O(NlogN) Curved-sky Weak Lensing Simulations

Matthew R. Becker and Andrey V. KravtsovUniversity of Chicago

1Monday, June 14, 2010

Why should you care about weak lensing (and simulating it)?Weak lensing measurements are an essential component of all current and upcoming large-area optical imaging surveys (e.g. DES, LSST, JDEM, etc.) aimed (at least partially) at understanding dark energy.

Specifically, weak lensing will be used for both cosmic shear measurements and to measure the masses of galaxies and clusters of galaxies.

Unfortunately, there are a number of systematic effects associated with the interpretation of weak lensing signals.

Large-area simulations of weak lensing applied to mock galaxy catalogs and based on N-body structure formation models are needed to address these issues.

2Monday, June 14, 2010

Multiple-plane Ray Tracing

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Pseudocode for a multiple-plane ray tracing algorithm:

initialize ray locations at the observerFOR i=1,# of lens planes DO

compute 2D mass density of the lens planesolve 2D Poisson equation for the lensing potential compute ray deflections and lensing Jacobian at ray positions from the lensing potentialusing ray deflections and the lensing Jacobian advance rays to next lens plane

ENDFOR

light cone from N-body simulation is divided into a set of nested shells to form lens planes

rays are traced from observer to source

3Monday, June 14, 2010

Computational Challenges in Weak Lensing Simulations

• requirements for weak lensing simulations (in order to make mock galaxy catalogs):

1) self-consistently handle large, contiguous areas of sky (e.g. DES, LSST, JDEM, etc. will cover ≥5000 deg2)

2) resolve both the relevant physical (~50 kpc/h) and angular (5-10 arcsecond) scales

3) produce proper redshift dependence of shear and magnification bias effects

• current algorithms either scale nicely with resolution (e.g. O(NlogN) as opposed to O(N3/2)) or handle a curved-sky, but cannot do both

• distributed or shared memory computers are a must - a full-sky map of the shear, convergence, and lensed positions at 12.9 arcsecond resolution in single precision takes up 60 GB of RAM

4Monday, June 14, 2010

The Essential Idea We want to divide the sphere into a union of patches which can be ray traced in parallel using the flat-sky approximation.

Why?

• flat-sky patches can be divided evenly between all of the processors

• thus the code can handle as many rays and N-body particles as you have RAM/processors for

• as I will show later, this scheme can reduce algorithmic complexity of the Poisson solver to approximately O(NlogN) from O(N3/2)

Potential problem:

How do you maintain continuity in the lensing quantities between the flat-sky patches and also resolve larger-scale modes in the shear power spectrum?

Order 4 HEALPix Map w/ Patches

5Monday, June 14, 2010

A Hybrid SHT+FFT Poisson Solver

Key Idea:Solution from SHT (via HEALPix) encodes the large-scale modes in the lensing potential, enforces the boundary conditions over the sphere, and enforces continuity across the flat-sky patches. The FFTs on the flat-sky patch resolve the small scale modes.

Other advantages:• SHT is expensive O(N3/2) and here

we only have to do 1 analysis + 1 synthesis operation as opposed to 1 analysis + 5 synthesis operations

• very easy to parallelize on distributed memory systems

• can use any adaptive finite-volume Poisson solver - I have chosen a set of nested FFTs which is approximately O(NlogN).

flat-sky potentialflat-sky density

HEALPix density HEALPix potential

BCs from SHT

Point Mass Test of the Poisson Solver

6Monday, June 14, 2010

Code Validation: A Point Mass

fractional error ≤ 0.2%

angle [arcsec]

fractional error

fractional error

deflection angle

tangential shear

discontinuities across patches exist but are

small

7Monday, June 14, 2010

Code Scaling with Resolution (extrapolated to a full-sky)

8Monday, June 14, 2010

Summary and Applications• Summary:

• this new algorithm enables accurate, efficient, high-resolution, full-sky lensing calculations on moderate sized machines (~100 processors/cores)

• very portable - written in C with MPI and standard libraries like HDF5, GSL, and FFTW

• Some example applications:

• code will first be used for the DES Blind Cosmology Challenge - will test if the DES can recover input cosmology from realistic mock galaxy catalogs

• optical sky survey simulations with realistic weak lensing information - cosmic shear, galaxy-galaxy lensing, cluster lensing, etc.

• also could be used for high-resolution CMB lensing calculations

9Monday, June 14, 2010