Transcript Document

Non-Gaussian signatures in
cosmic shear fields
Masahiro Takada
(Tohoku U., Japan)
Based on collaboration with
Bhuvnesh Jain (Penn) (MT & Jain 04, MT & Jain 07 in prep.)
Sarah Bridle (UCL) (MT & Bridle 07, astro-ph/0705.0163)
Some part of my talks is based on the discussion of WLWG
Oct 26th 07 @ ROE
Outline of this talk
• What is cosmic shear tomography?
• Non-Gaussian errors of cosmic shear fields
and the higher-order moments
• Parameter forecast including non-Gaussian
errors
• Combining WLT and cluster counts
• Summary
Cosmological weak lensing – cosmic shear
• Arises from total matter clustering
– Not affected by galaxy bias uncertainty
– well modeled based on simulations
(current accuracy, <10% White & Vale
04)
• A % level effect; needs numerous
(~108) galaxies for the precise
measurements
z=zs
past
z=zl
observables
a b
ab
 1   cos 2

present
z=0
 2   sin 2
Weak Lensing Tomography
• Subdivide source
galaxies into several
bins based on photoz derived from
multi-colors (e.g.,
Massey etal07)
• <zi> in each bin
needs accuracy of
~0.1%
• Adds some ``depth’’
information to
lensing – improve
cosmological paras
(including DE)
(e.g., Hu 99, 02, Huterer
01, MT & Jain 04)
＋m(z)
Tomographic Lensing Power Spectrum
• Tomography
allows to extract
redshift
evolution of the
lensing power
spectrum.
• A maximum
multipole used
should be like
l_max<3,000
Tomographic
Lensing Power
Spectrum (contd.)
zS
   m 0  dz L
0
d LS ( z L , z S )d L ( z L )
 ( zL , θ)
d S ( zS )
• Lensing PS has a less feature shape, not like CMB
– Can’t better constrain inflation parameters (n_s and
alpha_s) than CMB
– Need to use the lensing power spectrum amplitudes
to do cosmology: the amplitude is sensitive to A_s, de0
(or m0), w(z).
Lenisng tomography (condt.)
• WLT can be a powerful probe of DE energy density and its redshift evolution.
• Need 3 z-bins at least, if we want to constrain DE model with 3 parameters
(_de,w0, wa)
• Less improvement using more than 4 z-bins, for the 3 parameter DE model
An example of survey parameters
(on a behalf of HSCWLWG)
Area: ~2,000 deg^2
Filters: B~26,V~26,R~26, i’~25.8, z’~24.3
Nights: 150-300 nights

  (Cl ) 
2
1   

 

 Cl  (2l  1)lf sky  ng Cl
2
2




2
• PS measurement error  (survey area)^-1
• Requirements: expected DE constraints should be comparable
with or better than those from other DE surveys in same time
scale (DES, Pan-Starrs, WFMOS)
• Note: optimization of survey parameters are being investigated
using the existing Suprime-Cam data (also Yamamoto san’s talk)
Non-linear clustering
• Most of WL signal is from
small angular scales,
where the non-linear
clustering boosts the
lensing signals by an order
of magnitude (Jain &
Seljak97).
• Large-scale structures in
the non-linear stage are
non-Gaussian by nature.
• 2pt information is not
sufficient; higher-order
correlations need to be
included to extract all the
cosmological information
• Baryonic physics: l>10^3
Non-linear
clustering
l_max~3000
Non-Gaussianity induced by structure formation
• Linear regime O()<<1; all the Fourier modes of the
perturbations grow at the same rate; the growth rate D(z)
– The linear theory, based on FRW + GR, gives robust, secure predictions
 k ( z )  D( z ) k ( z  1000)
• Mildly non-linear regime O()~1; a mode coupling between
different Fourier modes is induced
– The perturbation theory gives the specific predictions for a CDM model
 ( z )   (1)   ( 2)   (3)  
 k( 2)   d 3k1  d 3k 2 F (k1 , k 2 ) k(1) k(1) (k  k1  k 2 )
1
2
• Correlations btw density perturbations
of
different scales
( 2)
) 4
 3  ( (1)of
) 2 non-linear
 ( (1structure
)  0 formation,
arise as a
consequence
originating from the initial Gaussian fields
• Highly non-linear regime; a more complicated mode coupling
• However, the non-Gaussianity is fairly accurately predictable
– N-body simulation based predictions are needed (e.g., halo model)
based on the
CDM modelL
NL
P (k , z )  f [ Pi (k ), z;  m , b ,....]
Aspects of non-Gaussianity in
cosmic shear
• Cosmic shear observables are non-Gaussian
– Including non-Gaussian errors degrades the
cosmological constraints?
– Realize a more realistic ability to constrain
cosmological parameters
– The dependences for survey parameters (e.g., shallow
survey vs. deep survey)
• Yet, adding the NG information, e.g. carried by the
bispectrum, is useful?
Covariance matrix of PS measurement
(MT & Jain 07 in prep.)
• Most of lensing signals are from non-linear scales: the errors are
non-Gaussian
• PS covariance describes correlation between the two spectra of
multipoles l1 and l2 (Cooray & Hu 01), providing a more realistic
estimate of the measurement errors
• The non-Gaussian errors for PS arise from the 4-pt function of
mass fluctuations in LSS
Cov[P(l1 ),P(l2 )]  P(l1),P(l2 )  P(l1 )P(l2 )

Gaussian errors 
Non-Gaussian errors 
l1

l2
l1

 

P(l1 )  
f sky l1l1 
n g 
l l
2
2
l2
1 2
1

4f sky
d 2l1'
 l1 2l l
1 1
l1
l2
d 2l 2'
 l2 2l l T(l1',l1',l2',l2' ) l1
2
2
l2
Correlation coefficients of PS cov. matrix
rij 
Cov[ Pi , Pj ]
Cov[ Pi , Pi ]Cov[ Pj , Pj ]
• Diagonal: Gaussian
Off-diagonal: NG, 4pt function
• 30 bins: 50<l<3000
• If significant
correlations, r_ij1
• The NG is stronger at
smaller angular scales
• The shot noise only
contributes to the
Gaussian (diagonal)
terms, suppressing
significance of the
NG errors
w/o shot noise
with shot noise
Correlations btw Cl’s at different l’s
• Principal
component
decomposition
of the PS
covariance
matrix
SiaC(li )S jbC(l j )  aab
Power spectrum with NG errors
(in z-space as
well for WLT)
• The band
powers btw
different ells are
highly
correlated (also
see Kilbinger &
Schneider 05)
• NG increases
the errors by up
to a factor of 2
over a range of
l~1000
• ell<100, >10^4,
the errors are
close to the
Gaussian cases
Signal-to-noise ratio: SNR
• Data vector: power spectra binned in multipole range,
l_min<l<l_max, (and redshifts)
D  P11(l1),P12 (l2 ), ,P(ns 1)ns (ln1),Pns ns (ln )
• In the presence of the non-Gaussian errors, the signal-tonoise ratio for a power spectrum measurement is
2
n


1
S

    Di CovPnm (l), Pn' m' (l')ij D j
N  i, j1
• For a larger area survey (f_sky ) or a deeper survey
(n_g ), the covariance matrix gets smaller, so the
 signal-to-noise ratio gets increased; S/N
Signal-to-noise ratio: SNR (contd.)
Gaussian
Non-Gaussian • Multipole range:
50<l<l_max
• Non-gaussian
50<l<l_max
errors degrade
S/N by a factor
of 2
• This means that
the cosmic shear
fields are highly
non-Gaussian
(Cooray & Hu 01;
Kilbinger &
Schneider 05)
The impact on cosmo para errors
• We are working in a multi-dimensional parameter space (e.g. 7D)
_de
error ellipse
w_0
_de
Non-Gaussian
Error
w_0
w_a
w_a
n_s
n_s
• Volume of the ellipse: VNG2VG
_mh^2
_mh^2
• Marginalized error on each parameter  length of the principal
_bh^2
axis: NG～2^(1/Np)G (reduced by the dim. of _bh^2
para space)
– Each para is degraded by slightly different amounts
– Degeneracy direction is slightly changed
An even more direct use of
NG: bispectrum
Bernardeau+97, 02, Schneider &
Lombardi03, Zaldarriaga &
Scoccimarro 03, MT & Jain 04, 07,
Kilbinger & Schneider 05
l
l
l1
l1
l1
l2
l2
l3
l3
z s1
 (i ) ( j )  C(ij ) (l )  W P :
2
2
GL 
l1
l3
l2
l2
l3
zs 2
given as a function of separation l
3
 (i ) ( j ) ( k )  B(ijk) (l1 , l 2 , l3 )  WGL
P4 : given as a function of triangles
A more realistic parameter forecast
MT & Jain in prep. 07
WLT (3 z-bins) + CMB
• Parameter errors: PS, Bisp, PS+Bisp
– G: (_de)=0.015, 0.014, 0.010  NG: 0.016(7%), 0.022(57), 0.013(30)
– (w0)= 0.18, 0.20, 0.13  0.19(6%), 0.29(45), 0.15(15)
– (wa)= 0.50, 0.57, 0.38  0.52(4%), 0.78(73), 0.41(8)
• The errors from Bisp are more degraded than PS
– Need not go to 4-pt!
• In the presence of systematics, PS+Bisp would be very powerful
to discriminate the cosmological signals (Huterer, MT+ 05)
WLT + Cluster Counts
MT & S. Bridle astro-ph/0705.0163
• Clusters are easy to find from WL survey itself: mass peaks
(Miyazaki etal.03; see Hamana san’s talk for the details)
• Synergy with other wavelength surveys (SZ, X-ray…)
– Combining WL signal and other data is very useful to discriminate real
clusters from contaminations
• Combing WL with cluster counts is useful for cosmology?
– Yes, would improve parameter constraints, but how complementary?
• Cluster counts is a powerful probe of cosmology, established
method (e.g., Kitayama & Suto 97)
Angular number counts:
w0=-1  w0=-0.9
d 2V
N cl   dz
dm n(m)S (m; obs)

ddz
Mass-limited cluster counts vs.
lensing-selected counts Hamana, MT, Yoshida 04
Convergence map
2 degrees
Halo distribution
• Mass-selected sample (SZ) vs lensing-based sample
Miyazaki, Hamana+07
QuickTimeý Dz
TIFFÅià•
èkÇ »ÇµÅj êLí£ ÉvÉçÉOÉâ ÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉǾå©ÇÈÇž ǽDžÇÕïKóvÇ­Ç•
ÅB
Mass
Light (galaxies)
Secure candidates
X-ray
A closer look at nearby clusters (z<0.3)
~30 clusters (Okabe, MT, Umetsu+ in prep.)
QuickTimeý Dz
TIFFÅià•
èkÇ »ÇµÅj êL í£ÉvÉçÉOÉâÉÄ
ǙDZÇÃÉsÉNÉ`ÉÉǾ å©ÇÈǞǽ Ç…ÇÕïKóvÇ­Ç•
ÅB
• Subaru is superb for WL measurement
• A detailed study of cluster physics (e.g. the nature of dark matter)
Redshift distribution of cluster samples
Cross-correlation between CC and WL
Cluster
A patch of the
observed sky
Shearing effect
on background
galaxies
• If the two observables are drawn from the same survey region, the two
probe the same cosmic mass density field in LSS
• Around each cluster, stronger shear signal is expected: up to ~10% in
induced ellipticities, compared to a few % for typical cosmic shear
• A positive cross-correlation is expected: Clusters happen to be
less/more populated in a given survey region than expected, the
amplitudes of <> are most likely to be smaller/greater
• Note that <  >: 2pt, cluster counts (CC): 1pt =>no correlation for
Gaussian fields
Cross-correlation btw CC and WL (contd.)
M/M_s>10^13
M/M_s>10^14
M/M_s>10^15
• Shown is the
halo model
prediction for
the lensing
power spectrum
• A correlation
between the
number of
clusters and the
ps amplitude at
l~10^3 is
expected.
Cross-covariance between CC + WL
• Cross-covariance between PS binned in l and z and the cluster
counts binned in z
• The cross-correlation arises from the 3-pt function of the cluster
distribution and the two lensing fields of background galaxies
– The cross-covariance is from the non-Gaussianity of LSS
• The structure formation model gives specific predictions for the
cross-covariance
SNR for
CC+WL
• The crosscovariance leads
to degradation
and improvement
in S/N up to
~20%, compared
to the case that
the two are
independent
Parameter forecasts for CC+WL
lensing-selected sample
mass-selected sample
WL
CC+WL
CC+WL with Cov
• Lensing-selected sample with detection threshold S/N~10 contains clusters with
>10^15Msun
• Lensing-selected sample is more complementary to WLT, than a mass-selected
one? Needs to be more carefully addressed
HSCWLS performance (WLT+CC+CMB)
• Combining WLT and CC does tighten the DE constraints, due to
their different cosmological dependences
• Cross-correlation between WLT and CC is negligible; the two are
considered independent approximately
Real world: issues on systematic errors
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E/B mode separation as a diagnostics of systematics
Non-gaussian signals in weak lensing fields
Theoretical compelling theoretical modeling of DE
Shape measurement accuracies vs. galaxy types, morphology, magnitudes…
Data reduction pipelines optimized for weak lensing analyses
Exploring a possibility to self-calibrate systemtaics, by combining different methods
Non-linearities in lensing; reduced shear needs to be included?
Intrinsic alignments
Source clustering, source-lens coupling
Usefulness of Flexions?
Develop a sophisticated photo-z code
Photo-z vs. color space?
Requirement on spec-z sub-sample; from which data?
N-body simulations (initial conditions, how to work in multi-dimenaional parameter space for N-body simulations,
the strategy…)
DE vs. modified gravity
Fourier space vs. real-space; explore an optimal method to measure power spectrum from actual data, with complex
survey geometry
Exploring a code of likelihood surface in a multi-dimensional parameter space (MCMC); how to combine with other
probes such as CMB, 2dF/SDSS, ….
Can measure DE clustering or neutrino mass from WL or else with HSC?
Defining survey geometry: a given total survey area, many small-patched survey regions vs. continuous survey region
Adding multi-color info for WL based cluster finding; color properties of member ellipticals would be useful to
discriminate real lensing mass peaks as well as know the redshift
How to calibrate mass-observable relation for cluster experiments? WL + colors + SZ + X-ray?
Constraining mass distribution within a cluster with HSC WL survey; mass profile, halo shape, etc
Strong lens statistics
Imaging BAO
Man power problem: who and when to work on these issues?
…
Issues on systematics: self-calibration
• If several observables (O1,O2,…) are drawn from the same
survey region: e.g., WLPS, WLBisp, CC,…
– Each observable contains two contributions (C: cosmological signal and
S: systematics)
O1i  C1i  S1i , O2 j  C2 j  S 2 j ,.....
• Covariances (or correlation) between the different obs.
– If the systematics in different obs are uncorrelated
Cov[O1i , O2 j ]  O1i O2 j  C1i C2 j
– The cosmological covariances are fairly accurately predictable
• Taking into account the covariances in the analysis could allow
to discriminate the cosmological signals from the systemacs –
self-calibration
– Working in progress
Summary
• The non-Gaussian errors in cosmic shear fields arise from
non-linear clustering in structure formation
– The CDM model provides the specific predictions, so the NG errors are
in some sense accurately predictable
• Bad news: the NG errors are very important to be included for
current and, definitely, future surveys
– The NG degrades the S/N for the lensing power spectrum measurement
up to a factor of 2
• Good news: the NG impact on cosmo para errors are less
significant if working in a multi-dimensional parameter space
– ~10% for 7-D parameter space
• WLT and cluster counts, both available from the same imaging
survey, can be used to tighten the cosmological constraints