Table Of ContentNoisy Spherical Resonant Detector of Gravitational Waves: Veto on the Longitudinal
Part of the Signal
Enrico Montanari
Department of Physics, University of Ferrara and INFN Sezione di Ferrara, Via Paradiso 12, I-44100 Ferrara, Italy
[email protected]
TheequationsofaresonantsphereininteractionwithN secondaryradialoscillators(transducers)
on its surface have been found in the context of Lagrangian formalism. It has been shown the
possibilitytoexertavetoagainstspuriouseventsmeasuringthelongitudinalcomponentofasignal.
Numericalsimulationshasbeenperformed,whichtakeintoaccountthermalnoisebetweenresonators
and thespheresurface, for a particular configuration of thetransducers.
PACS numbers: 04.20, 04.30.Nk, 04.80.Nn
8
9
9 I. INTRODUCTION
1
n The idea of a spherical detector of gravitational waves has been considered almost since the early seventies [1].
a Very accurate calculations have been performed [2] in order to use a self–gravitating object as a possible detector.
J Basing upon these results other authors [3] have solved the inverse problem for a free sphere in interaction with a
5 gravitational wave. In doing so they have shown an omnidirectional sensitivity and the possibility of testing the
presence of a longitudinal term in the gravitational signal. However a free spherical resonator is not a good device.
1 Infactasetofsecondarymechanicaloscillators(transducers)is neededinordertoconvertthe mechanicalsignalinto
v
anelectromagneticone(see[4,Thorne,p. 330]). Inthesphericalgeometryalmost5transducersareneededto obtain
6
all the informations contained in the motion of its 5 quadrupolar modes. The problem of the interaction of a sphere
0
0 with these secondary resonators has been solved in some particular configuration only in the nineties [5–7]. In the
1 configurationsenvisagedby these authorsthe omnidirectionalsensitivity andthe possibilityoftotalreconstructionof
0 the signalhasbeenshown,inthis wayrealizingacomplete observatorywithonlyone device(cfr [8–10]). However,in
8 the calculationsperformeduntilnow,notenoughattentionhas beenpaidto the possibility ofvetoesagainstspurious
9 events(seeforinstance[11])andtothe influenceofthermalnoiseinthereconstructionofthesignal. Inthispaperwe
/
c have focus our attention upon these two problems, solving them explicitly for the dodecahedron arrangement of the
q transducers first described by [5]. Moreover a Lagrangian which describes the interaction of a sphere with N radial
- transducersonthe surface has beenwritten. This is importantinorderto write the mostgeneralequations involving
r
g the motion of the modes of the sphere and of the secondary oscillators. In this way it will be possible to undertake
: a systematic study of other transducers’s configurations, enabling spherical detectors to improve their capabilities in
v
i the framework of gravitationalastronomy [11,12].
X
r
a
II. LAGRANGIAN OF A FREE ELASTIC BODY INTERACTING WITH A GRAVITATIONAL WAVE
An elastic body in interaction with a gravitationalwave could be described by ordinary elastic theory if its dimen-
sions are very small compared to the gravitational wavelength. Gravitation acts as an external field of density force
(e.g. [3])
fi =ρRi =ρE (t)xj (2.1)
ij
whereE arethecomponentsofthe“electric”partoftheRiemanntensorR . Ifu(x,t)representsthedisplacement
ij i0j0
from the equilibrium position in the point x then (e.g. [13])
∂2u
ρ =µ ∇2u+(λ+µ) ∇(∇ u)+ρR (2.2)
∂t2 ·
Using the standard approach the solution of (2.2) could be decomposed as
u(x,t)= A (t)ψ (x) (2.3)
n n
n
X
in which ψ (x) are the eigenfunctions with eigenvalue ω of
n n
1
µ (λ+µ)
∇2ψ (x)+ ∇(∇ ψ (x))= ω2ψ (x) (2.4)
ρ n ρ · n − n n
The eigenfunctions ψ (x) are a complete set of orthonormal functions with the scalar product defined as ( means
n ∗
complex conjugation)
1
<g(r,θ,φ)f(r,θ,φ)>= g∗ fρd3x (2.5)
| M ·
Z
where M is the mass of the elastic body. Therefore
1
M ψ∗n·ψn′ρd3x=δnn′ (2.6)
Z
Time dependent coefficients A (t) satisfies the following forced harmonic oscillator equations
n
A¨ +ω2 A =R (2.7)
n n n n
where
1
Rn(t)= M ψ∗n(x)·R(x,t)ρd3x (2.8)
Z
In this way a field equationhas been transformedin a set ofequations for the coefficients of a function series. Now
we can write the Lagrangianfrom which these equations are derived. This Lagrangianshould be a real expressionof
the generalized coordinates A (t) and A (t). As far as kinetic energy of the elastic body (EB) is concerned, one has
n ∗n
1 1
T = d3xρu˙2 = M A˙ A˙ (2.9)
EB 2 2 n ∗n
Z n
X
wherelastequalityfollowsfrom(2.6). Potentialenergyofboththeelasticbodyandtheexternalforcesareimmediately
written as
1
V = M ω2A A (2.10)
EB 2 n n ∗n
n
X
1
V = M [A R +A R ] (2.11)
ext −2 n n∗ ∗n n
n
X
III. LAGRANGIAN OF THE FREE SPHERE
If the elastic body is a sphere with radius R, then (see [2])
u(r,θ,φ,t)= A (t)ψ (r,θ,φ) (3.1)
nml nml
nml
X
where
ψ (r,θ,φ)=[a (r)n+b (r)R∇]Y (θ,φ) (3.2)
nlm nl nl lm
dj (q r) l(l+1)
l nl
a (r) =c R +d R j (k r) (3.3)
nlm nlm nlm l nl
dr r
d
b (r) =c j (q r)+d (rj (k r)) (3.4)
nlm nlm l nl nlm l nl
dr
and A (t)=( 1)mA (t).
Nownlwme briefl−y sketch∗nl−thme procedure used by [3] to take into account direction of the gravitational signal. In the
laboratoryframeO, withorigininthecentreofthe sphere,the gravitationalwaveis comingbyagivendirection. Let
us consider a reference frame O rotated with respect to O in such a way that z axis coincides with the direction of
′ ′
2
propagation. All the primed quantities refer to this last system. The radial component of the surface displacement
in the primed frame is then given by (see [3, eq. (7)])
n′·u′(R,θ′,φ′)= anl(R)A′nlm′(t)Ylm′(θ′,φ′) (3.5)
nlm′
X
Weareinterestedintheradialdisplacementinthelaboratoryframe. LetussupposethatO isrotatedwithrespectto
O by the hour angle H (the angle between the projection of z axis on the plane xy and x axis) and the declination
′ ′
δ (the angle between z axis and the plane xy, diminished by 2π if obtuse) [3]. In O we have
′
n·u(R,θ,φ)=nlmanl(R) m′ A′nlm′Dm(l)m′(H,δ)!Ylm(θ,φ) (3.6)
X X
where (l) (H,δ) are the rotation matrix for spherical harmonics (see e.g. [14]). Let us define
Dmm′
Fnlm = A′nlm′Dm(l)m′(H,δ) (3.7)
m′
X
which are the coefficients of decomposition (3.1) in O. These coefficients generalizes in a natural way coefficients F
m
(l)
as defined in [3, eq. (9)], for every order in indices n and l. More precisely one has F = F . (H,δ) are
m 12m Dmm′
unitary matrices, therefore it is possible to invert eq. (3.7) and find A as linear combinations of F
′nlm′ nlm
A′nlm′ = Dm(l)m∗′(H,δ)Fnlm (3.8)
m
X
The kinetic energy of the sphere (S) as a function of the generalized coordinates in the laboratory frame will be
given by (see (2.9))
1 1
TS = 2M A˙′nlm′A˙′n∗lm′ = 2M F˙nlmF˙n∗lm (3.9)
nlm′ nlm
X X
As far as the elastic potential energy of the sphere and the potential energy of the external gravitational force are
concerned one has
1 1
VS = 2M ωn2lA′nlm′A′n∗lm′ = 2M ωn2lFnlmFn∗lm (3.10)
nlm′ nlm
X X
1 1
Vext =−2M [A′nlm′Rn∗lm′ +A′n∗lm′Rnlm′]=−2M [G∗nlmFnlm+GnlmFn∗lm] (3.11)
nlm′ nlm
X X
where we have set
Gnlm = Dm(l)m′Rnlm′ (3.12)
m′
X
which are the coefficients of the decomposition of the gravitational force in O.
IV. COUPLING WITH TRANSDUCERS
A. General Case
Let us consider a set of N harmonic mechanical oscillators with mass m and resonant frequency ω , placed on
i i
the surface of the sphere in (R,θ ,φ ). Let q (t) be the generalized coordinates which describe radial motion of each
i i i
oscillator with respect to the surface of the sphere. We make the assumption that m are small with respect to
i
M, the mass of the sphere. In this way we suppose that the presence of secondary oscillators does not modify the
eigenfunctions of the sphere (in the same way one suppose that the transducer of a resonantbar does not modify the
eigenfunction of the cylinder). Purpose of this section is to write that part of Lagrangian relative to the motion of
the resonators and to their interaction with the surface of the sphere.
3
Concerningthepotentialenergyofthetransducers(t),thechoiceofgeneralizedcoordinatesbringstotheexpression
1
V = m ω2q2 (4.1)
t 2 i i i
i
X
Moreover,ifwecally thecoordinatesofthei–thoscillatorinthelaboratoryframe,thenkineticenergyoftransducers
i
is written as
1
T = m y˙2 (4.2)
t 2 i i
i
X
The relation between inertial and generalized coordinates is given by
(3.6),(3.8)
q =y n u(R,θ ,φ ,t) = y a (R)F Y (θ ,φ ) (4.3)
i i i i i nl nlm nlm i i
− · −
nlm
X
For convenience purposes, we define, for every l, the matrix P(l), whose components are
P(l) =Y (θ ,φ ) (4.4)
mi l∗m i i
The connection between these matrices and the matrix B of ref. [5] is given by B = P(2)∗ With these positions the
kinetic energy of the transducers is
1
Tt = 2 mi q˙i2+q˙i anl(R) Pm(l)i ∗F˙nlm+Pm(l)iF˙n∗lm +
Xi (cid:20) nXlm (cid:16) (cid:17)
anl(R)an′l′(R)Pm(l)i ∗Pm(l′′)iF˙nlmF˙n∗′l′m′ (4.5)
nlmn′l′m′ (cid:21)
X X
In conclusion the Lagrangian of an elastic sphere with N mechanical oscillators on its surface, undergoing an
external force is
L=T +T V V V (4.6)
S t S t ext
− − −
where terms involved have been defined in (3.9), (4.5), (3.10), (4.1) and (3.11) respectively. Generalized coordinates
are F , F and q , where l = 1,2,... , m = l, l+1,...,l , and i = 1,2,..,N . Since F = ( 1)mF
it is snulffimciennt∗lmto findiLagrangian e{quation}s from{q−an−d, say, F }. Therefor{e, to simp}lify the prnolb−lmem, w−e can tna∗lkme
i lm∗
suitablelinearcombinationsofthecoefficientsF insuchawaytoobtainrealquantities. Inparticular,foreverylit
nlm
existsa unitaryoperatorU(l) whichtransformsirreducibletensorsofrankl inrealvectorshaving2l+1components.
We will use lower case letters to indicate those real vectors corresponding to irreducible tensors. One can therefore
write
f =U(l)F (4.7)
nl nl
where f is a vector whose components are f (a 1,2,...,2l+1 ), while F (m l, l+1,...,l ) are the
nl nla ∈{ } nlm ∈{− − }
components of F . From now on indices a, b, c run from 1 to 2l+1. In this way, for instance, components of U(l)
nl
will be given by U(l). A possible choice for such unitary operators is the following
am
(l)
U =δ
1m m0
i
U2(ml)′m =−√2 (δm′m−(−1)mδm′−m) (4.8)
1
U2(ml)′+1m = √2 (δm′m+(−1)mδm′−m)
wherem 1,2,...,l . PerformingtheseunitarytransformationLagrangianisobtainedasafunctionoftherealand
′
∈{ }
independent generalized coordinates f and q . With this new set of variables, kinetic and elastic potential energy
nla i
of the sphere, kinetic and potential energy of the transducers and potential energy of the external gravitationalforce
acting on the sphere become respectively
4
1
T = M f˙2 (4.9)
S 2 nla
nla
X
1
V = M ω2 f2 (4.10)
S 2 nl nla
nla
X
1
T = m q˙2+2q˙ a (R)p(l)f˙ +
t 2 i i i nl ai nla
i (cid:20) nla
X X
+ anl(R)an′l′(R)p(ali)p(al′′i)f˙nlaf˙n′l′a′ (4.11)
nlan′l′a′ (cid:21)
X X
1
V = m ω2q2 (4.12)
t 2 i i i
i
X
V = M g f . (4.13)
ext nla nla
−
nla
X
where the matrix p(l) =(p(l)) is given by
ai
p(l) = U(l)P(l). (4.14)
ai am mi
m
X
Obviously the potential energy of the transducers is left unchanged. Lagrangianequations
d ∂L ∂L
=0
dt(cid:18)∂f˙nla(cid:19)− ∂fnla
d ∂L ∂L
=0
dt ∂q˙ − ∂q
(cid:18) i(cid:19) i
givethereforethefollowingequationsofmotionforthemodesofaforcedelasticsphereininteractionwithN harmonic
mechanical oscillators
Mf¨ +Mω2 f a (R)m ω2p(l)q =Mg
nla nl nla− nl i i ai i nla
i
X
(4.15)
m q¨ +m a (r)p(l)f¨ +m ω2q =0
i i i nl ai nla i i i
nla
X
B. Quadrupolar Interaction
In which follows we make the simplifying assumptions that secondary oscillators are identical, their common fre-
quency is equal to ω (the frequency of the mode n=1, l=2 of the sphere), their mass is m and the external force
12
acting onthe sphere hasonly l=2 components. Moreover,for highquality factors(τ is the relaxationtime ofthe
nlm
nlm mode)
1 n=1
|ω12−ωnl| >> τ τ = min{τ12m,τnlm′}; l=6 2
6
thereforeifthemotionofthe transducersisstudiedinthe frequencydomainnearthe frequencyω ,onlymodes with
12
n=1 and l =2 must be taken into account. To simplify notations we set
p(2) p f f g g ω = ω a (R) α
ai ≡ ai 12a ≡ a 12a ≡ a 12 0 12 ≡
With these assumptions we have
5
1
T = M f˙2 (4.16)
S 2 a
a
X
1
V = Mω2 f2 (4.17)
S 2 0 a
a
X
m
Tt = 2 q˙i2+2q˙i αpaif˙a+ α2paipa′if˙af˙a′ (4.18)
i (cid:20) a aa′ (cid:21)
X X X
mω2
V = q2 (4.19)
t 2 i
i
X
1
V = M g f (4.20)
ext a a
−2
a
X
by means of which the following equations of motions are obtained (t means transpose)
M 0 f¨ Mω2I mω2αp f I 0 Mg
mαIp5t mI q¨ + 00 5 −mω2I0 q = 05 I 0 (4.21)
(cid:18) N(cid:19)(cid:18) (cid:19) (cid:18) 0 N (cid:19)(cid:18) (cid:19) (cid:18) N(cid:19)(cid:18) (cid:19)
where I is the identity matrix in n–dimensional space.
n
If there are forces r(ns) between oscillators and the surface of the sphere (for instance thermal noise forces), then a
i
new potential energy term must be added to the Lagrangian
W = q r(ns) (4.22)
ns − i i
i
X
The equations of motion then become (cfr [7, eq. (25)])
MI 0 f¨ Mω2I mω2αp f I αp Mg
mαp5t mI q¨ + 00 5 −mω2I0 q = 05 −I r(ns) (4.23)
(cid:18) N(cid:19)(cid:18) (cid:19) (cid:18) 0 N (cid:19)(cid:18) (cid:19) (cid:18) N (cid:19)(cid:18) (cid:19)
The coefficient α = a (R) can be calculated from (3.3) solving eigenfunction equations (2.4). One find that α =
12
2.886.
−
Nowthesystemof5+N coupledoscillatorsmustbedecoupledthusfindingtheeigenfrequencies. Itisimportantto
checkifthefrequencyspreadduetothedecouplingisconsistentwiththeassumptionthatonlymodeswithn=1and
l=2 have been taken into account. Calling ω˜ these eigenfrequencies and τ˜ the relaxationtimes of the eigenmodes,
a a
then the following consistency relation should hold
1 n=1
|ω˜a−ωnl| >> τ τ = min{τ˜a,τnlm} l=6 2
6
Asitwillbeseen,theaboveconditionisfulfilledforhighqualityfactorsandm /M <<1. Toperformsuchaprogram
i
a particular arrangement of transducers must be chosen. It is what we are going to do in the next section.
V. DODECAHEDRON ARRANGEMENT
Now, the problem of the decoupling of the system made by the quadrupolar modes of the sphere interacting with
N harmonic oscillators is to be solved. In order to do so the number and exact position of the mechanical oscillators
on the surface must be known. This choice is of fundamental importance to keep isotropy in the sensitivity. Until
now,twoarrangementsinparticulararestudiedinview ofanexperimentalrealization. The firstone [5,7]isobtained
placing six radial resonators along the directions of a dodecahedron (this arrangement was yet investigated [8–10] in
theframeworkofthestudyofanetworkofresonantgravitationalantennae). Inthesecondone[6]5transducers(four
of which tangential) are placed in such a way that they could be excited by only one mode of the sphere; in this way
decoupling is realized at once. In this paper we restrict ourselves to the first case.
Let us consider therefore the dodecahedron arrangement. Five transducers are placed on the same parallel of the
sphere. Theircommonazimuthangleisθ =arcsin(2/√5);theirpolaranglesaregivenbyφ =2kπ/5(k 1,...,5 ).
k
∈{ }
Last transducer is placed at a pole. In this case matrix p become
6
1 c c c c 0
4 2 2 4
p=r53π −001 −−ssc24 −ssc24 −−ssc42 −ssc42 000 cs22a===csoins(cid:0)12255ππ(cid:1) cs44==csoins(cid:0)4455ππ(cid:1) (5.1)
2 4 − 4 − 2 1 2(cid:0)√3 (cid:1) (cid:0) (cid:1)
a a a a a 5a
− 1 − 1 − 1 − 1 − 1 1
Such a matrix has the following important properties (cfr [5,7])
3
ppt = I ; p =0 (5.2)
5 ai
2π
i
X
Decoupling system (4.23) means solving an eigenvalue problem. The eigenfunctions are the so called normal coor-
dinates, while the eigenfrequencies are called frequencies of free vibration (see [15]). In the case of this arrangement,
normalcoordinates are divided in three set: two quintuplet with eigenfrequencies ω =ω λ and ω =ω λ and a
+ 0 + 0
singlet with frequency ω where − −
0
1 3α2 m
λ =1 µ µ= (5.3)
± ± 2r 2π M
The normal coordinates ζp (p = 1,...,11) are therefore naturally divided in three groups: ζ = (ζ ) p 1,...,5 ,
p
ζ = ζ and ζ = (ζ ) p 7,...,11 . The equations of forced motion are (detailed ca−lculations a∈re{found i}n
6 + i ∈ { }
appendix A)
M
ζ¨ +λ2ω2ζ = λ g
− − 0 − r 2 −
ζ¨+ω2ζ =0 (5.4)
0
M
ζ¨ +λ2ω2ζ = λ g
+ + 0 + 2 +
r
Thereforeifg isgiven,onecanfindthemotionofcoefficientsf andtransducersq . Inparticularbecauseofthe form
a i
of matrix A (see eq. (A9)) we have
1
f = λ ζ +λ ζ
r2M − − + +
(cid:0) (cid:1) (5.5)
π ζ
q = pt λ ζ λ ζ + 1
r3m − −− + + √6m (6×1)
(cid:0) (cid:1)
VI. DECOMPOSITION OF THE GRAVITATIONAL WAVE FORCE
A gravitational wave, in the framework of a general metric theory of gravity, is described as a combination of six
matrices. A possible choice is the following (see also [16]):
1 0 0
2
E(S) = 0 1 0 (6.1)
√15
0 0 1
1 0 0 0 0 1 1 i 0
E(0) = 1 −0 1 0 E(±1) = 0 0 i E(±2) = i ±1 0 (6.2)
√3 − ± ± −
0 0 2 1 i 0 0 0 0
±
Matrix (6.1) accounts for the scalar part of the signal, while the others describe the quadrupolar components of the
wave. They are relative to m = 0, m = 1 and m = 2 respectively. One may argue that, once the direction of
± ±
propagationisknown,aspherewithdodecahedronarrangementcoulddetect,atleastinprinciple,allsixcomponents.
However, the scalar mode of the sphere has a principal resonant frequency ω , which is a little more than twice the
10
quadrupolaroneω (e.g.[16]). Currentexperimentalschemesneedsthat, inorderto measurethe scalarmode ofthe
12
7
sphere,this must be coupledwith asecondaryoscillatorhavingω as aresonantfrequency. Therefore,being instead
10
ω the resonant frequency of each transducer, scalar part of the signal cannot be detected. Besides, propagation
12
direction is in general unknown. One is led to the conclusion that with a dodecahedron arrangement it is possible
to get the direction of propagation and three components of the five quadrupolar possible ones. In the framework of
generalrelativity, a gravitationalwave could only have two possible states of polarization, the E(±2) ones. Therefore
one can think that a sphere could be used to test general relativity. However,the discovery of PSR1913+16[17] and
20 years of subsequent observations have put strong experimental limitations on other theory of gravitationdifferent
fromgeneralrelativity. This lastone passedall these new experimentaltests with complete success [18]showing that
“thecorrecttheoryofgravitymustmakepredictionsthatareasymptoticallyclosetothoseofgeneralrelativityovera
vastrangeofclassicalcircumstances”[19]. Thereforeanotherpossibilityistoconsidergeneralrelativityasthecorrect
metric theory of gravity and then to use the degree of freedom left as a veto against spurious signals. This is the
strategy of the present work. In which follows we consider a gravitationalwave in O having the form
′
E′(t)=ER(t)E(+2)+EL(t)E(−2)+El(t)E(0) (6.3)
where E = E and the connection with the usual polarization amplitudes is E (t) = E (t)+E (t) and E (t) =
L R∗ + R L
i(E (t) E (t)), while E is the eventual longitudinal polarization. ×
R L l
−
With this positionit is possible to find, for everyn, the vectorg (whose componentsare g ) whichentersin the
n n2a
equations of motions of the normal modes (5.4) as a function of E′(t) by means of eqs. (4.7), (3.12), (2.8) and (2.1).
Detailed calculations are found in appendix B. Here we give only the final result
γ 1
n
(g ) (t)= (3 cos2δ)cos2H E sinδsin2H E +
n 5 +
√2 4 − − ×
(cid:20)
√3
+ (1+cos2δ)cos2H E
l
4
#
γ 1
n
(g ) (t)= (3 cos2δ)sin2H E +sinδcos2H E +
n 4 +
√2 4 − ×
(cid:20)
√3
+ (1+cos2δ)sin2H E
l
4
#
γ 1 √3
n
(g ) (t)= sin2δcosH E cosδsinHE sin2δcosH E
n 3 + l
√2 2 − ×− 2 !
γ 1 √3
n
(g ) (t)= sin2δsinH E +cosδcosHE sin2δsinH E
n 2 + l
√2 2 ×− 2 !
γ √3 1
n
(g ) (t)= (1+cos2δ)E + (1 3cos2δ)E (6.4)
n 1 + l
√2" 4 4 − #
where γ is defined in appendix B.
n
Ithasbeenthereforeexplicitlyfoundthemotionofmechanicaltransducersonthesurfaceofthesphereasafunction
of directionand polarizationamplitudes of a signal(by means of (5.5), (5.4) and(6.4)), solving the direct problemof
motion.
VII. INVERSE PROBLEM
Inverseproblemliesinthedeterminationoftheincomingsignalcharacteristicsforagivenresponseofthedetection
apparatus. In this case what is measured is the 6 component vector q(t), that is the motion of transducers with
respectto the surfaceofthe sphere. Fromthis vectorone shouldobtainζ(t), describing the motionofnormalmodes,
andg,thegeneralizedforceforthenormalcoordinates. Finally,invertingsystem(6.4)oneshouldgetthosequantities
which describe the signal, that is direction, polarizations E and E and an eventual polarization with m = 1 or
+
m = 0. In which follows we consider only the possibility that the s×ignal could have a longitudinal polarizati±on E
l
(see (6.3)).
In Fourier space (f(ω)= (2π)−1/2 +∞f(t)exp( iωt) dt) using properties (5.2) of matrix p and eq. (5.4) one gets
−
−∞
R
8
2π ω2 ω2 2
g(ω)= 0− ptq(ω) (7.1)
−3α ω2
(cid:0) (cid:1)
This last equation implies, together with (5.2)
γ2
qt(ω)q(ω) gt(ω)g(ω)= 1 E2 +E2 +E2 (7.2)
∝ 2 + × l
(cid:0) (cid:1)
where last equality holds taking into account eqs. (4.7), (3.12) and (B1). Therefore the sum of the squares of the
detector responses are independent on the direction. This means that this transducer arrangement gives isotropic
sensitivity. The result could be also obtained calculating directly the sum of the squares of the components of g
in (6.4).
Inverse problem is then solvedwhen system (6.4) is inverted. To this aim it is useful to define an amplitude vector
a
√2
a= g, a2 =E2 +E2 +E2, (7.3)
γ1 + × l
whose length is the sum of the squares of the three polarization amplitudes of the signal. Performing this program
(results are found in appendix C) one obtains a six-th degree algebraic equation for x = tanH/2 with coefficients
dependingona. Thenarelationfortanδ asafunctionofH anda; finallythreerelationsforE ,E ,E asfunctions
+ l
of H, δ and a. ×
As it will be seen better in the following, since for a gravitational signal E must vanish, only two of the six
l
solutions for H can be accepted. In fact longitudinal components will be zero for only two values of H. These
acceptable solutions differ only because they describe signals coming along opposite directions. In fact the difference
between their hour angles is π. This property holds also for the discarded solutions: they are in pairs differing only
for the opposite direction of propagation.
Solutionof the inverseproblemlies thereforein the solutionofthe equationfor H. This couldbe done numerically
once vector a is known. To this purpose a code has been developed which solves the direct and the inverse problem.
The obtained calculation precisionon the reconstructionof the signal is very much higher than the errorsintroduced
by thermal noise which we are going to consider in next section.
VIII. THERMAL NOISE
Sofarthetheorydealtwithidealdetectors,thatistosaynoiselessones. However,asitiswellknown,realdetectors
arenotatrestbeforetheinteractionwithasignalbut,becauseoftheheatbathinwhichtheyarethermodynamically
in equilibrium, they have a Brownianmotion which give them a certain amount of energy. For this reasona detector
(a part from the quantistic aspects of the problem, aspects which come out indeed when the temperatures are lower
than those actually involvedat the present) couldonly revealsignals yielding an amountof energygreateror at least
comparableto the thermic one. Therefore,foranygiventemperature, therewill be a lowestsignaldetectable. Inthis
section we analyze how the behaviour of the system described until now is to be modified by the presence of thermal
noise.
Tothisaim,theproblemsthatmustbesolvedareatleasttwo. Thefirstoneistounderstandwhichisthenoisethat
one could expect in the transducers on the surface of the sphere at a certain temperature. The second one concerns
the knowledge of the decreasing in the precision with which not only the signal could be deconvolved (by means of
the determinationofthe arrivaldirectionandpolarizationamplitudes)but alsowith whichcouldbe usedthe vetoon
the longitudinal part of the signal. In other words a confidence level must be decided for which a certain value of E
l
(which could not be exactly zero) is to be considered equal to zero or not.
Letusbegintoconsiderthefirstproblemansweringtothequestion: whichnoiseistobeexpectedonthetransducers
at a given temperature? As a first estimate to be compared with the result of a more accurate calculation we may
suppose, if the sphere mass is very much greater than that of the oscillators,that the Browniannoise cause a motion
whose amplitude with respect the surface of the sphere is of the order of
2k T
q(0) = B (8.1)
i s mω02
This result shall be valid as an order of magnitude, allowing settlements which, in general, depend upon the position
oftheoscillatorsonthesurfaceofthesphere. Wewillseethat,inthedodecahedronconfigurationthisresultisindeed
correct being the same for every transducer.
9
Strictlyspeaking,theequipartitionenergytheorem,thatweusedforgivingtheorderofmagnitudeoftheamplitudes
q(0),isvalid,inclassicalmechanics,onlyforeveryindependentquadratictermintheenergyandforeveryoneofthese
i
the square–mean value is (1/2)k T. Therefore, when we consider interacting oscillators we can apply it just to the
B
normal coordinates. It must be observed,however,that the theory developed until now is only an approximation: in
facttogetherwiththelagrangianoftheelasticbodyininteractionwiththesecondaryoscillatorswehavenotconsider
the dissipation function [15, 1–5]. As it is well known it is a quadratic function of the velocities = (1/2) η˙iη˙j
ij
§ F F
and 2 is the rate of energy dissipation due to friction [15, 2.6]. The motion equations become [15, 6.5]
F § §
T η¨j + η˙j +V ηj = 0
ij ij ij
F
It is not in general possible to find normal modes for any arbitrary dissipation function. In other words it is not
possible to decompose the system in non–interacting modes diagonalizing simultaneously the three matrices T , V
ij ij
and . Therefore the equipartition theorem could not be applied even to the normal coordinates. If we make the
ij
F
assumptionofhighqualityfactorwecansupposethatnon–diagonalquadratictermsareverysmallwithrespecttothe
otherssowecansafelyneglectthem. Intheframeworkofthisapproximationwecanthereforeapplytheequipartition
theorem to the normal coordinates, whose Brownian motion will then be given by (see (5.4))
√2kT
ζ = n (t)
− λ ω0 −
−
√2kT
ζ = n(t) (8.2)
ω
0
√2kT
ζ = n (t)
+ λ ω +
+ 0
where n (t), n(t) and n (t) are eleven independent adimensional random functions (where “random” means that
+
their ave−rage values vanish) whose root–mean–square deviations are equal to one. The relative coordinates q(t) are
given as a function of the normal ones ζ(t) by the equation (5.5) which now could be written as
√2kT π n(t)
q = pt (n (t) n (t))+ 1 (8.3)
ω0 (cid:20)r3m − − + √6m (6×1)(cid:21)
Let us now define the following five functions
1
v(t)= (n (t) n (t)) (8.4)
+
√2 − −
Forthesefunctions (fromnowon<f >meansthe averageofagivenquantityf(t))oneobviouslyhasthat<v >=0
(that is to say they are random functions) and <v2 >=1. We can therefore write
a
1 2kT 1
q(t)= √2πptv(t)+ n(t)1 (8.5)
ω0r 3m (cid:20) √2 (6×1)(cid:21)
Let now be w(t) six functions defined as
2
w(t)= √2πptv(t) (8.6)
5
r
As it is easily seen these are random functions and we have also that <w2 >=1. This follows directly from the fact
i
that (ptp) =5/(4π). This property of the matrix p is of fundamental importance in order that all the transducers
ii
have the same noise amplitude. Introducing these new functions we obtain
1 2kT 5 1
q(t)= w(t)+ n(t)1 (8.7)
ω0r 3m "r2 r2 (6×1)#
putting now
1 5 1
z(t)= w(t)+ n(t)1 (8.8)
r3 r2 r2 (6×1)!
10
