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Cher Christian,
Voici ma version du latex weak lensing. J'ai fait pas mal
de modifs. J'ai aussi tente de remettre a plat les calculs de
magnitudes mais c'est tres tres difficile parce qu'on n'a pas
de donnees. Le tableau mag. limit. du sloan de la page web
megcam vs. cfht avec les filtres n'aide pas car il n'y a pas
de temps de pose. J'ai donc extrapole a partir de nos data
cft1k virmos. Je pense qu'il y aura au moins une iteration a
faire sur ces problemes de mag. limites.
Dis moi ce que tu en penses.
Amicalement
Yannick
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PLEASE NOTE: NEW OFFICE AND PHONE NUMBER: April 10, 2001 [NEW CHANGE!!]
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Yannick Mellier Tel : (33) 1 44 32 81 40
Institut d'Astrophysique de Paris Fax : (33) 1 44 32 80 01
98 bis boulevard Arago Switchboard: (33) 1 44 32 80 00
75014 Paris Office : 210A (2nd floor)
France
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\subsection{ Weak-lensing and large scale structure}
The analysis of gravitational lensing of background galaxies by
foreground structures has the capability of measuring many of the
important properties of the mass distribution of the universe with
great precision and minimal concerns about the problems of ``tracers''
such as galaxies or hot gas. The potential of such measurements was
recognized about 40 years ago but entered its modern phase about 15
years ago with the discovery of arcs in galaxy clusters. The CFHT has
played an important role in this work. Realizing the feasibility of
the regime of weak lensing lead to precise calculations of observables
in a series of theoretical papers (Blandford et al.
1991; Miralda-Escud\'e 1991; Kaiser 1992; Bernardeau van Waerbeke
\& Mellier 1997; Jain \& Seljak 1997; Kaiser 1998). At the same time
several groups began to
undertake the first analyses of weak lensing in clusters, where the
signal remains relatively strong. The availability of these data lead
to the development of observational techniques (Kaiser's imcat, for
instance), pipelines and data analysis centers (Terapix center at IAP) and
centers of strength (Hawaii, Paris and CITA in Toronto). These
centers have recently advanced weak lensing techniques to a level of
maturity where they are now poised to give us precise results for a
number of cosmological questions of great interest.
In the past year several groups published analyses of the measurement
of weak shear in the field, most of them coming from CFHT. The results
indicate that weak lensing is
now poised to deliver results on the general properties of mass
distribution in the Universe. Three major papers
based on CFHT observations have very recently illustrated the potential
of cosmological weak lensing analyses:
%%%
\begin{itemize}
\item Following the first cosmic shear detection with the VIRMOS-DESCART
survey (van Waerbeke et al
2000); van Waerbeke et al (2001) have published the
first tentative to ``break the degeneracy'' between the amplitude of the dark
matter fluctuation spectrum, $\sigma_8$, and the density parameter,
$\Omega_M$ (although some assumptions about other aspects of the mass
distribution needed to be made). This result was based on 6.5 square
degrees of imaging data. The way is now
clear for the next major step which will provide higher order statistics
to measure simultaneously $\sigma_8$ and $\Omega_M$.
\item Hoekstra et al (2001) using the Red sequence Cluster Survey have
for the first time being able to explore the biasing directly from
the cross-correlation between the dark matter, measured from weak
lensing analysis, and the galaxy distribution, measured from CFH12K optical
data. This important work shows that the linear biasing model
seems valid on scale ranging from 150 $h^{-1}$ kpc to 3 $h^{-1}$ Mpc.
Its amplitude
is still a matter of debate, because of the degeneracy between
the bias factor and the mass-light cross-correlation coefficient. But
recently, the VIRMOS-DESCART and the Canadian Cluster surveys joined
together and succeeded to address this issue, thanks to the
use of these two different samples. It turns out that the biasing seem
very close to one (assuming a flat $\Lambda$-dominated universe: Hoekstra,
van Waerbeke, Gladders, Mellier, Yee in preparation).
\item Finally, Pen, van Waerbeke \& Mellier (2001) have extracted
the $E$- (lensing) and
$B$- (curl) modes of the weak lensing signal and produced for the first time
a direct measurement of the power spectrum of the dark matter on scale
ranging from 1' to 15'.
\end{itemize}
%%%%
These results issued from
strong collaborations between
Canada (Pen, Hoekstra, Gladders, Yee, van Waerbeke)
and France (Mellier, Bernardeau, Le F\`evre, van Waerbeke, Bertin),
put CFHT at the forefront of cosmological
weak lensing studies. The amount of data and their
scientific interpretations are the most important ones obtained so
far from any wide field surveys. Moreover, they considerably helped to
define next realistic goals and critical limitations for a
larger survey. From a cosmological point of view,
the following objectives are envisioned for the CFH-LS:
\begin{enumerate}
\item Measurement of the projected
power spectrum of the dark matter on scale
between 30" to $<5$ degrees in order to reach 100 $h^{-1}$ physical
scales (beyond this, systematics are much larger than
signal, so this is not a reasonable goal,
even by assuming an improvement by a factor of 3 of weak lensing
signal extraction):
\item Measurement of the biasing factor with a 5\% accuracy
on scale ranging from
30" to $<5$ degrees. Exploring its dependence with
angular scale and lookback time in the redshift range $0.1<z<0.5$.
\item Measurement of $\Omega_M$ and $\sigma_8$ with a 10\%
accuracy from the join use of
second and third order statistics on the shear field,
\item Measurement of $\Lambda$ or any dark energy component from
large angular scale cosmic surveys coupled with the
CFH-LS SNIa surveys and with CMB data.
\item Measurement of galaxy halo properties from galaxy-galaxy
lensing analysis of $10^7$ galaxies.
\item Produce multi-source plane cosmic shear analysis in order to
study the evolution of biasing with redshift, to improve by a factor
of at least 3 the measurement of skewness of the convergence field and
to produce an estimate of the intrinsic ellipticy correlation
generated during the merging phase of galaxy halos.
\item Couple XMM survey, redshift survey and weak lensing data in
order to
analyze clusters of galaxies and the physics of both barynonic and
non-baryonic dark matter in clusters of galaxies as well as along
compact groups located inside large-scale structures.
\item Provide a mass statistic of clusters in the field in order
to infer the cluster radial mass profile,
\item Test alternative to dark matter, like MOND models, from
galaxy-galaxy lensing (see Hoeakstra et al 20001)
and cosmic shear studies (see White \& Kochankek 2001, who interpreted
CFHT results in a MOND context).
\bigskip
These scientific goals,
are all feasible on a short timescale,
as it is demonstrated from present-day results. Most studies will
use spectroscopic surveys carried out on 10m-class telescopes, like
VIRMOS. The join
use these data sets and numerical simulations show that
we need to cover 100 deg$^2$ (effective) in order to
reach these goals (van Waerbeke et al 1999).
An important use of the lensing data will be in conjunction with CMB
maps from upcoming experiments such as the MAP satellite. Lensing data
gives constraints on the cosmological parameters which are often
nearly ``orthogonal'' to the CMB constraints. Together these will give
very high precision measurements of the main parameters. A particularly
interesting possibility is to use the CMB itself as one set of sources
and cross-correlate it with the shear detected in the galaxies. This
will enable a number of important tests of the assumptions of the
analysis.
\subsubsection{Weak Lensing Data Requirements}
Weak lensing surveys have been extensively discussed in the published
literature. Kaiser and collaborators have been particularly active in
advancing the measurement techniques to new levels of precision with
phenomenally precise control over PSF variations and other systematics.
Mellier's group has done extensive simulation work to derive the survey
parameters required to make the next generation measurements
(see van Waerbeke et al 1999. Moreover, they have considered in detail
many of the known optical aberrations to show that
these no longer present a barrier (Erben et al 2001).
\bigskip
The specifications regarding the design of
a weak lensing surveys have to take
into account the following boundary conditions:
\begin{itemize}
\item We know from simulations and real data that we are now
able to measure a weak lensing signal of 1\% with a 10\% relative
accuracy for observations with seeing $<0.9"$, even with
optical/atmospheric distortion of 30\%. This means that, with
better optical design provided by the new
prime focus, better image quality provided by queue scheduling and
much larger sample, we can definitely
get a signal down
to 0.3\% . The variance of the shear averaged over an angular scale
$\theta$ can be measured to a 3-$\sigma$ confidence level ($c.l.$)
down to the following amplitude
\begin{equation}
<\gamma(\theta)^2>^{1/2} = 0.3\% \ \left[{A_T \over 100
\ deg^2}\right]^{{1 \over 4}} \times \left[{\sigma_{\epsilon_{gal}}
\over 0.4} \
right] \times
\left[{n \over 20 \ arcmin^{-2}}\right]^{-{1 \over 2}} \times
\left[{\theta \over 10'}\right]^{{-{1 \over 2}}} \ ,
\label{snshear}
\end{equation}
where $A_T$ is the total sky coverage of the
survey, $\sigma_{\epsilon_{gal}}$ the intrinsic dispersion of galaxy
ellipticities, $n$ the galaxy number density.
The systematics are presently a dominant factor beyond angular
scale of 2 degrees. Based on past
progresses on PSF anisotropy corrections and
on going works within the next five years, we can imagine
to reach 10 degrees, but it is very difficult to have clear views
whether technical issues will be solved beyond
this scale.
\item The measurement of the shape of the power spectrum up to 100
$h^{-1}$ Mpc, corresponds to angular scales of $\approx $ 5 degrees
at $z \approx 0.3$. This is the typical redshift of lenses
for surveys with limiting magnitude of $I \approx 25.$.
\item The contamination by intrinsic correlation of
ellpticities mimics weak lensing signal and can be a major source
of uncertainties for a shallow survey. Two effects play together to
reduce the lensing signal: (1) when the redshifts of sources
decreases, the gravitational shear amplitude decreases as well. (2)
In a deep surveys most galaxies projected along a line of sight are
not physically linked. Therefore, the intrinsic correlation of
elltipcities produced by tidal torque and angular moment generation
is obvisouly a small contribution. Conversely,
intrinsic correlation increases when the source redshift decreases as
for a shallow survey, and may even be a major contribution
when source redshift is lower than $z \approx 0.1$.
\item Because source clustering may significantly reduce the signal
of the skewness (see Hamana et al 2001), multi-source planes are needed,
which implies a need for spectroscopic follow up.
\end{itemize}
\medskip
From these conditions, we can now put more quantitative limits on
the weak lensing survey. In the following, magnitudes are estimated
in the Vega system, and limiting magnitude are established
from our experience on real data, based on the VIRMOS surveys.
\hbi CFHLS will need to go significantly deeper than $I = 22.5$. Even at
this magnitude, the amplitude of the shear is 6 times smaller than what was
measured with CFH12K. The shear is weaker because the redshift of galaxies is
too low, which decreases the shear signal. With a signal 5 times smaller, the
amplitude of the signal reaches an amplitude similar to that of the
systematics. Therefore the pure lensing signal extraction and its
cosmological interpretation are very difficult.
\hbi Photometric redshifts of very faint sources are not properly calibrated
if we go too deep: Beyond $I=24.$, there are basically no spectroscopic
surveys to calibrate photo-z. This is not acceptable since redshift will be
a major input beyond present-day WL surveys. The DEEP2 and VIRMOS
spectroscopic surveys plan to reach $I=24.$ for a large sample of
galaxies, so we are very confident that the redshift of sources will
be very well known. It would be even better if VIRMOS or DEEP fields
were priority targets for the survey.
\hbi When galaxies are too faint, many of them are at very high-z and
morphological evolution or merging processes affect the shape of galaxies
which may strongly depart from ellipse-like systems. This will considerably
increase the noise produced by intrinsic ellipticities of galaxies.
It is therefore not recommended to push the weak lensing survey far
beyond $z=1$.
\hbi As seen in Equation \ref{snshear}, the survey does need to be of
sufficient depth to include a large enough sample of galaxies per
solid-angle
and lower statistical noise.
From present-day surveys 30 galaxies$/arcmin^2$ is a very good target and
is achieved if $I=24.3$. For instance, for the VIRMOS-DESCART
survey, the {\sl effective} galaxy number density of the cosmic shear sample
(after masking and removing close-pairs and masking
which reduce by 30\% the galaxy density) is 20 galaxies$/arcmin^2$.
We can expect spectroscopic surveys and good photometric redshift
calibrations to soon be available at this depth. The average redshift will
be about 0.8, so we are also not concerned by morphological evolution.
Note that on real data (VIRMOS, over 20 nights integrating
all problems, including seeing variation, absorption, etc...), $I=24.3$
is obtained in 4150 sec. with the $i'$ filter
(1.15 hour after rescaling CFH12K/Megacam).
\noindent
\bigskip
The absolute minimal requirement for the next step in lensing
measurements is to obtain 100 square degrees of weak lensing field in
patches of at least $5\times 5$ degrees. Because of cosmic variance,
it is important to spread the surveys over uncorrelated patches.
The minimum is 4 patches. As far as the total area is conserved,
3 patches will be marginally acceptable.
The unavoidable problem of
very bright stars means that some of the area is lost. For example,
in the VIRMOS-DESCART survey, about 30\% of the total area has to be
masked in order to avoid bright stars, comet-like patterns,
noisy boudaries of CCDs. This leads to
functional minimal requirement of 130 square degrees in patches of
$6\times6$ square degrees in order to get an {\sl effective} area
of 100 square degrees.
{\bf $I=24.3$ with seeing $<0.9"$ over 4 patches covering 130 deg$^2$
is therefore the minimum required for the weak lensing. } In addition,
we need u'g'r'z' and at least $K$ band for photometric redshift.
Since the XMM field is unique and has a great potential for clusters
of galaxies, baryonic vs. non-baryonic matter, we also recommend to
carry out a complete follow up of the XMM field, extending the
$6^o \times 6^o$ XMM field to $10^o \times 10^o$. From a cosmic
shear point of view, this field would provide on field for which the
power spetrum could be evaluated up to 10 degrees.
It is not reasonnable to imagine having spectroscopic redshifts
for all galaxies up to $I=24.3$ in 130 square degrees. But on the
other hand, studies of weak distorted galaxies and dark matter
halos with look back time is an important goal for this survey.
In addition, numerical simulations done by Hamana et al (2001)
demonstrate that clustering of galaxies significantly affect the
measurement of the skewness (by 10 to 30\%). So it is absolutely
necessary to get photometric redshift for all galaxies without
spectroscopic data. Therefore, the survey location definitely needs
to cover area where massive spectroscopic follow up are planed in
order to calibrate and check photo-$z$.
With the expected redshift distribution of the weak lensing sample
($I=24.3$), an unambiguous determination of redshift is not
possible without photometric data in all Megacam filters, including
$z'$. Even in this optimal condition, $K$ (or $H$ data) is
also important. We computed the limiting magnitude in each band in
order to get photo-$z$ for all I-band selected galaxies. This
translates to $u'=25.3$, $B=25.9$, $V=25.6$, $R=25.6$, $z'=24.3$
(5$\sigma$3"). A comparison of VIRMOS/CFH12K data and Megacam throughput
finally leads to: 3700 sec. in $u'$, 3100 sec. in $g'$, 3100. sec.
in $r'$, 4150 sec. in $i'$, and 7200 sec. in $z'$.
For all filters, but the $I$-band (or the one primary used for
weak lensing analysis), median and worse seeing (provided it does not
go beyond about 1.0 arcsecond) is acceptable. Therefore, assuming
3 $6\times6$ + 1 $10 \times 10$ patches, the minimal function time
request for this program is 1227 hrs. , 189 nights
of 6.5 hours.
{\sl ?? The MSWG recommends
4 patches of $6\times6$ degrees for 208 square degrees at a cost of
720 queue
hours of 111 queue nights.}
It should be noted that the individual $1\times 1$ fields covered by
the mosaic need to be overlapped to build the
patches. The resulting area of these patches will be
$\approx 10\% $ smaller.
\subsection*{References}
Bernardeau, F.; van Waerbeke, L.; Mellier, Y.; 1997
{\em A\&A }{\bf 322}, 1.
Blandford, R.D., Saust, A.B., Brainerd, T.G., Villumsen, J.V.
1991. {\em MNRAS} {\bf 251} 600.
Erben, T.; van Waerbeke, L.; Bertin, E.; Mellier, Y.; Schneider, P.;
2001 {\em A\&A }{\bf 366}, 717.
Hoekstra, H.; Yee, H.;, Gladders, M.D. 2001
{\em ApJ }{\bf 558}, L11.
Jain, B.; Seljak, U.; 1997 {\em ApJ }{\bf 484}, 560.
Kaiser, N. 1992. {em Ap. J.} {\bf 388} 722.
Kaiser, N. 1998. {\em Ap. J.} {\bf 498} 26.
Miralda-Escud\'e, J. 1991. {\em Ap. J.} {\bf 380} 1.
Pen, U.; van Waerbeke, L.; Mellier, Y.; 2001
{\em preprint, }{astro-ph}/0109*** .
van Waerbeke, L.; Mellier, Y.; Erben, T.; et al.; 2000
{\em A\&A }{\bf 358}, 30.
van Waerbeke, L.; Bernardeau, F.; Mellier, Y.; 1999
{\em A\&A }{\bf 342}, 15.
van Waerbeke, L.; Mellier, Y.; Radovich, M.; et al.; 2001
{\em A\&A }{\bf 374}, 757.
White, M., Kochanek, C.S.; 2001. {\em preprint, }{astro-ph}/0105227.