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Weak Lensing
and
Dark Energy Cosmology
Tong-Jie Zhang[张同杰]
Department of Astronomy, Beijing Normal University
Cosmology Workshop
Institute of High Energy Physics, Chinese Academy of Sciences
2008/12/08
3-D : Accelerating Universe
WMAP
3-D Universe: 3 dark sides
(1). Our Universe—Dark energy
(2). Dark Matter [halo](暗物质[晕])
(3). Dark ages(黑暗时代)
Outline
• 0. Basic of Gravitational lensing
• 1. Dark Energy and Neutrino Mass Constraints
from WL, SN Ia and RGA
• 2. The signatures of BAOs on the convergence
power spectrum of weak lensing.
• 3. Application of wavelet on Weak lensing
0. Basic of Gravitational lensing
Schematic Diagram of Gravitational Lensing (引力透镜示意图)
Physics of Gravitational Lensing (GL)
Bending of Light under Gravity
Light will follow the straightest
possible path through flat space time.
If spacetime is curved near a massive
object, so the trajectory of light is also
curved.
Observational Event of of Gravitational Lensing
Einstein’s Cross
an Einstein ring
galaxy directly behind a galaxy
HST Image of a gravitational lens in galaxy cluster
Category of GL
• Strong gravitational
lensing
• Weak gravitational
lensing
Gravitational lens Theory—Sketch of a typical gravitational
lens system
Deflection angle
• General Relativity: for a point mass M
Lensing equation or ray-trace equation
Position of source
Position of image
Lensing equation
Position of source
Multiple images can be produced
if lens is strong
Position of image
Tong-Jie Zhang ApJ 602, L5-8(2004) [astro-ph@0401040]
Convergence and shear
K>=1  strong
K<<1  weak
Deflection potential
Distortion and Magnification
Shear:
Magnification:
Det
Critical curves in lens plane;
Caustics in source plane
Strong lensing
• Sources are close to the caustic lines.
• K >=1 and |r|>=1: The convergence and
shear are strong enough to produce giant arcs
and multiple images.
The probability for strong lensing
E(z) and f (M, z): dependent on cosmological model
CLASS observation
The Cosmic Lens All-Sky Survey CLASS):
An international (USA, UK and Netherlands) collaborative
project to map more than 10,000 radio sources in order to create
the largest and best studied statistical sample of gravitationally lensed
systems.
Sample:
Well-defined statistical sample: 8958
Multiply imaged sourses: 13
P_ob=N(>\theta)/8958
Lensing models
• SIS
• GNFW
Image separation probability for GNFW model
Tong-Jie Zhang ApJ 602, L5-8(2004) [astro-ph@0401040]
Constraint on potential
Kyu-Hyun Chae et al ApJ 607, L71-74(2004)
Weak lensing (cosmic shear)
Cosmic shear is the distortion of the shapes of background
galaxies due to the bending of light by the potentials
associated with large-scale structure in the universe.
Wek lensing regime:
K <<1 and |r|<<1
Distortion of background images: shape and correlation
Before lensed
After lensed
Measurement
• The ellipticity of galaxy
ellipticity
and shear
and the intrinsic
Weak lensing shear: spin-2 polarization field
y
a
The mean expectation of source ellipticities and alignment
b
Φ
x
Shear component
• The tangential shear and the 45 degree rotated
shear in the local frame defined by the line
connecting the pair of galaxies
b
a
aa
xi
b
xj
θ
Shear correlation function
Two-point cosmic shear statistics
1. shear correlation
2. the top-hat filtered variance of the shear
3. the variance of the aperture-mass
Power spectrum of convergence
OCDM
CDM
CDM
(linear)
Observational Constraint on cosmology
•H. Hoekstra, Y. Mellier, L. van Waerbeke, E. Semboloni, L. Fu et al, The Astrophysical Journal,
647:116–127, 2006
Joint constraint using WL and CMB
Contaldi et al, PRL, 90, 2003
1. Dark Energy and Neutrino Mass Constraints
from WL, SN Ia and RGA
• Yan, Gong, Tong-Jie Zhang, Tian Lan and Xue-Lei Chen
• (astro-ph@arXiv: 0810.3572) Sumitted to ApJ
The existence of non-zero neutrino masses
• has been established firmly by the experiments
detecting
• [1]. atmospheric neutrinos,
• [2]. solar neutrinos
• [3]. reactor neutrinos
• [4]. accelerator beam neutrinos
• The neutrinos were still relativistic at the decoupling
epoch.
• However, they are definitely non-relativistic at the
present epoch, as the neutrino oscillation experiments
have shown.
• Therefore, the matter density must contain the neutrino
contribution when they are non-relativistic,
Current constraints on neutrino mass:
F.D.Bernardis et al. 2008
WMAP5:
WMAP5 Results on neutrino
• WMAP5:
Weak Leasing and Neutrino Mass
Free streaming effect
,
W. Hu & D. J. Eisenstein, 1998, ApJ
• The massive neutrinos could suppress the matter
power spectrum on small scales, due to their free
streaming, thus reducing the convergence power
spectrum of the weak lensing, which is sensitive
to the small scale matter distribution.
• Weak lensing is therefore a powerful
measurement for both the dark energy and the
massive neutrinos.
The Likelyhood of WL:
Shear correlation function (Crittenden et al. 2002):
The likelyhood:
Other Likelyhood:
,
• SN Ia:
• RGA:
• BAO:
Data sets:
• Weak lensing data
CFHLST-wide, 22 deg^2 (Fu et al. 2008);
RCS, 53 deg^2 (Hoekstra et al. 2002)
• SN Ia data
SCP “Union” data, 307 samples (Kowalski et al. 2008)
• RGA (relative galaxy ages)
H(z) from GDDS, 9 samples (Simon et al. 2005)
• BAO data
A at z=0.35 (Eisenstein et al. 2005)
Results:
[1]. Weak Lensing Constraints on w:

Weak constraint on
w for current WL data

WL+SN+RGA+BAO:
w = -1.0 +0.19 -0.21
at 95.5% C.L.
(w = -1.0 +0.14 -0.11 for WMAP5)

wCDM
The similar degeneracy
direction and constraint
ability for SN Ia and RGA
[2]. Weak Lensing Constraints on Σmv
Σmv<=0.4eV
Σmv<=0.8eV
at 95.5% C.L.
[3]. Constraints on w and Σmv:

Weak degeneracy
between w andΣmv

Compatible and
comparable with the
results of WMAP5
2. The signatures of BAOs on the convergence power
spectrum of weak lensing
• In the early universe prior to recombination, the free electrons couple the baryons to
the photons through Compton interactions, so these three species move together as a
single fluid.
• The primordial cosmological perturbations on small scales excite sound waves in this
relativistic plasma, which results in the pressure-induced oscillations and acoustic peak.
•The memory of these
baryon acoustic
oscillations (BAOs) still
remain after the epoch
of recombination.
Two Effect of BAO after the epoch of recombination
• [1]. The BAOs leave their imprints through the propagating of photons
on the last scattering surface and produce a harmonic series of maxima
and minima in the anisotropy power spectrum of the cosmic microwave
background (CMB) at z=1000.
• [2]. Due to the significant fraction of baryons in the universe, BAOs can
also be imprinted onto the latetime power spectrum of the nonrelativistic matter.
Acoustic Oscillations in the
Early Universe and Today
Christopher J. Miller,1 Robert
C. Nichol,1 David J. Batuski2
22 JUNE 2001 VOL 292
SCIENCE
BAO on the latetime power spectrum of the non-relativistic matter
BAOs can give rise to the wiggles in the matter power
spectrum:
• (a). Correlation function of galaxies (z=0)
• (b). The power spectrum of 21 cm emission generated from
the neutral hydrogen from the epoch of reionization
through the underlying density perturbation
• (c). The power spectrum gravitaional lensing: strong and
weak
(a). BAO
on Correlation
galaxies(z=0):
• Each initial
overdensity
(in DM & function
gas) is an of
overpressure
that launches
Waves
in Matter
a spherical sound wave.Sound
This wave
travels
outwards at 57% of the
speed of light.
• Pressure-providing photons decouple at recombination. CMB travels
to us from these spheres.
• Sound speed plummets. Wave stalls at a radius of about 100 Mpc.
• Overdensity in shell (gas) and in the original center (DM) both seed the
formation of galaxies. Preferred separation of 100 Mpc.
100 Mpc
D. J. Eisenstein et al., Astrophys. J. 633, 560 (2005)
(b). BAO on the power spectrum of 21 cm emission
Xiao-Chun Mao and Xiang-Ping Wu, ApJ, 673: L107–L110, 2008
(c). BAO on the power spectrum gravitational
lensing: weak
The matter power spectrum
Tong-Jie Zhang, Qiang Yuan,
Tian Lan
astro-ph@arXiv:0812.0521
The convergence power spectrum of
weak lensing
Tong-Jie Zhang, Qiang Yuan, Tian Lan
arXiv:0812.0521
The statistical errors in the
measurements of weak lensing power spectrum
Tong-Jie Zhang, Qiang Yuan, Tian Lan
astro-ph@arXiv:0812.0521
Conlusions
• [1]. The BAOs wiggles can be found in both of the linear
and nonlinear convergence power spectra of weak lensing
at about 40 <= l<= 600, but they are weaker than that of
matter power spectrum.
• [2]. Although the statistical error for LSST are greatly
smaller than that of CFHT and SNAP survey especially at
about 30 < l < 300, they are still larger than the their
maximum variations of BAOs wiggles.
• [3]. Thus, the detection of BAOs with the ongoing and
upcoming surveys such as LSST, CFHT and SNAP survey
confront a technical challenge.
3. Application of wavelet on Weak lensing
• Construction of Convergence
Theoretical expression
N-body simulation parameters
•
•
•
•
Itanium Beowulf cluster at CITA
1024^3 mesh resolution
512^3 particles
output periodic surface density maps at 2048^2
resolution
• an initial redshift z_i=50, 1000 steps
• comoving box size L=200h^{-1} Mpc
Parameters
• a Hubble constant h=0.7
• A scale invariant n=1 initial power spectrum
• A flat cosmological model with \Omega_m +
\Lambda = 1
• \Omega_m=0.3
• \sigma_8=0.82
Stacking of map (here just a example)
Produce
Wavelet
• Pls see papers written by Prof.Fang Li-Zhi
such as:
1. Fang Li-zhi and W. Thews Wavelet in Physics.
Would Scientific Singapore
2. Fang Li-zhi et al’s papers appeared in ApJ.
Non-Gaussianity
• In the standard model of cosmology, fluctuations start
off small, symmetric, and Gaussian. Even in some
non-Gaussian models such as topological defects,
initial fluctuations are still symmetric: positive and
negative fluctuations occur with equal probability.
• As fluctuations grow by gravitational instability, this
symmetry can no longer be maintained: overdensities
can be arbitrarily large, while underdense regions can
never have less than zero mass. This leads to NonGaussianity in the distribution of matter fluctuations.
Non-Gaussianity using wavelet :
Skewness and Kurtosis
No significant non-Gaussianity can be identified
from the third and fourth order cumulants.
Jesús Pando, David Valls-Gabaud, and Li-Zhi Fang, PRL, Vol. 81, p. 4568-4571 ( 1998)
Weak lensing
R=3*60/2^j [arcmins];
J_max=11 for 2048
Tong-Jie Zhang, Ue-Li Pen, Li-Zhi Fang,
in preparation for submitting to ApJ
The significant non-Gaussianity can be identified on small scale.
My appeared papers related to Lensing
strong or weak
•
(1). Reconstruction of the One-Point Distribution of Convergence from Weak Lensing by LargeScale Structure
Zhang Tong-Jie; Pen Ue-Li
The Astrophysical Journal [ApJ], Volume 635, Issue 2, pp. 821-826 (12/2005)
•
(2) Gravitational Lensing by Dark Matter Halos with Nonuniversal Density Profiles
Zhang, Tong-Jie
The Astrophysical Journal [ApJ], Volume 602, Issue 1, pp. L5-L8.(02/2004)
•
(3). Optimal Weak-Lensing Skewness Measurements
Zhang, Tong-Jie; Pen, Ue-Li; Zhang, Pengjie; Dubinski, John
The Astrophysical Journal [ApJ], Volume 598, Issue 2, pp. 818-826. (12/2003)
•
(4). Detection of Dark Matter Skewness in the VIRMOS-DESCART Survey: Implications for
Omega0
Pen, Ue-Li; Zhang, Tongjie; van Waerbeke, Ludovic; Mellier, Yannick; Zhang, Pengjie;
Dubinski, John
The Astrophysical Journal [ApJ], Volume 592, Issue 2, pp. 664-673. (08/2003)
Thanks!