Effect of energy deposited by cosmic-ray particles on interferometric gravitational wave detectors
Kazuhiro Yamamoto, Hideaki Hayakawa, Atsushi Okada, Takashi Uchiyama, Shinji Miyoki, Masatake Ohashi, Kazuaki Kuroda, Nobuyuki Kanda, Daisuke Tatsumi, Yoshiki Tsunesada
aa r X i v : . [ g r- q c ] A ug APS/123-QED
Effect of energy deposited by cosmic-ray particles on interferometric gravitationalwave detectors
Kazuhiro Yamamoto, ∗ Hideaki Hayakawa, Atsushi Okada, TakashiUchiyama, Shinji Miyoki, Masatake Ohashi, and Kazuaki Kuroda
Institute for Cosmic Ray Research, the University of Tokyo,5-1-5 Kashiwa-no-Ha, Kashiwa, Chiba 277-8582, Japan
Nobuyuki Kanda
Department of Physics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, Osaka 558-8585, Japan
Daisuke Tatsumi and Yoshiki Tsunesada
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan (Dated: November 18, 2018)We investigated the noise of interferometric gravitational wave detectors due to heat energy de-posited by cosmic-ray particles. We derived a general formula that describes the response of a mirroragainst a cosmic-ray passage. We found that there are differences in the comic-ray responses (thedependence of temperature and cosmic-ray track position) in cases of interferometric and resonantgravitational wave detectors. The power spectral density of vibrations caused by low-energy sec-ondary muons is 100-times smaller than the goal sensitivity of future second-generation interferom-eter projects, such as LCGT and Advanced LIGO. The arrival frequency of high-energy cosmic-raymuons that generate enough large showers inside mirrors of LCGT and Advanced LIGO is one pera millennium. We also discuss the probability of exotic-particle detection with interferometers.
PACS numbers: 04.80.Nn, 95.35.+d, 95.55.Vj, 95.85.Ry
I. INTRODUCTION
Recent improvements of the sensitivity and operationalstability of gravitational wave detectors is remarkable.Observation runs have already been performed in sev-eral interferometer (LIGO [1], VIRGO [2], GEO [3],TAMA [4], CLIO [5]) and resonator (ALLEGRO [6], EX-PLORER [7], NAUTILUS [8], AURIGA [9], NIOBE [10],MARIO SCHENBERG [11]) projects. In order of gravi-tational wave detection, the reduction of noise and faketriggers is crucial, since the gravitational wave ampli-tude and number of events are expected to be small andrare. In 1969, it was pointed out that cosmic-ray particlescould cause fake triggers in resonators [12]. The interpre-tation of excitations of resonators by cosmic-ray particlesis follows: The heat energy deposited by cosmic-ray par-ticle passages induces temperature gradients around theirtracks, and thermal stress excites internal vibrations ofthe resonator. These phenomena have been investigated,for example, observations of excited resonator vibrationsby beams from accelerators [13, 14], simultaneous de-tection of resonator excitation and cosmic-ray particles[15], and studies of exotic events in super-conductive res-onators [16, 17, 18]. In some research, resonators werealso operated and treated as exotic-particle detectors[19, 20, 21, 22]. These studies suggest that cosmic-ray ∗ Electronic address: [email protected]; Present ad-dress: Max Planck Institute for Gravitational Physics, Albert Ein-stein Institute, Callinstrasse 38, D-30167 Hannover, Germany. heating is a possible noise source in interferometric de-tectors [23, 24, 25, 26] (other effects on interferometers,the momentum and electrical charge brought by cosmic-ray particles, are discussed in Refs. [23, 24, 25, 27]).We investigated details of this effect by cosmic-ray en-ergy deposition in interferometers. A formula that de-scribes the response of a mirror against a cosmic-raypassage was derived. This formula reveals differencesbetween the cosmic-ray responses of interferometers andresonators. We used it to evaluate the amplitude of vi-brations caused by cosmic-ray particles in typical cases ofinterferometers, and examined the effect in gravitationalwave detection. We also considered the probability ofexotic-particle detection with interferometers.
II. FORMULA OF EXCITED MOTION BY ACOSMIC-RAY PARTICLEA. Outline of derivation of the formula
In order to simplify the discussion, the mirror vibra-tion excited by a cosmic-ray particle is investigated. Avibration excited by many particles, like a shower, is asuperposition of that by one particle. The excitation bya particle is considered under the following assumptions.The particle goes straight and never stops in the mirror.Its speed is faster than that of sound in the mirror. Along and narrow heated volume appears at the instant ofparticle passage.The heat-conduction equation is solved in order to cal-culate the time evolution of the temperature gradient.The vibration of the mirror is examined using the equa-tion of motion of an elastic body with thermal stress,which is proportional to the thermal gradient.
B. Formula
Since the heated volume is smaller than that of themirror, itself [28, 29], the mirror and the initial heatedvolume are treated as an infinite body and a line, respec-tively. The direction of the cosmic-ray track is taken asthe z -axis. The heat-conduction equation is described as[30] ∂∂t δT − κρC ∆( δT ) = 1 ρC dEdl δ ( x ) δ ( y ) δ ( t ) , (1)where δT is the temperature difference caused by acosmic-ray particle. The quantities κ, ρ, C and dE/dl are the thermal conductivity, density, specific heat perunit mass, and energy loss of a particle per unit length,respectively. The solution is described as [31] δT = 14 πκt dEdl exp (cid:20) − ρC κt (cid:0) x + y (cid:1)(cid:21) . (2)The radius of the heated volume increases with time dueto conduction. The time when the heated area radiusbecomes a is τ a = ρCa κ . (3)The equation of motion of an elastic body with thermalstress is described as [30] ρ ∂ u ∂t − Y σ ) ∆ u − Y σ )(1 − σ ) grad div u = − Y α − σ div δT, (4)where u represents the displacement of a volume ele-ment in the elastic body. The quantities Y, σ and α are Young’s modulus, the Poisson ratio, and the linearthermal-expansion coefficient, respectively. By substi-tuting Eq. (2) for Eq. (4), we obtain the output of ainterferometer X , X = Z surface u opt ( r ) P ( r ) dS, (5)where u opt is the optical axis component of u and P isthe intensity profile of the laser beam, P ( r ) = 2 πr exp (cid:18) − r r (cid:19) . (6)The quantities r and r are the distance from the opticalaxis and the beam radius. We employ the modal expan-sion method [32, 33, 34] to calculate u and X . In this method, u and X are represented by a superposition ofthe resonant mode displacement, u ( r , t ) = X n w n ( r ) q n ( t ) , (7) X ( t ) = X n q n ( t ) , (8)where w n and q n represent the displacement and timedevelopment of the n -th resonant mode, respectively.These basis functions are normalized to satisfy a con-dition [33, 34], Z surface w n, opt ( r ) P ( r ) dS = 1 , (9)where w n, opt is the optical axis component of w n . Theequation of motion of each mode is the same as that of aharmonic oscillator, − m n ω ˜ q n ( ω ) + m n ω n [1 + i φ n ( ω )]˜ q n ( ω ) = ˜ F n ( ω ) , (10)in the frequency domain. The quantity φ n is the loss an-gle, which represents dissipation of the n -th mode [32].The force F n applied on the n -th mode is related to thethermal stress. The quantities m n and ω n are the effec-tive mass and the resonant angular frequency [33, 34].The effective mass is defined as m n = Z volume ρ w n ( r ) · w n ( r ) dV. (11)The quantities ˜ q n ( ω ) and ˜ F n ( ω ) are the Fourier compo-nents of q n and F n , respectively:˜ X ( ω ) = 12 π Z ∞−∞ X ( t ) exp( − i ωt ) dt, (12) X ( t ) = Z ∞−∞ ˜ X ( ω ) exp(i ωt ) dω. (13)The force F n is obtained from the modal decomposi-tion of the thermal stress on the right-hand side of Eq.(4). The decomposition procedure [33, 34] is as follows.The thermal-stress term is multiplied by w n . The inte-gral of this inner product over all the volume is F n . Thisforce F n decreases after the heated volume scale, a , be-comes larger than the n -th mode wavelength. In orderto simplify the discussion, it is assumed that the timeevolution of F n [19, 35] is expressed as F n ( t ) = ( F n (0) exp (cid:16) − tτ n (cid:17) ( t > t < . (14)The quantities F n (0) and τ n are the initial value and thedecay time of the force, respectively. The initial value, F n (0), is written as [20, 36] F n (0) = Y α − σ ρC (cid:18)Z div w n dl (cid:19) dEdl . (15)The integral along the cosmic-ray track represents thecoupling between the thermal stress and the n -th mode.The coefficient 1 / ( ρC ) is a factor used to transform theheat energy into the temperature gradient. The force F n is described as a product of the temperature gradient and Y α/ (1 − σ ). From Eq. (3), the time τ n when the heatedvolume radius becomes comparable to the wavelength ofthe n -th mode is expressed as τ n = ρCλ n κ = π ρCv κω n ∼ π Y Cκω n . (16)The quantities λ n and v are the wavelength and soundvelocity: λ n = 2 πvω n , (17) v ∼ s Yρ . (18)The Fourier component of F n in Eq. (14) is written inthe form ˜ F n ( ω ) = F n (0)2 π τ n ωτ n . (19)We now write down the formula of the mirror vibrationexcited by a cosmic-ray particle using Eqs. (8), (10), (15)and (19):˜ X ( ω ) = X n ˜ q n ( ω )= X n ˜ F n ( ω ) − m n ω + m n ω n (1 + i φ n )= 12 π Y α − σ ρC dEdl × X n − m n ω + m n ω n (1 + i φ n ) τ n ωτ n × (cid:18)Z div w n dl (cid:19) . (20) C. Frequency dependence of the formula
A schematic view of the frequency dependence of themodes ˜ q n ( ω ) in Eq. (20) is shown in Fig. 1. Here,we discuss the frequency dependence below the resonantfrequencies of the mirrors, because the observation bandof interferometers (around 100 Hz) is below the funda-mental mode (the order of 10 kHz). The cut-off fre-quency, 1 / (2 πτ n ), is extremely smaller than the resonantfrequency, ω n / (2 π ), as shown in Fig. 1, because soundis generally faster than heat conduction. The absolutevalue | ˜ q n ( ω ) | is inversely proportional to the frequencybetween 1 / (2 πτ n ) and ω n / (2 π ). Below the cut-off fre-quency, 1 / (2 πτ n ), | ˜ q n ( ω ) | is constant. -7 -6 -5 -4 -3 -2 -1 | q n | [ A r b i t r a r y U n i t] -5 -3 -1 Frequency [Arbitrary Unit] fundamental mode"highest" mode1/(2 πτ πτ r /(2 ) ω π )) "high""low" FIG. 1: A schematic view of the frequency dependence ofthe Fourier components of mode motion excited by a cosmic-ray particle, ˜ q n ( ω ) in Eq. (20). The absolute value | ˜ q n ( ω ) | isinversely proportional to the frequency between 1 / (2 πτ n ) and ω n / (2 π ). Below the cut-off frequency, 1 / (2 πτ n ), it is constant.The ”highest” mode is that with a wavelength comparableto the beam radius, r , and contributions of higher modesare negligible in the summation of Eq. (20) [37, 38]. Thecut-off frequency of the ”highest” mode, 1 / (2 πτ r ), is smallerthan the fundamental mode resonant frequency, ω / (2 π ), ingeneral. The cut-off frequency, 1 / (2 πτ n ), of the lower modeis smaller. In the range between 1 / (2 πτ r ) and ω / (2 π ) (the”high” frequency region in this graph), | ˜ X ( ω ) | = | P ˜ q n ( ω ) | is inversely proportional to the frequency. Below the cut-off frequency of the fundamental mode, 1 / (2 πτ ) (the ”low”frequency region in this graph), | ˜ X ( ω ) | is independent of thefrequency. The approximation formulae in the ”high” and”low” frequency regions, Eqs. (30) and (34), are derived usingEq. (20). The frequency dependence of ˜ X ( ω ) = P ˜ q n ( ω ) is asfollows. The ”highest” mode in Fig. 1 is that with awavelength comparable to the beam radius, r , and con-tributions of higher modes are negligible in the summa-tion of Eq. (20) [37, 38]. The thermal relaxation time, τ r , for this ”highest” mode is described as τ r = ρCr κ (21)from Eq. (16). The cut-off frequency of the ”highest”mode, 1 / (2 πτ r ), is smaller than the fundamental moderesonant frequency, ω / (2 π ), in general, as shown in Fig.1. The cut-off frequency, 1 / (2 πτ n ), of the lower mode issmaller from Eq. (16). In the range between 1 / (2 πτ r )and ω / (2 π ) (the ”high” frequency region in Fig. 1), | ˜ X ( ω ) | = | P ˜ q n ( ω ) | is inversely proportional to the fre-quency. Below the cut-off frequency of the fundamentalmode, 1 / (2 πτ ) (the ”low” frequency region in Fig. 1), | ˜ X ( ω ) | is independent of the frequency. From Eq. (16),the relaxation time, τ , is described as τ = ρCλ κ = ρCR κ , (22)because the wavelength of the fundamental mode, λ , iscomparable to the mirror diameter, 2 R .The typical values of | ˜ X ( ω ) | in the ”high” and ”low”frequency regions of Fig. 1 are evaluated using Eq.(20). In the ”high” frequency region, 1 / (2 πτ r ) < f <ω / (2 π ), it is approximated as ( | φ n | ≪ | ˜ X ( ω ) | ∼ π Y α − σ ρC dEdl ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n m n ω n Z div w n dl (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , / (2 πτ r ) < f < ω / (2 π ) . (23)The sign of the integral in Eq. (23) depends on themodes. The typical absolute value of the summation inEq. (23) is evaluated as the square root of a summationof squares of the terms. Equation (23) is rewritten as | ˜ X ( ω ) | ∼ π Y α − σ ρC dEdl ω × vuutX n m n ω n (cid:18)Z div w n dl (cid:19) , / (2 πτ r ) < f < ω / (2 π ) . (24)The integral along the cosmic-ray track in Eq. (24) isevaluated as follows. The average of the length of thecosmic-ray track is comparable to the radius of the mir-ror, R . The average of | w n | , (cid:10) | w n | (cid:11) , is related to Eq.(11), m n = Z ρ | w n | dV = M (cid:10) | w n | (cid:11) , (25)where M is the mass of the mirror. This equation gives p h| w n | i = r m n M . (26)The divergence of w n in Eq. (24) can be represented bythe product of w n and the wavenumber ω n /v , because w n is the basis of the solution of the wave equation. Con-sequently, Eq. (24) is described as | ˜ X ( ω ) | ∼ π Y α − σ ρC dEdl Rv √ M ω sX n m n ω n , / (2 πτ r ) < f < ω / (2 π ) . (27)The summation in Eq. (27) is the same as the responseof a mirror against a static force [39, 40], X n m n ω n = 1 − σ √ πY r . (28)In order to simplify the discussion, the relation M = πρR (29) is assumed. The radius of the mirror, R , is nearly equalto its thickness, H , in usual cases of interferometric grav-itational wave detectors. Using Eqs. (18), (28) and (29),Eq. (27) is written in the form | ˜ X ( ω ) | ∼ π / α √ − σ − σ ρC dEdl √ Rr ω , / (2 πτ r ) < f < ω / (2 π ) . (30)Equation (20) in the ”low” frequency band of Fig. 1is evaluated in the same manner as in the previous para-graph. Using Eq. (16), the result is written as | ˜ X ( ω ) | ∼ π Y α − σ ρC dEdl × vuutX n τ n m n ω n (cid:18)Z div w n dl (cid:19) ∼ √ π Y / α − σ ρκ dEdl √ R sX n m n ω n ,f < / (2 πτ ) . (31)It can be seen that only the fundamental mode is domi-nant, because of the frequency dependence of ω n . Thequantities of this mode are as follows [20, 37]: m ∼ M ∼ πρR , (32) ω ∼ πH s Yρ ∼ πR s Yρ . (33)Equation (31) is expressed as | ˜ X ( ω ) | ∼ √ π α − σ κ dEdl R, f < / (2 πτ ) . (34) D. Formula in time domain
Here, we discuss the excitation formula Eq. (20) inthe time domain. Only a contribution of the n -th modeis considered in order to simplify the discussion. If theQ-value, Q n = 1 /φ n ( ω n ), is larger than unity and 1 /τ n issmaller than ω n , q n in the time domain ( t >
0) is writtenin a form q n ( t ) ∼ F n (0) m n ω n exp (cid:18) − tτ n (cid:19) − F n (0) m n ω n cos( ω n t ) exp (cid:18) − ω n t Q n (cid:19) . (35)Figure 2 shows the fundamental mode, q , in the timedomain.The second term in Eq. (35) is dominated by ˜ X ( ω )near the resonant frequency. This is the excited resonantvibration and its decay. The outputs of resonant detec-tors are described with this term. The first term in Eq. -21 A m p li t ud e [ m ] q FIG. 2: Vibration of the fundamental mode caused by a low-energy cosmic-ray muon in the time domain, q in Eq. (35).In the calculation, the material values of sapphire at roomtemperature are used. Since the Q-value is extremely high,i.e. the decay time is longer than the period of the resonantmotion, we are not able to see the resonant motion of the oneperiod in this graph. (35) represents the drift of the center of the resonant vi-bration caused by the relaxation of thermal stress. This isdominated by ˜ X ( ω ) below the fundamental mode. Theoutputs of interferometric detectors are described withthis term.The initial amplitude of P n F n (0) / ( m n ω n ) is evalu-ated as Eq. (30), X n F n (0) m n ω n = 2 π / α √ − σ − σ ρC dEdl √ Rr = 6 . × − m (cid:18) α × − / K (cid:19) × . √ − σ − σ ! × (cid:18) . × J / kg / K C (cid:19) ×
12 MeV / (g cm − ) 1 ρ dEdl ! (cid:18)
25 cm2 R (cid:19) / × (cid:18) r (cid:19) / . (36)Here, we consider a sapphire mirror at room temperature. III. DISCUSSION ABOUT THE FORMULAA. Effect on low-temperature interferometers
In some future projects using interferometric detectorsas LCGT [41] and ET [42], mirrors will be cooled in order to reduce the thermal noise (for example, LCGT mirrorsat 20 K). In the quantities of the force, F n , which isrelated to the thermal stress, α, C and κ in Eqs. (15)and (16) strongly depend on the temperature [43]. Theinitial value of the force, F n (0) in Eq. (15), is propor-tional to α/C . The decay time of the force, τ n in Eq.(16), is proportional to C/κ . The Gr¨uneisen relation [44]predicts that the ratio α/C is independent of the temper-ature. The initial force, F n (0), and the initial amplitudeof the excited vibration do not depend on temperature.On the contrary, in the case of crystals, the decay time, τ n ( ∝ C/κ ), of the first term in Eq. (35) is extremelyshort at the cryogenic temperature, because of the small C and large κ . The cut-off frequency, 1 / (2 πτ n ), in thelow-temperature region is higher than that at room tem-perature (e.g. Ref. [26]). For example, the cut-off fre-quencies of sapphire at room temperature, obtained fromEqs. (21) and (22), are:12 πτ = 0 .
13 mHz / cm ρ ! (cid:18) . × J / kg / K C (cid:19) × (cid:18)
25 cm2 R (cid:19) (cid:18) κ
40 W / m / K (cid:19) , (37)12 πτ r = 9 . / cm ρ ! (cid:18) . × J / kg / K C (cid:19) × (cid:18) r (cid:19) (cid:18) κ
40 W / m / K (cid:19) . (38)The values at 20 K are:12 πτ = 58 Hz / cm ρ ! (cid:18) .
69 J / kg / K C (cid:19) × (cid:18)
25 cm2 R (cid:19) (cid:18) κ . × W / m / K (cid:19) , (39)12 πτ r = 4 . / cm ρ ! (cid:18) .
69 J / kg / K C (cid:19) × (cid:18) r (cid:19) (cid:18) κ . × W / m / K (cid:19) . (40)At low temperature, the cut-off frequencies are near theobservation band of gravitational wave detectors (around100 Hz). The ”high” frequency approximation of | ˜ X ( ω ) | in Eq. (30) is only appropriate for room-temperatureinterferometers, and not valid for cryogenic interferom-eters. In order to show the effect of the cooling mir-rors, | ˜ q ( ω ) | of a sapphire mirror at 300 K and 20 K areplotted in Fig. 3. In the low-frequency region, ˜ X ( ω ) be-comes much smaller due to cooling. The ”low” frequencyapproximation of | ˜ X ( ω ) | in Eq. (34) is proportional to α/κ = α/C × C/κ ∝ C/κ , which is small in the low-temperature region. Since the decay time of the force F n becomes shorter, it is difficult to excite the low-frequencycomponent. The Fourier components | ˜ q ( ω ) | in the high-frequency region of Fig. 3 are comparable in the cases -26 -25 -24 -23 -22 -21 -20 -19 | q n | [ m s ec ] -5 -4 -3 -2 -1 Frequency [Hz]
Typical fundamental mode motion of sapphire mirror 300K 20K
FIG. 3: Fourier components of the fundamental mode motion, | ˜ q ( ω ) | , of a sapphire mirror excited by a low-energy cosmic-ray muon at 300 K (solid line) and 20 K (dashed line). Thecut-off frequency, 1 / (2 πτ ), at 20 K is higher than that at300 K. The mirror cooling reduces the low-frequency compo-nent because the decay time of F , which is related to thethermal stress, is shorter. The higher frequency component isindependent of temperature because of the Gr¨uneisen relation[44]. of 20 K and 300 K. The ”high” frequency approximationof | ˜ X ( ω ) | in Eq. (30) is independent of temperature be-cause it is proportional to α/C and related to the firstterm of Eq. (35) at t ∼
0. From the cut-off frequencies inEqs. (39) and (40) and a comparison between Eq. (34)at 20 K and Eq. (30), it can be seen that the vibra-tion of a cooled sapphire mirror excited by a cosmic-rayparticle in the observation band (around 100 Hz) is afew-times smaller than that at room temperature. Thisis an advantage of cryogenic interferometers in additionto the suppression of the thermal noise [45, 46, 47], ther-mal lensing effect [48] and parametric instability [49]. Itmust be noted that motions excited by cosmic-ray par-ticles in resonant detectors are independent of tempera-ture [16, 17, 18]. This is because the initial amplitude,the second term of Eq. (35) at t ∼
0, does not dependon temperature.
B. Cosmic-ray track position dependence
In the calculation of Eq. (20), the signs of the inte-gral terms in the summation are important. The signdepends on the positions of a cosmic-ray track and thelaser beam spot, because the displacement of the mode w n is normalized to satisfy Eq. (9) [50]. If the particletrack is near the beam spot, the signs of the integrals ofmany modes are the same, because the basis functions, w n , on the track are similar. If the track is far from thespot, w n on the track and the integral signs are different InterferometerBar resonator A BA BLaser beam A>BA=BTransducer
FIG. 4: Track position dependence of the cosmic-ray heatingeffect of interferometers and bar resonators. In the outputs ofinterferometers, the vibration caused by a particle that goesalong track A near the beam spot is larger than that alongtrack B far from the beam spot. On the contrary, in the caseof bars, the vibration by track A is the same as that by track Bbecause the displacement of the resonant modes is symmetricor antisymmetric with respect to the center of the bar. Thedashed line shows the fundamental mode deformation. for various modes. Since the sign of q n in Eq. (20) belowthe fundamental mode is the same as that of the inte-gral, | ˜ X ( ω ) | below the first mode is larger and smaller ifthe cosmic-ray track is near and far from the beam spot,respectively. Another explanation about the cosmic-raytrack position dependence is as follows. The heated vol-ume on the particle track pushes around them. Sincethe center of the mirror does not move because of theconservation of momentum, a larger motion is observedif the track is near the beam spot. The track positiondependence of the cosmic-ray heating effect in interfer-ometers is different from that in bar resonators, as shownin Fig. 4. In the outputs of bars, the vibration causedby a particle that goes along track A in Fig. 4 is thesame as that along track B, because the displacement ofthe resonant modes is symmetric or antisymmetric withrespect to the center of the bar (the dashed line in Fig.4 shows the fundamental mode deformation). The dif-ference between interferometers and the bar resonatorsis related to the number of modes to be considered. Inthe case of bars, only the fundamental mode is taken intoaccount. On the other hand, in the case of interferome-ters, many modes contribute to the response of a mirror.The signs of these modes have an important role. Thediscussion above is the same as that about thermal noisebelow the fundamental mode caused by inhomogeneouslydistributed loss [33, 34, 51]. IV. APPLICATION I — LOW-ENERGYCOSMIC-RAY PARTICLESA. Low-energy cosmic-ray particles and interactionwith matter
Primary cosmic rays generate extensive air showersin the atmosphere. Cosmic-ray particles on the groundare secondaries from air showers. Three quarters of sec-ondary particles at sea-level are muons. The remainderare almost electrons [52]. Muons with small energy (lessthan about 0.22 GeV) and electrons can be neglected be-cause it is difficult to penetrate matter around the mir-rors, for example, walls of buildings, vacuum chambers[52]. The speed of muons that arrive at the mirrors iscomparable to that of light. The flux of these cosmic-raymuons at sea-level is about 2 × − / cm / sec [52, 53].Since the number of higher energy muons is smaller[54], the energy of most of the cosmic-ray muons is be-low 100 GeV. In this low-energy region, the dissipationprocess in material is dominated by ionization [54, 55],which is Coulomb scattering with electrons in atoms ofmatter [55]. The ionization loss is about1 ρ dEdl = 2 MeV / (g cm − ) , (41)and almost independent of the particle energies [54, 55].The typical loss per unit length, dE/dl , is severalMeV/cm.The effect of mirror excitation by cosmic-ray particlesdepends on the arrival frequency of particles and the de-cay time of the vibrations. If the decay time is longerthan the interval of the particle arrivals, the mirror vi-bration is maintained. If the next muon comes after thevibration has disappeared, the vibration can be treatedas a burst event. The number of muons, N , that hit amirror at sea-level per unit time is expressed as N = 2 × − / cm / sec × R × H = 8 / sec (cid:18) R
25 cm (cid:19) (cid:18) H
15 cm (cid:19) . (42)The average arrival interval of muons, 1 /N , is1 N = 0 .
13 sec (cid:18)
25 cm2 R (cid:19) (cid:18)
15 cm H (cid:19) . (43)The decay time of the fundamental resonant vibration isdescribed as Q πf = 8 × sec (cid:18)
40 kHz f (cid:19) (cid:18) Q (cid:19) . (44)Since the Q-values of mirrors used for gravitational wavedetectors are at least 10 , the decay time is extremelylarger than the expected arrival interval of cosmic-rayparticles. B. Power spectral density
The power spectral density, G cos ( f ), of vibrationscaused by low-energy cosmic-ray particles has been cal-culated (e.g. Refs. [23, 26]). It is assumed that arrivaltime of particles and track position in a mirror are atrandom. Since there are four mirrors in an interferome-ter, the one-side power spectral density of the noise of aninterferometer output is written in the form [56] G cos ( f ) = 4 L × π N D | ˜ X ( ω ) | E = 32 π NL D | ˜ X ( ω ) | E , (45)where L is the length of the interferometer arms. Thequantity D | ˜ X ( ω ) | E is the ensemble average of | ˜ X ( ω ) | ,which is the vibration caused by a muon. To evaluatethe power spectrum of room-temperature interferome-ters, the square of Eq. (30) is used as the ensemble aver-age, because this formula is appropriate to calculate thetypical | ˜ X ( ω ) | at 300 K and around 100 Hz, as shownin Sec. III A. From Eqs. (41), (42) and (45), the powerspectrum of room-temperature sapphire at sea-level iswritten as [57] p G cos ( f ) = 1 . × − / √ Hz (cid:18) L (cid:19) (cid:18) α × − / K (cid:19) × . √ − σ − σ ! (cid:18) . × J / kg / K C (cid:19) × (cid:18) R
25 cm (cid:19) / (cid:18) r (cid:19) / (cid:18)
100 Hz f (cid:19) . (46)The sensitivity of future second-generation interferome-ter projects, such as LCGT [41] and Advanced LIGO [58],is on the order of 10 − / √ Hz at 100 Hz. Therefore, theeffect of low-energy cosmic-ray particles is not a seriousproblem, even in these future projects.
V. APPLICATION II — SHOWER
High-energy cosmic-ray particles often generate manyparticles (showers). From Eqs. (41) and (63), if 1000shower particles pass in a mirror at the same time, theexcited vibration is large enough to be detected by fu-ture second-generation interferometers, such as LCGT[41] and Advanced LIGO [58] (e.g. Refs. [23, 24, 25]).Such excitations caused by cosmic-ray showers have beenobserved in a resonator [15].We investigated the effect of a shower generated bya high-energy muon inside a mirror with a Monte-Carlotechnique [59]. It was assumed that the material is sap-phire. We evaluated the probability that a high-energymuon that runs in a 30 cm thickness sapphire generatesmore than 1000 electrons. In this simulation, the flux ofmuons at sea-level was expressed as [61] I µ ( > E ) = 1 . × − / cm / sec (cid:18) E (cid:19) − . , (47)if E is more than 1 TeV. Our simulation showed that thenumber per unit time and per a mirror of muons thatgenerate more than 1000 electrons N ( > N ( > . × − / sec (cid:18) R
25 cm (cid:19) (cid:18) H
15 cm (cid:19) . (48)It must be noted that this average arrival number, N ( > N ( > . × year (cid:18)
25 cm2 R (cid:19) (cid:18)
15 cm H (cid:19) . (49)The effect of showers generated by high-energy muonsinside mirrors is not a serious problem.In the case of a shower that occurs near a mirror, theenergy of an original particle that generates 1000 parti-cles is about 1 TeV [25]. Since the spread of particlesin a TeV energy shower is quite large, the typical sizemirror of interferometers can not contain all of the en-ergy of a thousand particles. In order to know how oftenmore than 1000 particles go into a mirror, accurate sim-ulations about shower generation in apparatus aroundmirrors (for example, vacuum chambers and vibrationisolation systems) and the response of a mirror are nec-essary as resonators [66, 67, 68]. This is our future work. VI. APPLICATION III — EXOTIC-PARTICLESEARCH
The effect of cosmic-ray particles on gravitational wavedetectors suggests that the detectors are useful to searchfor exotic particles that dissipate a large amount of en-ergy in material. Ideas that resonators can be used asmagnetic monopole [19] or mirror dust particle [22] de-tectors were proposed. The upper limits of the flux ofnuclearite [69, 70] from the operation of resonators werereported [20, 21]. Here, we discuss interferometers asexotic-particle detectors in comparison with resonators(bars [6, 7, 8, 9, 10]).In order to detect exotic particles or other rare events,a larger aperture and higher sensitivity are required fordetectors. The cross section of a bar resonator is 10-timeslarger than that of an interferometer [71]. The area of abar is about 1.8 m (diameter, 0.6 m; length, 3 m) [21].The cross section of four mirrors of an interferometer isabout 0.15 m (diameter, 0.25 m; thickness, 0.15 m). Wediscuss the sensitivity of interferometers and bars for anexotic particle passage. A. Signal-to-Noise ratio of interferometers
Since the time evolution of an excited motion by anexotic particle is predicted from Eq. (20), the matchedfiltering method can be applied to the outputs of detec-tors. The output of a matched filter, the signal-to-noiseratio (S/N), is defined as [74]S / N = 4 π sZ ∞ | ˜ S ( ω ) | G det ( f ) df , (50)where ˜ S ( ω ) and G det ( f ) are the Fourier components ofthe signal and the one-side power spectral density of thenoise of gravitational wave detectors, respectively.In the case of interferometers, G det ( f ) and ˜ S ( ω ) of Eq.(50) are the strain noise, G int ( f ), and the ratio of theFourier component of the motion excited by an exoticparticle, ˜ X ( ω ), to the arm length, L , respectively. It issupposed that the temperature is 300 K. Here, we recallEq. (30) in the form | ˜ X ( ω ) | ∼ Aπ / √ − σ Y √ Rr f , (51) A = 14 π Y α − σ ρC dEdl . (52)S/N is expressed by using Eqs. (50) and (51):S / N int = 4 π / A √ − σ Y √ Rr L sZ dff G int ( f ) . (53) B. S/N of bar resonators
In the case of bar detectors, ˜ S ( ω ) and G det ( f ) of Eq.(50) are the force applied by an exotic particle to the n -thmode, ˜ F n ( ω ), in Eq. (19), and the tidal force, which cor-responds to the strain noise, G bar ( f ), respectively. Equa-tion (19) is rewritten as | ˜ S ( ω ) | = | ˜ F n ( ω ) | = A f (cid:18)Z div w n dl (cid:19) , (54)because the cut-off frequency, 1 / (2 πτ n ), is lower than theresonant frequency. Here, we take the fundamental modeof bars into account. Under the same approximation inthe derivation of Eqs. (30) and (34), ˜ S ( ω ) is expressedas | ˜ S ( ω ) | = | ˜ F ( ω ) | ∼ A π √ f . (55)The tidal force that corresponds to the strain noise, G bar ( f ), is obtained from Refs. [20, 75, 76], G det ( f ) = (cid:18) M b ω lπ (cid:19) G bar ( f ) , (56)where M b , ω and l are the mass, angular resonantfrequency of the fundamental mode and length of a bar.S/N is given by Eqs. (50), (55), and (56):S / N bar = 2 √ π AM b ω l sZ dff G bar ( f ) . (57) C. Comparison between interferometers and barresonators
Here, we discuss the effects of an exotic particle oninterferometers and bar detectors by using Eqs. (53) and(57). The integral term only depends on the sensitivity ofgravitational wave detectors. It must be noted that theweight, 1 /f , originates from the frequency dependenceof ˜ S ( ω ), i.e. Eqs. (51) and (54). The integral term inEq. (53) for future second-generation interferometers,e.g. the LCGT project [41], is sZ dff G int ( f ) = 3 . × . (58)The typical goal sensitivity of bar resonators [77] is p G bar ( f ) ∼ × − / √ Hz in the frequency range be-tween 850 Hz and 950 Hz. The integral term in Eq. (57)is sZ dff G bar ( f ) ∼ . × . (59)The integral term of interferometers is 1000-times larger,because interferometers have higher sensitivity and awider observation band. Since the observation band of in-terferometers is lower than that of resonators, the weight-ing function, 1 /f , increases the integral term of inter-ferometers.The factors, except for the integral term and A in Eqs.(53) and (57), are the ratios of the responses to an exoticparticle to that to the gravitational wave. If this factoris large, the detector is more suitable for exotic-particlesearches. This factor of interferometers is [78]4 π / √ − σ Y √ Rr L = 4 . × − / N √ − σ . ! × (cid:18) × Pa Y (cid:19) (cid:18)
25 cm2 R (cid:19) / × (cid:18) r (cid:19) / (cid:18) L (cid:19) . (60)In the case of bar resonators, this factor is2 √ π M b ω l = 1 . × − / N (cid:18) M b (cid:19) × (cid:18) × π rad / Hz ω (cid:19) (cid:18) l (cid:19) . (61) The factor of bars is extremely larger. The main reasonfor this difference comes from the sizes of the detectors, L and l . An exotic-particle detector must be a gooddisplacement sensor. A smaller size detector is a bet-ter displacement sensor, if the strain (gravitational wave)sensitivity is the same. The better strain sensitivity ofinterferometers shown by Eqs. (58) and (59) is canceledby their larger size. The factors in Eqs. (60) and (61),except for L and l , represent the mechanical responses ofa mirror and a bar. The response of a bar is typicallyabout 10-times larger.The amplitude of the force F n in Eq. (54) is propor-tional to A . This quantity depends on only the energyloss process of exotic particles and the material of themirrors and the bar resonators. This is evaluated as A = 7 . × − N (cid:18) Y × Pa (cid:19) (cid:18) α × − / K (cid:19) × (cid:18) . − σ (cid:19) (cid:18) . × J / kg / K C (cid:19) ×
13 GeV / (g cm − ) 1 ρ dEdl ! . (62)In the quantities of Eq. (62), only the linear thermal-expansion coefficient, α , strongly depends on the material[79]. The values of the coefficient α for fused silica andsapphire at 300 K, are 5 . × − / K and 5 . × − / K,respectively. The coefficient α of the alloy Al5056 [80],which is the most popular material of bar resonators [6,7, 8, 9], is 2 . × − / K.From the above discussion, the advantages of interfer-ometers, the higher strain sensitivity and wider observa-tion band, are canceled by their larger detector size, be-cause exotic-particle detectors must have good displace-ment sensitivity, not strain sensitivity. The larger me-chanical response (about 10 times) and linear thermal-expansion coefficient (several or several tens times) ofbar resonators enhance the sensitivity. The typical S/Nof interferometers is obtained from Eqs. (53), (58), (60),and (62):S / N int = 10 (cid:18) α . × − / K (cid:19) ×
13 GeV / (g cm − ) 1 ρ dEdl ! . (63)The S/N of bar resonators is evaluated from Eqs. (57),(59), (61), and (62):S / N bar = 3 × (cid:18) α . × − / K (cid:19) ×
13 GeV / (g cm − ) 1 ρ dEdl ! . (64)The sensitivity for an exotic particle of bars is a few tensor a few hundreds times better than that of interferome-ters [81]. The sensitivity of bars in the above discussion,0Eq. (59), is based on the goal sensitivity. The currentsensitivity is 10-times worse than it [77]. The current barresonators are the better exotic-particle detectors thanthe future second-generation interferometers as LCGT[41] and Advanced LIGO [58].It is difficult to improve the sensitivity of interferom-eters for exotic particles. One reason is that in order toenhance the signal, the mechanical response and the co-efficient of thermal expansion of a mirror must be larger.Equation (60) implies that a smaller mirror and beamyield a larger mechanical response. However, this strat-egy enhances the amplitude of the displacement noise,and the S/N does not increase. A smaller mirror in-creases the radiation-pressure noise ( ∝ R − ). A smallerbeam and a larger coefficient of thermal expansion in-crease the amplitude of the thermal noise caused by ther-moelastic damping in the mirror substrate ( ∝ α/r / )[83]. Although mirror cooling reduces the thermal noise[45, 46, 47], S/N does not become larger, because the ex-citation by an exotic particle becomes smaller than thatat room temperature, as shown in Sec. III A. VII. CONCLUSIONS
We obtained a general formula for a mirror vibrationcaused by a cosmic-ray particle, and studied the effectsin typical cases of interferometric experiments. This for-mula reveals differences in the responses of resonatorsand interferometers against cosmic-ray particles. In thecase of resonators, the contribution of the resonant vi-bration is dominant. On the contrary, in the case ofinterferometers, the motion of the centers of resonant vi-brations must be taken into account. Although the ef-fect of cosmic-ray particles of resonators is independent of the temperature, in the case of interferometers, vibra-tions caused by cosmic-ray particles can be reduced byusing cooling mirrors. In the case of bar resonators, theparticle track position dependence of the vibration by acosmic-ray particle is symmetric with respect to the cen-ter of a resonator, as shown in Fig. 4. On the other hand,in interferometers, larger motion is observed if the trackis near the laser beam spot on the surface of a mirror.The typical vibration amplitude of interferometerscaused by cosmic-ray particles was evaluated. The powerspectrum of vibrations by low-energy cosmic-ray muons(less than 100 GeV) is about 100-times smaller than thegoal sensitivity of the future second-generation projects,such as LCGT and Advanced LIGO. The arrival fre-quency of high-energy cosmic-ray muons that generateenough large showers inside the mirrors of LCGT andAdvanced LIGO is one per a millennium. If a showerthat occurs near a mirror brings more than a thousandparticles to the mirror (an original particle of the showerhas an energy that is more than 1 TeV), the vibrationwill be observed in LCGT and Advanced LIGO inter-ferometers. A detailed study on such shower events isour future work. We also discussed the possibility ofa use of gravitational wave detectors for exotic-particlesearches. Interferometers and bar resonators were com-pared as detectors for such an exotic-particle search. Thecross section of bars is 10-times larger than that of inter-ferometers. The sensitivity of bars for an exotic particleis (30 ∼ Acknowledgments
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