Numerical simulations for gravitational lensing

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Numerical simulations for gravitational lensing

• Lauro Moscardini
• Dipartimento di Astronomia
• Università di Bologna, Italy
• lauro.moscardini_at_unibo.it

INAF-COSMOCT School on Gravitational Lensing,
Acireale, November 2006
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GAS-strophysics

• From a GL point of view, gas acts as matter
  exactly like dark matter no differences in its
  treatment
• The gas component is only a small fraction of the
  total mass (about 15), it is subdominant
• Gas affects the total density profile mostly in
  the central regions of galaxy clusters however,
  effects are not so large and in first
  approximation can be neglected
• Useful for comparison with data in other
  wavelength X-ray, Sunyaev-Zeldovich effect,
  galaxy dynamics

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Hydrodynamics the fundamental equations
Euler equation
Continuity equation
First law of thermodyn.
Equation of state
(proper coordinates)
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Eulerian methods (I)
The mean values of the quantities are computed by
using a fixed (or adaptive) grid. The
conservation equations are solved by using finite
differences methods. Examples Lax,
Lax-Wendroff, Flux-Corrected Transport (FCT),
Total Variation Diminishing (TVD), Piecewise
Parabolic Method (PPM), Advantage a better
description of gradients. Disadvantage problems
of resolution and memory due to the use of a grid
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Lagrangian methods
The hydrodynamical quantities are known at the
particle positions. Example SPH (Smoothed
Particle Hydrodynamics) Advantage better
resolution where the density is
high Disadvantage necessity of artificial
viscosity
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Comparison of different methods (I)
DARK MATTER
Frenk et al. 1999
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Comparison of different methods (II)
GAS
Frenk et al. 1999
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Comparison of different methods (III)
TEMPERATURE
Frenk et al. 1999
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Comparison of different methods (IV)
X-ray Luminosity
Frenk et al. 1999
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Comparison of different methods (V)
Frenk et al. 1999
gas
X-ray Luminosity
DM
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Comparison of different methods (VI)
Frenk et al. 1999
X-ray
DM
Temp.
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Comparing AMR and SPH cosmological simulations (I)
Adiabatic test

• GADGET is a smoothed particle hydrodynamics
  (SPH) code adopting entropy conserving
  formulation (Springel et al. 2001)
• ENZO is an eulerian adaptive mesh refinement
  (AMR) code employing PPM or ZEUS-like schemes
  (Bryan et al. 2001)

OShea et al. 2004
Slab of size 3 x 3 x 0.75 Mpc/h
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Comparing AMR and SPH cosmological simulations
(II)
General agreement in the distribution functions
of temperature, entropy and density for gas of
moderate to high overdensity, as found inside
dark matter haloes
OShea et al. 2004
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Self-similar ICMgravity only at work (Kaiser
1986)
Hydrostatic eq. T(M,z) ? M2/3(1z) Bremss
emiss. LX ? M?T1/2 ?????????????
Entropy
LX ? T2(1z)3/2
S ? (T/?2/3) ? T (1z)-2
Borgani et al. 2002
Gas density
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Facts against a self-similar ICM
Entropy excess in groups
ST/ne2/3 ??Entropy ramp at 0.1R200. Ponman
et al. 2003 ??Entropy profiles relatively
enhanced in groups S ?T2/3
E(z)-4/3 Pratt Arnaud 04
Pratt Arnaud 04
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How to break self-similarity?
(1) Non gravitational heating
Evrard Henry '91 Bower '96 Cavaliere et
al. '98 Balogh et al. '99 Tozzi Norman '01
Bialek et al. '01 Borgani et al. '02 Babul
et al. '02

• Introduce a characteristic TX scale

• Place the gas on a higher adiabat

? Prevent it from reaching high density
? Suppress the X-ray luminosity
Sources SN energy feedback, AGN activity
(2) Radiative cooling
Pearce et al. '99 Bryan '00 Muanwong et al.
'01 Bryan Voit '01 Wu Xue '02 Voit et
al. '02 Dave et al. '02 Kay et al. '02

• Introduce a characteristic entropy scale

• Selectively remove low-S gas with tcoollttH

? Increase gas entropy in the hot phase
? Decrease the X-ray luminosity
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How much resolution do we need?

• Virial mass and global velocity dispersion need
  to resolve the
• overall depth of the gravitational potential
  ?v2 GM/Rvir
• ??NDM 103 ?
  0.1 Rvir
• Gas virial temperature need to resolve
  adiabatic gas
• compression and accretion shocks T M2/3
  
• ??Ngas few 103 ?
  0.05 Rvir
• Sunyaev-Zeldovich decrement gas pressure
  integrated along
• the line-of-sight (?T/T)CMB ?gas TX
• ??Ngas few 103 ?
  0.05 Rvir
• X-ray luminosity ?f-f ?gas2 T1/2
• need to resolve the central structure of the
  gas distribution
• ??Ngas few 104 ?
  0.01 Rvir
• Gas cooling runaway process
• ? mgas 107 M? ?
  5 kpc
• Star formation, SN feedback, metal enrichment
• subgrid physics!

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Conclusions of Part I

• Gravity (almost) under control, problems only in
  the very central regions of collapsed objects
  (but relevant for GL!)
• Convergence for different numerical schemes when
  non-radiative hydrodynamics is considered
• but true clusters are more complex necessity
  of including always more sophisticated physical
  modelling. In this case numerical convergence is
  still an open issue

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THE END

• (for part I)

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Outline of Part II

• Fast review of relevant lensing equations
• Ray-tracing simulations
• Finding images
• Realistic source distributions
• Lensing by LSS (multiplane lensing)

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Why numerical methods are important for
gravitational lensing?

• Lensing equations are not linear strong
  distortions, multiple images, etc.
• Analytic models often give a too simplified
  description of lenses on the contrary, numerical
  lenses consistently include ellipticity,
  substructures, asymmetries, etc
• Better representation of the source distribution

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Lenses are complex systems
An exanple of analytic model a NFW axially
symmetric model modified by introducing
ellipticity in the lensing potential
Meneghetti et al. 2006
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Simulations vs. analytic models
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Simulations vs. analytic models
Lensing properties of lenses modeled using
analytic models can significantly differ from the
actual simulation results
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Standard assumptions for GL

• The Newtonian gravitational potential of the lens
  is small ?ltltc2.
• True assuming M1015M?/h and R1 Mpc/h ?
  ??GM/R? (2000 km/s)2
• Velocities in the lens system are small vltltc.
• True vlt1000 km/s
• The extent of lenses along the line-of-sight are
  small w.r.t. distances.
• True cluster size is about 1-2 Mpc/h while
  distances are larger than several hundred Mpc

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Lens equation
observed position
true position
deflection angle
Lensing potential
Jacobian matrix
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Convergence and shear
convergence
Shear components
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Critical lines caustics
We obtain infinite magnification where

• This happens on the
• tangential critical line
• and on the
• radial critical line
• from sources on the corresponding caustics.

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Critical lines caustics
(lens plane)
(source plane)
Closed curves on which the points satisfy
They correspond to zones of different multiplicity
They correspond to regions of max. magnification
arcs
(elliptical lens)
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Ray-tracing simulations
A bundle of light rays is traced from the
observer through lens plane, where the lens mass
distribution is projected. Once its deflection
angle is computed, the path of each ligth ray
towards the source plane is followed. Those rays
which hit the source are collected and used for
reconstructing the lensed image.
Numerical simulations provide a realistic
description of the mass distribution in the lens
plane. They represent the best way for studying
the effects of asymmetries and substructures
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Realistic ray-tracing
Example of a code Meneghetti et al.

• Lens modelling
• use realistic mass models for the lenses
• ? numerical clusters LSS in spite of analytic
  models
• Source modelling
• realistic source morphologies use results of
  shapelet decompositions of real galaxies in the
  GOODS-HST data
• realistic colours, modeled using decompositions
  in different wavebands
• Simulate observations
• incorporate observational noises (sky
  background, photon noise, instrumental noises,
  etc.)
• flexible code (works with set-ups of several
  instruments)
• easily extendable

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Numerical simulations our lenses

• 1) Cosmological simulations
• different cosmological models (dark energy
  models, modified gravity, non-Gaussianity, etc.)
• 2) Individual galaxy clusters
• large database
• high-resolution re-simulations
• different physics (non-radiative, cooling,
  feedback, magnetic field, turbulence, thermal
  conduction, )
• variety of cosmological models

Simulations made by the Virgo Consortium
Springel et al.
Advantages non-linear clustering, robust
statistics, inclusion of projection effects,
etc.,
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Ray-tracing codes a general flow-chart

• Create surface density maps from a 3D density
  field (analytic or numerical model)
• Trace a bundle of rays through a regular grid in
  the lens plane, then compute for them the
  deflection angle
• All relevant quantities for lensing (i.e.
  convergence, shear, magnification, etc.) can be
  easily obtained from the deflection angle
• Distribute a large number of galaxy sources on
  the source plane, starting from a regular grid,
  then adapt it to improve computational efficiency
• Reconstruct the distorted images of the
  background sources by following the path of each
  light ray from the observer to the source
• Measure the shape properties of each distorted
  image (length, width, curvature, etc.)

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Creating surface density maps (I)
The problem we have a discrete distribution of
mass particle. How is it possible to obtain a
smooth density distribution on a grid? Actually
for lensing problems, we need the (2D) projected
density, i.e. convergence
In general
The kernel W can be decomposed into three factors
along the cartesian directions
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Creating surface density maps (II)
The interpolation schemes the same as for PM
method!

• Nearest grid point
• (NGP)

2) Cloud in cell (CIC)
3) Triangular shaped cloud (TSC)
Then, simple projection along a line of sight
gives convergence
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Building maps for deflection angles
From convergence to deflection angles direct
summation
Warning like the PP method, it can be very slow,
depending on the number of grid cells N (i.e.
resolution) in fact CPU grows like N2. How to
reduce CPU without loosing resolution?

• Fast-Fourier techniques (periodicity
  required)

Notice K can be determined and tabulated once

• Tree implementation scaling is N log N

• (Adaptive smoothing)

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Resolution issues how large the grid cells
should be chosen?

• They should be small enough for describing the
  most important features of the lens
  substructures, asymmetries, etc.

• They should be large enough for avoiding
  microlensing effects (graininess) from individual
  particles and for avoiding large Poisson
  fluctuations

There is not a general rule depending on the
problem, find the best balance between
resolution, smoothing and particle noise!
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Imaging
From lens to source plane via lens equation
blue lens plane red source plane

• In order to obtain images, mapping process must
  be reversed
• Build a regular grid on the source plane
• Find the mapped source point yij for each grid
  point in the lens plane xij
• Look for the nearest neighbours ykl surrounding
  yij in the source plane
• Interpolate source properties (known at ykl) at
  the position yij
• Assign it to xij
• In this way determine the surface brightness at
  all points in the lens plane

blue source plane red lens plane
Bartelmann 2004
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An example
Using numerical differentiation all other
quantities (shear, magnification, convergence,
etc.) can be obtained from the deflection-angle
field
Warning grid resolution must be sufficient to
describe variation of lensing quantities
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Finding critical curves (I)
Remind the definition of critical curve and
caustics
For critical curve we have to find roots for
Derivatives can be computed numerically
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Finding critical curves (II)
A grid point is close to the critical line if,
and only if, the sign of the Jacobian determinant
changes between it and one or more of its nearest
neighbours.
Defining
when
Si,j is next to a critical curve. Then use lens
equation for caustics!
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An example
caustics
critical line
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Source grids

• In principle you need
• to distribute sources in the source plane,
  assigning to them realistic observational
  properties (magnitude, elliticity, etc.)
• then to compute their images

But interesting lensing events are often rare,
but well localized! Example strongly lensed
sources are close to a caustic. ? adaptive grid
to save CPU and to have more robust
statistics (but results must be corrected using
suitable weights)
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Finding images
Easy! Given a source, find the grid points on the
lens plane which are mapped sufficiently close to
it. Problem the squared grid from the image
plane is mapped in a distorted figure in the
source plane. How is it possible to decide if a
point is inside or out side? Solution use
triangles! They remain triangles, having always a
well defined interiour.
A source is inside the mapped triangle if
Images closer than the resolution limit (grid)
will be not resolved!
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Image reconstruction
Extended (elliptical) sources given their
central position yc1 and yc2, 1) look for points
for which
2) then group points belonging to the same image
by using a friends-of-friends algorithm 3) define
image properties (e.g. length, width, curvature
of arcs)
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Arc statistics
Miralda-Escude 1993

• Find all image points
• Define the circonference passing through the
  boundary points
• Compute (1) its centre (2) the most distant
  boundary point from (1) (3) the most distant
  boundary point from (2).
• Length and curvature are defined using the circle
  segment connecting points (2) and (3)
• Width is defined fitting the image with simple
  geometric figures having the same area and length

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A realistic model for the sources shapelets
Shapelets are a scalable version of the
eigenfunctions of the quantum harmonic
oscillator. 1D shapelets Dimensionless basis
functions
Dimensional basis functions (Shapelets)
Orthonormality
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A realistic model for the sources shapelets
Extension to higher dimensions 2D shapelets
Separability of coordinates
Dimensional functions
with
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Why shapelets?
Galaxy 2D surface brightness can be decomposed
using these basis functions and expressed in
terms of coefficients

• Real galaxies are
• continuous
• peaked
• localized
• round

So are shapelets! ? very good for describing
galaxies (Refregier 2003)
nlt20 is usually sufficient for good decompositions
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Application to the GOOD HST/ACS data
approx. 1700 galaxies have been decomposed from
the GOODS HST/ACS data (Melchior, Meneghetti
Bartelmann 2006)
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Creating new galaxies
For each galaxy, the best-fitting coefficients
InGOODS and their errors have been saved New
galaxies are generated by slightly kicking the
coefficients
Shapelet (9x9) image
Original GOODS image
kicked image
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Including noises
Several noises must be added to the images to
resemble real observational conditions

• photon noise
• sky backgroud
• seeing
• instrumental noises

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Examples of possible applications (I)
Meneghetti, Melchior, Grazian, Moscardini, Dolag,
De Lucia Bartelmann, in prep.
a galaxy cluster _at_ z0.6 HST/ACS z-band texp3600
sec
1 arcmin2 deep field HST/ACS z-band texp10600 sec
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Examples of possible applications (II)
texp2000 sec airmass1
frame scale2.9
I-band LBT observation with 0.5 seeing
original image of a GOODS galaxy
I-band LBT observation without seeing
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Examples of possible applications (III)
a perfect observation
the original source
an LBT observation with 0.7 seeing
an HST observation
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Multiple lens planes
Weak lensing from LSS requires the cosmic volume
to be split into multiple lens planes. In this
case
reduced defl. angle
The lensing experienced by a light ray can be
computed as
Hamana Mellier 2001
where
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Resolution problems
Weakness of the lensing signal from LSS and
multiplicity of lens planes can introduce
problems of mass and angular resolution. 1) Small
resolution for lens planes close to the observer
(larger angles for a given physical scale) 2)
Shot-noise introduced by discretisation of mass
into particles 3) Different efficiency at
different redshifts, depending on both the
distance from the observer and structure growth
mp6.8 1010 M?/h
Pfrommer et al. 2004
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Examples of possible applications
1 square degree convergence map for a LCDM
model zs1
Pace et al. in prep
Resolution problems at larger l, becoming more
evident at lower redshifts in fact a given angle
corresponds to a larger physical scale at smaller
distances from the observer
Convergence power spectrum comparison simulation
results vs. analytic predictions
Pfrommer et al. 2004
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Conclusions

• Numerical simulations represent a very useful
  tool to test and calibrate lensing techniques on
  realistic mass distributions
• Moreover they provide useful fitting relations
  (and estimates of their dispersions) which can be
  applied in analytic models
• Realistic ray-tracing simulations can be used to
  evaluate expected results for present and future
  experiments and their robustness.