2D Mass Mapping - OU-LE3 Weak Lensing - Lausanne 2015 · 2D Mass Mapping OU-LE3 Weak Lensing - Lausanne 2015 François Lanusse Jean-Luc Starck, Adrienne Leonard, Sandrine Pires CosmoStat

2D Mass MappingOU-LE3 Weak Lensing - Lausanne 2015

François LanusseJean-Luc Starck, Adrienne Leonard, Sandrine Pires

CosmoStat LaboratoryLaboratoire AIM, UMR CEA-CNRS-Paris 7, Irfu, SAp, CEA-Saclay

Mass mapping with sparsity prior Cluster density mapping Substructure recovery using Flexion

Introduction

Main issues in Weak-Lensing mass-mapping:• Missing data (mask and limited number densities)• Shape noise• Reduced shear

=⇒ Most approaches avoid these issues by smoothing

Our goalWrite the mass-mapping as a single optimization problem witha multi-scale sparsity prior addressing all these issues.

François Lanusse (CEA-Saclay) 2D Mass Mapping 2/ 19


Optimization problem

minκ

1

2‖ (1− κ)g −Pκ ‖22 +λ ‖ Φtκ ‖1

• g : reduced shear• λ : regularization parameter

• P : lensing operator• Φ : wavelet dictionary

A few remarks:• Recovers the convergence from the reduced shear• P can be defined with and without binning the shear• P can be ill-posed in case of missing data• Sparse regularization of noise and missing data• We use isotropic wavelets for Φ

=⇒ Adapted to the recovery of cluster signal




minκ

1

2‖ (1− κ)g −Pκ ‖22︸︷︷︸Data fidelity term

+λ ‖ Φtκ ‖1




=⇒ Adapted to the recovery of cluster signalFrançois Lanusse (CEA-Saclay) 2D Mass Mapping 3/ 19



minκ

1

2‖ (1− κ)g −Pκ ‖22 +λ ‖ Φtκ ‖1︸︷︷︸

Sparsity term




=⇒ Adapted to the recovery of cluster signalFrançois Lanusse (CEA-Saclay) 2D Mass Mapping 3/ 19


Proximal minimization algorithmPrimal-dual splitting:{

κ(n+1) = κ(n) + τ(∇F (κ(n)) + Φα(n)

)α(n+1) = (Id− STλ)

(α(n+1) + Φt(2κ(n+1) − κ(n))

)adapted from Vu (2013)

• Fast and flexible algorithm

• Sparsity constraint λ estimated locally by noise simulations=⇒ Accounts for survey geometry, varying noise levels

• Same algorithm used for all the problems presented here



Handling missing data:

(a) Input (b) Galaxy catalogue with 30gal/arcmin2




(a) Input (b) Kaiser-Squires with 1’ bins




(a) Input (b) Kaiser-Squires with 0.05’bins




(a) Input (b) Kaiser-Squires with 0.05’bins + 0.1’ smoothing




(a) Input (b) Glimpse 2D


Lavf56.15.102

out4.mp4Media File (video/mp4)


Handling shape noise:

(a) Input (b) Kaiser-Squires + 0.5’smoothing




(a) Input (b) Kaiser-Squires + 1.0’smoothing




(a) Input (b) Glimpse 2D



• So far, we have only considered mapping from shear alone.

• The framework can be expanded to include additionalinformation, in particular redshifts.

=⇒ Allows cluster density mapping1 (assumingknowledge of lens and sources redshifts)

1Lanusse F., Starck J.-L., Leonard A., and Pires S. (2015), High ResolutionWeak Lensing Mass Mapping combining Shear and Flexion , in prep.



Our method is particularly well suited as it doesn’t need to binthe data.

Why avoid binning the shear catalogue ?=⇒ Allows to include individual galaxy redshifts :

minκ∞

1

2‖ (1− Zκ∞)g − ZPκ∞ ‖22 +λ ‖ Φtκ∞ ‖1

with Z = Σ∞critic/Σcritic(zi) and Σlens = κ∞Σ∞critic

Individual redshifts have two benefits:• Directly map the surface mass density of the lens

• Mitigate the mass-sheet degeneracy when κ becomessignificant (Bradac et al. 2004)



Simulation set:• Density maps from Bolshoi simulations (Kyplin et al. 2011)

• Virial masses [1014 h−1M�]: 10.9 - 3.02 - 2.70

• Cluster redshift: zclus = 0.30

• Field size: 10× 10 arcmin2 ∼ 2× 2 h−2Mpc2

4 2 0 2 4

4

2

0

2

4

Field 1

4 2 0 2 4

Field 2

4 2 0 2 4

Field 3

0.000.150.300.450.600.750.901.051.201.351.50

Figure : Convergence maps for sources at z →∞



Simulated deep space-based galaxy catalogue (HST/ACS):

• Uniform angular distribution with ngal = 80 gal/arcmin2

• Redshift distribution zmed = 1.75

• Strongly lensed sources (|g| > 1) excluded

• Gaussian photo-z errors σz = 0.05(1 + z)

• Gaussian shape noise σ� = 0.30

=⇒We generate 100 independent noise and galaxy distributionrealisations



0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0R [arcmin]

0.5

1.0

1.5

2.0 Halo 1

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0R [arcmin]

0.5

1.0

1.5

2.0 Halo 2

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0R [arcmin]

0.5

1.0

1.5

2.0 Halo 3

Field Input Map Measured Map1014 h−1M� 1014 h−1M�

1 4.08 3.91± 0.182 1.88 1.90± 0.203 1.59 1.60± 0.18

Table : Aperture mass in the central 2′ of each field.



• Shear is noise dominated on small scales=⇒ Substructure is lost when mapping from shear alone.

• Small-scale substructure can be recovered from stronglensing when available.

• Gravitational Flexion is useful in the intermediate regime.

Figure : Shear (left) and first flexion (right) (Bartelmann 2010)



10-3 10-2 10-1

k [arcsec]−1

10-4

10-3

10-2

10-1

100

101

102

103

104

N2

Shear aloneFlexion aloneFlexion (Pires et al. 2010)Combined

Figure : Noise power spectra for convergence estimator from shearand flexion



• Recent forward fitting method developed by Cain et al.(2011) and flexion mapping paper on substructurerecovery Cain et al. (2015).

• We can integrate flexion in our reconstruction framework=⇒ Jointly fit shear and flexion

minκ

1

2‖ (1− κ)

[gF

]−[PQ

]κ ‖22 +λ ‖ Φtκ ‖1

Adding flexion to our simulation set:• Flexion noise σF = 0.029 arcsec−1

=⇒ Reported by Cain et al. (2011) for Abell 1689

• Simulate reduced flexion, strongly flexed galaxiesexcluded (|F | > 1.0 arcsec−1)



Figure : Reconstruction from one realisation



Figure : Mean of 100 reconstructions



0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0R [arcmin]

0.0

0.5

1.0

1.5

2.0 Halo 1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0R [arcmin]

0.0

0.2

0.4

0.6

0.8

1.0 Halo 2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0R [arcmin]

0.0

0.2

0.4

0.6

0.8

1.0 Halo 3

Benefits of adding flexion:

• Improvement on the recovered profiles below 0.5 arcmin

• Recovery of small-scale substructure at the 10 arcsecscale



Conclusion:• New method to recover mass maps accounting for

• Reduced shear• Missing data• Shape noise

• Particularly efficient to recover small scale information(useful to detect and map galaxy clusters)

• Can be extended by including individual redshifts andflexion for high resolution cluster density mapping

• Can be trivially extended to 3D mapping (work in progress)


Mass mapping with sparsity priorThe optimisation problemPrimal-Dual algorithmHandling missing data and noise

Cluster density mappingIncluding individual galaxy redshiftsSet of cluster simulationsResults from shear only

Substructure recovery using FlexionWhy including flexion ?Our approachTests on simulationsConclusion

Documents

2D Mass Mapping - OU-LE3 Weak Lensing - Lausanne 2015 · 2D Mass Mapping OU-LE3 Weak Lensing - Lausanne 2015 François Lanusse Jean-Luc Starck, Adrienne Leonard, Sandrine Pires CosmoStat