Volume 128, number 5 PHYSICS LETTERS A 4 April 1988

ESTIMATE OF THE COSMIC RAY BACKGROUND IN AN INTERFEROMETRIC ANTENNA FOR GRAVITATIONAL WAVE DETECTION

Adalberto GIAZOTTO INFN, Sez. di Piss, Piss, Italy

Received 5 October 1987; revised manuscript received 4 December 1987; accepted for publication 8 February 1988 Communicated by J.P. Vigier

The effect of the cosmic ray interaction in the test masses of an interferometric antenna for gravitational wave (GW) detection is evaluated. In a 3 km antenna this background, ‘mainly due to muons gives a limit, for 1 ms GW pulses, of h - 8.5 x 10 -Z with a frequency of 2~ 10-l events/year and 8.5X 1O-26 with 4.1 X lo6 events/year. For periodic GW having frequency> 10 Hz the sensitivity limit is h - 1.7~ lo-“. This background seems to allow unshielded operation of the interferometer test masses.

It has been experimentally shown [ I] that charged particles traversing an aluminum rod can create me- chanical vibrations whose fundamental mode has amplitude

a! EL2 C(x)=---c0s(7tx/L),

C, M n (1)

where E is the energy lost by the particle in the bar, a! the thermal linear expansion coefficient, C,, the specific heat at constant volume, L and M are the bar length and mass respectively and x ( I xl <L/2) the distance of the particle hit point from the bar center. Amaldi and Pizzella [ 2 ] have shown that cosmic rays can create in a high sensitivity aluminum bar antenna for gravitational waves (GW) detection signals such as to require underground operation. It is then important to examine the effect of this background on the phase of a large base interferometric antenna [ 3 1. Let us consider a cosmic ray hitting one of the interferometers mirrors at an angle 8 (see fig. 1). From the energy-momentum conservation follows:

PM=P, EixE, (2)

where E, P are the cosmic ray energy momentum lost in the mirrors, P,,, is the mirror CMS momentum in

4 Y cosmic

mirror surface

\ry

Fig. 1. The mirror section with a traversing cosmic ray losing the

/ total momentum P, + P2. The momenta difference P, - P2 excites GLd the mirror’s resonance modes.

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Volume 128, number 5 PHYSICS LETTERS A 4 April 1988

the laboratory system, M the mirror mass, Ei the energy inelastically released to the mirror by the cosmic ray. In the second equation the rod CMS and vibrational kinetic energies have been neglected.

If P, and Pz are the momenta lost in the mirror for x> 0 and x< 0 respectively, it follows that the mirror first longitudinal mode is excited by the momentum

p, -p2 AP,= __ ( > 2 x.

For the sake of simplicity we can consider the mirror to be a simple harmonic oscillator, composed by a spring of stiffness K/2 and two masses M/2, having circular frequency o. = m and suspended in its CMS by means of a wire, whose pendular circular frequency is w,. The mirror surface displacement Ax due to the cosmic ray interaction is

A_x=~(x) exp[ - (t-&)/r01 sin[o,(t-t,)]B(t-t,)+~ I ew[-(t--v)/~~l sin[w(t--rl)lF, (rl) du

0 0

1 c - exp[--(~-rl)/~,l sin[o,(t-rl)lfi(q) dq,

+‘Mw, o I (4)

where F, and F2 are the impulsive forces

F, =mxS(t-tn ), FZ = (&)x&f-&) ,

and t, is the cosmic ray arrival time.

(5)

We can maximize Ax putting U,- ( PM)X- E/c (c is the speed of light) and x=0; with these conditions from eqs. (4) and (5 ) we obtain

A.% r(o)+& K > ew[--(t-t,)/~~l sin[~o(t-t,)l 0

+ &exp[-_(l--l,)lr,l sintw,(t-t,)l[8(t-t,)l. P

(6)

The AX Fourier transform, in the approximation ~,t, B 1, and wore B 1, is

Ax(s2, t,)&Yexp(iQt,) [(

ZaLw, 1 1 1 ~ C,Mx +z > -sZ2+2iS/ro+o~

+L 1 CM -Q2+2iQ/rP +og ’ (7)

For a quartz mirror having [3] M=400 kg, LcO.6 m, (r/C,=7x lo-lo kg “C/J, ~~=w~/2n=5x lo3 Hz, vP=wP/27c=0.5 Hz, eq. (6) gives

hx(Q, t,)=exp(iQt,)E (

3x10-* 8x lo-l2

> -.Q2+2is2/ro+w~ -Q2+2iQ/rP+wi ’ (8)

Since eq. (8 ) has poles for 52~ w. and 52~ w, we limit the frequency interval to the region wP < Qn< w,; however this is not a relevant limitation due to the high value of vo. Two experimental conditions are particularly im- portant; the search for GW impulsive and periodical signals.

For the first case we can compared the Fourier transform S(Q) of the mirror displacement produced by an impulsive GW signal having time length At, with Ax(sL, t,).

For QR< 1 /At, I S(Q) 1 becomes

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Volume 128, number 5 PHYSICS LETTERS A 4 April 1988

where A is the interferometer arm length and h the GW amplitude. The measurability condition for h is

3x10-~+8x10-1*

4 .@ > , W,<Q<O~ . (10)

We can choose a frequency, such that the GW impulse area is only slightly reduced, for example Q= 1 / 1 OAt;

with this value h becomes

h>1.6~10-~+

Assuming A=3x lo3 m, At= lop3 s, we obtain

(11)

h>8.5~ 10-26Eo,, , (12)

where EGeV is E measured in GeV. Eq. ( 12) shows that a cosmic ray event giving in the antenna pulses com- parable with those expected by the Virgo Cluster (h N 10 -*I ), should release 1 O4 GeV. These events are mainly produced by muons [ 21 releasing into the mirrors almost entirely their incident energy by means of four pro- cesses: knock-on electrons [ 41, bremsstrahlung [ 5 1, direct pair production [ 41 and photo nuclear interaction

[61. The number N of events per second which deposit into the mirror an energy > E is evaluated by means of

the double integral:

(13)

where p= 2.2 x lo3 kg/m3 is the mirror density, N, the Avogadro number, S- 3.6 X 10-I m2 the mirror proj- ected area, Ai and Zi are the atomic weight and number of the ith atomic component of the mirror, 0, is the cross section of the jth process, Wand E, are the deposited and the incident muon energies respectively, E,( W) is the minimum muon energy necessary to release the energy W by means of the jth process and dZ( E,) /dE, is the muon differential intensity spectrum at sea level [ 71 integrated over the solid angle.

In table 1 the sensitivity limits on h for 1 ms pulses for a few relevant values of EGeV together with N(ev/s) and NY (ev/year) are given. For periodical GW of circular frequency L&n,= 27ryg, ] S(Q) ] becomes

IS(s;$)l=+, (14)

where T is the measurement time. Comparing eq. ( 13) with eq. (8) we obtain the measurability condition

3x lo-‘+8x lo-‘*

w: a,2 >I T ew(iQt,)W, , q-Qg<oo, (15)

Table 1

E Ge” h ( 1 ms pulses) N (w/s)

1 8.5x 10 -26 1.4x 10 -’ 4.1x106

10 8.5X 10 -25 1.9x 10 --3 5.7x 10 4

lo* 8.5x 10 -B 7.2x 10 -6 2.3~10’

10 3 8.5x 10 -*l 7 x10-9 2 x10-l

lo4 8.5x 10 --22 7 x10-14 2 x10-6

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where W,, is the energy of the nth cosmic ray and C, is extended t, is a random variable we obtain

U*= T exp(iQt,) W, 2=x WfAni ) I

4 April 1988

to cosmic ray events having 0 < t, < T. Since

(16)

where Ani is the number of cosmic ray events having energy between Wi and W,, , . Substituting the sum with the integral we obtain

(17)

where W, is a low energy cut off that we have assumed to be equal to the total ionization energy loss in the mirror due to minimum ionizing particles i.e. E, -0.26 GeV. This choice is justified by the fact that energy losses lower than W, are taken into account by the ionization losses.

A numerical computation of eq. ( 17 ) gives U= 1.3 x 1O-6 J for T= 3 x 10’ s; with this value we obtain from

eq. (15)

h>L AT

3x lo-‘+8x lo-‘*

0s Q,z > Uz0.6~ l&-31 , (18)

where we have assumed vg= 10 Hz. The ionization losses can be evaluated putting in eq. ( 16) Wi=EI; since the charged cosmic ray flux at sea

level is [ 8 ] I, = 2.4 x 1 O2 ( m2 s) - ’ we obtain, assuming T= 3 X 10’ s, U= 2.2 x 10e6 J. With this value, eq. ( 15 ) evaluated at v,= 10 Hz, gives

h> 1.1 x 10-J’ . (19)

The sum of eqs. ( 18 ) and ( 19 ) gives, for periodic GW having vg > 10 Hz, the limit h > 1.7 x 10 -‘I, well below the sensitivity of the future interferometric antennas.

I am very grateful to G. Pizzella for having raised the problem, and to L. Bracci for illuminating discussions

References

[ I] A.M. Grassi Strini, G. Strini and G. Tagliaferri, J. Appl. Phys. 51 ( 1980) 948. [ 2 ] E. Amaldi and G. Pizzella, Nuovo Cimento 9 ( 1986) 6 12;

F. Ricci, Nucl. Instrum. Methods A 260 ( 1987) 49 1, [ 31 Antenna Interferometrica a grande base per la rivelazione di onde gravitazionali, Italian-French Collaboration, INFN PI/AE 87/ 1,

Pisa, 12 May 1987. [4] B. Rossi, High energy particles (Prentice-Hall, Englewood Cliffs, 1952). [ 51 A.A. Petrukhin and V.V. Shestakov, Can. J. Phys. 46 (1968) S377. [ 6 ] L.B. Bezrukov and E. Bugaev, Sov. J. Nucl. Phys. 33 ( 198 1) 635. [ 71 O.C. Allkofer and P.K.F. Grieder, Physics data, Fachinformationszentrum Energie-Physik-Matematik GMBH Karlsruhe, ISSN

0344-8407, No. 25-1 (1984). [ 81 Particle Data Group, M. Roos et al., Review of particle properties, Phys. Lett. B 111 ( 1982) 1.

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