Cosmic Shear : cosmological constraints

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Cosmic Shear cosmological constraints

• Ismael Tereno
• Institut dAstrophysique de Paris

Summary Cosmic shear response to cosmological
parameters What can be expected from the full
CFHT Legacy Survey Current limitations
Colloque PNC, November 2005
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1. Introduction
Large-scale structure induces correlations
between the observed shapes of background
galaxies.
Linearized lens equation
Amplification Matrix
x separation vector k convergence, ? shear,
r rotation
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Equation of geodesic deviation
T optical tidal matrix
Weakly inhomogeneous universe
Solution
Deflection potential
Poisson equation
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We are interested in second-order statistical
properties of the convergence and shear fields
It is sensitive to the cosmological parameters
through . Overall density factor .
Geometrical factor . Matter fluctuations -
Primordial Spectrum - Transfer function -
Linear growth factor
And also Non-linear fit
Source galaxies redshift distribution
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Example Response to Om
Increase in Pm higher in small scales
lower in large scales
Increase in Dist higher at all scales
The net effect is an overall rise more pronounced
at small scales
Measuring both linear and non-linear scales gives
extra information.
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2. Measurements
The two-point statistic we will use is the
variance of the shear in a top-hat window.
Estimated from ellipticities of galaxies.
(Weak lensing)
Current CFHTLS results Wide (Hoekstra et
al 05) 22 deg2 Deep (Semboloni et al
05) Wide (Fu et al , in prep) larger
scales All show low systematics good PSF
correction Wide Deep s8 0.85 0.07
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In order to predict the cosmological constraints
obtainable from cosmic shear measurements at a
broader range of scales and with smaller error
bars, we compute the expected signal from the
full CFHTLS-Wide, for a fiducial model .
Total area of 170 deg2 (724949)
Evaluated at 20 points between 0.6 arcmin and 2
deg. Due to the filter, the shear variance at
theta includes correlations until 2theta and
cosmic variance depends on the correlations until
2sqrt(2)theta.
Covariance Matrix
Not included Extra sources of uncertainties
arising from Redshift of sources Non-linear
fit PSF correction
Includes Poisson noise f (intrinsic
ellipticity, density of galaxies) Cosmic
variance f (correlations, theta_max / total
A) Overall dependence on the total area
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Size of the survey A 170 deg2 Density of
galaxies n 20 arcmin-2 Intrinsic ellipticity
s 0.4
Fiducial model
?cdm ?h2 0.114 ?b 0.022 ?_? 0.73 h
0.71
ns(k00.05) 0.93 ?s dns/dlnk -0.04 s8
0.9
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3. Likelihood evaluation
Markov Chain Monte Carlo (MCMC) method Models
mi generated (and not pre-determined on a grid)
with the Metropolis-Hastings algorithm
random starting model
0ltult1, random
Steps along the eigen directions of the
covariance matrix obtained
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Several chains of models were computed each
from a different starting point
Define W(p) the average of the m within
chain variances of the parameter p B(p) the
between chains variance of the m chains averages
of the parameter p
Convergence when B ltlt W R (BW) / W
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After this stage, the resulting merged Markov
chain of models starts to form a sample of the
probability distribution in the parameters space
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4. Results Cosmic shear
(Tereno et al 05)
7 Parameters space
Marginalized include the combined effect of all
7 parameters (the degeneracies)
Example of 3D plots of the sample
Knowledge of h narrows the s8/?c distribution
The well known s8/?m degeneracy
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Further information is available from the 7D
eigenvectors of the parameters covariance matrix
h ns ?? ?s
s8
1s
Best constrained directions
Y1 ? s8 , ?m Y2 ? h , ns Y3 ? s8 , ?m , ??
Amplitude of the spectrum
Slope of the spectrum
(from imposed narrow curvature range)
Curvature of the spectrum ?
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5. Results Cosmic shear CMB
4. Results Cosmic shear
7 parameters MCMC (?b, ?c,h,ns,?s,As,t) CMB
(WMAP CBI) and Cosmic shear CMB
(flatness imposed)
gain 1s(CMB) / 1s(cosmic shearCMB)
Near orthogonality between some of the contours
is responsible for large gains obtained, even
though most CMB contours are much smaller than
Cosmic Shear ones.
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Cosmic complementarity The ns / ?s case

The combined effect adds at one end of the
spectrum and cancels at the other end
ns tilts the spectrum ?s bends the spectrum
CMB
Cosmic shear MCMC for these 2 parameters, using
different scales
Small 0.6 6 Medium 8 30 Large 40
120
Cosmic shear
?ns
??s
kl (Peacock Dodds)
Orthogonal degeneracies for 2 experiments probing
opposite parts of the spectrum
The degeneracy is best determined with the signal
from the smallest scales
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6. Extra sources of uncertainties
Non-linear modeling
Cosmic shear
For scales lt 4, the new contribution dominates
the error bars. This may be a serious issue for
the future.
A change of 5 and -5 in the Pnl computed with
the halofit, causes a difference in the fiducial
shear variance that is quadratically added to the
data covariance matrix.
1d standard deviations change by 1.15 1.35
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Masking
At least 20 of the total area will be lost to
masking
PSF residuals in the modeling of the
high-frequency spatial variation of the PSF
approximations of the KSB method itself
Current CFHTLS results show good PSF correction
at all scales.
Source redshift
If no redshift information is available, zs must
be taken as a free parameter
1d standard deviations change by 1.15 1.40
Current CFHTLS results determine photometric
redshifts with 5 precision. With 5 bands
available, this should improve.
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Inclusion of other parameters
like dark energy. Current CFHTLS results
Quintessence study underway , aiming
to constrain, for example, Ratra-Peebles
parameters (instead of eq. of state) (Tereno,
Schimd, Uzan, Mellier et al, in
prep) and (Schimd, Tereno, Uzan, Mellier et al,
in prep) see Carlo Schimd talk at the EDEN
workshop, Paris, Dec 2005
Putting all together, we find an average cosmic
shear degradation factor of 1.9 per parameter
Use importance sampling w1 (m_cmb joint)
w (m_cmb cmb) L (m_cmb cov,cshear) w2
(m_cmb joint) w (m_cmb cmb) L (m_cmb
4cov,cshear)
The joint degradation factor is only of 1.25
1.45
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7. Conclusions
We studied the determination of cosmological
parameters by a cosmic shear CFHTLS-Wide type of
experiment and found the best constrained
parameters to be
and
We compared 2-dimensional cosmic shear
degeneracies with CMB ones and found evidence of
the cosmic complementarity, namely in, (s8,?m)
(ns,h) (ns,?s)
This led to predict a gain in the parameters
precision, when both experiments are combined, of
the order 1.5 2.5 for several parameters, such
as s8 , ?m ,h , ns , ?s which may help, for
example, to distinguish between inflationary
models.
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Observational limitations, like PSF correction
and knowledge of the redshift distribution, seem
to be under control. The main limitations come
now from the theoretical side. More precise
understanding of the non-linear collapse starts
to be critical at this point. The validity of
the approximations made in lensing thory may also
be reanalysed, like the Born approximation
or lens-lens coupling
This will allow cosmic shear measurements to
continue their evolution
VIRMOS (CFHT 12k)
CFHTLS-Wide (CFHT Megacam)
DUNE (Space)