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Stochastic Gravitational Wave Background Mapmaking using regularized deconvolution

Sambit Panda, Swetha Bhagwat, Jishnu Suresh, Sanjit Mitra

TAUP @Toyama: September 9-13, 2019 Jishnu Suresh

Stochastic Gravitational Wave Background

(no noise) (PRL 120, 091101, 2018)

individually undetectable (subthreshold) but detectable as a collectivity via their common influence on multiple detectors combined signal described statistically—stochastic gravitational-wave background

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PRL 120, 091101, 2018

Potentially detectable with advanced LIGO/Virgo

• Phys. Rev. D 100, 061101(R) (2019

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Estimation of SGWB Anisotropy

D = B.S + n

Difficult to get the Maximum Likelihood solution with 
 ill-conditioned beam

ˆ S = B−1 D

The observed data comes in the form of signal convolved with our detector response function with some additive noise. When this integral equation is visualized on a pixelized sky, we get the linear equations:

D(ˆ Ω) = Z

s2 dˆ

Ω0[B+(ˆ Ω, ˆ Ω0) + B⇥(ˆ Ω, ˆ Ω0)]S(ˆ Ω0) + n

BA(ˆ Ω, ˆ Ω0)

A = {+, ×}

are the corresponding beam response functions with polarizations; is the power in both polarizations.

S(ˆ Ω0)

Regularization helps in these cases

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• Phys. Rev. D 100, 043541 (2019)

TAUP @Toyama: September 9-13, 2019 Jishnu Suresh

Bayesian Regularization

Most likely solution Most probable solution

P(D|S, B) = 1 ZL e− 1

2 (D−BS)T N −1(D−BS)

P(S|D, B, µ, R(S)) = P(D|S, B) P(S|R(S), µ) P(D|µ, B, R(S))

M(S) := χ2(S) + µR(S)

Motivated by: S. H. Suyu, P. J. Marshall, M. P. Hobson, and R. D. Blandford, MNRAS, 371, 983 (2006)

5 TAUP @Toyama: September 9-13, 2019 Jishnu Suresh

Regularization - technique of incorporating additional information on the source map, while carrying out the process of deconvolution. This can be achieved by providing prior spatial information of source or incorporating a smoothness condition. P(S|D, B, µ, R(S)) = 1 ZP e−M(S)

Implementation

Compute the beam matrix of Hanford - Livingston baseline using PyStoch*.

* Python implementation of the stochastic anisotropic directional search: Phys. Rev. D 98, 024001 (2018)

Inject sources of different intensities, to create a source map. We generate a dirty map by convolving the injected map with the beam and add a noise map*. Use regularized and un-regularized deconvolution methods on the dirty maps for reconstruction of clean maps.

TAUP @Toyama: September 9-13, 2019 Jishnu Suresh 6

The noise map is generated by process- ing Gaussian noise in frequency domain corresponding to the two LIGO detectors.

* We follow the procedure described in Phys.Rev. D77, 042002 (2008), 0708.2728

Qualitative Improvements: Extended Source

Strong Weak Very Weak Injected map Dirty map No
 Regularization Grad
 Regularization

TAUP @Toyama: September 9-13, 2019 Jishnu Suresh

Slide: Sanjit

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Qualitative Improvements: Extended Source

Strong Weak Very Weak Injected map Dirty map No
 Regularization Norm
 Regularization

TAUP @Toyama: September 9-13, 2019 Jishnu Suresh

Slide: Sanjit

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Results: Quantitative Improvements

NSP = A · B p kAkkBk

NMSE = kA Bk2 kAk2

Quantitative measures of goodness of reconstruction in terms of NSP and NMSE of sky-maps. NSP (better measure for extended sources) and NMSE (better measure for point sources) are quoted for recovered maps obtain by no deconvolution (comparing dirty map to beam convolved injected map), unRegularized deconvolution, and norm & gradient Regularized deconvolution. The number of iterations for unRegularized deconvolution (iter_No-reg) and Regularization strength (λ) are also listed. Except for strong sources, incorporating Regularization significantly improves the quality of reconstruction

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Masked Maps

2 - Sigma Dirty map 3 - Sigma No
 Regularization Norm
 Regularization

TAUP @Toyama: September 9-13, 2019 Jishnu Suresh

Slide: Sanjit

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Stability of Deconvolution

Regularization stabilises the quality of deconvolution afuer 10 − 20 iterations

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Simulations

Not only Regularization improves the quality of deconvolution statistically, but it improves for every realisation.

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Mapping an anisotropic stochastic gravitational wave background using data from ground-based detectors is becoming progressively important as detectors are breaking sensitivity barriers and new cosmological results are being published. One fundamental hurdle is that the matrix that connects the source sky to the data is somewhat ill-conditioned, making it non-trivial to deconvolve the filtered cross-spectral data from pairs of detectors, a.k.a. the dirty map Regularized deconvolution provides a robust yet straightforward way to address this issue and the method can be readily applied to the current LIGO-Virgo analyses.

thank you

Summary

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Norm Regularization that tries to minimise power in the whole map, which is suitable for localised sources Types of Regularization Gradient Regularization that tries to reject small angular scale variations, which is suitable for extended sources Measure of quality of recovery No unique prescription to construct a quantitative estimator that measures the quality of a reconstructed map

Normalized Scalar Product, NSP , which quantifies the deviation of the source map and the recovered map through an inverse norm weighted Euclidean inner product Normalized Mean Squared Error (NMSE) as another independent measure to quantify the deviation of the reconstructed map from the source map.

We implement Bayesian Regularized deconvolution to reconstruct the source map from the dirty map. We vary the parameter λ over a range of 1 to 10^6 in logarithmic intervals and pick a value that (nearly) maximizes NSP or minimizes NMSE

A better recovery is indicated by the value of NSP being closer to one and the value of NMSE being closer to zero.

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Norm Regularization

Rnorm(S) = 1 2

npix

X

i=1

S2

i

Cij = rSirSjRnorm = δij

Gradient Regularization

incorporates a preference towards smooth source map reconstruction by penalizing the intensity difference between the neighboring pixels. The spatial fluctuations of the intensity in the noise is statistically more than the fluctuation in the signal. This is especially true if the signal has an extended pattern in the sky

Rgrad(S) = 1 2

npix

X

i=1 ni

X

k=1

(Si − Sjik)2

Cij = rSirSjRgrad(S) = δij 2niδij

introduces a preferential bias towards the solutions that minimizes the norm of the map. This is seen to have a noise suppression effect, especially in the case of point-like sources.

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Strong Source Weak Source Very Weak Source

Nearly insensitive to the choice of λ Although we pick a value for the Regularization constant λ for each injection separately such that Regularization produces optimal results, these figures demonstrate that a broad range of values for λ could produce similar results