We study lensing magnification of source galaxies by intervening galaxy groups and clusters using a halo model. Halos are modeled with truncated NFW profiles with ellipticity added to their lensing potential and propagated to observable lensing statistics. We present the formalism to calculate observable effects due to a distribution of halos of different masses at different redshifts along the l ine of sight. We calculate the effects of magnification on the number counts of high-redshift galaxies. Using BLAST survey data for submillimeter galaxies (SMGs), we find that magnification affects the steep, high flux par t of the counts by about 60%. The effect becomes much stronger if the intrinsic distribution is signi ficantly steeper than observed. We also consider the effect of this high-redshift galaxy population on contaminating the Sunyaev-Zel'dovich (SZ) signal of massive clusters using the halo model approach. We find that for the majority of clusters expected to be detected with ongoing SZ surveys, there is significant contamination from the Poisson noise due to background SMGs. This contr ibution can be comparable to the SZ increment for typical clusters and can also contaminate the SZ decrement of low mass clusters. Thus SZ observations, especially for the increment part of the SZ spectrum, need to include careful modeling of this irreducible contamination for mass estimation. Lensing further enhances the contamination, especially close to the cores of massive clusters and for very disturbed clusters with large magnification cross-section.
We present an analytical model which reproduces measured galaxy number counts from surveys in the wavelength range of 500 micron to 2 mm. The model involves a single high-redshift galaxy population with a Schechter luminosity function which has been gravitationally lensed by galaxy clusters in the mass range 10^13 to 10^15 Msun. This simple model reproduces both the low flux and the high flux end of the number counts reported by the BLAST, SCUBA, AzTEC and the SPT surveys. In particular, our model accounts for the most luminous galaxies detected by SPT as the result of high magnifications by g…
Galaxy-galaxy lensing is an essential tool for probing dark matter halos and constraining cosmological parameters. While galaxy-galaxy lensing measurements usually rely on shear, weak-lensing magnification contains additional constraining information. Using the fundamental plane (FP) of elliptical galaxies to anchor the size distribution of a background population is one method that has been proposed for performing a magnification measurement. We present a formalism for using the FP residuals of elliptical galaxies to jointly estimate the foreground mass and background redshift errors for a st…
We study how joint shear and magnification measurements improve the statistical precision of weak lensing mass calibration experiments, relative to standard shear-only analysis. For our magnification measurements, we consider not only the impact of lensing on the source density of galaxies, but also the apparent increase in source sizes. The combination of all three lensing probes - density, size, and shear - can improve the statistical precision of mass estimates by as much as 40% - 50%. This number is insensitive to survey assumptions, though it depends on the degree of knowledge of the para…
The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an a…
The magnification of galaxies in modern galaxy surveys induces additional correlations in the cosmic shear, galaxy-galaxy lensing and clustering observables used in modern lensing "3x2pt" analyses, due to sample selection. In this paper, we emulate the magnification contribution to all three observables utilising the SLICS simulations suite, and test the sensitivity of the cosmological model, galaxy bias and redshift distribution calibration to un-modelled magnification in a Stage-IV-like survey using Monte-Carlo sampling. We find that magnification cannot be ignored in any single or combined …
We introduce the use of the Fast Multipole Method (FMM) to speed up gravitational lensing ray tracing calculations. The method allows very fast calculation of ray deflections when a large number of deflectors, $N_*$, is involved, while keeping rigorous control on the errors. In particular, we apply this method, in combination with the Inverse Polygon Mapping technique (IPM), to quasar microlensing to generate microlensing magnification maps with very high workloads (high magnification, large size and/or high resolution) that require a very large number of deflectors. Using, FMM-IPM, the comput…
I present an estimator for the angular cross-correlation of two tracers of the cosmological large-scale structure that utilizes redshift information to isolate separate physical contributions. The estimator is derived by solving the Limber equation for a re-weighting of the foreground tracer that nulls either clustering or lensing contributions to the cross-correlation function. Applied to future photometric surveys, the estimator can enhance the measurement of gravitational lensing magnification effects to provide a competitive independent constraint on the dark energy equation of state.
We study the lensing magnification effect on background galaxies. Differential magnification due to different magnifications of different source regions of a galaxy will change the lensed composite spectra. The derived properties of the background galaxies are therefore biased. For simplicity, we model galaxies as a superposition of an axis-symmetric bulge and a face-on disk in order to study the differential magnification effect on the composite spectra. We find that some properties derived from the spectra (e.g., velocity dispersion, star formation rate and metallicity) are modified. Dependi…
Gravitational wave sources are a promising cosmological standard candle because their intrinsic luminosities are determined by fundamental physics (and are insensitive to dust extinction). They are, however, affected by weak lensing magnification due to the gravitational lensing from structures along the line of sight. This lensing is a source of uncertainty in the distance determination, even in the limit of perfect standard candle measurements. It is commonly believed that the uncertainty in the distance to an ensemble of gravitational wave sources is limited by the standard deviation of the…
Weak lensing is commonly measured using shear through galaxy ellipticities, or using the effect of magnification bias on galaxy number densities. Here, we report on the first detection of weak lensing magnification with a new, independent technique using the distribution of galaxy sizes and magnitudes. These data come for free in galaxy surveys designed for measuring shear. We present the magnification estimator and apply it to an X-ray selected sample of galaxy groups in the COSMOS HST survey. The measurement of the projected surface density Σ(r) is consistent with the shear measurements with…
Effect of gravitational magnification on the measurement of distance modulus of type Ia supernovae is presented. We investigate a correlation between magnification and Hubble residual to explore how the magnification affects the estimation of cosmological parameters. We estimate magnification of type Ia supernovae in two distinct methods: one is based on convergence mass reconstruction under the weak lensing limit and the other is based on the direct measurement from galaxies distribution. Both magnification measurements are measured from Subaru Hyper Suprime-Cam survey catalogue. For both mea…
We report lensing magnifications, extinction, and time-delay estimates for the first resolved, multiply-imaged Type Ia supernova iPTF16geu, at $z = 0.409$, using $Hubble\,Space\,Telescope$ ($HST$) observations in combination with supporting ground-based data. Multi-band photometry of the resolved images provides unique information about the differential dimming due to dust in the lensing galaxy. Using $HST$ and Keck AO reference images taken after the SN faded, we obtain a total lensing magnification for iPTF16geu of $μ= 67.8^{+2.6}_{-2.9}$, accounting for extinction in the host and lensing ga…
The effect of ionized gas on the propagation of radio signals is known as plasma lensing. Unlike gravitational lensing, plasma lensing causes both magnification and strong de-magnification effects to background sources. We study the cross section of plasma lensing for two density profiles, the Gaussian and power-law models. In general, the cross section increases with the density gradient. Radio sources can be used to measure the free electron density along the line of sight. However, plasma lensing de-magnification causes an underestimate of the electron density. Such a bias increases with th…
We present a concept study on weak lensing map reconstruction through the cosmic magnification effect in galaxy number density distribution. We propose a minimal variance linear estimator to minimize both the dominant systematical and statistical errors in the map reconstruction. It utilizes the distinctively different flux dependences to separate the cosmic magnification signal from the overwhelming galaxy intrinsic clustering noise. It also minimizes the shot noise error by an optimal weighting scheme on the galaxy number density in each flux bin. Our method is in principle applicable to all…
Spatial correlations of the observed sizes and luminosities of galaxies can be used to estimate the magnification that arises through weak gravitational lensing. However, the intrinsic prop- erties of galaxies can be similarly correlated through local physical effects, and these present a possible contamination to the weak lensing estimation. In an earlier paper (Ciarlariello et al. 2015) we modelled the intrinsic size correlations using the halo model, assuming the galaxy sizes reflect the mass in the associated halo. Here we extend this work to consider galaxy magnitudes and show that these …
Advances in upcoming weak lensing surveys pose new challenges for an accurate modeling of the lensing observables. The wide sky coverage of Euclid makes angular scales down to $l_\mathrm{min}=10$ accessible. At such large angular scales, general relativistic effects manifest themselves, and the lensing magnification cannot be correctly described by the standard lensing convergence only. The impact of line-of-sight velocities on the magnification angular power spectrum, referred to as the Doppler magnification, is already well recognized in literature. In particular, it was suggested that the D…
Weak lensing, Big Magnification: Recovering hidden lensing information from galaxy positions
Institute of Space Sciences (ICE, CSIC)（空间科学研究所）；Universitat Autònoma de Barcelona (UAB)（巴塞罗那自治大学）；Institut de Física d’Altes Energies (IFAE)（高能物理研究所）；The Barcelona Institute of Science and Technology（巴塞罗那科学技术研究院）
Cluster lensing has become an important tool in the search for high redshift galaxies through its ability to magnify sources. In order to determine the intrinsic properties of these galaxies, lensing mass models must be constructed to determine the magnification of the images. These models are traditionally two-dimensional, focusing on the mass within the cluster and either ignoring or approximating any contribution from line-of-sight galaxies. In this paper, we present the first full set of three-dimensional mass models of the six Hubble Frontier Fields and use them to test for systematic bia…
The weak lensing magnification of Type Ia Supernovae (SNe Ia) is sensitive to the clustering of matter, and provides an independent cosmological probe complementary to SN Ia distance measurements. The Nancy Grace Roman Space Telescope is uniquely sensitive to this measurement as it can discover high redshift SNe Ia and measure them with high precision. We present a methodology for reconstructing the probability distribution of the weak lensing magnification $μ$ of SNe Ia, $p(μ)$, from observational data, and using it to constrain cosmological parameters. We find that the reconstructed $p(μ)$ c…
We propose a new approach for measuring the mass profile and shape of groups and clusters of galaxies, which uses lensing magnification of distant background galaxies. The main advantage of lensing magnification is that, unlike lensing shear, it relies on accurate photometric redshifts only and not galaxy shapes, thus enabling the study of the dark matter distribution with unresolved source galaxies. We present a feasibility study, using a real population of z > 2.5 Lyman Break Galaxies as source galaxies, and where, similar to galaxy-galaxy lensing, foreground lenses are stacked in order to i…
Recent works have shown that weak lensing magnification must be included in upcoming large-scale structure analyses, such as for the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST), to avoid biasing the cosmological results. In this work we investigate whether including magnification has a positive impact on the precision of the cosmological constraints, as well as being necessary to avoid bias. We forecast this using an LSST mock catalog and a halo model to calculate the galaxy power spectra. We find that including magnification has little effect on the precision of the cosmo…
We describe a new method for measuring galaxy magnification due to weak gravitational lensing. Our method makes use of a tight scaling relation between galaxy properties that are modified by gravitational lensing, such as apparent size, and other properties that are not, such as surface brightness. In particular, we use a version of the well-known fundamental plane relation for early type galaxies. This modified "photometric fundamental plane" replaces velocity dispersions with photometric galaxy properties, thus obviating the need for spectroscopic data. We present the first detection of magn…
We present a joint shear-and-magnification weak-lensing analysis of a sample of 16 X-ray-regular and 4 high-magnification galaxy clusters at 0.19<z<0.69 selected from the Cluster Lensing And Supernova survey with Hubble (CLASH). Our analysis uses wide-field multi-color imaging, taken primarily with Suprime-Cam on the Subaru Telescope. From a stacked shear-only analysis of the X-ray-selected subsample, we detect the ensemble-averaged lensing signal with a total signal-to-noise ratio of ~25 in the radial range of 200 to 3500kpc/h. The stacked tangential-shear signal is well described by a family…
Aims. The main goal of this paper is to derive observational constraints on the halo mass fuction (HMF) by performing a tomographic analysis of the magnification bias signal on a sample of background submillimeter galaxies. The results can then be compared with those from a non-tomographic study. Methods. We measure the cross-correlation function between a sample of foreground GAMA galaxies with spectroscopic redshifts in the range $0.1 < z < 0.8$ (and divided up into four bins) and a sample of background submillimeter galaxies from H-ATLAS with photometric redshifts in the range $1.2 < z < 4.…
Measuring weak lensing cosmic magnification signal is very challenging due to the overwhelming intrinsic clustering in the observed galaxy distribution. In this paper, we modify the Internal Linear Combination (ILC) method to reconstruct the lensing signal with an extra constraint to suppress the intrinsic clustering. To quantify the performance, we construct a realistic galaxy catalogue for the LSST-like photometric survey, covering $20\,000\ °^2$ with mean source redshift at $z_s\sim 1$. We find that the reconstruction performance depends on the width of the photo-z bin we choose. Due to the…
We test the impact of some systematic errors in weak lensing magnification measurements with the COSMOS 30-band photo-$z$ Survey flux limited to $I_{auto}<25.0$ using correlations of both source galaxy counts and magnitudes. Systematic obscuration effects are measured by comparing counts and magnification correlations. We use the ACS-HST catalogs to identify potential blending objects (close pairs) and perform the magnification analyses with and without blended objects. We find that blending effects start to be important ($\sim$ 0.04~mag obscuration) at angular scales smaller than 0.1 arcmin. …
Lensing magnification and stacked shear measurements of galaxy clusters rely on measuring the density of background galaxies behind the clusters. The most common ways of measuring this quantity ignore the fact that some fraction of the sky is obscured by the cluster galaxies themselves, reducing the area in which background galaxies can be observed. We discuss the size of this effect in the Sloan Digital Sky Survey (SDSS) and the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS), finding a minimum 1 per cent effect at $0.1h^{-1}$Mpc from the centers of clusters in SDSS; the effect is an…
Weak lensing causes spatially coherent fluctuations in flux of type Ia supernovae (SNe Ia). This lensing magnification allows for weak lensing measurement independent of cosmic shear. It is free of shape measurement errors associated with cosmic shear and can therefore be used to diagnose and calibrate multiplicative error. Although this lensing magnification is difficult to measure accurately in auto correlation, its cross correlation with cosmic shear and galaxy distribution in overlapping area can be measured to significantly higher accuracy. Therefore these cross correlations can put usefu…
Weak lensing of Type Ia supernovae (SNe Ia) is a systematic uncertainty in the use of SNe Ia as standard candles, as well as an independent cosmological probe, if the corresponding magnification distribution can be extracted from data. We study the peak brightness distribution of SNe Ia in the Pantheon sample, and find that the high $z$ sub-sample shows distinct weak lensing signatures compared to the low $z$ subsample: a long tail at the bright end due to high magnifications and a shift of the peak brightness toward the faint end, consistent with findings from earlier work. We have developed …
We introduce a technique to measure gravitational lensing magnification using the variability of type I quasars. Quasars' variability amplitudes and luminosities are tightly correlated, on average. Magnification due to gravitational lensing increases the quasars' apparent luminosity, while leaving the variability amplitude unchanged. Therefore, the mean magnification of an ensemble of quasars can be measured through the mean shift in the variability-luminosity relation. As a proof of principle, we use this technique to measure the magnification of quasars spectroscopically identified in the Sl…
Weak lensing surveys exploit measurements of galaxy ellipticities. These measurements are subject to errors which degrade the cosmological information that can be extracted from the surveys. Here we propose a way of using the galaxy data themselves to calibrate the measurement errors. In particular, the cosmic shear field, which causes the galaxies to appear elliptical, also changes their sizes and fluxes. Information about the sizes and fluxes of the galaxies can be added to the shape information to obtain more robust information about the cosmic shear field. The net result will be tighter co…
Accurate measurement of galaxy cluster masses is an essential component not only in studies of cluster physics, but also for probes of cosmology. However, different mass measurement techniques frequently yield discrepant results. The SDSS MaxBCG catalog's mass-richness relation has previously been constrained using weak lensing shear, Sunyaev-Zeldovich (SZ), and X-ray measurements. The mass normalization of the clusters as measured by weak lensing shear is >~25% higher than that measured using SZ and X-ray methods, a difference much larger than the stated measurement errors in the analyses. We…
We present a derivation of a generalized optimally-weighted estimator for the weak lensing magnification signal, including a calculation of errors. With this estimator, we present a local method for optimally estimating the local effects of magnification from weak gravitational lensing, using a comparison of number counts in an arbitrary region of space to the expected unmagnified number counts. We show that when equivalent lens and source samples are used, this estimator is simply related to the optimally-weighted correlation function estimator used in past work and vice-versa, but this metho…
We present a statistical weak-lensing magnification analysis on an optically selected sample of 3029 \texttt{CAMIRA} galaxy clusters with richness $N>15$ at redshift $0.2\leq z <1.1$ in the Subaru Hyper Suprime-Cam (HSC) survey. We use two distinct populations of color-selected, flux-limited background galaxies, namely the low-$z$ and high-$z$ samples at mean redshifts of $\approx1.1$ and $\approx1.4$, respectively, from which to measure the weak-lensing magnification signal by accounting for cluster contamination as well as masking effects. Our magnification bias measurements are found to be …
We perform high resolution ray-tracing simulations to investigate probability distribution functions (PDFs) of lensing convergence, shear, and magnification on distant sources up to the redshift of $z_s=20$. We pay particular attention to the shot noise effect in $N$-body simulations by explicitly showing how it affects the variance of the convergence. We show that the convergence and magnification PDFs are closely related with each other via the approximate relation $μ=(1-κ)^{-2}$, which can reproduce the behavior of PDFs surprisingly well up to the high magnification tail. The mean convergen…
Many distant objects can only be detected, or become more scientifically valuable, if they have been highly magnified by strong gravitational lensing. We use EAGLE and BAHAMAS, two recent cosmological hydrodynamical simulations, to predict the probability distribution for both the lens mass and lens redshift when point sources are highly magnified by gravitational lensing. For sources at a redshift of two, we find the distribution of lens redshifts to be broad, peaking at z=0.6. The contribution of different lens masses is also fairly broad, with most high-magnification lensing due to lenses w…
Measuring and calibrating relations between cluster observables is critical for resource-limited studies. The mass-richness relation of clusters offers an observationally inexpensive way of estimating masses. Its calibration is essential for cluster and cosmological studies, especially for high-redshift clusters. Weak gravitational lensing magnification is a promising and complementary method to shear studies, that can be applied at higher redshifts. We employed the weak lensing magnification method to calibrate the mass-richness relation up to a redshift of 1.4. We used the Spitzer Adaptation…
Using data from the Pantheon SN Ia compilation and the Sloan Digital Sky Survey (SDSS), we propose an estimator for weak lensing convergence incorporating positional and photometric data of foreground galaxies. The correlation between this and the Hubble diagram residuals of the supernovae has $3.6σ$ significance, and is consistent with weak lensing magnification due to dark matter halos centered on galaxies. We additionally constrain the properties of the galactic haloes, such as the mass-to-light ratio $Γ$ and radial profile of the halo matter density $ρ(r)$. We derive a new relationship for…
We study the impact of lensing magnification on the observed three-dimensional galaxy clustering in redshift space. We used the RayGal suite of N-body simulations, from which we extracted samples of dark matter particles and haloes in the redshift regime of interest for future large redshift surveys. Several magnitude-limited samples were built that reproduce various levels of magnification bias ranging from s = 0 to s = 1.2, where s is the logarithmic slope of the cumulative magnitude number counts, in three redshift intervals within 1 < z < 1.95. We studied the two-point correlation function…
We report on the detection of gravitational lensing magnification by a population of galaxy groups, at a significance level of 4.9 sigma. Using X-ray selected groups in the COSMOS 1.64 deg^2 field, and high-redshift Lyman break galaxies as sources, we measure a lensing-induced angular cross-correlation between the samples. After satisfying consistency checks that demonstrate we have indeed detected a magnification signal, and are not suffering from contamination by physical overlap of samples, we proceed to implement an optimally weighted cross-correlation function to further boost the signal …
The existence of galaxy intrinsic clustering severely hampers the weak lensing reconstruction from cosmic magnification. In paper I \citep{Yang2011}, we proposed a minimal variance estimator to overcome this problem. By utilizing the different dependences of cosmic magnification and galaxy intrinsic clustering on galaxy flux, we demonstrated that the otherwise overwhelming galaxy intrinsic clustering can be significantly suppressed such that lensing maps can be reconstructed with promising accuracy. This procedure relies heavily on the accuracy of determining the galaxy bias from the same data…
We present a derivation of the cosmological distance-redshift relation up to second order in perturbation theory. In addition, we find the observed redshift and the lensing magnification to second order. We do not require that the density contrast is small, we only that the metric potentials and peculiar velocities are small. Thus our results apply into the nonlinear regime, and can be used for most dark energy models. We present the results in a form which can be readily computed in an N-body simulation. This paper accompanies arXiv:1207.2109, where the key results are summarised in a physica…
The detailed observation of the distribution of redshifts and chirp masses of binary black hole mergers is expected to provide a clue to their origin. In this paper, we develop a hybrid model of the probability distribution function of gravitational lensing magnification taking account of both strong and weak gravitational lensing, and use it to study the effect of gravitational lensing magnification on the distribution of gravitational waves from distant binary black hole mergers detected in ongoing and future gravitational wave observations. We find that the effect of gravitational lensing m…
Magnification changes the observed number counts of galaxies on the sky. This biases the observed tangential shear profiles around galaxies, the so-called galaxy-galaxy lensing (GGL) signal, and the related excess mass profile. Correspondingly, inference of physical quantities, such as the mean mass profile of halos around galaxies, are affected by magnification effects. We use simulated shear and galaxy data of the Millennium Simulation to quantify the effect on shear and mass estimates from magnified lens and source number counts. The former are due to the large-scale matter distribution in …
We develop a unified, analytic framework for gravitational lensing by Kerr black holes. In this first paper we present a new, general lens equation and magnification formula governing lensing by a compact object. Our lens equation assumes that the source and observer are in the asymptotically flat region and does not require a small angle approximation. Furthermore, it takes into account the displacement that occurs when the light ray's tangent lines at the source and observer do not meet on the lens plane. We then explore our lens equation in the case when the compact object is a Kerr black h…
Weak lensing can be observed through a number of effects on the images of distant galaxies; their shapes are sheared, their sizes and fluxes (magnitudes) are magnified and their positions on the sky are modified by the lensing field. Galaxy shapes probe the shear field whilst size, magnitude and number density probe the convergence field. Both contain cosmological information. In this paper we are concerned with the magnification of the size and magnitude of individual galaxies as a probe of cosmic convergence. We develop a Bayesian approach for inferring the convergence field from a measured …
Gravitational waves (GWs) may be magnified or de-magnified due to lensing. This phenomenon will bias the distance estimation based on the matched filtering technique. Via the multi-sphere ray-tracing technique, we study the GW magnification effect and selection effect with particular attention to the stellar-mass binary black holes (BBHs). We find that, for the observed luminosity distance $\lesssim 3~\mathrm{Gpc}$, which is the aLIGO/Virgo observational horizon limit, the average magnification keeps as unity, namely unbiased estimation, with the relative distance uncertainty $σ(\hat{d})/\hat{…
We present free-form gravitational lensing models derived with the WSLAP+ code for the six $\textit{Hubble Frontier Fields}$ clusters using the latest data available from the $\textit{Frontier Fields Lensing Models v.4}$ collaboration. We present magnifications maps in the lens plane and caustic maps in the source plane. From these maps, we derive the probability of magnification using two different, but related, methods. We confirm MACS 0717 as the cluster with the most complex structure, and having the largest lensing efficiency and Einstein radius. When comparing these results with the ones…
We address two selection effects that operate on samples of gravitationally lensed dusty galaxies identified in millimeter- and submillimeter-wavelength surveys. First, we point out the existence of a "size bias" in such samples: due to finite source effects, sources with higher observed fluxes are increasingly biased towards more compact objects. Second, we examine the effect of differential lensing in individual lens systems by modeling each source as a compact core embedded in an extended diffuse halo. Considering the ratio of magnifications in these two components, we find that at high ove…
