Eﬃcient methods to probe

Stochastic Gravitational Wave Background Anisotropy
With Ground-based Detectors!1

ICRR seminar, March 11,2019
Jishnu Suresh

To Map the GW Sky
http://mwmw.gsfc.nasa.gov/mmw_allsky.html

LISA~0.001 Hz??

DECIGO~0.1 HzLIGO/VIRGO/KAGRA??

??

~100 Hz????

??
slide: Sanjit Mitra!2

ICRR seminar, March 11,2019
Jishnu Suresh

Stochastic Gravitational Wave Background (SGWB)• individually undetectable(subthreshold)•but detectable as a collectivity via
their common influence on multipledetectors(PRL 120, 091101, 2018)

366

367

•

combined signal described
statistically—stochastic
gravitational-wave background368(no noise)

369

370

371

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375

376

377

378

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380

(~3 hr)ICRR seminar, March 11,2019

FIG. 2. We!3 present a simulated timeJishnu
seriesof duration 10Suresh

381

4 382

383

Stochastic Gravitational Wave Background (SGWB)•Unresolved astrophysical or cosmologicalsources

Characterizing the stochastic background• popcorn or continuous
3c2 H02⇢c == 7.8 ⇥ 10 9 erg/cm38⇡GN
Critical energy density to close the universe•Carry information not accessible inelectro-magnetic astronomy

Many models give power law spectra inour band• astrophysical sources- information on the anisotropic
local universe⌦GW (f ) = ⌦↵• primordial cosmological background(CGWB)- direct probe of inflation•Persistent unknown isotropic or
anisotropic sources!4✓ffref◆↵↵=0Flat (inflation, cosmic
strings in our band…)↵ = 2/3

Binary inspiral (BBH, BNS)ICRR seminar, March 11,2019
Jishnu SureshPotentially detectable with advanced LIGO/Virgo(ArXiv:1903.02886)(PRL 120, 091101, 2018)

ICRR seminar, March 11,2019
G. 1. The left panel shows the predicted median background(green) models described in!5 for the BNS (red) and BBHJishnu
Suresht, the total (combined) background (blue), and the Poisson error bars (grey shaded region) for the total background. We a

Types of Stochastic Gravitational Wave Background(i) Stochastic backgrounds can diﬀer in spatial distribution(statistically) isotropic

anisotropic(like cosmic microwave background)(galactic plane in equatorial coords)
Joe’s lectures;Les Houches Summer School 2018

!6ICRR seminar, March 11,2019
Jishnu Suresh

(ii) diﬀer in temporal distribution and amplitude
Stationary Gaussian

Non-stationary (non-gaussian)Foreground0(yr)1(e.g., from galactic whitedwarf binaries;
modulated by LISA’sorbital motion)
Joe’s lectures;Les Houches Summer School 2018

!7ICRR seminar, March 11,2019
Jishnu Suresh

(iii) diﬀer in power spectra depending on source

white noise

histograms
σ=1
2Δt σ2

BNS chirp

f-7/3

Joe’s lectures;Les Houches Summer School 2018

!8ICRR seminar, March 11,2019
Jishnu Suresh

Example: Rate estimates and signal durations imply BNS “confusion”& BBH “popcorn” for LIGO / Virgo
BNS (m1=m2=1.4 Msolar)

BNS “confusion”

histogram

BBH (m1=m2=10 Msolar)

BBH “popcorn”

histograJoe’s lectures;Les Houches Summer School 2018

!9ICRR seminar, March 11,2019
Jishnu Suresh

Example: Rate estimates and signal durations imply BNS “confusion”& BBH “popcorn” for LIGO / Virgo366
versitat d367
d’Educac368
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the Swiss370
sian Foun
371
Foundati372
Regional
373
ety, the S
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ties Phys
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Fund (O376
National377
Hungary378
Korea, I379
through380
novation,
381
Council
ICRR seminar, March 11,2019Research4 382!10
Joe’s lectures;Les Houches Summer School 2018

Jishnu Suresh
FIG. 2. We present a simulated time series of duration10

Cross-Correlation Search
Allen & Romano (2001)• Detector output = true signal + noise• Normally detector noise are uncorrelated for far away detectors and times:• Cross-correlation (CC) statistic is the best choice for unmodeled sources- one detector’s signal is the filter for other detector’s datah1 = n 1 + s
h2 = n 2 + ssignal and noise
are uncorrelated

signalnoise at H1 and L1 is uncorrelated!11

ICRR seminar, March 11,2019
Jishnu Suresh

There are three sources of diﬀerent strength (strong, medium and weak) marked in red.

cross-correlation is essentially a one dimensional map of the sky!12

ICRR seminar, March 11,2019
Jishnu Suresh

Overlap reduction function for H1/L1

Stochastic search
For a power law background…
✓◆↵f⌦GW (f ) = ⌦↵fref1⟨h̃1( f )h̃*2 ( f′)⟩ = δ( f − f′)Γ12( f )Sh( f )2Accounts for separation and orientation of detectors
Note we are most sensitive below 100 Hz

We can write an optimal estimator for theenergy density

Allen and Romano, Phys.Rev. D59 (1999) 102001

Choose Q to maximize SNR for fixed
spectral shape:Γ12( f )H( f )Q̃( f ) ∝P1( f )P2( f )!13expected signal spectrum

de-weight correlation whennoise is large or overlap is small
ICRR seminar, March 11,2019
Jishnu SureshRadiometer Algorithm
ZdtZ

dt0 s1 (t) s2 (t0 ) q(t, t0 )⌘Z

df se⇤1 (t, f ) se2 (t, f ) qe(t, f )• Essentially Earth Rotation Synthesis Imaging- Cross-correlate detector outputs in short time segments- map making: use time dependent phase delay

^
−Ω
^
−Ω

• Use spectral filters_^− Ω . Δ x (t)- to enhance signal power_Δ x (t)- to reduce noise power

Detector 1Detector 2
Lazzarini & Weiss (2004)
Ballmer (2006)
Mitra et al (2007)!14ICRR seminar, March 11,2019
Jishnu Suresh

Terrestrial Network
HHL

VI

KL

(credit: N. Cornish)

VIK

Sky coverage of individual detectors!15

ICRR seminar, March 11,2019
Jishnu Suresh

Terrestrial NetworkH

L(credit: N. Cornish)

VI

K

!16

ICRR seminar, March 11,2019
Jishnu Suresh

Terrestrial Network
HL(credit: N. Cornish)

V

I

K

!17

ICRR seminar, March 11,2019
Jishnu Suresh

Terrestrial Network
HL(credit: N. Cornish)

V

IK

!18

ICRR seminar, March 11,2019
Jishnu Suresh

Terrestrial Network
HIL(credit: N. Cornish)

VI

K

19!19

ICRR seminar, March 11,2019
Jishnu Suresh

Terrestrial NetworkHKL(credit: N. Cornish)

VI

K!20

ICRR seminar, March 11,2019
Jishnu Suresh

Radiometric Search for SGWB
• Observed data := Cross Spectral Density := product of SFT’s

CI

CfI t := sI1 (t; f ) sI2 (t; f )• Noise (in the small signal limit):nII
nf t:= nI1 (t; f ) nI2 (t; f )• Covariance matrix:

N2(T)
Cov(CfI t , CfI t ) ⇥4!21IIttffPI1 (t; f ) PI2 (t; f )ICRR seminar, March 11,2019
Jishnu Suresh

CSD Observed from an Anisotropic BackgroundP( ˆ ) :=• Anisotropic SGWB in some basis:

P e ( ˆ ); PP• Observed CSD = convolution of anisotropic background with additivenoise
CfI t :=

KfI t, P+ nIf tLow signal limit- the “kernel” or “beam”:
KI

KfI t,:=T H(f )I(f, t)✴ generalized overlap reduction function
I(f, t) :=A=+,⇥⇥

S2d ˆ FIA1 ( ˆ , t) FIA2 ( ˆ , t) e2⇥if ⇥·ˆ

!22x(t)/ce (ˆ)

ICRR seminar, March 11,2019
Jishnu SureshML Estimation of SGWB Anisotropy• ML estimate of SGWB anisotropy in any basis with a network of detectors:
P( ˆ ) :=P e ( ˆ ); P⇥ P̂ =P̂

- “Dirty” map (essentially filtered output):
X := K · N
†

1·C ⇥

X

=4TI,f t- Fisher information matrix:
1:= K · N†

1·K ⇥ [1]·X

H(f ) fI⇥t,s⇥⇥I1 (t; f ) s⇥I2 (t; f )PI1 (t; f ) PI2 (t; f )= 4I,f t!23

PH 2 (f )PI1 (t; f ) PI2 (t; f )I⇥(f, t)I(f, t)ICRR seminar, March 11,2019
Jishnu Suresh

Base Line Sidereal Rotation

The white circles indicate positions in the sky map which will have equal time/phasedelay when the signal from that part of the sky arrive in the LIGO Livingston andHanford detectors!24

ICRR seminar, March 11,2019
Jishnu Suresh

SGWB point source pre-processing!25

ICRR seminar, March 11,2019
Jishnu Suresh

!26

ICRR seminar, March 11,2019
Jishnu Suresh

SGWB point source processing!27

ICRR seminar, March 11,2019
Jishnu Suresh

SGWB point source post-processing!28

ICRR seminar, March 11,2019
Jishnu Suresh

SGWB extended source pre-processing!29

ICRR seminar, March 11,2019
Jishnu Suresh

SGWB extended source processing!30

ICRR seminar, March 11,2019
Jishnu Suresh

SGWB extended source post-processing!31

ICRR seminar, March 11,2019
Jishnu Suresh

Folding stochastic data to one sidereal day• No approximation, based on a mathematical symmetry

Sidereal Day• Incorporates all the dirty stuﬀ (quality cuts, overlapping window correction)

32
10
Time

Sidereal TimeAin, Dalvi & Mitra, PRD 92, 022003 (2015)

Folded full LIGO O2 CSD (abs)!32

ICRR seminar, March 11,2019
Jishnu Suresh

Folding Data• The estimator (“dirty map”)

X↵ =X

I⇤
K↵,f
t

If t=

X

If tsid2
IC
If t f tI⇤
K↵,ftsid

1 I
"t 1
2n

1 I
"t+1
2
Xhn

2
If t +

2
If t +2
If (t 1)

2
If (t+1)

oo

CfI (t 1)

CfI (t+1)

2IC
If (iday +tsid ) f (iday +tsid )

idayn
1 I
2
"iday +tsid 1+If(i+t)
daysid2n1 I
2"iday +tsid +1+If(i+t)
daysid2!33

2
If (iday +tsid 1)
2
If (iday +tsid +1)

oo

CfI (iday +tsid 1)
I
Cf (tsid +tsid +1)

ICRR seminar, March 11,2019
Jishnu Suresh• The Fisher information matrix (three data streams)

↵↵0=

X
If t=

X

I⇤
K↵,f
t

I⇤
IK↵,f
K
tsidf tsid ,↵0I⇤
IK↵,f
Ktsidf (tsid 1),↵0

If tsidXI⇤
IK↵,fKtsidf (tsid +1),↵02
If t +2
If t +2
If (t+1)o2
If (t 1)oKfI (t 1),↵0I
Kf (t+1),↵02
If (iday +tsid )

iday

X1

iday

If tsidXn

1 I"t+1
2

If tsidX

1 I"t 1
2

2
IK
f t,↵0
If tn

2

X1

iday

2

"Iiday +tsid 1"Iiday +tsid +1!34nn

2
If (iday +tsid ) +2
If (iday +tsid ) +2
If (iday +tsid 1)

2
If (iday +tsid +1)

ICRR seminar, March 11,2019
Jishnu Sureshoo(HV baseline)(LV baseline)• No data quality cuts has been incorporated to these SIDs
• One can incorporate either in folding code or SID code!35

ICRR seminar, March 11,2019
Jishnu Suresh

Validation of Folded Data

RMS Diﬀerences = RMS (folded - unfolded) / RMS (unfolded)
Real parts of FSID Fisher matrix= 4.51e-03
Imaginary parts of FSID Fisher matrix = 4.92e-03
Dirty SpH
= 2.38e-02
Clean SpH
= 2.44e-02
Dirty map
= 2.05e-02
Clean map
= 2.25e-02
Variance map= 2.12e-03
SNR map= 2.08e-02!36

ICRR seminar, March 11,2019
Jishnu Sureshrequired to calculate the narrowband maps using three pipel(previous)
Pipelines
theStochastic
standard pipeline,
the standard pipeline with folded data
PyStoch with folded data.

ulatedbottomng thened byctions.

Intermediate
data
Processing
time
Intermediate
results
Final results

Conventional
pipeline

Folding
pipeline

Folding an
PyStoch450 GB

1.5 GB

1.5 GB

10 CPUyears
800 TB

10 CPUdays
2.5 TB

40 CPU
minuteNot require

500 MB500 MB500 MBX ICRR seminar, March 11,2019HðfÞXX ¼!37Jishnu SureshAin, Suresh & Mitra, PRD 98 024001 (2018)

PyStoch - fast HEALPix based SGWB mapmaking• Folding + PyStoch = Few thousand times speed up- perform the whole analysis on a laptop in few minutes!
• Produces the narrowband maps as intermediate result- so separate search for diﬀerent frequency spectra becomes redundant!38

ICRR seminar, March 11,2019
Jishnu Suresh

Eﬃcient Overlap Reduction Function Computation
Breaking Down the ORFIf t,↵ :=XZAS2A ˆA ˆ2⇡ifˆd⌦FI1 (⌦, t)FI2 (⌦, t)eˆ⌦·xI (t)cˆe↵ (⌦)

GPS Time
1126615273

3314 pairs of seed maps (for one sidereal day) produced in a laptop in 20 seconds.

!39

ICRR seminar, March 11,2019
Jishnu Suresh

ORF from ORF seed maps

GPS Time
1126615273!40

ICRR seminar, March 11,2019
Jishnu Suresh

Parameter File(for folding)

SID Meta Data
(GPS tag & locations)

FoldingTime Series Data
from Hanford & Livingston

Job File(Each line is a job list of GPS tag stat & end)

Pre-Processing(produces CC spectra)

Parameter File(for SID)

Stochastic Intermediate Data (SID)

Frame locations

Gps tag start

Cache Files (named by job number)
Folded Stochastic
Intermediate Data (FSID)
Frame locations

Gps tag start

Cache Files (named by job number)

Processing(produces dirty map
and Fisher matrices)

Parameter File(for sph search)

Stochastic pipeline - PyStoch

Dirty Maps & Fisher Matrices
for each job
Post-Processing
Sph search

Data

Combined dirty map
and Fisher matrix

Information
Required forprocessing

Mapping program

Dirty Map

Variance Map(inv fisher matrix)

SNR Map

Pipeline
TaskClean Map

Input
ParametersSky Maps

Stochastic pipeline!41

ICRR seminar, March 11,2019
Jishnu Sureshsingle node of IUCAA computational facility) and storage(New)
StochasticPipeline- PyStochrequired
to calculate
the narrowband
maps using three pipelines:
the standard pipeline, the standard pipeline with folded data, andPyStoch with folded data.

tedom
thebyns.

Intermediate
data
Processing
time
Intermediate
results
Final results

Conventional
pipeline

Foldingpipeline

Folding andPyStoch

450 GB

1.5 GB

1.5 GB

10 CPUyears
800 TB

10 CPUdays
2.5 TB

40 CPU
minutes
Not required

500 MB500 MB

500 MB
Ain, Suresh & Mitra, PRD 98 024001 (2018)XHðfÞXf;pXp ¼
If !42

ð29Þ

ICRR seminar, March 11,2019
Jishnu SureshNarrowband Maps
1920 Narrowband maps (20 Hz to 500 Hz, 0.25 Hz bins) produced in a laptop inless than 10 minutes.

!43

ICRR seminar, March 11,2019
Jishnu Suresh

Multi-baseline studies

injected source
clean_map_HL
clean_map
detector network

Multi baseline improvements :Phys. Rev. D 83, 063002, 2011

• Sensitivity improvement• Sky coverage• Parameter accuracy• Quality of the sky mapsHow the stochastic analysis is going to improve with KAGRA coming online ?
!44

ICRR seminar, March 11,2019
Jishnu Suresh

Multi-baseline Maps : Broad Source!45

ICRR seminar, March 11,2019
Jishnu Suresh

Multi-baseline Maps : Clean Maps

• The GW radiometer algorithm is optimal to search for backgrounds- folding+PyStoch has made the algorithm few thousand times faster- PyStoch is handy to study multi baseline eﬃciency for stochastic searches- can also search for persistent unknown sources- can rapidly follow up the outliers!46ICRR seminar, March 11,2019
Jishnu Suresh

!47

ICRR seminar, March 11,2019
Jishnu Suresh

…and we expect many more weaker signals…
Stay tuned for..

thank you!48

ICRR seminar, March 11,2019
Jishnu Suresh