1. 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/ )
Some part of my talks is based on the discussion of WLWG
Oct 26th ROE

2. 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

3. 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
Large-scale structure
z=0
present

4. Weak Lensing Tomography
(e.g., Hu 99, 02, Huterer 01, MT & Jain 04)
Subdivide source galaxies into several bins based on photo-z 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)
＋m(z)

5. 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

6. Tomographic Lensing Power Spectrum (contd.)
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).

7. 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

8. 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: nights
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)

9. 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

10. 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
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
Highly non-linear regime; a more complicated mode coupling
N-body simulation based predictions are needed (e.g., halo model)
Correlations btw density perturbations of different scales arise as a consequence of non-linear structure formation, originating from the initial Gaussian fields
However, the non-Gaussianity is fairly accurately predictable based on the CDM model

11. 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?

12. 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 l1 l1 l2 l2 l2
Gaussian errors 
Non-Gaussian errors  l1 l2 l1

13. Correlation coefficients of PS cov. matrix
w/o shot noise
Diagonal: Gaussian Off-diagonal: NG, 4-pt 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
with shot noise

14. Correlations btw Cl’s at different l’s
Principal component decomposition of the PS covariance matrix

15. 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

16. Signal-to-noise ratio: SNR
Data vector: power spectra binned in multipole range, l_min<l<l_max, (and redshifts)
In the presence of the non-Gaussian errors, the signal-to-noise ratio for a power spectrum measurement is
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

17. Signal-to-noise ratio: SNR (contd.)
Gaussian
Multipole range: 50<l<l_max
Non-gaussian 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)
Non-Gaussian
50<l<l_max

18. The impact on cosmo para errors
We are working in a multi-dimensional parameter space (e.g. 7D)
_de
error ellipse
_de
w_0
w_a
n_s ….
_mh^2
_bh^2
Non-Gaussian Error
w_0
w_a ….
n_s
Volume of the ellipse: VNG2VG
Marginalized error on each parameter  length of the principal axis: NG～2^(1/Np)G (reduced by the dim. of para space)
Each para is degraded by slightly different amounts
Degeneracy direction is slightly changed
_mh^2
_bh^2

19. An even more direct use of NG: bispectrum
Bernardeau+97, 02, Schneider & Lombardi03, Zaldarriaga & Scoccimarro 03, MT & Jain 04, 07, Kilbinger & Schneider 05
An even more direct use of NG: bispectrum
given as a function of separation l
given as a function of triangles

20. 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,  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)

21. 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

22. Mass-limited cluster counts vs. lensing-selected counts
Hamana, MT, Yoshida 04
Halo distribution
Convergence map
2 degrees
Mass-selected sample (SZ) vs lensing-based sample

23. Miyazaki, Hamana+07
Mass
Light (galaxies)
Secure candidates
X-ray

24. A closer look at nearby clusters (z<0.3)
~30 clusters (Okabe, MT, Umetsu+ in prep.)
Subaru is superb for WL measurement
A detailed study of cluster physics (e.g. the nature of dark matter)

25. Redshift distribution of cluster samples

26. 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

27. Cross-correlation btw CC and WL (contd.)
M/M_s>10^13
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.
M/M_s>10^14
M/M_s>10^15

28. 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

29. SNR for CC+WL
The cross-covariance leads to degradation and improvement in S/N up to ~20%, compared to the case that the two are independent

30. 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

31. 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

32. Real world: issues on systematic errors
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? …

33. 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)
Covariances (or correlation) between the different obs.
If the systematics in different obs are uncorrelated
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

34. 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