#### Transcript Tackling LISA Data analysis challenge

```Heterodyne detection with LISA
for gravitational waves parameters
estimation
Nicolas Douillet
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1
Outline
• (1): LISA (Laser Interferometer Space
Antenna
• (2): Model for a monochromatic wave
• (3): Heterodyne detection principle
• (4): Some results on simulated data analysis
• (5): Conclusion & future work
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2
LISA motion during one Earth period
2
rotation symmetry in ecliptic longitude ( ) between
3
two consecutive spacecraft orbits (S1 , S2 , S3 )
LISA geometry 
2 

S 2  S1   

3 

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;
3
2 

S3  S 2   

3 

LISA configuration
- Heliocentric orbits,
free falling spacecraft.
- LISA center of mass
Follows Earth, delayed
from a 20° angle.
- 60° angle between LISA
plan and the ecliptic plan.
- LISA arm’s length: 5. 109 m to detect gravitational waves with frequency
in: 10-4    10-1 Hz
- LISA periodic motion -> information on the direction of the wave.
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Motivations for LISA
Existing ground based detectors such as VIRGO and LIGO are « deaf » in low
frequencies (  < 10 Hz).
Limited sensitivity due to « seismic wall » (LF vibrations transmitted by the
 A space based detector
allows to get rid of this
constraint.
 Possibility to detect
very low frequency
gravitional waves.
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Monochromatic waves
Sources: signals coming from coalescing binaries
long before inspiral step. Frequency  considered
as a constant.
H  h  h
h+ / h : amplitude following + / x polarization
+ /  : directional functions
Gravitational wave causes
perturbations in the metric
tensor.
Effect (amplified) of a
Gravitational wave on a ring
of particles:
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+ polarization
6
x polarization
Model for a monochromatic wave(1)
s  s  t , , ,  , ,, h,    s  i 1i N
Unknown parameters:
 (Hz): source frequency
h (-): wave amplitude
104 Hz    101 Hz
LISA response to the incoming GW:


 F  t  h sin  2  t  R sin  cos       
s  t   F  t  h cos 2  t  R sin  cos      

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
7

2 t
T
T : LISA period (1 year)
Model for a monochromatic wave (2)


 R

s  t   E  t  cos  2  t  sin  cos          t  
c




 h F 

h
F
  
  t    arctan 
With
2 1/ 2
E  t    h F    h F  


2
and
h  h 1  cos 2  
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;
Amplitude modulation (envelope)
Shape depends on source location: (, )
h  2h cos 
8
Pattern beam functions (1)
Change of reference frame for
Dn ,  and Dn , pattern beam functions.
Fn ,   t  
1
cos 2 Dn ,   t   sin 2 Dn ,   t  

2
Fn ,   t  
1
sin 2 Dn ,   t   cos 2 Dn ,   t  

2
: polarization angle
1  n  3
Spacecraft n° in LISA triangle.
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Pattern beam functions (2)
‘+’ polarization
Dn ,  t  
3


[36sin 2  sin  2  2  n  1 
64
3




 


  3  cos  2    cos 2  9sin  2  n  1   sin  4  2  n  1   
3
3 






 


 sin 2  cos  4  2  n  1   9 cos  2  n  1  
3
3 



 





 4 3 sin 2  sin  3  2  n  1     3sin    2  n  1    ]
3
3



 
4 sidebands
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
2 t
: LISA orbital phase
T
Pattern beam functions (3)
‘x’ polarization
Dn   t  
1




[ 3 cos   9 cos  2  n  1  2  
16
3






 cos  4  2  n  1  2 
3






 6sin   cos  3  2  n  1    
3






 3cos    2  n  1   ]
3


2 t

: LISA orbital phase
T
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Envelope heterodyne detection (1)
Principle:
(1): Fundamental frequency (0) search
Detect the maximum in the spectrum of the product between source signal (s)
and a template signal (m) which frequency lays in the range: [0   ;0   ]
Frequency precision is reached with a nested search.
max  S    M   
0
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Envelope heterodyne detection (2)
(3): Shift spectrum (offset zero-frequency) by heterodyning at 0 , then low-pass
filtering
s  t  e2i0t  S    S       0   S   0 
Y    S   H  
(Filter above 0 )
8 lateral bands: [0; 7] (empirical) -> compromise between accepted noise
level and maximum frequency needed to rebuild the envelope ( = 1/ T)
(2): Envelope reconstruction
Fourier sum
7
E n  Y k  e
2 j  n 1
 k 1
N
k 0
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Correlation optimization (1)
Correlation surface between template and experimental envelope
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Correlation optimization (2)
(1) Principle: correlation maximization between signal envelope end envelope
template (or mean squares minimization).
 E E
Corr  E, E0     i
i 1  Ei  E

2
N
  E E
0
   i0
  Ei 0  E0
 
2




(2) Method: gradient convergence and quasi-Newton optimization methods.
lim Corr  E, E0   0
j 
(3) Conditions: already lay on the convex area which contains the maximum.
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Signals and noises
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Spectrum and instrumental noises
Sn1    N 1 ,1 
0
  2
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Sn2    N 2 , 2 
Sources mix
Possible to distinguish between n
sources since their fundamental
frequencies are spaced enough
(sidebands don’t cover each other):
Sources
 j  i  15
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Envelope detection (1)

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Envelope detection (2)


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Symmetries & ambiguities
S  ,  
S     ,    
LISA main symmetry
Correlation symmetry
E(-, + ) = E(, )
Corr(, ) = Corr(-,+ )
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Symmetries (1)
Some parameters remains difficult to estimate due to the high number of the
envelope symmetries on the parameters  and .
Examples:
1/ 2
2
2
E  t    h F    h F  




Fi ,     Fi ,   ;
4



Fi ,      Fi ,   ;
4

Fi ,      Fi ,   ;


Fi ,     Fi ,   ;
4



Fi ,      Fi ,   ;
4

Fi ,      Fi ,   ;
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   0; 2 
Symmetries (2)


Fi ,      Fi ,   ;
2



Fi ,      Fi ,   ;
2



Fi ,      Fi ,   ;
2



Fi ,      Fi ,   ;
2

 
h     h   ;
2

h      h      h   ;
h     h   ;
   0; 2 
h     h   ;
   0;  
h      h      h  
Ie -> risks of being stuck on correlation secondary maxima in N dimensions
space (varied topologies resolution problem).
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How to remove sky location uncertainty (1)
Choice between (,) and (
-, +) depends on the sign of the product
2 R
k  r t  
sin  cos     
c
If  is the colatitude (ie 
 [0;  ] ), and when t=0


sgn k  r  t   sgn  cos   
From the source signal, we compute the quantity

hF
sgn   t  Arg  sa  t    Arc tan   
 h F

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
 

24
hence the sign of  and 
How to remove sky location uncertainty (2):
Source -> LISA, Doppler effect
  t   t 
 h F t  
sin  cos       arc tan   

c
h
F
t


  

R
1 
2 R
f t  
 
sin  sin     
2 t
cT
 h F  t   

 arc tan 
 
t 
h
F
t


  

;
 2 R

 v
f max   1 
sin     1      
cT


 c
;
  
 2 R

 v
f min   1 
sin     1      
cT


 c
;
  
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
2

2
0
How to remove sky location uncertainty (3):
Source -> LISA, Doppler effect
2 R

v

: LISA tangential speed

T

2 R : Ecliptic length
cT : Light year


f  t 
t
1  2  4 2 R


sin  cos     
2 t 2
cT 2
Moreover, if   0 (colatitude),
f  t 
t
f  t 
t
 0 si  -

2
 0 si  
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   

2

2
   
: frequency seen by LISA increases.
3
: frequency seen by LISA decreases.
2
26
Source localization
Simulated data from LISA data analysis community
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Statistics on sky location angles (,)
S  0     / 2,    / 3
Max error: polar source ( = /2)
 = f()
 = f()
Max sensitivity: source direction  to LISA plan
( ~ /6)
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Noise robustness tests (static source)



True value
Estimations (180 runs on the noise)
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Typical errors on estimated parameters
Average relative errors for   /3
i/i
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Ecliptic latitude 
5. 10-2

Ecliptic longitude 
1. 10-3

Polarization 
1.5 10-1
Inclination angle 
3. 10-1

~
Frequency 
8.5 10-6

Amplitude h
0.5 – 1
X
30
Compare two parameters estimation
techniques: template bank vs MCMC
(1): Matching templates (template bank and scan parameters space till reaching
correlation maximum -> systematic method)
● quite good robustness
- Limitations: ● N dimensions parameters space. (memory space and computation
time expensive)
● difficulties to adapt and apply this method for more complex
waveforms
(2): MCMC methods, max likelihood ratio: motivations
(statistics & probability based methods)
- Advantages: ● No exhaustive scan of the parameters space (dim N).
● much lower computing cost and smaller memory space
- Limitations: ● Careful handling: high number parameters to tune in the
algorithm (choice of probability density functions of the parameters)
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Conclusion and future work
- Encouraging results of this method (heterodyne
detection) on monochromatic waves. Could still to be
improved however.
- Continue to develop image processing techniques for
trajectories segmentation (chirp & EMRI) in timefrequency plan. (level sets, ‘active contours’ methods
import from medical imaging and shape optimization)
- Combining this methods (graphic first estimation of
parameters) with Monte-Carlo Markov Chains algorithms
(numeric finest estimation) allows in a way to ‘‘ logdivide’’ the dimensions of the parameters space (N5 + N2