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Title

Slide 1 of 13

Angular Resolution of PTA for GW Sources : Analytical Study

Linqing Wen (UWA)
October 17, 2012


Outline

Slide 2 of 13

Outline
Background Previous Study Our work Results and Implication Conclusion


What is Mathematica?

Slide 3 of 13

Motivation
Want to know directional uncertainty for individual GW sources detected by PTA analytical form helps provide insights assuming resolvable individual GW sources : possibly at high f (e.g., Sesana et al 2009)


What is Mathematica?

Slide 4 of 13

Method
Fisher Matrix its inverse provides lower bound for covariance of any unbiased parameter estimators (CramИ r­ Rao bound) represent intrinsic noise properties method independent at high SNRs, lower bound can be approached by maximum-likelihood based estimator caution raised by various researchers
e.g. Vallisneri, M. (2008), Lee, K. J. et al. (2011)


What is Mathematica?

Slide 5 of 13

Notation & Assumptions
Assuming: GW distance R >> pulsar distance D, = 0, pulsar sky direction precisely known independent noise between frequency bins and across different pulsars (OK for high f) Ignoring O x
2

curvature term in r x see however Deng, X . et al 2011 on the effect
GW n Pulsar k


Earth

x
0

Pulsar

p
r (x) R



0

Earth GW


What is Mathematica?

Slide 6 of 13

Response of Pulsar Earth Pair to GW
Response to two GW polarizations in frequency domain


D 1 cos 0 ei
t0

y


1 1 cos 0 cos 2 h 2i 1 1 cos 0 sin 2 h 2i

y Pulsar term


ei

t0

modulates both amplitude and phase


What is Mathematica?

Slide 7 of 13

Response of Pulsar Earth Pair to GW
pulsar term


dominates in response to varying GW direction 1 , best estimation for GW

If pulsar distance D is well known D 1 cos 0 direction ~diffraction limit SKA ' s ability to determin pulsar distance :

D 1 pc 1 kpc Smits et al.2011

OK assumption for 0

0


What is Mathematica?

Slide 8 of 13

Response of Pulsar Earth Pair to GW
If pulsar distance D is not well known pulsar distance need to be fit simultaneously with 0
in , D and 0 are degenerated the detected SNR for each pulsar, mainly 1 degeneracy

cos 0 term will help break the

Fisher (information) matrix can be used to quantify such statements


A Look At Workflow

Slide 9 of 13

Previous Work
Numerical simulation using MCMC for MBHBs Sesana & Vecchio 2010, CQG and PRD: Earth term only: 40 deg2 for 100 pulsars, network SNR = 10

Corbin V. & Cornish N. 2010, arXiv:1008.1782 (unpublished?) Consider pulsar term, fitting D also < 3 deg2 for 20 pulsars, network SNR = 20


A Look At Workflow

Slide 10 of 13

Previous Work
Analysing directly elements of Fisher matrix (Boyle L. and Pen. U. 2010, arXiv:1010.4337) known pulsars: diffraction limit of (1/SNR) /D poorly known distance: (1/SNR) D

Geometrical expression for interferometric GW detector network (Wen & Chen 2010 PRD) geometrical expression for angular resolution for ground-based interferometric GW detector network expression in terms of observables for best/worst cases.
1
IJLM IJLM

A2

,IJLM

similar technique applied to PTA in this work


A Look At Workflow

Slide 11 of 13

Manipulating Fisher Matrix ...
Fisher matrix
ij ni

d

n

j

d

= d a S 1 a d() is the data vector of length Np - timing residuals for Np pulsars at each frequency ni i 1, 2 : unknown GW sky direction, unknown parameters
ij

: discrete set of other -independent
1

=<

ni

d

n

j

d

-<

ni

d

d

<

d

d

<

d

n

j

d

Similar equation can be derived if
( Wen & Chen 2010)

is a continuous variable (can be dependent)


Mathematica in the Classroom

Slide 12 of 13

Results for PTA for arbitrary GW source
Area of error ellipses for GW sky direction 2 For known pulsar distances: n2
i

n2

i

ni nj

2

= 2



12

4

2 c2 A
2 ,LMJK

,

LMJKJkLM

A

,LMJK

rLM rLM rm

rJK . n , rL
1 pulsars Earth

L, M, J, K

1, 2, ... Np


rLM rm
L, M, J, K 1, 2, ... Np

rL
1 pulsars Earth

Geometrical view: Earth

1/weighted average of projected areas formed by pairs of pulsars and
Pulsar

Earth

GW

Pulsar-pulsar pair, pulsar-Earth pair have different weighting factor of 0 0 IJ LM pulsars only 0 JkLM M Earth IJ XL XI XL J , L Earth

0

IJ

, and XI respectively


For the best case (known GW signal waveform but unknown arrival time t0): ~ diffraction limit X
I

d
0 IJ I

0,I

I

d

cos2 2 d0,J I J I I dI 2 2 f 2 SNR2

For the worst case (unknown GW signal waveform h()): projection onto signal space involved: d P cross correlation of pulsar term involved : d
I IJ IJ dJ

p

X

I 0 IJ

d

p,I

d d
p,I

p,I

Pd PIJ d

pI

p,J


For unknown pulsar distances: no projected area involved, For the best case, (1+cos) term dominates d 1e
i

1 SNR2

) (1+cos ) d

00

D 1 cos 0
4 2 c2 XJ A
2 ,IJ

,

I, J

1, 2, ... Np pulsars

,

IJ XI

A XI

,IJ

rI d00,I

rJ . n sin
2

:

relative orientation of pair of pulsars wrt to GW direction

2 d00,I

-

d00,I Sin d00,I d00,I d00,I

2

For small (allowing poorly known D): no projected area involved,

1 SNR2 D


A Look At Workflow

Slide 13 of 13

Discussion and Future Work
Geometrical expression is given for PTA angular resolution weighted projected areas knowledge of pulsar distance is crucial to go beyond ~1/SNR wave perpendicular to pulsar planes gets better angular resolution, worst along the plane Next step: apply to IPTA data set with assumed distances