Continuous Waves Searches ULs

1
Continuous Waves Searches ULs

• B. Allen, S.Anderson, S.Berukoff, P.Brady,
  D.Chin, R.Coldwell, T.Creighton, C.Cutler,
  R.Drever, R.Dupuis, S.Finn, D.Gustafson,
  J.Hough,M.Landry, G. Mendell, C.Messenger,
  S.Mohanty, S.Mukherjee, M.A. Papa, B.Owen,
  K.Riles, B.Schutz, X. Siemens, A.Sintes,
  A.Vecchio, H.Ward, A. Wiseman, G.Woan, M. Zucker

www.lsc-group.phys.uwm.edu/pulgroup
2
Science activities
Plan of the talk

• first priority item UL on signals from known
  sources
• two search engines
• frequency domain
• time domain (presented by G. Woan , Glasgow)
• preliminary S1 investigations

• other tasks that might meet the end-of-the year
  deadline
• UL on Sco-X1 (Birmingham)
• Unbiased searches (Michigan)

3
Frequency domain search general overview

• core LAL routines LALDemod, LALBarycentering,
  LALComputeSky (AEI)
• given template parameters and data in frequency
  domain (SFTs) it computes the F statistic
• can be used for targeted searches, for area
  searches and as a module of a hierarchical search
• efficient algorithm to run in a (cheap)
  distributed computing environment
• there exists an LDAS DSO that runs on
  (LDAS-produced) SFTs, performs the search and
  write to the data-base table (G. Mendell)

4

• what is F and how we compute it
• how we go from F to an UL

F statistic from gr-qc/9804014, P. Jaranowski,
A. Krolak, B.F. Schutz
5
Signals model and parametersisolated rotating
pulsar emitting GWs at f02fr due to its
configuration not being perfectly axysimmetric

• (a,d) position in the sky
• f0 emission frequency (at SSB)
• fs spin-down parameters,
• h0 amplitude of signal (as received on Earth)
• i inclination angle
• y polarization angle
• f0 initial phase

6
How does the signal that we receive depend on
these parameters?
beam-pattern functions and depend on the relative
orientation of the detector w.r.t. the wave. They
depend on y and on the amplitude modulation
functions a(t) and b(t) that depend on the
relative instantaneous position between source
and detector.
the phase of the received signal depends on the
initial phase and on the frequency evolution of
the signal. The latter depends on the spin-down
parameters and on the Doppler modulation, thus on
the frequency of the signal and on the relative
instantaneous relative velocity between source
and detector.
7
What is the F statistic ?
We do not need to explicitly search over
h0,i,f0,y. Given the data, F is the likelihood
ratio for the quadruplet of values that best fits
the data. This maximization is done
analytically. As we will see, if a signal were
present, the actual value of F would depend, of
course, on the actual value of the signals
h0,i,f0,y parameters.
8
Why can we compute F efficiently ?

• the core of the calculation consists in
  computing
• Lets neglect for now the amplitude modulation.
  Its irrelevant for the point I am making now.

with fi being the phase of the signal (template)
and xi being the data.
9
Input data in frequency domain
TOTAL DATA SET Sampling rate 2kHz Tobs 4
months Size 80 GB Band 0-1 kHz at 1e-7 Hz
resolution.
Our data NM samples Can
also be divided in M chunks, each one having N
samples. Each of these chunks can be FFT-ed,
producing a set of short time-baseline FFTs
(SFTS)
MN
EACH CHUNK Duration of a chunk 1 h In 4 months
3000 chunks Each chunk 28 MB Band 0-1 kHz at
3E-4 Hz resolution.
10
The core of the calculation looks like this
11
Including the amplitude modulation
ai and bi are the amplitude modulation functions
that only depend on the template sky-position.
To construct F one has to combine Fa and Fb like
this
12
How is F distributed ?

• in the case of noise only, F is expected to be a
  c2 random variable with 4 degrees of freedom.
• if theres a signal, F is expected to follow a
  non-central c2 distribution with non centrality
  parameter proportional to inner product of signal
  with itself.

By studying how values of F(f0) computed for f0s
different from those of the targeted signal are
distributed, we can verify that our data follows
the expected noise-only distribution. If it does
not, we will take the measured distribution as
the actual one. Presumably we will obtain a value
of F from our search that lies well within the
noise curve we want to set an UL on h0.
13
F from 4 H2 S1 2048 s SFTs - IFO always locked -
GPS 714887312 714895504.
Preliminary results on F
fake pure gaussian noise passed through search
code
by G. Mendell and B.Cameron
14
According to our analysis, with YYY confidence,
this pulsar is not emitting GWs (of the type that
we are looking for) with h0 greater than XXX.
There are many ways to set upper limits
15
Most conservative upper limit - a curve not a
number !
F
It is a sample of a random variable drawn from a
certain distribution. Which distribution ? If
theres a signal, the distribution depends of the
signals i,y,f0 and h0 parameters. Let us choose
the triplet i,y,f0 that yields the smallest
non-centrality parameter iw,yw,f0w . Now the
distribution only depends on h0 pw (F h0). For
every values of h0 we can integrate the
corresponding pw (F h0) curve between F and
infinity, this yields P(h0), our final UL result
the result of our analysis.
20
80
with probability (confidence) P we can say that
if the data contained a signal with amplitude h0
(or greater), we would have measured a value of F
higher than the one that we have measured (F).
16
We could set more liberal UL by eliminating the
dependency from the i,y,f0 parameters
differently

• we could construct the p(Fh0) by integrating
  p(F h0, i,y,f0 ) over the values of i,y,f0 ,
  with suitable weights, i.e.prior. This method, in
  general, will produce a stronger (lower) UL than
  the previous one presented.
• It has also been suggested that we construct,
  just to provide a feel for the range of
  variability of our UL, also a least conservative
  UL. To compute this UL one follows the same
  recipe as for the most conservative UL, with the
  exception that the values of i,y,f0 are the ones
  for which the non-centrality parameter is
  maximum.
• all of the methods proposed up to here are
  essentially Frequentist ULs. The time-domain
  analysis will use a Baysian approach. In the case
  of known pulsars we do not expect that the two
  approaches will provide significantly different
  results.

17
Where do we stand now ?

• constructing SFT data and looking at it
• validating software pipeline to compute F
• developing software for extensive Monte Carlo on
  real data to assess detection efficiency and
  estimate pdfs (might not be necessary, but
  certainly it is necessary to check that thats
  the case)
• want to meet end of the year UL deadline

18
Power in 4 H2 S1 2048 s SFTs (IFO always locked
GPS 714887312 - 714895504)
2 Hz band centered on 1283.86 Hz The mean power
goes increases by 13 increase during this
interval
by G. Mendell
19
Power in 4 H2 S1 64 s SFTs (IFO always locked
GPS 714887296 - 714887552)
2 Hz band centered on 1283.86 Hz The mean power
appears to increase by a factor of 12 during 2048
seconds Low frequency power leaking out -gt
high-pass filtering or windowing needed with such
short time-baselines
by G. Mendell
20
Gaps in lock GPS 714162320 - 714170512
Power in 4 H2 S1 2048 s SFTs padded gaps in lock
Histograms
by G. Mendell
21
GEO - 10 SFTs - 60 s - 10 Hz band _at_ 1283 Hz
SFT2/ltSFT2gt
SFT2/ltSFT2gt with windowing before FFTing
22
LIGO H1 - 1800 SFTs - 64 s 1 - Hz band _at_ 1283 Hz
starting at 714151671
SFT2/ltSFT2gt
ltSFT2gt and s(SFT2)
SFT2/ltSFT2gt having removed the mean
B. Allens SFTs
23
More S1 investigations

• Unbiased CW Searches. Strategy (Michigan)
• Measure power in selected bins of averaged
  periodograms
• Bins defined by source parameters (f, RA, d)
• Estimate noise level statistics from
  neighboring bins
• Set upper limit on quasi-sinusoidal signal on top
  of empirically determined noise
• Scale upper limit by antenna pattern correction

They have started to study the average spectral
power distribution of the 2048s LDAS-produced
SFTs ? Probably not optimum, but fine for
exploration Upper limits will be based on excess
power summed incoherently over subsets of bins in
many SFTs
24
Study Range 659-661 Hz (early H1)

• Early H1 Average over 97 SFTs with gt90
  livetime

Bottom histogram now excludes region between bars
Same but with 5 mHz binning (typical ?f
searchband)
D. Chin, K. Riles
25
Study Range 659-661 Hz (late H1)
Bottom histogram now excludes region between bars
Late H1 Average over 146 SFTs with gt90
livetime
Cleaner than early H1 !!
Same but with 5 mHz binning (typical ?f
searchband)
D. Chin, K. Riles
26

• Preliminary conclusions from unbiased search team
  that in spirit apply to all S1 investigation
• At least some of the data looks okay
  Gaussian, locally white noise not a terrible
  approximation
• But non-Gaussianity and non-stationarity quite
  apparent and must be confronted
• But devil is in the details
• Optimization of SFTs (?T, windowing)
• Determining detection efficiency
• Correction for calibration drifts
• Expect analysis to evolve in iterative
  refinements but on a time scale that will allow
  us to meet our end-of-the-year deadlines, as
  promised.