Compact Gravity Wave detector

Munawar Karim

Department of Physics

St. John Fisher College

Rochester, NY 14618

karim@sjfc.edu

• 13th Midwest Relativity Meeting

• Windsor, ON, Oct. 2003• (gr-qc/0209015)

MICHELSON INTERFEROMETER

arms 10cm long• rigid mirrors

sampled interferometer• null detector configuration

• extension to 3-axis interferometer

• detector array

€

h ≈ 10−23 / Hz

Detector design

• Each sample independent, unlike Herriot delay line or Fabry-Perot cavity

• Reflected beams recorded by photo-diode at end of each round-trip

• Ensures independence of each sample• Rigid mirrors facilitate vibration isolation• Vacuum environment 10-7 T • Dark fringe operation• Frequency response

€

10−4 Hz to 104 Hz

GW

€

gμν = ημν + hμν

€

ds2 = gμν dxμ dxν = (ημν + hμν )dxμ dxν

0 = −c2dt 2 + 1+ h11 ωt − k ⋅x( )[ ]dx2

€

τ rt =2Lx

c+

12c

h11 ωt − k ⋅x( )0

Lx

∫ dx −12c

h11 ωt − k ⋅x( )Lx

0

∫ dx

Lx = L +ξ 1 ( proper length)

Excess time delay

Excess time delay in one round-trip

€

Δτ i = 2ξ 1 −ξ 2

c+ h(t)

2Li

c= 2

ξ 1 −ξ 2

c+ h(t)τ i

Wave surface

5 2.5 0 -2.5 -5

420-2-4

25

12.5

0

-12.5

-25

x

y

z

Response of platform

€

m∂ 2x1

∂t 2+ k x1 − L( ) = −mR1010

GW x1

m∂ 2x1

∂t 2+ k x1 − L( ) = m

12

∂ 2 ˜ h 11

∂t 2x1

€

ε i ≡x i − L

L=

ξ i

L

∂ 2ε i

∂t 2+ ω0

2 +12

ω2h11 sinωt ⎛ ⎝ ⎜

⎞ ⎠ ⎟ε i = −

12

ω2h11 sinωt

€

ξ1 −ξ 2

L≈ h11

ω2

ω02 −ω2

( )

ωω0

sinω0t − sinωt ⎛

⎝ ⎜

⎞

⎠ ⎟; ω0 ≠ ω

• Inhomogeneous Hill’s equation

Excess time delay

€

Δτ i = hτ i 1+ω2

ω02 −ω2

( )

ωω0

sinωτ i − sinωτ i

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

€

Δτ i =i=1

p

∑ pΔτ i

= h pτ i +ω2

ω02 −ω2

( )τ i

ωω0

sinω0τ i − sinωτ i

⎛

⎝ ⎜

⎞

⎠ ⎟

i=1

p

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

For p independent samples

€

ω2

ω02 −ω2

( )τ i

ωω0

sinω0τ i − sinωτ i

⎛

⎝ ⎜

⎞

⎠ ⎟

i=1

p

∑

€

→ω /ω0( )

2

1− ω /ω0( )2

( )dτ

0

0.01

∫ωω0

sinω0τ − sinωτ ⎛

⎝ ⎜

⎞

⎠ ⎟

€

ω =2π 800;ω0 = 2π104 stiff mount

€

ω /ω0( )2

1− ω /ω0( )2

( )dτ

0

0.01

∫ωω0

sinω0τ − sinωτ ⎛

⎝ ⎜

⎞

⎠ ⎟≈ −10−19

€

ω /ω0( )2

1− ω /ω0( )2

( )dτ

0

0.01

∫ωω0

sinω0τ − sinωτ ⎛

⎝ ⎜

⎞

⎠ ⎟≈ −0.25

€

ω =2π 800;ω0 = 2π soft mount

Shot noise uncertainty

• Time uncertainty due to shot noise for p round-trips

€

σδτ =±hλ

Pin 4πc1τ i

1p

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pΔτ i = σ δτEquate

Shot-noise limited sensitivity

• Result valid for generalized mirror mount

• Depends on pulse duration only

• Independent of arm-length

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h ≥hλ

Pin 4πc1

pτ i

1pτ i

≥10−22

Brownian noise

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σδτB /c = ±

4kBT

Qmω03

1pτ i

= ±2.72 ×10−28 sec s

€

σδτ =±10−24 sec s

as compared with shot-noise€

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