Thermal noise in GW detectors

How much can an object be at rest on Earth?

Geppo Cagnolicagnoli@fi.infn.it

INFN - FirenzeUniversity of GlasgowITIS Citta’ di Castello

UTB - Physics & Astronomy - 16 Sept. 2009

16 Sept 2009 UTB - Geppo Cagnoli - Thermal noise in GW detectors 2 of 22

Fixing the problem

• Earth is not an inertial reference frame

• Tidal effects

• Geological movements

Time scale or frequency rangehas to be defined

1Hz to 10 kHz

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Limiting the range

We could live the object on a tablebut

the Earth is noisy

Sfxx ( ). 012

mHz

Uncertainty PrincipleEt = h / 4

For a 10 kg mass:

Sfxx ( ) 10 18 m

Hz

1.E-22

1.E-18

1.E-14

1.E-10

1.E-06

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

[Hz]

m /

sqrt

(Hz)

10 orders of magnitude

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Mechanical Filteringof the Seismic Noise

Connectedto ground

Object

x

yA simple pendulumprovides a goodfiltering above theresonant frequency

20ff

)f(y)f(x

Possible improvements

• Use a spring to filter the vertical noise too

• With a multiple pendulum configuration is possible to fill the10 orders of magnitude

• We could use some damper to reduce the resonant peaks

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Virgo Superattenuator

THE REFERENCE POINT !!

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The Brownian motion

A challenge for the audience: how would you establish that this endless motion is not due to the activity of living organisms?

2 micron particles in water (left) and in concentrated DNA solution (right), 4 s of data

http://www.deas.harvard.edu/projects/weitzlab/research/brownian.html

Botanist Robert Brown, (1773-1858)

In 1832 the botanist Robert Brown observed a random motion of pollen and dust grains suspended in water

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Einstein’s Insight of 1905:A Way to Measure kB

average

diffusivity of big particle

Boltzmann's constant

absolute temperature (in kelvin)

3.14159...

viscosity of fluid

2

6

D

k

T

d Dt

kTD

r

radius of big particle r

Einstein’s specific prediction: in pure water at temperature 17o C (290 K or 63o F), a particle of diameter 1 m will move an average horizontal distance equal to 6 m in one minute.

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Is Einstein famous for Relativity?

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Random motion also in mechanical systems

R.K.PathriaStatistical MechanicsPergamon Press

Reducing the air pressure, the r.m.s.motion doesn’t change

butthe lower trace is almost monochromatic

whereas the higher is more random

It MUST vibrate ifthe equipartitiontheorem is right !

<E> = kT/2for each d.o.f.

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Non equilibrium thermodynamics

• Non isolated system shows uncorrelated fluctuations of volume and temperature

• Two independent fluctuating variables: T, V

TB

2

V

2B2

PV

TkVCTk

T

system

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Some comments

TB

2

V

2B2

PV

TkVCTk

T

• EASY TO JUSTIFY MECHANICAL VIBRATION FROM VOLUME FLUCTUATION

• NO SPECTRAL INFORMATION FROM THE PREVIOUS RELATIONS

• RESIDUAL GAS EFFECT IS HARD TO BE IMPLEMENTED

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The Fluctuation-Dissipation Theorem - 1

• H.B. Callen and T.A. Welton, Phys. Rev. 83, 34 (1951)

• R Kubo 1966 Rep. Prog. Phys. 29 255-284

• It applies to linear systems in thermal equilibrium

• It is used to predict the level of thermal noise of one observable x of the system 

Linear system

X(t)F(t)

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The Fluctuation-DissipationTheorem - 2

• It gives the amplitude of the fluctuations of force Sff() that is shaking the system, at each angular frequency

• As seen in the experiments, the noise spectrum is shaped by the “friction”

() =SPEED

FORCE=

v()

F()

)(Tk4)(S Bff

Coefficient ofviscosity =

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The double approachto thermal noise

• Direct

– 2 variables are fluctuating

– Intuitive

– The spectrum is hard to extract

• Indirect

– Dissipation replaces fluctuation

– Not intuitive

– Extremely powerful for noise level prediction: is “easy” to measure ()

TB

2

V

2B2

PV

TkVCTk

T

)(Tk4)(S Bff

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Our system

• 20 to 40 kg silica mirror

• Suspended by 4 fibres

• Dielectric coatings applied on the front faces for maximizing or minimizing reflection

• The reference is the mass front face, where the laser beam senses the position

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Volume fluctuations in solids

• The volume fluctuations (as the thermal ones) need to fulfil the boundary conditions

• Perfect solids (crystals) vibrates at their resonant frequencies

• The real solids have defects that move or change driven by the finite temperature of the solid:

– The vibration has a continuous spectrum rather than a discrete one

• NO DIRECT METHOD APPROACH:

– Mechanical losses of materials are investigates and thermal noise level is worked out through FDT

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How to measure the mechanical loss

A method widely used is to detect the free decay of the excited resonances of the system

In order to know the frequency distribution of noise, the viscosity constant has to be measured at all the frequencies of interest

A0/eA0

( )0

2m

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Sample

Vacuumtank

Sapphirehalf sphere

A new sample holding systemGeNS Dr. Elisabetta Cesarini

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The most severe limit for IFOs:thermal noise from the coatings

• Alternate layers of transparent materials with different index of refraction

• Impedance mismatch andinterference produce highcoefficient of reflectivity

• Its structure is not compact as the substrate

• 10 m of coating produces morethermal noise than 10 cm of substrate

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Asymmetrical thermal fluctuations are responsible of thermoelastic noise on silica fibres (DIRECT APPROACH)

In linear thermoelastic effect thermal expansion coefficient transforms thermal fluctuations in strain fluctuations

= ·T

Thermoelastic noiseEffect on suspension fibres - 1

Fibres bend and then the suspendedmass is shaken. The effect is small butrelevant in GW detectors

Same kind of deformations occur in mirrors

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Thermoelastic noiseEffect on suspension fibres - 2

• The heat transfer sets a characteristic time scale that makes the noise spectrum frequency dependent, like:

Debye peak:

2max

2 ff

f

Nois

e o

r fr

icti

on

in

ten

sity

frequency

• Fused silica facts:— Low — Low — High strength

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Fused silica fibre productionand testing

• The CO2 laser pulling machine was developed in Glasgow

• The machine was financed by EGO as well as PPARC and in 2006 it was delivered to Pisa

• The machine was then adapted to the Virgo necessities and thanks to the excellent work of Dr. Matteo Lorenzini, Francesco Piergiovanni, Dr. Filippo Martelli, Virgo now has fused silica suspensions of high precision and strength