Gravitational WavesTheory, Sources and Detection

Kostas Kokkotas

June 26, 2007

Kostas Kokkotas Gravitational Waves

GW: Linear Theory I

Weak gravitational fields can be represented by a slightly deformedMinkowski spacetime :

gµν ' ηµν + hµν + O(hµν)2 (1)

here hµν is a small metric perturbation. The indices will be raised andlowered by ηµν i.e.

hαβ = ηαµηβνhµν (2)

gµν = ηµν − hµν (3)

While Christoffel symbols and Ricci tensor will be given by the relations:

Γλµν =

1

2ηλρ (hρν,µ + hµρ,ν − hµν,ρ) (4)

Rµν = Γαµν,α − Γα

µα,ν =1

2

(hα

ν,µα + hαµ,να − hµν,α

α − hαα,µν

)(5)

R = ηµνRµν = hαβαβ − hα

α,β,β (6)

Finally, Einstein tensor gets the form:

Gµν =1

2

(hα

ν,µα + hαµ,να − hµν,α

α − hαα,µν

)− ηµν

(hαβ

,αβ − hαα

,β,β

)(7)

Kostas Kokkotas Gravitational Waves

GW: Linear Theory II

If we make the substitution 1:

hµν = φµν −1

2ηµνφ (8)

Einstein’s equations reduce to (how?):

−φµν,α,α − ηµνφαβ

,αβ + φµα,α,ν + φνα

α,µ = 0 (9)

Then by using the so called Hilbert gauge similar to Lorentz gauge in EM

φµα,α = φµα

,α = 0 (10)

we come to the following equation:

φµν,α,α ≡ φµν ≡

„∂2

∂t2−∇2

«φµν = 0 (11)

which is a simple wave equations describing ripples of spacetime propagatingwith the speed of light (why?). These ripples are called gravitational waves.

1Notice that φµν is traceless (why?)Kostas Kokkotas Gravitational Waves

GW: about Gauge conditions

By careful choice of coordinates the linearized Einstein equations can besimplified. We can fix ηµν = diag(−1, 1, 1, 1) can make small changes in thecoordinates that leave ηµν unchanged but induce small changes in hµν . Forexample lets consider a coordinate change of the form:

x ′α = xα + ξα (12)

where ξα are 4 small arbitrary functions of the same order as hµν . Then in thenew coordinate system we get

h′µν = hµν − ξµ,ν − ξν,µ (13)

This transformation is called gauge transformation . Since ξµ are 4 smallarbitrary functions we can choose them to simplify the form of the transformedhµν . For example the linearized form of Ricci tensor is:

Rµν =1

2[−hµν + Aµ,ν + Aν,µ] = 0 where Aµ = hλ

µ,λ −1

2hλ

λ,µ (14)

Then we need to find such conditions in the new coordinate system x ′α thatsimplify Rµν . The simplest is to get A′µ = 0 which leads to R ′µν = − 1

2h′µν=0.

Kostas Kokkotas Gravitational Waves

GW: about Gauge conditions II

In other words Einstein equations in linearized gravity for empty space

Rµν = 0

becomeshµν = 0 (15)

together with the gauge conditions

Aµ = hλµ,λ −

1

2hλ

λ,µ = 0 (16)

This last condition in GR is usually called Hilbert gauge condition and it issimilar to the Lorentz gauge condition in electromagnetism that is

Bµ,µ = 0

Kostas Kokkotas Gravitational Waves

GW: Properties I

GWs are periodic changes of spacetime curvature and for weak gravitationalfields far away from sources they described by a simple wave equations whichadmits a solution of the form:

φµν = Aµν cos (kαxα) , (17)

where Aµν is a symmetric tensor called polarization tensor includinginformation of the amplitude and the polarization properties of the GWs. Thekα is the wave-vector. This solution satisfies Hilbert’s gauge condition, that is:

0 = φµν,ν = −Aµνkν sin (kαxα)

which lead to the orthogonality condition

Aµνkν = 0 . (18)

while from the wave equation (11) we get

0 = φµν,α

,α = −Aµνkαkα cos (kαxα) ⇒ kαkα = 0. (19)

This relation suggests that the wave vector kα is null i.e. gravitational wavesare propagating with the speed of light.

Kostas Kokkotas Gravitational Waves

GW: Properties II

Based on (18), (19) the gauge choice that only the spatial components arenon-zero i.e. φµ0 = 0 and the symmetry of the polarization tensor we get for aGW propagating in the z direction i.e. it has a wave vector of the formkµ = (ω, 0, 0,−ω) where k0 = ω is the frequency of the wave that :

Aµν = h+εµν+ + h×εµν

× (20)

where εµν+ and εµν

× are the unit polarization tensors given by the relations

εµν+ ≡

0BB@0 0 0 00 1 0 00 0 −1 00 0 0 0

1CCA εµν× ≡

0BB@0 0 0 00 0 1 00 1 0 00 0 0 0

1CCA (21)

While h+ and h×, are the amplitudes of the gravitational waves in the twopolarizations.The GWs described in this spacific gauge are Transverse and Traceless, and wewill use the notation hTT

µν .

Kostas Kokkotas Gravitational Waves

GW: Effects...

We will study the effect of GWs on particles.A static or slowly moving particle has velocity vector uµ ≈ (1, 0, 0, 0) and onecan assume that τ ≈ t. Then in linearized gravity the geodesic equation will bewritten as:

duµ

dt= −1

2(hµα,β + hβµ,α − hαβ,µ) uαuβ (22)

leading toduµ

dt= −

„hµ0,0 −

1

2h00,µ

«. (23)

If we now use the T-T gauge (h00 = hµ0 = 0) we conclude that GWs do notaffect isolated particles!If instead we consider a pair of test particles on the cartesian axis Ox being atdistances x0 and −x0 from the origin and we assume a GW traveling in thez-direction then their distance will be given by the relation:

d`2 = gµνdxµdxν = . . .

= −g11(dx)2 = (1− h11)(2x0)2 = (1− h+ cos ωt) (2x0)

2 (24)

or approximatelly

∆` ≈„

1− 1

2h+ cos ωt

«(2x0) . (25)

Kostas Kokkotas Gravitational Waves

GW: Effects...

In a similar way we can show for two particles on the Oy axis that:

∆` ≈„

1 +1

2h+ cos ωt

«(2y0) . (26)

In other wards the coordinate distance of two particles is varying periodicallywith the time

Figure: The effect of a travelling GW on a ring of particles

Kostas Kokkotas Gravitational Waves

GW: Effects...

Figure: The effect of a travelling GW on a ring of particles

Kostas Kokkotas Gravitational Waves

GW: Tidal forces

Riemann tensor is a ”measure” of spacetime’s curvature and in linearizedgravity gets the form

Rκλµν =1

2(∂νκhλµ + ∂λµhκν − ∂κµhλν − ∂λνhκµ) , (27)

in the T-T gauge the Riemann tensor is considerably simplified

RTTj0k0 = −1

2

∂2

∂t2hTT

jk , for j , k = 1, 2, 3. (28)

Actually, the Newtonian limit of the Riemann tensor is:

RTTj0k0 ≈

∂2U

∂x j∂xk, (29)

where U is the Newtonian potential. In other words the Riemann tensor hasalso a pure physical meaning i.e. it is a measure of the tidal gravitationalacceleration. Then the distance between two nearby particles xµ(τ) willxµ(τ) + ξµ(τ) will be described by

d2ξk

dt2≈ −Rk

0j0

TTξj . (30)

Kostas Kokkotas Gravitational Waves

GW: Tidal forces II

The tidal force acting on a particle is

f k ≈ −mRk0j0ξ

j ≈ m

2

d2hTTjk

dt2ξj (31)

where m is particle’s mass. This means that

f x ≈ m

2h+ω2 cos[ω(t − z)]ξx

0 , and f y ≈ −m

2h+ω2 cos[ω(t − z)]ξy

0 . (32)

∇~f =∂f x

∂ξx0

+∂f y

∂ξy0

= 0 . (33)

Hence the divergence of the force ~f is zero, which tell us that the tidal forcecan be represented graphically by field lines.

Figure: The tidal field lines of force for a gravitational wave with polarization (+) (left panel) and (×) (right panel). The orientationof the field lines changes every half period producing the deformations as seen in Figure 1. Any point accelerates in the directions of thearrows, and the denser are the lines, the strongest is the acceleration. Since the acceleration is proportional to the distance from the center

of mass, the force lines get denser as one moves away from the origin. For the polarization (×) the force lines undergo a 450 rotation.

Kostas Kokkotas Gravitational Waves

GW: Properties

GWs, once they are generated, propagate almost unimpeded. Indeed, they areeven harder to stop than neutrinos! The only significant change they suffer asthey propagate is the decrease in amplitude while they travel away from theirsource, and the redshift they feel (cosmological, gravitational or Doppler).GWs are fundamentally different, however, even though they share similar waveproperties away from the source.GWs are emitted by coherent bulk motions of matter (for example, by theimplosion of the core of a star during a supernova explosion) or by coherentoscillations of spacetime curvature, and thus they serve as a probe of suchphenomena. By contrast, cosmic electromagnetic waves are mainly the resultof incoherent radiation by individual atoms or charged particles. As aconsequence, from the cosmic electromagnetic radiation we mainly learn aboutthe form of matter in various regions of the universe, especially about itstemperature and density, or about the existence of magnetic fields. StrongGWs are emitted from regions of spacetime where gravity is very strong andthe velocities of the bulk motions of matter are near the speed of light.Since most of the time these areas are either surrounded by thick layers ofmatter that absorb electromagnetic radiation or they do not emit anyelectromagnetic radiation at all (black holes), the only way to study theseregions of the universe is via gravitational waves.

Kostas Kokkotas Gravitational Waves

GW: Energy Flux

GWs carry energy. The stress-energy carried by GWs cannot be localizedwithin a wavelength. Instead, one can say that a certain amount ofstress-energy is contained in a region of the space which extends over severalwavelengths. The stress-energy tensor can be written as:

tµν =1

4

»2φαβ,µφαβ

,ν − φ,µφ,ν − ηµν

„φαβ,σφαβ,σ −

1

2φ,σφ,σ

«–(34)

which in the TT gauge of the linearized theory becomes

tGWµν =

1

32π〈“∂µhTT

ij

” “∂νhTT

ij

”〉. (35)

where the angular brackets indicate averaging over several wavelengths. For thespecial case of a plane wave propagating in the z direction, the stress-energytensor has only three non-zero components, which take the simple form

tGW00 =

tGWzz

c2= − tGW

0z

c=

1

32π

c2

Gω2

“h2

+ + h2×

”, (36)

where tGW00 is the energy density, tGW

zz is the momentum flux and tGW0z the

energy flow along the z direction per unit area and unit time .

Kostas Kokkotas Gravitational Waves

GW: Energy Flux

The energy flux has all the properties one would anticipate by analogy withelectromagnetic waves: (a) it is conserved (the amplitude dies out as 1/r , theflux as 1/r 2), (b) it can be absorbed by detectors, and (c) it can generatecurvature like any other energy source in Einstein’s formulation of relativity.

Estimate the energy flux in GWs from the collapse of the core of a supernovato create a 10 M black hole at a distance of ∼15 Mpc from the earth (at thedistance of the Virgo cluster of galaxies). A conservative estimate of theamplitude of the GWs on earth is of the order of h ≈ 10−22 (at a frequency ofabout 1kHz). This corresponds to a flux of about 3 ergs/cm2 sec. This is anenormous amount of energy flux and is about ten orders of magnitude largerthan the observed energy flux in electromagnetic waves!

The basic difference is the duration of the two signals; GW signal will last a fewmilliseconds, whereas an EM signal lasts many days. This example provides uswith a useful numerical formula for the energy flux:

F = 3

„f

1kHz

«2 „h

10−22

«2ergs

cm2sec, (37)

from which one can easily estimate the flux on earth, given the amplitude (onearth) and the frequency of the waves.

Kostas Kokkotas Gravitational Waves

GW: Nature of

EM radiation emitted by slowly varying charge distributions can be decomposedinto a series of multipoles, where the amplitude of the 2`-pole (` = 0, 1, 2, ...)contains a small factor a`, with a equal to the ratio of the diameter of thesource to the typical wavelength, namely, a number typically much smaller than1. From this point of view the strongest EM radiation would be expected formonopolar radiation (` = 0), but this is completely absent, because the EMmonopole moment is proportional to the total charge, which does not changewith time (it is a conserved quantity). Therefore, EM radiation consists only of` ≥ 1 multipoles, the strongest being the electric dipole radiation (` = 1),followed by the weaker magnetic dipole and electric quadrupole radiation(` = 2).For GWs, it can be shown that mass conservation (which is equivalent to chargeconservation in EM theory) will exclude monopole radiation. Also, the rate ofchange of the mass dipole moment is proportional to the linear momentum ofthe system, which is a conserved quantity, and therefore there cannot be anymass dipole radiation in Einsteins relativity theory. The next strongest form ofEM radiation is the magnetic dipole. For the case of gravity, the change of themagnetic dipole is proportional to the angular momentum of the system, whichis also a conserved quantity and thus there is no dipolar grav. radiation of anysort. It follows that gravitational radiation is of quadrupolar or higher natureand is directly linked to the quadrupole moment of the mass distribution.

Kostas Kokkotas Gravitational Waves

GW: Generation

Einstein (1918) derived the quadrupole formula for gravitational radiation bysolving the linearized form of his equations

φµν(t,~x) = −κTµν(t,~x) . (38)

The solution is:

φµν(t,~x) = − κ

4π

ZV

Tµν (t − |~x − ~x ′|,~x ′)|~x − ~x ′| d3x ′ , (39)

This solution suggests that φij is proportional to the second time derivative ofthe quadrupole moment of the source:

φij =2

r

G

c4QTT

ij (t − r/c) where QTTij (x) =

Zρ

„x ix j − 1

3δij r 2

«d3x (40)

where, QTTij is the quadrupole moment in the TT gauge, evaluated at the

retarded time t − r/c. This result is quite accurate for all sources, as long asthe reduced wavelength λ = λ/2π is much longer than the source size R.Using the formulae (35) and (36) for the energy carried by GWs, one can derivethe luminosity in GWs as a function of the third-order time derivative of thequadrupole moment tensor. This is the quadrupole formula

LGW = −dE

dt=

1

5

G

c5〈∂

3Qij

∂t3

∂3Qij

∂t3〉. (41)

Kostas Kokkotas Gravitational Waves

GW: Order of magnitude estimates

The quadrupole moment of a system is approximately equal to the mass M ofthe part of the system that moves, times the square of the size R of the system.This means that the 3rd-order time derivative of the quadrupole moment is

∂3Qij

∂t3∼ MR2

T 3∼ Mv 2

T∼ Ens

T, (42)

where v is the mean velocity of the moving parts, Ens is the kinetic energy ofthe component of the source’s internal motion which is non spherical, and T isthe time scale for a mass to move from one side of the system to the other.The time scale (or period) is actually proportional to the inverse of the squareroot of the mean density of the system (why?)

T ∼p

R3/GM. (43)

This relation provides a rough estimate of the characteristic frequency of thesystem f = 2π/T . The luminosity of GWs of a given source is approximately

LGW ∼ G

c5

„M

R

«5

∼ G

c5

„M

R

«2

v 6 ∼ c5

G

„RSch

R

«2 “v

c

”6

(44)

where RSch = 2GM/c2 is the Schwarzschild radius of the source. It is obviousthat the maximum value of the luminosity in GWs can be achieved if thesource’s dimensions are of the order of its Schwarzschild radius and the typicalvelocities of the components of the system are of the order of the speed of light.

Kostas Kokkotas Gravitational Waves

GW: Order of magnitude estimates II

The above formula sets also an upper limit on the power emitted by a source,which for R ∼ RSch and v ∼ c is:

LGW ∼ c5/G = 3.6× 1059ergs/sec. (45)

This is an immense power, often called the luminosity of the universe.Using the above order-of-magnitude estimates, we can get a rough estimate ofthe amplitude of GWs at a distance r from the source:

h ∼ G

c4

Ens

r∼ G

c4

εEkin

r(46)

where εEkin (with 0 ≤ ε ≤ 1), is the fraction of kinetic energy of the sourcethat is able to produce GWs. The factor ε is a measure of the asymmetry of thesource and implies that only a time varying quadrupole moment will emit GWs.Another formula for the amplitude of GW relation can be derived from the fluxformula (37). If, for example, we consider an event (perhaps a supernovaeexplosion) at the Virgo cluster during which the energy equivalent of 10−4Mis released in GWs at a frequency of 1 kHz, and with signal duration of theorder of 1 msec, the amplitude of the gravitational waves on Earth will be

h ≈ 10−22

„EGW

10−4M

«1/2 „f

1kHz

«−1 “ τ

1msec

”−1/2„

r

15Mpc

«−1

. (47)

Kostas Kokkotas Gravitational Waves

GW: Order of magnitude estimates III

For a detector with arm length of 4 km we are looking for changes in the armlength of the order of

∆` = h · ` = 10−22 · 4km = 4× 10−17!!!

These numbers shows why experimenters are trying so hard to buildultra-sensitive detectors and explains why all detection efforts till today werenot successful.Finally, it is useful to know the damping time, that is, the time it takes for asource to transform a fraction 1/e of its energy into gravitational radiation.One can obtain a rough estimate from the following formula

τ =Ekin

LGW∼ 1

cR

„R

RSch

«3

. (48)

For example, for a non-radially oscillating neutron star with a mass of roughly1.4M and a radius of 12Km, the damping time will be of the order of∼50msec. Also, by using formula (43), we get an estimate for the frequency ofoscillation which is directly related to the frequency of the emitted gravitationalwaves, roughly 2kHz for the above case.

Kostas Kokkotas Gravitational Waves

Example: Quadrupole Moment Tensor 3

We will calculate the mass quadrupole moment tensor of a homogeneous

triaxial ellipsoid x2

a2 + y2

b2 + z2

c2 = 1. By setting x ′ = x/a, y ′ = y/a and z ′ = z/athe volume integration over the ellipsoid reduces to that over the unit sphere

Q11 =

Z Z Zρ

“3x2 − r 2

”dxdydz =

Z Z Zρ

“2x2 − y 2 − z2

”dxdydz

=

Z Z Zρabc

“2a2x ′2 − b2y ′2 − c2z ′2

”dx ′dy ′dz ′

= ρabc“2a2 − b2 − c2

” Z Z Zz ′2dx ′dy ′dz ′

= ρabc“2a2 − b2 − c2

” Z 2π

0

Z π

0

Z 1

0

r 4dr cos2 θ sin θdθdφ

=m

5

“2a2 − b2 − c2

”(49)

where m = 43πabcρ is the mass of the ellipsoid. The other two non-vanishing

components of the mass quadrupole tensor are 2:

Q22 =m

5

“−a2 + 2b2 − c2

”and Q33 =

m

5

“−a2 − b2 + 2c2

”2By definition, the mass quad. moment tensor is traceless, Qjj = Q11 + Q22 + Q33.3From M. Carmeli, Classical Fields, World Scientific (1982)

Kostas Kokkotas Gravitational Waves

Example: Oscillating Quadrupole

The quadrupole moment tensor of a pair of equal masses m in distance b is:

Q(0)ij ≡

0@ − 23mb2 0 00 − 2

3mb2 0

0 0 43mb2

1A (50)

If the distance between the two masses varies periodically z = ±(b + a sin ωt)then the retarded value of the quadrupole tensor is

Q ij ≈»1 +

2a

bsin ω(t − r)

–Q(0)ij (51)

φij =2

r

G

c4Q ij (t − r) =

2

r

G

c4

2a

bω2 sin ω(t − r)Q(0)ij (52)

LGW = −dE

dt=

1

5

G

c5〈∂

3Q ij

∂t3

∂3Q ij

∂t3〉 =

32

15

G

c5〈mabω3 cos ω(t − r)〉2

≈ 16G

15c5(mab)2 ω6 (53)

and the damping time of the oscillator is :

1

τrad= − 1

E

dE

dt=

16

15

G

c5mb2ω4 where E =

1

2mω2a2 (54)

Kostas Kokkotas Gravitational Waves

GW: Binaries, an example

If we assume that the two bodies m1 and m2 making up the binary lie in thex − y plane at distances a1 and a2 from the center of mass, their orbits arecircular and rotating at angular frequency Ω.

Then the only non-vanishing components of the quadrupole tensor are :

Qxx = −Qyy =“a2

1m1 + a22m2

”cos2 Ωt =

1

2µa2 cos 2Ωt, (55)

Qxy = Qyx =1

2µa2 sin 2Ωt, (56)

where a = a1 + a2, a1m1 = a2m2 = aµ. Here µ = M1M2/M is the reducedmass of the system and M = M1 + M2 its total mass. And the GW luminosityof the system is (we use Kepler’s third law, Ω2 = GM/a3)

LGW = −dE

dt=

1

5

G

c5(2Ω)2

„1

2R2µ

«2

〈sin2 2Ωt + sin2 2Ωt + 2 cos 2Ωt〉

=32

5

G

c5µ2a4Ω6 =

32

5

G 4

c5

M3µ2

a5. (57)

Kostas Kokkotas Gravitational Waves

GW: Binaries, an example

The total energy of the binary system can be written as :

E =

„1

2M1a

21 +

1

2M2a

22

«Ω2 − GM1M2

a= −1

2

GµM

a(58)

As the gravitating system loses energy by emitting radiation, the distancebetween the two bodies shrinks at a rate

dE

dt=

1

2

GµM

a2

da

dt⇒ da

dt= −64

5

G 3

c5

µM2

a3, (59)

and the orbital frequency increases accordingly (T/T = 1.5a/a).If, the present separation of the two stars is a0, then the binary system willcoalesce after a time

τ =5

256

c5

G 3

a40

µM4(60)

Finally, the amplitude of the GWs is (why?)

h = 5× 10−22

„M

2.8M

«2/3 „µ

0.7M

« „f

100Hz

«2/3 „15Mpc

r

«. (61)

In all these formulae we have assumed that the orbits are circular. In general,the orbits of the two bodies are approximately ellipses, but it has been shownthat long before the coalescence of the two bodies, the orbits become circular,at least for long-lived binaries, due to gravitational radiation.

Kostas Kokkotas Gravitational Waves

GW: Binaries, an example

I The amplitude of the emitted GWs depends on the angle between the lineof sight and the axis of angular momentum; formula (61) refers to anobserver along the axis of the orbital angular momentum. The completeformula for the amplitude contains angular factors of order 1. The relativestrength of the two polarizations depends on that angle as well.

I If 3 or more detectors observe the same signal it is possible to reconstructthe full waveform and deduce many details of the orbit of the binarysystem.

I As an example, we will provide some details of the well-studied pulsarPSR 1913+16 (the Hulse-Taylor pulsar), which is expected to coalesceafter ∼ 3.5× 108 years. The binary system is roughly 5kpc away fromEarth, the masses of the two neutron stars are estimated to be ∼1.4Meach, and the present period of the system is ∼7h and 45min. Thepredicted rate of period change is T = −2.4× 10−12sec/sec, while thecorresponding observed value is in excellent agreement with thepredictions, i.e., T = (−2.30± 0.22)× 10−12sec/sec; finally the presentamplitude of gravitational waves is of the order of h ∼ 10−23 at afrequency of ∼ 7× 10−5Hz.

Kostas Kokkotas Gravitational Waves

GW: Binaries, an example: PSR 1913+16

Hulse & Taylor : Nobel 1993

Kostas Kokkotas Gravitational Waves

GW: Known Binary Systems as Sources of GWs

System Masses Distances Frequency Luminosity AmplitudeM pc 10−6 Hz 1030 erg/s 10−22

4 ι Boo (1.0, 0.5) 11.7 86 1.1 51µ Sco (12, 12) 109 16 51 2105 Am CVn (1.0, 0.041) 100 1900 300 5WZ Sge (1.5, 0.12) 75 410 24 86 Cyg X-1 (30,6) 2500 4.1 1.0 4PSR+16 (1.4,1.4) 5000 70 0.6 0.12

4Eclisping Binaries5Cataclysmic Binaries6Binary X-ray sources

Kostas Kokkotas Gravitational Waves

GW Detectors : Resonant I

Suppose that a GW propagating along the z axis with (+) polarization impingeson an idealized detector, two masses joined by a spring along the x axis

The tidal force induced on the detector is given by equation (32), and themasses will move according to the following equation of motion:

ξ + ξ/τ + ω20ξ = −1

2ω2Lh+e iωt , (62)

where ω0 is the natural vibration frequency of our detector, τ is the dampingtime of the oscillator due to frictional forces, L is the separation between thetwo masses and ξ is the relative change in the distance of the two masses. TheGW plays the role of the driving force, and the solution to the above equation is

ξ =12ω2Lh+e iωt

ω20 − ω2 + iω/τ

(63)

If the frequency ω of the impinging wave is near the natural frequency ω0 ofthe oscillator the detector is excited into large-amplitude motions and it ringslike a bell. Actually, in the case of ω = ω0 , we get the maximum amplitudeξmax = ω0τLH+/2 .

Kostas Kokkotas Gravitational Waves

GW Detectors : Resonant II

Since the size of our detector L and the amplitude of the gravitational wavesh+ are fixed, large-amplitude motions can be achieved only by increasing thequality factor Q = ω0τ of the detector. In practice, the frequency of thedetector is fixed by its size and the only improvement we can get is by choosingthe type of material so that long relaxation times are achieved.

Figure: A graph of NAUTILUS in Frascati near Rome. Nautilus is probably the most sensitive resonant detector available.

Kostas Kokkotas Gravitational Waves

GW Detectors : Resonant III

The cross section is a measure of the interception ability of a detector. Forresonance, the average cross section of our test detector, assuming any possibledirection of the wave, is

σ =32π

15

G

c3ω0QML2. (64)

This formula is general; it applies even if we replace our toy detector with amassive metal cylinder.Weber’s first detector. That detector had the following characteristics: MassM=1410 kg, length L=1.5 m, diameter 66 cm, resonant frequency ω0=1660Hz,and quality factor Q = ω0τ = 2× 105. For these values the calculated crosssection is roughly 3× 10−19cm2.The thermal noise is the only factor limiting our ability to detect gravitationalwaves. Thus, in order to detect a signal, the energy deposited by the GW everyτ seconds should be larger than the energy kT due to thermal fluctuations.This leads to a formula for the minimum detectable energy flux of gravitationalwaves, which, following equation (36), leads into a minimum detectable strainamplitude

hmin ≤1

ω0LQ

r15kT

M(65)

For Weber’s detector, at room temperature this yields a minimum detectable

strain of the order of 10−20.Kostas Kokkotas Gravitational Waves

GW Detectors : Resonant IV

In reality, modern resonant bar detectors are consisting of a solid metalliccylinder suspended in vacuo by a cable that is wrapped under its center ofgravity. The whole system is cooled down to temperatures of a few K or evenmK. To monitor the vibrations of the bar, piezoelectric transducers areattached to the bar. The transducers convert the bar’s mechanical energy intoelectrical energy. The signal is amplified by an ultra-low-frequency amplifier, byusing a device called a SQUID (Super-conducting QUantum InterferenceDevice) before it becomes available for data analysis.The above description of the resonant bar detectors shows that, in order toachieve high sensitivity, one has to:

1. Create more massive antennas.

2. Obtain higher quality factor Q. Modern antennas generally use aluminumalloy 5056 (Q ∼ 4× 107).

3. Lower the temperature of the antenna as much as possible. The resonantbar detectors are probably the coolest places in the Universe. Typicalcooling temperatures for the most advanced antennae are below thetemperature of liquid helium.

4. Achieve strong coupling between the antenna and the electronics and lowelectrical noise.

Kostas Kokkotas Gravitational Waves

GW Detectors : Resonant V

They have achieved sensitivities of a few times 10−21, but still there has beenno clear evidence of GW detection. They will have a good chance of detectinga GW signal from a supernova explosion in our galaxy (2-5 events per century).The most sensitive cryogenic bar detectors in operation are:

I ALLEGRO (Baton Rouge, USA) Mass 2296 Kg (Aluminium 5056), length3 m, bar temperature 4.2 K, mode frequency 896 Hz.

I AURIGA (Legrano, Italy) Mass 2230 Kg (Aluminium 5056), length 2.9 m,bar temperature 0.2 K, mode frequency 913 Hz.

I EXPLORER (CERN, Switzerland) Mass 2270 Kg (Aluminium 5056),length 3 m, bar temperature 2.6K, mode frequency 906Hz.

I NAUTILUS (Frascati, Italy) Mass 2260 Kg (Aluminium 5056), length3 m, bar temperature 0.1 K, mode frequency 908 Hz.

I NIOBE (Perth, Australia) Mass 1500 Kg (Niobium), length 1.5 m, bartemperature 5K, mode frequency 695Hz.

There are plans for construction of massive spherical resonant detectors, the

advantages of which will be their high mass, their broader sensitivity (up to

100-200 Hz) and their omnidirectional sensitivity. A prototype spherical

detectors are already in operation in Leiden, Italy and Brazil ( 1 m diameter

and mode frequency ∼3.2 kHz).

Kostas Kokkotas Gravitational Waves

GW Detectors : Laser Interferometers I

A laser interferometer is an alternative GW detector that offers the possibilityof very high sensitivities over a broad frequency band.

Figure: A schematic design of a Michelson interferometer.

Mirrors are attached to M1 and M2 and the mirror attached on mass M0 splitsthe light (beam splitter) into two perpendicular directions. The light isreflected on the two corner mirrors and returns back to the beam splitter. Thesplitter now half-transmits and half-reflects each one of the beams. One part ofeach beam goes back to the laser, while the other parts are combined to reachthe photodetector where the fringe pattern is monitored.

Kostas Kokkotas Gravitational Waves

GW Detectors : Laser Interferometers II

Let us consider an impinging GW with amplitude h and (+) polarization,propagating perpendicular to the plane of the detector 7. Such a wave willgenerate a change of ∆L ∼ hL/2 in the arm length along the x-direction andan opposite change in the arm length along the y -direction. The totaldifference in length between the two arms will be

∆L

L∼ h. (66)

For a GW with amplitude h ∼ 10−21 and detector arm- length 4 km (such as

LIGO), this will induce a change in the arm-length of about ∆L ∼ 10−16. If the

light bounces a few times between the mirrors before it is collected in the

photodiode, the effective arm length of the detector is increased considerably,

and the measured variations of the arm lengths will be increased accordingly.

This is a quite efficient procedure for making the arm length longer. The

optical cavity that is created between the mirrors of the detector is known as a

Fabry-Perot cavity and is used in modern interferometers.

7We will further assume that the frequency is much higher than the resonantfrequency of the pendulums and the wavelength is much longer than the arm length ofour detector

Kostas Kokkotas Gravitational Waves

GW Detectors : Laser Interferometers III

Photon shot noise. When a GW produces a change ∆L in the arm-length, thephase difference between the two light beams changes by an amount∆φ = 2b∆L/λ ∼ 10−9rad 8 for detectable GWs.The precision of the measurements, is restricted by fluctuations in the fringepattern due to fluctuations in the number of detected photons.The number of detected photons, N, is proportional to the intensity of the laserbeam N = N0 sin2(∆φ/2), here N0 is the no of supplied photons. Inversion ofthis equation leads to an estimation of the relative change of the arm lengths∆L by measuring the number of the emerging photons N.Statistical fluctuations in the number of detected photons imply an uncertainty

in the measurement of the arm length δ(∆L) ∼ λ

2b√

N0

Thus, the minimum GW amplitude that we can measure is :

hmin =δ(∆L)

L=

∆L

L∼ λ

bLN1/20

∼ 1

bL

„~cλ

τ I0

«1/2

, (67)

I0: intensity of the laser light (∼5-10W) & τ : the duration of the measurement.In laser interferometer the photon shot noise is the dominant for frequenciesabove 200 Hz, while its power spectral density Sn(f ) for frequencies 100-200Hzis of the order of ∼ 3× 10−23

√Hz .

8λ ∼ 10−8cm: the reduced wavelength of the laser light & b: the number ofbounces of the light in each arm

Kostas Kokkotas Gravitational Waves

GW Detectors : Laser Interferometers IV

Radiation pressure noise: According to (67), the sensitivity of a detector can beincreased by increasing the intensity of the laser. However, a very powerfullaser produces a large radiation pressure on the mirrors. Then an uncertainty inthe measurement of the momentum deposited on the mirrors leads to aproportional uncertainty in the position of the mirrors. Then, the minimumdetectable strain is limited by

hmin ∼τ

m

b

L

„τ~I0

cλ

«1/2

, (68)

where m is the mass of the mirrors.As we have seen, the photon shot noise decreases as the laser power increases,while the inverse is true for the noise due to radiation pressure fluctuations. Ifwe try to minimize these two types of noise with respect to the laser power, weget a minimum detectable strain for the optimal power via the very simplerelation

hmin ∼1

L

„τ~m

«1/2

(69)

which for the LIGO detector (where the mass of the mirrors is ∼100 kg and thearm length is 4 km), for observation time of 1 ms, gives hmin ≈ 10−23.

Kostas Kokkotas Gravitational Waves

GW Detectors : Laser Interferometers V

Quantum limit. An additional source of uncertainty in the measurements is setby Heisenberg’s principle, which says that the knowledge of the position andthe momentum of a body is restricted from the relation ∆x ·∆p ≥ ~. For anobservation that lasts some time τ , the smallest measurable displacement of amirror of mass m is ∆L; assuming that the momentum uncertainty is∆p ≈ m ·∆L/τ , we get a minimum detectable strain due to quantumuncertainties

hmin =∆L

L∼ 1

L

„τ~m

«1/2

. (70)

Surprisingly, this is identical to the optimal limit that we calculated earlier for

the other two types of noise. The standard quantum limit does set a

fundamental limit on the sensitivity of beam detectors. An interesting feature

of the quantum limit is that it depends only on a single parameter, the mass of

the mirrors.

Kostas Kokkotas Gravitational Waves

GW Detectors : Laser Interferometers VI

Seismic noise. At frequencies below 60 Hz, the noise in the interferometers isdominated by seismic noise. The vibrations of the ground couple to the mirrorsvia the wire suspensions which support them. This effect is strongly suppressedby properly designed suspension systems. Still, seismic noise is very difficult toeliminate at frequencies below 5-10 Hz.Residual gas-phase noise. The statistical fluctuations of the residual gas densityinduce a fluctuation of the refraction index and consequently of the monitoredphase shift. For this reason the laser beams are enclosed in pipes over theirentire length. Inside the pipes a high vacuum of the order of 10−9 Torrguarantees elimination of this type of noise.

Figure: Noise.

Kostas Kokkotas Gravitational Waves