provide the density of the element in at/cm2 and the peaks height is related to the atomic concentration of the element. The insert shows a zoom of the region between 1200 and 1600, where are visible the signals given by the Argon and the Titanium. The large peak at higher energy is due to the Tantalum.

Figure 3 .

 3 Figure 3. Map of the absorption of a Ti doped Ta2O5 coating measured over a 120 mm diameter.

2 O 5 .

 25 The same level of absorption was measured on a high reflectivity mirror made of 16 doublets of Nb 2 O 5 and SiO 2 . The measurement of the mechanical losses gives a value of about 4.6 10 -4 . This value is affected by an error of 20% due to the uncertainty on the Young modulus of Nb 2 O 5 which is used to deduce the coating mechanical losses from the quality factor of the coated cantilever. In conclusion the use of Nb 2 O 5 does not allow to improve the performances achievable with Ti doped Ta 2 O 5 .

Figure 4

 4 Figure4shows the result of the simulation. As expected the coating which performs best is the combination Ti:Ta 2 O 5 /SiO 2 . The combination Nb 2 O 5 /SiO2 2 gives promising results as well. This is due to the combination of two effects. First the higher index of refraction of Nb 2 O 5 allows obtaining the same reflectivity with less coating material. Second the Young modulus of Nb 2 O 5[START_REF][END_REF]22] better matches the one of the silica thus reducing the overall coating Brownian noise. It should be noted that the results for the Nb 2 O 5 can be improved if a higher index of refraction (corresponding to what found

Figure 4 .

 4 Figure 4. Mirror thermal noise expressed in m/sqrt(Hz) for different coatings deposited on fused silica substrate. The mirror displacement is readout with a beam having a size of 6 cm.

Table 1 :

 1 Main parameters of coating monolayers made of tantala doped with different materials

	Absorption (ppm)	Mechanical losses

Table 2 .

 2 2 O 5 . Measured coating parameters on monolayers of ZrO 2 , Ti doped ZrO 2 and W doped ZrO 2 . A negative stress is a compressive stress.

	Coating	Refraction index Absorption (ppm) Mechanical losses	Stress (MPa)
	ZrO 2	2.10	11	2.3 . 10 -4	-1780
	ZrO 2 : Ti	2.15	37	6.8 . 10 -4	-180
	ZrO 2 : W	2.12	10	2.8 . 10 -4	-600

Table 3

 3

	can be found in [20].
		SiO 2	Al 2 O 3 Ti:Ta 2 O 5	TiO 2	Nb 2 O 5	ZrO 2
	Mechanical losses	0.5 10 -4 2.4 10 -4	2 10 -4	6.3 10 -3 4.6 10 -4 2.3 10 -4
	Density (kg m -3 )	2200	3700	6425	4230	4590	6000
	Thermal conductivity (W m -1 K -1 )	0.5	3.3	0.6	0.45	1	1.09
	Specific Heat (J K -1 Kg -1 )	746	310	269	130	590	26

Table 3 .

 3 List of coating parameters used to evaluate the mirror thermal noise

	-5	10 -4


L Conti

M Bonaldi

L Rondoni

RareNoise: non-equilibrium effects in detectors of gravitational waves

Keywords: numbers: 04.80.Nn, 05.70.Ln, 05.40.-a, 07.05.Fb

The RareNoise project investigates non-equilibrium effects in Gravitational Wave detectors. We illustrate the Physics behind the project and the planned project development, involving experimental, numerical and theoretical research.

Introduction

Ground based detectors of Gravitational Waves (abbreviated with GW in the following) [1,2] are very sophisticated experiments, consisting of Michelson laser interferometers with arm length of up to a few kilometers, or mechanical oscillators with masses of up to several tons. They are most sensitive in the frequency band (10 ÷ 10 4 ) Hz; depending on the source type, the temporal evolution of GW signals expected in this frequency band changes widely, ranging from quasi-periodic to impulsive ( 1 ms) signals. Present detectors are expected to detect occasionally GW signals. Advanced versions of present laser interferometers will be operative from 2013; they will reach a sensitivity close to the quantum limit set by the uncertainty principle and are predicted to detect GW events possibly as often as daily. There is an ongoing theoretical research focusing on the prediction of signal waveforms for a wide class of phenomena that might happen in the universe and that might be detected [2]. In some cases GW emission is expected in connection with different astrophysical signals (such as Gamma Ray Bursts, neutrino bursts, X-rays). Apart from these studies, there is no anticipation or independent trigger on the signal arrival time: therefore the experimentalists try to run the detectors around the clock and look for any rare event, ie any fluctuation of the detector output that is out of the statistics by several standard deviations. In this context the detector quality is not only given in terms of its peak sensitivity but also of its duty cycle and stationarity of its noise: the latter is fundamental to lower the false alarm rate and assure a confident detection.

Making a detector with noise of pure stationary Gaussian statistics has proven to be as much difficult as making a very sensitive one, these two requirements often being contrasting. Up to now, any gravitational wave detector is characterized by noise that is neither stationary nor Gaussian, showing for instance non-Gaussian "tails" in the probability distribution. If these are large, they bias the statistics thus making more difficult the achievement of a given level of statistical confidence [3]. In the past some of these unexpected noise fluctuations have been claimed as true GW signals, falsified by subsequent analysis. Because of the difficulty in distinguishing a true GW signal from a non-modeled noise fluctuation, joint GW searches are mandatory that combine data from multiple detectors with uncorrelated noise. This brings the advantage not only of reducing the false alarm rate, but also of improving the sky coverage, the detection confidence and the estimation of the GW signal parameters.

From the thermodynamic viewpoint GW experiments are usually modeled as equilibrium systems but it is doubtful that this description is justified. For instance, in interferometric detectors the fibers which suspend the interferometer mirrors are driven out of equilibrium by the thermal gradient due to the light power dissipated into the mirror substrate and coating, extracted as heat through the fibers themselves. The instrument is further pushed towards a non-equilibrium state by the thermal compensation technique [4] which aims to overcome the mirror thermal lensing caused by the thermal expansion due to the laser power being absorbed not uniformly: interferometer's mirrors are subject to external, additional thermal load with the purpose of creating a thermal gradient that further deforms the mirror so to get the desired radius of curvature. In detectors based on massive mechanical resonators, temperature gradients which exist under operating conditions (below the superconductive transition) could drive the resonant-mass detector out of equilibrium. Feedback damping of the electromechanical resonators has already proved to cause a non-equilibrium steady-state [5]. Interferometric displacement sensors discussed in ref. [6] may also cause thermal gradients due to light absorption on mirror substrates and coatings. Laser-sources and Fabry-Perot optical resonators employed in both interferometric detectors and sensors are non-equilibrium systems. Due to the expected increase in the sensitivity of the next generation of the GW detectors favored also by the use of high laser powers and cryogenic operation, the influence of non-equilibrium effects on the statistics of fluctuations, is expected to rise. As a quantitative example, in the proposed cryogenic interferometric GW detector LCGT the fibers suspending the mirrors have one end attached to the mirrors at 20 K and the other to a mass at 10 K, with a heat flow of about 1 W [7]. Similarly a future, 3rd generation European GW interferometric detector [8] might employ cryogenic mirrors with possibly higher thermal gradients and comparable heat flow.

Research of non-equilibrium statistical mechanics is facing a renewed interest: even if a general theory is still lacking, for explaining the physical properties of a macroscopic non-equilibrium system, on the basis of the behavior of its microscopic constituents, several intermediate results have been proven and are collectively known as Fluctuation Theorems [9]. These theorems state a quantitative relation between the probability of observing a certain rate of entropy production or energy dissipation and the probability of observing an equal amount rate of entropy consumption or energy gain. While this result might appear as meager, it has aroused much interest because it stands up as one of the very few exact and parameter-free results obtained so far in non-equilibrium statistical mechanics: furthermore, it has recently been extended to cover a wide variety of phenomena and observables [9,10]. This theoretical research aims at describing the low probability, large fluctuations in the noise distribution, which are due to the interplay of the external driving, of the material properties and of the dissipation mechanisms. We found that these fluctuations differ from the Gaussian noise affecting equilibrium systems, usually assumed even in the analysis of systems such as ours. So far experimental efforts that test and stimulate these studies are very limited in number and sometimes their interpretation is difficult. Experiments involve mainly small-scale systems as molecular machines and nanostructured materials: in fact in large scale systems the large fluctuations become vanishingly rare. Only recently experiments based on non-equilibrium macroscopic systems have been performed: see for instance ref. [11] for the study of a standard metallic resistor crossed by a steady-state current and ref. [5] for the study of a macroscopic electromechanical oscillator cooled by an active feedback system.

If non-equilibrium effects are important in GW detectors, then the noise statistics is intrinsically different from the equilibrium Gaussian statistics. Large fluctuations are less rare: they increase the false alarm rate and worsen the detector reliability. On the other hand if non-equilibrium effects are demonstrated to be important, the experimental design of GW detectors must be reviewed so to minimize the occurrence of the large noise fluctuations and the theory of non-equilibrium fluctuations used in the data analysis processes. As a consequence, detectors might aim at a more confident detection at a given signal amplitude, or claim detection at a given confidence level but at lower signal amplitudes: in both cases detection chances are increased.

The RareNoise project

RareNoise [12] is a 5 year project started in July 2008 and funded by the European Research Council. The project is devoted to the study of non-equilibrium fluctuations in macroscopic systems; the study is performed via two mutually reinforcing approaches: numerical and analytical analysis and laboratory experiments. First application of the knowledge gained while performing the research is offered by groundbased GW detectors: the research will determine the extent to which detectors of GW are non-equilibrium systems.

The research focuses on the effect of temperature gradients in GW detectors. Situations similar to those that drive out of equilibrium mechanical subsystems of the detectors will be reproduced experimentally on smaller scale devices that can be realized and characterized more easily. The effects will be investigated of temperature gradients on the spontaneous vibration fluctuations of low-loss mechanical oscillators. The effects will be studied in a small frequency range around the oscillator resonance, set to be within the frequency range of interest for the GW detectors. In order to reproduce a situation similar to those experienced in GW detectors, it is mandatory that the oscillator is a very low loss one, ultimately limited by internal loss mechanisms. In contrast to previous experimental works on Fluctuation Theorems, where the system is often surrounded by a viscous fluid, this research focuses on material intrinsic damping, a situation common to high-precision experiments.

Since present theoretical understanding of the non-equilibrium fluctuations is not thorough, it is very difficult to predict how and how much the spontaneous vibrations of a mechanical oscillator under steady-state heat flux differ from those at the thermodynamic equilibrium: the latter are fully known, being distributed with a Gaussian law and quantified by the Fluctuation-Dissipation theorem [13]. A crude application of one of the formulae obtained through the recent extensions of the theory of non-equilibrium fluctuations [9] in macroscopic systems such as in GW detectors leads to a prediction regarding only exaggeratedly large fluctuations.

Since the first GW detectors, two materials have been identified for their low loss properties as best choices for construction of key detector parts: a metal, namely the aluminum alloy Al5056, and a semiconductor, namely single-crystal Silicon. The former is known to have intrinsic mechanical losses 3 × 10 -8 below 10 K [14]: along with the large availability and low cost, this feature motivates its usage as constituting material of most of the resonant mass, cryogenic GW detectors ever operated and of all running ones. At room temperature the intrinsic mechanical losses of Al5056 are at the level of 3 × 10 -6 [14]. Single crystals of Silicon can reach mechanical losses as low as 5 × 10 -10 at 3.5 K [15] and 10 -8 [16] at room temperature. These properties make Silicon the elective material for the next generation of gravitational wave detectors, while already being the choice for most MEMS and high-precision devices.

RareNoise focuses on these two materials, which will be modeled by means of molecular dynamics techniques. Given their different, temperature dependent levels of intrinsic mechanical losses, the experimental work is organized into three consecutive phases:

(i) an initial phase, with the losses expected to be in the range 10 -3 ÷ 10 -6 .

Experiments are performed at equilibrium temperatures of 300 K and 77 K with both Al5056 and Silicon. This is referred to as phase 1 in the following; (ii) an intermediate phase, with the losses expected to be in the range 10 -6 ÷ 10 -7 .

Experiments are performed at the equilibrium temperature of 4.2 K with both Al5056 and Silicon. This is referred to as phase 2 in the following; (iii) a final phase, with the losses expected to be in the range 10

Phase 1

Phase 1 will give first indications regarding the effects of thermal gradients on the spontaneous fluctuations of mechanical oscillators and therefore has a broad spectrum of investigation. Experiments are devoted to the study of the longitudinal and transverse vibration of oscillators when a thermal gradient is applied between the longitudinal extremes of the body. The study of transverse vibrations has direct application to the GW interferometers since it mimics the motion of the fibers supporting the mirrors, which will be subject to longitudinal thermal gradients; moreover it constitutes a new challenge for theoretical models that so far have not considered such a situation. After the room temperature campaign is completed, the first experimental results are expected: they will be compared with theoretical knowledge and used to develop the appropriate theoretical models. Then, the measurements will be repeated at the equilibrium temperature of 77 K: the temperature variation brings about material properties changes. Therefore it constitutes a unique training ground for developing a sound theoretical model of the spontaneous vibrations of an elastic body subject to thermal gradients. In particular it will be interesting to investigate the effect of varying the thermal expansion coefficient of the oscillator on the statistical properties of the non-equilibrium fluctuations: in this regard Silicon turns out to be of major interest since its coefficient of thermal expansion changes sign both at 124 K and 18 K. A prototype of the Phase 1 Al5056 oscillator has already been realized and successfully tested. The oscillator consists of a rod with cross cross-section fixed at one end and loaded by a mass (referred to as m l in the following) at the other end: the oscillator is kept with the rod along the vertical and the load mass at the bottom. The oscillator was designed with the aid of Finite Element Modeling (FEM) analysis so that its first longitudinal mode occurs at about 1.5 kHz while the first flexural modes occur at 317 Hz and 490 Hz. In figure 1 and figure 2 we show respectively the deformations corresponding to the first longitudinal mode and the second flexural mode, with respect to the shape of the oscillator at rest, as resulting from the FEM analysis. Main figures of the oscillator are listed in table 1.

The vibrations of the oscillator load mass are measured by a capacitive readout coupled to a low noise amplifier. The capacitive sensor is realized by facing an aluminum plate to the bottom surface of the load mass (see figure 3): the capacitor is thus realized between the mass and the facing plate. The capacitive sensor works in a constant charge configuration, once it is biased up to 500 V. The gap between the plate and m l is realized by a 50 µm teflon spacer: this is supported by two arms that extends along the vertical direction at two sides of the rod. The facing surfaces of the oscillator, of the plate supports and of the plate are finished via diamond turning so that a static electric field of about 1 × 10 7 MV/m can be applied. Using a commercial, low noise amplifier with noise voltage at the level of a few nV/ √ Hz (corresponding to a few 10 -16 m/ √ Hz), the oscillator resonance excited by the equilibrium thermal noise is expected to prevail in a frequency bandwidth of about 10 Hz, assuming losses at the level of 4 × 10 -4 at room temperature. At room temperature, in order to realize the thermal gradient along the oscillator, and thus to set the oscillator in a non-equilibrium steady-state, we make use of a thermal control system that stabilizes the temperature of the rod top end and that of the mass m l : the control acts on Peltier cells to cool the temperature of the rod top end while the rod load mass m l is heated via radiative heat exchange by an infrared thermal source which is faced to it (see figure 3). Being non-contacting, the radiative heating guarantees that no external losses are added to the oscillator. The temperature of m l is measured by a no-touching sensor, ie a IR thermometer faced to the mass (see figure 3). To improve the radiative coupling with both IR source and thermometer, the portion of the m l mass that is facing to them is hard anodized. A temperature gradient of about (10 -30) K will be applied at room temperature. Figure 3 shows the assembly of the oscillator, along with the capacitive sensor, the IR source and the thermopile. In figure 4 we show a picture of the Phase 1 prototype of Al5056 mechanical oscillator where the two thermometers used for the control of the thermal gradient are clearly visible.

Phase 2

The oscillators of phase 1 will be studied at the equilibrium temperature of 4.2 K: not only this offers the advantage of different material properties, in particular minimum losses, but also eases the establishment of larger relative thermal gradients. Given the lower temperature, this phase is more demanding on the vibration sensors both in terms of sensitivity and working environment: only the degrees of freedom will be monitored that can be adequately measured.

Phase 3

Phase 3 will concentrate on Silicon: a specially designed, monolithic oscillator will be produced to profit from the very low intrinsic losses of this material in the full temperature range from 300 K to 4.2 K. This will constitute the experimental situation most similar to interferometric GW detectors of third generation: therefore quantitative indications about non-equilibrium effects in such instruments are expected as final result, along with refinements of theoretical predictions. At present the experiments for phase3 are harder to plan than those of phases 1 and 2: for instance, decision on the kind of acoustic mode (whether longitudinal or transverse, given a longitudinal thermal gradient) to be monitored will be taken on the basis of the experimental and theoretical results of previous research.

Theoretical and numerical work

The theoretical work will be developed in parallel with the experiments, in order to identify the models and the parameters which can most realistically simulate the behavior of the experimental setup. Initially both mathematical and numerical investigations will be pursued of 1-dimensional chains of oscillators, on which a large literature exists, and which have recently found application in, e.g. the study of heat transport in carbon nanotubes [17]. Then, generalizations of such models, in The Teflon spacers and the cable carrying the capacitive sensor signal toward its preamplifier is clearly seen originating from the plate. The IR thermometer is also clearly visibile, facing an anodized spot on the mass m l . In this photo the oscillator is kept upside-down, with the mass m l at the top; moreover the mass is held in place by two spacers which are removed once the oscillator is reversed.

which several chains will be coupled to go beyond the 1-dimensional cases will be simulated by means of non-equilibrium molecular dynamics techniques. Particular attention will be devoted to the construction of the particles interaction potentials, in order to approximate as closely as possible the behavior of real rods, in the range of parameters accessible to experimental tests. Once the models are sufficiently refined, that they perform adequately in a range of parameters in which direct comparison with experiment is possible, they will be used to compute the observables which cannot be experimentally tested at the lowest temperatures, but which are still useful in the characterization of the non-equilibrium state of the system, and of its noise in particular. The work will end with refinements of the non-equilibrium theory, in particular by assessing the validity of the normal mode expansion formalism, and with novel applications to the interferometric GW detectors: non-equilibrium fluctuations will be estimated and their effect on the detection capabilities will be clarified.

In high-precision experiments, the sensitivity is often limited by the mechanical thermal noise: since at equilibrium any displacement of an elastic body can be expanded on the basis formed by the acoustic normal modes, the mechanical thermal noise can be modeled as the superposition of noise contributions from each acoustic mode [START_REF] Saulson | See for instance[END_REF]. It is actually matter of research of RareNoise to understand the extent to which the normal mode formalism is valid in non-equilibrium situations: this is crucial for a correct interpretation of the experimental data.

Figure 1 .

 1 Figure 1. Deformation of the oscillator assembly in correspondence of the first longitudinal mode of the mechanical oscillator, as predicted by the FEM analysis: the mode resonates at 1.5 kHz. The shape of the oscillator at rest is shown as black solid lines for comparison.

Figure 2 .

 2 Figure 2. Deformation of the oscillator assembly in correspondence of the second flexural mode of the mechanical oscillator, as predicted by the FEM analysis: the mode resonates at 490 Hz. The shape of the oscillator at rest is shown as black solid lines for comparison.

Figure 3 .

 3 Figure 3. Exploded view of the assembly of the Phase 1 Al5056 oscillator. The oscillator is machined from a single piece along with the supports for the second armature of the capacitor: the oscillator consists of a rod with cross cross-section with one end fixed and one end loaded by mass m l . A 50 µm thick Teflon spacer is interposed between the plate and its supports to realize the gap of the capacitive sensor. The plate supports hold also the IR thermometer that is faced to a hard anodized, ∼0.02 m in diameter spot on the mass m l . On the other side of the IR thermometer with respect to m l , a support holds the IR source in place, in front of a similar anodized spot on the mass m l .

Figure 4 .

 4 Figure 4. Photograph showing the Phase 1 Al5056 prototype oscillator. At the base of the rod a NTC thermistor is mounted on a copper plate: this is the thermometer used in the control loop to stabilize the temperature of the rod base.The Teflon spacers and the cable carrying the capacitive sensor signal toward its preamplifier is clearly seen originating from the plate. The IR thermometer is also clearly visibile, facing an anodized spot on the mass m l . In this photo the oscillator is kept upside-down, with the mass m l at the top; moreover the mass is held in place by two spacers which are removed once the oscillator is reversed.

Table 1 .

 1 Main figures of the Phase 1 prototype oscillator in Al5056.

	-6 ÷10 -8 . Experiments

Acknowledgments

We thank G. A. Prodi for useful discussions. The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n 202680. The EC is not liable for any use that can be made on the information contained herein.


Angela Di

Maria Allegrini

Jacopo Belfi

Nicolò Beverini

Filippo Bosi

Giorgio Carelli

Enrico Maccioni

Marco Pizzocaro

Alberto Porzio

Ulrich Schreiber

Salvatore Solimeno

Fiodor Sorrentino

Performances of G-Pisa: a middle size gyrolaser

Keywords: numbers: 04.80.Nn, 95.55.Yn, 42.55.F, 06.30.Gv, 42.79.Qx

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Inside the gravitational wave experimental community there is a large interest for inertial devices sensitive to rotations and tilts, necessary for active controls. AdVirgo plans to implement the tilts control for the Inverted Pendulum, which is the first stage of the SuperAttenuator suspension, designed to support and orientate the interferometer mirrors. The sensitivity requirements of AdVirgo for the IP control is usually expressed in power spectrum of the angle, and it is 10 -8 rad/

√

Hz above 30mHz [START_REF] Sorrentino | Advanced Virgo Baseline Design[END_REF]. Laser gyroscopes are devices sensitive to inertial angular motion, based on the Sagnac effect: in a closed cavity rotating at angular velocity Ω the two counter propagating beams complete the path at different times. The main advantage of such devices in comparison to mechanical based system is that the coupling among different degrees of freedom is very little, since they don't have moving parts. Different kinds of such devices have been developed mainly for navigation. We distinguish between passive (fiber optic gyros) and active (ring lasers) Sagnac interferometers. Passive device measures the phase shift between the two beams, while the active one measures the frequency difference, an inherently more accurate measurement. Small fiber gyros (FOG) are typically used for navigation and have so far a resolution of 2 × 10 -8 rad/sec/

√

Hz [START_REF] Velikosetzev | [END_REF], while the large ring laser gyros used in geophysics and geodesy reach the level of 10 -12 rad/s, and are still improving. The requirements of ADVirgo for tilt control espressed as angular velocity is 2 × 10 -9 rad/sec/ √ Hz, a factor 10 lower than available FOG devices, which could be likely improved in the near future. In the following we will focus on active ring laser (gyro). One application of large gyros is the monitoring of the variations of the Earth angular velocity vector. The orientation with respect to the Earth axis is important since the induced signal is proportional to the scalar product between the normal of the gyro area and the Earth axis, see equation 1. For horizontal gyros the signal is zero at the equator and maximum at the pole; at intermediate latitudes, both horizontal and vertical cavities work fine. Up to now the reached resolution is 10 -8 of the Earth rotation rate [START_REF] Stedman | Ring-laser tests of fundamental physics and geophysics[END_REF][START_REF] Schreiber | Earth tide and tilt detection by a ring laser gyroscope[END_REF][START_REF] Schreiber | Direct measurement of diurnal polar motion by ring laser gyroscopes[END_REF][START_REF] Rowe | Design and operation of a very large ring laser gyroscope[END_REF][START_REF] Dunn | Design and Initial Operation of a 367 m 2 Rectangular Ring Laser[END_REF].The Sagnac frequency, i.e., the beat signal between the two output beams, is:

δφ = 4A n • Ω/(λP ) + φ ρ
where A and P are the area and the perimeter of the cavity respectively, λ is the wavelength of the laser beam, n is the normal vector of the plane of the ring cavity and Ω is the induced vector of rotation. φ ρ is denoting additional, usually very small, contributions to the Sagnac frequency due to non-reciprocal effects in the laser cavity, such as Fresnel drag [START_REF] Aronowitz | Laser applications[END_REF] and shot noise. A laser gyro can monitor with high accuracy the orientation of the laboratory reference frame. The sensitivity of G (located in the Laser Ranging Station of Wettzell in Germany) is getting closer and closer to VLBI sensitivity providing independent measurement for the comprehension of the Earth reference frame; in poor words we can say that at the moment geodesy is the most successful application of large gyrolaser, and it is becoming a fundamental instrument for geophysics [START_REF] Lee | BSSA Special Issue on Rotational S eismology and Engineering Applications[END_REF] as well. But it can be used as well to improve the performances of future gravitational wave antennas [START_REF] Virgilio | About the use of gyro-lasers in gravitational waves interferometric detectors[END_REF][START_REF] Lantz | BSSA Special Issue on Rotational Seismology and Engineering Applications[END_REF][START_REF] Virgilio | The G-Pisa gyrolaser after 1 year of operation and considerations about its use to improve the Virgo IP control[END_REF][START_REF] Fiori | Siesta simulation of NE suspension tower Virgo note[END_REF]. Application for gravitational wave antennas detectors needs dedicated apparatus, the emphasis is more on small device, with very high duty cycle, working in an environment where large resonances due to the proper frequencies of the metallic structure are present, which in principle could spoil the sensitivity through up conversion produced by the non linear dynamic of the laser. The large ring laser gyros built by the joint ring laser working group in New Zealand and Germany have sides of the order of several meters. "G" [START_REF] Schreiber | Advances in the Stabilisation of Large Ring Laser Gyroscopes[END_REF][START_REF] Schreiber | Stabilitaetsbetrachtungen für grossereinglaser[END_REF] located at the Geodetic Observatory in Wettzell (Germany) is a monolithic square with 4 m sides, "UG2", a rectangle with 40 m by 20 m sides, is the largest ring [START_REF] Hurst | Experiments with a 834 m2 ring laser interferometer[END_REF]. In principle the larger the area the better the sensitivity, but the best gyro so far, is "G", which is operating very close to the quantum limit, and its measured noise power spectrum is of the order of 10 -12 rad/s/ √ Hz, with a duty cycle close to 100%. The Allan deviation has not shown any systematic effects on a time scale of about 3 hours, and the sensor drift is normally below 1.5 parts in 10 8 of the Earth rotation (1.1 • 10 -12 rad/s). Among the different instruments developed by the German NewZealand group, there is a very compact and flexible mechanical design, called GEOSENSOR, espressely design to be transportable. Existing GEOSENSORS are used mainly for geophysics, and have sides of 2 m, which is considered too large to be implemented inside a suspension. G-Pisa is based on the GEOSENSOR design, has been built by a INFN, University and CNISM of Pisa and Naples colaboration, financed by INFN commission V in order to learn the technique for future application in Virgo and for General Relativity tests. For G-Pisa, the GEOSENSOR design has been changed in order to be scalable down to 0.9 m, further scaling of the device is in principle feasible, but a new mechanical design is necessary. In the following section the limits in sensitivity due to shot noise are discussed. The third section describes the experimental apparatus. The fourth section discusses the results in sensitivity to be compared with the AdVirgo requirements. At the end the conclusions come, with considerations about our near future plan.

Sensitivity limits

In large gyros the shot noise of light gives the fundamental limit to sensitivity:

Ω sn = c 2π K L h ν µ T
2P t where c is the speed of light, L the side of the square ring, h the Plank Constant, ν the frequency of the light, µ the total cavity losses, T is the transmmision of each mirror, P is the total radiation power exiting from the cavity, t the observation time, and K is the scale factor of the instrument, which depends on geometry and wavelength, whose value in a square cavity at a wavelength of 632.8 nm is 1.58 × 10 6 × L. In figure 1 the shot noise limit is shown for various values of the parameters µ, T and P and an observation time of 1 s.

For the continuous line (good mirrors, but not the best ones) µ = 40 ppm, T = 0.2 ppm and P = 10 -8 W, for the dotted line absorption has been reduced to . 4 ppm, and for the point-dotted line the transmitted power has also been increased to 10 -7 W (top quality mirrors). It should be noted that with equivalent mirror quality the laser cavity losses increase by decreasing its length; as a consequence for very small rings it becomes more difficult to obtain reasonable high output power. As a fact, for a ring side of the order of 1 m or lower, it would be unlucky to have an output power of the order of 10 -8 W , independently from the mirror quality. In short, a device with L below 1 m could reasonably reach a sensitivity of 10 -9 rad/s/ √ Hz, while to reach 10 -11 rad/s/

√

Hz devices larger than 2 m are required. Backscattering-induced frequency pulling and lock-in pose severe limitation to the ring laser performances. It is usually avoided by introducing a bias, which separates the wavelengths of the two counter-propagating laser modes. The Earth rotation can be used in order to give the necessary bias. The parameter which sets the magnitude of the mode pulling is the lock-in threshold frequency l, where l ≈ c s λ /( π d P ), where c is the speed of light, s is the fraction of the laser field amplitude which is scattered by each mirror, λ is the laser wavelength, d is the beam waist and P is the ring perimeter.

In figure 2 the dashed lines show the lock-in limit as a function of the side of a square ring, for a beam waist of 0.5 mm and different values of of the scattering coefficient s. The continuous line shows the Sagnac frequency given by the Earth rotation for a device horizontally located at latitude 43 • as a function of the ring size. In general the lock-in problem, fig. 2, poses more severe limits at the size of a gyrolaser with the requirements given for AdVirgo, and a device with side at least 1 m is necessary, at least until the mirrors losses are not substantially decreased. Larger perimeters ring has several advantages: larger cavity provides more powerful laser, since the Earth bias depends on the relative alignment of the ring with respect to the Earth rotational axis, a larger ring allows higher flexibility, i.e. there is larger choice of how and were locate the instrument, a larger volume of gas provides a long term stability of the gas itself. For this reasons 1 m side seems a reliable choice, while 0.5 m side seems risky. The sensitivity requirements for our application is not as high as top quality gyrolaser, but, in term of sensitivity and bandwidth response, it is not in the range of ordinary commercially available inertial sensors. As it has been said in the introduction G-Pisa has been financed in order to learn the technique and study possible application, it has not been so far part of AdVirgo program, implementation of this instrument for AdVirgo is a purely engineering problem: given the available space and the relative orientation with the Earth axis it is possible to provide a mechanical design. In order to fit inside the available space such ring could be rectangular more than squared, the real parameter being the perimeter, but the square device is the optimal one, since in this case the ratio perimeter/area is minimum, and the device is more symmetric in general.

Experimental set-up

G-Pisa is a square cavity 5.60 m in perimeter and 1.96 m 2 in area. The mechanical design is flexible and each side of the square can be scaled from 1.40 m down to 0.90 m with minor changes. The mechanical system is mounted onto an optical table and has a stainless steel modular structure: 4 boxes, located at the corner of the square and containing the mirrors holders inside, are connected by tubes, so to form a ring vacuum chamber with a total volume of about 5 • 10 -3 m 3 . The vacuum chamber, is entirely filled with a mixture of He and a 50% isotopic mixture of 20 Ne and 22 Ne. The total pressure of the gas mixture is set to 560 Pa with a partial pressure of Neon of 20 Pa. In the center of one of the tubes there is a Pyrex insertion, a capillary with 4 mm internal diameter, approximately 15 cm long, which is the discharge tube. Capillary internal diameter has been chosen in order to favor selectively T EM00 cavity mode excitation. While the discharge in the laser medium of "G" and analog systems is excited by a RF source, where two coils couple the RF oscillator to the gas, in our system we choose a capacitive coupling. Two halves of a copper cylinder are used as electrodes and are part of the resonant circuit of the RF source. The laser medium in this way is included in the active circuit and the fluctuations in the plasma density do not affect the coupling of the RF source to the discharge, but only the oscillator frequency, about 115 MHz. This kind of discharge provides a very good passive stability and made possible to regulate the laser output power very close to the laser threshold since the first runs. The discharge is 5 cm long, but different lengths will be tested in the next months to minimize the effect of plasma intensity fluctuations around the laser threshold. The discharge tube has four micro-metric screws used to align the Pyrex capillary with the mirrors and the optical cavity.

Four spherical mirrors with 6 m radius curvature were chosen for the resonator, and two micro-metric lever arms acting on the tilts of each mirror, make it possible a fine tuning of the cavity alignment. Mirrors reflectivity is optimized for the emission line around 632.8 nm. The free spectral range of the cavity is 53.6 MHz, the horizontal beam waist is 0.68 mm, the sagittal beam waist 0.56 mm and the intra-cavity ring-down time is 20 µs. In order to achieve long term stability of the perimeter the laser gyro optical frequency will be locked to a reference laser. This will involve the measurement of the radio frequency beat note between the gyrolaser output and the reference laser. For such application the capacitive coupling will introduce less noise than the inductive one. The two outputs (clockwise and counter-clockwise sense of circulation) from one cavity mirror are combined by means of a 50% intensity beam splitter and detected by a photodiode. The photodiode current is voltage converted by a trans-impedance stage with a gain of 10 9 and 1 ms rise time. The two single beam outputs are also monitored by means of two fiber-coupled photomultipliers. The laser modal structure has been detected injecting the output beams in a high finesse linear cavity. The signals are acquired and analyzed off-line. Fig. 3 shows a picture of the apparatus.

Behavior of the gyrolaser signal and sensitivity measurements

In the present set-up our instrument is basically free running, this means that because of thermal expansion it exhibits different regimes: normal single-mode operation, split mode and chaotic regime. For thermal expansion the perimeter changes and the laser sometime has to change the number of wavelengths contained in the perimeter (mode jumps). Sometime the two counter propagating modes are separated by a free spectral range, which is 53.6 MHz for the present set up of G-Pisa. In this regime (split mode) the instrument could be as well useful, but requires a much different analysis and data acquisition since the beat note is as high as the free spectral range plus or minus the Sagnac frequency [START_REF] Hurst | Experiments with a 834 m2 ring laser interferometer[END_REF], and the clockwise or anticlockwise mode can be split. Increasing the discharge voltage it is possible to operate the laser in single mode or multi mode. It has been observed stable operation up to 4 longitudinal modes. A detailed description of the behavior of the apparatus is given in refs. [START_REF] Belfi | Rotational sensitivity of G-Pisa gyrolaser[END_REF][START_REF] Pizzocaro | Development of a ring laser gyro: active stabilization and sensitivity analysis[END_REF]. Mode jumps and splits can be easily avoided with thermal stabilization and with the perimeter active control, which we plan to implement in the near future. In the following the focus will be on the normal operation and relative sensitivity. G-Pisa is located in a very noisy environment without thermal control. Typically it works in normal operation for 10 -20 minutes before a mode jump takes place, and from time to time we observe a split mode, which persists usually for some minutes, until both beams return on the same longitudinal mode and then the standard operation and Sagnac signal are recovered. Fig. 4 shows a tipical raw signal, approximately 6 minutes time span. The rotational sensitivity is obtained reconstructing the phase of the beat note, and differentiating it, in order to obtain the angular velocity. The mean value of this signal gives the Earth rotational speed, and it has been checked that this mean value is compatible with the Earth rotation and the latitude in Pisa. To evaluate the sensitivity the mean value is usually subtracted. The power spectrum of the signal makes the estimate of sensitivity, which is the relevant parameter for the possible improvements of the gravitational waves interferometer suspension; since G-Pisa especially at low frequency is limited by environmental noise it has to be considered an upper limit, and is natural to take the best measurement taken in a quieter environment. Typical spectrums are reported in fig. 5. The horizontal line is the expected shot noise limit, the transverse line shows the requirement for the Virgo suspensions [START_REF] Sorrentino | Advanced Virgo Baseline Design[END_REF], expressed as angular velocity. Three different measurements are shown, taken at different time and in different conditions. The blue line was one of our first measurements, and the large structure around 7Hz was due to the resonance of the legs of the optical table supporting the ringlaser. After the first set of measurements the optical table legs have been fixed to the floor and the table has been as well seismically isolated with a set of pneumatic dampers. The green curve shows a typical spectrum, taken during the day, when there is a lot of activity around the laboratory right inside the Department of Physics of the University of Pisa, which is located in the town center. This spectrum shows a very large peak around 2Hz which has been proved to be due to a horizontal rotation resonance of the table itself. The level of this peak depends of course from the environmental activity. We have done a very long, unattended, measurement all the night long, with low acquisition rate; from this long run the pieces in which the system was optimally working have been extracted, analyzed, and among them the spectrum with the lowest profile below 1Hz is shown in figure in red. It correspond to 2 o'clock in the morning and shows that the measured sensitivity of G-Pisa is even better than the AdVirgo requirements [START_REF] Sorrentino | Advanced Virgo Baseline Design[END_REF]. We note that the set of mirrors used for these preliminary tests were not "top quality" mirrors (from ring-down measurement the mean loss of these mirrors have been evaluated to be of the order of 4 × 10 -5 ). A set of top quality mirrors will be installed in the near future, accordingly improvent of the performances of G-Pisa is expected. It is important to remind that the sensitivity measured with the 4 m side gyrolaser G, located in Wettzel (Germany), is more than a factor 1000 better than the measurements shown in fig. 5. By means of a simultaneous measurement of rotational noise, performed with the G-Pisa gyrolaser, and translational noise obtained by a three axial linear accelerometer, it was possible to set an upper limit to the correlation between rotations and translations to a level of 0.5% [START_REF] Virgilio | The G-Pisa gyrolaser after 1 year of operation and considerations about its use to improve the Virgo IP control[END_REF]. The perimeter control has been successfully closed by moving one of the 4 mirrors using a piezo, showing that controlling the perimeter G-Pisa runs continuosly in standard operation, mode jumps and splits have not been observed so far. Acting on a single mirror, the relative phase bteween the backward scattered radiation from the different mirrors is affected in different ways. Because the coupling between the two counterpropagating beam is given by the interferometric sum of the single backscattering sources, the lock in frequency can be strongly affected by this correction [START_REF] Pizzocaro | Development of a ring laser gyro: active stabilization and sensitivity analysis[END_REF].

Conclusions

The gyrolaser G-Pisa is operational since one year. Interruptions of the normal operation are due to thermal expansion induced mode jumps and split mode regime. The preliminary active perimeter control of G-Pisa has shown that the ringlaser can run continuously, whitout interruptions.

The low frequency measured power spectrum (below 1 Hz) is typically around 10 -8 (rad/s)/

√

Hz , which looks like to be real motion. In very quiet environment sensitivity measurements down to 10 -9 (rad/s)/

√

Hz have been obtained, which is much better than the limit needed to control the tilts of the Virgo suspensions, planned for AdVirgo. A device with perimeter 4 m is adequate for this purpose, a design with rectangular shape rather than square is feasible in order to fit inside the available space. A gyrolaser is a device more sensitive than the present requirements of AdVirgo, it cannot be excluded that in a very near future a relatively small and compact fiber baser Sagnac gyro (FOG) can fulfill the requirements.

2 ]Figure 1 .

 21 Figure 1. Shot noise limit as a function of the square cavity side. The three different traces correspond to sets of mirrors of different quality. Continuous line absorption µ = 40 ppm, transmission T = 0.2 ppm and power P = 10 -8 W, dotted line µ = 4 ppm, point-dotted line T = 10 -7

Figure 2 .

 2 Figure 2. Continuous line shows the Sagnac frequency, the three dashed lines show the lock-in threshold frequency for three different set of mirrors (from bottom to top: s=10 -4 , 5•10 -4 and 10 -3 ). Highlighted intersections represent the G-Pisa case, s = 10 -3 .

Figure 3 .

 3 Figure 3. The G-Pisa apparatus. The G-Pisa apparatus is on top of a standard breadboard, optical fibers are used to handle signals, and the laser used as reference to actively control the perimeter is visible.

Figure 4 .

 4 Figure 4. Standard operation Sagnac signal. After approximately 100 sec a mode jump of one of the two Laser mode takes place and for about 20 sec the instrument is blind. In the bottom figure, 0.1 sec time span is shown and the beat note is well visible.

Figure 5 .

 5 Figure 5.Comparison between the best low-frequency measurements. Light blue spectrum and green spectrum refer respectively to the case of rigid and passively airdamped optical table. The red curve shows the best measurement obtained at night, in a quiet environment. The horizontal dotted line is the expected G-Pisa shot noise, the transverse line shows the AdVirgo requirement for IP tilt control.

Acknowledgments

We have to acknowledge the encouragement given by Armando Reale, who has given great support to our work. We have to mention Antonio Pasqualetti and his crew at EGO, Roberto Cosci and Riccardo Macchia, for the help in setting up the whole vacuum circuit, and Francesco and Mario Francesconi, from the University of Pisa, for the help in the electronics development. Special thanks goes to Gabriele Balestri, Andrea Di Sacco, Pino Fausto and Alessandro Soldani, INFN-Pisa, for the help necessary to realize the mechanical support.


R Gold

B Brügmann

Radiation from low-momentum zoom-whirl orbits

Keywords: numbers: 04.25.Dm, 04.30.Db, 95.30.Sf

We study zoom-whirl behaviour of equal mass, non-spinning black hole binaries in full general relativity. The magnitude of the linear momentum of the initial data is fixed to that of a quasi-circular orbit, and its direction is varied. We find a global maximum in radiated energy for a configuration which completes roughly one orbit. The radiated energy in this case exceeds the value of a quasicircular binary with the same momentum by 15%. The direction parameter only requires minor tuning for the localisation of the maximum. There is non-trivial dependence of the energy radiated on eccentricity (several local maxima and minima). Correlations with orbital dynamics shortly before merger are discussed. While being strongly gauge-dependent, these findings are intuitive from a physical point of view and support basic ideas about the efficiency of gravitational radiation from a binary system.

Introduction

Compact binaries, especially black hole binaries, are expected to be primary sources for gravitational wave astronomy. The orbit of a black hole binary is often assumed to be a quasi-circular inspiral that arises after circularization of the orbit due to the emission of gravitational waves, since this is deemed the most likely astrophysical scenario. However, as a matter of principle we should be prepared to detect and recognize gravitational waves from all corners of the parameter space of binaries. So-called "zoom-whirl" orbits are of particular interest because of their distinct gravitational wave signature, and because in general eccentric orbits can be even more efficient than the quasi-circular inspirals in converting energy of a binary into gravitational waves. Zoom-whirl orbits of a binary are orbits that consist of tight and fast revolutions (the whirls) as well as a phase where the two objects move out to larger distances and back in (the zooms).

The basic features of zoom-whirl orbits were first discussed in the context of geodesics in a stationary black hole spacetime. This describes the limit of a small test mass moving around a large central black hole, see for example [START_REF] Chandrasekhar | The Mathematical Theory of Black Holes[END_REF][START_REF] Martel | Gravitational waveforms from a point particle orbiting a schwarzschild black hole[END_REF][START_REF] Levin | Dynamics of black hole pairs i: Periodic tables[END_REF][START_REF] Grossman | Dynamics of black hole pairs ii: Spherical orbits and the homoclinic limit of zoom-whirliness[END_REF][START_REF] Cutler | Gravitational radiation reaction for bound motion around a schwarzschild black hole[END_REF][START_REF] Glampedakis | Zoom and whirl: eccentric equatorial orbits around spinning black holes and their evolution under gravitational radiation reaction[END_REF][START_REF] Hughes | Gravitational radiation reaction and inspiral waveforms in the adiabatic limit[END_REF], some of which include radiation effects. The main question in full general relativity is how the classic, well-known picture of zoom-whirl geodesics changes for binaries with comparable masses in configurations where radiation damping becomes significant. A priori it is unclear how many whirls can be achieved by fine-tuning the initial orbital parameters (arbitrarily many in the case of geodesics) since the system is quickly losing energy and angular momentum. If little tuning is involved, then zoom-whirl orbits can be potential GW sources even for ground-based detectors, see for example [START_REF] Healy | Zoom-Whirl Orbits in Black Hole Binaries[END_REF][START_REF] Martel | Gravitational waves from eccentric compact binaries: Reduction in signal-to-noise ratio due to nonoptimal signal processing[END_REF][START_REF] Brown | The effect of eccentricity on searches for gravitational-waves from coalescing compact binaries in ground-based detectors[END_REF][START_REF] Wen | On the Eccentricity Distribution of Coalescing Black Hole Binaries Driven by the Kozai Mechanism in Globular Clusters[END_REF].

Recently, zoom-whirl orbits have become the subject of numerical simulations in full general relativity. In [START_REF] Pretorius | Black hole mergers and unstable circular orbits[END_REF], Pretorius and Khurana presented the first example of a whirl orbit of an equal mass black hole binary. In [START_REF] Sperhake | Eccentric binary black-hole mergers: The transition from inspiral to plunge in general relativity[END_REF][START_REF] Hinder | Circularization and final spin in eccentric binary black hole inspirals[END_REF][START_REF] Washik | Binary black hole encounters, gravitational bursts and maximum final spin[END_REF], several examples for the transition from inspiral to plunge, radiated energy and angular momentum have been studied. In [START_REF] Healy | Zoom-Whirl Orbits in Black Hole Binaries[END_REF], up to three whirl phases have been found. A first study of binaries with non-vanishing spin can be found in [START_REF] Healy | Final mass and spin of merged black holes and the golden black hole[END_REF], and the consequences for kicks have been studied in [START_REF] Healy | Superkicks in hyperbolic encounters of binary black holes[END_REF]. The focus in [START_REF] Sperhake | Cross section, final spin and zoom-whirl behavior in high-energy black hole collisions[END_REF][START_REF] Sperhake | The high-energy collision of two black holes[END_REF][START_REF] Shibata | High-velocity collision of two black holes[END_REF] is on high-energy collisions, where the kinetic energy becomes the dominant contribution to the total mass of the system.

In this paper we examine one particular type of zoom-whirl orbit for equal masses and vanishing spin. The magnitude of the linear momentum of the initial data is fixed to that of a quasi-circular orbit at the given initial separation, and its direction is varied. With our choice of initial data we probe a comparatively low-momentum regime aiming at astrophysically reasonably realistic binaries. Similar low-momentum cases have been studied in [START_REF] Healy | Zoom-Whirl Orbits in Black Hole Binaries[END_REF]. The novel aspect of the present work is that we perform a detailed investigation of how the energy radiated depends on the initial angle and find a physical interpretation of this dependence. Main result is that there is not just one maximum in the radiated energy as a function of the initial angle. Rather, several local maxima and minima can be identified, which should be compared to the variations in the mass and spin of the merger remnant noted in [START_REF] Healy | Final mass and spin of merged black holes and the golden black hole[END_REF]. We link these extrema to the motion of the punctures. Furthermore, for this low-momentum case, the amount of fine-tuning required is small; an angular resolution of about one degree suffices.

The term "zoom-whirl" has obvious meaning when several or more whirl orbits near an (unstable) quasi-circular orbit are found, but recent literature colloquially refers to orbits with one or two whirls also as "zoom-whirl" orbits. The physics behind the whirls is precession, and any eccentric orbit has some amount of precession. This begs the question at what point large precession becomes a whirl. We refer to an orbit as a zoom-whirl orbit if at least one whirl orbit is completed before the next zoom, i.e. if from one apoastron of a zoom to the next an additional angle of 2π or more is traversed. However, in order to distinguish large precession on the order of π that occurs for certain black hole mergers (as discussed in this paper) from perturbative precession effects on the order of arcseconds, we refer to precession effects on the order of π also as partial or fractional whirls.

Numerical Computations

We perform a parameter study of equal mass, nonspinning black hole binaries by varying the direction of the initial momentum (see below). We use the BAMcode [START_REF] Brügmann | Numerical simulation of orbiting black holes[END_REF][START_REF] Brügmann | Calibration of moving puncture simulations[END_REF] to solve the full Einstein equations in vacuum. The problem is cast as a Cauchy-initial-value problem by foliating the four dimensional spacetime into families of three-dimensional hypersurfaces labeled by a coordinate time t. Initial data for the initial hypersurface is obtained by the puncture method using a pseudo-spectral code [START_REF] Brandt | A simple construction of initial data for multiple black holes[END_REF][START_REF] Ansorg | A single-domain spectral method for black hole puncture data[END_REF] assuming the spacetime being conformally flat. The computational method uses adaptive mesh refinement (typically 7 -10 levels) together with Berger-Oliger timestepping. We use the χ-variant of the moving-puncture [START_REF] Campanelli | Accurate evolutions of orbiting black-hole binaries without excision[END_REF][START_REF] Baker | Gravitational wave extraction from an inspiraling configuration of merging black holes[END_REF] version of the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation [START_REF] Shibata | Evolution of three-dimensional gravitational waves: Harmonic slicing case[END_REF][START_REF] Baumgarte | On the Numerical integration of Einstein's field equations[END_REF]. The runs are performed with 6th order finite differencing in space [START_REF] Husa | Reducing phase error in long numerical binary black hole evolutions with sixth order finite differencing[END_REF] and a 4th order Runge-Kutta method. For the purpose of wave extraction we use an implementation of the Newman-Penrose formalism which is also 4th order accurate. We can demonstrate consistent, overall 4th order convergence in the 22-mode of rΨ 4 and the radiated energy E rad with typical relative errors of 1 -2% for E rad and slightly less than 1% for the 22mode. The error due to a finite extraction radius was estimated by extracting the waves at different radii r = 60M, 80M, 100M (where M is the total puncture mass) and computing the deviations from the expected 1/r fall-off. This error turned out to be roughly of the same order as the discretization error mentioned above.

Two punctures of mass m are placed on the x-axis at x 1 = -10M and x 2 = +10M , respectively, where M = 2m, so that the initial coordinate separation is D = 20M . The magnitude P of the linear momentum P of each puncture is set equal to the value for quasi-circular orbits, P = P qc = 0.061747M , using the methods of [START_REF] Walther | Numerical black hole initial data with low eccentricity based on post-Newtonian orbital parameters[END_REF]. We systematically vary the direction of P , which implies corresponding changes in the initial angular momentum and also the eccentricity of the orbit. This is parameterized by a shooting angle Θ defined as the angle spanned by the initial linear momentum vector P and the x-axis. Alternatively, we could e.g. vary the size of the initial momentum, which we leave for a future investigation. Varying the values of Θ from Θ = 30 • to Θ = 53 • we compute E rad and investigate how the energy might be related to the trajectories of the punctures. This study involves 24 binary black hole evolutions, plus convergence runs.

Results

We analyze the dependence of the radiated energy E rad on the shooting angle Θ and observe new features for eccentric, equal-mass and non-spinning binary black holes including zoom-whirl behaviour. The results on radiated energy are summarized in Fig. 1. Fig. 2 compares two of the waveforms, and the full orbital trajectories of three representative cases are shown in Fig. 3.

First, we note that for the size of the linear momentum considered here eccentric black hole binaries can be significantly (15%), but not drastically more efficient than quasi-circular binaries in converting initial energy into outgoing gravitational radiation. This can also be seen from Fig. 2, where we plot the l = 2, m = 2 mode of rRe(Ψ 4 ) for the Θ = 47 • run against a quasi-circular waveform. Especially at the onset of merger the amplitude of the eccentric waveform is significantly larger than in the quasi-circular case. This will be further explored in an upcoming work in the context of larger initial momenta. Second, for Θ in the range of 45 • -53 • , E rad fluctuates between 0.04M ADM and 0.05M ADM . The oscillations are much larger than the estimated relative numerical error of 1 -2%, and are therefore considered a feature of the physics rather than a numerical artefact. Analogous variations have been observed in the mass of the merger remnant as well as in the final spin in [START_REF] Healy | Final mass and spin of merged black holes and the golden black hole[END_REF]. However, these oscillatory features have not been observed by [START_REF] Sperhake | Cross section, final spin and zoom-whirl behavior in high-energy black hole collisions[END_REF][START_REF] Hinder | Circularization and final spin in eccentric binary black hole inspirals[END_REF], while the behaviour for low Θ looks very similar to [START_REF] Sperhake | Eccentric binary black-hole mergers: The transition from inspiral to plunge in general relativity[END_REF]. In the following we will explore the origin of the above findings in more detail.

We find strong evidence that the local minima and maxima are linked to different types of dynamics near the merger. In order to make a more precise statement we define the coordinate separation vector S pointing from one puncture to the other with length | S(t)| = D(t) for all t, and we denote the time when a common horizon is formed by t merger . We depict the following observation in Fig. 4. Whenever the tangent vector of the puncture orbit is closest to being orthogonal to S(t = t merger ) then the corresponding orbit turns out to maximize the radiated energy compared to other evolutions in the neighbourhood in parameter space. In contrast, those orbits whose tangent vectors point furthest inward at the onset of merger (e.g. Θ = 48 • , dotted line in Fig. 4) minimize the radiated energy. Fig. 5 shows the angle at merger time for a number of different runs. The location of the maxima and minima in Fig. 5 mimics the dependence of the energy on the shooting angle shown in Fig. 1.

A physical interpretation of this effect is intrinsically complicated due to the high degree of gauge-dependence in this highly non-linear region in spacetime. Nevertheless, the above mentioned effect might originate from the fact that initial eccentricities within a certain range can maximize or minimize the time the binary spends in the radiatively most efficient configuration, namely small separations and high velocities. It is rather intuitive that orbits with radial velocity v r ≈ 0 shortly before merger (e.g. Θ = 47 • , solid line in Fig. 4) spend comparatively more time in a radiatively-efficient zone than in the case where v r is larger. A large v r leads to a rapid rush through this crucial zone leaving the binary less time to radiate.

This interpretation is further supported by the data shown in Fig. 6. In this plot we compute a histogram of D = D(t), i.e. we count the number of time steps during which the coordinate separation of the binary lies within each bin of width ∆D, thereby visualizing how much time the binary spends at a given separation D. First, we observe that we can define the radius for innermost unstable circular orbits rather well for our gauge-choice, because in this plot they show up as a sharp and strong peak. Second, the comparison between the upper and lower panel confirm our interpretation, that more efficient runs spend more time in a radiatively-efficient zone than less efficient binaries. In this plot the emission zone corresponds to a band bounded on the left by D M erger , which is defined as the coordinate separation at which a common horizon forms. We find D M erger = (1.8 ± 0.1)M . We want to check to what degree the above hypothesis withstands the issue of gauge-dependence in an upcoming work. The idea is to further investigate the momentum dependence (which the gauge is sensitive to) and work out other diagnostics in order to clarify whether or not such arguments carry over to more general scenarios.

Concerning the phenomenology of the orbital dynamics and the radiated energy, we should emphasize that the maximum in radiated energy corresponds to an evolution which only completes about 1.3 orbits before merger, i.e. the orbit is highly eccentric. (A 3PN [START_REF] Memmesheimer | Third post-newtonian constrained canonical dynamics for binary point masses in harmonic coordinates[END_REF] estimate for Θ = 48 • gives e ≈ 0.8.) For 48 • ≤ Θ ≤ 48.5 • , i.e. near one of the minima and not at one of the maxima, the orbits exhibit a "whirl" phase slightly before merger. This whirl phase does not seem to be the most relevant moment for wave emission since we find that this almost circular motion happens at a much larger radius (which depends inversely on the eccentricity) than the coordinate separation at merger (see Fig. 6). This argument is supported by the observation that the orbits which manage to whirl, but do not exhibit a zoom minimize E rad . For Θ = 48.5 • the whirl phase is followed by a very small "zoom". Above that value for Θ the whirl phase becomes shorter and the zooms larger (note the prominent zoom of the Θ = 50 • run (dashed line) in Fig. 3), and for Θ > 52 • there are multiple subsequent close encounters (or fractional whirls) before the final merger.

In the strong field regime, the precise definition of eccentricity and precession is problematic. Furthermore, the precession of the orbit is no longer proportional to the eccentricity. For a large range of shooting angles (eccentricities) the precession becomes a significant fraction of 2π and for a small range it even exceeds 2π. This is one of the main results of our (and other) investigations. Prior to numerical simulations it was not clear whether zoom-whirl behaviour exists at all in non-spinning, equal mass binaries. Indeed, as we demonstrate, the range where full whirls occur is very limited, and the more typical phenomenology is rather governed by a strong form of precession. The transition from close encounters to zoom-whirls is completely smooth, and we especially want to highlight that there is no significant fine-tuning involved beyond about 1 degree in the shooting angle.

The expectation that a high degree of fine-tuning may be required originates with the search for geodesics that complete a large number of whirls. These are typically found near an unstable circular orbit (inside the radius of the innermost stable circular orbit), which suggests that due to the instability a high degree of fine-tuning is required. Also, the more time is spent in the whirl, the stronger the dependence on the initial parameters. In our case, only about one full whirl is found, so the sensitivity on the initial angle should be modest. Furthermore, although the whirl occurs near an unstable orbit with a diameter of 3.5M (Fig. 3), the degree of instability seems to be sufficiently mild to require only a small amount of fine-tuning.

These findings are very different from the high energy case [START_REF] Pretorius | Black hole mergers and unstable circular orbits[END_REF][START_REF] Sperhake | Cross section, final spin and zoom-whirl behavior in high-energy black hole collisions[END_REF], where furthermore evidence for critical scaling around the so-called threshold of immediate merger has been found. In this regime highly fine-tuned initial conditions are necessary. We plan to investigate larger momenta in future work.

Θ =47 • Θ =48 • Θ =50 • Figure 3.
Trajectories of three evolutions corresponding to one minimum and two maxima in E rad (compare Θ values in the legend with x-axis in Fig. 1).

The solid line represents an inspiral without noticable whirl or zoom, the dotted line shows a complete whirl before the plunge, and the dashed line starts with a fractional whirl (less than a full whirl orbit but strong precession), followed by a zoom phase and the plunge. Only one puncture trajectory for each evolution is shown for clearity.

Θ =47 • Θ =48 • Θ =50 • Figure 4
. Zoom into the inner part of three orbital evolutions corresponding to one minimum and two maxima in E rad (compare Θ values in the legend with x-axis in Fig. 1). The markers denote the onset of merger. A maximum in E rad occurs when the coordinate separation vector S (straight line) is closest to being orthogonal to the tangent vector of the puncture tracks at the time of merger. Shown is how much time the binary has spent at D ± 0.2. The histograms include data up until the merger time tmerger which read 137M , 184M and 432M , upper to lower, respectively. Less efficient binaries (middle panel) spent more time at a larger separation whereas efficient ones (upper panel) at lower separations. In the lower panel one can see that both whirl and zooms cause peaks in this diagram.

Figure 1 .

 1 Figure 1. Efficiency of gravitational radiation as a function of the shooting angle Θ. For the y-axis we normalize E rad by the ADM-mass of the initial time slice. The maximum at Θ ≈ 47 • corresponds to an initial orbital angular momentum L = DP sin(Θ) = 0.88M 2 .

Figure 2 .

 2 Figure 2. Comparison of 22-modes of the most efficient case (Θ = 47 • ) and the quasi-circular (Θ = 90 • ) case. The late-time, ring-down waveforms are quite similar. Differences are visible before the merger where the amplitude for the Θ = 47 • case is up to twice as large as for the quasi-circular binary, and in the inspiral part where the quasi-circular orbit results in more radiation.

Figure 5 .

 5 Figure 5. The angle between the tangent vector of the puncture orbit and the separation vector D as a function of the initial shooting angle. This plot looks similar in its general shape to the plot for the radiated energy in Fig.1. The location of the maxima and minima basically coincides.

Figure 6 .

 6 Figure 6. Another comparison regarding efficiency using a histogram of D.Shown is how much time the binary has spent at D ± 0.2. The histograms include data up until the merger time tmerger which read 137M , 184M and 432M , upper to lower, respectively. Less efficient binaries (middle panel) spent more time at a larger separation whereas efficient ones (upper panel) at lower separations. In the lower panel one can see that both whirl and zooms cause peaks in this diagram.

Acknowledgments

The authors wish to thank M. Campanelli, L. Gergely, J. Grigsby, D. Hilditch, J. Levin, D. Müller, G. Schäfer, D. Shoemaker and U. Sperhake for discussions. This work was supported in part by DFG SFB/Transregio 7 and DFG Research Training Group 1523. The computations were carried out on the HLRB2 system at the LRZ in Garching.