WSLAP Weak and Strong Lensing Analysis Package

J.M. Diego1, H. Sandvik2, P. Prototapas3, M. Tegmark4

1 IFCA (Santander)2 Max-Planck (Garching)3 Harvard (Smithsonian)4 MIT Rencontres de Moriond

La Thuile, March 2006

http://darwin.cfa.harvard.edu/SLAP/

What is this ?Method to find a combined solution of strong and/or weak lensing data.

Uses fast algorithms to invert the problem (a combined solution can be found in few seconds).

Speed also allows to find multiple solutions and to estimate their dispersion.

Non-parametric, i.e. No assumptions about the Mass profile are needed.

A1689

Two alternatives.

Parametric vs Non-parametric

Parametric methods

Big and smooth DM halo containing most of the mass.Many subhalos on top of the galaxies.Each halo contributes with ~ 7 parameters.

Non-Parametric methods

Starts with regular grid.Each cell contributes with 1 parameter.

Basics

Lens equation

DLS

DS

DL

DS

DS DLS

DLSDS

Deflection Angle

M’

M’

4 Gc2DL

’’|2

d’

’’

Approximation.

MDLSDS

Non-Parametric methods

Weak & strong lensing

dx/dx – dy/dydx/dy = dy/dx

xx+ x

The problemSystem of linear equations with 2Nd equations and (2Ns + Nc) unknowns.

X

yy+ y

WSLAP : A fast simulation tool.

Fast computation of the kernel .

Use linear algebra to compute theta positions.

Same technique used to calculate the shear distortions.

X

WSLAP : A fast analysis tool.

Take advantage of algebraic formulation of the problem.

Fast computational techniques allow to find the solution faster (bi-conjugate gradient, SVD, quadratic programming etc).

Cluster from Yago Ascasibar

X

Finding the solution

Conjugate Gradient

Singular Value Decomposition

Quadratic Programming

R = Xf(X) = RTC-1R = a – bx + (1/2)xAx

UT W VX = (VT W-1 U)

Min [f(X)], X > 0

Weight of the data sets.

Simulations

SL WL

Results

True SL

WL SL + WL

SL vs WL

SL

WL

SL + WL

Dispersion of the solution

Optimal Basis

Compact basis like the Gaussian render better results than for instance power laws or isothermal profiles.

Other more exotic basis can be used as well.

Extended basis like Legendre polynomia perform poorly.

Application to A1689 (SL only)

Application to A1689 (SL only)

http://darwin.cfa.harvard.edu/SLAP/WSLAP Beta version available at