A wideband and sensitive GW detector for khz frequencies: the dual sphere

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1 INSTITUTE OF PHYSICSPUBLISHING Class. Quantum Grav. 19 (2002) CLASSICAL ANDQUANTUM GRAVITY PII: S (02) A wideband and sensitive GW detector for khz frequencies: the dual sphere Livia Conti 1, Massimo Cerdonio 1, Michele Bignotto 1, Michele Bonaldi 2, William Duffy Jr 3, Paolo Falferi 2, Antoine Heidmann 4, J Alberto Lobo 5, Francesco Marin 6, Antonello Ortolan 7, Michel Pinard 4, Giovanni Andrea Prodi 8, Luca Taffarello 9, Stefano Vitale 8 and Jean-Pierre Zendri 9 1 INFN Padova Section and Department of Physics, University of Padova, via Marzolo 8, I Padova, Italy 2 CeFSA, ITC-CNR and INFN Gruppo Coll. Trento, I Povo, Trento, Italy 3 Department of Physics, Santa Clara University, Santa Clara, CA 95053, USA 4 Laboratoire Kastler Brossel, 4 place Jussieu, F75252 Paris, France 5 Departamento de Física Fonamental, Universidad de Barcelona, Diagonal 647, Barcelona, Spain 6 INFN Firenze Section and Department of Physics, University of Firenze, Lgo E Fermi 2, I Firenze, Italy 7 INFN, Laboratori Nazionali di Legnaro, via Romea 4, I Legnaro, Padova, Italy 8 INFN Gruppo Coll. Trento and Department of Physics, University of Trento, I Povo, Trento, Italy 9 INFN Padova Section, via Marzolo 8, I Padova, Italy conti@lnl.infn.it Received 25 September 2001, in final form 26 October 2001 Published 18 March 2002 Online at stacks.iop.org/cqg/19/2013 Abstract We discuss the new concept of a sensitive and wide-band detector, consisting of a solid sphere nested inside a hollow one: the dual sphere. The advantage is that it would be possible to keep both the omni-directionality and high sensitivity of the spherical geometry without giving up the wide band. In the few khz range the dual sphere would be complementary to advanced interferometers. We also discuss the main technological and scientific challenges that the construction of such a system poses, particularly regarding material choice, fabrication, cooling, suspension and readout. PACS numbers: 0480N, 9555Y 1. Introduction Gravitational wave (GW) detectors aim at measuring tiny strains δl/l of test masses where δl/l = h/2andh is the wave amplitude. In order to get a significant rate event,experimentalists /02/ $ IOP Publishing Ltd Printed in the UK 2013

2 2014 L Conti et al are involved in a worldwide effort to push the sensitivity of their ground-based GW detectors to h Basically, planned and operating detectors are naturally divided into two classes: the resonant detectors and the interferometric ones. The first class aims at detection in a frequency range centred on the internal mechanical resonance of the test mass. The latter is presently a bar, resonating at 900 Hz, but potentially more sensitive spherical detectors are being built in a worldwide network [1, 2]. The second class aims at detection in a frequency range which spans from frequencies larger than those of the system connecting the test masses (i.e. the suspensions) up to frequencies smaller than the first internal modes of the test masses themselves. The first km-long baseline interferometric detectors are now under construction. Interferometers are widely recognized as wide-band devices, in contrast to the narrow-band acoustic detectors; interferometers can also be operated in a narrow-band regime by peaking the sensitivity in a chosen frequency interval. In this paper, we consider a third kind of detector, which aims at detection in a frequency range close to and in between the resonances of two massive free bodies. In principle it can be constructed for either a wide-band or narrow-band operation. 2. Concept of the dual sphere 2.1. Broad band operation The proposed detector is based on two spherical masses nested together, which we call the dual sphere [3]: the inner mass is a full sphere while the other is a hollow sphere. The radii match so that only a small gap separates the two bodies. Let R and a be, respectively, the outer radius of the hollow sphere and the radius of the inner solid sphere. We consider non-mechanically-resonant displacement sensors which can be placed on the facing surfaces (i.e. the inner surface of the hollow sphere and the outer surface of the full sphere) to detect the radial relative motion of the two bodies. For the sake of the discussion of the concept, we assume in the following that the thermal noise is small enough to allow for standard quantum limited (SQL) operation. The minimum number of sensors and their location is set by the strategy chosen to reconstruct the sphere motion [4, 5]. By using the same material for both spheres and if a/r , it turns out that the first GW sensitive quadrupole mode of the inner full sphere occurs at a frequency f int which is a factor two or three larger than the first quadrupole mode f ext of the hollow one. The frequency region between these two modes is of particular interest as the displacements caused by the same driving GW are phase shifted by a π factor. Therefore, as the displacement sensors sense the differential displacement of the two surfaces, the two signals sum and the dual sphere system is sensitive not only in a frequency range around the two quadrupole modes but in the wider interval between them. At a frequency f >f int,f ext the two signals subtract from each other and the composite signal cancels out, with complete loss of the sensitivity to GW; this pattern repeats for the higher GW sensitive modes. The choice of the spherical geometry for the two main bodies has several advantages, which have already been widely discussed in the literature. To summarize some (see [3] and references therein): (i) the detector sensitivity is intrinsically independent of the incoming wave direction and polarization, thus offering a full sky coverage; (ii) the signal polarization and propagation direction can be reconstructed; (iii) the second quadrupole mode of the hollow sphere has a cross-section to GW similar to that of the first mode, so that, in principle, one could also use this second sensitive mode for extending the detection capabilities. For simplicity, we have made no use of this last possibility in the following.

3 A wideband and sensitive GW detector for khz frequencies: the dual sphere 2015 Table 1. Comparison of the main figures of different materials. ρ v s ρvs 5 Material (10 3 kg m 3 ) (10 3 ms 1 ) Q (10 21 kg m 2 s 5 ) Al at 1 K 14 Cu Be 10% at 4 K 24 Molybdenum at 2 K 93 Sapphire at 4 K 400 Beryllium at 4 K a 580 SiC at 4 K 640 a Currently under measurement for hot pressed, heat treated samples. The detector dimensions are a compromise between the frequency range of interest, the signal strength and practical constraints. By choosing an outer diameter for the hollow sphere of few metres, the sensitivity is peaked in the few khz range. The material choice should be guided by considering the integrated energy cross-section [6] which, apart from a geometric factor, scales as ρv5 s f0 3. (1) Here the material enters with its density ρ and sound speed v s and f 0 is the frequency of the sphere s first quadrupole mode. In table 1 we summarize the main properties of a few materials of interest. One very important material property is the mechanical quality factor Q which determines the thermal noise. In order to reduce this noise, we are considering operating the dual sphere at cryogenic temperatures T, for which we also quote the best values of Q reported in the literature [7]. From the figures of table 1 it is evident that molybdenum is a promising material. Even more interesting would be sapphire and beryllium. Sapphire has the largest known Q but the present difficulties of producing large samples make it non-viable. Therefore, we focus our discussion on molybdenum and beryllium (as evidenced for Mo, possibly the Q of hot pressed, heat treated Be is larger than that of the Be reported in table 1). We choose f ext = 1100 Hz and a/r = 0.6; this sets the dimensions as in table 2. Table 2. Dimensions of a dual sphere detector with f ext = 1100 Hz and a/r = 0.6. Beryllium Molybdenum Ext. Int. Ext. Int. Radius R (m) Radius a (m) Mass (10 3 kg) f (1st quadr.) (Hz) f (2nd quadr.) (Hz) In order to get the ultimate sensitivity of such a system we consider the back-action noise coming from the sensors and the sensor displacement additive noise to allow SQL operation [8]. We also assume that only the modes with radial component of the motion contribute to the total noise, i.e., we assume 100% efficiency of the sensors in rejecting non-radial motion of the spheres. We neglect any other noise contribution, for instance, those due to suspensions

4 2016 L Conti et al 1x10-19 f * 1x10-20 S hh [1/ Hz] 1x x x x Frequency [Hz] Figure 1. Sensitivity to GW of a dual sphere system of beryllium (solid line) and molybdenum (dash-dotted line) with SQL imposed at 1.3 khz. 1x x x10-21 S hh [1/ Hz] 1x x x x Frequency [Hz] Figure 2. Sensitivity to GW of a dual sphere system of beryllium (black solid line) and molybdenum (dash-dotted line) with SQL imposed at 1.3 khz, of LCGT (dashed line) and LIGO II (grey solid line). and cosmic rays. Under these assumptions, the expected displacement noise on each sensor is as shown in figure 1. Here we indicate the frequency f at which the signal cancels out. The other spikes that appear come from the thermal and back-action contribution of the non- GW-sensitive modes, with non-null radial component, of both the inner and the outer sphere. We assume one can suppress all these spikes using a method already developed for GW interferometric detectors [9]; therefore, in the following we will not consider the contribution of these modes. The ultimate sensitivity at a given frequency is set by imposing the standard quantum limit which consists in equating at the chosen frequency the contribution due to the sensor displacement additive noise and the back-action, while keeping the thermal noise negligible with respect to the latter. In figure 2 we plot the sensitivity expected for the beryllium and molybdenum systems with SQL imposed at 1.3 khz; this is reached when T/(Qr 0 ) Km 1 for Be and T/(Qr 0 ) Km 1 for Mo. Here r 0 is the

5 A wideband and sensitive GW detector for khz frequencies: the dual sphere 2017 Table 3. Dimensions of the mixed Be-Mo dual sphere detector. External beryllium Internal molybdenum Radius R 2.0 Radius a Mass (10 3 kg) f (1st quadr.) (Hz) f (2nd quadr.) (Hz) x x10-20 S hh [1/ Hz] 1x x x x x Frequency [Hz] Figure 3. Sensitivity to GW of a dual sphere system (black solid line) of Be and Mo with SQL imposed at 1.2 khz, of full spheres of Be and CuBe5% (angle-like lines) and of LIGO II operated in the narrow-band mode (grey solid line; the grey dashed line corresponds to the location of minimum for different tuning frequencies). characteristic linear dimension of the surface on the sphere whose radial motion is probed by the sensors [10, 11]. In a Fabry Perot cavity r 0 would be the beam waist at the mirror; in a capacitive transducer r 0 would be related to the dimension of the charged electrode. In figure 2 we also plot the sensitivity curves expected for advanced interferometric detectors like LCGT [12] and LIGO II [13] operated in the broad-band mode Narrow-band operation Like the interferometers, the dual sphere can also be operated in a narrow-band mode, which allows one to improve the sensitivity in a small frequency range. To this end one might consider a mixed system with the inner full sphere made out of Mo and the outer hollow sphere of Be: with this choice and with a/r = 0.6 one gets f int /f ext 1.2. We consider a system with main figures as in table 3. Imposing the SQL at 1.2 khz requires T/(Qr 0 ) Km 1 ; with these figures one gets the sensitivity shown in figure 3. Here we also show the sensitivity curves expected for narrow-band LIGO II [13] and two SQL-limited full spheres: one of Be and one of CuBe5% [1].

6 2018 L Conti et al 3. Discussion The ultimate sensitivities for the dual sphere are very attractive as with a dual sphere one could usefully complement the advanced long interferometers in the high-frequency region, but many issues should be addressed and a full feasibility study is required. To begin, we have investigated the possibility of cooling such a massive system. The beryllium case appears very promising as its cooling figures compare well (see table 4) with the experimental figures of AURIGA, which have been easily managed. A faster timescale could even be possible using recently developed techniques [1]. Even if the cryogenics seem not to cause a halt to the feasibility of a dual sphere, many other technical problems remain unsolved, such as machining, joining and fabrication of large samples and suspension. Table 4. Main figures for cooling a Be dual sphere of 70 ton (heat exchange surface of 100 m 2 ) and AURIGA (heat exchange surface of 6 m 2 ). Be dual sphere AURIGA to 77 K to 4 K to 77 K to 4 K Refrigerant (litres) LN LHe LN LHe Timescale (weeks) Heat extraction rate (W m 2 ) Moreover, a deeper and more detailed study should be performed on the expected thermal noise of the two spheres. We are now doing this by a direct application of the fluctuation dissipation theorem [10, 11], via both analytic calculations and finite elements modelling. The main difficulty comes from the fact that we are in a mixed frequency range, that is, at the same time below, above and at a few mode frequencies, and therefore the usual simplifications do not apply. Of course, whatever the readout, the off-resonant thermal noise will be dominated by the combination T/(Qr 0 ) for which we gave above the upper limits for SQL operation. Such values already show a big difficulty: for T/Q 10 8 Kther 0 needed is 3cm. The usage of short ( 1 cm) Fabry Perot cavities with r 0 1 mm, as under development for the optomechanical transducer of Auriga [14], appears unsatisfactory; different optical configurations should be devised which would allow a larger r 0, as a confocal Fabry Perot cavity. Alternatively a capacitive transducer may be a much easier way to the needed r 0 ;here the problem appears at present to be related to the discharge electric field. A capacitive readout could also prove interesting in the sense that it would likely allow the achievement of lower T/Qratios as the heat input due to the laser power dissipated in the mirrors is removed. References [1] De Waard A et al 2002 Proc. of the 4th Edoardo Amaldi Conf. on Gravitational Waves (Perth, Western Australia, 8 13 July 2001) Class. Quantum Grav [2] Aguiar O et al 2002 Proc. of the 4th Edoardo Amaldi Conf. on Gravitational Waves (Perth, Western Australia, 8 13 July 2001) Class. Quantum Grav [3] Cerdonio M et al 2001 Phys.Rev.Lett [4] Johnson W W and Merkowitz S M 1993 Phys.Rev.Lett Merkowitz S M and Johnson W W 1997 Phys. Rev. D and references therein [5] Lobo J A 2000 Mon. Not. R. Astron. Soc [6] Misner C W, Thorne K S and Wheeler J A 1973 Gravitation (New York: Freeman) p 1024 [7] Duffy W Jr and Bassan M 1998 Cryogenics and references therein [8] Heffner H 1962 Proc. IRE

7 A wideband and sensitive GW detector for khz frequencies: the dual sphere 2019 [9] Finn L S and Mukherjee S 2001 Phys. Rev. D [10] Levin Y 1998 Phys. Rev. D [11] Liu Y T and Thorne K S 2000 Phys. Rev. D [12] Kuroda K et al 1999 Int. J. Mod. Phys. D [13] LIGO document 1999 number M A-M [14] De Rosa M et al 2002 Proc. of the 4th Edoardo Amaldi Conf. on Gravitational Waves (Perth, Western Australia, 8 13 July 2001) Class. Quantum Grav