Mathematical Challenges of the Euclid
Spatial ProjectJ.-L. Starck
CEA, IRFU, Service d'Astrophysique, France
jstarck@cea.fr
http://jstarck.free.fr

on behalf of the Euclid Consortium & ESA.
,

Euclid
Euclid, ESA Cosmic Vision: launch in 2020:• 800 members (450 researchers).
• 107 laboratories• 13 countries
Euclid is the result from the fusion in 2008 oftwo missions:• DUNE• SPACE

Understand the origin of the Universe’s accelerating expansion
probe the properties and nature of dark energy, dark matter, gravity
and distinguish their effects decisivelyby tracking their observational signatures on the• geometry of the universe: Weak Lensing + Galaxy Clustering• cosmic history of structure formation: WL, z-space distortion, clusters of galaxies
Gains in space:
Stable data: homogeneous data set over the whole sky
Systematics are small, understood and controlledHomogeneity : Selection function perfectly controlled- Observe 15 000 deg2 during 7 years in optical and nearinfrared wavelength. (forme) dans le visible ET- 1,2m Telescope, 4 bands- Photometric redshift (distance) for 1 000 000 000 galaxies.
- Spectroscopic IR measurement of 50 000 000 galaxies.
==> 850 Gbits of data per day.

http://www.euclid-ec.org/

Weak Lensing

Observer

Gravitational lens

Background galaxies

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Tomographic Weak Lensing

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Euclid Red Book

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Detection + Classification stars/galaxies

Galaxies

Stars

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Shape Parameters
The shear map (γ1, γ2)

γ1 = deformation along the x-axis,and γ2 at 45 degrees from it.
γ = γ1 + iγ 2 = γ e 2iθ

€γ2γ1Where the modulus represents theamount of shear and the phase represents
its direction.

PB 1: We need accurate measurements from noisy data

Point Spead Function
Galaxies are convolved by an asymetric PSF
Convolution with an isotropic PSF circularises galaxies.
Convolution with an anisotropic PSF also changes their shapes... coherently!
Worst from ground (large PSF, with unpredictable spatial / temporal variation).

PB 2: Shape measurements must be deconvolved

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Space Variant PSF

PB 3: We need to interpolate the PSF shape !

Intrinsic Ellipticities✓ Galaxies have an intrinsic ellipticipty

PB 4: We need to correct the measurements from the intrinsicellipticity
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From shear measurements
to shear map
Shear map

PSF Deconvolution

To Build a Shear Catalog
We need to solve a triple inverse problem !!!
1) Determine the PSF at any position from the measured PSF.
2) Measure the galaxy shear and correct it from the PSF.
3) Correct the shear from intrinsic ellipticities+ noise and missing data!!!

Missing dataShape measurement techniques:chronology of challenges~3 more challenges should beorganised before 2020

In GREAT10, there were 3 sub-challenges:
1.the main (galaxy) challenge: to measure the ellipticity of galaxies, assuming the PSF isknown
2.a star challenge: to estimate the PSF and interpolate it at the position of galaxies
3.a 'light' challenge (named 'kaggle') to attract more people

Shear Field and Mass Map�Ê(k) = κ̂(k)
B̂(k)

�

1
=|k|2
�

�

��
�
γ̂1 (k)k12 − k22
2k1 k2γ̂2 (k)2k1 k2 −k12 + k22���Aκ

PB 5: A simple reconstruction is very noisy

Typical Weak Lensing Pipeline
1) Take an image of the sky2) Measure galaxy shapes and distances3)c)

Bin the data ina) Create a 3D shear
field

redshift slicesd) Correlation function
tomographyb)

Reconstruct thematter distribution4) Constrain cosmological parameters

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Cosmological Parameters Constraints andHigh Order Statistics

Model3 (σ8=0.8, Ωm=0.36)

Model1 (σ8=1, Ωm=0.23)

Model2 (σ8=0.9, Ωm=0.3)

Model4 (σ8=0.7, Ωm=0.47)

Model5 (σ8=0.6, Ωm=0.64)

Cosmological Parameters Constraints andHigh Order Statistics- Aperture mass map = wavelets, but wavelets calculation is between 10
and 1000 times faster.
- A. Leonard, S. Pires, J.-L. Starck, "Fast Calculation of the Weak Lensing Aperture Mass Statistic", MNRAS, 423, pp 3405-3412, 2012.

- Wavelet Denoising + Wavelet Peak Counting is the most efficient
statistical tool to discreminate Cosmological Models.
- S. Pires, A. Leonard, J.-L. Starck, "Cosmological Parameters Constraint from Weak Lensing Data", MNRAS, 423, pp 983-992, 2012.

model 1model 2model 3model 4model 5wavelet peak counting On MRLENS Filtred MAPS (At scale OF about 1 arcmin)

Pseudo-3D Weak Lensing

R. Massey et al, Maps of the Universe’s Dark matter scaffolding,, Nature, Vol. 445,
pp. 286-290, 2007

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κ

Tomographic 3D Weak Lensing

Q=M measurements:Nδ+M x N (M > N)

number of bins in the source plane

N redshift bin for the density contrast

δ is sparse.

δQ spreads out the information in along
PB6: More unkown than measurements

κ bins.
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3D Weak Lensing
The convergence κ, as seen in sources of a given redshift bin, is the linear
transformation of the matter density contrast, δ, along the line-of-sight (Simonet al 2009):

κ = Qδ + N
�(i)3H02 ΩM w�+1
W (w)fK (w)Qi� =,dw
2c2a(w)
w�

with
W(i)

(w) =� w(i)0δ(r) ≡ ρ(r)/ρ − 1�
� fK (w − w )dw
fK (w� )�dzp(z)dw�

z=z(w� )where H0 is the hubble parameter, ΩM is the matter density parameter, c isthe speed of light, a(w) is the scale parameter evaluated at comoving distancew, and
−1/2
sin(K 1/2 w),
K>0KfK (w) = w,
K=0 ,(−K)−1/2 sinh([−K]1/2 w) K < 0gives the comoving angular diameter distance as a function of the comoving
distance and the curvature, K, of the Universe.
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Compressed Sensing* E. Candès and T. Tao, “Near Optimal Signal Recovery From Random Projections: Universal
Encoding Strategies? “, IEEE Trans. on Information Theory, 52, pp 5406-5425, 2006.
* D. Donoho, “Compressed Sensing”, IEEE Trans. on Information Theory, 52(4), pp. 1289-1306, April 2006.
* E. Candès, J. Romberg and T. Tao, “Robust Uncertainty Principles: Exact Signal Reconstruction
from Highly Incomplete Frequency Information”, IEEE Trans. on Information Theory, 52(2) pp. 489 - 509, Feb. 2006.

A non linear sampling theorem“Signals with exactly K components different from zero can be recovered perfectlyfrom ~ K log N incoherent measurements”

Reconstruction via non linear processing:⇒Application: Compression, tomography, ill posed inverse problem.

3D Weak Lensingκ=M measurements:Nδ

Q+M x N (M > N)

number of bins in the source plane

N redshift bin for the density contrast

δ is sparse.

Q spreads out the information inδ along κ bins.
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3D Weak Lensing
12min � δ �1 s.t. � κ − Qδ �Σ−1 ≤ �δ2
A. Leonard, F.-X. Dupe, J.-L. Starck, "A compressed sensing approach to 3D weaklensing", Astronomy and Astrophysics, arXiv:1111.6478, A&A, 539, A85, 2012.

Reconstructions of two clusters along the line of sight,
located at a redshift 0.2 and 1.0 (data binned into Nsp = 20 redshift bins, but aim toreconstruct onto Nlp = 25 redshift bins).

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Conclusions
Euclid will provide:– tight constraints over the broadest range of DE, MG models ever explored.
- Weak Lensing directly measures the mass (as opposed to light)- But require tight control of systematic.
- Algorithms need clearly to be improved in order to meet Euclid scientific requirements.
* Psf measurements* Shear on individual galaxies* Lensing statistics.
* 2D convergence map and 3D density contrast map.
- Recent developments in Math&Statististics (sparsity concept, compressed sensing,proximal optimization theory, etc) will be extremely useful to optimize Euclid Science.

http://www.euclid-ec.org
http://www.cosmostat.org
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