Weak lensing Study in VOICE Survey I: Shear Measurement
The VST Optical Imaging of the CDFS and ES1 Fields (VOICE) Survey is a VST INAF Guaranteed Time program proposed to provide deep optical imaging covering two 4 deg cosmic windows. We present the cosmic shear measurement over four 1 deg fields in the CDFS region in the -band using LensFit. Each tile has more than one hundred exposures to reach deep imaging. More than 50 exposures of each tile passed image quality selection criteria for weak lensing study. The -band co-added image reaches limiting magnitude for point sources, which is 1.2 mag deeper than KiDS The photometric redshifts are estimated using the VOICE together with near-infrared VIDEO data . The mean redshift of the shear catalogue is 0.87 considering shear weight. The effective number density is 16.35 gal/arcmin, which is nearly double of that of KiDS. The performance of LensFit on deep imaging was calibrated using VOICE-like simulation (Liu et al., 2018). Furthermore, we analyze the reliability of the shear catalogue by calculating the star-galaxy cross-correlations, the tomographic shear correlations of two redshift bins and the contaminations of the blended galaxies. As a further sanity check, we constrain cosmological parameters by exploring the parameter space with Population Monte Carlo sampling. For a flat CDM model we obtain .
keywords:gravitational lensing: weak - methods: data analysis - survey - cosmology: observations
Gravitational lensing effect is due to the differential deflection of light caused by foreground masses. The induced coherent shape distortion of background galaxy image is referred to as weak lensing shear, and is typically much smaller than the intrinsic ellipticity of galaxy itself. Such signals can only be measured in a statistical way by averaging over a large sample of galaxies. Weak lensing effects depend sensitively on the growth of large-scale structures and the expansion history of the Universe, thus representing a complementary probe to other observables in order to constrain cosmological models (e.g. Hinshaw et al., 2013; Planck Collaboration et al., 2016). Particularly, the gravitational nature of weak lensing effects makes them uniquely important in probing the dark side of the Universe (e.g. Bartelmann & Schneider, 2001; Fu & Fan, 2014; Kilbinger, 2015; Mandelbaum, 2017).
The advance of weak lensing cosmological studies relies on the developments of wide-field imaging surveys. Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS, Heymans et al., 2012a) has shown that cosmic shear is a powerful cosmological probe (Kilbinger et al., 2013a; Benjamin et al., 2013; Fu et al., 2014; Liu et al., 2016). On-going surveys, such as Dark Energy Survey (DES, Becker et al., 2016; Jarvis et al., 2016), Kilo-Degree Survey (KiDS, Kuijken et al., 2015; Hildebrandt et al., 2017) and Hyper Suprime-Cam (HSC) survey (Aihara et al., 2018; Mandelbaum et al., 2018) are enlarging the sky coverage to a few thousands square degrees. In the coming years, next-generation weak-lensing projects such as the Euclid mission111http://sci.esa.int/euclid, the wide Field Infrared Survey Telescope (WFIRST222https://wfirst.gsfc.nasa.gov/) and the Large Synoptic Survey Telescope (LSST333https://www.lsst.org) will present yet another large increase in survey volume, making high-precision weak lensing studies possible.
While large-sky coverage is essential to minimize the cosmic variance, deep weak lensing surveys are important to study the evolution of large-scale structures over a wide redshift range. However, deep imaging surveys are challenging. The higher number density of background galaxy (few tens to hundred galaxies per square arcminute) causes crowding problems, so that object blending becoming a non-negligible issue, particularly for ground-based observations. There are much less deep surveys for weak lensing compared to the development of wide surveys due to the more stringent observing conditions. CFHTLS Deep (Semboloni et al., 2006) released a 4 deg shear catalogue with the depth of , as the first generation of deep survey. Recently, Deep Lens Survey (DLS, Jee et al., 2013, 2016) successfully derived cosmological constraints using cosmic shear catalogue with a limit of mag and a mean source redshift of over 20 deg. Schrabback et al. (2010) presented the space-based galaxy shape measurements Hubble Space Telescope Cosmic Evolution Survey (COSMOS) and found evidence of the accelerated expansion of the Universe from weak lensing tomography. Although the galaxy number density is 76 arcmin with limit mag, the COSMOS field of view is only 1.64 deg.
The VLT Survey Telescope (VST) Optical Imaging of CDFS and ES1 (VOICE, co-PIs: Giovanni Covone & Mattia Vaccari, Vaccari et al., 2016) is a GTO survey preformed with the VST. VOICE collaborated with the SUpernova Diversity And Rate Evolution (SUDARE), the other VST GTO survey, agreed to join efforts to cover the CDFS sky region (Cappellaro et al., 2015; Botticella et al., 2017). SUDARE has covered the common fields in the , optimizing the strategy in order to search and characterize supernovae at intermediate redshift ( ). VOICE has observed these fields in the band. While SUDARE has observed also in less than optimal sky conditions, the total amount of data is more than enough to obtain stacked images with the depth and image quality required by VOICE science objectives, including weak lensing. The two fields, VOICE-CDFS and VOICE-ES1, have been observed by several facilities in different bands, including Spitzer SWIRE (IR), VISTA-VIDEO (NIR), Spitzer-SERVS (MIR), Herschel-HerME (FIR), GALEX (UV) and ATLAS (radio). Adding optical data from VOICE makes the fields extremely valuable in astrophysical studies. One of the science drivers for VOICE is to detect clusters of galaxies at relatively high redshifts, and to study their mass distributions using weak lensing signals of the observed galaxies.
The VOICE survey uses the same telescope, detector (OmegaCAM) and optical filters as KiDS. The -band data are used for weak lensing measurements. Differently from KiDS, where each pointing is observed only in one epoch consisting of five consecutive exposures, VOICE survey does multiple-epoch observations for each pointing of the -band with total number of exposures over a hundred. For the data used for weak lensing shear measurements, the limiting magnitude for point source in -band co-added images reaches mag within 2″aperture diameter, which is about magnitude deeper than KiDS data.
Same as for the KiDS (Kuijken et al., 2015, hereafter K15), we use LensFit (Miller et al., 2007; Kitching et al., 2008; Miller et al., 2013) for galaxy shape measurements. We note that this is the first time that LensFit is applied to data with such a large number of exposures. Since the observing conditions vary significantly from epoch to epoch, we need to first examine the observational data carefully for quality control. We note that this is the first time that LensFit is applied to data with such a large number of exposures and posed a few challenges. First, the observing conditions vary significantly from epoch to epoch and we needed to go through a severe quality control of the individual exposure. Second, we needed to adapt the LensFit parameters for the peculiar dataset, because VOICE data are deeper than CFHTLenS and KiDS (de Jong et al., 2017). To validate the setup and calibrate the shear measurement, we made use of dedicated simulations which have been presented in a companion paper Liu et al. (2018).
The structure of this paper is organized as follows. In Section 2, we describe VOICE data and data reduction. The shape measurement procedures, the calibration from VOICE-like simulation and the photometric redshift are presented in Section 3. Two-point correlation analyses and null tests for shear systematics are presented in Section 4. To further demonstrate the quality of our shear measurements, in Section 5, we show the cosmological constraints of and derived from cosmic shear two-point correlations. The summary is given in Section 6.
2 The survey
This paper focuses on the VOICE-CDFS field, which covers about 4.9 deg. It is composed by four tiles (CDFS1, CDFS2, CDFS3, CDFS4), about 1 deg each. The pixel scale of the OmegaCAM CCDs is . The center of the VOICE-CDFS field is RA and DEC. The observations started from 2011, and ended in 2015. Each tile was observed in four optical bands with exposure time of 600s (), 360s ( and ) and 400s (). The -band data were used, in addition to the weak lensing study presented here, for variability based search of supernovae (Botticella et al., 2017) and Active Galactic Nuclei (Falocco et al., 2015). For each tile, more than one hundred exposures (each one for an exposure time of 360 seconds) were taken in the -band. Same as KiDS, a single epoch consists of five consecutive exposures obtained in dithering mode in order to cover the detector gaps. All epochs then repeat the same initial position. The cumulative exposure time ranges from 15.3 to 20.9 hours for the four fields. The total exposure time for the other three bands is less as shown in Table 1. VOICE has much more than one hundred exposures for each tiles distributed over years. Thus the image quality and the point spread function (PSF) of the individual exposures could be very different from exposure to exposure.
2.1 Exposure selections
The raw data reduction was performed using the pipeline VST-Tube (Grado et al., 2012). As described in detail in Cappellaro et al. (2015), VST-Tube can perform over scan correction, flat fielding, CCD gain harmonization, illumination correction, and cosmic ray removal.
For shear measurements, because weak lensing signals are very weak, about an order of magnitude smaller than the intrinsic ellipticity of galaxies, we apply very strict image selection criteria. Since VOICE -band observations were carried out over 4 years with large number of exposures, the observing conditions show significant variations among epochs, although the r-band has strict constraints on the seeing and illumination imposed. To homogenize the observations and maximize the quality of our shear measurements, we filter our data according to seeing and its variations within the field of view before further data processing (i.e. image co-adding, object detection and shape measurements).
The PSF full width at half maximum (FWHM) of different exposures in the -band ranges from 0.4 to 1.5 as shown in the top panel of Fig. 1. The median value is 0.86. Weak lensing studies focus on background galaxies which are mostly faint and small. Because seeing smears galaxy images if they are significantly smaller than the seeing disc, we select only those exposures with seeing smaller than .
The sky background brightness can also affect object detection and shape measurement. The background values calculated by SExtractor (Bertin, 2011) spread in an extremely wide range from a few hundreds to a few thousands Analog-to-Digital Units (ADUs). We assign the median background value of the 32 CCDs as the reference background flux value of each exposure. As shown in the bottom panel of Fig. 1, the rms value is strongly correlated with the background flux. Most of the exposures have relatively small background flux and small variations from CCD to CCD. We apply a cut on background rms dispersion in order to have a homogeneous background noise. After several iterations of examining B-mode in shear two-point correlations, the exposures with background rms dispersion over 20 are rejected in the shear analysis, corresponding to a background flux cut of 900 ADU.
In order to have a uniform depth from epoch to epoch, we further review the remaining exposures, and only keep those epochs with at least four exposures passing the selection criteria. In condition, about one-third of total exposures are used for weak lensing analysis, as shown in red in Fig. 1. The number of useful exposures for the four tiles is 62, 54, 79 and 62, respectively. The final mosaic reaches a 5 limiting magnitude of within 2 aperture diameter for point sources. The average limiting magnitude for bands is 25.3, 26.4, 25.2.
2.2 Astrometric calibration
We perform the astrometric calibration of each tile separately using the software SCAMP444https://www.astromatic.net/2010/04/20/scamp-1-7-0-release. Only exposures that passed our selection criteria are used simultaneously for the calibration to improve the internal accuracy. The external accuracy depends on the choice of reference catalogue. We perform two sets of calibrations using 2MASS (Skrutskie et al., 2006) and GAIA (Gaia Collaboration et al., 2016), respectively. The calibrated exposures are co-added by SWarp555https://www.astromatic.net/2010/09/04/swarp-2-19-1-release to produce the final stacked image used for source detection. We match the objects between the reference catalogue and the VOICE deep image, and compare coordinates difference of the two calibrations. As shown in Fig. 2, the astrometric dispersion based on GAIA (0.056) is about four times smaller than the that from 2MASS (0.19). Thus, GAIA is chosen as the absolute reference for VOICE astrometric calibration.
Saturated stars and their surrounding areas have to be masked because the flux measured in those regions can be affected by strong systematic errors. Those areas are identified by the automatic mask software Pullecenella (Huang et al., 2011; de Jong et al., 2015), which has been created specifically to treat the VST images. For LensFit, the galaxy model fitting is performed on each individual exposure. Thus the mask should not be produced from the deep co-added images to avoid over masking. Instead, we masked the affected areas of the individual epochs, i.e., the stacked images over five consecutive and dithered exposures. Fig. 3 shows an example of masked regions near saturated stars with a large reflection halo. The remaining unaffected area after masking is 84% of the original VOICE-CDFS area.
2.4 Photometric redshift catalogue description
For each tile of VOICE-CDFS, all the high-quality, astrometric calibrated exposures are co-added using SWarp to produce the deep stacked image. Source positions and star-galaxy classification are performed on the stacked image. The SExtractor software (Bertin & Arnouts, 1996) is run to generate the final source catalogue. The star-galaxy classification is done in the magnitude-size diagram (Huang et al., 2011), where magnitude and size are represented by the SExtractor parameters MAG_AUTO and MU_MAGMAG_AUTO. Sources with size smaller than the stellar one are defined as spurious and removed from the catalogue. As shown in Table 2, about 2800 stars are selected from each tile, which are used to measure the PSF. More than galaxies per tile are selected. This galaxy catalogue is used for photometric redshift estimates (photo-) and also as an input to the shape measurement software LensFit (Miller et al., 2007; Kitching et al., 2008; Miller et al., 2013).
For photo- measurements, we employ the optical observations in from VOICE, and the near-infrared data obtained by the VIDEO survey (Jarvis et al., 2013) performed with the VISTA telescope: these bands cover 80% of the VOICE images. We did not include the VIDEO band since it covers a negligible fraction ( 50%) of each VOICE field. The VOICE and VIDEO stacks were produced selecting exposures with a similar cut in the seeing ( 1.0 arcsec). We therefore decided to base our photometric redshift estimate on magnitudes measured on apertures of the same size in all bands. To this end we use the SEP Python library (Barbary, 2016): the SEP library implements algorithms from the SExtractor software (Bertin & Arnouts, 1996) as stand-alone functions and classes. We use it to measure aperture magnitudes (6 diameters) centered on the source positions in the -band catalogue. Compared to the so-called dual-mode in SExtractor, the SEP library allows to perform a list-driven photometry on images with different size, scale or center: WCS coordinates from the catalogue are converted to pixel positions in the image using functions available in the astropy python library and then passed to SEP. The background subtraction is also available within SEP.
The next step is the removal of residual errors in the calibration of the photometric zero point. To this end, we benefit from the overlap of the CDFS fields with the APASS survey666https://www.aavso.org/apass. We match unsaturated stars (15 16 ) in the . Non-negligible offsets ( mag) are found in (CDFS3 and CDSF4) and (CDFS3).
Photo- are finally derived using the BPZ software (Benítez, 2011): BPZ adopts a Bayesian approach, where the likelihood that a template fits the colours of a galaxy at a given redshift is combined with a prior defining the probability to find a galaxy of that type, as a function of magnitude and redshift. This allows to reject those solutions which would maximize the likelihood, but that would be unphysical according the known prior distributions. The BPZ library consists (Benítez et al., 2004) of four modified Coleman, Wu and Weedman types (Coleman et al., 1980), and two Kinney, Calzetti & Bohlin (Kinney et al., 1996) star burst galaxy templates. The derived photo- will be discussed in Sect. 3.4.
3 LensFit Shape measurement
The shear measurement accuracy depends sensitively on the data quality and on the data processing steps, such as the observing conditions, the quality of the camera, the PSFs shape and stability, the background noise, etc.. It is also crucial to use a reliable shape measurement algorithm optimized for the considered survey. Image simulations specifically made for the survey are normally needed to validate the optimizations and also to quantify the possible biases of the shear measurements.
KiDS data analyses (e.g., Hildebrandt et al., 2017) proved that LensFit (Miller et al., 2013, hereafter M13) is a suitable shape measurement algorithm for OmegaCAM images, with an accuracy reaching 1%. The instrument, single exposure time and dithering pattern of VOICE are the same as in KiDS.
We therefore also adopt LensFit for the shape measurement. LensFit constructs a seven-parameter galaxy model fit including the galaxy position, flux, scale-length, bulge-to-disc ratio, and galaxy ellipticity. Although the signal-to-noise ratio of an individual galaxy detected from co-added image is high, it is problematic for high-precision galaxy shape measurement using the co-added image mainly because the co-addition of PSFs of different shapes and orientations of different exposures may result in a complex stacked PSF. Furthermore, the co-adding procedures, particularly the interpolation of individual exposures to a common pixel grid introduces noise correlation between pixels, which can affect the shape measurement. Thus in LensFit, the model fitting is done on individual exposures, and the probabilities of the parameters derived from different exposures for a galaxy are statistically combined to derive its final shape measurement. The details of the algorithm of LensFit are referred to Miller et al. (2007); Kitching et al. (2008) and M13. In the following, we describe the key issues particularly relevant to the VOICE data.
3.1 PSF fitting
The observational campaign of VOICE was distributed over several years. The PSF patterns of the same tile are very different from month to month, even night to night. We show in Fig. 4 a few examples of PSF ellipticity patterns at different epochs in the CDFS1 tile constructed by co-adding PSFs from five exposures within an epoch. The four epochs were observed at different times, from summer to winter. Strong temporal variations of PSF are clearly seen. Furthermore, any sub-optimal optical configuration of the telescope contributes significantly to the PSF. As discussed in K15, any primary mirror astigmatism of the curved focal plane of the VST results in an increasing ellipticity in the center of the field (top-right panel of Fig. 4), while a tilt of the secondary mirror causes the increase of ellipticity near one edge of the field (bottom-left panel of Fig. 4).
Therefore the PSF model fitting is made for each single exposure. Nevertheless as shown in Fig. 4, the PSF varies not only over the full field of OmegaCAM, but also from CCD to CCD, thus two different polynomial fitting models are applied: a 4th order polynomial fit for the full field-of-view and a 1st order chip-dependent polynomial for individual CCDs, as done by K15 for the KiDS survey.
3.2 Exclusion of galaxies
LensFit fits each single galaxy in a postage stamp with a size of 48 48 pixels, which is a compromise between large enough to obtain a correct model fit while small enough for a fast handling and fitting. The center of the postage stamp is chosen to be the position of the galaxy detected from the deep co-added image. Before the model fitting, LensFit performs a few quality checks. We give a short summary here, and refer to M13 for more details about the fitting algorithm.
Galaxies larger than the size of the postage stamp are excluded from the analysis.
To deblend the neighboring galaxies, if more than one object are found within the same postage stamp, the algorithm checks whether the neighbour galaxy can be masked by replacing the pixel values of the background without contaminating the isophotes of the target galaxy. Comparing the Gaussian smoothed isophotes of the neighbour galaxy measured from the co-added image to the smoothed pixel noise, if the signal-to-noise ratio is larger than a defined threshold, this neighbour galaxy will be masked out. Since VOICE is deeper than CFHTLenS and KiDS, in order to retain enough galaxies while still suppress most of the neighbour contaminations, we optimized this threshold from two (M13 for CFHTLenS) to five. Imaging simulations of Liu et al. (2018) show that this choice does not introduce significant bias to the VOICE shear measurements. More details are discussed in Sect. 3.5 and Liu et al. (2018).
If masked pixels are outside the target galaxy’s isophote on single exposure, the pixels are replaced by the background values and the process continues. If the masked pixels are within the isophote, then that exposure will not be used in the joint analysis.
If the weighted centroid of a galaxy is more than 4 pixels away from its stamp center, it implies that there may be blended objects existed within the stamp. Thus this galaxy is excluded as well.
As shown as in Table 2, the fraction of excluded galaxies from the above criteria is about 19%.
3.3 Shear catalogue
LensFit defines the galaxy weight taking into account both the shape-noise variance and ellipticity measurement-noise variance (M13). About 17% of total galaxies fail in galaxy model fitting although they have passed the exclusion selection. They are given a weight of zero, and their numbers are shown as in Table 2. Faint galaxies are much noisier than bright ones, and their weights are much lower as presented in Fig. 5. The magnitude histogram of the non-zero weight galaxies is shown in Fig. 6. The peak magnitude of the weighted distribution is about 24.2 mag, which is about 1 mag deeper than the LensFit selected galaxies in KiDS.
In order to have continuous coverage of CDFS fields, an overlap of 37 arcmin has been taken among the four tiles. Thus galaxies from the overlapping regions have to be dealt with separately, if they are detected more than once. Due to astrometric errors, some galaxy positions may be slightly different in the overlap region of different exposures. If a pair of galaxies has a separation of less than pixels, we regard them as a single galaxy and only keep the higher signal-to-noise measurement result.
The final shear catalogue has over galaxies with non-zero weight, corresponding an effective weighted galaxy number density 16.35 arcmin, which is about double of the density in the KiDS survey.
3.4 The photometric redshift distribution
The shear catalogue is matched to the photo- catalogue (Sect. 2.4). We choose the peak value of the Probability Density Function as an estimate of its photo-. The mean and median values of the photo- of the shear catalogue (non-zero weight) are 0.87 and 0.83, respectively. We fit the redshift distribution using the following formula:
where the best fit values of the parameters are 0.50, 0.39, 4.66, 0.60, respectively. The histogram and the fitted photo- distributions are shown in Fig. 7. The fitted Eq. 1 is used as the redshift distribution of the VOICE galaxies for theoretical shear two-point correlation predictions in Sect. 4.3. The normalized histogram of photo- is used for cosmological constraints (Sect. 5) to avoid the possible bias due to the model fitting.
We note that this paper focuses on presenting the VOICE shear measurement results. The photo- distribution of the background galaxies are needed for cosmological constraints. We check the photo- measurements by comparing with a subsample with spectroscopic redshifts (spec-). We matched the galaxies to the spectroscopic redshift sample (Vaccari et al., 2010; Vaccari, 2015) and found 24239 galaxies. As shown in Fig. 8, the photo- has generally good agreement with spec-. The median value of photo-spec-spec- is with Median Absolute Deviation (MAD) 0.06. We separate the full sample into two redshift bins according to the median value 0.83 of the whole shear catalogue. The matched galaxies in low and high bins are 19791 and 4291, respectively. The sub-samples of two redshift bins show opposite as compared to the spectroscopic redshift. We found and for the low and high bin. The MAD values are 0.06 and 0.11, respectively.
3.5 VOICE-like simulation
VOICE is about one magnitude deeper than CFHTLenS and KiDS, composed of a few tens usable exposures for each field. We need to optimize LensFit to consider the dense background galaxies and check its shear measurement accuracy of dealing with such a large number of exposures simultaneously for each galaxy shape measurement.
To validate our optimization and calibrate the measured shear, we perform image simulations in accord with the observed -band images. We briefly summarize the simulation results here and refer to the paper by Liu et al. (2018) for more details. In the simulation, we use the sources detected in the stacked images as the input parent sample, and fix many observing conditions, such as the dithering pattern, background noise, celestial positions and brightness of the detected objects, to mimic the real observation. In this case, galaxy clustering and blending effect are included naturally. The PSFEx package (Bertin, 2011) is used to model the spatially varied PSF for every exposure. For each galaxy, a randomly sampled intrinsic ellipticity value and a constant shear with modulus of the reduced shear = 0.04 is assigned. In total, four different shear combinations (, ) are used, which are (0.0283, 0.0283), (0.0283, 0.0283), (0.0153, 0.0370), and (0.0370, 0.0153), respectively. The simulated single exposure images are then generated by the Galsim toolkit (Rowe et al., 2015), and the galaxy shapes are also measured by LensFit. Overall, our simulation presents good agreements with the observation, especially the distributions of the PSF properties. We apply the bin-matching method to the SNR and size plane to calibrate the bias of the simulation data. The final residual multiplicative bias after calibration reaches to an accuracy of 0.03 with negligible addictive bias in different SNR and size bins.
The sensitivity of the bias calibration to the undetected and neighboring objects is also discussed in Liu et al. (2018). The undetected objects are likely to skew the background noise so that they can potentially bias the shape measurements of galaxies, especially those with low SNR. Taking the depth and noise level into account, we find that the impact of the undetected galaxies is negligible for the VOICE survey. Additionally, the bias results from galaxy blending effect are also analyzed. Further analyses show that their impact on the 2PCF can be securely neglected due to the small fraction they account for (Sect. 4.7).
4 Shear two-point correlation analyses
Cosmic shear is the weak lensing effect caused by the large-scale structures in the Universe. We briefly summarize the theoretical relations between second-order weak lensing observables and cosmological quantities in Sect. 4.1, and then present the correlation analyses of the VOICE shear catalogue. For details on the theoretical foundation of weak gravitational lensing we refer to the literature (e.g. Bartelmann & Schneider, 2001; Fu & Fan, 2014; Kilbinger, 2015; Mandelbaum, 2017).
4.1 Theoretical background
Weak lensing induced by the large-scale structures measures the convergence power spectrum through two-point correlation statistics. It is a projection of the total matter density fluctuation power spectrum under the Limber approximation (Kaiser, 1992):
The projection integral is carried out over comoving distances , from the observer out to the limiting distance of the survey. The lensing efficiency is given by
where is the Hubble constant, is the speed of light, is the present total matter density, and is the scale factor at comoving distance . The cosmology-dependent co-moving angular diameter distance is denoted by .
Cosmic shear two-point correlation functions (2PCFs) are the Hankel transforms of the convergence power spectrum , which can be written as the linear combinations of the E- and B-mode spectra, and , respectively
where and are the first-kind Bessel functions of order 0 and 4, corresponding to the components and , respectively.
In real observations, the most direct measurement of weak gravitational shear signal is derived from galaxy ellipticity measurements. The unbiased 2PCFs and are estimated by averaging over pairs of galaxies (Schneider et al., 2002b),
Here, the sum is performed over all galaxy pairs with angular separation within some bin around . and are the tangential and cross-components of the galaxy ellipticity, respectively, with respect to the line connecting the two galaxies. is the weight for the -th galaxy, obtained from the LensFit.
Weak gravitational lensing only contributes to an E-mode power spectrum and therefore, a non-detection of the B-mode is a way to check the quality of shear measurement of the data. The E-/B-mode shear correlations , the aperture-mass dispersion and the shear top-hat rms are the most popularly used second-order shear correlations. The decomposed E- and B-mode estimators in an aperture of radius can be written as integrals over the filtered correlation functions of and (Crittenden et al., 2002; Schneider et al., 2002a), as follows:
where is the bin width varying with . The estimators and are only sensitive to the E- and B-mode, respectively, with suitable filter functions and . The detail expressions of other two-point correlations are referred to Table 1 and Appendix A of Kilbinger et al. (2013b).
4.2 Multiplicative Bias Correction
As shown in Eq. (5), given an unbiased shear measurement, 2PCFs and can be estimated, from an observational point of view, by averaging over pairs of galaxies. However, data reduction and shear measurement methods can generate possible biases. Thus a shear calibration (Heymans et al., 2012b) is usually applied to describe the relation between the observed shear and the true signal, which accounts for a potential additive bias and a multiplicative bias for the -th component of the galaxy ellipticity (),
In our analyses, the additive bias is estimated from observational shear catalogue, and found to be consistent with zero, on average at the level of and for and , respectively. However, the multiplicative biases are non-negligible. We derive them based on our image simulations (Liu et al., 2018). Particularly, we obtain the values in multiple two-dimensional bins of the galaxy S/N and the size from simulation analyses. We then apply them to the galaxies in the observed shear catalogue according to their SN and size. We find different values for and . We then need to take into account the multiplicative bias for and separately when calculating the shear 2PCFs, which is different from previous studies, such as CFHTLenS and KiDS. We drive the corresponding 2PCFs components taking into account different values as follows.
Considering a pair of galaxies located at and , respectively, their tangential and cross components with respect to the pair separation are given by
where is the polar angle . 2PCFs (Eq. 5) can then be expressed in terms of a complex ellipticity quantity composed of two components, ,
Therefore, we need to introduce four calibration factors ( and ) here
where considering the pair symmetry. The final calibrated 2PCFs are then obtained by
4.3 Shear two-point correlation estimations
Based on the above analyses, we calculate the shear 2PCFs using the combined VOICE shear catalogue of the four tiles of CDFS. The results are shown in Fig. 9. The upper left panel shows (red full dots) and (black open diamonds), respectively. The upper limit of the angular separation considered here is taken to be , in accord with the survey area of . For the lower limit, although we show the results from in Fig. 9, we actually calculate starting from , which corresponds to the LensFit postage stamp size 48 pixels.
The other three panels of Fig. 9 show the results of (top-right), (bottom-left) and (bottom-right), respectively. They are derived from by performing integrations with different filters. To avoid introducing artificial B-mode due to the finite integration range, we limit to consider the three quantities up to the angular scale , the radius of an aperture with maximum separation of galaxy pairs. It is seen that the B-mode is consistent with zero for all the three derived quantities. The multiplicative biases of have been corrected (Eq. 12 and 13). The amplitudes of the corrections are a few present on 2PCFs.
The different filter functions of three derived second-order functions lead to different sensitivities on smoothing scales. For instance, is the one with the highest correlation between data points, thus the E-/B-mode components look smoother than those of the other two quantities. The error bars are the squared root of the diagonal terms of the covariance matrix measured from VOICE-like ray-tracing simulations to be described in Sect. 4.4.
The results are compared to the theoretical predictions using the cosmological parameters derived from KiDS (Hildebrandt et al., 2017) and Planck15 (Planck Collaboration et al., 2016) cosmology, where and respectively, with the same angular scale range for . The redshift distribution used for the theoretical predictions is the solid line shown in Fig. 7, which is the fitting result to the photo- distribution of the VOICE shear catalogue (Eq. 1). The corresponding predictions are shown with solid green lines in different panels of Fig. 9.
4.4 Covariance Estimation
To model and interpret the observed 2PCFs, we need to estimate the error covariance. To do so, we use N-body simulations described in Liu et al. (2015) to account for the non-Gaussianity of the cosmic shear field on small and medium angular scales, and perform ray-tracing calculations to construct the shear and convergence maps. The cosmology involved is the flat cold dark matter (CDM) model with , , , , and , where , , and are the present dimensionless densities of the total matter, cosmological constant, and the baryonic matter, respectively, is the rms of linearly extrapolated density perturbations over , is the power index of the power spectrum of initial density fluctuations, and is the Hubble constant in units of . In order to cover a large redshift range up to in ray-tracing calculations, we pad 12 independent simulation boxes, with 8 small boxes each with a size of to and 4 larger boxes each with a size of from to , and use in total 59 lens planes. From one set of padded boxes, we can generate 4 sets of lensing maps each with an area of sampled on pixels. For each set, we have 59 shear and 59 convergence maps at 59 different redshifts corresponding to the far edges of the 59 lens planes. In total, we run 24 sets of simulations, and generate lensing maps with the total area of . The more detailed descriptions for our N-body simulations and ray-tracing calculations can be found in Liu et al. (2015) and Liu et al. (2014).
With these lensing maps, we then generate 384 VOICE-like mock catalogues to estimate the error covariance. The generating procedure for each mock is as follows.
(i) We place the 4 continuous VOICE tiles randomly over the simulated map area, with the positions, photo-, galaxy weights and the mask information preserved in the analyses. The amplitudes of ellipticities of the galaxies are also preserved, but with their orientations being randomized.
(ii) For each galaxy in the catalogue, its reduced shear is calculated by interpolating the signals from the pixel positions on simulated maps to the galaxy position. The interpolation is also done in the redshift dimension. Regarding the randomized ellipticity obtained in (i) as its intrinsic ellipticity , the mock observed ellipticity can then be constructed from
(iii) The 2PCFs analyses are then carried out for each mock, with the same procedures for the observed data, the error covariance can be further estimated with these 2PCFs results from the whole 384 mocks. These covariance matrices are used to give error bars shown in Fig.9, and also applied to derive cosmological constraints to be presented in Section 5.
4.5 The star-galaxy cross-correlation function
The results in Fig. 9 show that our VOICE shear catalogue exhibits no detectable B-mode. To further check the data quality, we analyze the level of PSF-related systematics by measuring the star-galaxy cross correlation , where is the observed shear estimators, is a complex dimensional vector of PSF ellipticity at the position of the galaxy in each of the dithered exposures of the field. For these analyses, star-galaxy pairs with the angular separation in the range of are taken into account, and they are divided into 6 evenly-distributed log-normal bins. The zero-lag star-galaxy correlation , hereafter , which indicates the primary systematics, is derived using the model of PSF ellipticity to determine at the location of each galaxy, with
If the PSF model and correction are exact such that observed shear estimator are uncorrelated with PSF, should be consistent with zero.
Following some arguments discussed in Heymans et al. (2012b), with a measure of the zero-lag star-galaxy correlation , we can make a prediction of the star-galaxy correlation at any angular scale using
where is the measured covariance matrix of PSF ellipticities between exposures at zero-lag and is the same PSF measurement but for sources at separation . Here we only consider the case using weighted PSF ellipticities in the final shear catalogues. Thus, Eq. (16) reduces to
where and indicate objects separated by a distance .
Fig. 10 shows the star-galaxy cross-correlation function measured in CDFS1-4 fields. Generally speaking, the whole star-galaxy cross-correlation function is consistent with zero and is well within the range as indicated in the KiDS survey.
4.6 Tomography check
The reliability of shear measurement of VOICE can be further tested by considering the tomographic shear signals. We separate the full shear sample into two photo- bins divided by the median photo- of 0.83. The results of (left) and (right) are shown in Fig. 11. As expected, the shear correlation of the high redshift bin is higher than that of the low redshift bin. There are no obvious B-modes in all angular scales for both of cases. The solid green lines are the theoretical predictions from the KiDS and Planck15 cosmology with the redshift distributions for the two bins directly from the photo- measurements. We can see that our VOICE results are in good agreements with the theoretical predictions.
As this paper mainly focuses on the shear measurement of VOICE, the tomographic results presented here are only for checking the reliability of the shape measurement. Being our next task, we will perform cosmological studies using the tomographic correlations from VOICE. For that, we will consider carefully the impacts of galaxy intrinsic alignments and photo- errors.
4.7 Blending Effect
For VOICE observations, the final mosaic reaches to limiting magnitude of 26.1 mag with aperture diameter for point sources. Over 488,000 galaxies are detected with number density of 32.85 arcmin after rejecting the mask regions. Following Chang et al. (2013), we define the neighbors simply by their separation on the celestial sphere. We find that only 0.04% of galaxies are neighbors within separation, while the fraction increases dramatically to over 16% within 3.0 separation. These galaxies can be either physically related neighbors which have similar shear or projected close pairs but with different redshifts and shape distortions. Though LensFit has encoded an algorithm to deal with them (Miller et al., 2013), potential bias is still inevitable in the measured shear due to the inappropriate modeling of the surface brightness distributions in the overlapped regions.
Although most of the neighbors have been excluded by LensFit, about 31.6% of the neighboring galaxies within separation still have shape measurements. The ellipticity dispersion of these remainders is 3.4% larger than the overall dispersion. The weighted number density of them is about 1.28 arcmin. We compare the shear two-point correlation functions of the full sample and that after rejecting neighbors. The results are shown in Fig. 12. We find that the differences are within the error bars given the relatively large statistical uncertainties of the VOICE shear sample. For future large surveys with dramatically reduced statistical errors, the neighboring contaminations need to be carefully accounted for.
From our image simulations (Liu et al., 2018), we further quantify the impact of the close neighbors on the multiplicative biases. It is found that the SNR of these galaxies are systematically overestimated by LensFit due to the contamination of the neighboring galaxy. As a result, these close neighbors do contribute extra multiplicative bias, especially at high SNR. The weighted average bias resulting from these neighbors is about 0.002 from our simulation analyses. Although this can be safely neglected for VOICE analyses, it can be a serious concern for future large surveys that need the multiplicative bias to be controlled at the level less than 0.001.
5 Cosmological Constraints
The most sensitive constraints from weak lensing alone are the cosmological parameters of the matter density and the linear amplitude of mass fluctuations . In this section, we present the marginalized constraints for and of flat CDM cosmological model. We note that the main focus of the paper is to present the shear measurements. The cosmological constraints shown here are for reliability check again in addition to the 2PCFs presented in the previous sections. Considering also the relatively large statistical uncertainties of the VOICE shear catalog, here we do not include different possible systematics, such as galaxy intrinsic alignments, baryonic effects, photo- errors, etc.. We will do more careful cosmological analyses as our next task.
5.1 Sampling the posterior
We use the open source code Cosmo_PMC777http://cosmopmc.info (Kilbinger et al., 2011) to sample the VOICE weak-lensing constraint posterior with Population Monte Carlo (PMC). For the flat CDM model, the base parameters are , , , and . The prior ranges are summarized in Table 3.
The perplexity parameter of Cosmo_PMC is a value between 0 and 1, where 1 means the perfect agreement between importance function and the posterior. Generally, reaches 0.7 after 10 runs of iterations, enough satisfied for us to stop the iterations. We use 30,000 sample points in each iteration. For the last iteration, larger samples with 300,000 points are used to reduce the Monte-Carlo variance.
|Total matter density|
|Spectral index of prim. density fluct.|
5.2 Choice of second-order estimators
We mainly use the aperture mass dispersion for deriving cosmological constraints, for the following reasons. 1) The filter function of is much narrower compared to the one of top-hat shear rms . Thus of different smoothing scales are less correlated. 2) For , only the lower angular limit is problematic and causes leakage of the B-mode into the E-mode signal on small smoothing scales.
Anderson (2003) and Hartlap et al. (2007) have shown that the inverse covariance calculated directly from the covariance matrix constructed from simulations is biased, resulting a biased maximum likelihood (ML) estimator. We correct the ML estimator by multiplying with the Anderson-Hartlap factor (Hartlap et al., 2007). The bias depends on the number of simulations , and the number of data bins . Here we have and . Thus the correct factor is .
Before presenting the main constraints, we first check the consistency by comparing the constraints from and those from the 2PCFs for the flat CDM model. The results are shown in Fig. 13. It is seen that the two second-order quantities give rise to very similar contours in the plane of and . It demonstrates that the B-mode of has negligible impact on the cosmological parameters constraints.
As the goal of this paper is to present the VOICE shear catlaog measurement, we only show the marginalized constraints of and for flat CDM cosmological model in Fig. 14. The degeneracy direction of these two parameters is approximately a power law. Its amplitude is given by the parameter . In order to compare to the results from KiDS analyses, we fix and derive the constraints of parameters and . The results are and in the case of CDM model. By fixing as KiDS-450 analyses did (Hildebrandt et al., 2017), we have . The result is in broad agreements with that from KiDS-450 and from other studies, showing that our shear measurements are not contaminated significantly by systematics comparing to the statistical uncertainties.
To compare with constraints derived from CMB measurements, we also show the results from WMAP9888https://lambda.gsfc.nasa.gov/product/map/dr5/parameters.cfm (green) and Planck15999https://wiki.cosmos.esa.int/planckpla2015 (TT + lowP, red) in Fig. 14. The VOICE constraints are in broadly agreements with them due to the relatively large statistical uncertainties. On the other hand, although statistically comparable, a mild tension with PLANCK15 can still be seen. Similar tension is shown if we compare with Planck polarization data (TT TE EE lowP) despite the large statistic error of VOICE shear 2PCF. The above analyses mainly show the validity of our shear catalog. The detailed cosmological studies taking into account different systematics will be presented in our forthcoming paper.
We present the cosmic shear measurement of the 4.9 deg CDFS field from VOICE -band images obtained using VST/OmegaCAM. Each tile has been observed by more than 100 exposures. After stringent selections for high quality data, including cuts on seeing, sky background brightness variation, about one-third exposures are used for shear measurement. The corresponding -band co-added images reaches limiting magnitude for point sources, which is 1.2 mag deeper than KiDS. We use the LensFit shape measurement software, which has been successfully applied on CFHTLenS and KiDS. However, it is the first time that LensFit is applied to a deep survey with more than a few tens exposures. We checked the accuracy of the shear measurement from VOICE-like imaging simulations in the companion paper by Liu et al. (2018). From those, we obtain the multiplicative bias calibration values at different galaxy SNR and size bins. After the calibrations, the final residual multiplicative bias of LensFit shear measurement is measured with an accuracy of 0.03 with negligible addictive bias. The final VOICE-CDFS shear catalogue contains more than galaxies with non-zero weight, corresponding to the effective number density of galaxies 16.35 arcmin, about double of the KiDS one. The photo- of each galaxy is estimated using the VOICE together with the near-infrared VIDEO data. The mean redshift of the shear catalogue is 0.87 considering shear weights.
To check the reliability of our VOICE shear catalogue, we calculate the star-galaxy cross-correlations. Generally speaking, the whole star-galaxy cross-correlation function is consistent with zero. We further calculate the 2D shear 2PCFs and the derived second-order statistics, and those with two tomographic redshift bins divided by the median redshift 0.83 of the sample. The results are compatible with the theoretical predictions using the cosmological parameters derived from KiDS and Planck15. VOICE is a deep imaging survey, and it is important to assess the impact of possible blending effect. As detail discussed in Liu et al. (2018), although most of the neighbors have been excluded by LensFit, about 31.6% of the neighboring galaxies within separation still have shape measurements. By comparing the shear two-point correlation functions between the full sample and that after rejecting neighbors, we find that the impact of these neighboring galaxies on the shear correlations is within the VOICE statistical uncertainties. This can be a serious concern, however, for future large and deep surveys.
To further validate our shear measurements, we derive cosmological constraints from the second-order shear statistics . We show the marginalized constraints for and of flat CDM cosmological model, resulting . The results are fully consistent with those from other weak lensing studies.
Having tested the quality of our shear catalogue, next, we plan to carry out detailed cosmological studies with different systematics carefully accounted for. Furthermore, our results will allow us to detect galaxy clusters over a broad redshift range, and constrain their mass distribution from VOICE shear catalogue.
We thank Jun Zhang for helpful comments on shear measurement. L.P.F. acknowledges the support from NSFC grants 11673018, 11722326 & 11333001, STCSM grant 16ZR1424800 and SHNU grant DYL201603. Z.H.F. acknowledges the support from NSFC grants 11333001 and 11653001. X.K.L. acknowledges the support from YNU Grant KC1710708 and General Financial Grant from China Postdoctoral Science Foundation with Grant No. 2016M591006. Support for G.P. is provided by the Ministry of Economy, Development, and Tourism’s Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS. M.R. acknowledges the support from PRINM IUR 2015 “Cosmology and Fundamental Physics: illuminating the Dark Universe with Euclid”. M.V. acknowledges support from the European Commission Research Executive Agency (FP7-SPACE-2013-1 GA 607254), the South African Department of Science and Technology (DST/CON 0134/2014) and the Italian Ministry for Foreign Affairs and International Cooperation (PGR GA ZA14GR02).
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