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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work A fast parameter space search for continuous gravitational waves from known binary systems C Messenger University of Glasgow

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work

A “fast” parameter space search for continuous gravitational waves from known binary systems

C Messenger

University of Glasgow

December 19, 2006 GWDAW 11

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work

Outline

1 Introduction

Overview of the data analysis problem

2 The data analysis challenge

The sources

3 Current Solutions

The fully coherent approach The Stack-Slide approach The Radiometer approach

4 A New Solution

A New Solution :The sideband search The proposed search Following up candidates

5 Conclusions and future work

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work Overview of the data analysis problem

Introduction

Non-axisymmetric spinning neutron stars are thought to be candidates for continuous gravitational waves sources. “Targetted” searches for known isolated and binary pulsars have been/are being done at an unprecented sensitivity. “Parameter space” searches fall into 2 catagories

“blind” searches eg. all sky search [Col06] “semi-targetted” searches where the parameter space is constrained by previous EM observations eg. Sco X-1 search [Col06]

Parameter space searches for continuous gravitational waves sources are typically computationally limited. eg. Coherent S2 Sco X-1 search limited to observation time of ∼ 6 hours. New search strategies are required (proposed method adapted from [RCE03]).

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work The sources

LMXB’s

LMXB’s consist of a neutron star (NS) (or black hole) in orbit with a lower mass companion star (either main sequence, white dwarf or evolved star). The lower mass companion has filled its Roche Lobe and material is being transferred into an accretion disk around the

  • NS. accretion disk.

These are not seen as pulsars so the frquency is unknown (although some exhibit type 1 X-Ray bursts). GW emission mechanism Asymmetries in the NS crust caused and sustained indirectly by infalling material [Wag84, UCB00].

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work The sources

Millisecond accreting X-Ray pulsars

An accreting binary system where pulses in the X-Ray emission are observed at the NS rotation frequency. The pulses are generated by infalling material being channelled onto “hotspots” on the NS surface. Small subset (7 currently known) have spin periods of order 10−3 sec and are known as millisecond (or recycled) pulsars. GW emission mechanism Asymmetries in the NS crust caused and sustained indirectly by infalling material [Wag84, UCB00, MP05].

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work The sources

Binary Radio Pulsars

Radio pulsars in binary systems typically have very well defined orbital and phase parameters but not all of them. The work by [PW] and consequent results [Col] leave ∼ 40 radio pulsars as unsuitable for the single filter time domain analysis. GW emission mechanism Long term asymmetries in the NS crust [Bil98, UCB00, Cut02]

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work The fully coherent approach The Stack-Slide approach The Radiometer approach

The fully coherent approach

Using matched filtering, we perform a search over a bank of templates. A metric approach is used to optimally place templates. The F-statistic is then computed for each template [JKS98]. This approach was used for the S2 analysis [Col06]. Accreting binary pulsar

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Observation Time Span, Tspan (sec) Number of Templates

projected metric u n p r

  • j

e c t e d m e t r i c u n c

  • r

r e l a t e d − u n p r

  • j

e c t e d m e t r i c frequency templates total templates

  • rbital templates

Key Problem for LMXB’s This approach is computationally prohibitive Tcomp ∝ T 7 (for Sco X-1 T < 105 sec).

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work The fully coherent approach The Stack-Slide approach The Radiometer approach

The Stack-Slide approach

t This is an incoherent search. The data are split into M contiguous chunks. A coherent search is performed on each chunk. The search products are summed (stacked) as a function of source frequency (slid) [BCCS98]. Stack-Slide example

time frequency frequency frequency normalised stacked power time frequency time

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work The fully coherent approach The Stack-Slide approach The Radiometer approach

The Radiometer approach

t The radiometer approach has been developed initially for the analysis of the GW stochastic background [Bal06, Col04]. It can be used to target a particular sky location. It cross corrolates data between 2 detectors and uses the signals (if present) as filters. Radiometer example

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work A New Solution :The sideband search The proposed search Following up candidates

A toy model example

Let us define a toy model binary signal as h(t) = g(t)h0 cos

  • 2πf0
  • t + a sin

2π P (t − t0)

  • + φ0
  • ,

where g(t) is the time domain window function. This can be decomposed in the frequency domain to give ˜ h(f ) = ˜ g(f ) ∗ h0 2

m

  • n=−m

Jn(2πf0a)e−i(nt0+φ0)δ(f − fn) where M = 2m + 1 ≈ 4πf0a. The power is then |˜ h(f )|2 ≈ |˜ g(f )|2 ∗ h2 4

m

  • n=−m

J2

n(2πf0a)δ2(f − fn)

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work A New Solution :The sideband search The proposed search Following up candidates

The frequency domain signal

For T 3P the signal power |˜ h(f )|2 is localised in M ≈ 4πf0a “spikes”. t Each “spike” is seperated by 1/P Hz The relative amplitude of each “spike” is defined by the f0 and a. The power/F-statistic is independent of the orbital phase parameter (usually the time of periapse passage tp). Example signal profile

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.5 1 1.5 2 2.5 3 3.5 x 10

4

frequency (Hz) |h(f)|2

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work A New Solution :The sideband search The proposed search Following up candidates

Dealing with gaps

In the (very likely) situation where the data contains large and frequent gaps in time we deal with this by computing the Fourier transform of the window function g(t). t Example g(t)

T T g(t) time + T s

Example ˜ g(f )

−1 −0.5 0.5 1 x 10

−5

−5 5 10 15 20 x 10

5

frequency (Hz) Re(g(f))

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work A New Solution :The sideband search The proposed search Following up candidates

Orbital eccentricity

We can use the Fourier series expansion of the sin and cos of the eccentric anomoly E(t) (k ∈ −∞, . . . , −1, 1, . . . ∞) eg. cos E = −e 2 + 1 k Jk−1(ke) cos kM. t Non-zero eccentricity acts to spread power amonst existing sidebands. Results in change in relative amplitude of spikes not location. Eccentric signal example

199.9996 200 200.0004 2000 4000 6000 8000 frequency (Hz) 2F

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work A New Solution :The sideband search The proposed search Following up candidates

Searching with a “comb”

t First compute F-statistic demodulating for sky position

  • nly.

Use a finite sized flat comb c(f )

  • f unit amplitude teeth each

spaced by 1/P Hz as a template → SNR loss < 35% The comb will have M = 4πf0a teeth. The comb

f0 f0 SIGNAL COMB SIGNAL * COMB 2F(f) C(f) frequency frequency

The detection statistic C(f ) = c(f ) ∗ 2F(f ) (1)

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work A New Solution :The sideband search The proposed search Following up candidates

The statistics

The mean and variance of the detection statistic C(f ) are given by C = d2

  • pt + 4M,

σ2

C = 4d2

  • pt + 4M

where M = 4πf0a and is fixed for a given source and d2

  • pt = h2

0T/2Sh. The SNR of the detection statistic C(f ) is

therefore ρC ≈ C − C(h0 = 0)

  • σ2

C

≈ h2

0T/2Sh

2

  • h2

0T/2Sh + M

The 1σ sensitivity is therefore approximately h(1σ) ≈ 2M1/4 Sh/T d2

  • pt ≪ M,

2

  • 2Sh/T

d2

  • pt ≫ M.

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work A New Solution :The sideband search The proposed search Following up candidates

The F-statistic likelihood

The F-statistic components Fa and Fb are calculated using Fa =

  • x(t)a(t)e−iΦ(t)dt

Fb =

  • x(t)b(t)e−iΦ(t)dt.

Let Fa =

  • 1 + i
  • 3 and Fb =
  • 2 + i
  • 4
  • 1 = T (AA1 + CA2) /4,
  • 2 = T (CA1 + BA2) /4,
  • 3 = −T (AA3 + CA4) /4,
  • 4 = −T (CA3 + BA4) /4,

and the covariance matrix governing

  • is

σ2(

) =

C O O C

  • C = 1

8TSh A C C B

  • ,

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work A New Solution :The sideband search The proposed search Following up candidates

MCMC parameter estimation

An MCMC is performed over the frequency, orbital and nuisance parameters using only

  • data located at the sideband frequencies.

MCMC chains

50 100 fi 199.9 200 200.1 f0 (Hz) 0.008 0.01 0.012 a (sec) −5000 5000 tp − T0 (sec) 2 4 6 ω (rad) 0.05 0.1 e 0.05 0.1 h0 −1 1 cosι −1 1 ψ (rad) 2 4 6 φ0 (rad) −1600 −1400 −1200 log(L)

MCMC posteriors

74 75 76 0.5 1 fi p(fi) 200 200 200 10 x 10

5

f0 (sec) p(f0) 0.009 0.01 0.011 2000 a (sec) p(a) ω tp − T0 2 4 6 −2000 2000 0.05 0.1 20 e p(e) 0.1 0.2 20 40 h0 p(h0) −1 1 1 2 cosι p(cosι) −0.5 0.5 0.5 1 ψ p(ψ) 5 0.2 0.4 φ0 p(φ0)

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work A New Solution :The sideband search The proposed search Following up candidates

Comparison of methods

Search method comparison (VERY ROUGH)

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Observation Time (sec) h0/Sh

1/2 (Hz1/2)

Coherent Stack−Slide SideBand Radiometer 4 X Period Spin−Down Templates Period Templates ~6 days

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work A New Solution :The sideband search The proposed search Following up candidates

How fast is this search?

The following scaling is true when not searching over sky position, period or spin down (3GHz Desktop): F-statistic : TF ∼ 500 sec

  • T

106sec

2

fband 1Hz

  • C(f ) : TC ∼ 2 sec
  • T

106Sec fband 1Hz

  • ln
  • T

106Sec fband 1Hz

  • .

Stack-Slide run time for a single template is similar. For stack-slide to compete it needs to search over many orbital

  • templates. ie. slowing it down by the number of templates.

At present the majority of the search time is spent in the MCMC stage and is strongly dependent upon the threshold set on C(f ).

C Messenger A “fast” GW search for CW’s from binary systems

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work

Conclusions and future work

Pipeline improvements

Extend the analysis to make use of the multi-IFO F-statistic. Incorporate spin up/down into the MCMC. Look at applicability to LISA as an all sky search for white dwarf binaries.

Analysis plans

We plan to analyse all LMXB’s with known period and sky position. To do a simple single frequency MCMC search for the accreting millisecond X-Ray pulsars (T < 106 sec) (coherent). Assess which of the surplus radio pulsars are suitable and perform the MCMC search on these.

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Introduction The data analysis challenge Current Solutions A New Solution Conclusions and future work

References

Stefan W. Ballmer. A radiometer for stochastic gravitational waves. Classical and Quantum Gravity, 23:S179, 2006.

  • P. R. Brady, T. Creighton, C. Cutler, and B. F. Schutz.

Searching for periodic sources with LIGO. prd, 57:2101–2116, February 1998.

  • L. Bildsten.

Gravitational Radiation and Rotation of Accreting Neutron Stars. apjl, 501:L89+, July 1998. The LIGO Scientific Collaboration. Upper limits on gravitational radiation from 78 radio pulsars in preparation. The LIGO Scientific Collaboration. Analysis of first LIGO science data for stochastic gravitational waves. prd, 69(12):122004–+, June 2004. The LIGO Scientific Collaboration. Coherent searches for periodic gravitational waves from unknown isolated sources and scorpius x-1: results from the second ligo science run, 2006.

  • C. Cutler.

Gravitational waves from neutron stars with large toroidal B fields. prd, 66(8):084025–+, October 2002.

  • P. Jaranowski, A. Kr´
  • lak, and B. F. Schutz.

C Messenger A “fast” GW search for CW’s from binary systems