Introduction to

Gravitational Wave Detection

Ronald W. HellingsMontana State University

PTA WorkshopPenn State7/20/05

space

What is a gravitational wave?

• A 2-D analogy

motion in thisdimension ismeaningless

2 free masses

The masses trackeach other with lasers

The gravitational wave is a wave of curvature

each slice is a section ofan arc of constant radius

the free masses remain fixed at their coordinate points

As a gravitational wave passes through the space...

while the distance between them

increases due to the extra space in the curvature wave.

The laser signal has to cover more distance and is delayed

Why are gravitational waves called “a strain in space”?

points that are close have little space injected between them

points that arefurther away have morespace injected between them

h

Quadrupole Gravitational Waves

a ring of free test masses h+

mor

e sp

ace

less space

Quadrupole Gravitational Waves

a ring of free test masses

h

Let’s do the math

Geometry

Earth

plane wave

elliptical polarization

polarizationangle

propagationvector

n̂

pulsars

2 ( ) i jij ijds h dx dx

The Gravitational Wave Metric Tensor

ˆ ˆ( ) ( ) ( ) ( ) ( )ij ij ijh t h t h t n n

e.g. choose the z-axis along and the x-axis so = 0.n̂

1 0 0 0 1 0

0 1 0 and 1 0 0

0 0 0 0 0 0

Then

The path of the radio signal from the pulsar to the Earth is a null path, so

2 2

2 2

0

( ) 1i j

i jij ij ij

dt ds

dx dxdt h dx dx ds h

ds ds

1

2

e e e i j

ijp p p

dx dxdt ds h ds

ds ds Approximate

and integrate

1 1ˆ ˆ ˆ ˆ ( ) ( )

2 2

ei j i j

ij ij ijp

s s s h ds s s s H e H p

where ( ) ( )ij ijH t h t dt

1ˆ ˆˆ ˆ ( ) ( )

2i j

ij e e ij p ps s s H t H t n x n x

ˆ( ) ( )ij ijh t h t n xhij is a wave, so

reception occurs at t = t, x = 0 emission occurs at t = t s, ˆsx s

1ˆ ˆˆ ˆ (1 )

2i j

ij ijs s H t s H t n sso

The change in distance is proportional to the integral of the wave amplitude.

So let’s get an observable that is proportional to the wave

( ) 1ˆ ˆˆ ˆ (1 )

2i j

ij ijd

s s h t s h tdt

n s

Gravitational waves are proportional to the time derivative of pulsar arrival time residuals. But...

in the long wavelength limit (s<), ( ) ( ) ( )h t h t h t

( ) 1ˆ ˆˆ ˆ (1 )

2i j

ijd

s s sh tdt

n s

and

or 1ˆ ˆˆ ˆ (1 )

2i j

ijs s h t n s

LIGO

Low band of LISA

The Gravitational Wave Spectrum

Type Range Run Time Sources Instrument

HF 10 Hz 1000 Hz

compactstars

bars,LIGOs

MF 0.1 Hz 10Hz

10 Hz 10 mHz

1 nHz 10 Hz

10 nHz 0 Hz

? MAGGIE,lunar LIGO

LF binariesSMBHs

LISA

one per day

one per a few days

one per year

VLF

ULF

once in alifetime

cosmic astrophysics PTA

snapshotsonly

cosmicstructure

COBE, MAPPlanck, etc.

The Gravitational Wave Spectrum

Type Range Run Time Sources Instrument

HF 10 Hz 1000 Hz

compactstars

bars,LIGOs

MF 0.1 Hz 10Hz

10 Hz 10 mHz

1 nHz 10 Hz

10 nHz 0 Hz

? MAGGIE,lunar LIGO

LF binariesSMBHs

LISA

one per day

one per a few days

one per year

VLF

ULF

once in alifetime

cosmic astrophysics PTA

snapshotsonly

cosmicstructure

COBE, MAPPlanck, etc.

Long wavelength limit

Long and short regimes

Long and short regimes

Short wavelength only

( ) 1ˆ ˆˆ ˆ (1 )

2i j

ij ijd

s s h t s h tdt

n s

The Pulsar Limit

~1000 yearsnow

Every pulsar in every direction has correlated timingnoise due to this term. This allows a weighted correlationanalysis to optimally use data from multiple pulsars.

The correlated part of the timing noise

( )ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )i j i j i j

ij ij ijd

s s h t s s h t s s h tdt

For the nth pulsar in the direction sn, this may be written

( )( ) ( ) ( )n

n n nd

h t h t n tdt

(This generalizes the result of Hellings & Downs, 1983, which assumed plane-polarized gravitational waves.)

The cross-correlation of data from 2 pulsars will produce

2

mn m n m n

m n

C h h h h

h h O nh

If are isotropic, and uncorrelated, then and h h

2mn mnC h O h h O nh

where 1

4mn m n m n d

But should be uncorrelated? and h h

IT DEPENDS ON THE SOURCE!

Needs

• Calculation of for plane polarization mn done

• Calculation of and for general polarizationmn mn

• Thought on sources of stochastic gravitational background