A&A 374, 757-769 (2001)
DOI: 10.1051/0004-6361:20010766
L. Van Waerbeke^{1,2} - Y. Mellier^{1,3} - M. Radovich^{4,1} - E. Bertin^{1,3} -
M. Dantel-Fort^{3} - H. J. McCracken^{5} -
O. Le Fèvre^{5} - S. Foucaud^{5} - J.-C. Cuillandre^{6,7} - T. Erben^{1,3,8} - B. Jain^{9,10} -
P. Schneider^{11} -
F. Bernardeau^{12} - B. Fort^{1}
1 - Institut d'Astrophysique de Paris, 98bis boulevard
Arago, 75014 Paris, France
2 -
Canadian Institut for Theoretical Astrophysics, 60 St
Georges Str., Toronto, M5S 3H8 Ontario, Canada
3 -
Observatoire de Paris, DEMIRM, 61 avenue de
l'Observatoire, 75014 Paris, France
4 -
Osservatorio Astronomico di Capodimonte, via Moiariello, 80131m Napoli, Italy
5 -
Laboratoire d'Astrophysique de Marseille, 13376 Marseille Cedex 12,
France
6 -
Canada-France-Hawaii-Telescope, PO Box 1597, Kamuela, Hawaii 96743,
USA
7 -
Observatoire de Paris, 61 avenue de l'Observatoire, 75014 Paris, France
8 -
Max Planck Institut für Astrophysiks, Karl-Schwarzschild-Str. 1,
Postfach 1523, 85740 Garching, Germany
9 -
Dept of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA
10 -
Dept. of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218,
USA
11 -
Universitaet Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
12 -
Service de Physique Théorique, CE de Saclay, 91191 Gif-sur-Yvette Cedex, France
Received 2 February 2001 / Accepted 25 May 2001
Abstract
We report a measurement of cosmic shear correlations
using an effective area of 6.5 deg^{2} of the VIRMOS deep
imaging survey in progress at the Canada-France-Hawaii Telescope.
We measured various shear correlation functions, the aperture
mass statistic and the top-hat smoothed variance of the shear
with a detection significance exceeding .
We
present results on angular scales from 3 arcsec to half a degree. The lensing origin of the signal
is confirmed through tests that rely on the scalar nature of
the gravitational potential. The different statistical measures
give consistent results over the full range of angular scales.
These important tests of the measurements demonstrate that the
measured correlations could provide accurate constraints on cosmological
parameters, subject to the systematic uncertainty in the source
redshift distribution.
The measurement over more than two decades of scale
allows one to evaluate the effect of the shape of the power spectrum on
cosmological parameter estimation.
The degeneracy on
can be broken
if priors on the shape of the
linear power spectrum (parameterized by )
are assumed. For instance,
with
and at the
confidence level, we obtain
and
for open models, and
and
for flat (-CDM) models.
We discuss how these results would scale if the assumed source
redshift distribution needed to be modified with forthcoming measurements
of photometric redshifts.
From the tangential/radial mode decomposition we can set an upper limit
on the intrinsic shape alignment, which has recently been suggested as a
possible contribution to the lensing signal. Within the error bars, there is
no detection of intrinsic shape alignment for scales larger than 1'.
Key words: cosmology: theory - dark matter - gravitational lensing - large-scale structure of the Universe
Cosmological gravitational lensing produced by large-scale structure (or cosmic shear) has been advocated as a powerful tool to probe the mass distribution in the universe (see the reviews from Mellier 1999; Bartelmann & Schneider 2001 and references therein). The first detections reported over the past year (Van Waerbeke et al. 2000; Bacon et al. 2000; Kaiser et al. 2000; Wittman et al. 2000; Maoli et al. 2001; Rhodes et al. 2001) confirmed that the amplitude and the shape of the signal are compatible with theoretical expectations, although the data sets were not large enough to place strong constraints on cosmological models. Maoli et al. (2001) combined the results from different groups to obtain constraints on the power spectrum normalization and the mean density of the universe : Their result is in agreement with the cluster abundance constraints, but they were not yet able to break the degeneracy between and .
The physical interpretation of the weak lensing signal can be made more securely using detections of cosmic shear from different statistics and angular scales on the same data set (as in Van Waerbeke et al. 2000). Unfortunately, their joint detection of the variance and the correlation function using the same data was not fully conclusive: the sample was too small to enable a significant detection of the cosmic shear from variances with different weighting schemes and 2-point statistics over a wide range of scales. The use of independent approaches is nevertheless necessary and it is an important step to validate the reliability of cosmic shear, to check the consistency of the measurements against theoretical predictions and to understand the residual systematics. A relevant example is the aperture mass statistic (defined in Schneider et al. 1998). It is a direct probe of the projected mass power spectrum, and it is not sensitive to certain type of systematics (like a uniform PSF anisotropy) which may corrupt the top-hat smoothed variance, or the shear correlation function. Even the shear correlation function can be measured in several ways, by splitting for instance the tangential and radial modes.
In this paper we report the measurement of the top-hat smoothed variance, the aperture mass, the shear correlation function, and the tangential and radial shear correlation functions on a new homogeneous data set covering an effective area of 6.5 deg^{2}. The depth and the field of view are well suited for a comprehensive analysis using various statistics. We show that the amplitude of residual systematics is very low compared to the signal and discuss the consistency of these measurements against the predictions of cosmological models.
We also discuss alternative interpretations. It has been suggested recently that intrinsic alignments of galaxies caused by tidal fields could contribute to the lensing signal (Pen et al. 2000; Croft & Metzler 2000; Heavens et al. 2000; Catelan et al. 2000; Crittenden et al. 2000a; Crittenden et al. 2000b). This type of systematic is problematic because its signature on different 2-points statistics mimics the lensing effect. A mode decomposition in electric and magnetic types (or E and B modes), similar to what is performed for the polarization analysis in the Cosmic Microwave Background, can separate lensing from intrinsic alignment (see Crittenden et al. 2000a; Crittenden et al. 2000b). The E and B mode analysis is the subject of a forthcoming paper; the aperture mass statistic presented in this paper is a similar analysis to the E and B mode decomposition, and allows us to put an upper limit on the contamination of our survey by the intrinsic alignments.
This paper is organized as follow: Sect. 2 describes our data set, and highlights the differences in the data preprocessing from our previous analysis (Van Waerbeke et al. 2000). The measurement of the shear from this imaging data is discussed in Sect. 3. Section 4 summarizes the theoretical aspects of the different quantities we measure, and lists the statistical estimators used. The results and comparison to a few standard cosmological models are shown in Sect. 5. In Sect. 6 we perform a maximum likelihood analysis of cosmological models in the parameter space. The results on very small scales are shown separately in Sect. 7, and we conclude in Sect. 8.
The DESCART weak lensing project^{} is a theoretical and observational program for cosmological weak lensing investigations. The cosmic shear survey carried out by the DESCART team uses the CFH12K data jointly with the VIRMOS survey^{} to produce a large homogeneous photometric sample which will eventually contain a catalog of galaxies with redshifts as well as the projected mass density over the whole field (Le Fèvre et al. 2001). In contrast to Van Waerbeke et al. (2000), the new sample presented in this work only uses I-band data taken with the CFH12K camera and is therefore more homogeneous. It is worth noting that only half of the data of the previous CFHT12K sample is reused in our new sample. A comparison of the results will also allow checking the consistency and the robustness of the cosmic shear analysis.
The CFH12K data was obtained during dark nights
in May 1999, November 1999 and April 2000 following
the standard observation procedure
described in Van Waerbeke et al. (2000).
The fields are spread over 4 independent deg^{2}
areas of the sky identified as F02, F10, F14 and F22. Each
field is a compact mosaic of 16 CFH12K pointings named P[n] with
n=1-16.
Once the survey is completed, each of them will cover 4 deg^{2}. Currently,
of the final 16 deg^{2}, only 8.38 deg^{2} is available for
the analysis - most of
the pointings are located in three different fields (F02, F10, F14 listed
in Table 1). This total
field of view gets significantly reduced by the masking and
selection procedures described below. A summary of
the data set characteristics are listed in Table 1.
Target | Used area | Exp. time | Period | Image quality |
F02P1 | 980 | 9390 s | Nov. 1999 | 0.75'' |
F02P2 | 1078 | 7200 s | Nov. 1999 | 0.90'' |
F02P3 | 980 | 7200 s | Nov. 1999 | 0.90'' |
F02P4 | 1078 | 7200 s | Nov. 1999 | 0.80'' |
F10P1 | 882 | 3600 s | May 1999 | 0.65'' |
F10P2 | 882 | 3600 s | May 1999 | 0.75'' |
F10P3 | 490 | 3600 s | May 1999 | 0.75'' |
F10P4 | 882 | 3600 s | May 1999 | 0.65'' |
F10P5 | 882 | 3600 s | May 1999 | 0.75'' |
F10P7 | 1176 | 3600 s | Apr. 2000 | 0.75'' |
F10P8 | 1176 | 3600 s | Apr. 2000 | 0.70'' |
F10P9 | 98 | 3600 s | Apr. 2000 | 0.65'' |
F10P10 | 784 | 3600 s | Nov. 1999 | 0.80'' |
F10P11 | 294 | 3600 s | Nov. 1999/Apr. 2000 | 0.90'' |
F10P12 | 1176 | 3600 s | Apr. 2000 | 0.80'' |
F10P15 | 686 | 3600 s | Apr. 2000 | 0.85'' |
F14P1 | 882 | 3600 s | May 1999 | 0.80'' |
F14P2 | 882 | 3600 s | May 1999 | 0.85'' |
F14P3 | 686 | 3600 s | May 1999 | 0.75'' |
F14P4 | 1078 | 3600 s | May 1999 | 0.75'' |
F14P5 | 980 | 3600 s | May 1999 | 0.70'' |
F14P6 | 686 | 3600 s | May 1999 | 0.80'' |
F14P7 | 686 | 3600 s | May 1999 | 0.70'' |
F14P8 | 882 | 3600 s | May 1999 | 0.85'' |
F14P9 | 1078 | 3600 s | Apr. 2000 | 0.75'' |
F14P10 | 784 | 3600 s | May 1999 | 0.85'' |
F14P11 | 882 | 3600 s | Apr. 2000 | 0.80'' |
F14P12 | 784 | 3600 s | Apr. 2000 | 0.80'' |
F14P13 | 882 | 3600 s | Apr. 2000 | 0.85'' |
F14P14 | 882 | 3600 s | May 1999 | 1.0'' |
F14P15 | 882 | 3600 s | Apr. 2000 | 0.90'' |
F14P16 | 1176 | 2880 s | Apr. 2000 | 0.65'' |
F22P3 | 686 | 3600 s | May 1999 | 0.75'' |
F22P4 | 980 | 3600 s | Nov. 1999 | 0.75'' |
F22P6 | 588 | 3600 s | Apr. 2000 | 0.80'' |
F22P11 | 294 | 2880 s | Apr. 2000 | 0.75'' |
The data reduction was done at the TERAPIX data center^{}. More than 1.5 Tbytes of data were processed in order to produce the final stacked images. The reduction procedure is the same as in Van Waerbeke et al. (2000), so we refer the reader to this paper for the details. However, in order to improve the image quality prior to correction for the PSF anisotropy and to get a better signal-to-noise ratio on a larger angular scale than in our previous work, all CFH12K images were co-added after astrometric corrections.
The astrometric calibration and the co-addition were done using the MSCRED package in IRAF. Some tasks have been modified in order to allow a fully automatic usage of the package. For each pointing, we first started with the images in the I band. An astrometric solution was first found for one set of exposures in the dither sequence using the USNO-A 2.0 as reference, which provides the position of sources with an RMS accuracy of 0.3 arcsec (that is 300-500 objects per field). The astrometric solution was then transferred to the other exposures in the sequence. All object catalogs were obtained using SExtractor (Bertin & Arnouts 1996)^{} and a linear correction to the world coordinate system was computed with respect to the initial set. Finally, all images were resampled using a bi-cubic interpolation and then stacked together.
At this stage, each stacked image was inspected by eye and all areas which may potentially influence the later lensing analysis signal were masked (see Van Waerbeke et al. 2000 and Maoli et al. 2001). Since we adopted conservative masks, this process had a dramatic impact on the field of view: we lost 20% of the total area and ended up with a usable area of 6.5 deg^{2}.
The photometric calibrations were done using standard stars from the Landolt catalog (Landolt 1992) covering a broad sample of magnitude and colors. A full description of the photometric procedure is beyond the scope of this work and will be discussed elsewhere (Le Fèvre et al., in preparation). In summary, we used the SA110 and SA101 star fields to measure the zero-points and color equations of each run. From these calibrations, we produced the magnitude histograms of each field in order to find out the cut off and a rough limiting magnitude. Although few fields have exposure time significantly larger than 1 hour, the depth of the sample is reasonably stable from field to field and reaches (this corresponds to a 5 detection within a 3 arcsec aperture). Up to this magnitude, 1.2 million galaxies were detected over the total area of 8.4 deg^{2}, and the final number density of galaxies over the usuable area of 6.5 deg^{2} is . This is about two times less than the number of detected galaxies because of the filtering processes described in the next section.
The details of our shape measurement procedure and Point Spread Function (hereafter PSF) correction have been extensively described in two previous papers (Van Waerbeke et al. 2000; Maoli et al. 2001), and tested against numerical simulations (Erben et al. 2001). Therefore we will not reproduce these details here, but only give a short overview of the procedure. The shape measurement pipeline uses the IMCAT software (Kaiser et al. 1995)^{} combined with the SExtractor package. The different steps in the procedure are as follows:
Figure 1: Top and third panels: averaged and component versus the ellipticity of the PSF at the galaxy location, without the anisotropic PSF correction. Second from top and bottom panels: averaged e and e including the anisotropic PSF correction. A residual bias is the corrected ellipticity is present (shown by the two straight solid lines), as in our previous analysis, whose origin is still unclear. It is a constant bias which can be easily corrected for. | |
Open with DEXTER |
The raw ellipticity
of a galaxy is measured from the second moments I_{ij} of the surface
brightness
:
(1) |
We summarize the different statistics we shall measure, and how they depend on cosmological models. We concentrate on 2-point statistics and variances, since higher order moments are more difficult to measure, and will be addressed in a forthcoming paper.
Let us assume a normalised source redshift distribution parameterized as:
Figure 2: The thin solid line shows our redshift distribution model given by Eq. (2) with . Two other models will also be used: one is (thin dashed line) and one is (thin dot-dashed line). The thick solid line corresponds to the model used in Wilson et al. (2000) for the galaxies. All the distributions are normalised. | |
Open with DEXTER |
We define the power spectrum of the convergence as
(following the notation in Schneider et al. 1998):
(5) |
For each galaxy, we
define the tangential and radial shear components (
and
)
with respect to the center of the aperture:
From the shear
and its projections defined in
Eq. (7) we can also define
various galaxy pairwise correlation functions related to the
convergence power spectrum.
Note that the tangential and radial shear projections in what follows
are performed using the relative location vector of the
pair members, not from an aperture center.
The following correlation functions can be defined (Miralda-Escudé 1991; Kaiser 1992):
where is the pair separation angle. The cross-correlation is expected to vanish for parity reasons (there is no preferred orientation on average).
It is easy to see that the Eqs. (4), (6), (10)-(12) are different ways to measure the same quantity, that is the convergence power spectrum . Ultimately the goal is to deproject in order to reconstruct the 3D mass power spectrum from Eq. (3), but this is beyond the scope of this paper. Here we restrict our analysis to a joint detection of these statistics, and show that they are consistent with the gravitational lensing hypothesis. We will also examine the constraints on the power spectrum normalization and the mean density of the universe .
Let us now define the estimators we used to measure the quantities given in Eqs. (4), (6), (10)-(12).
The variance of the shear is simply obtained by a cell averaging
of the squared shear
over the cell index i. An
unbiased estimate of the squared shear for the cell i is:
The
statistic is calculated from a similar
estimator, although the smoothing window is no longer a top-hat but
the Q function defined in Eq. (9). An unbiased estimate of
in the cell i is:
(16) |
The shear correlation function
at separation
is obtained by identifying all the pairs of galaxies
falling in the separation interval
,
and calculating the pairwise shear correlation:
Figure 3: Top-hat smoothed variance of the shear (points with error bars). The three models correspond to for the short-dashed, solid and long-dashed lines respectively. The power spectrum is a CDM-model with . The error bars correspond to the dispersion of the variance measured from 200 realizations of the data set with randomized orientations of the galaxy ellipticities. | |
Open with DEXTER |
Figure 4: The aperture mass statistic for the same models as in Fig. 3. The lower panel plots the R-mode, obtained by making a 45 degree rotation as described in the text. There is no significant detection for (corresponding to an effective angular scale of 1', as discussed in the text), which shows the low level of contamination by galaxy intrinsic alignment and/or residual systematics. | |
Open with DEXTER |
Figure 5: Shear correlation function . The models are the same as in Fig. 3. The lower panel uses a log-scale for the x-axis to highlight the small scale details. | |
Open with DEXTER |
Figure 6: Top panel: tangential shear correlation function . Bottom panel: radial shear correlation function . The models are the same as in Fig. 3. | |
Open with DEXTER |
In this section we present our measurements of the 2-point correlations of the shear using the different estimators defined above. Figures 3 to 8 show the results for the different estimators: the variance in Fig. 3, the mass aperture statistic in Fig. 4, the shear correlation function in Fig. 5, the radial and tangential shear correlations in Fig. 6, and the cross-correlation of the radial and tangential shear in Fig. 8. Along with the measurements we show the predictions of three cosmological models which are representative of an open model, a flat model with cosmological constant, and an Einstein-de Sitter model. The amplitude of mass fluctuations in these models is normalized to the abundance of galaxy clusters. The three models are char- acterized by the values of and as follows:
It is reassuring that the different statistics agree with each other in their comparison with the model predictions. These statistics weight the data in different ways and are susceptible to different kinds of systematic errors. The consistency of all the 2-point estimators suggests that the level of systematics in the data is low compared to the signal. A further test for systematics is provided by measuring the cross-correlation function , which should be zero for a signal due to gravitational lensing. It is shown in Fig. 8 that it is indeed consistent with zero at all scales. The figure also shows the cross-correlation obtained when the anisotropic contamination of the PSF is not corrected - clearly such a correction is crucial in measuring the lensing signal.
The lower panel of Fig. 4 shows the R-mode of the mass aperture statistic. As this statistic uses a compensated filter, the scale beyond which the measured R-mode is consistent with zero (5' on the plot) corresponds to an effective angular scale . This places an upper limit on measured shear correlations due to the intrinsic alignment of galaxies, given the redshift distribution of the sources. The vanishing of for effective angular scales larger than 1' strongly supports our conclusion that the level of residual systematics is low: this is a very hard test to pass, as it means that the signal is produced by a pure scalar field, which need not be the case for systematics. We checked that is Gaussian distributed with a zero average all over the survey, which is what we would expect from a pure noise realisation. For scales below 5' on the plot, the R-mode is not consistent with zero at the 2- level. Since the cross-correlation is consistent with zero at this scale, the source of the R-mode is probably not a residual systematic caused by an imperfect PSF correction. Rather, it might be due to the effect of intrinsic alignments (Crittenden et al. 2000b).
The error bars shown in Figs. 3 to 8
are calculated from a measurement
of the different statistics in 200 realizations of the data set,
with randomized orientations of the galaxies. We measured the
sample variance from ray-tracing simulations (Jain et al. 2000) and find
that it is smaller than
of the noise error bars shown here
(see Van Waerbeke et al. 1999 where the sample variance has been calculated for surveys with
similar geometry), therefore we have not included it in our figures.
Figure 7 shows an estimate of the
sample variance for the rms shear using a compact 6.5 deg^{2}ray-tracing simulation (Jain et al. 2000). This figure shows that
the sample variance is about one order of magnitude smaller than
for the range of scales of interest. Hence our errors are not
dominated by sample variance, as was the case in the first detections
of cosmic shear. As the probed
angular scales approach the size of the fields (which is
with the CFH12K camera) the sample variance becomes larger.
This could be responsible for the small fluctuations in the measured
correlations in Figs. 5 and 6 for scales larger
than 24'.
Figure 7: Shear rms (solid line) measured in a ray-tracing simulation (Jain et al. 2000) for the open model. The dashed line is the sample variance of the shear rms measured from 7 different realisations of the mass distribution for a survey of 6.5 deg^{2}. | |
Open with DEXTER |
Figure 8: Shear cross-correlation function . The signal should vanish if the data are not contaminated by systematics. As a comparison, the open circles show the same cross-correlation function computed from the galaxy ellipticities where the anisotropic correction of the PSF has been skipped. | |
Open with DEXTER |
As noted elsewhere (e.g. Bernardeau et al. 1997; Jain & Seljak 1997), the parameters which dominate the 2-point shear statistics are the power spectrum normalization and the mean density . We investigate below how the statistics measured in Figs. 3 to 6 are consistent with each other when constraining these parameters. Our parameter estimates below rely on some simplifying assumptions; a more detailed analysis over a wider space of parameters will be presented elsewhere. In particular, as discussed later, the choice of the slope of the power spectrum is weakly known, and may significantly affect the parameter estimate. As pointed out in Sect. 4, the uncertainty on the redshift distribution is also a concern, but we partialy address this point by constraining the parameters using three different redshift distributions. A complete analysis involving marginalisation over and the redshift distribution using tight priors is left for a future work.
We assume that the data follow Gaussian statistics and neglect sample variance
since it is a very small contributor to the noise for
our survey, as discussed above. We compute the likelihood function :
Figure 4 (bottom panel) shows that for effective scales
smaller than 1' there is a non-vanishing R-mode which could come either
from a residual systematic, or from an intrinsic alignment effect. Therefore
it is safer to exclude this part from the likelihood calculation:
for the top-hat variance, we excluded the point at 1', for the correlation
functions the points below 2', and for the
statistic the points
below 5'. For the correlation function, we also excluded the points at
scales larger than 20' because of the small fluctuations in the measured
correlations. The constraints
on the cosmological parameters are not significantly affected whether
these large scale points are excluded or not.
Figure 9: Likelihood contours in the plane from the top-hat smoothed variance shown in Fig. 3. The first point in Fig. 3 was not included in the likelihood calculation to avoid the small scale systematic shown in Fig. 4 (bottom panel). The cosmological models have , with a CDM-type power spectrum and . The redshift of the sources is given by Eq. (2). with . The confidence levels are (0.68, 0.95, 0.999). | |
Open with DEXTER |
Figures 9 to 13 show the
constraints for each of the statistics shown in
Figs. 3 to 6. The contours show the ,
and
confidence levels. The agreement between the
contours is excellent, though the
statistic and the radial
correlation function do not give as tight constraints as the other
statistics. The correlation function measurements below
2' may be considered by using error bars that include a possible
systematic bias: this is equivalent to adding a systematic
covariance matrix
to the noise covariance
matrix
in Eq. (18). The new contours
computed with the enlarged error
bars^{} are shown
in Fig. 14.
The maximum of the likelihood in the variance and correlation function
likelihood plots is at
and
.
Note that the results are in very good agreement^{}
with a similar plot in Maoli et al. (2001)
(Fig. 8), here the contours are narrower, and are obtained
from a homogeneous data set. Moreover, the degeneracy
between
and
is broken.
Figure 10: As in Fig. 9, but using the statistic of Fig. 4 (top panel) instead of the top-hat variance. The first five points in Fig. 4 were not included in the likelihood calculation in order to avoid the small scale systematic shown in Fig. 4 (bottom panel). | |
Open with DEXTER |
Figure 11: Likelihood contours as in Fig. 9, but using the shear correlation function (Fig. 5) instead of the top-hat variance. The first two points and scales larger than 20' in Fig. 5 were not included in the likelihood calculation to avoid the contribution from the small scale systematic shown in Fig. 4 (bottom panel). | |
Open with DEXTER |
Figure 12: As in Fig. 9, but using the tangential shear correlation function (Fig. 6) instead of the top-hat variance. The first two points and scales larger than 20' in Fig. 6 were not included in the likelihood calculation in order to avoid the contribution from the small scale systematic shown in Fig. 4 (bottom panel). | |
Open with DEXTER |
Figure 13: As in Fig. 9, but using the radial shear correlation function (results in Fig. 6) instead of the top-hat variance. The first two points and scales larger than 20' in Fig. 6 were not included in the likelihood calculation in order to avoid the contribution from the small scale systematic shown in Fig. 4 (bottom panel). | |
Open with DEXTER |
The partial breaking of degeneracy between
and was expected from the fully non-linear calculation of shear correlations
(Jain & Seljak 1997). In the non-linear regime the dependence of the
2-points statistics on
and
becomes
sensitive to angular scale.
For example, as shown in Jain & Seljak (1997),
the shear rms measures
on scale between 2'-5',
and
on scales
.
Therefore a low universe should see a net decrease of shear power at large scale compared to
a
universe (for a given shape of the power spectrum),
as is evident in Fig. 3. Note that the aperture mass
is still degenerate with
and (Fig. 10) because it probes effective scales up to
only, which is not enough to break the degeneracy.
Figure 14: Likelihood contours as in Fig. 11, but all the points in Fig. 5 on scales smaller than 20' were used. In order to account for the small scale systematic shown in Fig. 4 (bottom panel) the error bars on the first two points were increased to include the systematic amplitude. | |
Open with DEXTER |
It seems that the aperture mass (Fig. 10) gives
a slightly larger for a large
compared to the other statistics, while they all
agree for
.
This could be an indication
for a low
Universe, however in practice,
the probability contours for the different statistics cannot be
combined in a straightforward way because they are largely redundant.
The best strategy here is to concentrate on one particular
statistic.
We expect the best constraints from the
shear correlation function (since it contains all the information
by definition), and therefore base our parameter estimates
on the likelihood contours obtained from it.
The contours in the
plane in
Fig. 14 closely follow
the curve
.
This allows us
to obtain the following measurement of
(from this figure alone):
If we choose a strong prior for ,
we can constrain
the two parameters separately; for
we get, at the
confidence level:
and
for open models
and
and
for flat (-CDM) models.
However, this result is clearly sensitive to
the prior choosen for .
In particular, if we use the
relation
for a cold dark matter model,
then some extreme combinations of ,
and
cannot be ruled out from lensing alone. The degeneracy between
and
is broken only if we take
to lie in
a reasonable interval. Such interval can be motivated by
galaxy surveys for instance, which give
at
confidence level for the APM
(Eisentsein & Zaldarriaga 2001). For instance the choice
would make
consistent with the data.
The second source of uncertainty comes from the
redshift distribution, kown only approximately. As discussed in Sect. 4.1 and shown in
Fig. 2 we have a rough idea of this distribution, but until we
obtain the information on the photometric or spectroscopic redshifts (which is in progress) we
cannot guarantee a precise cosmological parameter estimation here. Figures 15 and 16 show
the confidence contours as calculated in Fig. 14
but with the two other redshift distributions defined in Sect. 4.1. Despite
the large differences of the distribution, in particular for the number of
galaxies at z>1.5, it is reassuring that the contours are in fact only
slightly modified. The detailed
analysis involving a marginalisation over
and over the redshift
distribution of the sources (constrained using photometric redshifts)
is left for a forthcoming study. However,
for the reasonable values of ,
the degeneracy-breaking for the high
models
is not affected by the present uncertainty on the redshift distribution
of the sources.
Our result is consistent with the rough guide given by the scaling
(Jain & Seljak 1997).
Figure 15: Likelihood contours as in Fig. 14, but the source redshift distribution is assumed to be lower, with . | |
Open with DEXTER |
Figure 16: Likelihood contours as in Fig. 14, but the source redshift distribution is assumed to be higher , with . | |
Open with DEXTER |
Our correlation function measurements extend to much smaller scales
than shown in the figures above. The limit is set only by the fact
that we reject one member of all pairs closer than 3 arcsec.
Figures 17 and 18
show the tangential, radial and total shear correlation functions. The pair
separation bins are much smaller than in Figs. 5 and 6, which explains why the error bars are larger.
Even at the smallest scales, the shear
correlation function
is consistent
with the model predictions.
Figure 17: Tangential (top panel) and radial (bottom panel) shear correlation functions and down to 3''. The solid, long-dashed (hidden by the solid line) and short-dashed lines are predictions from the same models as in Fig. 3. | |
Open with DEXTER |
The surprising result for the small scale correlations is the behavior of the tangential and radial shear correlation functions: at scales smaller than 5'' we find an increased amplitude for , and a . Though surprinsing, a negative is not unphysical: for instance in Kaiser (1992) (Table 1), a shallow mass power spectrum (n>-1) implies such an effect. In terms of halo mass profile, it corresponds to a projected profile steeper than -1.5. However, regardless of the nature of this signal, it is important to note that this is a very small scale effect which has no impact on the statistics discussed in preceding sections. The contribution of the increased signal from to the variance at 1' is less than ; moreover since is not affected at all, the variance is also unaffected. As an explicit test, we checked that by removing one member of the pairs closer than 7'' the measured signal in Figs. 3-6 is unchanged. In a similar cosmic shear analysis using the Red-sequence Cluster Survey (Gladders & Yee 2000) another group finds a similar small scale behavior, though at lower statistical significance (H. Hoekstra, private communication).
The cross-correlation vanishes down to 3'', therefore no obvious systematic is responsible for this effect. The effect is unlikely to be caused by overlapping isophotes, or close neighbors effects because : if it were a close neighbor alignment we would expect that (the average tangential ellipticity for all the pair members in each pair separation bin ) carries all of the signal, which is not the case. In fact we find , which means that a close neighbor effect can hardly exceed of the small scale signal.
A forthcoming paper using the same data set will be devoted to the measurement
of E and B modes (as defined in Crittenden et al. 2000a), and we will study this
small scale signal in more detail. At this stage of the analysis we
cannot exclude a possible residual systematic.
However, a preliminary analysis shows that the B mode down to 3'' is much
smaller than the E mode, which is hard to understand if the signal comes
entirely from residual systematics.
Figure 18: Same as Fig. 17 but for the shear correlation function . The open circles correspond to the cross-correlation . | |
Open with DEXTER |
The final stage of the VIRMOS survey is to accomplish 16 deg^{2} in patches of 4 deg^{2}, 4 colors each, thus allowing the possibility to use the photometric redshifts of the galaxies. The use of photometric (or spectroscopic) redshifts will be useful to put robust constraints on the cosmological parameters and improve the scientific interpretation of cosmic shear (e.g. doing tomography as in Hu 1999) but also to measure the intrinsic alignments itself (which can be used to constrain galaxy formation models for instance).
The constraints obtained so far are within a framework of structure formation through gravitational instability with Gaussian initial conditions and Cold Dark Matter. As the amount of observations increases and the measurement quality improves, the first hints of the shape of the power spectrum will be soon available. It opens new means of really testing the formation mechanisms of the large-scale structure and the cosmological parameters beyond the standard model (Uzan & Bernardeau 2000).
Over the last two years, we have seen the transition from the detection of the weak lensing signal to the first measurements of cosmological parameters from it. The agreement between theoretical predictions and observational results with such a high precision indicates that the measurement of cosmic shear statistics is becoming a mature cosmological tool. Many surveys are under way or scheduled for the next 5 years. They will use larger panoramic cameras than the CFH12K, and will cover solid angles 10 to 100 times wider than this work. The results of this work give us confidence that cosmic shear statistics will provide valuable measurements of cosmological parameters, probe the biasing of mass/light, produce maps of the dark matter distribution and reconstruct its power spectrum.
Acknowledgements
We are grateful to S. Colombi, U.-L. Pen, D. Pogosyan, S. Prunet, I. Szapudi and S. White for useful discussions related to statistics. We thank H. Hoekstra for sharing his results prior to publication. This work was supported by the TMR Network "Gravitational Lensing: New Constraints on Cosmology and the Distribution of Dark Matter'' of the EC under contract No. ERBFMRX-CT97-0172, and a PROCOPE grant No. 9723878 by the DAAD and the A.P.A.P.E. We thank the TERAPIX data center for providing its facilities for the data reduction of CFH12K images.